Di r ecci ó n:Di r ecci ó n: Biblioteca Central Dr. Luis F. Leloir, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Intendente Güiraldes 2160 - C1428EGA - Tel. (++54 +11) 4789-9293

Co nta cto :Co nta cto : digital@bl.fcen.uba.ar

Tesis Doctoral

Compositos elastoméricosCompositos elastoméricosestructurados: estudio de los efectosestructurados: estudio de los efectos

magneto-piezoresistivos y sumagneto-piezoresistivos y suaplicación en dispositivosaplicación en dispositivos

piezométricospiezométricos

Mietta, José Luis

2016-07-01

Este documento forma parte de la colección de tesis doctorales y de maestría de la BibliotecaCentral Dr. Luis Federico Leloir, disponible en digital.bl.fcen.uba.ar. Su utilización debe seracompañada por la cita bibliográfica con reconocimiento de la fuente.

This document is part of the doctoral theses collection of the Central Library Dr. Luis FedericoLeloir, available in digital.bl.fcen.uba.ar. It should be used accompanied by the correspondingcitation acknowledging the source.

Cita tipo APA:

Mietta, José Luis. (2016-07-01). Compositos elastoméricos estructurados: estudio de los efectosmagneto-piezoresistivos y su aplicación en dispositivos piezométricos. Facultad de CienciasExactas y Naturales. Universidad de Buenos Aires.

Cita tipo Chicago:

Mietta, José Luis. "Compositos elastoméricos estructurados: estudio de los efectos magneto-piezoresistivos y su aplicación en dispositivos piezométricos". Facultad de Ciencias Exactas yNaturales. Universidad de Buenos Aires. 2016-07-01.

❯♥rs ♥♦s rst ♥s ①ts ② trs

♣rt♠♥t♦ í♠ ♥♦rá♥ ♥ít ② í♠ ís

♦♠♣♦st♦s st♦♠ér♦s strtr♦s st♦ ♦s t♦s ♠♥t♦♣③♦rsst♦s ② s ♣ó♥

♥ s♣♦st♦s ♣③♦♠étr♦s

ss ♣rs♥t ♣r ♦♣tr tít♦ ♦t♦r ❯♥rs ♥♦s rs ♥ ár í♠ ♥♦rá♥ ♥ít ② í♠ ís

t♦r

♦sé s tt

rt♦rs ss

r♦ rtí♥ rP♦ ♥♦ ♠♦r♥

♦♥sr st♦

r ♠s

r tr♦

♣rt♠♥t♦ í♠ ♥♦rá♥ ♥ít ② í♠ ís♥sttt♦ í♠ ís trs ♠♥t ② ♥r

t ♥s ①ts ② trs❯♥rs ♥♦s rs

tó♥♦♠ ♥♦s rs ♥s ♦

s♦s ♦s q sr♠♥ ♥ ts♦♦ ♣r rr ♦♠♦ ♥♦♥

♠♣♦rt♥t t♥ s ♥♦t t♦ st♦♣ qst♦♥♥ r♦st② s ts ♦♥rs♦♥ ♦r ①st♥ ♥ ♥♥♦t ♣ t ♥ ♥ ♦♥t♠♣tst ♠②strs ♦ tr♥t② ♦ ♦ t ♠r♦s strtr ♦ rt② t s♥♦ ♦♥ trs ♠r② t♦ ♦♠♣r♥ tt ♦ ts ♠②str② ②

r ♦s ♦② r♦st② ♦♥t st♦♣ t♦ ♠r

rt ♥st♥

r♠♥t♦s

r rt♥ r ♣♦r ♣r♦♣♦♥r♠ r st tr♦ ♣♦r ♦s ♦♥s♦s♥tr♦ ② r ♦rt♦r♦ ② ♣♦r ♥sñr♠ q ♦ ♠♣♦rt♥t ♥♦ s srs♥♦ ♥t♥r é ♠ tr♥♦ r♠♥t♦

r P♦ ♠♦r♥ ♣♦r ♣tr♠ ♦♠♦ tsst ② ♣r♠tr♠♥trr ♠♥♦ ♣r♦r♠ó♥ ♥ s♥tr ♣♥♥t

♠ ♦♥sr st♦ r r ♠s ♣♦r ♦♥trr ♦♠♣tt ♠ ♦r♠ó♥ ♦t♦r

♠s ♣rs ♣♦r s ♠♦r ♥♦♥♦♥ ♣♦r s sr♦ ② ♣♦r ♥sñr♠ ♥♦ r♥♥r ♦s sñ♦s

♦♥♥ ♠ ♦♣♦t♦ ♣♦r ♠♥r♠ ♦♥ s s♦♥rs ② ②r♠ sr ♥ ♠♦r ♣rs♦♥

♠ r♠♥♦ ♠s ♦s r♠♥ ♣♦r str s♠♣r ♦s ♦s ♦rt♦r♦ r♥♦ ② ♣♦r s ♣♦②♦ ♦s ♠♠r♦s r♦ ♣♦r ss ♦rs sr♥s rsó♥

♦r♥ ♠♥srt♦ t ♥s ①ts ② trs ❯♥rs

♥♦s rs ♣♦r ①♥t ♦r♠ó♥ r

t♦♦s q♦s ♦s q ♣r♥í

s♠♥

♦♠♣♦st♦s st♦♠ér♦s strtr♦s st♦ ♦s t♦s ♠♥t♦♣③♦rsst♦s ② s ♣ó♥

♥ s♣♦st♦s ♣③♦♠étr♦s

❯

♦s ♦♠♣♦st♦s st♦♠ér♦s strtr♦s ♣♦r ss ss ♥♥és stá♥ ♦r♠♦s ♣♦r s♣rs♦♥s ♠tr ♥♦rá♥♦ ♥ ♥ ♠tr③st♦♠ér ♥ s ♥♥ ♣r♦♣s ♥s♦tró♣s ♣rs♥tss ♦♥st ♥ st♦ ①♣r♠♥t ② tór♦ sst♠s

♥ ♦s sst♠s ①♣r♠♥ts st♦s ♣rtís q s♦♥ s♠tá♥♠♥t ♠♥éts ② ♦♥t♦rs tr s ♥♥tr♥ s♣rss ♥♣♦♠ts♦①♥♦ P strtró♥ ♠tr ♣♦r ♦r♠ó♥ ♥s ♠tr ♥♦rá♥♦ ♥ ♠tr③ ♣♦♠ér s ♦r r♥♦ s♣rsó♥ ♥ ♣rs♥ ♥ ♠♣♦ ♠♥ét♦ ♥t♥s♦

st♦ ①♣r♠♥t ♦♠♥③ó ♦♥ sí♥tss ② rtr③ó♥ ♥♥♦♣rtís ♠♥tt Ps ② ♦r♠ó♥ r♦s ♠r♦♠étr♦sPs♣t q stá♥ ♥ st♦ s♣r♣r♠♥ét♦ ② s♦♥ ♦♥t♦rs ó♠♦s sst♠ ♦t♥♦ ♦ r♦ ♣rs♥t ♣③♦rsst rsst étr ρ s ♥ó♥ t♥só♥ ♠á♥ ♣ ② ♠♥t♦rsst♥ ρ s ♥ó♥ ♠♣♦ ♠♥ét♦ ♣♦ ♦ r♦ ♠tr ♥ st ss s srr♦♦ t♠é♥ ♥ s♥s♦r t♥só♥♠á♥ s♦ ♥ ♦ ♥②♥♦ ♠♣♠♥tó♥ ♦♥tt♦s♥♣s♠♥t♦ ② ró♥ s rs♣st

st♦s sst♠s ♣♥ ♣rs♥tr ♥s♦tr♦♣í étr ♦t ♣♦rss ss ♥ ♥és s r ♦♥t étr ♣r ú♥♠♥t♥ ♥ ró♥ s♣ ♣ó♥ ♠♣♦ ♠♥ét♦ r♥t ♣r♣ró♥ st♦ tór♦ ♦♠♥③ ♦♥ ♥áss ♠♥t s♠♦♥s ♦♥t r♦ ó♠♦ ♦s ♣rá♠tr♦s strtrs ♦s sst♠s ♥②♥ ♥ ♣r♦ ♦t♥r ♥ s s♠♦♥s s ♦♠♣tó ♣r♦ ♣r♦ó♥ ♥ ♥ s r♦♥s rtrísts ♠tr strtr♦ ♥ s♥ t♣ tór s srr♦ó ♥♠♦♦ ♦♥sttt♦ rs♣st ♣③♦rsst ♥s♦tró♣ ♦ ♦♥ó♥ ♥♠♥t s ①t♥ó ♦ ♠♦♦ ♦♥ ♥ ♣rr rs♣st ♠♥t♦rsst ♦sr

Prs s P③♦rsst ♥t♦rsst♥ Pr♦ó♥ ♦♥t r♦ ♦♠♣♦st♦ ♥s♦tr♦♣í

s♠♥ ①

trtr st♦♠r ♦♠♣♦sts st② ♦♠♥t♦♣③♦rsst ts ♥ tr ♣♣t♦♥

♥ ♣③♦♠tr s

st♦♠r trtr ♦♠♣♦sts r ♦r♠ ② ♥♦r♥ ♠tr s♣rs♦♥s ♥ ♥ st♦♠r ♠tr① ♥ ♥s♦tr♦♣ ♣r♦♣rtsr ♥ s ss ♣rs♥ts ♥ ①♣r♠♥t ♥ t♦rt st② ♦ s②st♠s

♥ t st ①♣r♠♥t s②st♠s ♣rts r s♠t♥♦s②♠♥t ♥ tr② ♦♥t r s♣rs ♥ ♣♦②♠t②s♦①♥P strtr♥ ♦ t ♠tr ② ♦r♠t♦♥ ♦ ♥s ♦ ♥♦r♥♠tr ♥ t ♣♦②♠r ♠tr① s ② r♥ t ♥ t ♣rs♥ ♦♥ ♥t♥s ♠♥t

①♣r♠♥t st② strt t t s②♥tss ♥ rtr③t♦♥♦ ♠♥tt ♥♥♦♣rts Ps ♥ rt ♦r♠t♦♥ ♦ ♠r♦♠trPssr r ♥ s♣r♣r♠♥t stt ♥ r ♦♠ ♦♥t♦rs s②st♠ ♦t♥ tr r♥ s ♣③♦rsstt② t tr rstt② ρ s ♥t♦♥ ♦ ♣♣ strss ♥ ♠♥t♦rsst♥ ρ s ♥t♦♥ ♦ ♠♥t ♣♣ tr r♥ ♥ ts ss t s ♥♦♣ s♦ ♠♥ strss s♥s♦r s ♦♥ tt ♥♥ t♠♣♠♥tt♦♥ ♦ ♦♥tts ♥♣st♦♥ ♥ rt♦♥ ♦ t rs♣♦♥s

s s②st♠s ♠② ♣rs♥t ♦t tr ♥s♦tr♦♣② tt s♣♣r tr ♦♥tt② ♦♥② ♥ ♦♥ s♣t rt♦♥ t ♦♥ ♦♣♣t♦♥ ♦ t ♠♥t r♥ ♣r♣rt♦♥ t♦rt st②strts t t ♥②ss ② ♦♥t r♦ s♠t♦♥s ♦ ♦ t strtr♣r♠trs ♦ s ♥♥ t ♣r♦t② ♦ ♥ ts s♠t♦♥st ♣r♦t♦♥ ♣r♦t② ♥ ♦ t rtrst rt♦♥s s ♦♠♣t ♥ t s♦♥ t♦rt st ♦♥sttt ♠♦ ♦ t ♥s♦tr♦♣♣③♦rsst rs♣♦♥s ♥r ♦♥t♦♥s ♦ s ♥ ♦♣ ♥②ts ♠♦ s ①t♥ ♥ ♦rr t♦ ♣rt t ♦sr ♠♥t♦rsst♥rs♣♦♥s

②♦rs P③♦rsstt② ♥t♦rsst♥ Pr♦t♦♥ ♦♥t r♦♦♠♣♦st ♥s♦tr♦♣②

P♦♥s ② Pt♥ts

♦s ♣r♥♣s rst♦s ♣rs♥t♦s ♥ st ss ♦t♦r r♦♥ ♣♦s ♥ ♦s s♥ts rtí♦s

tt ♦r ② r ♠rt trs ♥ trtrs

tt r ② P ♠♦r♥ ♦r♥ ♦ P②s♠str②

tt P ♠♦r♥ ② r ♦t ttr

s♠s♠♦ s♦tó s♥t ♣t♥t ♥♥ó♥

tt r ② ♦r rr♦ s♥s♦rs ♣rsó♥② ♠♣♦ ♠♥ét♦ s♦s ♥ ♠trs ♠♥t♦r♦ó♦s ást♠♥t ①s ② ♥s♦tró♣♦s ♦r♠♦s ♣♦r ♥ó♥ ♥♥♦♣rtís írs ♠♥t♦♠tás ❳P P rá♠t

①

❮♥

♥tr♦ó♥

trs ② ét♦♦s ①♣r♠♥ts

í♥tss ♠tr r♥♦ í♥tss ♥♥♦♣rtís ♠♥tt í♥tss ♠r♦♣rtís ♠♥tt♣t

Pr♣ró♥ ♦♠♣♦st♦s P34❬❪ Pr♦♣s ♠á♥s P③♦rsst♥ ② ♠♥t♦rsst♥ ♦♠♣♦st♦ P

34❬❪ t♦ H s♦r ♦♥t r♥♦ 34❬❪

♥s♦r t♥só♥ ①

ró♥ s♥s♦r t♥só♥ ♠♣♠♥tó♥ ♦♥tt♦s étr♦s ♠♣qt♠♥t♦ rr♦ s♥s♦rs

rtr③ó♥ ♦s ♦♥tt♦s étr♦s ♥s♦r rs♣st ② rs ♠ért♦

s♣st ♣③♦rsst ♦♠♣♦rt♠♥t♦ ♥á♠♦ s♥s♦rr♠s ♦s ♦♥♠tó♥ ♦rr♥t♣♦t♥ t♦ tr♠♦rsst♦ sst♥ qí♠

♥s♦tr♦♣í étr ♦t r♦ ór♦

♦♥♣t♦s Pr♦ ② stíst ♦rí Pr♦ó♥

♥tr♦ó♥ sr♣t♦rs ♥ sst♠ ♣r♦t♦

Pr♦ó♥ ♥ ♠trs ♦♠♣st♦s ♥s♦tró♣♦s ♠♣ó♥ ♦♠étr sst♠ ♥ st♦

①

① ❮♥

♥s♦tr♦♣í étr ♦t trs ② ♠ét♦♦s

♥♠ér♦s

st♠ó♥ ♣r♦ ♣r♦ó♥ ♦rt♠♦

♦♥stró♥ ♦♠♣t♦♥ sst♠ s♠♥t♦s rts ♣r♦♥ts

st♦ ♦♥t ó♠♣t♦ ♣r♦ ♣r♦ó♥ ❱ó♥ ♦rt♠♦

♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

❱ó♥ ♦rt♠♦ ② ♣r♦♥s tórs ♥ ♥♠ér ♣r♦♥s tórs

♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

♠♦♥s ♥ sst♠s ♥s♦tró♣♦s♥s ♦♥sr♦♥s

t♦ s♠trí sst♠ t♦ ♦s ♣rá♠tr♦s strtrs

t♦ σθ t♦ σℓ t♦ 〈ℓ〉

t♦ t♠ñ♦ sst♠

s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ② s♠

♦♥s

♥tr♦ó♥ sst♥ étr ♦♠♥s ♦♥t♦rs ♦r♠s ♣♦r

♣s♦♥s s♥s sst♥ ♣♦r t♥♦ tró♥♦ Rtunnel ♦♦ x(P ) rtr③ó♥ ♠á♥

♠tr③ ♣♦♠ér sts ♦s rst♦s ①♣r♠♥ts ♣r r

r♥ ♥♥ st ♠tr E rrr t♥♦

r♥♦♣♦í♠r♦ ϕ ② st♥ ♠ t♥♦ xo s♦rRtunnel

s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ② s

♠♦♥s

♥tr♦ó♥

❮♥ ①

♦♦ ♦r♠s♠♦ rt

st♦s ② ssó♥ ♦♠♣♦rt♠♥t♦ ①♣r♠♥t ♠♥ét♦ ② ♠♥t♦

rsst♦ ♠tr rr♥ 34❬❪P

♠♦♥s rs♣st ♠♥t♦rsst ♣r s♥ st♦ s♣r♣r♠♥ét♦

①t♥só♥ ♠♦♦ sst♠s ♦♥ ♠♥t③ó♥♦q

♦♥s♦♥s ♥rs

♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦

♥rs s♦ ♣r♦ ♣r♦ó♥

♦rí r♦s

♥♦♥s st♦ ♦♥t ♥ r♦s

♦♦s st

♦♦ ♦♦♦ ♦♦ ♦♦ ♦♦ ♦♦ r♥ ♦♦ ♦♦♥②♥

♦rí

st ró♥♠♦s

❮♥ rs

♠♣♦ rr♦♦ ♦♠♣ró♥ sq♠át ♦♠♣♦st♦s s♦tró♣♦s ② ♥s♦

tró♣♦s strtr ♥tr♥ ♥s♦tr♦♣í étr ♦t ② ♦

sq♠ ①♣r♠♥t t③♦ ♣r sí♥tss ♥♥♦♣rtís ♠♥tt

sq♠ sí♥tss ♠r♦♣rtís ♠♥tt♣t q♣♦ t③♦ ♣r r♦ ♦♠♣♦st♦ P

34❬❪ sq♠ ♦r♠ó♥ ♥ r ♥trr③ trés

ró♥ r♦só♥ ♥ ♠trs s♦s ♥ P ♥③♦r t①tr s♣♦st♦ ①♣r♠♥t t③♦ ♣r ♠ó♥ rs

♣st ♣③♦rsst ♦s ♦♠♣♦st♦s sq♠ sst♠ ①♣r♠♥t t③♦ ♣r ♠ó♥

rs♣st ♠♥t♦rsst ♦s ♦♠♣♦st♦s sq♠ sst♠ ①♣r♠♥t t③♦ ♣r ♠ó♥

t♦ t♥só♥ ② ♠♣♦ ♠♥ét♦ s♦r ♦♥t ♣♦♦ 34❬❪

ró♥ s♥s♦r ① t♥só♥ ♠á♥ ♦♠♣♦rt♠♥t♦ ♣③♦rsst♦ ♦s ♦♥tt♦s étr♦s ró♥ s♥s♦r sró♥ rt s♣ró♥ r♥♦♠tr③ ♣♦♠ér ♥s♦rr♠ s♥s♦r t♥só♥ ♦s ♦♥♠tó♥ s♥s♦r t♦ t♠♣rtr s♦r rs♣st s♥s♦r t♥só♥

♦♥r♦ strí♦ Pr♦ó♥ ♥ ♥ ♥ tts r♣♦s st♦s ♥ sst♠ ♣r♦t♦ st♦s

①

① ❮♥ rs

t♦ ró♥ ♦♣ó♥ ♥ ♣r♦ó♥ st♦ Pr♦ó♥ ♥ ♦♥t♥♦ s♦s t♦r♦s strtr rt ♥ sst♠s ♣r♦t♦ st♦ stró♥ ♥r ♣r s ♣s♦♥s ♥

rr♥ str♦♥s ♦♥ts ② á♠tr♦s ♠♦s ♠t

♦♥t ♣r s ♣s♦♥s ♥ rr♥ sq♠ ♦♥♣t ♣r♦①♠ó♥ s ♣s♦♥s

♣♦r s♠♥t♦s rts

rr♦r st♠♦r ♣r♦ ♣r♦ó♥ sst♠ ♦s s♠♥t♦s rts q s ♥trst♥ sq♠ ♦rt♠♦ t③♦ ♣r st♦ ♣r♦t♦

s♠♥t♦s rts ♦ stró♥ s♦tró♣ ♦♠♣♦♥♥ts ♦r③♦♥t ② rt ♥ s♠♥t♦ rt

♦♥r♦♥s ♣r sst♠s s♦tró♣♦s r♦s s♠♥t♦s rts ♣r♦♥ts

stró♥ ♦rs ♥s♦tr♦♣í ♠r♦só♣ ♣r ♥sst♠ r♦

stró♥ ♦rs ♣r♦ ♣r♦ó♥ ♦♥r♥ st♠ó♥ ♣r♦ ♣r♦ó♥ rs ♣r♦ ♣r♦♦♥ ♥ sst♠s r♦s

s♦tró♣♦s s♠♥t♦s rts ♣r♦♥ts s♦ ♥s ♣r♦t rít ② sí♦ stá♥r tr♠♥ó♥ 〈Φ〉∞,r=1,ℓ ♣r r♥ts ♦rs ℓ ♥ ♥♠ér ♣♥♥ ♥s rít

♦♥ t♠ñ♦ ♦s s♠♥t♦s rt t♦ s ♥trs♦♥s ①♦♣r♠trs ♥ s rs

♣r♦ ♣r♦ó♥ ♠ñ♦ ♠♦ str ♥ ♥ó♥ ♥s s

♠♥t♦s rt ♦♠♣♦rt♠♥t♦ PSC ② ℘ ❱♦rs ℘H ℘V ℘HX ② ℘U ♣r ♥ sst♠ s♠♥t♦s

rts ♣r♦♥ts r♦ s♦tró♣♦ ♦♥ ♦♥ ℓ = 1000 ②L = 1750

♣♥♥ s r♥ts ♣r♦s ♣r♦ó♥ ♥s rít ♦♥ t♠ñ♦ sst♠

t♦ s♠trí sst♠ ♥ t♦ σθ ♥ s str♦♥s ♥rs ♦s s♠♥t♦s

rt

❮♥ rs ①①

♠♣♦ r③♦♥s ♦♥t r♦ ♣r r♥ts ♦rs σθ

t♦ σθ s♦r ♥s♦tr♦♣í ♠r♦só♣ t♦ σθ ♥ s rs ♣r♦ ♣r♦ó♥ t♦ σθ ♥ ♣r♦ ♣r♦ó♥ só♦♦r③♦♥t t♦ σℓ ♥ s str♦♥s ♦♥t ♦s s♠♥t♦s

rt ♠♣♦ r③♦♥s ♦♥t r♦ ♣r r♥ts ♦rs

σℓ t♦ σℓ s♦r ♥s♦tr♦♣í ♠r♦só♣ t♦ σℓ ♥ ♣r♦ ♣r♦ó♥ só♦♦r③♦♥t t♦ 〈ℓ〉 ♥ s str♦♥s ♦♥t ♦s s♠♥t♦s

rt t♦ 〈ℓ〉 ♥ ♣r♦ ♣r♦ó♥ só♦♦r③♦♥t ♠♣♦ r③♦♥s ♦♥t r♦ ♣r r♥ts ♦rs

〈ℓ〉 t♦ 〈ℓ〉 s♦r ♥s♦tr♦♣í ♠r♦só♣ t♦ t♠ñ♦ sst♠ ♥

② ♥ sst♠s ♣rs♥tó♥ ♦s ♣rá♠tr♦s ♠♦♦ ♣③♦rsst♥ ♥s♠♦s ♣r♥♣s ♥♦r♦s ♥ rs♣st ♣③♦r

sst ♠tr ♥♦ tró♥♦ trés s♣rs ♠sérs s♣st ást ♠tr③ ♣♦♠ér P s♣st ♣③♦rsst ♠tr ② st ♠♦♦

♦♥sttt♦ r♠♥t♦♥s ♥tr♥s s ♣s♦♥s t♦ ♦r γ s♦r ♣rá♠tr♦s strtrs r♣r♦s t♦ E ② xoγ ♥ rs♣st ♣③♦rsst

t♦ ♦♠♣rsó♥ ② ♠♣♦ ♠♥ét♦ ♥ ♦♥t étr r♥♦ 34❬❪

♦r♠ó♥ ♠♥t♦ást ♥ ♦q ♠tr st♦♠ér♦

r ♠♥t③ó♥ r♥♦ 34❬❪ ó♥ ást ② ♠♥t♦rsst s♣st ♠♥t♦rsst 34❬❪P 4.2%

rs s♠s Rtunnel/R

⋆ ♥ ♥ó♥ H ♣r s♥ st♦ s♣r♣r♠♥ét♦

①① ❮♥ rs

t♦ ♠♣♦ ♦rt♦ ♥ rs s♠s Rtunnel/R⋆

♥ ♥ó♥ H ♣r s rr♦♠♥ét♦s t♦ rtr ♥ rs s♠s Rtunnel/R

⋆ ♥♥ó♥ H ♣r s rr♦♠♥ét♦s

t♦ K ♥ rs s♠s Rtunnel/R⋆ ♥ ♥ó♥

H ♣r s rr♦♠♥ét♦s t♦ (x⋆γ) ♥ rs s♠s Rtunnel/R

⋆ ♥ ♥ó♥ H ♣r s rr♦♠♥ét♦s

r♥s♦r♠ó♥ r♥♦r♠③ó♥ ♣r p < p∞ r♥s♦r♠ó♥ r♥♦r♠③ó♥ ♣r p > p∞ ♦♥r♦♥s ♣r♦♥ts ♥ ♥ ♦♥ b = 2

r♦s ② sst♠s ♣r♦t♦s s♠♥t♦s rts ♦rt♠♦

♦r♠ó♥ í♥ ♥ ♠trs ♣rást♦s

❮♥ s

♥ó♥ ♦s sí♠♦♦s s♦s ♥ ♠♦♦ ♣③♦rsst♥ ♥ ♦s sst♠s

♥ó♥ ♦s sí♠♦♦s s♦s ♥ ♠♦♦ ♠♥t♦rsst♥ ♥ ♦s sst♠s

①①

♣ít♦

♥tr♦ó♥

❲ ♥ s♦t② ①qst② ♣♥♥t♦♥ s♥ ♥ t♥♦♦② ♥ r② ♥②♦♥ ♥♦s ♥②t♥ ♦t

s♥ ♥ t♥♦♦②

r ♥

s♠♥ ♥ st ♥tr♦ó♥ s sr st♦ rt t♠át tr♦ ② s ♥♥♥ ♦s ♦t♦s ♣r♥♣s srr♦r♥ ♦s ♣ít♦s s♥ts

♥ ♥ ♠trs r♥ ♥♦♠r ♠trs ♦♠♣st♦s♦ ♦♠♣♦st♦s q♦s ♦r♠♦s ♣♦r ♥ó♥ ♦s ♦ ♠s ♠trs ♣r ♦♥sr ♦♠♥ó♥ ♣r♦♣s q ♥♦ s ♣♦s ♦t♥r ♥♦s ♠trs ♦r♥s st♦s ♦♠♣st♦s ♣♥ s♦♥rs ♣r ♦rr♦♠♥♦♥s ♣♦♦ ss r③ rsst♥ ♣s♦ rsst♥ sts t♠♣rtrs rsst♥ ♦rr♦só♥ r③ ② ♦♥t étr ♥tr ♦trs ♣r♦♣s ❬❪ t♠♥t s ♣t q ♥ ♠trs ♦♠♣st♦ ♥♦ ♠♣ s s♥ts rtrísts sr stá♦r♠♦ ♣♦r ♦s ♦ ♠ás ♦♠♣♦♥♥ts st♥s ís♠♥t ♣rs♥trs ss qí♠♠♥t st♥ts ♥s♦s ♥tr sí ② s♣rs ♣♦r ♥♥trs ♠♥♦s ♥ ss ♣r♦♣s r s♠♣ s♠ s ♣r♦♣s ss ♦♠♣♦♥♥ts s♥r ♦ ♣rt♥♥ ♦s ♠trs ♦♠♣st♦s q♦s ♠trs ♣♦ás♦s ♥ ♦s q ♠♥t ♥trt♠♥t♦ tér♠♦ ♥♦ rt♦ s ♠♥ ♦♠♣♦só♥ s ss♣rs♥ts ♦♥s ♠tás ❬❪

♣ró♥ ② srr♦♦ ♦s ♠trs ♣♦♠ér♦s ♣ ♦♥srrs ♥ s r♥s r♦♦♥s s♦ ♣s♦ ♥ ♦ rr♥t ♥ ② t♥♦♦í ♠trs ♦s ♥s ♦♥s♦s ♥ ♦s ♠ét♦♦s ♣r♦ó♥ st♦s ♠trs ♥♦s ss ①♥ts rtrísts r♦♥ q r♥ rá♣♠♥t ♣t♦s ② ♠♣♦s ♣♦r st♦r ♥str

♣ít♦ ♥tr♦ó♥

② ♦♥stró♥ ♣r♦♥♠♥t♦ sts rtrísts s♥t ♣s♦ ♥ ♦♥só♥ ♠trs ③ ♠ás ♦s s♥ss ♣r♦ts st ♦r♠ ♣r♣ró♥ ♠trs ♦♠♣st♦s s♦s ♥ rr③♦ ♠trs ♣♦♠ér♦s ♦♥ rs ② r♥♦s♥♦rá♥♦s s s♥ ♥ ①♥t ♠♣♦ ♦ ♣r♦♥♠♥t♦❬ ❪

❯♥ s♦ ♣rtr ♦s ♦♠♣♦st♦s s♦s ♥ ♠trs ♣♦♠érs s♦♥q♦s ♥ ♦s q r♥♦ ♣rs♥t ♥ rqttr ♥ Pr rs ♣♦♠♦s ♦♥srr s rs♥s r♦r③s ♦♥ r♦♥ ♥♦ts ♥♦t♦s r♦♥♦ st♦s ♠trs s♦♥ ♠♣♠♥t ♦♥♦♦s ②t③♦s ♣♦r s ①♥t rsst♥ ♠á♥ ❬❪ s ú♥ tr♥♦ ♣r♦s♦ ♣r♣ró♥ ♦♠♣♦st♦ ♠♥t ①trs♦ ♠á♥♦ ❬❪♦ ó♥ ♦♥ ásr ❬❪ ♣ ♥rs ♥ ♦r♥tó♥ ♣rr♥ srs r♦♥♦ st ♦r♠ ♠tr ♣rs♥t ♥s♦tr♦♣í strtrq ♣ ♥r ♥s♦tr♦♣í ♠á♥ étr ó♣t t ❬ ❪

tr strt ♣r ♦t♥ó♥ strtrs ♥s♦tró♣s ♦♥sst ♥ t③ó♥ r♥♦s ♠♥t♦t♦s Pr ♠♣r♦ ♣♦♥srrs ♦s rr♦♦s q ♦♥sst♥ ♥ ♥ ♠tr③ íq s♦♥t ♦rá♥♦ ts ♥ r♥♦ ♥♥♦♣rt♦ ♠♥ét♠♥t t♦ rr♦ ♠♥tt ♠tt ② ♥ srt♥t á♦ ♦♦ ró①♦ ttr♠t♠♦♥♦ á♦ ítr♦ ♣r tr s♠♥tó♥ r♥♦P♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ r♥♦ s ♠♥t③ ②s ♦r♥③ s♥♦ s ♥s ♠♣♦ ♠♥ét♦ r ♠str ♥ rr♦♦ ♠♥tt ♥ s♦r ♥ s♣r r♦ ♦♥♥ ♠á♥ ♣r♠♥♥t ♦ ♦♥ró♥ qr♦ ♣rs♥t ♣♦s♦♥ ♦r♥tó♥ r ♦rrs♣♦♥♥t s ♥s ♠♣♦ ♠♥ét♦ qtr ♠♣♦ ♠♥ét♦ ② r r♥♦♠trí r♥♦ ♠♦♠♥t♦ r♦♥♥♦ ♠♣s s♣rsó♥ ést ♦♥♦ ♦♥ró♥ s♦tró♣ ♦r♥ ❬ ❪ ♦s rr♦♦s r♦♥ srr♦♦s♣♦r t P♣ ♥ ♥ ♦s ñ♦s s ❬❪ ② s ss ♣r♦♣s♠♥t♦r♦ós ♠♦ ss ♣r♦♣s r♦ós ♣♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ q s ♣r♠t st♦s ♠trs s♦rr♠♣t♦ ② r♦♥s t♠♥t s♦♥ t③♦s ♣r♥♣♠♥t ♥ ♥♥rí ♠á♥ ② ♥str♠♥tó♥ ♥ít ② ♦♥♥rí srr♦♦ ♣rótss ♦rt♦♣és ❬ ❪

P♥ r③rs ♦s ♠♦♦♥s ♦♥♣t♠♥t ♥♠ts s♦rsst♠s rr♦í♦s sr t③ó♥ r♥♦s ♠r♦♣rt♦② r♠♣③♦ ♠tr③ ♣♦r ♥ ♠tr③ rí r♦ rátr ást♦ ♣r♠r ss ♠♦♦♥s ♠♣ q tó♥tér♠ sr ♥ ♥ó♠♥♦ r♥t qtr st♠♦ ♠♥ét♦①tr♥♦ ♥ ♠r♦ ♥ ♦♥tr♣rt sró♥ rtt♦r ♥ sró♥ ss P♦r ♦♥s♥t ♣r ♠♥t♥r strtró♥ r♥♦ r♠♦r stí♠♦ ♠♥ét♦ ①tr♥♦ s rrr s♥

♠♦ó♥ ♣r ♦ ♣♥ srs ♣r♥♣♠♥t ♦s ♣r♦t♦♦♦s ♣r♣ró♥ ♣r♠r♦ ♦♥sst ♥ s♦ó♥ ♠tr③ ♣♦♠ér♥ ♥ s♦♥t ♦ s♣rsó♥ r♥♦ strtró♥ ♠s♠♦ ②r♠♦ó♥ s♦♥t ♣♦r ♣♦ró♥ ❬ ❪ s♥♦ ♥ ♠♦ ss ♥ s♣rsó♥ r♥♦ ♥ ♥ ♠tr③ ♦ r♦ ♥trr③♠♥t♦ strtró♥ r♥♦ ② ♣♦r út♠♦ ♥r♠♥t♦ r♦ ♥trr③♠♥t♦ ♣r♦s♦ ♦♥♦♦ ♦♠♦ r♦ ♠tr③ ❬ ❪

r rr♦♦ ♠♥tt ♥ s♦r s♣r r♦ ♦♥♥ ♠á♥ ♣r♠♥♥t ♦

♦♠♦ s trá ♠s ♥t ♦s sst♠s st♦s ♥ ♣rs♥ttr♦ ♦♥sst♥ ♥ ♦♠♣♦st♦s ♦r♠♦s ♣♦r s♣rs♦♥s ♣rtís ♦♥t♦rs ② ♠♥ét♠♥t ts s♣rss ♥ ♥ ♣♦í♠r♦ st♦♠ér♦♥♦ ♦♥t♦r ♣r♣r♦s s♥♦ s♥♦ ♦s ♣r♦t♦♦♦s ♠s rrsrt♦s ♦s stó♠r♦s s♦♥ q♦s ♦♠♣st♦s ♣♦♠ér♦s q ♠str♥♥ ♦♠♣♦rt♠♥t♦ ást♦ ♠r♦ tér♠♥♦ q ♣r♦♥ ♦♥♥só♥ ♣♦í♠r♦ ② ást♦ s s ♥tr♠ ♦♥ tér♠♥♦ ♦♠q ♥ r♦r s ♠ás ♦ ♣r rrrs ♣r♦t♦s ♥③♦s

♦ q ♠tr③ ♣♦♠ér qí t③ P ♣♦②♠t②s♦①♥ ♣♦♠ts♦①♥♦ s ♥♦ ♦♥t♦r st♦s sst♠s ♥♥ só♥ ♠② r♥t q♦s ♦r♠♦s ♣♦r ♠trs ♦♥t♦rs tí♣♠♥t P ♣♦②♥♥ ♣♦♥♥ ♦♥ ♦♥ó♥ stá ♣♦r ♠tr③ ❬ ❪ st♦ ♥r r♥ts t♣♦s ♦♣♦rt♥s ② ♠t♦♥s ♣r s ♣ó♥ ♥ tr♥sst♦rs ♣ ❬ ❪♠r♦tró♥ ① ❬❪ sst♠s ♠r♦tr♦♠á♥♦s ❬ ❪ s♥s♦rs qí♠♦s ❬❪ ♣s tró♥s ❬❪ ② ♦♥stró♥ tr♦♦str♥s♣r♥ts ♣r s♣♦st♦s ♦t♦♦t♦s ② ♦♣t♦tró♥♦s ❬ ❪ P♦r♠♣♦ ♥♦ sr ♥sr♦ q ♠tr③ s ♦♥t♦r s ♣♦s ①t♥r s♣tr♦ ♣♦ss ♥ ♥t♦ ♥tr③ qí♠ ♦s ♣♦í♠r♦s ♠♣♦s ♦♥ ♦♣♦rt♥ ♦♣t♠③r ♣♦s r♥tr st♥ts ♠trs sú♥ ♦s rqst♦s ♥sr♦s ② ♣♦♥s ss ♦♠♦ ♠♣♦ ♦ ♥tr♦r st ♦♠♣rr s st♥ts ♣♦ss♦rs ♣♦r ♦s sst♠s s♦s ♥ st②r♥t♥ rr

♣ít♦ ♥tr♦ó♥

♦ str♥♦t♥♦ ② P ♦♥ r♥ts ♣r♦♣s ♠á♥s ②sts ís♦qí♠s ❬ ❪

P♦r ♦tr ♣rt ♦ q ♠tr③ s s♥♠♥t ♥ s♥t♠♣ ♥s ♥♦r♣♦rr ♣rtís ♦♥t♦rs ♥ ♣r♦♣♦r♦♥ss♣r♦rs ♥ ♦r rít♦ ♣♦r ♥♠ ② ♦♥t étr ♣r s♣r♦rs ♠tr③ ú♥ ♥tr③ qí♠ ② ♠♦r♦♦í s ♣rtís ♠♣s ② ♠tr③ s ♦♥♥tr♦♥s ríts ♣♥ sr rt♠♥t ts ♣♦r ♠♣♦ 50% ♣♣ ♥ s♦ ♦♠♣♦st♦s s♦tró♣♦s ♥♦ strtr♦s ♣qts rt♦♥ ♥ ♠tr③ P ❬❪ st♦ ♠♣ rts ts ♥♦♥♥♥ts P♦r ♠♣♦ ♥ t♦r ♠♣♦rt♥t ♦♥srr s ♠♥t♦ ♥ ♦s♦st♦s t♥t♦ ♠trs ♦♠♦ tr♦ sí♥tss ♥ s♦ srs♥♥♦♠trs sñ♦s ♦ ♣r ♣ó♥ s tr ts q s♦ t♦s ♥s ♣rtís r♥♦ ♥ ♠♦s ♠♣♦rt♥ts ♥ s ♣r♦♣s ásts ② tr♦ós ♦♠♣♦st♦ ♦♥ rs♣t♦ ♠tr③ ♣♦♠ér srtá♥♦s ♥t♦♥s s ♥ts ♦rs ♣♦r ♣♦s r ♣♦í♠r♦ sts ts s♦♥ s♦s②s ♣♦r s♦ sst♠s strtr♦s ♦♠♦ ♦s srt♦s ♠s rr ② t③♦s ♥ sttr♦ r ♠str sq♠át♠♥t r♥ strtr ♦♠♣♦st♦s s♦tró♣♦s ② ♥s♦tró♣♦s ♦♠♥rs

r ♦♠♣ró♥ sq♠át ♥ ♦♠♣♦st♦ s♦tró♣♦ ② ♥s♦tró♣♦ ♦♥ stró♥ ♦♠♥r r♥♦

♥ t♦ ♦r♠ó♥ rqttrs ♥s♦tró♣s ♥s ♥ ♥ró♥ s♣í r ♦♥♥tró♥ rít ♣r ♦srr ♦♥ó♥étr ♣s ♦r ♣r♦ó♥ trés ♠tr ❬❪ st♦r♠ t♥t♦ ♥ ♣rs♥t tr♦ ♦♠♦ ♥ ♦tr♦s ♥tr♦rs ♥str♦ r♣♦ ❬ ❪ s ♦t♥♦ ♦♥t étr ♥ sst♠s ♦♥t♥♥♦♣r♦♣♦r♦♥s ♣rtís ∼ 4%

♦rrs♣♦♥ sñr qí q ♥ ♦r♠ s♠r ♦ ♥tr♦♦ rs♣t♦ s rs♥s r♦r③s ♦♥ rs r♦♥♦ ♦r♠ó♥ strtrs

♥s ♦r♠s ♦r ♣♦r ♥ ♥ú♠r♦ ♦ ♣rtís r♥♦♣ ♥r rss ♣r♦♣s ♥s♦tró♣s ♥ ♠tr ♥ tr♦s♥tr♦rs ♥str♦ r♣♦ s srr♦♥ ♣r♦♣s ♥s♦tró♣s ♦♥r♥♥ts ♦♥ st ❬ ❪ rs ♠♥t③ó♥ ② rs♦♥♥s♠♥éts ❬ ❪ ♦♥♠♥t s r♥♦ s ♦♥t♦r étr♦ ♦♠♦ ♥ ♦s sst♠s q ♦♥r♥♥ ♣rs♥t tr♦ ♦ s ♦♥♦♥s ♣r♣ró♥ s ♠tr ♦♠♣st♦ ♣rs♥trá ♦♥ó♥ étr ♥s♦tró♣ ❬ ❪ ♠♥♦r rsst étr ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦

♦r srr ♥ t ♦♠♣♦st♦ strtr♦ ♥ q s♥trrá t♥ó♥ r♥ ♥ ♣rs♥t tr♦ ♣r ♦ ♥♥r ♦s♦t♦s q srá♥ srr♦♦s ♦ r♦ ♦s ♣ít♦s s♥ts ♦s♦♠♣♦♥♥ts q ♦♥stt②♥ ♠tr ♦♠♣st♦ s♦♥ sr

♥♦ ♠r♦♣rtís ♠♥tt♣t ♥ ♥t rrs ♥st♥t♠♥t ♦♠♦ 34❬❪ ♦ µPs

tr③ ♣♦♠ér P

P♦r ♦ s ♥♦trá ♦♠♦ P34❬❪ ♦ ♠tr ♦s ts sí♥tss ♠tr r♥♦ ② ♣r♣ró♥ ♦♠♣♦st♦ s rá♥ ♥ ♣ít♦ P♦r ♠♦♠♥t♦ ♥♦s ♠tr♠♦s ♣♥t③r strtr♥tr♥ ♠s♠♦ ② s ♣r♦♣s rtr③s ♥ tr♦s ♥tr♦rs ♠♦♦ st♦ rt rtr③ó♥ ♦♠♣t ♠tr r♥♦ r③ ♣r♠♥t ♠♥t ♥ r trí té♥s♥tr s P❳ P♦r ❳② rt♦♥ ró♥ ②♦s ❳ P♦♦s ❱ ❱rt♥ ♠♣ ♥t♦♠tr ♥t♦♠trí str❱r♥t ♥♥♥ tr♦♥ r♦s♦♣② r♦s♦♣♦ tró♥♦ rr♦ r♥s♠ss♦♥ tr♦♥ r♦s♦♣② r♦s♦♣♦ tró♥♦ r♥s♠só♥ tt♥t ♦t t♥ ♦rr r♥s♦r♠ ♥rr ♣tr♦s♦♣② s♣tr♦s♦♣í ♥rrr♦ ♣♦r r♥s♦r♠ ♦rr ♥ t♥ ♦t t♥ ❯❱❱s s rt♥❯❱❱s ♣tr♦s♦♣② s♣tr♦s♦♣í ❯❱❱s ♣♦r t♥ s ❯ ♣r♦♥t♥ ♥t♠ ♥trr♥ s ♥t♦♠tr♥tó♠tr♦ s♣♦st♦s ♣r♦♥t♦rs ♥trr♥ á♥t② ❱♦t♠♣r♦♠trí trs ♣♥ts ♦s rst♦s s rtr③♦♥s t♦s ♥ tr♦s ♣r♦s ♥str♦ r♣♦ ❬ ❪ sr♥q strtr ♥tr♥ s µPs ♦♥sst ♥ ♥ ♠tr③ ♣t ♠tár♥r ♦♥ r♣♦♥s ♥♥♦♣rtís ♠♥tt ♥ st♦ s♣r♣r♠♥ét♦ rrs ♦♠♦ Ps s♣rss ♥ st ♥♥♦strtr s µPs ♦♥r ♦s ♣r♦♣s r♥ ♦♠♣♦rt♠♥t♦ ó♠♦ ♦♥ ♠② rsst étr s♣r♣r♠♥ts♠♦ t♠♣rtr s♣r♦r t♠♣rtr ♦q♦ TB ≈ 175 P ♦srrsq ♣r♠r s rtrísts ♠♥♦♥s stá s♦ tr♥s♣♦rt étr♦ trés ♠t ♥♦ ♠♥trs q s♥ rtríst

♣ít♦ ♥tr♦ó♥

♣ s♦rs ♣rs♥ s Ps ♣st♦ q ésts ♣rs♥t♥ ♦♦♠♣♦rt♠♥t♦ ♠♥ét♦ ♠♥trs q ♦♠♣♦♥♥t ♠tá♦ ♣rs♥t♠② ♣♦r ♦♠♣♦rt♠♥t♦ ♣r♠♥ét♦

♥♦ ♣r♦♠♥t♦ srt♦ r♠♥t ♠s rr ② q srát♦ ♥ ♣ít♦ r♥♦ s ♠③ ♦♥ ♠tr③ ♣♦♠ér ú♥ ② ♣♦str♦r♠♥t s r ♥ ♣rs♥ ♥ ♠♣♦ ♠♥ét♦ ♥♦r♠ ①tr♥♦ st ♣r♦s♦ ♣♦r ♦ ♦ ♥ ♥s♦tr♦♣í ♥tr♥ ♥ ♦♠♣♦st♦ ♥♦ s♦ ♣t r♦s♦♣② r♦s♦♣í Ó♣t ♠♥ó♥ s ♦sr♥ strtrs r♥♦ ♦r♥ts ♥ ♦r♠♣rr♥ ♥ ró♥ ♠♣♦ ♠♥ét♦ ♣♦ r♥t r♦ ♠tr ♦ r♦ ♣rs♥t tr♦ ♥♦s rrr♠♦s ss strtrs♦♠♦ ♦♠♥s ♥r♠♥t♥♦ ♠♥ó♥ ♦ ♥ t③♥♦ ♠♥ó♥ s s③ ♥ ♠②♦r t strtr ♥tr♥ sts rqttrs ♦♠♥rs s ♠s♠s stá♥ ♦♥♦r♠s ♣♦rstrtrs t♥rs ♠s ♥s ♥ ♠♦ ② ♠s ♥s ♥ ♦s ①tr♠♦sq s ♦♥t♥ ♥ st ♦♥t①t♦ ♥♦♠♥r♠♦s ♦♠♦ ♣s♦♥s s strtrs ② s rá rr♥ ♠tr ♦♠♣st♦ srt♦ ♦♠♦ trtr t♦♠r ♦♠♣♦st ♦♠♣♦st♦ st♦♠ér♦ strtr♦ r sq♠t③ strtr ♥tr♥ ♦ ♥áss

r strtr ♥tr♥

s ♠♣♦rt♥t str q s strtrs s ♦sr♥ s♦♦ ♦ rt♦s ♣rá♠tr♦s ①♣r♠♥ts ♣r♣ró♥ ♠tr ♦♠♣st♦ P♦r

r③♦ ♦s rs ♠tr② ♦r♥ ② ♦♠s ♥r♠♥♥ ♥s ❯♥rstätrs♥ ♠♥ ❳♥♦♥ ♦♥ ② ❲♥ ❨♥ ♥t♥t trs ♥ ❱rt♦♥♦♥tr♦ ♦rt♦r② ♥ ♣♦r s ♦r♠♦rs rs s♦r strtr ♠trs♦♠♣st♦s ♥s♦tró♣♦s

♠♣♦ s ♦♥♥tró♥ r♥♦ s ♠② ♥♦ s ♦sr srt③ó♥ ♥ rqttr ♥tr♥ r♥♦ s♥♦ q ♠s♠♦ ♦r♠♥ r ♦♥t♥ ♥♦ ♦♠♦é♥ ♦♥ ♦r♥tó♥ ♣rr♥ ♥ ró♥ ♣ó♥ ♠♣♦ ♠♥ét♦ ♦ ♠s♠♦ r③♦♥♠♥t♦ s t♠♣♦ ①♣♦só♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ② ♥t♥s ♠s♠♦ s♦♥ s♥t♠♥t ♦s s rrá r♠át♠♥t ♥t ♦♠♥s♦r♥ts st♦s ♦s ♥tt♠♥t r③♦♥s ♥ s♦ ♦sr♦s ②st♦s t♥t♦ ①♣r♠♥t ♦♠♦ tór♠♥t ♥ sst♠s rr♦í♦s❬ ❪ ♦♠♦ sí t♠é♥ ♥ ♦♠♣♦st♦s s♦s ♥ rs♥s ♣st♦♠érs❬❪ ② st♦♠érs ❬ ❪

♥ ♥ tr♦ ♥tr♦r s ♣r♣rr♦♥ ♦♠♣♦st♦s P34❬❪♦♥ rs ♦♥♥tr♦♥s r♥♦ ❬❪ ♦♦s ♦s r♦♥ rtr③♦s♣♦r ❱ ❯ ①tr♦♠trí ② ❱♦t♠♣r♦♠trí trs♣♥ts ♣rs♥t♥♦ ♠② rsst étr ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ ♠♥♦r ♦♥♥tró♥ r♥♦ 4.2% ♣rs♥t ♠②♦r ♥s♦tr♦♣í ♥ t♦s s ♣r♦♣s s ♥♣rtr t♦s s ♠strs ♣r♣rs s♥♦ ♦ ♦♥t♥♦ r♥♦♦ s ♦♥♦♥s ①♣r♠♥ts srts ♥ ó♥ ♣rs♥tr♦♥♦♥t étr ♣r s♦♠♥t ♥ ró♥ ♦r♥tó♥♣rr♥ r♥♦ ♦♥ó♥ q ♠♦s ♦ ♥ ♠r ♦ttr ♥s♦tr♦♣② ♥s♦tr♦♣í étr ♦t s rs ② ♠str♥ sq♠át♠♥t ♥ó♠♥♦ ♥ st t♣♦ ♠trs ❯♥ ♣r♦♣ ú♥ ♠ás ♥trs♥t ♣ srr ♥áss stó♥ ①♣r♠♥t ♠♦str ♥ s rs ② ♦ ♦♥♦♥s ①♣r♠♥ts ú♥ ♠ás rstr♥s ♠tr ♣ ①r ♦ ♠é♥♦s ♦rr♥t étr ♥♦ ♥ s♦♦ s ♦s tr♦♦s s ♥♣rt♠♥t ♥♦s ♥ s rs ♦♣sts ♦♠♣♦st♦ ♥s♦tró♣♦♠♥t ♦♥ó♥ ♦ ♠♣ ♣r♦ ♥♦ rs

♥ ♦tr♦ tr♦ ♣r♦ ♥str♦ r♣♦ ❬❪ ♣r♦♥♦ ♥ s t③ó ♣r ♠♣♠♥tó♥ ♦♥t♦rs ást♦s ♠♥s♦♥s t♣♦ ❩r © ♣r ♦♥①ó♥ rt♦s ♥tr♦s ♥ ♣r♦ ♦s♦♥t♦rs srr♦♦s ♣r♠t♥ r③r ♦♥①♦♥s ♥ ♦s ♠♥s♦♥s ♦q ♦♥stt② ♥ r ♠♦r rs♣t♦ ♦s ♦♥t♦rs ♥♠♥s♦♥str♦♥s ❬ ❪

♦s ♠trs ♦♠♣st♦s ♦r♠♦s ♣♦r ♠trs ♥♦ ♦♥t♦rs rá♠s ♦ ♣♦♠érs ♦♥ r♥♦s ♦♥t♦rs tr ♣♥♣rs♥tr ♦ s ♦♥♦♥s ♣r♣ró♥ s rs♣st ♣③♦rsst ♥♦ ♥ s r ♠♦ ♥ rsst♥ étr ♠tr ♣♦r♣ó♥ t♥só♥ ♠á♥ rs♣t♦ t♦r ♣ ♥♦♥trr ♥ trtr ♥♠r♦s♦s ♠♣♦s r♥ts r♥♦s ♦♠♦ sr ♣qts rt♦ ❬ ❪ s ❬❪ ♥♥♦s ♠ts ♥♦ ❬❪ ♣♦♦s♠tá♦s ❬ ❪ r♥♦ ❬❪ ② ♣♦♦s ó①♦s ♦♥t♦rs ❬ ❪

♣ít♦ ♥tr♦ó♥

r ② ♥s♦tr♦♣í étr ♦t s ♠ ♦rr♥t ♥♦ ♥s♦♦ s ♦s tr♦♦s s ♥ ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ ② ♥s♦tr♦♣í étr ♦t ♦ strtr r♥♦s s ♠ ♦rr♥t ♥♦ ♥ s♦♦ s ♦s tr♦♦s s ♥♣rt♠♥t ♥♦s ♥ s rs ♦♣sts ♦♠♣♦st♦ ♥s♦tró♣♦

♥ ♥t♦ s ♣♦♥s t♥♦ós ♦s ♠trs ♦♠♣st♦s ♦♥r♥♦s ♦♥t♦rs tr ♥♠rs♦s ♥ ♥ ♠tr③ st♦♠érr♣♦rt♦s st ♠♦♠♥t♦ ♣rs♥t♥ r♥s ♥♦♥♥♥ts q ♥ ♦ ♠♣rt srr♦♦ s♥s♦rs t♥só♥ ♣rtr ♦s ♠s♠♦s♠♣♥♦ s ♣ó♥ ♥ ♠♣♦ s♥s♦rs ♣r s ♦♠r③ó♥ sr ♥♦ s♦♥ rrss s r q rs♣st ♣③♦rsst rí ♥ ♦r♠ rrrs ♠ q s s ♠tr ♥♦ ♥t♦♦ t♣♦ ró♥ t♥♥ rt ♣♥♥ ♦♥ t♠♣rtr tr♦ ♥ sr ♠tó♥ ♣r s ♣ó♥ ♣rát t③♥r♥ ♣♦r♥t ♣rtís r♥♦ ♥ ♦♠♣♦st♦ ♦ q ♣rss ♣r♦♣s ásts r③á♥♦♦ ♥ ♠♦s s♦s s ♣rtís r♥♦ t③s ♣rs♥t♥ rtrísts rr♦ ♦ rr♠♥éts ♦♥♦ t♦ ♥ ♣♦s ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣rr ú♥ s♣és ♠♥r ♦ stí♠♦ ♥♦ s♦♥ ♠♥ét♠♥t rrss ♣rs♥t♥ rs♣st s♦tró♣ ♦ ♦♥ r ♥s♦tr♦♣í ♠♣♦st♥♦ srr♦♦ s♥s♦rs ♠♣♦ t♥s♦♥s rst t♥♦ó♠♥t í ♥♦r♣♦rr tr♦♦s ♠tr s♥s s♥ q ést♦ tr♦r rs♣st ♠s♠♦ ♦♥ s♦

♦♠♥♥♦ ♣③♦rsst♥ ♦♥ ♦s sst♠s ♣♥♦t♥rs ♠trs ♦♠♣st♦s ♦♥ rs♣st ♣③♦rsst ♥s♦tró♣

♦srá♥♦s ♥ ♠♦ rsst♥ étr ♠tr ♣♦r ♣ó♥ t♥só♥ ♠á♥ ♥ ró♥ ♥s♦tr♦♣í strtr r♥♦ ♦♠♦ rstrá ♥t ♣r t♦r ♦ ♦♥ó♥ ♥ ró♥♦rt♦♦♥ ♦r♥tó♥ r♥♦ ♠tr s ♦♠♣♦rt ás♠♥t♦♠♦ ♥ s♥t étr♦ ♦♥ ♠② ts rssts étrs s♦s ♠tr③ s♥t ♦♥ ♥ rs♣st ♣③♦rsst

♦ ♦ s s♣r♥ q ♥ ♦r♠ s♣rr ♦s ♣r♠r♦s ♥♦♥♦♥♥♥ts t♦s ♠s rr s t③ó♥ ♠trs q♣rs♥t♥ ♦♥ r♥ r♥♦♠tr③ ♠trs ♣♦♠érs ♦♥r♦ ♦♥t tó♥ tér♠ ② t③ó♥ sst♠s♥s♦tró♣♦s r♥♦s ♥ st♦ ♣r♠♥ét♦ ♦ s♣r♣r♠♥ét♦ t♠♣rtrs ♦♣ró♥ s♥s♦r ♦♦s st♦s t♦rs s♦♥ sts♦s♣♦r P34❬❪ 4.2% ♥ t♦ ♣ít♦ srr♦ ♣r♠r ♦t♦ ést ss sñ♦ ró♥ ② rtr③ó♥ ♥ rr♦ s♥s♦r t♥só♥ ♠á♥ ♦♥ rs♣st ♣③♦rsst ♦♠♣t♠♥t ♥s♦tró♣ ② rrs s♦ ♥ ♦ sst♠

tr ♣r♦♣ q s♦ ss♠♥t st s ♦t♦s ♣♦r ♥ss t♥♦ós s♥♦ ♦t♦ ♣r♥♣ ♣rs♥ttr♦ ♦♥sst ♥ r á s ♥♥ ♦s r♥ts ♣rá♠tr♦sstrtrs ♠tr ♥ ♣r♦ ♦t♥r sst♠s q①♥ ♦r t③♦ ♣r t ♥ stá s♦ ♥ st♦ sst♠s ♣r♦t♦s rt♥rs s♠♥t♦s rts ♦♥sr♥♦ ♦♠♦ ♦♥s♥ rt ♥s♦tr♦♣í ♣r♦t t♦t ♣r♦♦♥ ♦s ♦t♦s s♦♦ ♥ ró♥ ♦r♥tó♥ ♣rr♥ ést♦s s♣t♦ ♦s ♥t♥ts t ❬❪ ♥ st♦ ♦♥t étr sst♠s ♦♠♣st♦s ♦♥ ♠♦á♥♦♦s ♦♠♦s♠♥t♦s rts ♦♥t ② s♣rsó♥ ♥r ♥♦r♠ ♦③♥♦ s ♥trés ♥ sst♠s r♦s ás r♥t♠♥t ❩♥ t ❬❪r♣♦rt♥ s♠♦♥s ♥♠érs ♥ sst♠s ♣r♦t♦s tr♠♥s♦♥s rs í♥rs ♦♥ stró♥ ♥r s♦tró♣ ② ♥s♦tró♣ ♥ srtí♦ s ú t♦ ró♥ s♣t♦ ♦s ♦t♦s ♣r♦♥ts♦♥srá♥♦♦s é♥t♦s ♥ s♦ sst♠s ♥s♦tró♣♦s ♦♥sr♥♥ t♣♦ ♣rtr stró♥ ♥r ♣r♦♦ ♣♦r ♥tró♥ ♦s ♦t♦s ♣r♦♥ts ♦♥ ♥ ♦ ❲t ② ♦♦r♦rs ❬❪ ♦♥sr♥sst♠s ♣rtís ♥tr♣♥trs ♦♥ stró♥ ♥r ♥s♦tró♣ ♣♦ tr♥s♦♥s ♣r♦ts ♥ sst♠s tr♠♥s♦♥s ú♦ss á ♥tr③ tr♠♥s♦♥ ♦s sst♠s st♦s ② sr♥ts ♦♥s s str♦♥s ♥rs ♠②♦r r♥ rs♣t♦ ♣rs♥t st♦ r ♥ q ést♦s tr♦s ♥♦ s ♦③♥ ♥ s♥♦ ♥ s♣t♦s ♥rs ♣r♦ó♥ r♥ó♥ str♥s♦♥s ♣r♦ts ♥ s r♥ts r♦♥s ♠tr ♥ q♥♦s ♥♦r♠♦s ♥ ♦s ♣ít♦s ② s♦ r♥t♠♥t ♦♥sr ♣r ♣r♦ó♥ s♠♥t♦s rts ② ♦tr♦s ♦t♦s ♠♥s♦♥s

♣ít♦ ♥tr♦ó♥

♣♦r rt♥s ② ♦♦r ❬❪ ♣r♥♣ ♦t♦ ♦ tr♦ s ♥tr♦r ♥ ♥ té♥ s♠ó♥ ♣r ♠♦♦s ♣r♦ó♥ ♥ ♦♥t♥♦ ② ♣r s♦ sst♠s ♣r♦ó♥ s♠♥t♦s rtssó♦ s ♦♠♣t ♠r ♣r♦ó♥ ♣r sst♠s r♦s ♠r♦só♣♦s ♦♥ stró♥ ♥r s♦tró♣ ♥ ♠♦ ♥ ♣rs♥t sss ú ♣r♦ó♥ s♠♥t♦s rts ♥ sst♠s rt♥rs♦♥ str♦♥s ♥rs ② ♦♥t ♦t♥s ①♣r♠♥t♠♥t ♣rtr ♠strs rs sst♠s ①♣♦r♥♦ t♦ sstr♦♥s ♥ ♣r♦ ♦t♥r sst♠s ♦♥

P♦r ♦tr ♣rt ①st♥ ♠♦♠♥t♦ ♠② ss♦s ♥♦r♠s r st♦ ①♣r♠♥t rs♣st ♣③♦rsst ♠trs ♥tr♦s s st♥ ♦s tr♦s ♣♦♥r♦s ♦sss ② ♦♦r♦rs ❬❪s ss♦s ú♥ s♦♥ ♦s ♦rs tór♦s ♦ ♥ó♠♥♦ sí s ♦ss trr ♦t♦ ♣r♥♣ ♣rs♥t ss s srr♦♦ ♥ ♠♦♦♦♥sttt♦ ♣r rs♣st ♣③♦rsst t♦t♠♥t ♥s♦tró♣ st♦ssst♠s ♦♥ ♥ ♣rtr ♦♠♦ s ♣rs♥trá ♥ ♣ít♦ s rá ♠♦♦ srr♦♦ ♣r s♦ P34❬❪4.2% ♥trés t♥♦ó♦♥♥r

P♦r út♠♦ ♠r♥ s♣ t♥ó♥ q♦s ♦♥ r♥♦s ♦♥t♦rs ♦rr♥t étr q ①♥ ♦♥♠♥t t ♠♥ét ♣r s♣r♣r rr ♦ rr♦♠♥ts♠♦ ♦♠♦ ♣♦r♥t♠♥t ♣♦r ♦tr♦s ♥st♦rs ❬ ❪ ② ♣♦r ♥str♦ r♣♦ ❬ ❪ st♦s ♠trs ♣♥ ①r rs♣st ♠♥t♦rsst ♥t ♣r ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ s♦r ♠tr strtr♦ ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ s ♦sr ♥ s♠♥ó♥ rsst♥ étr ♥ ró♥ ♥ ♠ ♥str♦ ♦♥♦♠♥t♦ ♠♦♠♥t♦ ♥♦ s s♣♦♥ ♥ trtr ♥ ♠♦♦ ♦♥sttt♦♣r ♦♠♣r♥só♥ ést ♥ó♠♥♦ ♥ s ♦ q ♦♥stt② út♠♦♦t♦ ♣r♥♣ ♣rs♥t tr♦ st út♠♦ s♣t♦ srr♦ ♥ ♣ít♦

t♦r ♥♦trá q ♥ ♠s rs ♠♦strs ♥ ♣rs♥t sss ♠♣♥ tít♦s s ② ♥♦♠♥tr ♥ ♥és ♣s ♥ ♦s s♦s s♠♥tr♦♥ s rs ♦r♥s t③s ♥ ♦s rtí♦s ♣♦s ♥rsts ♥tr♥♦♥s

♣ít♦

trs ② ét♦♦s①♣r♠♥ts

s♥tst ♥ s ♦rt♦r② s ♥♦t ♠r t♥♥ s s♦

♦♥r♦♥t♥ ♥tr ♣♥♦♠♥ tt♠♣rss ♠ s t♦ t② r r②

ts

r ♦♠ ♦♦sr

s♠♥ ♥ ést ♣ít♦ s t♥ ♦s ♠trs ② ♠ét♦♦s ①♣r♠♥ts t③♦s ♣r ♣r♣ró♥ ② rtr③ó♥ ♦s♦♠♣♦st♦s strtr♦s st♦s

í♥tss ♠tr r♥♦

í♥tss ♥♥♦♣rtís ♠♥tt

♥ r♣♦rt♦ ♥♠r♦s♦s ♠ét♦♦s ♣r♣ró♥ ♥♥♦♣rtís ó①♦s ♥ ♣rtr ♣r s♦ ♠♥tt s ♥ srt♦ ♣r♥♣♠♥t tr♦ r♣♦s té♥s s♥téts sr r♦tr♠ ❬❪ ♣♦rs♦♠♣♦só♥ tér♠ ♣rrs♦rs ♦rá♥♦s ❬❪ ♥ sst♠s ♠r♦♠s♦♥♦s ❬❪ ② ♣♦r ♦♣r♣tó♥ ❬❪ str♦ r♣♦ t③♦ ♠ét♦♦ ♦♣r♣tó♥ á♦s ♣r ♦t♥r ♥♥♦strtrs ♠♥tt ❬ ❪ ♦tt ❬ ❪ ♦tts sstts♦♥ s♠r♦ ❬❪ ♥íq ❬❪ ② ó①♦s rr♦ sstt♦s ♦♥ s♠t♦ tr♦ ❬❪ ♥ ♣rs♥t tr♦ s ♦♣tó ♣♦r té♥ t s♦ ♣r ♦t♥ó♥ ♥t ② ♣r③ ♣r♦t♦ ♥srs♣r ró♥ ② ♦♣t♠③ó♥ s♣♦st♦s té♥ ♣

♣ít♦ trs ② ét♦♦s ①♣r♠♥ts

sí♥tss ♥♥♦♣rtís ♠♥tt 34 Ps ♦ s♠♣♠♥t Pss srt♦ ♥ rtí♦s ♣r♦s ❬ ❪ ② s t ♦♥t♥ó♥

♦t ♥♥♦♣rtís ♠♥tt s ♣r♣r s♥♦ 22.25 ♠ s♦ó♥ ♦s 3 · 62 0.450 ② 2 · 42 0.225 2 : 1 ♥á♦ ♦rír♦ 0.4 ñ ♦t ♦t s♦r 200 ♠ s♦ó♥ ♦s 1.5 ♣r♠♥t st ♣ = 14 ♦ t ♦ tó♥ t♠♣rtr rt♦r 60 s ♦♥tr♦ ♠♥t ró♥ tr♠♦stt③ ♣♦r ♠s ♠s♠♦ ♦sr ♥♠t♠♥t ♣r♣tó♥ ♥ ♣♦♦ ♥♦ ♠rró♥ ♦sr♦ ♦rrs♣♦♥♥t s Ps 34 t♠♣rtr ② tó♥ s ♠♥t♥♥ r♥t ♦só♥ t♦t s♦ó♥ tó♥ ♦♥t♥ó♥ s r③ ♥ stó♥ ♣r♣t♦ ♥ ♠③ ró♥ ♠♥t tó♥ ♠s♠ t♠♣rtr r♥t 2 ♣r♦♠♥t♦ s♥tét♦ srt♦ sr③ ♥ t♠ósr ♥tró♥♦ ♣r tr ♦①ó♥ ♦s ♦♥s sq♠ ①♣r♠♥t s ♠str ♥ r s 34 Ps ss♣r♥ s♦r♥♥t ♣♦r tr♥tró♥ tr♥tr 12000 ♣♦r 20 ♠♥t♦s ♣r♣t♦ ♦t♥♦ s ♦♥ r♣t♥♦ ♦s ♦s ♦♥tró♥ st q ♣ s 7 ♣r♦①♠♠♥t ♦s ♦♥tró♥ ♦♥r♠ ♣♦r P❳ q s sñs s♦s ♠♣r③ sí♥tsss ♠♥♥ ♦ ♥♦s ♣♦♦s ♦s ♦♥tró♥ ❬ ❪♥♠♥t s 34 Ps s s♥ ♥ st í♦ 40 ♣♦r 24 ♦rs♦s ♥áss r③♦s ♣♦r ♠str♥ q ♦r♠ s Ps s sér♦♥ ♥ stró♥ ♦♥♦r♠ ♠♦♥♦♠♦ á♠tr♦s ♦♥ s ♠á①♠♦ ♥13 ♥♠ ♥ ①♥t ♦♥♦r♥ ♦♥ t♠ñ♦ ♦s ♦♠♥♦s rst♥♦s♦s ♠♥t ró♥ ②rrr ♣rtr rt♦r♠s♦t♥♦s ♠♥t P❳ (14± 2) ♥♠ ❬ ❪

í♥tss ♠r♦♣rtís ♠♥tt♣t

♠r♦s♦s ♠ét♦♦s s ♥ ♣♦ ♣r ♦t♥r ♦♠r♦s ♥♥♦♣rtís ó①♦s ♦♥ ♠ts ❬❪ ♥ ♣rs♥t tr♦ ♦t♦ ♦t♥r ♦♠r♦s ♦♥t♦rs tr q ♣rs♥t♥ st s 34 Ps t♥♦ s ♦①ó♥ Pr t ♥ s ♦♣tó♣♦r ♣t ♠tá ♦ s ♦st♦ ♥♦ ♠② ♦ r ♣r♠ ♦s ♦①♥ts t♠♦sér♦s ② s á ♠♥♣ó♥ ♣r♦t♦♦♦ sí♥tss♦ ♣r♦♠ ró♥ ♦♥s s♦r♦s ♥ s♣r s 34 Ps ♣t♦ ♦s tr♦s ♥ t ❬❪ ❨♥ t ❬❪ ss t ❬❪ té♥ ♣ s srt♦ ♥ rtí♦s♣r♦s ❬ ❪ ② s t ♦♥t♥ó♥ ♣r♣r ♥ s♣rsó♥ s Ps ♥ s♦ó♥ 3+2 ♥ ró♥ ♠♦r 1 : 10 ss♣♥só♥ s s♦♥ r♥t 30 ♠♥t♦s 20/25 ② ♦ s ♥t 50 ♠♥t♥ t♠♣rtr ♣♦r 20 ♠♥t♦s ♦ tó♥ ♥t ♥ s♥t ♣s♦ s ñ s♦r ss♣♥só♥ 34+ ♥ s♦ó♥

í♥tss ♠tr r♥♦

r sq♠ ①♣r♠♥t t③♦ ♣r sí♥tss ♥♥♦♣rtís ♠♥tt ♥r ♥tr ♥tró♥♦ ♠♣♦ r♦♦♥ s♦♦♥s ② 2 : 1 rt♦r r♦ ♥♠s♦ ♣♥ t♦r ♠♥r tr♠♦stt③

♦s ♠♦♥♦rt♦ 0.4 ♣♦r ♦t♦ r♠♥♦ r♦ s ♠♥t♥ t♠♣rtr ② tó♥ ♥t r♥t ♦r P♦r r♠♥t♦ s 34

Ps ♣♦♦ ♠rró♥ ♦sr♦ s t♦r♥ ♠rró♥ r♦ s ♠r♦♣rtís ♠♥tt♣t ♦r♠s rrs ♥st♥t♠♥t ♦♠♦ 34❬❪ ♦ µPss s♣r♥ ♠③ ró♥ ♣♦r ♠♥t③ó♥ s♥♦ ♥ ♠á♥ ♣r♠♥♥t ② ♦ ♣♦r tr♥tró♥ ♣r♦①♠♠♥t ss ♦s ♦♥tró♥ s♦r♥♥t ♦t♥♦ ♦ s♣ró♥ s♦♠♣t♠♥t tr♥s♣r♥t ♥♠♥t s µPs s s♥ ♥ st í♦ 40 ♣♦r 24 ♦rs s ♠r♦♣rtís ♦♠♣sts sí ♦t♥s ♣rs♥t♥ ♥ stró♥ ♠♦♥♦♠♦ t♠ñ♦ ♦♥ ♠á①♠♦ ♥ d = 1.3 µ♠tr♠♥♦ ♣♦r ② ❬❪

♦♠♦ s ♠♥♦♥ó ♥ ♣ít♦ rtr③ó♥ ♦♠♣t sPs ② µPs r③ ♠♥t ♥ r trí té♥s ♥trs P❳ ❱ ❯❱❱s ❯ ② ❱♦

♣ít♦ trs ② ét♦♦s ①♣r♠♥ts

t♠♣r♦♠trí trs ♣♥ts ♦s rst♦s t♦s ♥ tr♦s ♣r♦s❬ ❪ sr♥ q strtr ♥tr♥ s µPs ♦♥sst ♥ ♥ ♠tr③ r♥r ♣t ♠tá ♦♥ r♣♦♥s Ps ♠♥tt st♥♥♦strtr r♣rs♥t ♥ r ♦♥r ♦s ♣r♦♣s r♥ sr ♦♠♣♦rt♠♥t♦ ó♠♦ ♦♥ ♠② rsstétr s♣r♣r♠♥ts♠♦ t♠♣rtr s♣r♦r TB ≈ 175

r sq♠ sí♥tss 34❬❪

Pr♣ró♥ ♦♠♣♦st♦s P34❬❪

Pr ♦t♥ó♥ ♦s ♦♠♣♦st♦s st♦♠ér♦s strtr♦s s② ♥t ♥trr③♥t P ②r ♦♦r♥♥ s ♠③♥♥ ♣r♦♣♦r♦♥s 1 : 10 ♣♣ 20/25 P♦str♦r♠♥t s r ♠③♦♥ s ♠r♦♣rtís 34❬❪ 4.2% ♦♥♥tró♥ ó♣t♠ ♥tér♠♥♦s ② ♣r♦♣s ♠á♥s ♦s ♦♠♣♦♥♥t ♣♦♠ér♦s ② r♥♦ ♥♦rá♥♦ s ♣s♥ rt♠♥t ♥ ♥③ ♥ít t ♠③ s ♦♠♦♥③ ♠á♥♠♥t ♣r ♦rr s♣rsó♥ r♥♦② s ♦♦ ♥ á♠r í♦ 20/25 r♥t 2 ♦rs st ♠♥r♦♠♣t♠♥t s rs r sí ♠③ ♦♠♦♥③ t♦í s ♦♦ ♥ ♥ ♠♦ í♥r♦ ♠♥♦ ♦ 3 ♠ r♦② 1 ♠ á♠tr♦ ♦ ♠♦ s ♣rt ♥ s♣♦st♦ srr♦♦ ♣♦r♥str♦ r♣♦ q ♣r♠t r♦ ♠str ♥ t♠♣rtr♥ ♣rs♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦Hcuring ♠♥trs ♠str sr♦t ♦ ♦♥t♥t ♣r♦①♠♠♥t 30 r♣♠ ♦r♠ t tr ♥tó♥ r♥♦ és r st ♦r♠ ♠③ sr Tcuring = (75± 5) ♥ ♣rs♥ µoHcuring = 0.35 s r♥t3 ♦rs ♥♦r♠ ♠♣♦ ♠♥ét♦ Hcuring s ♥③ t③♥♦

Pr♣ró♥ ♦♠♣♦st♦s P34❬❪

♥ tr♦♠á♥ stá♥r ♠♣♥ ❱r♥ ❱ ♣r♦st♦ ♦s ♣♦♦s 10 ♠ á♠tr♦ ♥♦

r q♣♦ t③♦ ♣r r♦ P34❬❪ ♦ ♠♥♦ t③♦ ♣r r♦ ♦♠♣♦st♦ s ♣③s í♥rs ♣r♠t♥ str ♦♥t ♥ ♠str ♥ ①tr♠♦ s♣r♦rr♦ ♣♥ s ♠str ♥ ♠str P s♥ r♥♦ ♥♦rá♥♦r ♥s♠♦ ♣r s♣♦st♦ r♦ ♥s♦r t♠♣rtr ♠♦ r♣t♦r ♠♦ rr ♦♥tr♦♦r t♠♣rtr ♠♦t♦r ♦rr ♦ ó♥ ♠♣ rr ♥ ♣♥ s♥s♦r t♠♣rtr ♦ ♥ s ♣♦só♥♥ rt ♠♥♦ ♠á♥s ♦rtsí r r♥♦ ③

qí♠ s♦ ♦r♠ó♥ stó♠r♦ ♥trr③♦ ♣♦r ♠♦ r♦ tér♠♦ s rs♠ ♥ r ♠♦s ♦♠♣♦♥♥ts P s ② ♥trr③♥t s♣♦♥s ♦♠r♠♥t ♦♥t♥♥ ♦ó♠r♦s s♦①♥♦ ♦♥ r♣♦s ♥í♦s tr♠♥s ♥t r♦ ♦♥trr③♥t t♠é♥ ♥② ♦ó♠r♦s s♦①♥♦s ♥trr③♠♥t♦ ♥♦ ♦s s ♣rs♥t ♠♥♦s trs ♥s s♦rr♦ ♦♠♣♦♥♥t s ♥② ♥ t③♦r ♣t♥♦ q ♣♦st r♦

♣ít♦ trs ② ét♦♦s ①♣r♠♥ts

stó♠r♦ ♣♦r ♠♦ r♦♥s ♥trr③♠♥t♦ ♦r♥♦♠tá❬ ❪ ♥♦ ② t③♦r s ② ♥t r♦ s ♠③♥ t③♦r ♣rt♣ ♥ r♦ stó♠r♦ ♣s t③ ó♥ ♦s ♥s ♥ s♦r ♦s r♣♦s ♥í♦s ♥ ♦r♠á♥♦s ♣♥ts22 βó♥ ♥s ♥ ♥í♦ st ♣r♦s♦s ♥♦♠♥ r♦só♥ ♦s ♦s ♥s ♦s ♠út♣s st♦s rt♦s ♥ ♥ ♦r♠ó♥ ♥trr③♠♥t♦s tr♠♥s♦♥s ❯♥♥t st t♣♦ r♦♥s ó♥ s q ♥♦ s ♥r♥ ♣r♦t♦srss ♠ás ♠♦♥♦ ró♥ ♥tr ♦s ♦♠♣♦♥♥ts ♥t r♦s ♣♥ ♠♦rs s ♣r♦♣s ♠á♥s stó♠r♦♦t♥♦ ♥ t♦ ♥r♠♥t♥♦ s ró♥ s ♦t♥ ♥ stó♠r♦♠s rí♦ ♦♥ ♠②♦r ♠ó♦ ❨♦♥ ♠♥♦♥r q ♥r♠♥t♥♦ t♠♣rtr r♦ s r t♠♣♦ r♦ ❬ ❪

r sq♠ ♦r♠ó♥ ♥ r ♥trr③ trés ró♥ r♦só♥ ♥ ♠trs s♦s ♥ P

♦ r♦ ② s♠♦♦ ♦♠♣♦st♦ s ♦rt♥ ♠strs s♥♦ ♥ s♣♦st♦ s♣♠♥t sñ♦ ♣r t ♥ ♦♠♣st♦ ♣♦r ♥s♦♣♦rt ♠♥♦ q ♣♦só♥ ♥r♦ ♦♠♣♦st♦ s♦♣♦rt♣rs♥t ♥ r♥r q ♣r♠t r ♥ ♦r♠ t ♣♦r♦rtr ♣r♦♠♥t ts ♠tr Pr♦♠♥t♦s s♠rs ♥ s♦t③♦s ♣♦r ♥str♦ r♣♦ ♣r ♦t♥ó♥ ♦♠♣♦st♦s strtr♦s P ♦♥ r♥♦s ♥♥♦strtr♦s ♥íq ❬❪ ♦tts ❬❪ ②

Pr♦♣s ♠á♥s

♦tts sstts ♦♥ s♠r♦ ❬❪ ♠②♦rí ést♦s s ♥ rtr③♦ ♥ t ♠♥t ❱ ❯ rr♦♠♥ts♦♥♥ s♦♥♥ rr♦♠♥ét ② ①tr♦♠trí

Pr♦♣s ♠á♥s

♥áss s rs rs♣st ást ♠tr③ ♣♦♠ér s♥r♥♦ ② ♦♠♣♦st♦ P34❬❪ s r③ó t③♥♦ ♥ ♥③♦r t①tr t r♦s②st♠s ❳ r r ♥ ♣rs♥ttr♦ ♥③♦r t①tr s t③ó ♣r ♠r r③ ♥sr ♣r♠♥t♥r ♥ ♦ ♦♠♣rsó♥ ♦♥st♥t ♥ ①♣r♠♥t♦s ♦♠♣rsó♥ ♥① ♦ ♦♠♣rsó♥ s ó rtrr♠♥t ♥ 100µ♠s−1 s♦♥ ♣r t③ P s♦♥ í♥r ♣♥ 36 ♠♠ á♠tr♦ ♦s ①♣r♠♥t♦s s r③r♦♥ 20/25 ♦♠♣r♠♥♦ s ♠strs ♥tr 8 ② 30% s s♣s♦r ♥ 2.5 ♠♠ ár ♠str ♦♥ ♥♦r♠ ♣♦r ró♥ ♦♠♣rsó♥ s ∼ 0.8 ♠2 ♥t st ♣r♦♠♥t♦ ①♣r♠♥t r③ ♣ s ♠ ♥ t♠♣♦r ♦r♠ t q ♥♦r♠ó♥ ♣r♠r s ♦t♥ ♦♠♦ r③ rsst♠♣♦ tr♥srr♦ s ♣r♦♣s ásts r♥ts s ♠strs sr♣r♥ ♣rtr s♦s rá♦s s♣s♦r ♥ s ♠strs s tr♠♥♦ ♠♥t s♦tr ♥str♠♥t♦ ♦ ♣r♦♠♥t♦ ró♥

r ♥③♦r t①tr t r♦s②st♠s ❳ t③♦♣r rtr③ó♥ ♠á♥ ♠tr③ ♣♦♠ér s♥ r♥♦ ② ♦♠♣♦st♦ P34❬❪

♣ít♦ trs ② ét♦♦s ①♣r♠♥ts

P③♦rsst♥ ② ♠♥t♦rsst♥ ♦♠♣♦st♦ P34❬❪

Pr ♠ó♥ rs♣st ♣③♦rsst ♦♠♣♦st♦ s♠strs í♥rs 1 ♠ á♠tr♦ ② 2.5 ♠♠ s♣s♦r s ♥ ♥tr ♦s tr♦♦s ♦r♦ ♥tr♦ ♥ s♦♣♦rt ② s ♦♥t sst♠ ♥ ♣♦t♥♦stt♦ r♥t♥ ♥♦ ♦♠♦ ♥ r ② r♣rs♥t♦ ♦♠♦ ♥ ♠♣rí♠tr♦ ② ♥ ♥t ♦♥t♥ ♣♦t♥ ♥ r ♣ t♥só♥ ♥♦r♠ s♣r ♠str ♠♥t ♥ t♦r♥♦ s♥ ♥ ② s ♠ r③ ♣ ♠♥t ♥s♥s♦r r③ ♥③ tró♥ rstr♥ s rs rtrísts♣♦t♥♦rr♥t tí♣♠♥t ♥tr ±3 ❱ ♣r t♥só♥ ♣P ♣rtr sts rs s ♦t♥ rsst♥ étr sst♠ ♥♥ó♥ t♥só♥ ♠á♥ ♥① ♣ s r R(P )

r s♣♦st♦ ①♣r♠♥t t③♦ ♣r ♠ó♥ rs♣st ♣③♦rsst ♦s ♦♠♣♦st♦s P♦t♥♦stt♦ s♣② s♥s♦r r③ s♦♣♦rt st③♦r r♦♥s t♦r♥♦ s♥♥ s♦♣♦rt ♠str s♥s♦r r③ ♥③ tró♥

P③♦rsst♥ ② ♠♥t♦rsst♥ ♦♠♣♦st♦ P34❬❪

s♣t♦ rs♣st ♠♥t♦rsst s ♠strs í♥rs ♦♠♣♦st♦ 1 ♠ á♠tr♦ ② 2.5 ♠♠ s♣s♦r s ♥ ♥tr ♦str♦♦s ♦r♦ ♦♥t♦s ♥ ♣♦t♥♦stt♦ r♥t♥ st s♣♦st♦ s ♥tr ♦s tr♦♠♥s stá♥r ♠♣♥s♦s ♣r r♦ s ♠strs ❱r♥ ❱ ♠♥t st♦str♦♠♥s ♣r♦♥ ♥ ♠♣♦ ♠♥ét♦ ♥♦r♠ H ♥ ró♥ ♥trq ♦s s♣r r ♦♥ s s♣♦st♦ ①♣r♠♥t Pr srr ♥ ♦♥tt♦ étr♦ ♥tr ♠str ② ♦s tr♦♦s ♦r♦ s ♣♥ t♥só♥ ♥ P ⋆ ≈ 75 P sq♠ ①♣r♠♥t s ♠str ♥ r Pr ♠♣♦ ♠♥ét♦ ♣♦ ♠♦ ♦♥ ♥ ssí♠tr♦

r sq♠ sst♠ ①♣r♠♥t t③♦ ♣r ♠ó♥ rs♣st ♠♥t♦rsst ♦s ♦♠♣♦st♦s

r♦♣ t s♠tr s rstr r rtríst♣♦t♥♦rr♥t étr tí♣♠♥t ♥tr ±3 ❱ ♣rtr stsrs s ♦t♥ rsst♥ étr sst♠ ♥ ♥ó♥ ♠♣♦♠♥ét♦ ①tr♥♦ ♣♦ s r R(H)

st③ó♥ ♠♣♦ ♠♥ét♦ ♥♦ ést s ♠♦ ♥♦ ♦r ♦tr♦ ♦rr ♥ s♥♦s s♦s ♦rs stá♥ ♦s ♣♦r t♠♣♦ rs♣st ssí♠tr♦ ♥♦ ♠♣♦ ♠♥ét♦ ①tr♥♦s ♠♦♦ rsst♥ étr R ♠♣③ ♠r ② s♦r s rstr♦ ♥ ♥ó♥ t♠♣♦ st q ♥③ ♥ ♦r st t♠♣♦ tr♥só♥ rtríst♦ s♦♦ st③ó♥ rsst♥ étr s ♥♦s s♥♦s ♣♥♥♦ ♦r s♣í♦ H ② P ⋆ P♦r st r③ó♥ s ♦♥sró ♥ t♠♣♦ ♥tr ♠♦♥s ♠♥t♦s ♦♥ ♦t♦ srr ró♥ t♦t sst♠ ♦ ♥áss str qí q t♠♣♦ ró♥ rtríst♦ s♦♦ st③ó♥ R s ♠♦ ♠②♦r q st③ó♥ H st♦sr q t♠♣♦ ró♥ rtríst♦ s♦♦ st③ó♥ R ♥♦ stá r♦♥♦ ♦♥ t♦rs ♥str♠♥ts s♥♦ ♦♥ ♣r♦s♦ ró♥ ♥

Pr ♠ó♥ r ♠♥t③ó♥ r♥♦ 25 st③ó ♥ ♠♥tó♠tr♦ ❱ ♦r

♣ít♦ trs ② ét♦♦s ①♣r♠♥ts

t♦ H s♦r ♦♥t r♥♦34❬❪

♠ó ♦♥t étr ♣♦♦ 34❬❪ r♥ts ♦♠♣rs♦♥s ② ♠♣♦s ♠♥ét♦s t③♥♦ sq♠ ①♣r♠♥t♠♦str♦ ♥ r

r sq♠ sst♠ ①♣r♠♥t t③♦ ♣r ♠ó♥ t♦ t♥só♥ ② ♠♣♦ ♠♥ét♦ s♦r ♦♥t ♣♦♦34❬❪ tr♦♦s ♦r ♣♦♦ ♠str ♠♦ ♦♥t♥♦r rs♥ rí

st s♣♦st♦ ♠② s♠r t③♦ ♣♦r ♦♥ts ② ♦♦r♦rs♣r ♠ó♥ ♦♥t étr ♣♦♦s ❬❪ ♦♥sst ♥ ♥♠♦ rs♥ rí ♦♥ ♥ ♣r♦ró♥ í♥r ♦♥t♥ ♥tr♦ s ♦♦ ♣♦♦ ♥③r ♥tr ♦s tr♦♦s í♥r♦s ♦r♦s s s ♦♥t♥ ♥ ♣♦t♥♦stt♦ r♥t♥ st rr♦s s♦r ♥ s♥s♦r r③ q ♣r♠t ♠r r③ ♣ ♣♦r♥ t♦r♥♦ s♥ ♥ ♥ ♦r♠ ♠② s♠r ♦ srt♦ ♣r ♠ó♥ rs♣st ♣③♦rsst ♦♠♣♦st♦ Pr ♥s t♥s♦♥s s ♠ó ♦♥t ♥ ♥ó♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥♦ rr♦ r ♥tr ♦s tr♦♠♥s ♥tr♦r♠♥t ♠♥♦♥♦s

♣ít♦

♥s♦r t♥só♥ ①

♥♦♦② s st t♦♦ ♥ tr♠s ♦tt♥ t s ♦r♥ t♦tr ♥

♠♦tt♥ t♠ t tr s t ♠♦st♠♣♦rt♥t

ts

s♠♥ sr sñ♦ ró♥ ② rtr③ó♥ ♥rr♦ ① s♥s♦rs t♥só♥ ♠á♥ ♥s♦tró♣♦ ② ♣♦rtát ♦♥ rs♣st ♦♠♣t♠♥t rrs ♥ ♥tr♦ 0−550 P♥ ♠tr s♥s ♦s ♦♥tt♦s étr♦s ② ♠tr ♥♣s♠♥t♦ stá♥ s♦s ♥ ♦♠♣♦st♦s ♣♦♠ts♦①♥♦ P ♠tr s♥s s ♥ t ♦r♠♦ ♣♦rstrtrs ♦r♥ts ♦♥stts ♣♦r ♠r♦♣rtís ♠♥ttrts ♦♥ ♣t 34❬❪ ♠♣♠♥tr♦♥ ♥ sr ♦♥tt♦s étr♦s s♦s ♥ ♦♠♣♦st♦s P♣♥tr ♣t P♦rút♠♦ rr♦ s♥s♦rs ♥♣s♦ ♦♥ P

♦♥t♥♦ st ♣ít♦ ♣r♠♥t ♣♦ ♥ rtí♦ tt t ♠rt trs ♥ trtrs ♦

♥♦ ♣ s♣♦st♦ srr♦♦ ♣rs♥t ♣t♥t ♣♥♥t ♥ r♥t♥ ❳P P rá♠t

ró♥ s♥s♦r t♥só♥

♠♣♠♥tó♥ ♦♥tt♦s étr♦s

♦♠♦ s sr ♥ ♣ít♦ s♦ sst♠ P34❬❪ ♥ ♥ s♥s♦r t♥só♥ ♠á♥ rqr ♠♣♠♥tó♥ tr♦♦s q ♥♦ tr♥ rs♣st ♠tr ② q ♥♦ tr♦r♥ rs♣st ♠s♠♦ ♦♥ s♦ ss♦

♣ít♦ ♥s♦r t♥só♥ ①

Pr ♦t♥r ♦♥tt♦s ♦♥ ♦♠♣♦rt♠♥t♦ ♠tá♦ ♦s rt♠♥t ♥ s♣r ♥ t P34❬❪ s ♣r♣ró ♥♦♠♣♦st♦ ♠③♥♦ ♣♥tr ♣t ♦♠r P ♦♥t rP♥t P ♣♣s ❯ ② P ♦♥ ró♥ s ♥trr③♥t ♣♣ ♥ ♥t s rrrá st ♦♠♣♦st♦ ♦♠♦ P ♣♥t ♣♥tr ♣t t③ ♦♥sst ♥ ♥ s♣rsó♥ ♠r♦♣rtís ♣t♠tá (43± 3) % ♣♣ ♥ ♥ rs♥ rí Pr ♣r♣rr ♦♠♣♦st♦P ♣♥ s r ♥ ♣qñ ♥t P 2% ♣♣ ♣♥tr ♣t 98% ♣♣ ♠③ sí ♦r♠ s ♦♠♦♥③♠á♥♠♥t ② ♦ s ♣♦st ♥♦ s♦ ♥ r♥ s♦r s♣r t P34❬❪ ♦r♠♥♦ trs ♣rs ♦♥tt♦s rrs s♣r s ♠♣ó ♣r♠♥t ♦♥ t♥♦ ♦ q ♣rs♥t tr♥s♣r♥ ♦ tr♥s♣r♥ ♠tr③ ♣♦♠ér P ② ♦ ♦♥t♥♦ r♥♦ ♣r♠t ♦rrt ó♥ ♦s ♦♥tt♦s ♣♦st♦s s ③ ♣r♠t ♦♥tr♦r ár ♦s ♠s♠♦s ∼ 2 ♠♠ á♠tr♦ t♦ s♦ ♥ ♥♦ ♦s♦♥tt♦s ♣♦st♦s s ♥srtó ♥ ♦r 30 ❲ P♦str♦r♠♥t sst♠ ♦♥tt♦s s s ♦♦ó ♥ st 100 ♣♦r 30♠♥t♦s ♦ t♠ósr r ♦♥ ♥ ♦rr ♣♦ró♥ s♦♥t ♣r♦♥♥t ♣♥tr ♣t ② ♣r♦♠♦r r♦ P ♥t st ♣r♦♠♥t♦ s ♦t♥♥ ♦♥tt♦s ♦♥ ♠② rsst étr < 0.032Ω♠ ② ①♥t r♥ ♣r ♦♥ t ♣♦s♠♥t ♦r♠ó♥ ♥s ♥tr ♦s ♦♥tt♦s② ♦s r♣♦s ♥í♦s ♦ rs ♥ t

♥ ♦r♠ ♦♥ s ♣r♣rr♦♥ ♦♠♣♦st♦s Prt♦ ② P❲ t r♦♥ ♥♦ts ♥♦t♦s r♦♥♦ Prsút♣s ♦♥ rs♦s ♦♥t♥♦s r♥♦ 20−80% ♣♣ ♦♥ ♥ r ② ♦♠♣rr ss ♣r♦♣s ♦♥ s ♦s ♦♥tt♦s P♣♥t s♦ t♦♦s s♦s ♦♠♣♦st♦s ♦♠♦ ♣♦ss ♦♥tt♦s étr♦s s♦r s srtó ♣♦r s ♦♥t étr ② ss ♣r♦♣sr♦ós ♥ss st♦s rst♦s s♦♥ ♦♥sst♥ts ♦♥ ♦s r♣♦rt♦s♥ st♦s ♣r♦s ❬ ❪ ♥ ♥r♠♥t♦ ♥t r♥♦ stá s♦♦ ♦♥ ♥ ♥r♠♥t♦ ♥ ♦♥t étr t♠é♥s ♥r♠♥t r ♦♠♣♦st♦ ♠tr r♦ s rtr ♦♥ ② s ③ r ♥♦r♠♠♥t r♥ ♦♠♣♦st♦ ♦♥tt♦ ♦♥ ♠tr

♠♣qt♠♥t♦ rr♦ s♥s♦rs

♥♠♥t ② ♦s trs ♣rs ♦♥tt♦s q ♦♥stt②♥ rr♦ s♥s♦rs r♦♥ rt♦s ♦♥ P ♣♣ ② ♥♠♥t

t♦r r♦rr q ♦♠♦ s srt♦ ♥ t ♥ ♣ít♦ ♦s♦♥tt♦s ♥ str ♥♦s rt♠♥t ♠♦s ♦s t ♣rq ①st ♦♥t étr ♥tr ♦s

rtr③ó♥ ♦s ♦♥tt♦s étr♦s

r♦ 100 ♣♦r 230 ♠♥ ♥t st ♥♣s♠♥t♦ s ♦r s♠♥t♦ étr♦ t♦♦ sst♠ ♥②♥♦ ♦s ♦♥tt♦s q s ③ ♣r♦ rsst♥ r♥t ♥ts ís♦s ② qí♠♦s ó♠♦ ssrrá ♠s ♥t

r ♠str sq♠t③ó♥ ♣r♦s♦ ró♥ rr♦ s♥s♦rs t♥só♥ s♦ ♥ ♠tr ♥ r s ♠str ♥ ♦t♦rí rr♦ s♥s♦r ♥t♦ ♥ ♠♦♥ ♥t♦s r♥t♥♦s ♠♥trs q ♥ r s r♣rs♥t ♥♦rt ♦♥t♥ rr♦ s♥s♦r ♦♥ q♥ ♥ ♥t♦s ♦s♠♥t♦s ♦♥stt②♥ts s ♦♥tt♦s ② rt P

r sq♠t③ó♥ ♣r♦s♦ ró♥ rr♦ s♥s♦rs t♥só♥ ♠á♥ s♦ ♥ ♠tr ♦t♦rí rr♦ s♥s♦r ♥t♦ ♥ ♠♦♥ ♥t♦s r♥t♥♦s s ♠tr s♥s P34❬❪ ② ♦♥tt♦s P♣♥t ♣rs♥tó♥ ♦rts ♦♥t♥s s♥s♦r

rtr③ó♥ ♦s ♦♥tt♦s étr♦s

Pr rtr③r ♣♦rt ♦s ♦♥tt♦s étr♦s ♥ rsst♥étr ♦ s♥s♦r rr♦ s ♣r♦ó tr♠♥r rs♣st ♣③♦rsst ♠tr P ♣♥t Pr ♦ ♠③ s ♦♦ó ♥ ♥ ♠♦ ♦ 1 ♠2 ár ② 12 ♠♠ tr q♣♦str♦r♠♥t s ró ♦ s ♦♥♦♥s ♣r♠♥t sts 100

30 ♠♥ ♥ st ♦ t♠ósr r ♦t♥é♥♦s ♥ ♣st ♦ ♦♠♣♦st♦ ♣st ♦t♥ s ó ♥ s♣♦st♦ ①♣r♠♥tt③♦ rtr③ó♥ rs♣st ♣③♦rsst r r

r③♦ ♥ ♦ ♦r ♣♦r s ♦♦ró♥ ♥ s str♦♥s rr♦ s♥s♦r

♣ít♦ ♥s♦r t♥só♥ ①

r ♠str rs♣st ♣③♦rsst ♦♠♣♦st♦ P ♣♥t ♥ tér♠♥♦s ρo/ρ ♥ ♥ó♥ t♥só♥ ♥① ♣P s♥♦ ρ rsst étr ♠tr ② ρo ≡ ρ (P = 0) = 3.2Ω♠ ♦r♠ r ♦rrs♣♦♥ ♦♠♣♦rt♠♥t♦ ♥ ♦♥♦♦ ♦s ♦♠♣♦st♦s ♠tr③ ♣♦♠ér ♦♥ stró♥ s♦tró♣ ♣rtís ♦♥t♦rs s♦♠t♦s t♥só♥ ♠á♥ ♥① ♥ st♦s sst♠s ♣r♦ ♣r♦ó♥ ♥tr ♣rtís ♥s ♥♦ s ♥♦♠♣♦rt♠♥t♦ ♥r♥t s s♥♦ q ♥③ s ♦r stró♥ ♣r t♥s♦♥s ♠♦rs stró♥ ♠♥♦s ♣r♦t♦s trés ♥ ♣♥♥ s♠♦ st ♣♥♥ ①♣ ♥ ♠♦♦s♣r♠♥t srr♦♦s ❬ ❪ q ♣r♦♣♦♥♥ ♥ sr♣ó♥ ♣tór ♥tt♠♥t r③♦♥ ♦♥ ♦s ♦♥tt♦s ♦ ♥s ♥tr ♣rtís r♥♦ ♥s s r♥ ② s str②♥ ♥♦ ♥ t♥só♥ ♠á♥ s♣ s♦r ♦♠♣♦st♦

r ♦♠♣♦rt♠♥t♦ ♣③♦rsst♦ ♦♠♣♦st♦ ♦♥stt②♥t ♦s ♦♥tt♦s étr♦s t③♦s ♥ rr♦ s♥s♦r P ♣♥t ♠♥ ♥srt sr ♣tór♠♥t ♦r♠ó♥ ♠♥♦s ♣r♦t♦s ①t♥s♦s ♦♥t♦rs ♦rr♥t étr ♣♦r ♣ó♥ t♥só♥ ♠á♥ ♥①

s ♠② ♠♣♦rt♥t ♥♦tr q ♦♥t st ♠tr ♥③s ♦r ♠á①♠♦ ♣r P ∼ 200 P ♦♥ s♦♦ ♥ 16% ♠♦ ♣♦r♥t♥ s rsst ∆ρ ♥♦ ♦♠♦

∆ρ ≡ 100× ρo − ρ∞ρ∞

♦♥ ρ∞ ≡ ρ (P = ∞) st ♦r s ♠♦ ♠♥♦r q ♦sr♦ ♣r ♠tr ♥s♦tró♣♦ P34❬❪ ♣r ∆ρ = 635% st ♦r♠ q ♠♦str♦ q ♠tr t③♦ ♣r ♦♥ó♥

♥s♦r rs♣st ② rs ♠ért♦

♦s ♦♥tt♦s étr♦s ♣rs♥t ♥ rs♣st ♣③♦rsst s♣r r♥t ♠tr ♥s♦tró♣♠♥t strtr♦ st ♣r♦♣♣r str r♦♥ ♦♥ st ♦s ♦♥tt♦s ♥ ♦♠♣ró♥ ♦♥ ♦t♦ r ♣ótss s ♠♣♠♥tó ♥♣r♦t P ♣♥t ♦♥ts s♥trr③♥t ♣♣ st♣r♦t ①ó ♥ ♠♦ ♣♦r♥t ♠á①♠♦ ♥ s rsst étr ∆ρ = 48% ♥♦t♦r♠♥t ♠②♦r 16% ♦sr♦ ♣r ♦♠♣♦st♦ ♦♥♣r♦♣♦ró♥ s♥trr③♥t t③♦ ♣r ♠♣♠♥tó♥ ♦s♦♥tt♦s s♦r st ♦sró♥ ①♣r♠♥t s ♦♥sst♥t ♦♥ ♣ótss s♦st♥ ♣s ♥ ♠♥♦r ♣♦r♥t ♥trr③♥t ♦♥r♠②♦r st ♦♠♣♦st♦ ♠♥♦r ♠♦♦ ❨♦♥

♥s♦r rs♣st ② rs ♠ért♦

s♣st ♣③♦rsst

♥ st só♥ s sr s♠♣ñ♦ ♣③♦♠étr♦ rr♦ s♥s♦r ♠tr ♥♣s♦ ♥t♦ ♦s ♦♥tt♦s ♥ ♣r♠r r rs♣st ♣③♦rsst ♥♦ ♦s trs s♥s♦rs rtr③rs rtríst ♦rr♥t♣♦t♥ r♥ts t♥s♦♥s ♠á♥s 25 s ♠str♥ ♥ r t③♥♦ ♥ ♦ rr♦5 ♠❱s−1 ♥ r ♥ ♥r♠♥t♦ ♥ t♥só♥ ♠á♥ ♣ ss rs rtrísts I − V ♠str♥ r♠♥t qt♦♦s ♦s s♥s♦rs ①♥ ♦♠♣♦rt♠♥t♦ ó♠♦ ♥ t♦♦ ♥tr♦ t♥s♦♥s ♥♦ s ♦sr♥ ♠rs ♥ t♦s rt♦rs ♥ s♦s ♥tr♦s ♣♦t♥ ② t♥só♥ ♠á♥

r ♠str rs♣st s♥s♦r ♦ t♥só♥ ♠á♥♥ tér♠♥♦s rsst♥ étr R ♥ ♥ó♥ t♥só♥ ♠á♥♥① P ♣r ♥ s♥s♦r ♣rtr rr♦ ♦s r♥ts s♥s♦rs rr♦ t♥♥ rs♣st ♠② s♠r s rr q r rs♣st♣③♦rsst ♥♦ s ♠♦ ♦ ①♦♥s ♦ q ♠♥st r♥ rsst♥ rr♦ s♥s♦r r♥t st t♣♦ ♦r♠ó♥♠á♥ s str q rs♣st ♥♦ ♦s trs s♥s♦rs rr♦ s ♥♣♥♥t rst♦ ♦s s♥s♦rs ♥ ♦trs ♣rs r rs♣st ♣③♦rsst ♥♦ ♦s s♥s♦rs ♥♦ s t ♣♦r ♣ó♥ t♥só♥ ♠á♥ s♦r qr ♦s ♦tr♦ss♥s♦rs st ♦ ♦♥s♥ ♦ ♥ ♠tr rr♥ t ♠♣♦ t♥s♦♥s ♠á♥s ♥ ♥ s♣r rt♠♥t ①t♥ ♣r ♦ t③rs ♥ rr♦ s♥s♦r ♦♥ ♥♥ú♠r♦ ♠②♦r s♥s♦rs ②♦ ①t♥r ár ♠♣♦ ♥ ♦r♠ s♠rs rs ♣③♦rsst♥ ♥♦ s r♦♥ ♠♦s ♦ ♦s ♦♠♣rsó♥s♦♠♣rsó♥ ♦ q ♦♥stt② ♥ ♠♦r s♥t s♦r ♣r♦t♦t♣♦s s♥s♦rs s♦s ♥ ♠trs srr♦♦s ♣♦r ♦tr♦s

♣ít♦ ♥s♦r t♥só♥ ①

r ❱♦t♠♣r♦♠trí í r♥ts t♥s♦♥s ♣s t♠♣rtr ♠♥t ❱♦ rr♦ 5 ♠❱s−1 ♥ ♥r♠♥t♦ t♥só♥ P P1 = 40 P P2 = 110 P P3 = 207 PP4 = 301 P P5 = 563 P r ró♥ s♣s ♦♠♦rsst♥ étr R rss P í♥ só r♣rs♥t ♠♦♦ ♠♣ír♦ ♦ ♣♦r ♥srt♦ log (R−R∞) rss P ♣r P > 40P í♥ só r♣rs♥t st ♦ ♣♦r ♣♥♥ts S = (5.43 ± 0.02) × 10−3 é × P−1 ② ♦r♥ ♦r♥ s(1.53± 0.01) ♦♥ R2 = 0.99947

t♦rs ❬ ❪ ♥á♦♠♥t ♦ q s ♦♥ s ♣r♦♣sásts ♠tr rrs ♣③♦rsst ♣ strs ♣♦r s♥ t♦ ♥s ❬❪ ♥ t♦ ♥ ♦s ♠trs ♣r♠♥t srr♦♦s ♣♦r ♦tr♦s ♥st♦rs ② ♥ rt ♥tró♥ ♥tr r♥♦ ② ♠tr③ ♣♦♠ér s ② s ♣♦r rt③ó♥ s♣r r♥♦ ♦♥ ♦ s♥ ♥ qí♠♦ ♥ ♠tr③ ♦ ♥ ♣♦r ♥tr♦♥s íss rts s♥♦ ♦r♥ ♥ ①♣r♠♥t sr q s♦♠tr ♦s sst♠s ♥ ♦♠♣rsó♥ ♠á♥ ♣♦s♠♥t s♣r♦♥ r♦r♥♠♥t♦s strtrs rrrss ♥ ♥tr③ r♥♦♠tr③ ♦r♠ t q s♦♠♣r♠r ♠tr ② r♦ s ♦♥ó♥♥ ♥♦ s ♦t♥ ♠s♠ strtr ♥tr♥ st r♠♥tó♥ r♥ ♥ sr♣ó♥ tt s♥ ♥ó♠♥♦ stérss ♠á♥② ♣③♦rsst ♠tr ♥ sst♠ srt♦ ♥ ♣rs♥t tr♦ ♥tr③ ís ② qí♠ ♣ r♥♦♠tr③ t③s ♥♦s ♣rs♥t s ♥tr♦♥s rts ♥trs s ú♥ ①st ♥♣qñ s♣ró♥ ♥tr s strtrs r♥♦ ② ♠tr③ st♦♠ér s♣ró♥ ♣ ♦srrs ♠♥t ♠r♦s♦♣í ♥ ♦rts ♣r♣♥rs s strtrs t♥rs r♥♦ ♦♠♦ s ♠str ♥ srs ② s s♣ró♥ ♣r ♥♦ sr ♥ ♦♥s♥ ♦rt ♠tr ♣s s ♦sr♥ s♣r♦♥s ♠② s♠rs ♥ ♠♦r♦♦í② t♠ñ♦ ♠♥t r♦r♠♥tó♥ ♠tr ♥r♥♦ ♠tr

♥s♦r rs♣st ② rs ♠ért♦

♥ ♥tró♥♦ íq♦ ② ♦ s♠♥tá♥♦♦ rá♣♠♥t

r r♦rís P34❬❪ 4.2% P♥♦ ♦♥t♥ P♥♦s r♥srss s ♥s r♥♦ ♥♦s q s ♦sr ♥ ♦r♠ rt ró♥ s♣ró♥ r♥♦♠tr③♥ ♦♥ ♥ rs ♥ t♦♦s ♦s ♣♥s µP ♥ r♥♦ ②P ♠tr③ ♣♦♠ér

tr rtríst str s q s rs rs♣st ♣③♦rsst♣r♠♥♥ ♥trs ♦ ♠♥♦s ♠ss st♦ s ♥ ♦♥s♥ rt ♥r qí♠ r♥♦ t③♦ rs♣t♦ ♦①ó♥♥ r ② ♥♣s♠♥t♦ rr♦ s♥s♦r ♣r♠r sts r♠♦♥s s r♦r③ ♣♦r ♦ q ♥ rr♦ s♥s♦r r♦♦♥ ♠tr r♥♦ ♣r♣r♦ ♦♥ ♠ss ♥t♣ó♥ ♣rs♥t ♠s♠♦ s♠♣ñ♦ q ♥♦ ♣r♣r♦ ♦♥ r♥♦ r♥t♠♥t s♥tt③♦❯♥ ③ ♠s st♦ ♦♥stt② ♥ ♥t s♥t s♦r ♦tr♦s ♠trssrt♦s ♥ trtr ❬ ❪

r♥♦ ♥á♠♦ t♦♦s ♦s s♥s♦rs rr♦ s 0550 P ♦t♥ stró♥ ♣r♥t ♥ rs♣st ♣♦r ♥♠ ♦s 350 P rs♣st ♣③♦rsst qr ♦s s♥s♦rs rr♦ ♣ srst ♠♣ír♠♥t ♣♦r ♥ ♠♦♦ ①♣♦♥♥ ♥ r♥♦ 0550P r♦ ♦♥ ó♥

R (P ) = R∞ +A1 exp

(−Pξ1

)+A2 exp

(−Pξ2

)

♣ít♦ ♥s♦r t♥só♥ ①

♦♥ R∞ ≡ R (P = ∞) r♣rs♥t rsst♥ étr P ≫ 550 P♦r ♦♥r♥ t♥s♦♥s ♠á♥s ♠② r♥s Pr t♦♦s ♦ss♥s♦rs s ♦t♥ ♥ ①♥t st s♥♦ ó♥ ♥♦♥t♥ ♥ r ♦rrs♣♦♥ ♦ st ♣r rs♣st♣③♦rsst ♥ s♥s♦r ♣rtr rr♦ ♦s ♣rá♠tr♦s str♣r♦s s♦♥ ♦s s♥ts R∞ = (17.1± 0.2) Ω A1 = (34± 1) Ω ξ1 =(81± 3) P A2 = (61± 2) Ω ② ξ2 = (2.9± 0.1) P ♦♥ R2 = 0.9982r♦ ♦♥t ♦rró♥ Prs♦♥

st ♥áss ♣ s♠♣rs ♥ ♥tr♦ 40− 550 P r③♥♦♥ st ♠♦♥♦①♣♦♥♥

R (P ) = R∞ +A exp

(−Pξ

)

♥ s♦ s♥s♦r ♦♥sr♦ ♥ r ♦s ♣rá♠tr♦s r♣r♦s ♣♦r ♦ st s♦♥ R∞ = (17.1± 0.2) Ω A = (34± 1) Ω ②ξ = (80± 4) P ♦♥ R2 = 0.996 st ♦r♠ ♥ ♥tr♦ t♥só♥♠á♥ t♦ rs♣st ♣③♦rsst ♠t ♥ ♥③ó♥ ♥s s♠♦rít♠ ♥ t♦ ♣rtr ♠♥s♦♥③♥♦ rst tr

log

(R (P )−R∞

Ω

)= log

(A

Ω

)− log e

ξP

rá♦ ♥srt♦ ♥ r ♠str r ró♥♥ ♥ s ♦t♥ ♣♥♥t S = (5.43± 0.02)× 10−3 é ×P−1 ♦rrs♣♦♥♥t s♥s ♣rs♥t s♥s♦r

♦♠♣♦rt♠♥t♦ ♥á♠♦ s♥s♦rr♠s

r ♠str rs♣st ♥ tér♠♥♦s ♥t♥s ♦rr♥tétr I ♥ ♥ó♥ t♠♣♦ t ♥ ♣♦t♥ étr♦ ♦ ♣r ♥s♥s♦r ♣rtr rr♦ s♦♠t♦ ♦s t♥s♦♥s ♠á♥s r♥tsss rs I − t ♦♥stt②♥ ♦s s♥s♦rr♠s s♦♦s ♥ s♥s♦r s♣í♦ ♠②♦r t♥só♥ ♣ s ♦t♥♥ ♠②♦rs ♦rr♥ts étrsst ③♦ s ♦♥sst♥t ♦♥ ♦♠♣♦rt♠♥t♦ ♣③♦rsst♦ ♥t♦srt♦ ♣r♠♥t ♥r♠♥tr t♥só♥ ♠á♥ ♣ s♦r ♦ss♥s♦rs rr♦ s♠♥② s rsst♥ étr ♠é♥ r♦♥ ♦s ♦s t♠♣♦s rtríst♦s ró♥ ② rs♣st t♠♣♦ ró♥ τrelax s ♥♦ ♥ ♣rs♥t ♦♥t①t♦ ♦♠♦ t♠♣♦ rqr♦ ♣r r♣rr ♦rr♥t s ♦ qtr stí♠♦ t♥só♥♠á♥♦ ♠♥trs q t♠♣♦ rs♣st τresp s ♥ ♦♠♦ t♠♣♦ q s rqr ♣r ♥③r sñ ♥t♥s ♦rr♥t s♦ ♥ stí♠♦ ♦ t♥só♥ ♠á♥ ♦s t♠♣♦s ró♥ ♦sr♦s❬♣♦r ♠♣♦ τrelax (300 P) = 450 ♠s ② τrelax (67 P) = 220 ♠s❪ s♦♥ ♠♥♦rs ♦s t♠♣♦s rs♣st ♦rrs♣♦♥♥ts ❬τresp (300 P) = 935 ♠s

♥s♦r rs♣st ② rs ♠ért♦

② τresp (67 P) = 1100 ♠s❪ ♥ ♣rtr t♠♣♦ ró♥ qr s♥s♦r rr♦ s s♠r t♠♣♦ ró♥ P34❬❪ ♦sr♦ ♥ s rs ró♥ rs♣st ást ❬ ❪st rst♦ ♦♥stt② ♥ ♥ ♦♥ q ♣rs♥ ♦s♦♥tt♦s étr♦s ② ♥♣s♠♥t♦ ♠tr ♥♦ t s♥t♠♥t ró♥ ♠tr s♥s t♥só♥ ♠á♥

r rs ♥t♥s ♦rr♥t ♥ ♥ó♥ t♠♣♦ s♥s♦rr♠ ♣r ♥ s♥s♦r rr♦ ♦ ♦s t♥s♦♥s ♣s ♣♦t♥♣♦ V stá ♦ ♥ 300 ♠❱

♦s ♦♥♠tó♥ ♦rr♥t♣♦t♥

r ♠str ♦s ♦♥♠tó♥ ♣r ♥♦ ♦s s♥s♦rs rr♦ ♦ s tt stérss étr ♥ ♦s ♦s ② ♥♦ s ♦sr ss ♥tr ♦rr♥t étr tt ② ♣♦t♥ étr♦♣♦ ♥tr♦ s rs♦t t♠♣♦r t③ ♥♦s 50 ♠s♣r♦①♠♠♥t st ♦sró♥ s ♦♥sst♥t ♦♥ ♦♠♣♦rt♠♥t♦ ♦♠♣t♠♥t ó♠♦ ♦s s♥s♦rs st♦s str♦ ♥ r

t♦ tr♠♦rsst♦

tr ♣r♦♠át rrr♥t ♥ ♦s ♠trs ♦♠♣st♦s s♦s ♥♠trs ♣♦♠érs s♥ts ♦♥ r♥♦s ♦♥t♦rs s sr ♠♣♦rt♥t t♦ tr♠♦rsst♦ ♣♦r s ♦sr♥ r♥s ♠♦s ♥ rsst♥ étr ♠tr ② ♣♦r ♦ t♥t♦ ♥ rs♣st ♣③♦rsst ♣♦r ♣qñ♦s ♠♦s ♥ t♠♣rtr ♠♥t Pr r ♠♦ ♥ rs♣st s♥s♦r ♦♥ t♠♣rtr s ♠ó rsst♥étr ♥♦ ♦s s♥s♦rs rr♦ ♥ ♥ó♥ t♠♣rtr

♣ít♦ ♥s♦r t♥só♥ ①

r ♦s ♦♥♠tó♥ ♣r ♥ s♥s♦r rr♦ ♦ s ♦srstérss étr ♥ ss ♥tr ♦rr♥t tt ② ♣♦t♥♣♦ ♥tr♦ rs♦ó♥ t♠♣♦r ♥str♠♥t ∼ 50 ♠s

♦ t♥s♦♥s ♠á♥s ♥①s rtrrs Pr r③r ♠ó♥s rstrr♦♥ s rs rtrísts I−V ♥♦ rr♦ s♥s♦r ♥♥ st ♦♥ t♠ósr r ♣♥♦ ♥ t♥só♥ ♠á♥ 54 P② r♥♦ t♠♣rtr ♥tr♦ ♥tr♦ 25−155 ♦s rst♦s s ♠♦♥s s rs♠♥ ♥ r ② r♠♥t ♠str♥ q t♠♣rtr ♥♦ ♠♦ s♥t♠♥t rs♣st ♦s s♥s♦rs srr♦♦s ♦ q rsst♥ étr s ♥r♠♥t s♦♠♥t 6.3% ♠♥tr t♠♣rtr 25 155 st ♠♦ s ♠♦ ♠♥♦rq q♦s ♣r♠♥t r♣♦rt♦s ♣r ♦tr♦s ♠trs s♦s ♥r♥♦s ♠tá♦s ❬ ❪ ♦r ♦t♥♦ ♣r ♦♥trsst♦ tér♠♦ s α ≡ 1

R

(∂R∂T

)H,P

= 5.0 × 10−4 ♣♦st♦ ② ♣qñ♦ ♦r ♣♦st♦ st ♣rá♠tr♦ stá s♦♦ tó♥ tér♠ ♠tr③ ♣♦♠ér ♥ t♦ ♥r♠♥trs t♠♣rtr sst♠s ♣r♦ s♣ró♥ s r♦♥s ♦♥t♦rs r♥♦ ♣♦r tó♥ tér♠ ró♥ st♦♠ér q s♣r s ♠s♠s ❬ ❪ ♦ q ♦r ♦t♥♦ ♣r α s ♣qñ♦ ♣ strs á♠♥t♥ tér♠♥♦s tt♦s ♦♥sr♥♦ ♦s t♦rs r♦ ♦r ♦♥t tó♥ tér♠ ♠tr ♣♦♠ér♦ P ② st♠♣rtrs s ♠tr③ s ♥♥tr ♦s s t♠♣rtr tr♥só♥ ítr Tg ≈ −120 ♣r P ②r ♣♣ ❬❪st♦s t♦rs ♣r♥ ♠♥t♦s ♣qñ♦s ♥ s s♣r♦♥s ♥tr r♦♥s r♥♦ ♣♦r ♥t♠♥t♦ sst♠ ② ♣♦r ♦ t♥t♦ ♥r♠♥t♦s ♦s rsst♥ étr

♥s♦r rs♣st ② rs ♠ért♦

r ♠♦ ♥ rs♣st s♥s♦r ♦♥ t♠♣rtr ♥ ♥tr♦ 25 − 155 t♥só♥ P = 54 P ró♥ ♥ t♦♦ ♥tr♦ t♠♣rtr s ♥r♦r 6%

sst♥ qí♠

♥♠♥t rst r♥ ♥trés r rsst♥ qí♠ rr♦ s♥s♦r r♥t rs♦s s♦♥ts ♦rá♥♦s ② s♦♦♥s ♦ssPr ♦ rr♦ s♥s♦r s s♠ró ♥ ♦s ♠♦s r♥t 48 P♦str♦r♠♥t s♣♦st♦ ♦ ♦♥ ♦♥③♦ s♦ ♥s♣♦♥♦ ♦s s♦♥ts t③♦s r♦♥ t♦♥♦ ♦r♦♠t♥♦♦r♦♦r♠♦ t♥♦ ♠t♥♦ s♦♣r♦♣♥♦ t♦♥ t♦♥tr♦ ♥♥♦ ②t♦♥♦ s♣t♦ s s♦♦♥s ♦ss s t③r♦♥ s♦♦♥s ② s♦ó♥ str ♥ t♦♦s ♦s s♦s s♥s♦r♥♦ sr ñ♦ ♥♦ ♣♦r st trt♠♥t♦ ♥s♣ó♥ s ♥♦ ♠strñ♦s ♥ s♣r ♠tr s ú♥ s rs rs♣st ♣③♦rrssts t♦♦s ♦s s♥s♦rs rr♦ ♥♦ ♠str♥ r♥s ♥♦t♦rsrs♣t♦ r str ♥ r

•

♥ st ♣ít♦ s sró sñ♦ ró♥ ② rtr③ó♥ ♥ ♣r♦t♦t♣♦ rr♦ s♥s♦r ♣③♦♠étr♦ s♦ ♥ P34❬❪ ♦ró s♣rr ♥ ♠♣♦rt♥t sí♦ t♥♦ó♦ ♠♣♠♥tó♥ tr♦♦s q ♥♦ tr♥ rs♣st ♠tr ② q ♥♦ tr♦r♥ rs♣st ♠s♠♦ ♦♥ s♦ ss♦

r♥♦ ♥á♠♦ qr ♦s s♥s♦rs rr♦ s 0 − 550P ①♥♦ ♥ rs♣st ♣③♦♠étr ①♣♦♥♥ rá♣ ② ♦♠♣t♠♥t rrs ♦ s ♦sr ♦♥ó♥ étr tr♥srs ♣r♥♥♥♦ ♦s s♥s♦rs rr♦ st♦ s rr♦ s♥s♦r ♣rs♥t rs

♣ít♦ ♥s♦r t♥só♥ ①

♣st ♥s♦tró♣ P♦r ♦tr♦ ♦ ♦♥ó♥ étr ♦♥t♥ ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ ♥ ♣rs♥t ♦♠♣♦rt♠♥t♦ ó♠♦ ♥ t♦♦ ♥tr♦ t♥só♥ rr♦ ♥♦ s tr♦ ♥ñ♦ ♣♦r ①♣♦só♥ s♦♥ts ♦rá♥♦s ♦♠♥s ② s♦♦♥s ♦ssrs ♦ s ♦sr ♦①ó♥ ♣♦r ①♣♦só♥ ♠♥t

♦s sts ♣r♦♣s qí ♠♥♦♥s ♦♥rt♥ rr♦ s♥s♦rsrr♦♦ ♥ ♥ tr♥t s♥s♦r ♥trs♥t ♣r sr t③♦ ♥ ♠♥ts rs♦s ♦♥ ♦s s♥s♦rs tr♦♥s ♦♥ ♦ ♦♥t♥♦ ♠t ♦ ♦♠♣♦rt♠♥t♦ s♠♦♥t♦r ♥♦ ♣♥ sr t③♦s s ③ rr♦ srt♦ ♣♦rí ♦♣t♠③rs ♣r ró♥ ♣s rts② s♥s♦rs ♠♣♦ t♥só♥ ♠á♥ ♥ ♠♣♦ ♦♠étr♣♦rí sr s♦ ♣r ♠ó♥ rt♠♦ rí♦ ♥ t♠♣♦ r r♥t♣r♦♦s ①t♥s♦s t♠♣♦ ♦ s ♦rt♦ t♠♣♦ rs♣st ♣s♦ r♦ ♣♦t ② ♣♦r sr ♥♦ ♥s♦ ♥ st s♦ s ♥sr♦ ♦♣t♠③r í♠t tó♥ s♣♦st♦ ♦ q ♣ rs ♠♦♥♦ srs s♥téts

str qí q s♦ ♥ r♥♦ ♥ st♦ s♣r♣r♠♥ét♦ r ♣♦s t③r s♣♦st♦ srr♦♦ ♦♠♦ ♥rr♦ ♠♥s♦♥ s♥s♦rs ♠♣♦ ♠♥ét♦ s♦ ♥ t♦s♠♥t♦rsst♦s s♥ stérss ♠♥ét ♣r♦♣ q s strá ♥ ♣ít♦ ♥ s s♦ s♥s♦ ♠♣♦s ♠♥ét♦s ①tr♥♦s ♦rr♣♦t♥♠♥t s♥ ♥s ♠r ♥♥ú♥ t♦r ♥ ♥str♠♥tó♥tr♥só♥ t③♦s ♥ s♥s♦ t♥s♦♥s ♠á♥s

♣ít♦

♥s♦tr♦♣í étr ♦t r♦ ór♦

tr♥ ② t rt ♥ ♦rr♦r r♦♠ts ♠♥t ♦ ♥t♦♥s ♦

♥♦t rts

rs r♠t

s♠♥ ♥ st ♣ít♦ s ♥tr♦♥ ♥♦s ♦♥♣t♦s tór♦s♥sr♦s ♣r ♦♠♣r♥só♥ ② srr♦♦ ♦s rst♦s ♦♥r♥♥ts st♦ ♣r♦t♦ ♦s ♦♠♣♦st♦s st♦♠ér♦s strtr♦s ♠♦str♦s ♥ ♦s ♦s ♣ít♦s s♥ts r③ ♥ sr♣ó♥ stíst strtr ♥tr♥ ♦♠♣♦st♦ strtr♦s♦ ♥ ró♥ rr♦ s♥s♦r ♣③♦♠étr♦ srt♦ ♥ ♣ít♦ ♣r ♦ s rs♠♥ ♥♦s ♦♥♣t♦s stístt♠át ♠♥t ② s r③ ♥ r ♥tr♦ó♥ ♦rí Pr♦ó♥ ♣

♦♥♣t♦s Pr♦ ② stíst

♦♥t♥ó♥ s rs♠♥ ♥♦s ♦♥♣t♦s stíst t♠át ♠♥t q srá♥ ♥srs ♣r r③r ♥ sr♣ó♥ stíst strtr ♥tr♥ rr♥ P34❬❪ ♦♥ ♥ tr ♥ st♦ ♥s♦tr♦♣í étr ♦t ♥♥ ♦♥t①t♦ ♣r♦t♦ ♦s ♦♥♣t♦s s ♣rá♥ ♣♦str♦r♠♥t ♥ ♦♥ó♥ ♥tr♣rtó♥ s rs ♣r♦ ♣r♦ó♥♦♠♦ sí t♠é♥ ♥ sr♣ó♥ stíst♦strtr ♠tr rr♥ str q ♥tr♦ó♥ qí r♥ ♥♦ ♣rt♥

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

sr ♥ sr♣ó♥ ♦♠♣t ② rr s♦r t♠át s♥♦ só♦ r♥r ♦s ♦♥♦♠♥t♦s ás♦s rqr♦s ♣r ♦♠♣r♥só♥ ♦s t♠s♣r♥♣s tr♦

♠♣♠♦s ♥t♦♥s ♥♦ q ♥ ①♣r♠♥t♦ s t♦r♦ ♥♦ rst♦ ♥ r③ó♥ ♦ ①♣r♠♥t♦ s ♥rt♦ ♣r♦ sr♥s rts ♦s rst♦s ♣♦ss t♥♥ ♥ ♣tró♥ stró♥ rr ♥ ♥ r♥ ♥ú♠r♦ r♣t♦♥s ♥ st ♦♥t①t♦ s♥ s♣♦ ♠str ♥ ①♣r♠♥t♦ t♦r♦ (Ω) ♦♠♦ ♦♥♥t♦ t♦♦s ss rst♦s ♣♦ss

r③r ♥ ①♣r♠♥t♦ t♦r♦ ♥r♠♥t s stá ♥trs♦ ♥♥ ♥ó♥ rst♦ ♠ás q ♥ rst♦ ♥ sí ♠s♠♦ sí ♣♦r♠♣♦ rr♦r ♥ ♦ ♦s s s ♣♦rí str ♥trs♦ só♦ ♥ s♠ ♦s ♣♥t♦s ♦t♥♦s ② ♥♦ ♥ ♣r ♦rs q ♦ ♦r♥ s ♦r s♠ s ♥t ♥trés ♦ ♠ás ♦r♠♠♥t s ♥ó♥ ♦rs rs ♥ s♦r s♣♦ ♠str s ♥♦♠♥ rt♦r ❱r ♣♦rq t♦♠ st♥t♦s ♦rs ② t♦r ♣s ♦r ♦sr♦ ♥♦ ♣ sr ♣r♦ ♥ts r③ó♥ ①♣r♠♥t♦♥q sí s s ás s♦♥ ss ♣♦ss ♦rs ♦r♠♠♥t s ♥ ♥ r t♦r X ♦♠♦ ♥ ♥ó♥ ♦s ♣♦ss rst♦s ①♣r♠♥t♦ t♦r♦ ♦♥ ♦♠♥♦ ♥ Ω ♠♥ ♥ s♣♦ ♦s ♥ú♠r♦srs s r

X : Ω −→

♥ ♣rs♥t sr♣ó♥ ♦s ♦rs r t♦r s ♥♦trá♥ ♦♥trs ♠♥úss x ♥ st s♦

♠♦♦ ♠♣♦ ♦♥srr s♥t ①♣r♠♥t♦ t♦r♦ srr♦ ♦s s ♥ ♦ qr♦ ② s rstr ♦r rr♦♦ ♣♦r ♥♦ st♦s s♣♦ ♠str s♦♦ s

Ω =(x1, x2) /xi ∈ 1, 2, 3, 4, 5, 6

② ♣♦ss rs t♦rs s♦s ♦♥ st ①♣r♠♥t♦ s♦♥X ♥ú♠r♦ rs ♣rs Y ♠á①♠♦ ♣♥t Z s♠ ♣♥t♦s t ♦♥♥t♦ ♦rs rs q t♥♥ s♦♦ ú♥ ♠♥t♦ s♣♦ ♠str s♥♦♠♥ r♥♦ r t♦r

ΩX = x ∈ : ∃s ∈ Ω, X (s) = x

ΩX r♥♦ r t♦r X s ♥ ♦♥♥t♦ ♥t♦ ♦ ♥♠r♥t♦♥s r t♦r s ♥♦♠♥ srt ♥ s♦ q s ♥♥tr♦ ♥t♦ ♦ ♥♥t♦ ♥t♦♥s r t♦r s ♥♦♠♥ ♦♥t♥ P♦r ♠♣♦ ♣r r t♦r Z ♥tr♦r♠♥t ♥ s t♥ΩZ = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

♦ q ♦r ♥ r t♦r s tr♠♥♦ ♣♦r rst♦ ♥ ①♣r♠♥t♦ ♣♦r♠♦s s♥r ♣r♦s ♦s ♣♦ss ♦rs

♦♥♣t♦s Pr♦ ② stíst

♦ ♦♥♥t♦s ♦rs r ♥ tér♠♥♦s ♣r♦ ♣r srr s♥ó♥ ♥ ♣r♦ P ♦s ♦rs r♥♦ ♥r t♦r sú♥ s ♥tr③ s t③♥ ♦s ♥♦♥s r♥ts sr

r t♦r s srt s t③ ♥ó♥ ♠s ♦♥ó♥ stró♥ ♣♥t

r t♦r s ♦♥t♥ s t③ ♥ó♥ ♥s♥ó♥ stró♥ ♦ ♥ó♥ ♥s ♣r♦

♥ó♥ stró♥ ♣♥t ♥ r t♦r srt s♥ ♥ó♥ p q r♣rs♥t ♣r♦ q X t♦♠ ♥♦ ♦s♣♦ss ♦rs srt♦s xi, i = 1, 2, . . . s r

p : ΩX −→ [0, 1]

xi −→ p(xi) = P(X = xi) = P(w ∈ ΩX/X(w) = x

)

P♦r ♦ t♥t♦ ♥ó♥ p(x) sts s s♥ts ♣r♦♣s

(i) 0 ≤ p(x), ∀x(ii)

∑

xi∈ΩXp (xi) = 1

P♦r ♠♣♦ ♦♥srr ♥♠♥t ①♣r♠♥t♦ t♦r♦ ♦rrs♣♦♥♥t rr♦r ♦s s ♥ ♦ qr♦ ♦♠♦ s ♦ ♥tr♦r♠♥t s♣♦ ♠str s♦♦ s Ω =

(x1, x2) /xi ∈ 1, 2, 3, 4, 5, 6

r t♦r X ♥ú♠r♦ rs ♣rs rr♦r ♦s s ♥ ♦qr♦ st ♥t q r♥♦ r t♦r sΩX = 0, 1, 2 sí ♣r♦ ♦t♥r 2 rs ♣rs ♣ rs♠② á♠♥t

p(2) = P(X = 2) = P(x1, x2) ∈ ΩX/x1, x2 ∈ 2, 4, 6

= 9/36 = 1/4

♦r♠ s♠r ♥ó♥ ♥s ♣r♦ ♥ rt♦r ♦♥t♥X s ♥ ♥ó♥ f : Ω −→ q sr ♣r♦ s♥t ♠♥r s t♥♠♦s ♥ s♦♥♥t♦ ♥ú♠r♦s rs A ⊂ ♣r♦ q r t♦r ♦♥t♥X t♦♠ ♥ ♦r ♥ ♦♦♥♥t♦ s P (x ∈ A) =

∫A f (x) dx ♦♥ s s♥ts ♣r♦♣s

(i) f(x) ≥ 0, ∀x ∈

(ii)

∫

f (x) dx = 1

tr ♥ó♥ q rtr③ stró♥ ♣r♦ ♥ rt♦r X s ♥ó♥ stró♥ ♠ F (x)

F (x) = P (−∞ < X ≤ x) = P (s ∈ Ω : X(s) ≤ x) , ∀x

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

♥ ♣rtr ♣r ♥ r t♦r ♦♥t♥ s t♥

F (x) =

x∫

−∞

f(x′)dx′

P♦r ♦ F (x) s ♥ ♥ó♥ ♦♥t♥ ♥♦ t♥ st♦s ② t♦♦s ♦s ♦♥♥t♦s♦r♠♦s ♣♦r ♥ s♦♦ ♣♥t♦ t♥♥ ♣r♦ r♦ s♠s♠♦ ♥ó♥ ♥s ♣r♦ f(x) ♣ rs ♣rtr F (x) r♥♦s r

f (x) =dF (x)

dx

s♣r♥③ ② r♥③ ♥ r t♦r ♣r ♥ r t♦r srt X ♦♥ ♦rs ♣♦ss x ∈ x1, x2, . . . , xN ② ss ♣r♦sr♣rs♥ts ♣♦r ♥ó♥ ♣r♦ p(x) s ♥ s♣r♥③ X ♦♠♦

E [X] =

N∑

i=1

xip (xi)

② s ♥ r♥③ X ♦♠♦

V ar [X] = σ2X =N∑

i=1

(xi − µ)2p (xi)

♦♥ µ ≡ E [X]♥á♦♠♥t ♣r ♥ r t♦r ♦♥t♥ X s♣r♥③ s

♠♥t ♥tr t♦♦s ♦s ♦rs ② ♥ó♥ ♥sf(x)

E [X] =

∞∫

−∞

xf (x) dx

② r♥③ s ♦♠♦

V ar [X] =

∞∫

−∞

(x− µ)2f (x) dx

♦♥♠♥t s ♥ só♥ tí♣ ♦ stá♥r ♦♠♦ rí③ r ♣♦st r♥③ DSX = σX =

√V ar [X]

♥ ♠♦s s♦s ♥tr♣rtó♥ ♥tt ♦ s♥♦ s♣r♥③s ♦rrs♣♦♥ ♦♥ ♦r ♠♦ tór♦ ♦s ♣♦ss ♦rs q ♣t♦♠r r t♦r 〈X〉 ♦ t♠é♥ ♦♥ ♥tr♦ ♠s ♦s♦rs r s♠♥♦ q ♦r t♥ ♥ ♠s ♣r♦♣♦r♦♥ ♥ó♥ ♥s ♥ ♦s ♥ ♠♦ r♥③ ♥t s♣rsó♥ ♠t♠át ♦s t♦s ♦♥ rs♣t♦ ♠

♦♥♣t♦s Pr♦ ② stíst

st♠ó♥ ♣♥t s ♦♥♦r ♦r ♠♦ stró♥ ♦sá♠tr♦s ♥♥♦♣rtís ♠♥tt ♦t♥s ♠♥t ♥ ♥♦ ♠ét♦♦ sí♥tss ♦r♠ rr♦s r♦ s ♠r á♠tr♦ t♦ss ♥♥♦♣rtís s♥tt③s ② ♦ r ♦r ♠♦ á♠tr♦♦ q ♦♥stt② ♥ ♣r♦s♦ r♦ ② t♦s♦ ♥ ♠r♦ stíst♣r♦ té♥s q ♣r♠t♥ ♦t♥r ♦♥s♦♥s ♥rs ♣rtr ♥♦♥♥t♦ ♠t♦ ♣r♦ r♣rs♥tt♦ t♦s ♠str ♥ stíst t♠ñ♦ ♠str s ♥ú♠r♦ ♠♥t♦s q ♦♠♣♦♥♥ ♠str ①trí ♥ ♣♦ó♥ ♥sr♦s ♣r q ♦s t♦s ♦t♥♦ss♥ r♣rs♥tt♦s ♣♦ó♥ ♥♦ s st♠ s ♥r ♥♦ s t♥ r♥tí q ♦♥só♥ q s ♦t♥ s ①t♠♥t ♦rrt♣r♦ stíst ♣r♠t ♥tr rr♦r s♦♦ st♠ó♥ ♦t♦ st♠ó♥ ♣♥t s sr ♥ ♠str ♣r ♦t♥r ♥ú♠r♦sq ♥ ú♥ s♥t♦ s♥ ♦s q ♠♦r r♣rs♥t♥ ♦s rr♦s ♦rs ♦s ♣rá♠tr♦s ♥trés ♦ s♦s tér♠♥♦s ♥ st♠♦r ♣♥t ♥ ♣rá♠tr♦ ϕ s ♥ ♦r q ♣ sr ♦♥sr♦ r♣rs♥tt♦ ϕ② s ♥rá ϕ ♣é♥♦s ①♣rsr ♠s♠♦ ♣rtr ♥ ♥ó♥ ♠str ♥ ♣rtr ♥ st♠♦r ♣♥t ♥ss♦ s ♥ st♠♦r♣♥t ♣rá♠tr♦ q sts

E [ϕ] = ϕ,

♦♥ E [ϕ] ♥ s♣r♥③ ϕ st♠♦r ϕ ♥♦ s ♥ss♦ s♥♦♠♥ ss♦ ϕ

b (ϕ) = E [ϕ]− ϕ.

♥ ♠r♦ st♠ó♥ ♣♥t s ♠♣♦rt♥t rr ♦♥♣t♦ ♥♦♥s rs t♦rs s♦s ♠str♦ ♣♦♥r q ss♦♥ ♥ ♠str t♠ñ♦ N ♥ ♣♦ó♥ ♥ts ♦t♥r ♠str ♥♦ s s á srá ♦r ♦sró♥ sí ♣r♠r♦sró♥ ♣ sr ♦♥sr ♥ r t♦r X1 s♥ ♥r t♦r X2 t P♦r ♦ t♥t♦ ♥ts ♦t♥r ♠str s ♥♦trá X1, X2, . . . , Xn s ♦sr♦♥s ② ♥ ③ ♦t♥ ♠str ♦s♦rs ♦sr♦s s ♥♦trá♥ ♦♠♦ x1, x2, . . . , xn ♠s♠♦ ♠♦♦ ♥ts ♦t♥r ♥ ♠str qr ♥ó♥ srá ♥ r t♦r❬ ❪ ♣♦r ♠♣♦ X maxXj t ❯♥ ③ ♦t♥ ♠str ♦s♦rs ♦s srá♥ ♥♦t♦s ♥ tr ♠♥ús x maxxj t

Pr r s rt♦♠r♠♦s ♠♣♦ stró♥ á♠tr♦s s ♣rtís ♠♥tt ♠strr ♣rtís ② ♠r á♠tr♦ ♥ s ♥ s rs t♦rs Xj á♠tr♦ ♥♥♦♣rtí jés♠ ♦♥ j = 1, . . . , 500 ♦ ♦s ♦rs ♦t♥♦s ♣r♦s á♠tr♦s ♦s ♥♦t♠♦s ♦♠♦ xj ♦♥ j = 1, . . . , 500 s s st♠r á♠tr♦ ♠♦ ♣♦ó♥ ♠ ♣♦♦♥ µ ♣ rs♠♥t ♦r ♠♦ s ♦sr♦♥s r③s s♦r ♠str xN

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

s♦♦ r t♦r XN =1

500

∑N=500j=1 Xj ♣s XN ♦rrs

♣♦♥ ♥ st♠♦r ♥ss♦ ♠♥t ♥trés s ③ s sr♥ ♦ ♦trs ♥♥♦♣rtís ♦r ♦t♥♦ xN rs♦ ♣r♦r r♥t qí s ♥♦t rátr t♦r♦ r XN

♦r ♥tó♥ ♥r♠♥tr t♠ñ♦ ♠str t♦♠ ♦r xN ♦♥rrá ♦r ♠ ♣♦♦♥ µ s r

xNN→Np−−−−→ µ,

s♥♦ Np t♠ñ♦ ♣♦ó♥ ② ♦s r♥s ú♠r♦s ♦♠♦♦r♦ ❬ ❪

Pr tr♠♥r st só♥ ♦♥srr ♥ r t♦r Y ♥ ♦♠♦ ♥ó♥ N rs t♦rs X1, . . . , XN s r Y =G (X1, . . . , XN ) ♦♥♦s s ♥♦♥s ♥ss ♣r♦ ♥ s N rs st♠♦s ♥trs♦s ♥ tr♠♥r ♥ó♥ ♥s ♣r♦ r t♦r ♦♠♣st Y

s N rs t♦rs s♦♥ ♥♣♥♥ts ♦s ♦rs q t♦♠♥ s ♥♦ t♥ ♦s qr ♦tr ♥ ss ♣r♦s ♥ó♥ stró♥ ♠ ❨ ♣ srrs ♦♠♦

FY (y) =

∫

N∏

k=1

fXk (xk) dx1 . . . dxN ,

♦♥ =(x1, . . . , xN ) ∈ N /G (x1, . . . , xN ) ≤ y

②

fX1,...,XN (x1, . . . , xN ) ≡N∏

k=1

fXk (xk)

s ♥♦♠♥ ♦♠ú♥♠♥t ♥ó♥ ♥s ♣r♦ ♦♥♥t t♦r t♦r♦ (X1, . . . , XN ) ❬ ❪ P♦r út♠♦ ♦ ♦♠♣trFY (y) ♠♥t ①♣rsó♥ ♣ ♦t♥rs ♥ó♥ ♥s ♣r♦ r t♦r Y r♥♦

fY (y) =dFY (y)

dy.

♦rí Pr♦ó♥

♥tr♦ó♥

♠♣③r♠♦s ést só♥ sr♥♦ r♠♥t r♦s ♦♥♣t♦s ás♦s s♦♦s ♦s ♥ó♠♥♦s rít♦s ♥ ♦♥t①t♦ ♣r♦ó♥

♦rí Pr♦ó♥

r♥ ♣rt trt♦ ♦s ♠♦♦s ♣r♦ó♥ rs ♥ ss s♣t♦s ú♦s ② s s♠♣ ♥tt ② t♦♦ ♦ q s rqr s rt♦♥t♥♠♥t♦ ♦♠trí ♠t♠át ♠♥t ② ♣r♦ ♠ásst♦s ♠♦♦s sr♥ ♦♠♦ ♥ ①♥t ♥tr♦ó♥ ♦s ♠♦♦s ♥♦r♠át♦s srt♦s ② ♥áss rá♦ ❯♥ ás♦ ♠♣♦ ♣r ♦♠♣r♥r ♦♥♣t♦ ♣r♦ó♥ s ♦♥r♦ strí♦ ❯♥ ♦♥r♦ ♣rt♥♦♥r ♥s tts Pr ♦ ①t♥ ♠s ♣r♣r s♦r ♥ ♣♥♠♥t ♥ ♦r♠ ♦ts ♣♦♥♠♦s q ♦t ♠s ♣rts ♣ ①♣♥rs ♠♥trs q s ts s ♦r♥♥ ♥ ♥ ♦r♥♦♦♠♦ s ♦♠ú♥♠♥t ♣r♣rr s tí♣s ♦♦s ♠r♥s ♦sts s t♦♥ s ♥♥ ♣r ♦r♠r ♥ t ♠②♦r t♠ñ♦ ♦♥r♦ ♦♦ s ♦ts ♠s ♠② r ♥ ♦tr ♦ ♦ó♥ ♣♦♠♦s ♥♦♥trr ♥ r♥ t q s ①t♥ s ♥ ♦r ♣ st ♦r ♦♣st♦ r

r ts s ♥ ♥ ♣ ♣r ♦r♥♦ ♦tr q ①st ♥♠♥♦ ♦r♠♦ ♣♦r ts s♣r♣sts q ♦♥t s rsts rts♦♣sts ♣ ♦ ♠♥♦ ①st s q s ts ♣r♦♥ ♣ ♦ ♥ q ①st ♥ r♣♦ ♦♠♣t♠♥t ①t♥♦ ♦ r♣♦♣r♦♥t s♣♥♥♥ str ♥ ♥és

①st ♥ t t q r t♦ ♣ sst♠ s q s ♣r♦♦ tr♥só♥ ♣r♦ó♥ ♥♦ ①st t t ♥ ♦r♥♦ stá ♣♦r ♦ ♠r ♣r♦ó♥

♦♥sr♠♦s ♥ ♠♣♦ ♠ás strt♦ ♣r rr ♦♥♣t♦ ♣r♦ó♥ ♣rs♥t♠♦s ♥ ♣r ♦r♥♦ ♣♦r ♥ ♥tr♠♦ tt♦♥ st♦ ♣rs♥t ♥♦ ♦s st♦s ♦♣♦s ♦ í♦ ❯♥ ♥tr♠♦ r♦ ♦r♠♦ ♣♦r N×N st♦s s q t♥ t♠ñ♦ L = N st♦ s ♦♣♦ ♦r♠ ♥♣♥♥t ss ♥♦s ♦♥ ♣r♦ p ♦q♥t♠♥t s ♦♣ ♥ ró♥ p st♦s ♥ ♦r♠ t♦r st♠♦♦ ♣r♦ó♥ s ♥♦♠♥ ♣r♦ó♥ st♦ st ♣r♦t♦♥♦s st♦s ♦♣♦s stá♥ s♦s ♦ ♥ stá♥ ♦r♠♥♦ r♣♦s ♦♥ ss ♥♦s ♠ás r♥♦s ♥ str ♦ r♣♦ ♥ ♦♥♥t♦ st♦s r♠ás r♥♦s ♦♣♦s r r p s ♣qñ s s♣r q s♦♦♣r③♥ ♣qñ♦s strs s♦s r p = 0.2 p s r♥

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

r ♠♣♦ ♥ str r♣♦s st♦s ♥ ♠♦♦ ♣r♦ó♥ st♦ ♣r ♥ sst♠ r♦ t♠ñ♦ ♥ L = 2 ♦s ♦sst♦s ♥♦s ♠ás r♥♦s ♦♣♦s ♣♥t♦s ♥ s♦♥ ♣rt ♠s♠♦str ♦s ♦s st♦s ♦♣♦s ♥ ♥♦ s♦♥ ♥♦s ♠ás r♥♦s ♣♦r ♦q ♥♦ ♦r♠♥ ♣rt ♠s♠♦ str

♥ s♣r♠♦s q ♠②♦rí ♥tr♠♦ r sté ♦♣♦② q ♦s st♦s ♦♣♦s ♦r♠♥ ♥ r♥ str q s ①t♥ s ♥rst ♦♣st r p = 0.8 ♦ str s ♥♦♠♥ r♣♦①t♥♦ r♣♦ ♦♠♣t♠♥t ①t♥♦ ♦ r♣♦ ♣r♦♥t s♣♥♥♥ str s ♥ ♦♥ ♥ r t♠ñ♦ ♥♥t♦ ♦♠♣t♠♥ts♦♣ p = 0 ② s ♥ ♥♥♦ ♥♦ ② ③r ♦s st♦s r ♣st♦ q ♥♦ ①st ♥ r♣♦ ①t♥♦ ♣r ♦rs p ♣qñ♦s ②q s stá ♣rs♥t ♥♦ p s r♥ ♥ ①str ♥ ♦r♥tr♠♦ p ♣r③ r♣♦ ①t♥♦ ♣♦r ♣r♠r ③ sst♠ s ♥♥t♦ L −→ ∞ ①st ♥ ♣r♦ ♠r p∞ ♥ s♥t ♦r♠

p < p∞, ♥♦ ①st ♥♥ú♥ r♣♦ ♣r♦♥t ② t♦♦s ♦s strs s♦♥ ♥t♦s

p ≥ p∞, ①st ♠♥♦s ♥ r♣♦ ♣r♦♥t

♦♥ ♦r ♠r ♣r♦♦♥ p∞ ♣♥ rs rtrísts sst♠ ♦♠♦ sr s s♠trí ② ♠♥só♥ t♦♣♦ó ❬❪

Pr r s s srr♦s ♦♥srr s♥t ♠♣♦ ís♦qí♠♦ ❬❪ ♦♥srr q ♦s st♦s ♦♣♦s ♦rrs♣♦♥♥ ♠tr♦♥t♦r ♦rr♥t étr q ♦s st♦s s♦♣♦s stá♥ ♦♠♣st♦s ♣♦r ♥ s♥ts étr♦s ② q ♦rr♥t étr s♦♦ ♣s♣③rs trés ♦s st♦s ♦♥t♦rs ♠ás r♥♦s ♥tr sí ♥♥ ①♣r♠♥t♦ ♦♥ ♥ r ♥ st♦s s♥ts ♦ ♥ ♦r♠ ♣r♦rs ② t♦r s r♠♣③♥ ♥♦ ♦s st♦s s♥ts ♣♦r ♦♥t♦rss♣t♦ ♦r♠ó♥ r♣♦s st♦s ♦♥t♦rs s♥♦ s sts ♠ás rr ♦s st♦s ♦♥t♦rs ♣rt♥ ♠s♠♦ r♣♦ sstá♥ ♦♥t♦s ♣♦r ♥ ♠♥♦ ♥♦s ♠ás r♥♦s ♦rr♥t étr rá trés ♦ r♣♦ st ♦r♠ ♣r s r♦♥s st♦s ♦♥t♦rs p ést♦s strá♥ ♦♠♣t♠♥t s♦s ♦ ♥ ♦r♠rá♥ ♣qñ♦s r♣♦s ♦♥ ss ♥♦s ♠ás r♥♦s ② ♣♦r ♦ t♥t♦ ♠③

♦rí Pr♦ó♥

r ♠♣♦s ♦♥r♦♥s ♥ sst♠ ♣r♦ó♥ st♦♦♥♦r♠♦ ♣♦r ♥ ♥tr♠♦ r♦ t♠ñ♦ L = 16 ♣r trs ♦rs ró♥ ♦♣ó♥ r♥ts p = 0.2 0.59 ② 0.8 ♦s st♦s ♦♣♦s s♠str♥ ♦sr♦s ♦tr q ♥ st ♠♣♦ ①st ♥ r♣♦ ♣r♦♥t♣r p = 0.59 ② 0.8

s ♦♠♣♦rt ♦♠♦ ♥ s♥t ② q ♥♦ ①st ♥ ♠♥♦ ♦♥t♦r q♦♥t rsts ♦♣sts ♠tr Pr ♦rs p ♠♦ ♠②♦rs ♠③ s ♦♠♣♦rt ♦♠♦ ♥ ♦♥t♦r étr♦ ♣s ①st♥ ♠út♣s ♠♥♦s ♦♥t♦rs q ♦♥t♥ rsts ♦♣sts ♠tr t♠ñ♦ sst♠ s ♠♦ ♠②♦r q t♠ñ♦ st♦ ♥♦♠♥♦ sst♠ ♥♥t♦ ①st ♥ ró♥ ♥tr♠ p∞ ♦rr♥t étr♣ ♠♣③r ♣r♦r s ♥ rst st s ♦♣st ♦ p∞ sst♠ ♥♥t♦ s ♦♠♣♦rt ♦♠♦ s♥t ♠♥trs q ♣♦r ♥♠ p∞ sst♠ ♥♥t♦ s ♦♠♣♦rt ♦♠♦ ♦♥t♦ ♦rr♥t étr ♦ tr♥só♥ s ♦♠étr ♥ ♦♠♣ñ ♦♥ ♥ ♠♦ rást♦♥ s ♣♦r♣s étrs ♠tr ② ♣♥t♦ rít♦ ♦ ♠♦stá rtr③♦ ♣♦r ♠r ♣r♦ó♥ p∞

st ♠♦♠♥t♦ s ♥ ♦♥sr♦ ♣r♦♠s ♣r♦ó♥ st♦♦♥ ♦s st♦s ♥ ♥tr♠♦ s♦♥ ♦♣♦s t♦r♠♥t ♥ ♠r♦♦s ♠♣♦s ♠ás ♥trs ♣r♦ó♥ ♣rt♥♥ ♣r♦ó♥ ♥ ♦♥t♥♦ ♦♥ s ♣♦s♦♥s ♦s ♦s ♦♠♣♦♥♥ts ♥ ♥ ♠③ t♦r♥♦ stá♥ rstr♥s st♦s srt♦s ♥ ♥ ♥tr♠♦ rr P♦r ♠♣♦♣♦♠♦s r s♦s rrs ♥ ♦r♠ t♦r ♥ ♥ r♠♥s♦♥ ♦♠♦ s ♠str ♥ r ♦s s♦s ♦r♠♥ ♣rt ♠s♠♦ r♣♦ s s t♦♥ ♦ s♦♣♥ ② s t♥ ♥ r♣♦ ♣r♦♥t s ①st♥ r♣♦ q ♦♥t ♦s rsts ♦♣sts ♦r♠♦ ♣♦r s♦s q s t♦♥♦ s♦♣♥

♥ s ♦♥t①t♦ ♣r ♥r ♠♥t r♥t q♥t p t♦♠rs ♥ ♥t ♠♥só♥ t♦♣♦ó sst♠ ② ♦s ♦t♦s♣r♦♥ts q ♦ ♦♠♣♦♥♥ ♥ s♦ sst♠s tr♠♥s♦♥s ♠♥t q ♥♦tr♠♦s ♦♠♦ Φ s ró♥ ♥ ♦♠♥ ♦s ♦t♦s ♣r♦♥ts ♠♥s♦♥ ♦ ♥ ♥t ♦t♦s ♣r♦♥ts ♣♦r♥ ♦♠♥ ♥s ♦t♦s ♦ ♦♥♥tró♥ ♦t♦s ♠

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

r Pr♦ó♥ ♥ ♦♥t♥♦ s♦s ♦s t♦r♠♥t ♥♥ r ♠♥s♦♥

♥ ♥s L−3 ♦♥ L t♠ñ♦ ♥ sst♠ ♥ ♥t♦ ♦s sst♠s ♣r♦ó♥ ♥ ♦♥t♥♦ ♠♥s♦♥s s st♥♥ ♦ssst♠s ♦♥ ♦t♦s ♣r♦♥ts ♠♥s♦♥s q ♣♦s♥ ár ♣r♦♣ q♦s ♦♥ ♦t♦s ♣r♦♥ts ♥♠♥s♦♥s q♦s q ♥♦ t♥♥ár ♣r♦♣ ♥ ♣r♠r♦ ♦s s♦s ♠♥t r♥t ♥ tér♠♥♦s ♣r♦ó♥ ♣ sr ró♥ ♥ s♣r ♦s ♦t♦s ♣r♦♥ts♠♥s♦♥ ♦ ♥ ♥t ♦t♦s ♣r♦♥ts ♣♦r ♥ ár♠ ♥ ♥s L−2 ♦♥ L t♠ñ♦ ♥ sst♠ ❯♥ ♠♣♦ st t♣♦ sst♠ s♦♥ ♦s sst♠s ♠♥s♦♥s s♦s ♣r♦♥ts♦♠♦ ♠♦str♦ ♥ r ♥ ♠♦ s ♦s ♦t♦s ♣r♦♥ts ♥♦♣rs♥t♥ ár s♦♦ s s ♦♥srr ♦♠♦ ♠♥t ♣r♦t rtríst ♥s s♣r ♦t♦s ❬❪ ♦ ♣r sst♠s ♦ tr♠♥s♦♥s ♥♥t♦s s t♥rá

Φ < Φ∞, ♥♦ ①st ♥♥ú♥ r♣♦ ♣r♦♥t ② t♦♦s ♦s strs s♦♥ ♥t♦s

Φ ≥ Φ∞, ①st ♠♥♦s ♥ r♣♦ ♣r♦♥t

♦♥ Φ∞ ♥ ♦r rt♦ ró♥ ♥ ♦♠♥ ♦ s♣r ♦♥♥tró♥ s♣r ♦ ♥ ♦♠♥ ♦ ♥s ♦t♦s ♣r♦♥ts ♣♦r♥♠ ①st ♠♥♦s ♥ r♣♦ ♣r♦♥t ♦♠♣t♠♥t ①t♥♦ ♦ tr♠♥s♦♥ rs♣t♠♥t

♥ ♥str ①♣r♥ r st♠♦s ♠r③♦s ♦♥ ♦s ♠♦s ss ♠tr ❯♥♦ ♦s ♠♣♦s ♠ás ♦♠♥s s q

♦♠♦ s sr ♥ ♣é♥ ♦r♠ ♦rrt rtr③r t♠ñ♦ ♥sst♠ ♣r♦t♦ s ró♥ ♥tr t♠ñ♦ ♥ sst♠ L ② ♥ ♠♥só♥rtríst ♦s ♦t♦s ♣r♦♥ts ℓ ró♥ ss ♠s q t♠ñ♦ s♦t♦ sst♠

♦rí Pr♦ó♥

♣ ①str ♦♠♦ só♦ íq♦ ♦ s s s♦ q ♠ ♥ s ♦tr ♥ ♣rsó♥ ② t♠♣rtr ♥ ♥s P♦r ♠♣♦ tr♥só♥ ♦ q ♦rr 0 r s t♣♦ ♠♦s ♥ ♠♣♦ ♥ tr♥só♥ s tr♠♦♥á♠ ♠②♦rí ssst♥s t♠é♥ ①♥ ♥ ♣♥t♦ rít♦ ♣♦r ♥♠ ♥ ♣rsó♥ ②t♠♣rtrs ♣rtrs ② ♥♦ s ♣♦s st♥r ♥tr s s ② íq

tr♦ ♠♣♦ ♥ ♣♥t♦ rít♦ ♦rr ♥ ♦s sst♠s rr♦♠♥ét♦s t♠♣rtr r Tc ② ♠♣♦ ♠♥ét♦ ♥♦ ♠♦s q s t♠♣rtrs ♥s sst♥s ①♥ rr♦♠♥ts♠♦ ♥ ♠♥t③ó♥ s♣♦♥tá♥ ♥ s♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ s♥r♠♥t t♠♣rtr ♥ ♠tr rr♦♠♥ét♦ ♠♥t③ó♥s♣♦♥tá♥ ♠tr s♠♥② ♥ ♦r♠ ♦♥t♥ ② s♣r ♣♦r ♦♠♣t♦ t♠♣rtr Tc Pr T > Tc ♠tr s ♣r♠♥ét♦

♦s sst♠s ♣r♦t♦s t♠é♥ ♣rs♥t♥ ♥ ♣♥t♦ rít♦ s trt ♥ tr♥só♥ s ♦♠étr ♥ st♦ ♦♥t ♥♦ t ♦♥t ②♦ ♣♥t♦ rít♦ stá ♦ ♣♦r ♠r ♣r♦ó♥ ❯♥ sst♠ ♣r♦t♦ ♥♥t♦ ♥ ♠r ♣r♦ó♥ p∞ ♦ Φ∞sú♥ ♦rrs♣♦♥ t♥ ♥ ♣r♦♣ ♥trs♥t ♥♦♠♥ ♥r♥③ s ss ♣r♦♣s t♦♣♦ósstrtrs ♥♦ ♣♥♥ s ♦sró♥ t③ st ♦r♠ ♦♥♣t♦ ♥r♥③ s ♠r ♣r♦ó♥ ② rt t♦s♠r stá♥ í♥t♠♠♥t r♦♥♦s r ♠str rqttr t♦s♠r ♥r♣♦ ♣r♦♥t ♥ ♥ sst♠ ♣r♦ó♥ st♦ r♥ t♠ñ♦ ♥ ♠r ♣r♦ó♥ rs s ♠á♥s s♦♥ ♠♣♦♥s s ③♦♥s♠rs ♦♥ rr♦s ♥♦s ❬❪

sr♣t♦rs ♥ sst♠ ♣r♦t♦

♦♥srr ♥ sst♠ ♦♥ N ♦t♦s ♣r♦♥ts st ♦r ♥strssó♥ s ♥tr♦ ♥ ♠r ♣r♦ó♥ ② ♥ ♣ró♥ ♠♥♦s ♥ r♣♦ ♣r♦♥t ♣♦r ♥♠ ♦ ♠r tr ♥t qrtr③ ♣r♦ó♥ s Psc (Φ) ♣r♦ q ♥ ♠♥t♦ sst♠ ♣rt♥③ r♣♦ ♣r♦♥t s r

Psc =♥t ♠♥t♦s ♥ r♣♦ ♣r♦♥t

N

♦♥sr♠♦s q ♦s ♦t♦s ♣r♦♥ts stá♥ r♣♦s ♥ M strsM s ♥t t♦t strs ♥ sst♠ Ms ♥t

♠♦♠♥t♦ srtr st ss ♥ s♦ srts ♦♦ strtrsrst♥s ♦ ❬❪ ♠ás s ♠♦str♦ q s r③s r♦ ♥s♦s ♣rs♥ ♥ s♣r ró ♥ ♦♥tt♦ ♦♥ ♥ ♥rqttr ♠♥r rr ♥ ♦r♠ ♣♥ s q s♦ ♥♦♠♥ ♣♦r♠♦s t♦rs rt♦ st♦ ró♥ ❬ ❪ P♦r ♦tr♦ ♦ ①st♥ ssó♥ ♠② t s♦r ♥ó♥ ♥ strtr r ♦ ♥♣♦r ♠♥s r♥♦ r ♣♦r ♠♣♦ r ❬❪

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

r rqttr t♦s♠r ♥ r♣♦ ♣r♦♥t ♥ ♥ sst♠ ♣r♦ó♥ st♦ r♥ t♠ñ♦ ♥ ♠r ♣r♦ó♥ rs s ♠á♥s s♦♥ ♠♣♦♥s s ③♦♥s ♠rs ♦♥ rr♦s ♥♦s

strs t♠ñ♦ s sstrs ② s ≡ Ms/M ró♥ sstrs♠é♥ s ♥ ♥t

ws ≡sMs∑ssMs

=sMs

N

q ♦rrs♣♦♥ ♣r♦ q ♥ ♠♥t♦ ♣r♦♥t ♣rt♥③ ♥ sstr ♥ ♦r♠s♠♦ ♦rí Pr♦ó♥ rst ♦♥♥♥t♥r t♠ñ♦ ♠♦ str ♦♠♦

S ≡

∑ss2Ms

∑ssMs

♦♥ ♣r tr r♥ s♠ ①② r♣♦ ♣r♦♥t

st ♦♥♥♥t s♦r sst♠ ♥ ♠♥só♥ ♥ ♦♥t ♦ ♦♥t ♦♥t ξ (Φ) ❯♥ ♠♥r r♦ s ♥r

♦tr q N S Ms M ② s s♦♥ ♥ó♥ ♥s s♣r ♦t♦s Φ

♦rí Pr♦ó♥

r♦ r♦ ♥ sstr ♦♠♦

Rs ≡1

s

s∑

i=1

(ri − r)2

♦♥

r ≡ 1

s

s∑

i=1

ri

② ri s ♣♦só♥ iés♠♦ ♠♥t♦ ♣r♦♥t ♥ ♠s♠♦ r♣♦ st♦r♠ ♥t r ♦♥ ♦♥ ♥ó♥ ♠r ♥tr♦ ♠s r♣♦ ❯s♥♦ st ♥ó♥ ♦♥t ♦♥t ξ s ♥ ♦♠♦

ξ2 ≡

∑s(s− 1)wsR

2s

∑s(s− 1)ws

♦③r♠♦s ♦r ♥ ♠ás ♥str t♥ó♥ ♥ sst♠s ♣r♦♦♥ ♥ ♦♥t♥♦ st ♦r ♠♦s st♦ q ♥ ♦s sst♠s♣r♦t♦s ♦♥ t♠ñ♦ ♦s ♦t♦s ♣r♦♥ts s s♣r r♥t t♠ñ♦ sst♠ tr♥só♥ ♣r♦ó♥ ♦rr ♥ ♥ ♦r ♥♥♦ ♠r ♣r♦ó♥ ♥♦t♦ ♦♠♦ Φ∞ ℘L (Φ) = ℘ (Φ;L) ♣r♦ q ♥ sst♠ ♣r♦ó♥ ♥ ♦♥t♥♦ t♠ñ♦ L♣r♦ ♥ ró♥ ♦t♦s Φ ró♥ ♥ ár ♦ ♦♠♥ ♦ ♥s ♦t♦s ♣♦r ♥ ár ♦ ♦♠♥ sú♥ ♦rrs♣♦♥ ♥♦♠♥r ♣r♦ó♥ ② s ♥♦t ♦♠♦ ℜ s♥♦ ♥♦tó♥ ②♥♦s ②♦♦r♦rs ❬❪ ♦♥ó♥ q stsrs ♥ ♥ sst♠ ♣rq ①st ♣r♦ó♥ P♦r ♠♦♠♥t♦ ♣r tr ♥áss t♦♠♠♦s♦♠♦ r ♣r♦ó♥ ♦r♠ó♥ ♥ r♣♦ ♣r♦♥t q ♦♥t s rsts rts ♥ sst♠ ♠♥s♦♥ r♦ ♣r♦ó♥♦r③♦♥t st ♦r♠ ℘L (Φ) ♥♦♠♥ ♦♠ú♥♠♥t ♥ó♥ ♣r♦ ♣r♦ó♥ ♣r♦ q ①st ♠♥♦s ♥ r♣♦♣r♦♥t ♥ ♥ sst♠ ♦♥ L ② Φ ♦s

♥ r ♣r♦ó♥ ♥ ♣ ♠♦strrs ❬❪q ♣r♦ ♣r♦ó♥ ♥ ♠r ♣r♦ó♥ ♦♥r ♥ ♦r ♥♦ ♥ sst♠s t♠ñ♦ ♥♥t♦ s r

lımL→∞

℘L (Φ∞) = c,

♣r ♥ ♦♥st♥t r 0 ≤ c ≤ 1 st ♦r♠ ♣♦r ♦ srt♦ ♥ ó♥ ♣r ♥ sst♠ ♥♥t♦ s t♥

℘L→∞ (Φ) = ℘∞ (Φ) =

0, Φ < Φ∞c, Φ = Φ∞1, Φ > Φ∞.

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

sí ℘∞ (Φ) s ♥ ♥ó♥ só♥ ♥ Φ = Φ∞tr ♥ó♥ ♥trés s ♥ó♥ stró♥ ♣r♦

♣r♦ó♥ ΓL (Φ) ♥ ♦♠♦

ΓL (Φ) ≡ d℘LdΦ

(Φ) = lım∆Φ→0

℘L (Φ +∆Φ)− ℘L (Φ)

∆Φ

sí s ♦ss ♣rt♥♦ ♥ sst♠ í♦ ② r♥♦ ♥♦ ② ♥ ♦r♠t♦r ♠♥t♦s ♣r♦♥ts ΓL (Φ) r♣rs♥t ♥r♠♥t♦ ♥ ♣r♦ ♣r♦ó♥ ♥♦r♠③♦ ♣♦r ♥r♠♥t♦ ♥ ♦♥♥tró♥ ♦t♦s Φ s♦♦ ♠♥t♦ ♥ ♥s ♦t♦s ∆Φ ❬❪

♥tr♠♦s ♦r ♥str t♥ó♥ ♥ sst♠s ♣r♦t♦s ♥t♦s ♣r♥tr♥♦s ó♠♦ s tr♥só♥ ♣r♦ó♥ ♥ ♦s sst♠s ♣rt♠♦s ♥ sst♠ ♥t♦ ♥♠♥t í♦ ♥♦r♣♦r♠♦s ♦t♦s ♣r♦♥ts ♥♦ ② rstr♠♦s ♦r Φ ♣r ♣r ♣r♠rr♣♦ ♣r♦♥t rst ♥t q s ♣♦♦ ♣r♦ ♦rr ♣r♦ó♥♦♥ ♥ ♦♥♥tró♥ ♦t♦s ♣qñ s ③ s ♣♦♦ ♣r♦ t♥rsst♠s ♦♥ Φ ♠② r♥s s♥ r ♦r♦ ♣r♦ó♥ ♥ ♠♦ ♥ ♠②♦rí ♦s s♦s ♦srr♠♦s ♣ró♥ ♥ r♣♦ ♣r♦♥t♣r ♦rs Φ ♥tr♠♦s t♠át♠♥t s t♥♥ s ♦♥♦♥s ♦♥t♦r♥♦

lımΦ→0

ΓL (Φ) = lımΦ→∞

ΓL (Φ) = 0

P♦r s ①♣rs♦♥s ② s t♥ q ℘L (Φ) rí ♣♦♦ ♥ s♦♥♦♥s r♦♥tr Φ → 0 ② Φ → ∞ s ③ ΓL (Φ) s ♥ó♥ ♥s ♣r♦ r t♦r Φ ♦♥ Φ ♥ st ♦♥t①t♦s rr ♦♥♥tró♥ ♦t♦s ♣r♦♥ts ♣r ♣r♠rr♣♦ ♣r♦♥t ② ℘L (Φ) s ♥ó♥ stró♥ ♠ s♦①♣t♥♦ ♦♥ó♥ tr ℘L (Φ = 0) = 0, ∀L

♦ q sst♠ s ♥t♦ ①st ♥ ♣r♦ ♥t ♥♦♥trr ♥ r♣♦ ♣r♦♥t ♣r qr ♦♥♥tró♥ ♦t♦s ♥t♦♠♦ qr ♦tr ♥ó♥ stró♥ ♣r♦ ΓL (Φ) ♣rs♥t ♦s ♣rá♠tr♦s rtríst♦s ♥s ♠ ♦t♦s ♣r ♥ r♣♦ ♣r♦♥t s♣r♥③ 〈Φ〉L ② s r♥③ ∆2

L ♦ ♥ sí♦ stá♥r ∆L s♣t♦ ó♠♣t♦ ♥ít♦ s♦s ♣rá♠tr♦s〈Φ〉L ♦rrs♣♦♥ s♣r♥③ r t♦r Φ r ó♥ ② s ♦♠♦

〈Φ〉L =

∞∫

0

ΦΓL (Φ) dΦ

♠♥trs q r♥③ s ♦♠♦ ∆2L = 〈Φ2〉L − 〈Φ〉2L ♦♥ 〈Φ2〉L =∫∞

0 Φ2ΓL (Φ) dΦ♦s ♥ sst♠ ♣r♦t♦ ② ♥ r ♣r♦♦♥ ♥s

st♦♥r á s ♣♥♥ 〈Φ〉L ② ∆L ♦♥ t♠ñ♦ sst♠ L

Pr♦ó♥ ♥ ♠trs ♦♠♣st♦s ♥s♦tró♣♦s

② ó♠♦ s ♠♦ r ℘L (Φ) ♠♦r t♠ñ♦ ♥ sst♠♣r♦t♦ ♥ ♠r♦ t♦rí Pr♦ó♥ st♦ s ♦♥♦ ♦♠♦ s♦ s ♠♥ts ♣♥♥ ♦s ♣rá♠tr♦s rtríst♦s ♥ sst♠ ♣r♦t♦ ♦♥ s t♠ñ♦ ♣ ♦rrs ♠♥t ♠ét♦♦ s♦ sst♠s ♥t♦s ❬ ❪ ♥ ♠r♦ ♦r♠s♠♦ ♠s ♥r t③♦ ♣r s s ♥áss s sst♥t ♥ ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦ ♣♦rss ss ♥ ♥s q ♦♥st ①♠♥r s ♥ts íss r ♣♥t♦ rít♦ ② ♣r♦♠♥t s ♥♦ ♦s ♠ét♦♦s ♠s ♠♣♦rt♥ts srr♦♦s ♥ ís tór s út♠s trs és ♦♠♦ s sr ♥ ♣é♥ ♦♠♥♥♦ s ①♣rs♦♥s q sr♥ ♦♥ s♠♦♥s ♦♥t r♦ ♠ét♦♦ r♦ ♣♥ ♦t♥rs♦s ♣rá♠tr♦s rtríst♦s ♦s sst♠s ♣r♦t♦s

Pr♦ó♥ ♥ ♠trs ♦♠♣st♦s ♥s♦tró♣♦s

♦♠♦ s strá ♥ ó♥ ♦ s ♦♥♦♥s ♣r♣ró♥s ♥ sst♠s ♣ strs ♥ tér♠♥♦s ♣r♦ó♥ s♠♥t♦s rts ♥ sst♠s ♠♥s♦♥s s♦s sst♠sq♥ rtr③♦s ♥ s t♦t ♣♦r ♥ ♦♥♥t♦ ♥t♦ ♣rá♠tr♦sstrtrs Pr str ró♥ ♥tr ♦s ♣rá♠tr♦s strtrs② ♣r♠r♦ srr♦rs ♥ ♦rt♠♦ ♦ ♣r rs s♦tró♣s s♥ ♦r♥tó♥ ♣rr♥ ♦s s♠♥t♦s rts rsq r♣r♦③ ♦s rst♦s tór♦s ♣rs♥t♦s ♥ tr♦s ♥tr♦rs ②♦ ♠♦r♦ ♠♥t ♣r str sst♠s ♥s♦tró♣♦s ♣r♦s s ss♥ ♥♦t♦r♠♥t ♦s rst♦s s♣♦♥s ♥ ♦rí❬ ❪ Pr ♣r♠r s♦ rst ♦♥♥♥t str ♦♠♣♦rt♠♥t♦ s♦ ♦s ♣rá♠tr♦s ♣r♦t♦s 〈Φ〉L,r=1,ℓ ② ∆L,r=1,ℓ♦♥ ℓ s ♦♥t ♦s s♠♥t♦s rt ② ró♥ s♣t♦r s ♥ ♥ ♥ sst♠ ♠♥s♦♥ rt♥r ♦♠♦ r = Lp/Ln ♦♥Lp ♦♥t sst♠ ♦♥sr ♥ r ♣r♦ó♥ ② Ln

♦♥t sst♠ ♥ ró♥ ♦rt♦♦♥ Lp sí r = 1 s rr sst♠s r♦s ♦♠♦ s t ♥ ♣é♥ ♣r sst♠s♣r♦t♦s ♠♥s♦♥s r♦s s ♦♥t ♥♦r♠ ℓ sr s♦ ♦s ♣rá♠tr♦s 〈Φ〉L,r=1,ℓ ② ∆L,r=1,ℓ ♦♠♦

〈Φ〉L,r=1,ℓ = Φ∞,r=1,ℓ + aℓL−1/ν−ϑ

r③♦ ♦s rs ♦r t♥♦ ♥sttt ♦ P②ss r ♣ú r rst♦♣r ♦♦r ♥t ♥sttt st♦s ❯♥♦s t♣♥ rt♥s tt♦♦♥r ❯♥rst② ♠♥ ② ♥t ♥sttt ② ♥t♦♥ ♥ ♥sös♦♥ ♣♦r s s♥trs ② ♥ t♠át s♦ ② ♦♥t♥ sst♠s ♣r♦t♦s

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

②∆L,r=1,ℓ = cℓL

−1/ν

s♥♦ Φ∞,r=1,ℓ ♠r ♣r♦ó♥ ♣r sst♠s ♥♥t♦s ♥srít L ♦♥t rst sst♠ aℓ ② cℓ ♦s t♦rs ♣r①♣♦♥♥s ν ①♣♦♥♥t ♦♥t ♦rró♥ 1/ν s ♥♦♠♥ ①♣♦♥♥t s♦ ② ϑ ♦♥t ♦rró♥ ①♣♦♥♥t s♦ ♣r sst♠s ♠♥s♦♥s r♦s ❬ ❪

sí♦ stá♥r ♥ó♥ stró♥ ♣r♦ ♣r♦ó♥ ∆L,r=1,ℓ stá r♦♥♦ ♦♥ ♥♦ stró♥ s♥t ♦r♠ ♥t♦ ♠②♦r s ♦r ∆L,r=1,ℓ ♠②♦r s ♥tr♦ ♦rs Φ ♣r s ♣r♦ ♦♥ ♣r♦ ♣r tr♥só♥♣r♦t ♦♠♦ s sró ♥ ó♥ ♣r ♥ sst♠ ♥t♦①st ♣r♦ ♥♦ ♥ ♦t♥r ♥ str ♣r♦t♦ ♥ ♥♦Φ < Φ∞,r=1,ℓ ♥ ♠♦ ♣r ♥ sst♠ ♥♥t♦ ♥ó♥ ♣r♦ ♣r♦ó♥ s ♥ ♥ó♥ só♥ ♥ Φ∞,r=1,ℓ s r ∆∞,r=1,ℓ = 0 ∀ℓ♦ ∆L,r=1,ℓ sr ♥ ♥ó♥ ♠♦♥ót♦♥ r♥t rs♣t♦ L② ♥♦♥ stá srt ♣♦r ♦ ♦tr♦ ♠♦♦ tr♥só♥ ♣r♦t s ♠ás ② ♠ás ♥♦st ♠♥tr t♠ñ♦ sst♠

♥ ♥t♦ tr♦s ♥tr♦rs ♥ ♠♦str♦ q♦s sst♠s ♣r♦ó♥ st♦ ② sst♠s ♣r♦ó♥ s♠♥t♦s rts ♥ ♦♥t♥♦ ♣rt♥♥ ♠s♠ ♥rs ♣r♦ó♥ sr t♥♥ ♦s ♠s♠♦s ①♣♦♥♥ts rít♦s ❬❪ s ú♥ st♦s ♣r♦s♠str♥ q t♦♦s ♦s sst♠s ♣rt♥♥ ♠s♠ ♥rs s ♠♥só♥ t♦♣♦ó sst♠ r ♣r♦ó♥ ♦♥♦♥s ♦♥t♦r♥♦ ② ró♥ s♣t♦ r s♦♥ s ♠s♠s ❬❪ ❯t③♥♦ r♠♥t♦s♥rs s♦ s s♣r q ♦♥r♥ t♦♦s ♦s sst♠s♥t♦s sté ♣♦r ①♣♦♥♥t −1/ν ♦♥ ν = 4/3 ♥ sst♠s ♠♥s♦♥s ❬❪ ♥ ♠r♦ s♥♦ ♣ó♥ ♥ ❩ ❬❪♦ ② r♦♥② ❬ ❪ ♠♦strr♦♥ trés q ♠♥t♦ ♣♦r s♦ ♦r ♠♦ ♥s ♦t♦s ♣r♦♥ts〈Φ〉L stá rtr③♦ ♣♦r ①♣♦♥♥t −1/ν − ϑ ♥t♠♥t ❬❪ s ♥♦♥tr♦ q ♣r sst♠s r♦s r = 1 s ♣r♦♥tsϑ = 0.83 ± 0.02 ♦♥sst♥t ♦♥ ♦s ♦rs ♣r♠♥t r♣♦rt♦s 72/91❬❪ 0.85 ❬❪ ② 0.90±0.02 ❬❪ ♠♥trs q ♣r sst♠s rt♥rsr 6= 1 s t♥ ϑ = 0 ❬❪ ó♥ s ①♣rs♦♥s ② s t ♥ ♣é♥

s♣t♦ ♦r ♠r ♣r♦ó♥ r♥t♠♥t ② ❩♥❬❪ r♣♦rtr♦♥ ♣r sst♠s s♦tró♣♦s r♦s s ♣r♦♥tsΦ∞,r=1,ℓ=1 = 5.63726 ± 0.00002 ♦♥sst♥t ♦♥ ♦r 5.71 ± 0.24 r♣♦r

♦ s♠r s ♦♥ s tr♥s♦♥s s tr♠♦♥á♠s q ♦rr♥ ♥ ♥s♦♥♦♥s ♥ ♥s ♥ sst♠s ♠r♦só♣♦s ② ♥ ♥ ♥tr♦ ♦♥♦♥s ♥sst♠s t♠ñ♦ r♦

♠♣ó♥ ♦♠étr sst♠ ♥ st♦

t♦ ♣♦r P ② r tr♦ és trás ❬❪ P♦str♦r♠♥t ➎t♥♦ ② ❬❪ r♦♥ s♦ 〈Φ〉L,r,ℓ=1 ♣r r♥ts♦rs ró♥ s♣t♦ ② ♠♦strr♦♥ q ♥ sst♠s rt♥rsΦ∞,r,ℓ=1 ≈ 5.63726 ∀r ♠♥trs q rt♥s ② ♦♦r ❬❪ st♠r♦♥ ♦♥♠②♦r ♣rsó♥ ♥s rít r♣♦rt♥♦ Φ∞,r=1,ℓ=1 = 5.6372858(6)

♠♣ó♥ ♦♠étr sst♠ ♥ st♦

♦♠♦ s sró ♥ ♣ít♦ t♦s s ♠strs ♦♠♣♦st♦sP34❬❪ 4.2% ♣r♣rs t③♥♦ s ♦♥♦♥s ①♣r♠♥ts ts ♥ ó♥ ①r♦♥ ♦♥t étr♣r s♦♠♥t ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦♦ s

st r♥ ♥trés t♥♦ó♦ r r♣r♦ ♦t♥ó♥ sst♠s q ①♥ ó♥ ♦♥ ♥s tr♠♥r ó♠♦ ♥② strtr r♥♦ ♥ ♣r♦ ♦t♥r ♥ st rs♣♦♥r s ♥trr♦♥t s ó ♦ ♥ ♥áss ①t♥s♦ ♠r♦rqttr ♠tr ♠♥♦♥♦ r♥t r♦ ♠tr ♦♠♣st♦ ♣ó♥ ♠♣♦♠♥ét♦ ①tr♥♦ Hcuring ♥ ♠♥t③ó♥ s ♠r♦♣rtísírs ♠♥tt♣t ♣♦r ♠♥t③ó♥ s Ps ♠♥ttq ♦♥♦r♠♥ s ♥tr♦♥s ♦♣♠♥t♦ ♠♥ét♦ ♥tr sts♠r♦♣rtís r♥♦ ♥r ♥ ♦r♥♠♥t♦ s♣♦♥tá♥♦ r♥♦ ♥♦r♠ ♣qñ♦s s♠♥t♦s ♥s ♥♦ ♦r♠♦ ♣♦r ♥ r♥♥ú♠r♦ µPs P♦str♦r♠♥t ♥tró♥ ♠♥ét ♣♦r t♣♦③♦ ♥tr sts ♦♠r♦♥s µPs ♥r ♦♣♠♥t♦ ésts ♦♥ ♦r♠ó♥ strtrs t♥rs ♠②♦rs q ♥♦♠♥♠♦s♣s♦♥s ♦♥t♥♦ r♥♦ s ♠② r♦ ♥ ♠trstrtr♦ ♥ s ♦srrá♥ sts ♣s♦♥s s♣rs ♥s s♦trs ♣♦r st♥s rtrísts ♠s♠♦ ♦r♥ q t♠ñ♦ s♣s♦♥s ♦ ú♥ ♠②♦rs st ♠tr ♣ ♥♦ ♣rs♥tr ♦♥t étr ♣r ♥ ♥♥♥ ró♥ P♦r ♦tr♦ ♦ s ♦♥t♥♦ r♥♦ s ♠② ♦ ♥ ♠tr strtr♦ ♥ s ♦srrá ♥stró♥ ♥♦♠♦é♥ r♥♦ ♦r♥t♦ ♣rr♥t♠♥t ♥ ró♥ ♣ó♥ ♠♣♦ ♠♥ét♦ r♥t r♦ st strtr t♥♣r ♦♥ts étrs s ♥ ♠s r♦♥s ♠tr ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ ② ♥ ró♥♦rt♦♦♥ ♦ ♦sr♦ ♣r ♦♠♣♦st♦ ♥trés ♦rrs♣♦♥ ♥ stó♥ ♥tr♠ ♣♦r ♥tró♥ t♦r t♣♦ ③♦ ♥tr s♣s♦♥s s st♥ r♣♦♥s r♥♦ ♠②♦r t♠ñ♦ qt♠é♥ stá♥ ♦r♥ts ♣rr♥t♠♥t ♥ ró♥ Hcuring ♥♦♠♥r♠♦s sts strtrs ♦♠♥s ♥ ♦rrs♣♦♥ ♥ str

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

♦r♠♦ ♣♦r rs ♣s♦♥s ♦s s ♦♥♦♥s ♣r♣ró♥s sts strtrs ♦r♥ ♠②♦r ♦♥t♥ ♦s ♦♣st♦s ♥ ♥ ró♥ ♣ó♥ Hcuring ② s ♥♥tr♥ s♣rs ♥ s ♦trs ♥ ró♥ ♦rt♦♦♥ ést st ♥s♦tr♦♣í strtr ♠♣ q só♦ ♣♥ strs ♥s ♦rr♥t étr ♥ ró♥ Hcuring ♣♦r ♦ q ①rá s ♣rs♦ t♦♠r ♥ ♥t q♥ ♠r♦rís ♠tr P34❬❪ 4.2% ♣♥srt③rs r♠♥t s ♣s♦♥s ♥s q ♦♥♦r♠♥ sr♥ts ♦♠♥s s ♣s♦♥s ♣r♥ ♦♠♦ strtrs ♦♦♥s ♠s rss ♥ ♥tr♦ ② ♠s s ♥ ♦s ①tr♠♦s ♥t♦♥s ♥ ♣s♦♥ s st♥ ♦♥st ♣♦r ♦s rtrísts á♥♦ ♣qñ♦ ♥tr s ♣s♦♥s ② ♦sró♥ ♥ ③♠♥t♦ ♥ ♥♦ ② ♥ ♥ s ♥ st st♦ ♣rtr s ♠r♦rís ♣ tr♠♥rs stró♥ ♦♥ts á♠tr♦s ② ♦r♥tó♥ s ♣s♦♥s s r á♥♦ ♦♠♣r♥♦♥tr s ♣s♦♥s ② Hcuring

r str♦♥s ♥rs s ♣s♦♥s ♥ P34❬❪ 4.2% í♥ ♦♥t♥s ♦rrs♣♦♥ stró♥ ♥♦r♠ st ♣♦r ó♥ ♦s ♣rá♠tr♦s st r♣r♦ss ♠str♥ ♥srt♦s

r ♠str st♦r♠ ♦t♥♦ ♣r stró♥ ♥r s ♣s♦♥s ♣r rr♥ ♦t♥♦s ♣♦r ♠r♦s♦♣í ♦ st♦r♠ ♣ strs ♠♣ír♠♥t ♠♥t ♥ ♥ó♥ stró♥ ss♥ ♥ ♦♥t♥ ♥ r ♣♦r ①♣rsó♥

fθ = p(θ; 〈θ〉, σθ) =1

σθ√2π

exp

[−(x− 〈θ〉)2

2σθ2

]

♥tr ♥ ró♥ Hcuring s r 〈θ〉 = 0 ♦♥ ♥ sí♦ s

♠♣ó♥ ♦♠étr sst♠ ♥ st♦

tá♥r σθ = (4.65± 0.02) ♦♥t♦ r③♦ s♦r 389 ♣s♦♥s ①♥t r♦ st stró♥ r ♥ t♦r s ♥tr♦♥s ③♦ ♥tr r♣♦♥s ♣rtís r♥♦ r♥t strtró♥ ♠tr s ③ r s♣rsó♥ ♥r♣qñ♦ ♦r σθ s ♦♥s♥ ♥t♥s♦ ♠♣♦ ♠♥ét♦ t③♦r♥t r♦

r ♠str ♦s st♦r♠s s♦♦s s str♦♥s ♦♥ts ℓ ② á♠tr♦s ♠♦s ♠t ♦♥t d s ♣s♦♥s ♦t♥s ♦♠♣t♥♦ 364 ② 311 ♣s♦♥s rs♣t♠♥t♥ ♠♦s s♦s s ♦t♥ ♥ ①♥t r♦ st t③♥♦ ♥ ♥ó♥ stró♥ ♦♥♦r♠

fλ = p (λ; 〈λ〉, σλ) =1√

2πσλλexp

[− (lnλ− ln〈λ)2

2σ2λ

]

stró♥ s ♦♠ú♥♠♥t t③ ♣r rtr③ó♥ strtr sst♠s ♥s♦tró♣♦s t♣♦ ♥t♦r♦♦ st♦♠rst♦♠r♦ ♠♥t♦r♦ó♦ ② rr♦í♦s ❬❪ Pr st rt♦ s ♦♥ts s t♥ λ = ℓ ♦s ♣rá♠tr♦s st 〈ℓ〉 = (1.35±0.01)♠♠② σℓ = (0.26± 0.01)♠♠ ♦♥ R2 = 0.9965 Pr st rt♦ ♦s á♠tr♦s s t♥ λ = d ♦s ♣rá♠tr♦s st 〈d〉 = (10.40 ± 0.02)µ♠② σd = (0.30 ± 0.01)µ♠ ♦♥ R2 = 0.9977 ♦♥♠♥t ó♠♣t♦ ♥s ♠ ♥s ♥ ♣♥♦s ♦♥s Hcuring rr♦ ♦rΦ ≈ 12 ♣s♦♥s×♠♠−2

r str♦♥s ♦♥ts ② á♠tr♦s ♠♦s ♠t ♦♥t ♣r s ♣s♦♥s ♥ P34❬❪ 4.2% s í♥s ♦♥t♥s ♦rrs♣♦♥♥ str♦♥s ♦♥♦r♠s sts ♣♦r ó♥ ♦s ♣rá♠tr♦s st r♣r♦s s ♠str♥ ♥srt♦s ♥ ♣♥

♦rrs♣♦♥ str qí q s ♦sr♥ st♦r♠s ♠② s♠rs♣r r♥ts ♠strs rr♥ ♣r♣rs ♦ s ♠s♠s

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

♦♥♦♥s ①♣r♠♥ts ② ♠s♠♦ ♦♥t♥♦ r♥♦ ♥t♦ r♦st③ ♣r♦s♦ ró♥ rr♥

♥ rqttr ♥tr♥ ♦s sst♠s s ♦♠♣ ♥ s♦ ♦ st♦ ♣ ♠♦rs ♠s♠ ♦ s♥t ♦♥sró♥♦♠♦ ♠str r á♠tr♦ s ♣s♦♥s s ♠♦♠♥♦r q ♦♥t s strtrs P♦r ♦♥s♥t ♥ ♣rs♥t ♥áss ♣s♦♥ srá♥ ♠♦ ♦♠étr♠♥t ♦♠♦ ♥s♠♥t♦ rt ♦♥t ♣s♦♥

♥♦ s♠♣♦ s♣t♦ ♦♠étr♦strtr sst♠ ♦ ♥áss q ♦♥t♠♣r ♥ó♠♥♦ tr♥s♣♦rt étr♦ ♥ ♦ssst♠s ♦♠♣st♦s ♦ ♥ó♠♥♦ srá ♦ ♥ t ♥ ♣ít♦ P♦r ♠♦♠♥t♦ st ♦♥srr rst♦ ♣r♥♣ q rr♦ ♦♥áss ♠♥s♠♦ ís♦ q ♦r♥ tr♥s♣♦rt étr♦ ♥ ♠tr ♥ st♦ s t♥♦ tró♥♦ ♥tr ♦s r♦♥s r♥♦s♣rs ♣♦r st♥s ♥♥♦♠étrs ♦ s♥♥♦♠étrs ♥tr s r♦♥s r♥♦ ♣ r ♦ ♥♦ ♠tr③ ♣♦♠ér ♥ ♦trs ♣rs①strá ♥ ♦ étr♦ ♣r ♥tr ♦s r♦♥s ♦♥t♦rs r♥♦s ② s♦♦ sí s ♠s♠s s ♥♥tr♥ s♣rs st♥s ♠② rsP♦r ♦ t♥t♦ ♦ s♠♣ó♥ ♦♠étr ♥tr♦r♠♥t srr♦♥ ♣rs♥t ♦r♠s♠♦ s ♦♥srrá q ①st ♦ étr♦ ♣r♥tr ♦s s♠♥t♦s rt s♦♦ s ést♦s s ♥trst♥ ♠t♠♥t r sr ♥ ♦r♠ sq♠át ♠♦♦ ís♦ ② strtr ♥ó♠♥♦

♠♣ó♥ ♦♠étr sst♠ ♥ st♦

r sq♠ ♦♥♣t ♣r♦①♠ó♥ s ♣s♦♥s ♣♦r s♠♥t♦s rts ♥ s ♠str♥ ♥ r ♦s ♦♥♥t♦s♣r♦♥ts q ♦♥t♥ s rsts rts ♦♥sr q ①st♦♥tt♦ étr♦ ♥tr ♦s s♠♥t♦s rts s♦♦ s ést♦s s ♥trst♥

♣ít♦ ♥s♦tr♦♣í étr ♦t r♦ ór♦

s ♣rs♦ t♦♠r ♥ ♥t qí ♦s ss ♥ ♣r♠r r strt♠♥t ♥♦ tr♥s♣♦rt étr♦ r♦ ♣♦r t♥♦ tró♥♦♥tr r♦♥s ♦♥t♦rs r♥♦ s ①t♥s♦ ♥ t♦♦ ♠tr ♣st♦q ♥ó♠♥♦ ♥♦r♦ t♥♦ tró♥♦ ♥♦ t♥ s♦ ♥ st♥ ♠á①♠ ♦rt t♦ st♥ ♣rtr ♣r♦ t♥♦ s strt♠♥t ♥ ♥ ♠r♦ ♦♠♦ ♠str♥ ♦s rst♦ssrt♦s ♥ ♣ít♦ ♥ ♦r♠ ♦♥sst♥t st♦s ♣r♦s r③♦ss♦r sst♠s ♦♠♣st♦s ♥♦ strtr♦s ❬ ❪ só♦ ①st♦♥t étr ♣r ♥tr r♦♥s ♦♥t♦rs s♣rs st♥s ♣qñs ♦r♥ ♥♥ó♠tr♦ ♦ ♠♥♦rs ♥ s♥♦ r♦♠♦ t♠é♥ s trá ♥ ♣ít♦ s ♣s♦♥s r♥♦♣rs♥t♥ ♠r♦ ② ♥♥♦ r♠♥t♦♥s ♥tr♥s ♦♥ ♦ s♥ ♥tr♣♥tró♥ ♠tr③ ♣♦♠ér st r♠ó♥ stá sst♥t ♠♥t ♥①♣r♠♥t ♥rt ♠♦♦ rs♣st ♣③♦rsst ② ást ②rt ♦sró♥ s r♠♥t♦♥s ♠♥t ♠r♦rís s st s♦ ♦ ♠♦♦ ♦♠étr♦ ♦♥t♠♣♦ s♠♥t♦ rtrí sr ♦♥sr♦ ♦♠♦ ♥ ♦♥♥t♦ s♠♥t♦s ♦♥t ♠♥♦r♦♥t♦s ♦♥♠♥t ♥ ♦r♠ ③♦ ♥ ♠r♦ ♦ ♥strs♠♣ó♥ tr♥s♣♦rt étr♦ ♦ ♦♥♥t♦ ♣ ♦♥srrs ♦♠♦ ♥ ú♥♦ s♠♥t♦ rt s♥ ♣ér ♥r st ♦r♠♦ ♠♦♦ srt♦ ♥♦ s ♥sr♦ r ♥ ♦r♠ ①st sr♠♥t♦♥s s ♣s♦♥s ♣r str ♥s♦tr♦♣í étr s♥♦ q rst s♥t s ♦sró♥ s ♠s♦só♣ P♦r ♦tr♦♦ ♥ trtr s♥ t③rs ♦♠♦ srá ♥str♦ s♦ s♠♦♥s♥ sst♠s ♠♥s♦♥s ♣r st♠r ♣r♦♣s ♣r♦ts sst♠s tr♠♥s♦♥s ♥ ♠r♦ ♥♦ s r♥rs ♥ r♠♥tó♥t ③ s♠♣ó♥ t♦♣♦ó ♣r♣ró♥ ♠strs sst♠s í♥rs ② s ♦♥♦♥s ♣r♣ró♥ t③s ♣r t ♥ ♥♥ s♠trí r♦t♦♥ ♥ t♦r♥♦ r♦tó♥ ♠♦ r♦ s♠trí í♥r ❬ ❪ ♥♣♥♥t♠♥t stró♥ ♦s ♦t♦s ♣r♦♥ts s ♦sr♦ q ♥s ♣r♦ ♣ró♥ ♦s r♣♦s ①t♥♦s s♣♥♥♥ strs ss♣♠♥t ♥♦r♠ ♥ ♦s sst♠s ♠r♦só♣♦s ❬ ❪♥ ♣rtr s stró♥ ♦s ♦t♦s ♣r♦♥ts s s♦tró♣ ♣r♦ ♦t♥r ♥ ú♥♦ str ♣r♦♥t s s ú♥ ♣r♦ q ést str ♣r♦♥t ♣rs♥t ♠s♠ s♠trí q sst♠ ♣rt♥ t♠é♥ s st ♦r♠ s s ♦sr♥ r♣♦ ①t♥♦ ♥ sst♠ tr♠♥s♦♥ s♠trí í♥r ss♣r ♦♥ ♣r♦ q t♠é♥ s ♦sr ♥ r♣♦ r♣♦①t♥♦ ♥ sst♠ ♠♥s♦♥ s♦♦ ♦rrs♣♦♥♥ts ♦rts ♥♣♥♦s rs ♥tr♦s ♥ ♦♥t♥ ♥r♦ ♠♦ ♦rst r③♦♥ ♦♥srr á ró♥ t♦♣♦ó ♥ ♦s sst♠s ♦ st♦ ♠♥t st♦ ♦s ♦rts ♥♦ ést♦ ♣♥③rs ♠s s♠♣♠♥t s ♦♥t étr ② ♦♥t ♣r

♠♣ó♥ ♦♠étr sst♠ ♥ st♦

♦t st ♦r♠ ♥ sst♠ ♠♥s♦♥ s♠♥t♦s rt ♦♥ s

tró♥ ♥s♦tró♣ s r ♦♥ stró♥ ♥r ♥♦ ♥♦r♠ qrtr③ ♣♦r ♥s s♣r ♦t♦s ♣r♦♥ts Φ s ♠♥s♦♥s sst♠ Lx ② Ly ② ♥ ♦♥♥t♦ ♥t♦ ♣rá♠tr♦s strtrsψk = σℓ, 〈ℓ〉, σθ

•

♦♥ ♦t♦ ①trr ♣rá♠tr♦s rsts ♣r s s♠♦♥s♦♠♣t♦♥s s ó ♦ ♥ rtr③ó♥ strtr stíst ♠strs P34❬❪ ♦♥ r♥♦ ♠r♦♣rtís 34❬❪ s ♥♥tr ♦r♥③♦ ♦r♠♥♦ ♣s♦♥s ♥srts ♥ ♠tr③ st♦♠ér P ♥♦♥tró q s ♣s♦♥s ♣♥ ♣r♦①♠rs ♦♠♦ s♠♥t♦s rts ② q ♣♦ó♥ ♣s♦♥s ♣rs♥t ♥ stró♥ ♦♥ts ♣r♦①♠♠♥t ♦♥♦r♠ ♥tr r♦r 1.35♠♠ ♣r s♦ rr♥ ♥ st ss ② ♥ stró♥ ♥r ♣r♦①♠♠♥t ss♥st út♠ stá ♥tr ♥ ♣rr♥ ♦ ♣♦r ró♥ ♣ó♥ ♠♣♦ ♠♥ét♦ r♥t ♣r♣ró♥ ♠str ♦♥ ♥sí♦ stá♥r ♣r♦①♠♦ ♥♦ r♦s

♦s ♦♥♣t♦s qr♦s ♥ st ♣ít♦ srrá♥ s♦♣♦rt ♦♥♣t♣r ♦♠♣r♥só♥ ♥áss ♣r♦t♦ q s r③rá ♥ ♦s trs ♣ít♦s s♥ts s ③ ♦rí t ♣rt♥ srr rr♥♣r t♦r q s ♥ ♥ ár ♦rí Pr♦ó♥ ♣ sñ♦ ♠trs ♦♠♣st♦s s♥t♦♥t♦r

♣ít♦

♥s♦tr♦♣í étr ♦ttrs ② ♠ét♦♦s♥♠ér♦s

♦ ♥♦t ♦s ②♦r t ♠t② ♦rtrsss ♦r ♠t♠ts t♠ts rs t♦ t ♥ s t ②s s

t♥s ❯♠

s♠♥ ♥ ést ♣ít♦ s ♣♥ ♦s ♦♥♣t♦s ♥tr♦♦s ♥ ♣ít♦ ② ♥ ♦s ♣é♥s ② ♣r srr ♥ t ♥♦ ♦s ♣r♦♠♥t♦s ♦rít♠♦s ② té♥s ♥♠érs t③s ♥ ó♥ ♥áss ♥ ♠trs ♦♠♣st♦sstrtr♦s ♥ tér♠♥♦s sst♠s ♣r♦t♦s ♠♥s♦♥s s♠♥t♦s rts

st♠ó♥ ♣r♦ ♣r♦ó♥

♠♣③r♠♦s ♣♦r srr stíst ♥♦r ♥ st♠ó♥ ♣r♦ ♣r♦ó♥ ♣r ♦ srr ♥ t ♠♥s♠♦♦rít♠♦ srr♦♦

♦♥sr♠♦s s♥t ①♣r♠♥t♦ t♦r♦ q ♦♥sst ♥ n ♣rs s ♥r♥ n sst♠s rt♥rs t♠ñ♦ ♥ L ② ró♥ s♣t♦ r ♥♦ ♦♥ N s♠♥t♦s rt ♦♥t ♣♦só♥ ② ♦r♥tó♥ ♦s s♠♥t♦s rt q♥ srts ♣♦r ♦♥♥t♦ ♣rá♠tr♦s strtrs ψk rstr ♥t sst♠s ♣r♦♥ts♦t♥♦s m ♥t m ≤ n

♣ít♦ ♥s♦tr♦♣í étr ♦t trs ② ♠ét♦♦s

♥♠ér♦s

❯s♥♦ s s ♥tr♦s ♥ ó♥ rs♣t♦ s ♥♦♥s rs t♦rs ♥ ♠str♦s ♥♠♦s s n rs t♦rs

Xj =

1, sst♠ jés♠♦ t♥ ♠♥♦s ♥ r♣♦ ♣r♦♥t

0, s♦ ♦♥trr♦.

♦♥ j = 1, 2, . . . , n ♥ ♥ s jés♠s r③♦♥s ♦♥sr♠♦s♦♠♦ é①t♦ t♥r ♠♥♦s ♥ r♣♦ ♣r♦♥t ♣r♦ t♥r é①t♦ ♥ qr s r♣t♦♥s s ℘

L,r,ψk(Φ) ② rst ♥t

q s rs X1, . . . , Xn s♦♥ t♦rs ♥♣♥♥ts é♥t♠♥tstrs ♦s ss rs t♦rs t♥♥ ♥ stró♥ r♥♦♥ ❬❪ s r

Xj ∼ Be (p)

♦♥Be ♥ stró♥ r♥♦♥ ♦♥ j = 1, 2, . . . , n ② p ≡ ℘L,r,ψk

(Φ)P♦r ♣r♦♣s stró♥ r♥♦ rst

E [Xj ] = p

V ar [Xj ] = σ2Xj = p (1− p)

♦♥sr♠♦s r t♦r X = 1n

∑nj=1Xj r♣t♠♦s H s

①♣r♠♥t♦ t♦r♦ srt♦ ♥♦ ♦♥ n r③♦♥s ♦t♥r♠♦sH♦rs X ♣r♦r r♥ts ♣♦r ♥tr③ t♦r X ♥ ♥ s♦s H ①♣r♠♥t♦s ♣r♦ ♦t♥r ♥ ♦r x = m/n♣r st♠ó♥ ℘

L,r,ψk(Φ) stá ♣♦r stró♥ t♣♦

♥♦♠

P(X =

m

n

)=

(n

m

)pm(1− p)n−m, m ≤ n

P♦r ♦r♠ ♥tr í♠t ❬❪

X∼−−−→

n→∞N

(p,p (1− p)

n

),

♦♥ ∼−→ ♥ ♦♥r♥ ♥ stró♥ ② N ♥ stró♥ ♦r♠♦ ss♥ sí rst

E[X]= p = ℘

L,r,ψk(Φ)

V ar[X]=p (1− p)

n.

st ♦r♠ s r③♠♦s ♥ ♥ú♠r♦ r♥ r③♦♥s n stró♥ ♦s ♦rs X s ss♥ ② ♥♦st r♥③s ♣r♦♣♦r♦♥ 1/n P♦r s ♣r♦♣s X ♦rrs♣♦♥ ♥ st♠♦r ♥ss♦ ℘

L,r,ψk(Φ) ② ♣♦♠♦s st♠r ℘

L,r,ψk(Φ) ♦♥ ♥

♦rt♠♦

①tt ② ♣rsó♥ ♥r♠♥t♥♦ ♥ú♠r♦ r③♦♥s ts❬ ❪ ♣s

x =m

n−−−→n→∞

℘L,r,ψk

(Φ) .

rr♦r st♠ó♥ r③ ♣ rs ♦♠♦ ε =√V ar

[℘L,r,ψk

]

♥ ♠r♦ s♠♦s ①♣rsr ♦ rr♦r ♥rt③ ♥ tér♠♥♦s ♠strs ② ♥♦ ♣♦♦♥s ❬❪ r③ó♥ ♣♦r s t③ st♠♦r ε

ε =

√(m/n) (1−m/n)

n

r ♠str ♦♠♣♦rt♠♥t♦ st♠♦r ε rs♣t♦ ♥ú♠r♦ r♣t♦♥s ♥ ①♣r♠♥t♦ n ② rs♣t♦ ♦r st♠ó♥ m/n

0 400 800

0.03

0.06

0.09 b m/n = 0.1 m/n = 0.2 m/n = 0.5

n

a

0.0 0.5 1.00.0

0.5

m/n

r ♦♠♣♦rt♠♥t♦ st♠♦r ε rs♣t♦ ♥ú♠r♦ r♣t♦♥s ♥ ①♣r♠♥t♦ n ② rs♣t♦ ♦r st♠ó♥ ℘ = m/n

♦rt♠♦

♦ st♦ ♥ ó♥ ♥tr♦r ♦♥stt② s ♣r ♦♥♦♥r srs ♥ó♥ Pr♦ Pr♦ó♥ ♣r ♥str♦ sst♠ ♥trés ♠♥t r③ó♥ s♠♦♥s ♦♥t r♦

s s♠♦♥s ♦♥t r♦ s r③r♦♥ ♣♦r ♠♣♠♥tó♥ ♦rt♠♦s srt♦s ♥ ♥ sst♠ s♦tr ♠t♠áts rtt♦ ó♦ rt♦ ♦ ♥ P q ♦♠♥ ♣♦t♥ ♠♦s♣qts ó♦ rt♦ ①st♥ts ♦♥ ♥ ♥tr③ ♦♠ú♥ P②t♦♥♠ás ♣rs♥tr ♥ t ♦♠♥ ♣r♦r♠♦rs ♥ s st♦ s♠t♦r ①st♥ ♥♠r♦s♦s r♦s rr♥ ❬ ❪

♣ít♦ ♥s♦tr♦♣í étr ♦t trs ② ♠ét♦♦s

♥♠ér♦s

♦ q ♦♥rt ♥ ♥ rt ② tr♥t ó♦ rt♦ ♠ © ♣ © t♠t © ② t © ♠ás ♥t♦♥ ♥ ♣t♦r♠ ó♥ ♥ t♠♣♦ r ♦rt♠♦s s♦s♥ ♦ sst♠ ♥♦♠♥ ♦ s♣♦♥ ♥ ♦r♠ rtt ♥♦s♠t♦♠

♦♥stró♥ ♦♠♣t♦♥ sst♠ s♠♥t♦s rts ♣r♦♥ts

♥ s♦ sst♠s r♦s s♠♥t♦s rts ♣r♦♥ts ♦♥t ♥♦r♠ ℓ ② ♦r♥tó♥ t♦r sst♠s sótr♦♣♦s ♦rt♠♦ ♦♠♥③ ♦♥ ♥ r♦ í♦ t♠ñ♦ L ♦♥t♥ó♥ s♥r♥ ♥tr♦ sst♠ N ♣♥t♦s Aj ♦♥ j = 1, 2, · · · ,N ♦r♠t♦r s r s♥ ♦rró♥ s♣ ♦ ♥ st♥ ℓ s♦s ♣♥t♦s s ♥r♥ ♦s N ♣♥t♦s ①tr♠♦s ♦s ♠♥t♦s ♣r♦t♦s Bj ♦♥ j = 1, 2, · · · ,N ♣♦r ♥ró♥ t♦r á♥♦ ♦s♠♥t♦s rs♣t♦ ♦r③♦♥t θ P♦r trtrs sst♠s sótr♦♣♦s s t♥ −π ≤ θ ≤ π ♦♥ ♥ stró♥ ♥r ♦♥t♥ ② ♥♦r♠ Pr ♥ró♥ t♦r á♥♦s ② ♣♥t♦s ♦ ♥ stró♥ ♣r♦ ♦♥t♥ ② ♥♦r♠ s t③ó ♠ó♦ ♥t♦ s♣②stts♥♦r♠ ♣r♦s♦ ♦♥stró♥ sst♠s rt♥rs ♥sótr♦♣♦s s ♦♠♣t♠♥t ♥á♦♦ ♦rt♠♦ ♦♠♥③ ♦♥ ♥rtá♥♦ í♦ t♠ñ♦ L ② ró♥ s♣t♦ r ♦♥t♥ó♥ s♥r♥ ♥tr♦ sst♠ N ♣♥t♦s Aj ♦r♠ t♦r ② ♣♦r út♠♦ s str♦♥s ♥r ② ♦♥ts ♦♣ts s ♥r♥ ♦s N♣♥t♦s ①tr♠♦s ♦s s♠♥t♦s rts

st♦ ♦♥t

r♠♥ ♦♥stró♥ sst♠ ♥trés s ♣r♦ ①♣♦rr ♥trsó♥ ♥tr ♦s r♥ts ♠♥t♦s ♣r♦♥ts ♦♥sr♠♦s ♥♣r♠r r ♥ ssst♠ ♦♥stt♦ ú♥♠♥t ♣♦r ♦s s♠♥t♦s rts ♦♠♦ ♠♦str♦ ♥ r

♥♦t♠♦s ♦♠♦ Aj =(xAj , yAj

)② Bj =

(xBj , yBj

) ♦♥ j = 1, · · · ,N

♦s ♣♥t♦s tr♠♥s ♦s s♠♥t♦s rt ♦♥ ♣r♦♣óst♦s ♦rít♠♦s♦s s♠♥t♦s rts ♣♥ ♦♥srrs ♦♠♦ ♦s t♦rs ♥ 3

♦♥ trr ♦♦r♥ ♠♦s s ♥ s r

−−−→A1B1 = (xB1 − xA1) .x+ (yB1 − yA1) .y + 0.z−−−→A2B2 = (xB2 − xA2) .x+ (yB2 − yA2) .y + 0.z

st ♥t q A1B1 ∩ A2B2 6= ∅ ♠♦s s♠♥t♦s rt s ♥trst♥ s ♦s ♣♥t♦s A1 ② B1 q♥ ♥ ♦s ♦s ♦♣st♦s s♠♥t♦

♦rt♠♦

r sst♠ ♦s s♠♥t♦s rts q s ♥trst♥

rt A2B2 ② s ③ ♦s ♣♥t♦s A2 ② B2 q♥ ♥ ♦s ♦s ♦♣st♦s s♠♥t♦ rt A1B1 ♥ ♦r♠ q♥t ♦s t♦rs

−−−→A1A2 ②−−−→

A1B2 ♥ t♥r s♥t♦s r♦tó♥ ♦♣st♦s rs♣t♦ t♦r−−−→A1B1 ②

♠ás ♦s t♦rs−−−→A1B1 ②

−−−→A2A1 ♥ t♥r s♥t♦s r♦tó♥ ♦♣st♦s

rs♣t♦ t♦r−−−→A2B2 ❯t③♥♦ s ♣r♦♣s ♣r♦t♦ t♦

r ② sr ♣ ♠♦strrs á♠♥t ❬❪ q ♠s ♦♥♦♥s s♦♥♠t♠át♠♥t q♥t

(−−−→A1B1 ×

−−−→A1A2

)·(−−−→A1B1 ×

−−−→A1B2

)≤ 0

(−−−→A2B2 ×

−−−→A2B1

)·(−−−→A2B2 ×

−−−→A2A1

)≤ 0

st ♦r♠ s ♠s ♦♥♦♥s s sts♥ ♦s s♠♥t♦s rts♥ stó♥ s ♥trst♥ P♦r ♣ó♥ s ①♣rs♦♥s ♥tr♦rs s♣r♦ ①♣♦rr ♥trsó♥ ♥tr t♦s ♦s s♠♥t♦s rt sst♠ ♥ ♥ sst♠ ♦♥ N s♠♥t♦s rts ♥ ①♣♦rrs N 2

♣♦ss ♥trs♦♥s ♥ ♠r♦ s♠trí ♣r♦♠ s s♠♥t♦ i s ♥trst ♦♥ s♠♥t♦ j ♥sr♠♥t s♠♥t♦ j s♥trst ♦♥ s♠♥t♦ i ② ♦ q qr s♠♥t♦ ♥♦ s ♥trst♦♥s♦ ♠s♠♦ ♥ú♠r♦ ♥trs♦♥s ①♣♦rr s N 2/2−N

st♦ s ♥trs♦♥s ♥tr s N s♠♥t♦s rt rst♥ ♠tr③ ♥trsó♥ J N ×N ♠♥t♦s ♦♥

Ji,j = Jj,k =

1, ♦s s♠♥t♦s rts i ② j s ♥trst♥

0, s♦ ♦♥trr♦.

♦ ♥ó♥ J ♣ srs ♦♠♦ ♠tr③ s♦ ♥ r♦ ♥trsó♥ ♠♣♠♥trs ♦s ♦rt♠♦s st♦ ♦♥t ♥tr♥♦♦s ♥ r♦ ♣r r stró♥ t♠ñ♦s r♣♦s s♠♥t♦s rts ♦♠♦ sí t♠é♥ ♣r str st♦ ♣r♦t♦ sst♠ s r r ♣rs♥ ♥ r♣♦ ♣r♦♥t ♥ ♣rs♥ttr♦ s ♦♥♦♥ó ♠♣♠♥tó ♥ ♦rt♠♦ t♣♦ ♠♥♦♥

♣ít♦ ♥s♦tr♦♣í étr ♦t trs ② ♠ét♦♦s

♥♠ér♦s

♦ ♥ ♣é♥ ♦♥ ♠♥♦r t♠♣♦ ó♠♣t♦ ♥ ♦♠♣ró♥ s♣♦♥ ♥ ♦r♠ ♥t ♥

ó♠♣t♦ ♣r♦ ♣r♦ó♥

Pr ó♠♣t♦ ♣r♦ ♣r♦ó♥ s♦ ♥ ♦ sst♠ rt♥r ♠♥s♦♥ t♠ñ♦ L ② ró♥ s♣t♦r ♦♥ s♠♥t♦s rts ♣r♦♥ts ② ♦♥t ♣♦só♥ ② ♦r♥tó♥ q♥ srts ♣♦r ♦♥♥t♦ ♣rá♠tr♦s strtrs ψk℘L,r,ψk

(Φ) ♣r♠r♦ s ♥ ♦s ♦rs Φ ♣r ♦s s s qr tr♠♥r ♦r ♣r♦ ♥♦tr♠♦s ♦s ♦rs ♦♠♦Φi = Ni/L

2 s♥♦ Ni ♥t iés♠ s♠♥t♦s rt ♥ sst♠ Pr ♥♦ s♦s ♦rs Φi s ♥r♥ n sst♠s ♣r♦t♦s♦♥ ♦s ♣rá♠tr♦s strtrs ♦s ② t③♥♦ ♦rt♠♦ srt♦s ú ♥ ♥ n r③♦♥s ♣rs♥ ♠♥♦s ♥r♣♦ ♣r♦♥t ♦ r ♣r♦ó♥ mi ♥t sst♠s ♣r♦♥ts ♦t♥♦s ♥ s n r③♦♥s ♣r ♦r iés♠♦ Φ P♦r ♦ srt♦ ♠ás rr s t♥ ℘

L,r,ψk(Φi) = mi/n ♦♥t♥

ó♥ s r♣t ♣r♦♠♥t♦ ♣r ♦s r♥ts ♦rs Φi ♦t♥♥♦ st ♦r♠ ♦rró♥ s

Φi −→℘L,r,ψk

(Φi)

rá♦ ♦rró♥ ♦♥stt② r ♣r♦ ♣r♦ó♥ Pr srr ♥ ♥ st♠ó♥ ℘

L,r,ψk(Φ) s t③ó ♥

♥ú♠r♦ r♣t♦♥s n ♥♦ ♠♥♦r 103 ♣r t♦♦s ♦s ①♣r♠♥t♦s r③♦s q ♦rrs♣♦♥ ♥ ♥ú♠r♦ ♠②♦r t③♦ ♣♦r ♦tr♦s t♦rss♦r t♠áts s♠rs ❬ ❪

❱ó♥ ♦rt♠♦

ó♥ ♦rt♠♦ ②♦s rst♦s s ♠♦strrá♥ ♥ ♣ít♦s♥t ♦♥sst ♥ ♦♥ó♥ rs ♣r♦ ♣r♦ó♥♣r sst♠s r♦s s♠♥t♦s rt ♣r♦♥ts ♣r r♥ts♦rs L ② ℓ ♥ s rs ℘

L,r=1,ℓ(Φ) ①♥ ♥ ♦♠♣♦rt

♠♥t♦ str♠♥t♦ ①♣♦♥♥ ♥ ss ♦s ❬❪ ♦♥ ♥s ♣rát♦ss ♦♥srrs ♥ó♥ stró♥ ♣r♦ ♣r♦ó♥ΓL,r=1,ℓ (Φ) ó♥ ♦♠♦ ss♥ ❬ ❪ sí ♣rtr ♦s ♦rs ♦♠♣t♦s ℘

L,r=1,ℓ(Φ) ♣ st♠rs ♥s ♣r♦t

rít ♣♥♥t t♠ñ♦ sst♠ 〈Φ〉L,r=1,ℓ ② sí♦ stá♥r

♥ó♥ str♠♥t♦ ①♣♦♥♥ t♠é♥ ♦♥♦ ♦♠♦ ♥ó♥ ♦rs❲♠s❲tts ❲❲ s t③ r♥t♠♥t ♣r srr ♠♣ír♠♥t tss ró♥ sst♠s ís♦s ♣♦♠ér♦s ♦♠♣♦s ② ♦rrs♣♦♥ ♥ ♥♦♥ t♣♦ fβ,k(Φ) = k exp

(

Φβ)

♦rt♠♦

♥ó♥ stró♥ ∆L,r=1,ℓ ♣♦r st ♠♥t ♥ ♥ó♥ rr♦rerf(x) ≡ 2√

π

∫ x0 e

−t2 dt r♦ ♦♥ ó♥

℘L,r=1,ℓ

(Φ) =1

2

[1 + erf

(Φ− 〈Φ〉L,r=1,ℓ

∆L,r=1,ℓ

)]

r ♠str sq♠ ♦rít♠♦ t③r

Se elije la longitud de

agujas

Se elije el tamaño del

sistema (L)

Se fija la cantidad de

agujas percolantes (Nj)

Se construyen n

sistemas

Se registra la cantidad

de sistemas

percolantes (mj)

Se estima la

probabilidad de

percolación como

mj /n

¿ j = Nmax ?

j = j+1

NO

SI

Se confecciona la

curva de probabilidad

de percolación

Se determina la

esperanza y la varianza

Ajuste

r sq♠ ♦rt♠♦ t③♦ ♣r st♦ ♣r♦t♦ s♠♥t♦s rts rrs ♦♠♦ ❵s ♦ stró♥ s♦tró♣

♦ ♥ ♦r ℓ ♣rtr ♦s ♦rs 〈Φ〉L,r=1,ℓ ② ∆L,r=1,ℓ ♣rs s♦ s ♠♥ts ♦♥ t♠ñ♦ sst♠ L ②

♣ít♦ ♥s♦tr♦♣í étr ♦t trs ② ♠ét♦♦s

♥♠ér♦s

♦♠♣rr♦ ♦♥ ♦♠♣♦rt♠♥t♦ r♣♦rt♦ ♥ ♦rí ② s ♦r♠r ♦rt♠♦ srr♦♦

tr♦ ♥♦q ♦♠♣♠♥tr♦ t③r ♣r ó♥ ♦rt♠♦s s ♥ ♣♥♥ ♥s rít ♦♥ t♠ñ♦ ♦s s♠♥t♦s rt ♦rí r ① ② ❬ ❪sr♥ q

ℓ2Φ∞,r=1,ℓ = Φ∞,r=1,ℓ=1 = 5.6372858(6).

sí ♦s ♦rs st♠♦s Φ∞,r=1,ℓ ♣r r♥ts ♦rs ℓ ♠♥t♥ ♦rt♠♦ ♦ ♥ stsr ró♥ ♥tr♦r

P♦r ♦tr♦ ♦ ♣r r s♦tr♦♣í sst♠ s ♦♠♣tr♦♥ s♦♠♣♦♥♥ts ♦r③♦♥t ② rt ♥ ♦s s♠♥t♦s rts♥r♦s ℓx ② ℓy ♥ r rs♣t♠♥t ♥ ♣rtr ♥ ♦rt♠♦ ó♥ ♣r sst♠s s♦tró♣♦s s t③r♦♥ s♠♥t♦s rt ♦♥t ℓ

r ♦♠♣♦♥♥ts ♦r③♦♥t ℓxj ② rt ℓyj ♥♦ ♦ss♠♥t♦s rt ♥rs ♣rtr ♥ ♣♥t♦ ♥ Pj ♥ á♥♦ ♦r♥tó♥ rs♣t♦ ♦r③♦♥t θj ② ♥ ♦♥t ♥ s♠♥t♦ rt ℓj

♥♥♦ ♥s♦tr♦♣í ♠r♦só♣ sst♠ N s♠♥t♦s rts ♦♠♦

A ≡

N∑j=1

ℓj |cos θj |

N∑j=1

ℓj |sin θj |

s t♥ A = 1 ♣r ♥ sst♠ ♣rt♠♥t s♦tró♣♦ ♦♥ ♦s s♠♥t♦s rt ♥♦ t♥♥ ♥ ♦r♥tó♥ ♣rr♥ ♥ sst♠ A = +∞♣r ♥ sst♠ ♦♥ s♠♥t♦s rt ♦r♥t♦s ♣rt♠♥t ♥ ró♥ ♦r③♦♥t ② A = 0 ♣r ♥ sst♠ ♦♥ s♠♥t♦s rt ♦r♥t♦s♣rt♠♥t ♥ ró♥ rt

♦rt♠♦

qr r♦ ♣r t♦r q sí ♥ ♥s♦tr♦♣í ♠r♦só♣ sr ♠étr r♦ ♥s♦tr♦♣í stró♥ ♦s s♠♥t♦s rts ♥ ♥áss ♣r♦t♦ r③♦ ♥ ♦s ♣ít♦ss♥ts

•

♥ ést ♣ít♦ s ♦rr♦♥ ♣r ♦s ♦♥♣t♦s ♥tr♦♦s ♥ ♣ít♦ ② ♥ ♦s ♣é♥s ② ♣r srr♦♦ ♥ ♠t♦♦♦í♦rít♠♦♥♠ér t③r ♣r ♦♥ó♥ s rs ♣r♦ ♣r♦♦♥ ♥ sst♠s ♠♥s♦♥s s♠♥t♦s rts♣r♦♥ts ♦♠♦ sst♠ ♠♦♦ ♣r st♦ ♥ ♠trs♦♠♣st♦s strtr♦s

♥ ♣ró①♠♦ ♣ít♦ s ♠str♥ ♦s rst♦s s♦♦s ó♥ ♦rt♠♦ srr♦♦

♣ít♦

♥s♦tr♦♣í étr ♦t❱ó♥ ♦rt♠♦

r③ ♦♥strr ♥ s ♥ rqtt♦

rstóts

s♠♥ st ♣ít♦ ♠str ♦s rst♦s s♦♦s ó♥ ♦rt♠♦ srt♦ ♥ ♣ít♦ ♣r st♦ s♣r♦♣s ♣r♦ts sst♠s s♠♥t♦s rts r♣r♦♥♦ rst♦s ♣r♠♥t r♣♦rt♦s ♣r ést t♣♦ sst♠s s ③ s r♣♦rt ♣♦r ♣r♠r ③ ♥ ♥♠ér ♣r♦♥s tórs ♣r st♦s sst♠s ♣r♦t♦s ♦ r♥ts rs ♣r♦ó♥

❱ó♥ ♦rt♠♦ ② ♣r♦♥s tórs

Pr r s s srr♦s ♥ ♣ít♦ ♥ r s♠str♥ ♥s ♦♥r♦♥s ♣r sst♠s r♦s t♠ñ♦ L = 5♦♥ s♠♥t♦s rts ♦♥ stró♥ s♦tró♣ ② ♦♥t ♥♦r♠ ℓ = 1♦rrs♣♦♥♥ts rs♦s ♦rs Φ ♠ ♥ ♥s L−2 ♦♥ L t♠ñ♦ ♥ sst♠ ♦♥strs s♥♦ ♣r♦♠♥t♦ srt♦♥ ó♥ r♣♦ t♦t♠♥t ①t♥♦ ♦ r ♣r♦ó♥ ♦r③♦♥t s rst ♥ ♦♦r r ♠♥trs q ♦s s♠♥t♦s rts q ♥♦ ♦r♠♥ ♣rt r♣♦ ♣r♦♥t s ♠str♥ ♥ ♦♦r ③

♦♠♦ s ♠♦strrá ♠s ♥t ♥ st ♠s♠♦ ♣ít♦ s ♦♥r♦♥s ♠♦strss♦♥ q♥ts s ♦rrs♣♦♥♥ts sst♠s ♦♥ ♣rá♠tr♦s ℓ′ L′ ② Φ′ = Φ/(ℓ′)2

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

♦tr q ♥r♠♥tr ♥s ♦t♦s ♣r♦♥ts ♠②♦r s ró♥ st♦s q ♣rt♥ r♣♦ t♦t♠♥t ①t♥♦ s ♠♣♦rt♥tstr q ú♥ ♥♦ ♦♥ró♥ ♠♦str ♣r Φ = 5.8 ♦rrs♣♦♥ ♥ sst♠ ♦♥ ♣r♦♦♥ ♦r③♦♥t ♥tr③ t♦r ♥ró♥ st♦s sst♠s s s t♥ rs r③♦♥s sst♠s♦♠♦ ♠♦str♦ ♥ r ♣r ♦ ♦r Φ r♦s ést♦s ♥♦♠♦strrá♥ r♣♦s t♦t♠♥t ①t♥♦ ♥ t♦ tr ♥ ♥ú♠r♦s♥t♠♥t r♥ r③♦♥s ♦s sst♠s n ♥t sst♠s ♣r♦♥ts ♦t♥♦s srá n℘

L=5,r=1,ℓ=1

r ♥s ♦♥r♦♥s ♣r sst♠s r♦s t♠ñ♦ L =5 ♦♥ s♠♥t♦s rts ♦ stró♥ s♦tró♣ ② ♦♥t ♥♦r♠ℓ = 1 ♦rrs♣♦♥♥ts rs♦s ♦rs Φ ♠ ♥ ♥s L−2♦♥ L t♠ñ♦ ♥ sst♠ r♣♦ t♦t♠♥t ①t♥♦ ♦ r ♣r♦ó♥ ♦r③♦♥t s rst ♥ ♦♦r r ♠♥trs q ♦ss♠♥t♦s rts q ♥♦ ♦r♠♥ ♣rt r♣♦ ♣r♦♥t s ♠str♥♥ ③ ♦tr q ♥r♠♥tr ♥s ♦t♦s ♣r♦♥ts ♠②♦rs ró♥ ♦s q ♣rt♥ r♣♦ ♣r♦♥t

♥ r s ♠str stró♥ ♣♥t st♦r♠ ♦rs ♥s♦tr♦♣í ♠r♦só♣ ❬ó♥ ❪ ♣r ♥ sst♠ r♦

❱ó♥ ♦rt♠♦ ② ♣r♦♥s tórs

♦♥ L = 4500 ② ℓ = 1000 ♣r ♥ t♦t 10500 r③♦♥s ♦♥ N = 103

s♠♥t♦s rts ♥ ♦♠♦ s ♦sr ♥ r ♦s ♦rs ♥s♦tr♦♣í r♦só♣ s str②♥ ♦r♠ ♣r♦①♠♠♥t ss♥ ♥tr♦s ♥ A = 1.0000 ♦♥ rt♠♥t ♣♦ s♣rsó♥(σA/〈A〉)× 100 ∼ 3% ♦ q ♠str q ♥ró♥ sst♠s s♦tró♣♦s ♣♦r ♣rt ♦rt♠♦ s ♥t ♦r ♥ st♠♦s ♥trs♦s♥ ♦t♥r ①♣rsó♥ ♥ít ♥ó♥ stró♥ ♥s♦tr♦♣í r♦só♣ fA P♦r s ♥ó♥ ♥s♦tr♦♣í r♦só♣ s♥ r t♦r ♥ó♥ 2N rs t♦rs N ♦♥ts s♠♥t♦s rts ℓj ② N á♥♦s θj ♦♥ j = 1, · · · ,N ♦ ♦r N ② ♥♦♥ ♥s♦tr♦♣í r♦só♣ ❬ ♣♦r ó♥ ❪ ①♣rsó♥ ♥♦ s ♥tr ♥ ♦r♠ ①♣ít st♦r♠ ♣r ♦t♥r fA s rrr ♠ét♦♦s ♥♠ér♦s s ♣r♦ r③rs♠♦♥s ♦♥t r♦ ♦♠♣t♥♦ sts ♦rs ♦♥ts ② á♥♦s s ♣♦r ss ♥♦♥s ♥s ♣r♦ ② ♣rtr s ♥s♦tr♦♣í ♠r♦só♣ ❬❪ ♦ s ♣ ♦♥strr♥ st♦r♠ stró♥ ♦rs ♥s♦tr♦♣í ♠r♦só♣ qs ♣rs♠♥t ♦ ♠♦str♦ ♥ r

r stró♥ ♣♥t st♦r♠ ♦rs ♥s♦tr♦♣í ♠r♦só♣ ♣r ♥ sst♠ r♦ ♦♥ L = 4500 ② ℓ = 1000 ♣r ♥ t♦t 10500 r③♦♥s ♦♥ N = 103 s♠♥t♦s rts ♥ í♥♦♥t♥ ♦rrs♣♦♥ stró♥ ss♥ ♣r♦①♠ s♦

♥ r s ♠str stró♥ ♣♥t ♦s rst♦s s♠ó♥ ♥ sst♠ s♦tró♣♦ r♦ ♦♥ L = 1 ℓ = 1 ② Φ = 12r③♥♦ 673 ó♠♣t♦s n = 103 r③♦♥s ♥ í♥ ♦♥t♥♦rrs♣♦♥ stró♥ ss♥ ♣r♦①♠ s♦ ♥ ①♣r♠♥t♦ ♦rrs♣♦♥ ♥ ①♣r♠♥t♦ t♦r♦ ♥♦♠ ♦r♦ st r rst♦ ♣r♦ ♣♦r ♦r♠ ♥tr í♠t ♣r♦①♠ó♥ ♥♦♠ ♣♦r ♥♦r♠ ó♥

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

r stró♥ ♣♥t st♦r♠ ♦s rst♦s s♠ó♥ ♥ sst♠ s♦tró♣♦ r♦ ♦♥ L = 1 ℓ = 1 ② Φ = 12 r③♥♦673 ó♠♣t♦s n = 103 r③♦♥s ♥♦ í♥ ♦♥t♥ ♠♦str♥ r ♦rrs♣♦♥ stró♥ ss♥ s♦

r ♠str ó♠♦ s ♠♦ ♦r ℘L,r,ℓ

♣r r = 1sst♠ r♦ L = 370 ℓ = 100 ② Φ = 5.884 × 10−4 ♠ qs ♠♥t ♥ú♠r♦ r③♦♥s ①♣r♠♥t♦ ♦♥t r♦ n r③ ♥ ♣r♠r r③ó♥ ② s ú s ② ♦ ♥♦ ♣r♦ó♥ s r♣rs♥ ♦ s♥ ♥♦ ♦ ♠s r♣♦s ♦♠♣t♠♥t ①t♥♦s ♦♥s ú♥ r③ó♥ s t♥ ℘

L,r,ℓ= 1 s s ♦sró ♣r♦ó♥ ② ℘

L,r,ℓ= 0

♥ s♦ ♦♥trr♦ ♦♥t♥ó♥ s r③ ♥ s♥ r③ó♥ ② st♥ ℘

L,r,ℓ∈ 0, 1/2, 1 sú♥ s♦ ♦♥t♥♥♦ ♦♥ st r③♦♥♠♥t♦

♥r♠♥tr ♥ú♠r♦ r③♦♥s ℘L,r,ℓ

♦♥r ℘L,r,ℓ

(Φ) Prn = 103 s t♥ ε

(℘L,r,ℓ

)≤ 1.58% ♥ ♥ ♣rsó♥ ♣t

❱ó♥ ♦rt♠♦ ② ♣r♦♥s tórs

r ❱♦r ℘L=370,r=1,ℓ=100

♦♥ Φ = 5.884 × 10−4 ♠ q s♥r♠♥t ♥ú♠r♦ r③♦♥s ①♣r♠♥t♦ ♦♥t r♦ n

s rs ② ♠str♥ ♥♦s ♦rs ♣r♦ ♣r♦ó♥ ♣r sst♠s r♦s r = 1 s♠♥t♦s rts ♣r♦♥ts ♦♥ stró♥ ♥r s♦tró♣ ♣r r♥ts ♦rs t♠ñ♦ sst♠ ♦♥ ♦♥ts s♠♥t♦s rts ♥♦r♠ ℓ = 1 10 ②100 rs♣t♠♥t ♦s sts ♥♦ ♥s s♦♦s ó♥ rst♦ ♦♥srr ss♥ ♥♦♥ ♥s ♣r♦ s r ΓL,r=1,ℓ ∼ N (〈Φ〉L,r=1,ℓ,∆L,r=1,ℓ) s ♠str♥ ♦♠♦ í♥s♦♥t♥s ♥ s rs ♦s ♠s♠♦s s r③r♦♥ ♦♥sr♥♦ st♠ó♥ rr♦r ♦r ℘

L,r=1,ℓ ♦ ♣rtr ó♥

♠♣♠♥t♥♦ ♦rt♠♦ ♥rrqrt ♥ t♦♦s♦s s♦s s ♦t♦ ♥ ①♥t r♦ st ♦♥ R2 ≥ 0.99985 ♦♠♦s s♣r ♣♦r ♦ st♦ ♥ ó♥ ♥ s rs s ♦sr r♠♥t q tr♥só♥ ♣r♦ó♥ s ♠♥♦s r♣t s♠♥r t♠ñ♦ sst♠ L ♦♥sr♥♦ stró♥ ♣r♦①♠♠♥t ss♥ ΓL,r=1,ℓ ♥s rít ♠ ♣ st♠rsrá♣♠♥t ♦♠♦ ♦r Φ ♣r ℘

L,r=1,ℓ= 1/2 ♥♦ ♦ ♣♥

t♦ ♥ ó♥ ♦ ♣rá♠tr♦ s s♣③ ♦rs ♠②♦rs♦♥ s♠♥ó♥ t♠ñ♦ sst♠ s ú♥ r ♦♥ L = 13♥ r ♠str ♥ tr♥só♥ r♣t ♣r ♥ ♥s rít♠ 〈Φ〉L=13,r=1,ℓ=1 = 5.754 ♥ ♥ ♦♥♦r♥ ♦♥ ♦r ♣t♦♣r Φ∞,r=1,ℓ=1 = 5.6372858(6) ❬❪

♣rs♥t♥♦ ♣r♦ ♣r♦ó♥ ♣r r♥ts ♦rs

♦rt♠♦ ♥rrqrt ♣r sts ♥♦ ♥s ♦♠♥ ♠ét♦♦ sst♦♥ ♦♥ ♥ ①t♥só♥ ♠ét♦♦ ♣ ♣r ♣r♦①♠ó♥ ♥♥tr ❬ ❪

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

r Pr♦ ♣r♦ó♥ ♣r sst♠s r♦s r = 1 s♠♥t♦s rts ♣r♦♥ts ♦♥ stró♥ ♥r s♦tró♣ ♣rr♥ts ♦rs t♠ñ♦ sst♠ ♦♥ ♦♥ts s♠♥t♦s rts ♥♦r♠ ℓ = 1 10 ② 100 ♦s sts ♥♦ ♥s s♦♦s ó♥ rst♦ ♦♥srr ss♥ ♥ ♥s ♣r♦ Pr♦ ♣r♦ó♥ ♣r r♥ts ♦rs ℓ ♥♥ó♥ Φℓ2 ♦♥♥ s rs ♦♥ ♠s♠♦ ♦r ró♥L/ℓ ♠str q ♥t ♠ás r♥t s ró♥ L/ℓ ♠ás q L ② ℓ♥ ♦r♠ ♥♣♥♥t

ℓ ♥ ♥ó♥ Φℓ2 ♥ ③ r♦ s♠♣♠♥t ♦♠♦ ♥ ♥ó♥ Φs ♦sr ♦♥♥ s rs ♦♥ ró♥ L/ℓ st♦ ♠strq ♦♠♦ s ♥tó ♥ ♦s ♣ít♦s ♥tr♦rs ② s sr ♥ ♣é♥ ♥t ♠ás r♥t s ró♥ L/ℓ ♠ás q L ② ℓ ♥ ♦r♠♥♣♥♥t ♥ tér♠♥♦s s♠♣s ést♦ ♥ q ♥ ♥ó♥ ♥s s♠♥t♦s rts ♥t r♥t s t♠ñ♦ sst♠ ♠♦ ♥ ♥s t♠ñ♦ rtríst♦ ♦s ♦t♦s ♣r♦♥ts♥ ♥str♦ s♦ ℓ ♦ ♦tr ♦r♠ ♣r ♥ ♦sr♦r ♦ ♥st♥ ♥ sst♠ t♠ñ♦ L ♦♥ N s♠♥t♦s rts ♦♥t ℓ s ♥st♥ ♥ sst♠ t♠ñ♦ L′ ♦♥ N s♠♥t♦s rts t♠ñ♦ ℓ′ = (L′ℓ)/L ♥ r s r♥ ♣r♦ ♣r♦ó♥ ♣r trs ♦rs r♥ts ℓ = 1, 10, 100 ♣r ♦s

❱ó♥ ♦rt♠♦ ② ♣r♦♥s tórs

♦rs t♠ñ♦ sst♠ L ts q L/ℓ = 2, 4, 8 ①♥t ♦♥♥ s rs ♦♥ ♦♥t L/ℓ rt ③ ♥áss ♣♦rr♣♦ r♥♦r♠③ó♥ s♦♦ ó♥

♥♦ sq♠ ♦rít♠♦ ♣♥t♦ ♥ r ♣rtr s rs ♣r♦ ♣r♦ó♥ r③♥♦ ♦s sts ♦rrs♣♦♥♥ts ①♣rsó♥ s ♦t♥♥ ♦s ♦rs 〈Φ〉L,r=1,ℓ ②∆L,r=1,ℓ s s♠♦♥s r③s r♣r♦♥ s ②s s♦ s♣rs ♣r ♥s rít ♦t♦s ♣r♦♥ts ② sí♦ stá♥r ♦♠♦ s ♦sr ♥ r s rs s♦ s ♦t♥ϑ = 0.83±0.04 ② ν = 1.33±0.03 ♦r ♦s ♦rs ♣r♠♥t r♣♦rt♦s❬ ❪

r ♥s ♣r♦t rít 〈Φ〉L,r=1,ℓ=1 ② sí♦ stá♥r∆L,r=1,ℓ=1 rss t♠ñ♦ sst♠ L ♣r ♥ sst♠ s♦tró♣♦ s♠♥t♦s rts ♣r♦♥ts ℓ ② L stá♥ ♠♦s ♥ s ♠s♠s ♥srtrrs

♦♥t♥ó♥ s ♦ró ó♥ ♦rt♠♦ ♣♦r ♥ ♠ét♦♦♦♠♣♠♥tr♦ srt♦ ♥ ó♥ s♦ ♥ ró♥ ♣♥♥ ♥s rít ♦♥ t♠ñ♦ ♦s s♠♥t♦s rt P♦r ♣♥♥t rá♦ 〈Φ〉L,r=1,ℓL

1/ν+ϑ rssL1/ν+ϑ ♦rrs♣♦♥ 〈Φ〉∞,r=1,ℓ Pr♠r♦ s r③r♦♥ ♦s rá♦s ♣rr♥ts ♦rs ℓ t③♥♦ ♦r ϑ = 0.83(4) r ♠str♦s sts ♣r ℓ = 1, 10, 100 ② 103

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

r rá♦s 〈Φ〉L,r=1,ℓL1/ν+ϑ rss L1/ν+ϑ ♣r r♥ts ♦rs

ℓ ♥ sst♠s r♦s s♠♥t♦s rts ♣r♦♥ts ♦♥ stró♥ ♥r s♦tró♣ ♣♥♥t ♦s ♠s♠♦s ♦rrs♣♦♥ 〈Φ〉∞,r=1,ℓ

♥ t♦♦s ♦s s♦s s ♦t♦ ♥ ①♥t r♦ st ♦♥ R2 ≥0.99997 ♦ ♦♠♦ s ♦sr ♥ r s rr♦♥ ♦s ♦rs 〈Φ〉∞,r=1,ℓ ♦t♥♦s ♥ ♥ó♥ t♠ñ♦ s♠♥t♦s rts ℓ ♥s ♦rít♠ ♦t♥é♥♦s

γ = ℓηΦ∞,r=1,ℓ

♦♥ η = 1.99993(2) ② γ = 5.6373(1) ♦♥ ♥ ①♥t r♦ stR2 = 0.99996 ♦♥sst♥ts ♦♥ s ♣r♦♥s tórs srt♦s ❬ó♥ ❪

st ♦r♠ ♦rt♠♦ srr♦♦ q ♦ ♠♥t ♦s♣r♦♠♥t♦s ♦♠♣♠♥tr♦s ♣♥♦ sr t③♦ ♣r ♦tr♦s ♥áss

♥ ♥♠ér ♣r♦♥s tórs

r♥ ♦ r③♦ ♣♦r ♦tr♦s t♦rs ❬ ❪ ② ♦♥ ♥ t③r s s♠♦♥s ♦♥ ♠♥♦r ♥t ♠♥t♦s ♣r♦♥ts ♣r ♦t♥r rst♦s ①tr♣♦s sst♠s ♥♥t♦s s sst♠r♦♥ s♥trs♦♥s ♥tr ♠♥t♦s ♣r♦♥ts q s♥ ♣♦r r sst♠

♥ ♥♠ér ♣r♦♥s tórs

r ♣♥♥ ♥s rít ♦♥ t♠ñ♦ ♦s s♠♥t♦s rt

r ♠str s rs ♥ó♥ ♣r♦ ♣r♦ó♥♣r ℓ = 1 ② ♦rs ♣qñ♦s L ♦t♥s t♦♠♥♦ ♥ ♥t s ♥trs♦♥s ♥tr ♠♥t♦s ♣r♦♥ts q s♥ ♣♦r r sst♠ ②sst♠♥♦ s ♠s♠s

♦♠♦ s s♣r ♣r ♥♦s ♦rs ℓ L ② Φ ♦s s ♦t♥♥ ♦rs♠②♦rs ℘

L,r,ℓs s t♦♠♥ ♥ ♥t s ♥trs♦♥s ♥tr ♠♥t♦s

♣r♦♥ts q s♥ ♣♦r r sst♠ s♠s♠♦ ♥r♠♥tr t♠ñ♦ sst♠ L s♠♥② r♥ ♥tr ♠♦s t♣♦s rs♣st♦ q s♠♥② ♦♥t ♣rí♠tr♦ár sst♠ ∼ L−1 ② ♣♦r♦ t♥t♦ s♦♥ ♠♥♦s r♥ts s ♥trs♦♥s ①♦♣r♠trs ♣♦r r ♣rí♠tr♦ sst♠ ♥ ♣rtr ♣r ℓ = 1 ② L ≥ 3.75 s t♥ ♥r♥ ♣♦r♥t ♠á①♠ ♥tr ♦s ♦rs ℘

L,r,ℓ♥r♦r 1.4%

P♦r ♦tr♦ ♦ s stó ♦♠♦ rí t♠ñ♦ ♠♦ str S ♥♦ ♣♦r ó♥ ♠ q s ♥r♠♥t ♥t ♠♥t♦s♣r♦♥ts ♥ ♥ sst♠ ♦♥ ℓ ② L ♥♦s r ♠str srs SL,r=1,ℓ=10 (Φ) ♣r r♥ts ♦rs L P ♦srrs q ♦r ♠á①♠♦ S s♠♥② ♥r♠♥trs t♠ñ♦ sst♠ L♠ás ♥s s♠♥t♦s rts s ♥③ ♦ ♦r♠á①♠♦ Φmax ♠♥t ♠♥tr t♠ñ♦ sst♠ ♥ ♦♥sst♥♦♥ s ♣r♦♥s ♦rí Pr♦ó♥ ❬ ❪

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

r rs ♣r♦ ♣r♦ó♥ ♣r ℓ = 1 ② ♦rs ♣qñ♦s L 0.6 0.8 1.25 ② 3.75 s rs ③ s ♥r ② ♦t ♦rrs♣♦♥♥ s s♠♦♥s r③s sst♠♥♦ s ♥trs♦♥s ♥tr♠♥t♦s ♣r♦♥ts q s♥ ♣♦r r sst♠ ①♦♣r♠trs rst♦ s rs t♦♠ ♥ ♥t s ♥trs♦♥s ♥ t♦♦s ♦ss♦s s ♦t♥♥ ♦rs ♠②♦rs ℘

L,r,ℓs s t♦♠♥ ♥ ♥t s

♥trs♦♥s

r ♠ñ♦ ♠♦ str ♥ ♥ó♥ ♥s s♠♥t♦s rts ♣r sst♠s r♦s ♦♥ stró♥ ♥r s♦tró♣

♥ ♥♠ér ♣r♦♥s tórs

s ♠♣♦rt♥t rr q s ♥ st ♦♠♣♦rt♠♥t♦ ♣ t③rs♣r ♦t♥r ①♣♦♥♥ts s♦ ♠♥t té♥s s♦ sst♠s♥t♦s rqr r③r ♥ ♠②♦r ♥ú♠r♦ r♣t♦♥s n ♣r♦t♥r rst♦s r♣r♦s ② s ❬ ❪

♦ q s t♥ ♥ r♣♦ t♦t♠♥t ①t♥♦ ♥ sst♠ ♥♦♠♣ q t♦♦s ♦s ♠♥t♦s ♣r♦♥ts ♣rt♥③♥ ♠s♠♦ s♦♥ ♦♥♦♥s ♥s s♠♥t♦s rts ♠♦ ♠②♦rs q ♥s rt ♥ s q s t♠♥t ♣r♦ q t♦♦s ♦s ♠♥t♦s♣r♦t♦s sst♠ ♦r♠♥ ♣rt ♠s♠♦ str ♥ tér♠♥♦s ♣r♦ t♦♠♥♦ ♥ ♥t ♥ó♥ PSC ♣♦r ó♥ s t♥

PSC (Φ) ≤ ℘ (Φ) , ∀Φ

②PSC (Φ) ≈ ℘ (Φ) ≈ 1, ♣r Φ ≫ 〈Φ〉L,r=1,ℓ.

♥ t♦ r ♠str ♦♠♣♦rt♠♥t♦ PSC (Φ) ② ℘ (Φ) ♣r♥ sst♠ r♦ ♦♥ ℓ = 1 ② L = 5 ♦♥ s r s ♦tr q ♥r♠♥tr ♥s ♦t♦s ♣r♦♥ts ♠②♦r s

r ♦♠♣♦rt♠♥t♦ PSC ② ℘ ♣r ♥ sst♠ r♦ ♦♥s♠♥t♦s rts ♣r♦♥ts ♦♥ ℓ = 1 L = 5 ② stró♥ ♥rs♦tró♣ ♦tr q PSC (Φ) ≤ ℘ (Φ) , ∀Φ

ró♥ ♦s q ♣rt♥ r♣♦ ♦♠♣t♠♥t ①t♥♦ ♦r ♦str♦ ♥ r ♦♠♦ s ♦sr ♥ r ♣r qr s ♦♥r♦♥s ♦♥ ①st ♥ r♣♦ ♣r♦♥t ést ♥② ♥♣♦r♥t ♦ ♦s s♠♥t♦s rts q ♦♠♣♦♥♥ sst♠♥ ♦♥sst♥ ♦♥ ♥r♠♥t♦ r♣t♦ ♥ó♥ PSC (Φ) ♥ ♦trs

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

♣rs ♣r♦ q ♥ r♣♦ ♣r♦♥t sté ♦r♠♦ ♣♦r ♠②♣♦♦s s♠♥t♦s rts s

s ♥t ♥ ést ♣ít♦ str♠♦s sst♠s ♣r♦t♦s ♥s♦tró♣♦s sr♣ó♥ st♦s sst♠s s ♣♥t♦ st ♦rí Pr♦ó♥ rqr ♥tr♦ó♥ ♣r♦s ♣r♦ó♥ ♦♥s q ♦♥t♠♣♥ r♦♥ sst♠ ♥tr♦r♠♦s s ♥♦♥s qí ♣r t③rs ♥ ♦t♥ó♥ ♥ ♥♠ér ♥s ♣r♦♥s tórs ♥trs♥ts ♥♦t♠♦s s ♣r♦s t♥r r♥ts t♣♦s r♣♦s ♣r♦♥ts ♦♠♦ s ♦r③♦♥t♠♥t ℘H rt♠♥t ℘V só♦ ♦r③♦♥t♠♥t ℘HX só♦ rt♠♥t℘V X ♥ qr s ♦s r♦♥s ℘U ② ♥ ♠s r♦♥s s♠tá♥♠♥t ℘HV ss ♣r♦s ♥♦ s♦♥ ♥♣♥♥ts ♥ s♦trs ② q sts♥ s ①♣rs♦♥s trs ℘U = ℘H + ℘V − ℘HV ②℘U = ℘HX + ℘V X + ℘HV ❬ ❪

♦♠♦ s sró ♠ás rr ♥ s s♠♦♥s ♠♦strs st ♦rs t③ó ♦♠♦ r ♣r♦ó♥ ①st♥ ♥ ♠♥♦ ♣r♦ó♥q ♦♥t s rsts rts ♦♣sts sst♠ ♣r♦ó♥ ♦r③♦♥t s♠trí sst♠ sst♠ r♦ r = 1 ♦♥ s♠♥t♦s rts s♦tró♣♠♥t str♦s s s♣r ♦t♥r ♠s♠ r ♣r♦ ♣r♦ó♥ s ♥ ♠♦ s ♦♥sr ①st♥ ♥♠♥♦ ♣r♦ó♥ q ♦♥t s rsts ♦r③♦♥ts ♦♣sts sst♠ ♣r♦ó♥ rt s ①♣rs♦♥s ♥tr♦rs rst ♥t q

℘UL,r=1,ℓ

(Φ) ≥ ℘HL,r=1,ℓ

(Φ) = ℘VL,r=1,ℓ

(Φ) ,

℘HVL,r=1,ℓ

(Φ) ≤ ℘HL,r=1,ℓ

(Φ) = ℘VL,r=1,ℓ

(Φ) ,

℘UL,r=1,ℓ

(Φ) = 2℘HL,r=1,ℓ

(Φ)− ℘HVL,r=1,ℓ

(Φ) ,

②℘UL,r=1,ℓ

(Φ) = 2℘HXL,r=1,ℓ

(Φ) + ℘HVL,r=1,ℓ

(Φ) ,

♣r qr ♦♥♥t♦ ♦rs Φ, ℓ, L

♦♥ ①♣ó♥ ♦s sst♠s ♦♥ t♠ñ♦ rtríst♦ ♦s ♦t♦s ♣r♦♥ts s ♦r♥ t♠ñ♦ sst♠ ♥ st♦s s♦s s ♦r ♣r♦ó♥ ♦♥ ♠②♣♦♦ ♠♥t♦s ♣r♦♥ts ② r♣♦ ♦♠♣t♠♥t ①t♥♦ q s ♦r♠ ♦rr♣r♦ó♥ stá ♦r♠♦ ♣♦r ♠② ♣♦♦s ♠♥t♦s

♥ ♥♠ér ♣r♦♥s tórs

r ♠str s rs ℘H ℘V ℘HV ② ℘U ♣r ♥ sst♠r♦ ♦♥ ℓ = 1000 ② L = 1750 ♦s ♦rs sts♥ s r♦♥s

r ❱♦rs ℘H ℘V ℘HV ② ℘U ♣r ♥ sst♠ s♠♥t♦s rts ♣r♦♥ts r♦ s♦tró♣♦ ♦♥ ♦♥ ℓ = 1000 ② L = 1750

♦♠♦ s sr ♥ ó♥ ℘H∞,r,ℓ

(Φ∞) = 1/2 ♣r qr sst♠ ♣r♦t♦ r♦ ♦♥ stró♥ s♦tró♣ ♦t♦s ♣r♦♥t♥ rst♦ ♥tt♠♥t s♣r ♥ r s ♠str♥ s ♣r♦s ℘H ℘HV ℘HX ② ℘U ♥s rít Φ∞,r=1,ℓ = 5.6372858rss t♠ñ♦ sst♠ L ♣r sst♠s r♦s ♦♥ ♦♥t s♠♥t♦ rt ♥♦r♠ ℓ = 1 ② stró♥ s♠♥t♦s rtss♦tró♣ ré♠♥ s♥tót♦ L → ∞ s ♥③ ♣r L ≈ 30 ② ♦♥t♥ó♥ s t♥ ♦s rst♦s ♣r L = 100 ❱r♠♦s q ♦s♦rs s♥tót♦s stá♥ ♥ ♠② ♥ ♦♥♦r♥ ♦♥ ♦s ♥♦♥tr♦s ♥trtr ♥ t♦ ♣r ℘H s ♦t♦ 0.51 ± 0.03 s♥♦ ♦r ①t♦0.5 ❬❪ Pr ℘HV t♥♠♦s 0.33 ± 0.03 ♠♥trs q ♦r ①t♦s 0.32212045 ❬❪ Pr ℘U t♥♠♦s 0.69± 0.03 s♥♦ 0.67787955 ♦r s♣r♦ ♣♦r ó♥ ℘U = ℘H + ℘V − ℘HV ♥♠♥t ♣r℘HX s ♦t♦ 0.18 ± 0.02 s♥♦ 0.17787955 ♦r ♦t♥♦ ♠♥t ①♣rsó♥ ℘HX =

(℘U − ℘HV

)/2 st♦ ♦♥stt② ♣r♠r ó♥

♥♠ér ♥ ♣r♦ó♥ s♠♥t♦s rts s ♣r♦♥s tórs♣r s ♣r♦s ℘HX ℘HV ② ℘U ♥ í♠t sst♠s ♥♥t♦s ♥s rít

♣ít♦ ♥s♦tr♦♣í étr ♦t ❱ó♥ ♦rt♠♦

r ♣♥♥ ♦♥ t♠ñ♦ sst♠ s r♥ts ♣r♦s ♣r♦ó♥ ♥s rít Φ∞,r=1,ℓ=1 ♦sr r♠♥t í♠t s♥tót♦ L→ ∞

•

♦s rst♦s ♠♦str♦s ♥ ♣rs♥t ♣ít♦ ♠str♥ q ♦rt♠♦ srr♦♦ s ♦ ♣r str s ♣r♦♣s ♣r♦ts sst♠s ♠♥s♦♥s s♠♥t♦s rts s ③ ♦rt♠♦ srt♦ ♣r♠t r t♠ñ♦ ♦s strs ♣r♦t♦s r♣r♦♥♦rst♦s ♣r♠♥t r♣♦rt♦s ♣r ést t♣♦ sst♠s r♣♦rt ♣♦r♣r♠r ③ ♥ ♥♠ér ♣r♦♥s tórs ♣r♦s♣r♦ts sst♠s r♦s s♠♥t♦s rts ♦ r♥tsrs ♣r♦ó♥ str q ♥áss qí ♠♦str♦ ♣①t♥rs s♥ t ♥ ♣r r③r ♥ sr♣ó♥ ♦♠étr♣r♦t t sst♠s ♦♥ ♠♥t♦s ♣r♦♥ts ♦♠trír♥t ♦♠♦ sí t♠é♥ sst♠s t♦♣♦♦í ♦r♥ ♠②♦r

♥ ♣ít♦ s♥t s t③rá ♦rt♠♦ ♦ ♣r r t♦ ♦s ♣rá♠tr♦s strtrs rtríst♦s ♥ ♦t♥ó♥ ♥s♦tr♦♣í étr ♦t ♥ s ♠♥t s ♠♦♦ ♦♠♦ sst♠s ♠♥s♦♥s rt♥rs s♠♥t♦s rts ♣r♦♥ts

♣ít♦

♥s♦tr♦♣í étr ♦t♠♦♥s r♦♥s ♦♥♦s ♠trs s ♣r♣r♦s

❲ r r tt ②♦r t♦r② sr③② qst♦♥ tt s s str t s r③② ♥♦ t♦

♥ ♦ ♥ ♦rrt

s ♦r

s♠♥ ♥ ést ♣ít♦ s ♠str♥ ♦s rst♦s s♠♦♥s ♥♠érs ①t♥ss s♦r sst♠s ♠♥s♦♥s rt♥rs♦♥ stró♥ ♥s♦tró♣ s♠♥t♦s rts s♦s ♦♠♦ ♠♦♦s s♠♣s s ♠strs ①♠♥s ①♣r♠♥t♠♥t ♥③ t♦ t♠ñ♦ sst♠ s ró♥ s♣t♦ ② ♦s♣rá♠tr♦s strtrs q rtr③♥ stró♥ ♦♥t ② ♥r s ♣s♦♥s r♥♦ ♥ ♦sró♥ st♦ s r③ ♦♠♣t♥♦ rs ♣r♦s s♣♠♥t ♣r♦ ♣r♦ó♥ só♦♦r③♦♥t ℘HX ② ♥③♥♦ ♥s♦tr♦♣í r♦só♣ q ♥t r♦ ♦r♥tó♥♣r♦♠♦ ♣♦♦♥ ♦s ♦t♦s ♣r♦♥ts

♦♥t♥♦ st ♣ít♦ ♣r♠♥t ♣♦ ♥ rtí♦ tt t ♦r♥ ♦ P②s ♠str②

♦ ♥♦ ♣

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

♠♦♥s ♥ sst♠s ♥s♦tró♣♦s♥s ♦♥sr♦♥s

st ♣ít♦ r♦♥♦ ♦♥ st♦ ♥ sst♠s ♥ ♥ ♦♥t①t♦ strt♠♥t ♣r♦t♦ s ♦rá ♣r♥♣ ♦t♦♣♥t♦ rs♣t♦ st♦ ♥ sst♠s s♠♥t♦s rts ♦♠♦ ♠♦♦ ís♦ s♥♦ q ♣r♠t st♠r ♣r♦ ♦t♥r sst♠s ♦♥ Pr ♦ ♥♥t♦ s♣♦ ♦♥r♦♥s ♣♦ss só♦ s ♦♥t♠♣rá♥ sst♠s q s ♥♥tr♥ ♥ r♥ístrtr♠♥t ♥♦ sst♠ ♣r♣r♦ ② rtr③♦ ①♣r♠♥t♠♥t s♥♦ st r♠ó♥ s rá ♠s r♦ ♥ ss♦♥s s♥ts ♥ ♦r♠ rtrr ② s♥ ♣ér ♥r só ♣r s s♠♦♥s rts♥♦ x ♦♠♦ ró♥ ♣ó♥ Hr♥ s r q ♦ ♦rrs♣♦♥ ró♥ ♦r♥tó♥♣rr♥ r♥♦ ② ♣♦r ♦ t♥t♦ ♦s s♠♥t♦s rts ♦ ó♥ s ♠♥s♦♥s sst♠ rt♥r t③s ♣r t♦ss s♠♦♥s ts r♦♥ Lx = 3 ♠♠ ② Ly = 4 ♠♠ ó♥ s ♠♥s♦♥s rq ♥ q s ♠strs t③s ♣r ó♥ ♥ s ♠strs r♦♥ ♠strs ♣rs♠áts ♠♥s♦♥s3♠♠× 4♠♠× 4♠♠ ♦♥ ♣r♠r s ♠♥s♦♥s ♦rrs♣♦♥ ♣ó♥ Hr♥

♦ s♠♣ó♥ ís♦strtr srr♦ ♥ ó♥ s ♦♥srrá q ♥ sst♠ ①rá s ♣rs♥t ♥ str♣r♦t♦ ①t♥♦ s♦♠♥t ♥ ró♥ ♦r③♦♥t ♦♥t♥♦ srsts rts ♦♣sts sst♠ rt♥r t♠át♠♥t ♦ ♠♦♦ srt♦ s t♥

pTEAL,r,ψk

(Φ) = ℘HXL,r,ψk

(Φ)

♦♥ pTEAL,r,ψk

(Φ) s ♣r♦ ♦t♥r ♦ s ♦♥♦♥sstrtrs L, r, ψk,Φ ♥ st st ró♥ rst ♥t q♦ ♠♦♦ ♣r♦♣st♦ ♠②♦r ♣rt ♦s sst♠s ♣r♣r♦sq ♣rs♥t♥ ♦s ♣rá♠tr♦s strtrs L, r, ψk ② ♥s ♦t♦s♣r♦♥t Φ ①rá♥ s ℘HX

L,r,ψk(Φ) ≈ 1 ①♣r♠♥t♠♥t ♥

♦♥♥t♦ ♣rá♠tr♦s s♦♦s ♦♥ó♥ ♦♥stt② ♦ q ♠♦s♥♦♠♥♦ ♥ ③♦♥ sr ♣r♦ó♥ ♥s♦tró♣ ♦r③♦♥t ② ♣♦r♦ t♥t♦

♣r♥♣ ♦t♦ ♣rs♥t ♣ít♦ s str ♥ ♠t♦♦♦í q ♣r♠t str s ♦♥♦♥s strtrs s♦s ③♦♥ sr s♥ ♠♣♦rt♥ ♣r srr♦♦ s♣♦st♦s q sí ♦ ♠♥♥ s♥s♦rs ♠♣♦ ♣rsó♥ ♦♥t♦rs ❩r♠♥s♦♥s t

t♦ s♠trí sst♠

t♦ s♠trí sst♠

♥ ♣r♠r r s ♥sr♦ r s ♣ ♥③rs ♦♥ó♥ ♥ sst♠s ♦♥ stró♥ ♥r s♦tró♣ ♦s ♦t♦s ♣r♦♥ts st♦ s s♥ ♥ ♣rr♥ ♦r♥tó♥ ♣♦r ♠♦ó♥♥♠♥t ró♥ s♣t♦ sst♠ r = Lx

Ly= LH

LV Pr ♦

s tr♦♥ r③♦♥s ♦♥t r♦ ♣r sst♠s ♦♥ ár q s♠strs rtr③s ①♣r♠♥t♠♥t S = LHLV = 12 ♠♠2 ② r♥t ró♥ s♣t♦ r sts r③♦♥s r♦♥ r③s ♦ stró♥ ♥r ♥♦r♠ stró♥ s♦tró♣ s♠♥t♦s rt② ♥ stró♥ ♦♥♦r♠ ♣r s ♦♥ts ♦s ♦t♦s ♣r♦♥ts t③♥♦ ♦s ♣rá♠tr♦s strtrs ♦t♥♦s ①♣r♠♥t♠♥t♣r ♥str♦ sst♠ 〈ℓ〉 = 1.35 ♠♠ ♠♠ ② σℓ = 0.26 ♠♠ ♥t st ♠t♦♦♦í ①♣♦r♥♦ ♦♥t ♦s sst♠s ♥r♦s♦♠♣t♦♥♠♥t s ♦♥str②r♦♥ s r♥ts rs ♣r♦ ♣r♦ó♥

r ♠str s rs ♣r♦ó♥ rt ℘V (Φ) ♦r③♦♥t ℘H (Φ) ♦r③♦♥t ② rt ℘HV (Φ) ♦r③♦♥t ♦ rt℘U (Φ) ② só♦♦r③♦♥t ℘HX (Φ) ♣r s ♦♥♦♥s strtrs s rs ♣r♦♦♥ só♦rt ℘V X ♥♦ ♠♦strs ♣♥sr á♠♥t s ♣♥♥ ♠t ♥tr s r♥tsrs ♣r♦ó♥

℘U = ℘H + ℘V − ℘HV

℘U = ℘HX + ℘V X + ℘HV

♥ s rs ♣r♦ó♥ só♦♦r③♦♥t s tr♠♥ó ♠á①♠♦♥③♦ ℘HX

max ♦s rst♦s ♦t♥♦s s str♥ ♥ r r ♠str q ℘HX ≈ 1 s r pTEA ≈ 1 s♦♦ ♣r r♦♥s s♣t♦ r∗ < 0.152 q ♦rrs♣♦♥♥ ♣r♦①♠♠♥t L∗

H =√r∗S <

1.35 ♠♠ ♦ ♠r t♠ñ♦ ♦r③♦♥t L∗H ♦rrs♣♦♥ ♦♥t

♠ ♦s ♠♥t♦s ♣r♦♥ts 〈ℓ〉 ♥t♦♥s ♦ ♥ stró♥s♦tró♣ ♦s s♠♥t♦s rt ♣ ♥③rs s t♠ñ♦ sst♠ ♥ ró♥ ♥trés ♥ ♥str♦ s♦ ♦r③♦♥t s ♠♥♦rq t♠ñ♦ rtríst♦ ♦s ♦t♦s ♣r♦♥ts sts ♦♥♦♥s ♥♦s ♥♥tr♥ ♦♠ú♥♠♥t ♥ sst♠s rs ♦ ♣r s ♣♦♥st♥♦ós ts ♥ s q s s t③r sst♠s ♦♥ t♠ñ♦♠②♦r ♦s ♦t♦s ♦♥t♦rs ♦rr♥t étr s ♥sr♦♠♣♦♥r ♥ ró♥ ♦r♥tó♥ ♣rr♥ ♦s ♦t♦s ♣r♦♥ts s♣rsó♥ ♥r ♣qñ σθ

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

r Pr♦ ♣r♦ó♥ rt ℘V (Φ) ♦r③♦♥t℘H (Φ) ♦r③♦♥t ② rt ℘HV (Φ) ♦r③♦♥t ♦ rt ℘U (Φ) ② s♦♠♥t ♦r③♦♥t ℘HX (Φ) ♣r ♥ sst♠ rt♥r ♦♥ árS = LHLV = 12 ♠♠2 ♦ ♥ stró♥ ♥r ♥♦r♠ s♦tró♣② stró♥ ♦♥t ♦♥♦r♠ s♠♥t♦s rts ♦♥ 〈ℓ〉 = 1.35♠♠ ② σℓ = 0.26 ♠♠ ♥ ♦s ♦♦rs ♥ rr♥ s r♥tsr♦♥s s♣t♦ ♠♦strs ♥

t♦ ♦s ♣rá♠tr♦s strtrs

t♦ ♦s ♣rá♠tr♦s strtrs

t♦ σθ

♦♥ rátr sr♣t♦ ♥ r s r♣rs♥t♥ str♦♥s ♥rs ♣r ♦s ♠♥t♦s ♣r♦t♦s srts ♦♠♦ str♦♥sss♥s ♥trs ♥ ♦r♥ ♣r r♥ts ♦rs σθ

r str♦♥s ♥rs ♣r ♦s ♠♥t♦s ♣r♦t♦s srts♦♠♦ str♦♥s ss♥s ♥trs ♥ ♦r♥ ♣r r♥ts ♦rs σθ

r ♠str ♠♣♦s sst♠s rt♥rs ♦♥ s♠♥t♦s rts ♣r♦♥ts rsts Lx = 3♠♠ ② Ly = 4♠♠ ♣r r♥ts♦rs ♥s ♦t♦s Φ ①♣rs♦s ♥ s♠♥t♦s rts♠♠2♦ ♥ stró♥ ♥r ss♥ ♥s♦tró♣ ♦♥ σθ = 2.5, 7.5② 15 ② ♥ stró♥ ♦♥ts ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.35 ♠♠ ②σℓ = 0.26 ♠♠ ♥ r s r♣rs♥t♥ ♦s ♦t♦s ♣r♦♥ts q ♦r♠♥♣rt ♦s strs q ①♥ ♣r♦ó♥ ♦r③♦♥t ♦♠♦ s s♣r ♣r ♥ ♦ ♦r σθ ♥r♠♥trs ♥s ♦t♦s ♣r♦♥ts♠②♦r s ♥t s♠♥t♦s rts q ♣rt♣♥ ♥ str ①t♥♦ ♠é♥ s ♦sr q ♣r ♥ ♦ ♦r ♥s ♦t♦sΦ s ♥r♠♥t ró♥ ♦s ♦t♦s q ♦r♠ ♣rt str♣r♦♥t ♠♥tr σθ

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

r ♠♣♦s sst♠s rt♥rs ♦♥ s♠♥t♦s rts ♣r♦♥ts rsts Lx = 3♠♠ ② Ly = 4♠♠ ♣r r♥ts ♦rs ♥s ♦t♦s Φ ①♣rs♦s ♥ s♠♥t♦s rt♠♠2 ♦ ♥stró♥ ♥r ss♥ ♥s♦tró♣ ♦♥ σθ = 2.5, 7.5 ② 15 ② ♥stró♥ ♦♥ts ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.35♠♠ ② σℓ = 0.26♠♠ ♥r s r♣rs♥t♥ ♦s ♦t♦s ♣r♦♥ts q ♦r♠♥ ♣rt ♦s strsq ①♥ ♣r♦ó♥ ♦r③♦♥t

t♦ ♦s ♣rá♠tr♦s strtrs

r ♠str st♦r♠s ♥s♦tr♦♣í r♦só♣A ♥ ♥ ó♥ ♦t♥♦s ♣r trs ♦rs r♥ts σθ15 10 ② 7.5 ♥♦ ♦t♥♦ ♣♦r ó♥ r♣t♦♥s ♦♥N = 1000 t♦s s

r st♦r♠s ♥s♦tr♦♣í ♠r♦só♣ A ♦t♥♦s♣r trs ♦rs r♥ts σθ 15 10 ② 7.5 ♥♦ ♦t♥♦ ♣♦ró♥ r♣t♦♥s ♦♥ N = 1000 t♦s s ♣r ♥ sst♠rt♥r ♦♥ ró♥ s♣t♦ r = Lx/Ly = 3/4 ② stró♥ ♥s♦tró♣ s♠♥t♦s rt ♦♥ ♣rá♠tr♦s stró♥ ♦♥♦r♠〈ℓ〉 = 1.35♠♠ ② σℓ = 0.26♠♠ ♥s♦tr♦♣í ♠r♦só♣ ② s sí♦stá♥r ♥ ♥ó♥ σθ Pr ♦rs ♣qñ♦s σθ s ♦t♥ ♥ ♦♠♣♦rt♠♥t♦ ①♣♦♥♥ r♦ −1 ♣r ♠s ♥ts

Pr t♦♦s ♦s ♦rs σθ stró♥ s ♣r♦①♠♠♥t ss♥ ♦♥ ♥ ①♥t r♦ st R2 ≥ 0.99926 ❬í♥s ♦♥t♥s ♥❪ Pr ss str♦♥s ♦r ♠♦ ② sí♦ stá♥r ♥s♦tr♦♣í ♠r♦só♣ 〈A〉 ② σA rs♣t♠♥t s♥ ♥ ♦♠♣♦rt♠♥t♦ ♠♦♥ót♦♥♦ r♥t ♦♥ σθ ♦♠♦ s str ♥ r s ♥♦t♦r♦ q ♣r ♦rs ♣qñ♦s σθ σθ < 55 ①st ♥ ró♥♥ ♥tr ln〈A〉 ② lnσθ ♦♠♦ t♠é♥ ♥tr lnσA ② lnσθ ❬í♥s ♦♥t♥s ♥ ♦♥ R2 = 0.9998 ♣♥♥t = −1.02(7) ② ♦r♥ ♦r♥= 4.32(1) ♣r ln〈A〉 ② R2 = 0.99897 ♣♥♥t = −0.97(6) ② ♦r♥

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

♦r♥ = 0.48(3) ♣r lnσA❪ ♦♠♦ s tó ♥ ♣ít♦s ♥tr♦rs strtr③ó♥ A ♦rrs♣♦♥ ♥ sr♣ó♥ s♠♣ ♠♥t ♠♣♦s r♥r ♥ ①♣rsó♥ ♥ít ①♣ít♣r s ♣♥♥ ♥♦♥ ♦♥ ♦s ♣rá♠tr♦s strtrs sst♠♣r♦t♦

r ♠str t♦ σθ ♥ s rs ♣r♦ ♣r♦ó♥ ℘H ℘V ② ℘HX ♥ ♥ó♥ ♥s s♠♥t♦s rts♣r ♥ sst♠ rt♥r ró♥ s♣t♦ r = Lx/Ly = 3/4 ② stró♥ ♥s♦tró♣ ♦t♦s ♦♥ stró♥ ♦♥ts ♦♥♦r♠♣r♠tr③ ♣♦r 〈ℓ〉 = 1.35♠♠ ② σℓ = 0.26♠♠

r t♦ σθ ♥ s rs ♣r♦ ♣r♦ó♥ ℘H

r ℘V r♦♦ ② ℘HX ③ ♥ ♥ó♥ ♥s s♠♥t♦s rts ♣r ♥ sst♠ rt♥r ró♥ s♣t♦ r = Lx/Ly = 3/4② stró♥ ♥s♦tró♣ ♦t♦s ♦♥ stró♥ ♦♥ts ♦♥♦r♠ ♣r♠tr③ ♣♦r 〈ℓ〉 = 1.35♠♠ ② σℓ = 0.26♠♠

♦♠♦ s sró ♥ ó♥ t♦s s ♠strs P34❬❪ 4.2% ♣rs♥tr♦♥ Φ ≈ 12 ♣s♦♥s♠♠2 ② ♦♥t étr ♣r s♦♠♥t ♥ ró♥ ♣ó♥ ♠♣♦ ♠♥ét♦ ♣♦ r♥t r♦ ♠tr ♣♥ ♥r♦r③qr♦ r ♠str q ♣r s ♦r Φ ② ♦s ♣rá♠

t♦ ♦s ♣rá♠tr♦s strtrs

tr♦s σθ = 4.65 〈ℓ〉 = 1.36♠♠ ② σℓ = 0.26♠♠ ♣rá♠tr♦s ①♣r♠♥ts♦sr♦s ♣r P34❬❪ 4.2% s t♥ ♥ ♠② t ♣r♦ ♣r♦ó♥ só♦♦r③♦♥t ♦ ♦rrs♣♦♥ ♦ ♥str♦♥ ♣r♦①♠ó♥ ♥ ♠② ♣r♦ ♦srr ♥ st sst♠ ♠tr t♥ sí ♥ ♠② ♥ ♦rró♥ ♥tr ss♠♦♥s ts ② ♦♠♣rt♠♥t♦ ♦sr♦ ①♣r♠♥t♠♥t

r ♠str ℘HX(Φ) ♦♠♦ ♥ r♠ ♦♥t♦r♥♦ ♥ ♥ó♥ Φ ② σθ ♦sr q ② ♦rs Φ ♣r ♦s s ℘HX(Φ) = 1s♦♦ s σθ < 15 P♦r ♦tr♦ ♦ ♣ rs r♠♥t q ♥tr♦ ♦rs Φ ♣r ♦s s ℘HX(Φ) = 1 s ♥r♠♥t rt♠♥t s♠♥r σθ ② s ③ ♥t♦ ♠♥♦r s ♦r σθ ♠②♦r s ♦r Φrqr♦ ♣r ♥③r ③♦♥ sr

r t♦ σθ ♥ ♣r♦ ♣r♦ó♥ ℘HX ♥ ♥ó♥ ♥s s♠♥t♦s rts ♣r ♥ sst♠ rt♥r ró♥ s♣t♦ r = Lx/Ly = 3/4 ② stró♥ ♥s♦tró♣ ♦t♦s ♦♥stró♥ ♦♥ts ♦♥♦r♠ ♣r♠tr③ ♣♦r 〈ℓ〉 = 1.35♠♠ ②σℓ = 0.26♠♠

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

t♦ σℓ

♥ r s ♠♣♥ str♦♥s ♦♥t ♣r ♦s ♠♥t♦s ♣r♦t♦s srts ♦♠♦ str♦♥s ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.35♠♠ ♣r r♥ts ♦rs σℓ ♦tr ♦♠♦ ♣♦só♥ ♠á①♠♦ stró♥ s s♣③ ♦rs ♠♥♦rs ♠♥tr ♦r σℓ

r str♦♥s ♦♥t ♣r ♦s ♠♥t♦s ♣r♦t♦s srts ♦♠♦ str♦♥s ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.35 ♠♠ ♣r r♥ts♦rs σℓ

♥♦ ♥ ♣r♦♠♥t♦ s♠r srt♦ ♥ ó♥ ♣r ♣rr t♦ σℓ s tr♦♥ s♠♦♥s t③♥♦ ♥ stró♥♥r ss♥ ♣r♠tr③ ♣♦r 〈θ〉 = 0 ② σθ = 7.5 ② ♥ stró♥ ♦♥t ♦♥♦r♠ ♦♥ ♣rá♠tr♦s 〈ℓ〉 = 1.35♠♠ r r♥ ♠♣♦s sst♠s ♥r♦s ♦♠♣t♦♥♠♥t ♣r trs ♦rsr♥ts sí♦ stá♥r stró♥ ♦♥♦r♠ ♦♥ts ♦s s♠♥t♦s rt σℓ ② r♦s ♦rs ♥s ♦t♦s Φ ♦♦s ♣rá♠tr♦s strtrs 〈ℓ〉 = 1.35♠♠ ② σθ = 7.5

t♦ ♦s ♣rá♠tr♦s strtrs

r ♠♣♦s sst♠s rt♥rs ♦♥ s♠♥t♦s rts ♣r♦♥ts rsts Lx = 3♠♠ ② Ly = 4♠♠ ♣r r♥ts ♦rs ♥s ♦t♦s Φ ①♣rs♦s ♥ s♠♥t♦s rt♠♠2 ♦ ♥stró♥ ♥r ss♥ ♥s♦tró♣ ♦♥ σθ = 7.5 ② ♥ stró♥ ♦♥ts ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.35 ♠♠ ② σℓ = 0.02, 0.45 ② 6.00♠♠ ♥ r s r♣rs♥t♥ ♦s ♦t♦s ♣r♦♥ts q ♦r♠♥ ♣rt ♦sstrs q ①♥ ♣r♦ó♥ ♦r③♦♥t

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

r ♠str st♦r♠s ♥s♦tr♦♣í r♦só♣ A♦t♥♦s ♣r ♦s ♦rs r♥ts σℓ 0.30 ♠♠ ② 5.00 ♠♠ ♥♦♦t♥♦ ♣♦r ó♥ 10500 r♣t♦♥s ♦♥ N = 1000 Pr t♦♦s ♦s

r st♦r♠s ♥s♦tr♦♣í r♦só♣ A ♦t♥♦s♣r ♦s ♦rs r♥ts σℓ ♦♥ N = 1000 ♣r ♥ sst♠ rt♥r ró♥ s♣t♦ r = Lx/Ly = 3/4 ② ♣rá♠tr♦s strtrs 〈ℓ〉 =1.35♠♠ ② σθ = 7.5 ♥s♦tr♦♣í ♠r♦só♣ ♠ ② s sí♦stá♥r rss σℓ

♦rs σℓ ♥ ♣rtr ♣r q♦s ♠♦str♦s ♥ r stró♥ ♣ sr st ♥ ♦r♠ ♣r♦①♠ ♣♦r ♥ stró♥♦♥♦r♠ ♦♥ ♥ ①♥t r♦ st R2 ≥ 0.99511 ❬í♥ ♦♥t♥♥ s rs ❪ Pr ♦rs ♦s σℓ stró♥ s ♣r♦①♠♠♥t ss♥ ♣r♦♣♦ ♦♠♣♦rt♠♥t♦ ♥♦♥ stró♥ ♣r♦ ♦♥♦r♠ ♥s♦tr♦♣í r♦só♣ ♠ 〈A〉 ② ssí♦ stá♥r σA s♥ ♥ ♥r♠♥t♦ ♠♦♥ót♦♥♦ ♦♥ σℓ ♦♠♦ s str♥ r

♠♥t strt t③ ♣r str ♥♥ σℓ s♦r ♠tr s r rs ℘HX(Φ) ♣r r♥ts ♦rs σℓ r ♠str ♣r♦ ℘HX(Φ) ♦♠♦ ♥ r♠ ♦♥t♦r♥♦ ♥ ♥ó♥ Φ ② σℓ P ♦srrs q ♥tr♦ ♦rs Φ ♣r ♦s s ℘HX(Φ) ≈ 1 rí ♠② ♣♦♦ ♦♥ ♣rá♠tr♦ st♦♦♠♥t s ♦sr ♥ ♥r♠♥t♦ ♦ ♥tr♦ ♠♥tr σℓ 0

t♦ ♦s ♣rá♠tr♦s strtrs

♠♠ 1♠♠ P♦r ♦ s♦s ♦rs ♣rát♠♥t ♥♦ s ♦sr ró♥♦♥ σℓ ② ♣♦r ♦ t♥t♦ ♣♦só♥ ② t♠ñ♦ ③♦♥ sr s st♥t ♥s♥s σℓ

r t♦ σℓ ♥ ♣r♦ ♣r♦ó♥ ℘HX ♥ ♥ó♥ ♥s s♠♥t♦s rts ♣r ♥ sst♠ rt♥r ró♥ s♣t♦ r = Lx/Ly = 3/4 ② stró♥ ♥s♦tró♣ ss♥ ♦t♦s♦♥ stró♥ ♦♥ts ♦♥♦r♠ ♣r♠tr③ ♣♦r 〈ℓ〉 = 1.35♠♠② σθ = 7.5

t♦ 〈ℓ〉♥ st s♦ s ♦♥sr ♥ stró♥ ♥r ss♥ ♦♥ ♣

rá♠tr♦s 〈θ〉 = 0 ② σθ = 7.5 ② ♥ stró♥ ♦♥♦r♠ ♣r sstr♦♥s ♦♥t ♦s ♦t♦s ♣r♦♥ts ♦ ♦s ♣rá♠tr♦sσℓ = 0.26♠♠ ② r♥ts ♦rs 〈ℓ〉 ♦♥ ♥t♥ó♥ ♠r♠♥t sr♣t ♥ r s ♠str♥ str♦♥s ♦♥t ♣r ♦s ♠♥t♦s ♣r♦t♦s srts ♦♠♦ str♦♥s ♦♥♦r♠ ♦♥ σℓ = 0.26♠♠ ♣r r♥ts ♦rs 〈ℓ〉 ♦tr s♣③♠♥t♦ ♠á①♠♦ stró♥ ♦rs ♠②♦rs ♠♥tr ♦r 〈ℓ〉

r ♠str ♣r♦ ℘HX(Φ) ♦♠♦ ♥ r♠ ♦♥t♦r♥♦ rss Φ ② 〈ℓ〉 P ♥♦trs q ♥♦ ♥tr♦ ♦rs Φ ♣r ♦s s ℘HX(Φ) ≈ 1 ♠ ♠② ♣♦♦ ♦♥ ♣rá♠tr♦ st♦ ♠♥trs q ♦r Φ rqr♦ ♣r ♥③r ③♦♥ sr s ♥r♠♥t rt♠♥t s♠♥r 〈ℓ〉

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

r str♦♥s ♦♥t ♣r ♦s ♠♥t♦s ♣r♦t♦s srts ♦♠♦ str♦♥s ♦♥♦r♠ ♦♥ σℓ = 0.26 ♠♠ ♣r r♥ts♦rs 〈ℓ〉

r t♦ 〈ℓ〉 ♥ ♣r♦ ♣r♦ó♥ ℘HX ♥ ♥ó♥ ♥s s♠♥t♦s rts ♣r ♥ sst♠ rt♥r ró♥ s♣t♦ r = Lx/Ly = 3/4 ② stró♥ ♥s♦tró♣ ss♥ ♦t♦s ♦♥ stró♥ ♦♥ts ♦♥♦r♠ ♣r♠tr③ ♣♦rσℓ = 0.26♠♠ ② σθ = 7.5

t♦ ♦s ♣rá♠tr♦s strtrs

♥ ♦r♠ q♥t ♣r ♥ ♦r Φ ♦ ♥t♦ ♠②♦r s ♦r 〈ℓ〉 ♠②♦r s ℘HX ② s ③ ♠②♦r s ♣r♦♣♦ró♥ ♦t♦s ♣r♦♥tsq ♦r♠♥ ♣rt strs q ①♥ ♣r♦ó♥ ♦r③♦♥t r r

r ♠♣♦s sst♠s rt♥rs ♦♥ s♠♥t♦s rts♣r♦♥ts rsts Lx = 3♠♠ ② Ly = 4♠♠ ♣r r♥ts ♦rs ♥s ♦t♦s Φ ①♣rs♦s ♥ s♠♥t♦s rt♠♠2 ♦ ♥stró♥ ♥r ss♥ ♥s♦tró♣ ♦♥ σθ = 7.5 ② ♥ stró♥ ♦♥ts ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.00, 1.65 ② 3.00 ♠♠ ② σℓ = 0.26♠♠ ♥ r s r♣rs♥t♥ ♦s ♦t♦s ♣r♦♥ts q ♦r♠♥ ♣rt ♦sstrs q ①♥ ♣r♦ó♥ ♦r③♦♥t

r ♠str ♥ st♦r♠ tí♣♦ ♣r ♥s♦tr♦♣í r♦só♣ ♦t♥♦ ♣r 〈ℓ〉 = 1.22♠♠ ♣♦r ♥ró♥ 10500 r♣t♦♥s♦♥ N = 1000 Pr t♦♦s ♦s ♦rs 〈ℓ〉 stró♥ s ♣r♦①♠♠♥t ss♥ ♦♥ ♥ ①♥t r♦ st R2 ≥ 0.99977 í♥♦♥t♥ ♥ ♦♥trr♠♥t ♦ ♦sr♦ ♣r ♦s ♦tr♦s ♦s ♣rá♠tr♦s strtrs σθ ② σℓ ♥ st s♦ ♦s st♦r♠s ♥♦ ♠♥ ♦r♠ stíst♠♥t s♥t ♣r ♦s r♥ts ♦rs 〈ℓ〉 sí♣r t♦♦s ♦s ♦rs 〈ℓ〉 stró♥ ♥s♦tr♦♣í ♠r♦só♣s ♣r♦①♠♠♥t ss♥ ♦♥ 〈A〉 ≈ 9.55 ② σA ≈ 0.23

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

r st♦r♠ tí♣♠♥t ♦t♥♦ ♣r ♥s♦tr♦♣í r♦só♣ A ♣r sst♠s rt♥rs ró♥ s♣t♦ r = Lx/Ly = 3/4N = 1000 ② ♣rá♠tr♦s strtrs σθ = 7.5 ② σℓ = 0.26♠♠

t♦ t♠ñ♦ sst♠

♠♣ír♠♥t t♦s s ♠strs ♦♥ r♥ts t♠ñ♦s ♥ ♦s♥tr♦s 1.5 ♠♠ ≤ LH ≤ 3 ♠♠ 4 ♠♠ ≤ LV ≤ 10 ♠♠ ♣rs♥tr♦♥ Pr r t♦ t♠ñ♦ sst♠ ♥ ♣r♦ ♦t♥r s tr♦♥ r③♦♥s ♦♥t r♦ ♥r♥♦ sst♠s rt♥rs ♦♥ ♦rs LH ② LV ♥tr♦ s♦s ♥tr♦s ①♣r♠♥ts ♦s♣rá♠tr♦s strtrs r♦♥ ♦s ♥ ♦s ♦rs ①♣r♠♥ts stró♥ ♥r ss♥ ♥s♦tró♣ ♦♥ σθ = 4.65 stró♥ ♦♥ts ♦♥♦r♠ ♦♥ 〈ℓ〉 = 1.35 ♠♠ ② σℓ = 0.26 ♠♠ ② ♥ ♥s ♦t♦s ♣r♦♥ts Φ = 12 ♠♠−2 ♥ú♠r♦ s♠♥t♦s rts ♣♦r♥ ár ♥ ♣♥♦s ♦♥t♥s s ♣s♦♥s ♣rtr s r♥ts rs ♣r♦ó♥ só♦♦r③♦♥t ℘HX ♣r r♥ts t♠ñ♦s sst♠ ♦s ♣♦r LH ② LV s ♦♥str②ó r♠ ♦♥t♦r♥♦♠♦str♦ ♥ r ♦ ♦s ♣rá♠tr♦s strtrs

str♦ ♠♦♦ s♥♦ ♠str q ♣r♦ ♦srr ♥ ♦s sst♠s srt♦s s ♠② ♣r t♦♦s ♦s t♠ñ♦s ② ♦♠♦ r ♥tt♠♥t s♣r s ♠②♦r ♣r sst♠s ♦♥ ♠♥♦r ró♥ s♣t♦ r st♦s rst♦s ♠str♥ ♥ ♦♠♣♦rt♠♥t♦ ♦♥ ♥ ♠♣ó♥ t♥♦ó s♥t ♦♥trr♠♥t ♦ q s ♥ sst♠s♦♠♦é♥♦s ② rsst étr ♥♦ ♣♥ t♠ñ♦ ♠str♦♥ ①♣ó♥ sst♠s ♥♥♦só♣♦s ♦s sst♠s ♠r♦ strtr♦s♦♠♦ ♦s trt♦s qí ♣♥ ♦ ♥♦ ♣rs♥tr ♣♥♥♦ t♠ñ♦ ♠str

t♦ t♠ñ♦ sst♠

r r♠ ♦♥t♦r♥♦ ♣r ♣r♦ ♣r♦ó♥ só♦♦r③♦♥t ℘HX ♥ ♥ó♥ t♠ñ♦ sst♠ ♦ ♣♦r LH ② LV ♦♦s ♣rá♠tr♦s strtrs ①♣r♠♥ts stró♥ ♥r ss♥♥s♦tró♣ ♦♥ σθ = 4.65 stró♥ ♦♥ts ♦♥♦r♠ ♦♥〈ℓ〉 = 1.35 ♠♠ ② σℓ = 0.26 ♠♠ ② ♥ ♥s ♦t♦s ♣r♦♥tsΦ = 12 ♠♠−2

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❯♥ ♣r♠r ♦♥só♥ st♦ r③♦ s q ♦♣ó♥ sst♠s ♦r♠ rt♥r ♦♥ ♥ ró♥ s♣t♦ ♠♦r ♥♦ ss♥t ♣r ♥③r ♥ ♦r♠ t ♣♦r ♦ q rst ♥sr♦♥tr♦r ♥s♦tr♦♣í ♥tr♥ ♠♥t ♥ stró♥ ♥r ♥s♦tró♣ ♦s ♦t♦s ♣r♦♥ts ♥ ♦♥♦r♥ ♦♥ s ♦sr♦♥s ①♣r♠♥ts ♥♦♥trr♦♥ ♣rsr♣♦♥s ♣r ♥③r ♦♥♦♥s strtrs srs ♦♥ ①♣tt r ①♣r♠♥tsts ②t♥ó♦♦s ♥ ó♥ ♦♥♦♥s ①♣r♠♥ts ♥srs ♣r ró♥ s♣♦st♦s ♦♥ r♦ s♦ ♥s♦tr♦♣í étr ♥tr♦trs ♦ss s ♠str q ①st ♥ rt ♣♥♥ ♣r♦ ♦t♥r ♦♥ s♣rsó♥ ♥r σθ ② ♦♥ ♦♥t ♠ ♦s ♦t♦s ♣r♦♥ts ♠♥trs q s♣rsó♥ ♥ ♦♥t t♥ ♣♦♦t♦ s♣t♦ ó♥ t♦ t♠ñ♦ sst♠ ♣r♦t♦ ♠♦♦ srt♦ ♠str q ♦s sst♠s ♦♠♣st♦s ♦♥ r♥♦s♠r♦ ♦ ♥♥♦strtr♦s ♦♠♦ ♦s trt♦s qí ♣♥ ♦ ♥♦ ♣rs♥tr ♣♥♥♦ t♠ñ♦ ♠str st♦ s ♥ ♦♥s♥ rt r♥ ♠♣t♦ q t♥ t♠ñ♦ ♥ sst♠ ♥ ♣r♦ ♦srr ♥ ést ♥ ♦♥♥t♦ ♦♠♣t♠♥t ①t♥♦

♥♠♥t rtr♦ ♦♠étr♦♣r♦t♦ ♥♦♥tr♦ ♣r

♣ít♦ ♥s♦tr♦♣í étr ♦t ♠♦♥s r♦♥s ♦♥

♦s ♠trs s ♣r♣r♦s

①♣ s♠♥ttt♠♥t s ♦sr♦♥s ♠♣írs r③s s♦r♥ r♥ ♥t ♠strs sts ①♣r♠♥t♠♥t st♦♣r♦ ♥ ♥ ♦♥r♠ó♥ q ♦s ♠♦♦s s♦s ♥ ♣r♦♦♥♠♥s♦♥ s♠♥t♦s rts s♦♥ ♦s ♣r sr♣ó♥ sst♠s strtr♦s t♣♦ ② sst♠s ♦♠étr♠♥t ♥á♦♦s ♣♦r♠♣♦ ♦♥ ♦tr♦s ♦t♦s ♥♠♥s♦♥s ♦♠♦ ♥♥♦t♦s ♥♥♦rs♦ ♥♥♦s

♣ít♦

s♣st ♣③♦rsst①♣r♠♥t♦s ♠♦♦ ②s♠♦♥s

tr♦♥ s t♦r② s t s s♦s ♥ ♥rst♥♥ t ② ♥tr♦rs tt ♥ ♠♦st t r

r ②♥♠♥

s♠♥ ❯♥ s♣t♦ ♠② ♥trs♥t ♦s ♠trs ♦♠♣st♦s st♦♠ér♦s strtr♦s ♦♥ stró♥ ♥s♦tró♣ r♥♦s ♦♥t♦rs étr♦s s ♣♦s ①r ♣③♦rsst ♣rs♥t qí ♥ ♠♦♦ ♦♥sttt♦ ♣r ♣③♦rsst ♥s♦tró♣ rrs ♥ st t♣♦ ♠trs s♦ ♥ á♦ ♦♥t trés strtrs ♦♠♥rs ♠trs ♥♦rá♥♦s ♥♦s ♥ ♥ ró♥ s♣í ♥tr♦ ♥♠tr③ ♣♦♠ér s♥t ❱r♦s t♦s ♦♠♦ t♥♦ tró♥♦ ♦♥t étr ♥tr♦ s ♣s♦♥s ② ♦♥tt♦étr♦ ♥tr s strtrs r♥♦ ② ♦s tr♦♦s s♦♥ ♥♦s♥ ♠♦♦ ♦s rst♦s ①♣r♠♥ts ♠ó♥ rsst♥ étr sst♠ R ♥ ♥ó♥ t♥só♥ ♠á♥ ♥①♣ ♥ ró♥ ♦s ♣s♦♥s P s♦♥ st♦s ♠②♥ ♣♦r ♠♦♦ ♣r s♦ ♠r♦♣rtís 34❬❪ ♥st♦ s♣r♣r♠♥ét♦ ♥s r♥t r♦ P s♥s r③ r♥ts ♣rá♠tr♦s ♣♦r ♠♣♦ rrr ♣♦t♥ ♣r t♥♦ tr♦♥s s ♥♠♥ts ♣rs♥t♥ s♠♦♥s rs♣st ♣③♦rsst s♣r ♣r♦tr♦s sst♠ q ♥ ♣rtr ♠str♥ ♥♥ ♠ó♦ ❨♦♥ ♠tr③ ♣♦♠ér

♦♥t♥♦ st ♣ít♦ ♣r♠♥t ♣♦ ♥ rtí♦ tt t ♦t ttr ♦ ♥♦ ♣

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥tr♦ó♥

♦♠♦ s ♥ó ♥ ♣ít♦ s ♥♦♠♥ ♣③♦rsst♥ ♥ó♠♥♦ ís♦ s♦♦ ♠♦ ♥ rsst♥ étr ♥ ♠tr ♣♦r♣ó♥ t♥só♥ ♠á♥ s♦r ♠s♠♦ ♥ ♣rtr s rsst♥étr s♠♥② ♥ó♠♥♦ s ♥♦♠♥ ♣③♦rsst♥ ♥t

♥ á♠t♦ ♦s ♠trs ♦♠♣st♦s strtr♦s ss♥ ♦sr♣♦rts ♣③♦rsst♥ ♥t P♦r ♠♣♦ ♦q ② ♦sss ❬❪r♣♦rt♥ ♣③♦rsst♥ ♥t ♥ sst♠s ♣rtís rr♦ s♣rss ♥ stó♠r♦s s♦①á♥♦s ♠ás ♥♦s q q♦s ♦♥sr♦s ♥ ♣rs♥t ss ♥ ♦ rtí♦ ♥♦ s ♠♦ ♥ ♠♣ír s♠♠♣ír♦ tór♠♥t ♦s t♦s rsst♥ étr ♠tr R ♥ ♥ó♥ t♥só♥ ♠á♥ ♥① ♣ P t ② ♦sss ❬❪ ♥♥ tr♦ ♣♦♥r♦ ♥ t♠át ♣rs♥t♥ st♦s ♣③♦rsst♥ ♥sst♠s ♦♠♣st♦s strtr♦s s♦s ♥ ♠r♦♣rtís ♥íq s♣rss ♥ P ♦♥ ♦ r♦ ♥trr③♠♥t♦ ♥tr s r♥s♥tr ♦ st♦ ② ♣rs♥t♦ qí ♣♥ strs ♠♥♦r r♦ ♥trr③♠♥t♦ ♠tr③ ♠②♦r ♦♥♥tró♥ r♥♦ r♥ 20−30% ② ♠♥♦rs ♦r♠♦♥s r♥s 2.5% ♥ s♦s tr♦s r♥♦ stá ♦r♥③♦ ♦♠♦ ♥ ♦r ♣rs s♥ strtr ♥tr♥♦♥ ♣r q r♣rs♥t ♥ ♣rtí r♥♦ stá ♦r♥♥ ♦r♠ ♦♥st s♣r ss ♥s ♠♥t ♥ ♥ ♣ ♠tr ♣♦♠ér♦ s♥t ❬❪

♦♠♦ s sró ♥tr♦r♠♥t ♥ ♣rs♥t st♦ s ♦♥sr♥strtrs ♦♠♣t♠♥t r♥ts ♦s tr♦s ♠♥♦♥♦s ♥♥♦♣rtís ♠♥éts ♦r♠♥ r♣♦♥s ② rs sts r♣♦♥sstá♥ rts ♣♦r ♣t ♠tá ♠♦♦ ♠tr③ r♥r ♦r♠♥♦♠r♦♣rtís írs q ♦ s r♣♥ ♣♦r ♣ó♥ Hcuring

♦r♠♥♦ ♣s♦♥s sts s ③ s ♥♥tr♥ ♦r♠♥♦ strs ♦♠♥rs ♦r♥t♦s ♥ ró♥ Hcuring ♠♦♦ tór♦ rs♣st ♣③♦rsst s♦ sts strtrs s ♦r♦ qí♣♦r ♣r♠r ③ ♠♦♦ ést sst♠ rqr ♥ ♣r♥♣♦ t♦♠r ♥♥t ♦♥t á♠tr♦ ② ♦r♥tó♥ rt s ♣s♦♥ss♠s♠♦ t♦rs ♦♠♦ rsst♥ étr s ♠r♦♣rtís sr ♦♥sr ♥ ♥str♦ ♠♦♦ ♠♥trs q ♥ ♦s rtí♦s ♣♦s♣♦r ♦tr♦s r♣♦s s ♣rtís r♥♦ s♦♥ ♦♥srs ♦♠♦ ♦♥t♦rs étr♦s ♣rt♦s ❬❪ ♦♥♠♥t ♥ ♦s sst♠s ♦r ♣rs ② ♣rá♠tr♦s s♣í♦s q ♥ sr ♣r♠♥t st♠♦s t ❬❪ ♦♠♦ sr s♣s♦r ♣ ♣♦♠ér ♥tr ♣rtís② ♥ú♠r♦ s♣r♦♥s ♥tr r♦♥s r♥♦ s♠s♠♦ s s♣♦♥♥r♠♥t q s ♣r♦ ♦r♠ó♥ ♣ást s♣r s♣rtís r♥♦ ♥♦ s ♠s♠s ♥tr♥ ♥ ♦♥tt♦ rt♦ ♣♦r ♣ó♥ t♥só♥ ♠á♥ r③ó♥ ♣♦r r♦s s♣r

♥tr♦ó♥

s ♣rtís r♥♦ sr tr♠♥ ♥ s ♠r♦ tr♦ ú♥ ♥ ♥ s♣♦♥ ♦s sst♠s ♦r ♣rs♣r♥ ♦r♠rs ♠s á♠♥t ♥ ♠trs ♦♥ rt♠♥t ♦ r♦ ♥trr③♠♥t♦ ①♥♦ r♥s ♦rs ♣③♦rsst♥ ♥t rt♠♥t s ♦r♠♦♥s P♦r ♦tr♦ ♦ ♥ ♦s sst♠s ♦♥s ♦r♠♥ ♣s♦♥s ♦♠♦ ♥str♦ s ♦sr♥ ♦r♠♦♥s ♠tr ♠②♦rs 5% ② rqr♥ ♥ ♦r tór♦ r♥t

sí s ♦ss ♣r♥♣ ♦t♦ ♣rs♥t ♣ít♦ s ♣rs♥tr ♥♠♦♦ ♦♥sttt♦ ♣r rs♣st ♣③♦rsst ♥t ♦sr ♣r ♠tr ♦ ♦♥ó♥ s r ♥ sr♣ó♥ ♠t♠át♠♥t rr ♦ ♥ó♠♥♦ ♦♠♦ sí t♠é♥ r♥r ♣r♦♥s♣r ♦tr♦s ♠trs s♠rs

♦s sí♠♦♦s t③♦s srá♥ ♥♦s ♦ r♦ ♣ít♦ ② ♠②♦rís ♠str♥ ♥

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥ó♥ ♦s sí♠♦♦s s♦s ♥ ♠♦♦ ♣③♦rsst♥ ♥ ♦s sst♠s

í♠♦♦ ♥ó♥P ♥só♥ ♠á♥ ♥① ♣ ♥ ró♥ ♦r♥tó♥

♣rr♥ r♥♦R sst♥ étr t♦tRE−CH sst♥ étr ♦♥tt♦ r♥♦tr♦♦RL sst♥ étr ♥ strtr ♦♠♥r r♥♦RCH sst♥ étr ♥ ró♥ ♦♥t♦rRtunnel sst♥ étr t♥♦ tró♥♦ ♥tr r♦♥s ♦♥

t♦rs s♣rs ♣♦r ♣♦í♠r♦N ♥t ♠ ♦♠♥s ♥ ♠str n ♥t ♠ r♦♥s ♦♥t♦rs ♥ ♥ ♦♠♥ r♥♦〈D〉 á♠tr♦ ♠♦ s ♣s♦♥s〈DµP 〉 á♠tr♦ ♠♦ s ♠r♦♣rtís ♠♥tt♣t〈ℓ〉 ♦♥t ♠ s ♣s♦♥sJtunnel ♦ t♥♦ tró♥♦ ♥tr ♦s r♦♥s ♦♥t♦rsVtunnel r♥ ♣♦t♥ ♥tr ♦s r♦♥s ♦♥t♦rs ②♥tsItunnel ♥t♥s ♦rr♥t t♥♦ tró♥♦ϕ tr rrr ♣♦t♥ rt♥r ♥ t♥♦ tró♥♦m s t tró♥ ♥ r♥♦Lo s♣s♦r t♥só♥ ♠á♥ ♥ ♠str s ♥

♠ó♥ ♣③♦rsst♥ L (P = 0)y s♣s♦r ♠str ♠tr③ ♣♦♠ér s♥ r♥♦ ♥ ♣rt

r ♦rrs♣♦♥ s♣s♦r ♠str P s♦ rr♥

x ♣ró♥ ♠ ♥tr r♦♥s ♦♥t♦rsl ♦♥t t s r♦♥s ♦♥t♦rsη ♥t s♣♦ts ♥ ③♦♥ ♦♥tt♦ tr♦♦♣s♦

♥〈a〉 ♦ ♣r♦♠♦ ♦s s♣♦tsρµ sst étr ♠tr r♥♦ ♥ r♥í

③♦♥ ♦♥tt♦ ♦♥ tr♦♦ρE sst étr tr♦♦ ♥ r♥í ③♦♥

♦♥tt♦ ♦♥ r♥♦H r③ ♠tr r♥♦

sst♥ étr ♦♠♥s ♦♥t♦rs♦r♠s ♣♦r ♣s♦♥s s♥s

s ♣s♦♥s stá♥ ♥s ♥ ró♥ Hcuring ♦♥ ♥♠② ♣qñ s♣rsó♥ ♥r ♦♠♦ s sró ♥tr♦r♠♥t ♠tr rr♥ ♣rs♥t strtrs ♦♠♥rs r♥♦ ♥♦r♠ ♣♦r ♥♠r♦ss ♣s♦♥s q trs♥ ♠tr s♣♦♥rá q ①st♥ N ss ♦♠♥s t♦s s ♦♥ ♠s♠ rsst♥étr RL ♥ú♠r♦ ♦♠♥sN ♣♥ t♦rs ①♣r♠♥ts

sst♥ étr ♦♠♥s ♦♥t♦rs ♦r♠s ♣♦r♣s♦♥s s♥s

s♣í♦s ② ♠♥só♥ ♠str ② s♠♥t stá ♥ ♥tr♦102 − 103 ❬ ❪ ♣r♦♥♦ r♥ ♥s♦tr♦♣í strtr sst♠s s♣♦♥rá q s strtrs ♦♠♥rs r♥♦ ♦♥sst♥ ♥ ♣s♦♥s s♥s ♥ trs ♦tr ♥ ♥ rr♦ t♣♦ ③♦♥ r♦r rs st♥ó♥ ♥tr ♦s ♦ ♦♥t strs r♣♦s ♦♥t♦rs ♦s ② ♦s t ♦♥t strsr♣♦s ♦♥t♦rs t♦s ♥ tr♠♥ó♥ N ❬❪ ♦♠♦ s r♣rs♥t ♥ r ♥ rr♦ ①♣r♠♥t t③♦ ♣r rtr③ó♥ rs♣st ♣③♦rrsst r ♦s s s ①t♥♥ ♥ tr♦♦ ♦♣st♦ ♦♥tr②♥♦ ♦♥t étr ♠tr ♥ ♠♦ ♦s s stá♥ ♦③♦s ♥tr♦ ♠tr t♦♥♦ ♥♦ ♦ ♥♥♥♦ ♦s tr♦♦s ② ♥♦ ♦♥tr②♥ ♦♥tétr ♠tr ♦♠♣♦rtá♥♦s ♦♠♦ ♠♥t♦s ♦♥t♦rs s♦s♥ ♣rs♥t ♦r♠s♠♦ s s♣♦♥ q t♦s s strtrs ♦♠♥rs♦srs ♥ ♠r♦rís ♦rts tr♥srss s ♦♠♥s ♦rrs♣♦♥♥ strtrs s r NECC = N = NSEM ♠é♥ ss♣♦♥ q ♥♦ ② ró♥stró♥ s r♥t ♦♠♣rsó♥ ♠tr ♣♦r ♣ó♥ t♥só♥ ♠á♥ ♥① ♥ r s ♥t ♣r♦①♠ó♥ strtr r♦♥s ♦♥t♦rs r♥♦s♥s

r ♦s s s ①t♥♥ ♥ tr♦♦ ♦♣st♦ ♦♥tr②♥♦ ♦♥t étr ♠tr ♥ ♠♦ ♦s s stá♥ ♦③♦s ♥tr♦ ♠tr t♦♥♦ ♥♦ ♦ ♥♥♥♦ ♦s tr♦♦s② ♥♦ ♦♥tr②♥ ♦♥t étr ♠tr ♦♠♣♦rtá♥♦s♦♠♦ ♠♥t♦s ♦♥t♦rs s♦s

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

rr qí q sr♣ó♥ ♦♠étr s ♣s♦♥s♥ tér♠♥♦s s♠♥t♦s rts t③ ♥ ♦s ♣ít♦s ♥♦ s♥♦♠♣t ♦♥ sr♣ó♥ qí ♥tr♦ r♦♥s ♦♥t♦rs r♥♦ s♥s s♥♦ q ♦rrs♣♦♥♥ ♦s ♥s s♠♣ó♥r♥ts ♥ t♦ ♣r st♦ ♥s♦tr♦♣í étr ♦t ♣ít♦s s t③ó ♥ ♥ t ♠r♦♠s♦só♣♦ r♣rs♥t♥♦ s♣s♦♥s r♥♦ ♦♠♦ s♠♥t♦s rts ♣r♦♥ts ② s ♦♥sró ①st♥ ♦♥t étr ♥♦ ♥ ♥tr ♦s ①tr♠♦s sst♠ s♦♦ s ①st ♥ s♣♥♥♥ str q ♦♥t ♦s ①tr♠♦s♦r♠♦ ♣♦r ♦♥t rt ♥tr ♦s s♠♥t♦s rt ♦t♦ ♣rs♥t ♣ít♦ s ♦tr♦ r t♦ ♣ó♥ t♥só♥♠á♥ ♥① ♥ ró♥ ♦r♥tó♥ ♣rr♥ s ♣s♦♥s s♦r ♦♥t étr sst♠ ♥ ró♥ Pr♦ ♦♥srrs strtr ♥tr♥ s ♣s♦♥s ② s♣ró♥ ♥tr s ♥ s s♥t♦ s ♥r♠♥t♦ ♥ tstrtr ♦♥t♠♣♦ ♥ ♠r♦ ♣♦r ♦tr♦ ♦ s r③ ♥ s♠♣ó♥ strtr ♠②♦r s ①♣♦t♥♦ ♠② ♣qñ s♣rsó♥♥r s ♣s♦♥s r♥♦ rs♣t♦ ró♥ ♣ó♥ ♠♣♦ ♠♥ét♦ r♥t r♦ ♠tr③ st♦♠ér s ♦♥sr q s strtrs ♦♥t♦rs s ♥♥tr♥ t♦s s♥s♦r♠♥♦ strtrs ♦♠♥rs q ♦♥t♥ ♦s tr♦♦s ♦♣st♦s

♦♠♥ r♥♦ ♦♥tt♦ ♦♥ ♦s tr♦♦s s ♦♥tt♦étr♦ stá s♦♦ ♥ rsst♥ étr q ♥♦tr♠♦s ♦♠♦RE−CH ♦♥sr ♠s♠ ♣r ♠♦s tr♦♦s ♣r t♦ss ♦♠♥s r♥♦ s í♥s ♦rr♥t ♥ ró♥ ♣r♣♥r s ♦♠♥s r♥♦ s♦♥ s♣rs ♦ rsstétr ♠tr③ ♣♦♠ér r 1014 Ω♠ ♣r P ❬❪♦♥sr♥♦ sí ♥ ♦r♠ ①♣ít ♦♥ó♥ ♥t♦♥s rst♥t q rsst♥ étr t♦t R ♣ sr ①♣rs ♥ tér♠♥♦s rt♦s q♥ts ♠♥t ①♣rsó♥

R =2RE−CH

N+RL

N

tér♠♥♦ ♣r♥♣ sr ♠♦♦ s RLN ú♥ ♥♦ ♦♥tró♥

tér♠♥♦ 2RE−CH

N ♣ sr t♦♠ ♥ ♥t ①♣ít♠♥t ♥ ♠♦♦♦♠♦ ♥ ♥ó♥ t♥só♥ ♠á♥ ♥① ♣ P s ♠♦strráq s s♣r

rsst♥ ♦♥tt♦ ♥tr ♦s s♣rs ♦♥t♦rs ♣♥ ♠♦s t♦rs ♦♠♦ t♦♣♦♦í s s♣rs ♥ ♦♥tt♦ ♣rs♥ ♦♥t♠♥♥ts ♥②♥♦ s ss rst♥s ♦♥tétr s ó①♦s ss t st rsst♥ ♣♦r ♦♥tt♦ s s ♣♦r ♦♥stró♥ ♥s ♦rr♥t trés st♦s ♦ s♣♦ts ár ♣qñ q tú♥ ♦♠♦ ③♦♥s rr♦ ♦♥tt♦ étr♦ ♥tr

sst♥ étr ♦♠♥s ♦♥t♦rs ♦r♠s ♣♦r♣s♦♥s s♥s

s s♣rs ♥♦rs ♥♦♠♥s s♣♦ts ♥ ρµ ② ρE s rssts étrs ♠tr r♥♦ ② ♠tr q ♦♠♣♦♥ ♦str♦♦s ♥ r♥í s ③♦♥s ♦♥tt♦ rs♣t♠♥t ♦♥srr♠♦s q t♦♦s s♦s s♣♦ts s♦♥ rrs ♥ ♥t♦♥s η ♥ú♠r♦ s♣♦ts ② ai i = 1, . . . , η ♦s r♦s ♦s s♣♦ts ♦♥srr♠♦ss s♥ts ♣ótss s s♣r t♦ ♣♦ss ♦♥t♠♥♥ts ♦s s♣♦ts t♥♥ s♣s♦r ♥♦ ♣rs♥t♥ ♦♥t ① ♥ ♥ ró♥ ♦ ♦rr♥t ♦s ♠trs ♥ ♦♥tt♦ s ♦♠♣♦rt♥♦♠♦ ♠trs ♠r♦só♣♦s ♥ s♥t♦ q ss ♠♥s♦♥s ♥ ró♥ tr♥srs rs♣t♦ ♦ ♦rr♥t s♦♥ ♥♥ts s ♠r♦ríst♦♠s s♦r s♣rs ♠tás q ♣rt♣♥ ♦♥tt♦s étr♦s♠str♥ q ♥ ró♥ ♥tr s s♣rs ♥ ♦♥tt♦ s ♦r♠♥ r♣ó♥ s♣♦ts ú♥ t♥s♦♥s ♠á♥s rs ♦ stss♣♦s♦♥s ♦♥s ♣ ♣rs ♠♦♦ ♦♠ ❬❪

RE−CH =ρµ + ρE

2

(1

2η 〈a〉 +1

2α

)

♦♥ 〈a〉 = 1η

∑ηi=1 ai ♦r ♣r♦♠♦ r♦ ♦s s♣♦ts ② α s

r♦ r♣♦ ♥tr s♣♦ts ♥♦♠♥♦ r♦ ♦♠ ♦ s♦♥♦♥s ①♣r♠♥ts ts η s ♠② r♥ ❬ ❪ ♣♦r ♦q ♣r tr♠♥r rsst♥ ♣♦r ♦♥tt♦ étr♦ s s♥t ♦♥♦♥♦r α

RE−CH =ρAg + ρAu

4α

♦♥ ♠♦s t♦♠♦ ρE = ρAu = 2.44 × 10−8Ω♠ ② ρµ = ρAg = 1.59 ×10−8Ω♠ ♦rs 20 ❬❪

❯♥ ró♥ ♥tr α ② t♥só♥ ♥♦r♠ ♣ s♦r ♠str ♣ sr á♠♥t ♦t♥ ♦♥sr♥♦ q ár r ♦♥tt♦Ac s ♣♦r Ac = πα2 = F

H ♦♥ F s r③ ♦♥tt♦ ② H s r③ ♠tr ♠s ♥♦ ♥♦r♦ ♥ ♦♥tt♦ étr♦ ♥♥str♦ s♦ H = HAg = 250 P ❬ ❪ st♠♥♦ r③ ♦

♦♠♦ F = P×ár ♥ ♦♠♥ = P

[π(〈D〉2

)2] s♥♦ 〈D〉 á♠tr♦

♠♦ ♥ ♣s♦♥ s ♦t♥

RE−CH (P ) =ρAg + ρAu

2 〈D〉

√HP

st ó♥ srá t③ ♣r ♠♦r ♣♥♥ RE−CH ♦♥ t♥só♥ ♥♦r♠ ♥① P

♦♥sr♥♦ q t♥só♥ ♠á♥ ♥① ♣ ♣r s ♦♥♠trs r♦ ♥trr③♠♥t♦ rt♠♥t ♦ s ♦♠ú♥♠♥t♠②♦r 1 P P ≥ 1 P ② 〈D〉 ∼ 10µ♠ ♥t♦♥s s ♣rRE−CH ≤ 1Ω♣♦r ♠♣♦ RE−CH ≈ 1Ω s st♠♦ P = 100 P ♦♠♦ ♥ú♠r♦

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♦♠♥s ♥s ♦r♠s ♣♦r ♣s♦♥s N s tí♣♠♥t ♦r♥ 102− 103 s t♥ q ♦♥tró♥ rsst♥ ♦♥tt♦étr♦ tr♦♦r♥♦ RE−CH

N rsst♥ t♦t R s st♠ ♥ ♥tr♦ (10−2−10−3)Ω ♦ q ♦rrs♣♦♥ ♥♦ ♦ ♦s ór♥s ♠♥t♠♥♦r q rsst♥ ♦sr R Ps q ó♥ ♣sr t③ ♣r st♠r ♦♥tró♥ RE−CH ♦♠♦ ♥ó♥ P sr♠♥t s♣r st♦ ♣ ♦♥r♠rs ♣♦r st ♦s rst♦s①♣r♠♥ts R (P ) s♥♦ ♠♦♦ ♦♥sttt♦ srr♦♦

s♥t ♦t♦ s r RLN ♥ ó♥ RL t♥ ♦s

♦♥tr♦♥s sr tr♥s♣♦rt tró♥♦ ♥ r♦♥s ♠♦tró♥ rt♠♥t t ♥tr♦ ♥ ♣s♦♥ ② s rrrs s♦s s s♣r♦♥s ♥tr s r♦♥s ♦♥t♦rs RCH r♣rs♥t rsst♥ ♥ ③♦♥ ♠♦ étr n s ♥ú♠r♦ s ③♦♥s Rtunnel s rsst♥ s♦ ♦ étr♦ ♥tr sr♦♥s ♦♥t♦rs s♣rs ② s♣♦♥♥♦ q s ③♦♥s stá♥ ♣rt♠♥t ♥s ♥ tér♠♥♦s rts rst

RL = nRCH + (n− 1)Rtunnel (P )

♥ú♠r♦ n r♣rs♥t ♥t ♠ s♣r♦♥s ♥ ♥ ♦♠♥ r♥♦ ♥tr r♦♥s ♦♥ ♠♦ étr s rt♠♥t sí ♥ ♣r♠r ♣r♦①♠ó♥ s r③♦♥ ♦♥srr ss ③♦♥s ♦♠♦♥ ♣s♦♥ ♥q ♣♦s ♣rs♥tr s♣r♦♥s ♥tr♦ ♣s♦♥s ♥♦ s srt ② ♦ s ♦♥t③ ♣♦r ♣rá♠tr♦ n

s♣t♦ RCH ♥ú♠r♦ µPs q ♦r♠ ♣rt ♥ ♣s♦♥ ② ♣♦r ♦ t♥t♦ ♥ ♦♠♥ s ♠② ♦ r③ó♥ ♣♦r s♣ ♥ ♣rs♥t st♦ srr♦♦ ♥ ♠♦♦ ♦♥sttt♦ ♣r RCH ♥ tér♠♥♦ rs rsst♥s étrs Pr r s ♥ú♠r♦ µPs ♠♥tt♣t ♥ ♥ ♣s♦♥ χ ♣ st♠rs ♦♥sr♥♦ ♥ ♣s♦♥ ♦r♠ í♥r á♠tr♦♣r♦♠♦ 〈D〉 ② ♦♥t ♣r♦♠♦ 〈ℓ〉 ♦r♠ ♣♦r ♠♣qt♠♥t♦ χ♠r♦♣rtís írs á♠tr♦ ♠♦ 〈DµP 〉 ♦♥ ♥ ♥ ♠♣qt♠♥t♦ f sts ♠♥ts stá♥ ♦♠étr♠♥t r♦♥s ♣♦r

π(〈D〉2

)2〈ℓ〉f = χ4

3π

(〈DµP 〉

2

)3

χ ♣ sr st♠♦ ♥ s♦ ♦♠♦

rr♥ ♥ st tr♦ 〈D〉 = 10.4µm 〈ℓ〉 = 1.3 ♠♠ ② 〈DµP 〉 = 1.3µ♠Pr ♥ ♠♣qt♠♥t♦ ♦♠♣t♦ µPs s t♥ f = 0.74 t♦s♦r♠s strtr ♥tr♥ s ♣s♦♥s ♥♦ stá ♠♣qt ♥♦r♠ ♦♠♣t ♦♠♥♦ f = 0.5 ② f = 0.74 ♥ ♥tr♦ ró♥r③♦♥ s ♦t♥ χ = 0.5 × 105 f = 0.5 ② χ = 8 × 105 f = 0.74♥ qr ♦s s♦s rst ♥t q s ♣s♦♥s stá♥♦r♠s ♣♦r ♥ r♥ ♥ú♠r♦ ♣rtís ♥♦ ♠② t♦s ♥st♠ó♥ t RCH P♦r t♦s s r③♦♥s rr ♠♥♦♥ ♦s

sst♥ étr ♦♠♥s ♦♥t♦rs ♦r♠s ♣♦r♣s♦♥s s♥s

♣rá♠tr♦s n ② RCH s ♦♥srrá♥ ♥♣♥♥ts t♥só♥ ♠á♥ ♣ P ② r♠♥t sts sr r♣r♦s ♣♦r st ♦srst♦s ①♣r♠♥ts rs♣st ♣③♦rsst ♦s ♣rá♠tr♦s ♥♦r♦s ♥ s ♦♥s s ♠str♥ ♥ r♣rs♥tó♥sq♠át r

r ♣rs♥tó♥ ♦s ♣rá♠tr♦s ♠♦♦ ♣③♦rsst♥

sst♥ ♣♦r t♥♦ tró♥♦ Rtunnel

t♦r Rtunnel ♥ ó♥ s ♠♦♦ ♥ s ♦s ♣ró①♠s♦♥s ♦s ♠♥s♠♦s ♥♦r♦s ♥ ♦ étr♦ ♥tr ♦s ♦t♦s ♠tá♦s ♠r♦♠tr♦s s♣r♦s ♣♦r ♥ ♣ ♠tr s♥t♥ ♥str♦ s♦ ♠tr③ ♣♦♠ér s♦♥ t♥♦ tró♥♦ ♠só♥ étr ♥ ♣♦r ♠♣♦ tr♥s♠só♥ rs♦♥♦tt② ♦♥ó♥ P♦r♥ ② ♦tr♦s ♠♥s♠♦s ♦♥ó♥ s♥♦♦r♥ ♥ ♠r♦ st♦s ♣r♦s ♥ ♠♦str♦ q ♦♥tétr trés s ♣ s♥t st ♦r♥ ♣♦r t♥♦ tró♥♦ ❬ ❪ ♣♦r ♦ q ♥ ♣rs♥t ♠♦♦ s♦♦ s♦♥srr st út♠♦ ♠♥s♠♦ P♦r ♦♥s♥t Rtunnel r♣rs♥t rsst♥ étr s♦ t♥♦ tró♥♦ ♥tr ♦s r♦♥s ♦♥t♦rs s♣rs ♣♦r ♥ ♣ ♠t③ ♣♦♠ér s♥t ♦♥srqí q s s♣rs ♥♦rs ♦rrs♣♦♥♥ ♥♠♥t s r♦

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥s ③ ② ♦ s r♦♥s ♦♥ts ②♥ts s ③s ♦♥srrá q s rtr s ♣r♦①♠♠♥t sér ♦♥ á♠tr♦ á♠tr♦ ♠♦ s ♣s♦♥s 〈D〉 r str♦s ♦s ♠♥s♠♦s ♣r♥♣s ♥♦r♦s ♥ rs♣st ♣③♦rsst ♠tr

r ♥s♠♦s ♣r♥♣s ♥♦r♦s ♥ rs♣st ♣③♦rsst ♠tr

t♥♦ tró♥♦ ♥tr ♦s s♣rs ♦♥t♦rs ♣♥s ♣rs♦ ♥ r♥ ♣♦t♥ ♠♦r ♥tr s s♣rs Vtunnel♣ ①♣rsrs s♥♦ ♦r♠s♠♦ ♠♠♦♥s ❬ ❪ ♥ ♦ ♦r♠s♠♦ s ♦♥sr q s s♣rs ♦♥t♦rs s ♥♥tr♥s♣rs ♣♦r ♥ ♣ s♥t s♣s♦r x ♦♥ s ♣rs♥t ♥ rrr ♣♦t♥ rt♥r tr ϕ ② s s♣r♥ s r③s ♠♥❬❪ ♦ ♦r♠s♠♦ ♠♥♦♥♦ ♦rr♥t étr ♣♦r ♥ ár s r ♦ étr♦ Jtunnel s ♦ ♥ ré♠♥ ó♠♦ ♣♦r

Jtunnel = Bγ

xexp (−γx)Vtunnel

sst♥ étr ♦♠♥s ♦♥t♦rs ♦r♠s ♣♦r♣s♦♥s s♥s

♥ ♦r♠s♠♦ ♠♠♦♥s γ ② B ♥♥ ♦s ♣♦r ❬❪

γ =4π

h

√2mϕ

B =3e2

8πh

♦♥ e s r tró♥ m s ♠s t tró♥ ♥ ♠tr r♥♦ ② h s ♦♥st♥t P♥ ♦r ♥ s ♥sr♦♥r③r rst♦ ♦t♥♦ ♣♦r ♠♠♦♥s ♦♥ ♥ ♥r s♦ ♦s s♣rs ♠sérs ♦♠♦ s ♠str ♥ r s♣ró♥ ♥tr s s♣rs ♠sérs ♥ á♥♦ β stá ♣♦r

X (β) = x+ 〈D〉 (1− cosβ)

r ①tr♠♦s r♦♥s ♦♥t♦rs ♠♦s ♦♠♦ s♣rs♠sérs á♠tr♦ 〈D〉 s♣rs ♣♦r ♥ st♥ x

❯s♥♦ ést♦ ♦rr♥t ♣♦r t♥♦ tró♥♦ ♥tr s ♦s s♣rsItunnel s ♦t♥ ♣♦r ♥tró♥ ♦ étr♦ ♥ ró♥ XJX (X (β)) = J (X) X

Itunnel =

∫∫

G

JX · dA

♦♥ G ♦rrs♣♦♥ s♣r ♠t sr ②

JX = Bγ

X (β)exp [−γX (β)]VtunnelX

①♣♦t♥♦ s♠trí ♣r♦♠ ♠♥t♦ r♥ dA ♣ sr①♣rs♦ s♥ t ♥ ♦♠♦ ♥ó♥ β ♠♥t s♦ ♦♦r

♥s sérs ♠♥t dA =(〈D〉2

)2sinβdφdβr ❯t③♥♦ ró♥

♠Rtunnel =

VtunnelItunnel

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

s ♦t♥

Rtunnel =4

〈D〉2∫ 2π0 dφ

∫ π20

BγX(β)exp [−γX (β)] sinβ cosβ dβ

♦ ♥ ♥ ♦r♠ ♠ás ♦♠♣t

Rtunnel =2

πBγ∫Wx

1z exp [−γz] (W − z) dz

♦♥ W ≡ x+ 〈D〉 ♣rá♠tr♦ γ ② ♦♥st♥t B ♥ s ♦♥s ② s

rá♥ ♦♥sr♦s ♦s ♣♦r s ①♣rs♦♥s ② ♥tr ó♥ ♥♦ ♣rs♥t s♦ó♥ ♥ít ❬❪ ② ♥♠ér♠♥t s♥♦ ♠ó♦ ♠r ♥trt♦♥ t P♦srrs q ①♣rsó♥ ♥tr ♥ ó♥ ♣♥ st♥ x q ♥ ♥str♦ ♠♦♦ s s♣♦♥ r♥t ♦♥ t♥só♥ ♠á♥ ♥① ♣ P ♥t♦♥s ♦t♦ s♥t ó♥ s♠♦r ♣♥♥ x rs♣t♦ stí♠♦ P

♦♦ x(P ) rtr③ó♥ ♠á♥ ♠tr③ ♣♦♠ér

♦ q ♠ó♦ ❨♦♥ r♥♦ s ♠♦ ♠②♦r q ♠tr③ st♦♠ér s ♦♥srrá q ♣ó♥ P ♥♦ ♥tr♦ ♥st♦rsó♥ strtr s r♦♥s ♦♥t♦rs

♥ ♥ s♥t♦ ♣tór♦ ♣rá♠tr♦ x(P ) s st♥ ♠ q♦s tr♦♥s ♥ trsr ♥ ♥ ♣s♦ t♥♦ t♥♥ st♣ s r③♦♥ ♥ ♥ ♣r♠r ♣r♦①♠ó♥ ♦♥srr ♦♠♦ s♣ró♥ ♥tr♣s♦♥s ♥ ♠r♦ ♦ q s ♣s♦♥s s♦♥ strtrs♦♠♣s ♦♥ ♦s r♣♦s ♠r♦♣rtís r♥♦ ♣♥ str s♣r♦s t♠é♥ ♣♦r ♣s ♣♦í♠r♦ ♥t♦♥s s s♣r♦♥s t♠é♥♣♥ str ♣rs♥t ♥tr♦ s ♣s♦♥s

s♣♦♥rá q ♣ ♠tr③ ♣♦♠ér q s♣r ♦s r♦♥s♦♥t♦rs r♥♦ ♣rs♥t ♠s♠♦ ♦♠♣♦rt♠♥t♦ ást♦ q ♥♠str ♠r♦só♣ ♦ ♣♦í♠r♦ st stó♥ s ♣ ♥ sst♠s ♦♥ r♦ ♥trr③♠♥t♦ rt♠♥t ♦ ♦♠♦ ♦♥sr♦ ♣♦r ♥str♦ r♣♦ ② ♦tr♦s P♦r ♠♣♦ ♥②♦ t ❬❪ srt♦r♥t♠♥t q ♥ ♦♠♣♦st♦s ♦♥ ♠trs ♣♦♠érs q ♣rs♥t♥ ♦s r♥s ♥trr③♠♥t♦ ♥♦ s ♦sr ♠♦♠♥t♦ r s♣rtís r♥♦ s♥♦ q ést s ♠ ♥ ♦r♠ í♥ ♦♥ ♠tr③st ♥ó♠♥♦ s ♥♦♠♥ ♦r♠ó♥ í♥ ♥ ♥és s rr♥ ♦♠♦♥ ss♠♣t♦♥ ② s s♣♦♥ ♥ ♣rs♥t tr♦ ♦ t♦ r♦ ♥trr③♠♥t♦ P t③♦ ♥ ♥str♦ sst♠ t♦s s♦♦s ♦r♠♦♥s ♠②♦rs ♠tr③ ♥ ♥ r♥♦s ♠②

sst♥ étr ♦♠♥s ♦♥t♦rs ♦r♠s ♣♦r♣s♦♥s s♥s

rí♦s ♦♠♦ ♦s srt♦s ♣♦r ♦♠rt t ❬❪ ♥♦ s♦♥ ♦♥t♠♣♦s♥ rs♠♥ ♥ ♥str♦ ♠♦♦ s r♦♥s ♦♥t♦rs ♦♠♣ñ♥ ♠♦♠♥t♦ ♦♠♣rsó♥ ♠tr③ ♥♦ s ♣ ♥ t♥só♥ ♠á♥♦ q ♠♣ ♦ q s ♠strs s♦♥ ♥♦♠♣rss sts♣♦só♥ s ①♣rs ♠t♠át♠♥t ♦♠♦

x (P ) = xoλ (P )

♦♥ λ (P ) ≡ y(P )yo

♦♥ y(P ) yo s♦♥ s♣s♦r ♠str ♠tr③♣♦♠ér s♥ r♥♦ ♣♦r ♠♣♦ ♥ t♦♠♦ qí ♦♠♦ rr♥ P ♣r♦ ♦ ♥ t♥só♥ ♠á♥ P ② P = 0 rs♣t♠♥t λ r ♥♦♠r ♦♥ó♥ r ♣é♥ P♦r ♦tr♦ ♦ ♣rá♠tr♦ xo s s♣ró♥ ♣r t♥♦ ♥ P = 0

♦ ♠♦♦ qí ♣rs♥t♦ rqr ♥ ①♣rsó♥ ♣r λ ♥♥ó♥ P ♣r sr ♥tr♦ ♥tr♦ ♣♥♥ Rtunnel ♦♥x s♥♦ s♣♦só♥ ♦r♠ó♥ í♥ ró♥ ①♣r♠♥t λ ♦♥ P s ♦t♥ ♣rtr ♥s②♦s ♦♠♣rsó♥ ♥① ♠strs ♠tr③ ♣♦♠ér s♥ r♥♦ ♦♠♦ ♠♦str ♥ r ♥ sr P r♣rs♥t ♥♦♠♥ t♥só♥ ♥♥r ♥♥r♥ strssP = (r③ ♣) / (r r③=0) ♣♥♥ P − λ s st♦♠ú♥♠♥t t③♥♦ ♦s ♠♦♦s ♦♦♦ ♦♦♥②♥ ♦♦ ♦ ♦♦ r♥ ❬ ❪ st♦s ♠♦♦s♣r♦♥ ①♣rs♦♥s ♥íts ♣r P ♥ ♥ó♥ λ q r♦♥ ss♣r str ♦s rst♦s ①♣r♠♥ts ♥ ♣rtr ♥ r s♠str♥ ♦s sts s♦♦s ♦s ♠♦♦s ② ♦s rst♦s♦t♥♦s ♣r ♥♦ ♦s ♠♦♦s s sr♥ ♦♥t♥ó♥ t♦r ♥trs♦ ♣ ♥♦♥trr ♥ sr♣ó♥ ♠s t ♦s♠♦♦s st ♠♥♦♥♦s ♥ ♣é♥

♦♦ ♦♦♦

s♠♥♦ q ♦♠♥ ♠tr ♥♦ ♠ ♥ ♦r♠ ♣rr♥t ♦♠♣rsó♥ s♦♦ ♥♦♠♣rs t♥só♥ ♠á♥ ♥♥r P ♥ r♦ ♦♥ ♠♦♦ ♦♦♦ ♣♦r ❬ ❪

P = 2CN−H1

(1

λ2− λ

)

♥ ♦ ♠♦♦ ♣rá♠tr♦ st CN−H1 s r♦♥ ♦♥ ♠ó♦

❨♦♥ ♠tr③ ♠♥t CN−H1 = E/6 ❬ ❪ ♠♦♦ ♦

♦♦ ♣r♦ ①♣rs♦♥s rrs ♣r λ (P ) s♦♦♥s ♥ ♣♦♥♦♠♦

r③♦ r ♦rt ♥t② rr ♥♥r♥ sr ♥tr ❯♣♦r s rs s♦r á♥ sst♠s ♦♥t♥♦s

❱és ♣é♥ ♣r ♠ás ts

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

0.8 0.9 1.0

0

100

200

300 Experimental data

Neo-Hooke

Mooney-Rivlin

Str

ess (

kP

a)

λ

r ♥só♥ ♥♦r♠ ♥① P ♥ ♥ó♥ λ (P ) ≡ y(P )yo

♣r♥ ♠str P ró♥ s♥trr③♥t ♥♦ r♦ ♦♥♣rtís r♥♦ ♥ ♦♥t♥ r♦ ♦rrs♣♦♥ st ♠♦♦ ♦♦♦ ❬ó♥ ❪ ♦♥ CN−H

1 = (146±1) P í♥ ♦♥t♥③ ♦rrs♣♦♥ st ♠♦♦ ♦♦♥②♥ ❬ó♥ ❪♦♥ CM−R

1 = (−692± 5) P ② CM−R2 = (711± 4) P

ú♦

λj

(P)= ajξ

(P)− P + bj

P 2

ξ(P)

♦♥

ξ(P)≡ 1

21/3

[√1− 4P 3 + 1− 2P 2

]1/3

♦♥ P ≡ P/E aj = −12

(i√3 + 1

)−1

2

(−i

√3 + 1

) 1 ② bj = −1

2

(−i

√3 + 1

)

−12

(i√3 + 1

) 1 ♣r j = 1 2 3 i s ♥ ♠♥r

♦sr r♠♥t ♥ r q ♣r s♦ ♥ st♦ ♠♦♦ ♦♦♦ ♥♦ st r ①♣r♠♥t rs♣st ♠á♥ ♠tr③ ♣♦♠ér P♦r ♦ t♥t♦ s ♦♥sr ♠♦♦ ♦♦♥②♥

♦♦ ♦♦♥②♥

①♣rsó♥ ♣r t♥só♥ ♥♥r ♣r♦st ♣♦r ♠♦♦ ♦♦♥②♥ s ❬ ❪

P =

(2CM−R

1 +2CM−R

2

λ

)(1

λ2− λ

)

sts ♦s rst♦s ①♣r♠♥ts ♣r rr♥

♦t♥ ♥ ①♥t st ♦s rst♦s ①♣r♠♥ts ♣r P ♠♥t st ♠♦♦ r r ♦♥ CM−R

2 = (711± 4) P ②CM−R1 = (−692± 5) P s ①♣rs♦♥s rrs ♣r λ ♥ ♥ó♥

P s♦♥ ♣r st ♠♦♦ s♦♦♥s ♥ ♣♦♥♦♠♦ árt♦ ♣sr q ♣♥ ♦t♥rs ①♣rs♦♥s ♥íts ♣r st s♦ t♠é♥ ❬❪ s♠s♠s sr♥ r tr♦③♦s ♣♦r ♦ q s ♦♣t♦ ♣♦r ♦t♥rs♦♦♥s ♥♠érs ② srs ♥tr♦ ①♣rsó♥ ♣r R ❬♦♥s ② ❪ trés ♣♥♥ ♣r♦♣st x♦♥ λ ❬ó♥ ❪

❯♥ ③ ♥tr♦ ♥tr♦ s ♦♥s ♦♥sttts ♠♦♦ ró♥ λ = λ (P ) ♣r ♠tr③ ♣♦♠ér ♥♦ ♣r ♠♦♦♣ sr s♦ ♣r str ♦s rst♦s ①♣r♠♥ts R = R (P ) ♥ ♦r srt♦ st♦ rqr tr♠♥r ♦s ♣rá♠tr♦s ♠♦r♦ó♦s ♥ú♠r♦ ♠♦ strtrs ♦♠♥rs ♥ ♠str ♦♥sr N ② á♠tr♦ ♠♦ s ♦♠♥s ♦♥sr♦ á♠tr♦ ♥ ♣s♦♥ 〈D〉 ♦ ♠♦♦ qí srr♦♦♣rs♥t trs ♣rá♠tr♦s r♣rs ♣♦r st rr♦s rtrr♠♥t♦♠♦ A1 A2 ② A3 ② ♥♦s ♦♠♦

A1 =(n− 1)xo

Bγ

A2 = γxo

A3 =n

NRCH

♥ ♣rá♠tr♦ B s ♥ ♦♥st♥t ♥rs ❬r ❪ ♣rá♠tr♦ n ② xo ♣♥ ♦t♥rs A1 ② A2 ♥ tér♠♥♦s γ ♣r♠t st♠r γ s rrr t♥♦ ϕ ② ♠s t tró♥ m s♦♥ ♦♥♦s

s ♦♥s ② r♣rs♥t♥s r♦♥s ♦♥sttts ♣rs♥t ♠♦♦ ♦s sts ♦s t♦s♦t♥♦s ①♣r♠♥t♠♥t ♣r rs♣st ♣③♦rsst R = R (P )♣r rr♥ ♦♥sr♦ s ♣rs♥t ② ♥③ ♥ ó♥s♥t

sts ♦s rst♦s ①♣r♠♥ts ♣r rr♥

s r♥ts ♠strs ♣r♣rs ♣♦r ♥str♦ r♣♦ ①♥ ♦s rtrísts r♥ts ♣③♦rsst♥ ♥s♦tró♣ ② rrs❬ ❪ s♣t♦ ♥s♦tr♦♣í s ♦♥♦♥s ①♣r♠♥ts ♣♥

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

sr ♦♣t♠③s ♣r ♦t♥r ♥s♦tr♦♣í étr ♦t s r♦♥t étr ♣r ♥ ♥ ró♥ ①s♠♥t ♦♠♦ s♥ ♥ ♣ít♦s ♥tr♦rs

rs♣st ♣③♦rsst s ♦♠♣t♠♥t rrs s♥ stérss ♦ ♦s ♦♠♣rsó♥s♦♠♣rsó♥ s ③ rs♣st ♥♦ s ♠♦ ♣♦r t♦rsó♥ ♦ ①ó♥ ♠tr ♥ ró♥♣r♣♥r s ♣s♦♥s ❬ ❪ st♦s rst♦s sr♥ q♥♦ ② ♠♦s rrrss ♥ rqttr ♥tr♥ s ♣s♦♥s♣♦r t♦ s ♣rtr♦♥s ♠á♥s ♦ ♦r♠ó♥ ♣ást s ♠r♦♣rtís r♥♦ q ♦r♠♥ s ♥s s r③♦♥♠♥tsrt

①st♥ ♦s t♦rs ♣r♥♣s q s♠♥t ♥♥ rrrs♥ rs♣st ♣③♦rsst r ♠♦ s ♣rtís r♥♦ ♥ ♠♦ ♠tr③ ♣♦♠ér ♣rs t♥só♥ ♠á♥①tr♥ st ♥♦ s s♦ ♦s ♦♠♣♦st♦s ♦♥ ♦ r♦ ♥trr③♠♥t♦ ♦♠♦ s sró ♥ ó♥ ①st♥ ♥s qí♠♦s ♥tr ♣♦í♠r♦ ② r♥♦ ♥ s s♦ ♦s ♥s ♣♥ r♦♠♣rs ♥ ♦r♠ rrrs r♥t ♦s ♦s ♦♠♣rsó♥s♦♠♣rsó♥♥♥♦ t♦s stérss st♦ t♠♣♦♦ ♣r sr s♦ qí ♦♥sr♦ ♣st♦ q ♦sró♥ rt ③♦♥s ♥♦só♥ ♥tr r♥♦ ② ♠tr③ sr♥ q ♥♦ stá♥ ♦r♦ ♦s ♥s qí♠♦s♥tr sts ♦s ss r r

s ♥ rs rtrísts I − V ♥♦ ♠♦strs ♠s♥ ♥ó♥ t♥só♥ ♣ P ♥♥ q st♦ ①♦♠♣♦rt♠♥t♦ ó♠♦ ♥ t♦♦ ♥tr♦ ♣♦t♥s étr♦s ② t♥s♦♥s ♠á♥s ♥rs ♣♥♥t s rs ♦rrs♣♦♥♥ ♦s ♦rs R(P ) rsst♥ étr ♠ t♥só♥ ♣r

qr r♦ ♣r t♦r q ♣r♦♠♥t♦ s♦ ♣r ♦t♥ó♥ ♥tr♣rtó♥ s rs ♠r♦só♣s s ♦♠♣t♠♥t ♥r ♥ t♦ ést ♣r♦♠♥t♦ s ♣ ♥ t♦♦ sst♠ ♦♥ rtrísts strtrs s♠rs rr♥ t♦s♦r♠s ♦s ♦rs qí ♥♦r♠♦s ♦rrs♣♦♥♥ ♠str rr♥ 34❬❪P 4.2% ♠tr r♥♦ ró♥ s♥trr③♥t µoHcuring = 0.35 t♠♣rtr r♦ Tcuring =(75± 5) t♠♣♦ ①♣♦só♥ 3 ♦rs Lo = 2.50 ♠♠ Pr ♠str rr♥ s ♦t♥ A1 = (2.2 ± 0.2)Ω A2 = γxo = (11.1 ± 0.5) ②A3 = (0.47± 0.07)Ω ♦♥ ♥ ♥ r♦ st R2 = 0.995

♦♠♦ s ♠♥♦♥ó ♥tr♦r♠♥t n ② xo ♣♥ sr ♦s ♥ ♥ó♥ γ q ♦ ♣♥ rrr ♣♦t♥ s♦ t♥♦tró♥♦ ϕ ② ♠s t tró♥ ♥ ♠tr r♥♦m rrr ϕ s ♦t♥ ♠♦♥s ♥ó♥ tr♦ s♦ ♣r ♠ts♥t Pr ♥♥♦s ♣t ♥ P s r♣♦rt♦

sts ♦s rst♦s ①♣r♠♥ts ♣r rr♥

0 200 400

1

2

3

4

5R

()

Stress (kPa)

Compression Decompression Model

r st rsst♥ étr ♠tr rr♥t③♥♦ ♠♦♦ ♦♥sttt♦ srr♦♦ r t①t♦ í♥ ♦♥t♥r♦ ♦rrs♣♦♥ st ♠♦♦ t③♥ ♦s ♣rá♠tr♦s ♥tr〈D〉 = 10.4µ♠ ②N = 660±50 s♣s♦r ♠str s Lo = (2.50± 0.01)♠♠

r♥t♠♥t ϕ = 1 ❱ ❬❪ ❯s♥♦ ♦ ♦r ϕ ② ♠s tró♥r m = me ♦♥♦♥s q s ♥rá♥ ♣♦r s♣rí♥ † s ♦t♥γ† = 1.0× 1010 ♠−1 n† = (98± 9) ② x†o = (1.08± 0.05) ♥♠ ♣r rr♥

❯s♥♦ N = (660± 50) s r♣r R†CH = (3.2± 0.3)Ω ♦s ♦rs

n† ② x†o ♣♥ sr s♦s ♣r st♠r ♦♥t t s r♦♥s ♦♥t étr l q ♥♦ ♦♥♥rs ♦♥ ♦♥t s ♣s♦♥s ℓ q ♦rrs♣♦♥ st♥ ♠ ♥tr s♣r♦♥s t♥♦ tró♥♦ s♣♦♥♥♦ á ♦♥ó♥ í♥♦♦♠étr♦

Lo = (n− 1)xo + nl

Pr ♠str rr♥ s t♥ Lo = (2.50±0.01) ♠♠ ♦ q♦♥ l† = (26± 4) µ♠ ♥ ♦r ♥s 50 s ♠♥♦r q ♦♥t♠ s ♣s♦♥s ♦srs ♣♦r 〈ℓ〉 = 1.35 ♠♠ strst♦ s ♦♥sst♥t ♦♥ ♦sró♥ r♠♥t♦♥s ♥tr♥s ♥s ♣s♦♥s s♦s ♥tr♣♥tró♥ ♣♦í♠r♦ ②♦ ♦♥str♦♥s rrrs strtrs ♦♠♦ s ♦srs ♥ r r str ♠r♦rís q str♥ ♥tr♣♥tró♥ ♠tr③ ♣♦♠ér ♥tr♦ s ♣s♦♥s r♥♦ ♥

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

rr♥ 34❬❪P 4.2%

r ♠á♥s q str♥ ♥tr♣♥tró♥ ♠tr③ ♣♦♠ér ♥tr♦ s ♣s♦♥s r♥♦ ♥ rr♥34❬❪P

st♦s rst♦s ♠str♥ t♦t ♦♥♦r♥ ♦♥ ♦sr♦♥s ♣rs ♠ó♦s ❨♦♥ ♦s s E ♠♥♦rs ♦s ♣r♦s ♠♥t ♦s♠♦♦s ♠á♥♦s ♠trs ást♦s ♦♠♣st♦s ♦♠ú♥♠♥t t③♦s ♦♠♦ sr ♠♦♦ ♣♥s ♠♦♦ ❬❪ ② ♠♦♦ ❬ ❪ ♥♦ ♦r 〈ℓ〉 s ♥tr♦♦ ♥ ♦s ♠♦♦s❬❪ st ♦r♠ r♥ ♥tr♦r♠♥t ♦sr ♥tr ♦s ♦rs♦sr♦s ② st♠♦s E sr ♠s♠ ♦♥só♥ q qí ♦t♥ s ♣s♦♥s stá♥ r♠♥ts ② ♦s r♠♥t♦s ♣rt♣♥♥ t♥♦ tró♥♦

♠♦♦ srr♦♦ ♣r♠t st♠r ♦♥tró♥ tér♠♥♦♥ s s ② rsst♥ étr t♦t R P♦r ♠♣♦ ♣r P = 100 P s ♦t♥

(n−1N

)Rtunnel ≈ 1.17Ω

(2N

)RE−CH ≈

3 × 10−4Ω ②(nN

)RCH ≈ 0.5Ω ts t♥s♦♥s ♠á♥s R ♦♥r

♦♥tró♥ ♣♦r rsst♥ étr ♥trí♥s s r♦♥s♦♥t♦rs r♥♦

(nN

)RCH ♦r r♣r♦ RCH s ♠②♦r

q s♦♦ ♥ ♥ ♦♥t♥ ② ♦♠♦é♥ ♣t ♠tá ♥♦♥♦r♥ ♦♥ ♠♥ ♣s♦♥s ♦r♠s ♣♦r r♣♦s ♠r♦♣rtís ② rsst♥ ♦♥tt♦ ♥tr s

♦s rst♦s st sr♥ q RE−CH s s♣r st s♥ ♦♥só♥ ♥♦ ♥ ♣ótss s r RE−CH s ♠♥t♥ ♥ ①♣rsó♥ ♥r s ♣r ♠♦r ♦s rst♦s ①♣r♠♥ts ♦ q ♣rs s♥ ♦♠♣ó♥ ♥ ♦ r③r st ♦rrs♣♦♥♥ts ♦♥r♠ ♣ró♥ RE−CH s ♠② ♣qñ ② s♣r ♥ ♦♠♣ró♥ ♦♥ s ♦trs ♦♥tr♦♥s R

♦♠♦ s ♠♥♦♥ó ♥tr♦r♠♥t ♦r γ stá ♥♦ ♣♦r ♠st tró♥ m ② rrr t♥♦ tró♥♦ ϕ s♦s ♣rá♠tr♦s s♦♥ rt♠♥t ♣♥♥ts ♥tr③ qí♠ ② strtr

sts ♦s rst♦s ①♣r♠♥ts ♣r rr♥

r♥♦ ② ♠tr③ ♣♦♠ér Pst♦ q γ tr♠♥ ♦s ♦rs r♣r♦s n xo ② l s tr♦♥ s♠♦♥s ♦s ♣rá♠tr♦s ♥ ♥ó♥ γ r r m r ♥ ♥tr♦ (0.5 − 1.5)me

♠♥trs q ϕ r ♥tr (0.5− 1.5) ❱ Pr s♦s ♥tr♦s ró♥ γ rí ♥tr (0.5− 1.5)× 1010 ♠−1 ♦sr ♥ r ql ② n rí♥ ♥ ♦r♠ ♥♦t♦r r 2 ór♥s ♠♥t P♦r ♦tr♦♦ xo ♣r♠♥ ♥tr 0.6 ② 2.1 ♥♠ ♣r♦①♠♠♥t st♦ ♥ q♥q ♥ st♠ó♥ ♣rs n ② l rqr ♠♦♥s rtrs γ ♠♦♦ ♣r ♦rs xo q s ♥♥tr♥ ♥tr♦ s s tí♣♣r t♥♦ tró♥♦ ❬❪

0.6 0.9 1.2 1.5

1

2

c

b

x o (nm

)

a

0.6 0.9 1.2 1.50

100

200

n

0.6 0.9 1.2 1.50

50

100

(1010m-1)

(1010m-1)

l (m

m)

(1010m-1)

r ♠♦♥s xo n ② l ♥ ♥ó♥ γ s♥♦ ♦s ♦rsA1 = (2.2± 0.2)Ω A2 = γxo = (11.1± 0.5) ② A3 = (0.47± 0.07)Ω ♠str rr♥ ♥ γ†

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥♥ st ♠tr E rrr t♥♦ r♥♦♣♦í♠r♦ ϕ ② st♥ ♠ t♥♦ xo s♦r Rtunnel

♥ st ó♥ s ①♣♦r ♠♥t á♦s ♥♠ér♦s ♥♥ ♥♦s ♣rá♠tr♦s ís♦s s♥s ♣r sñ♦ s♥s♦rs t♥só♥♠á♥ s♦s ♥ s ♣③♦rsst♦s Pr s♠♣r ♥áss s ♦♥sr q s s♣rs ♣rt♣♥ts ♥ t♥♦ tró♥♦ s♦♥ ♣r♦①♠♠♥t ♣♥s ♣♦r ♦ q Rtunnel (P ) stá ♦ ♣♦r ①♣rsó♥ ♦r♥ ♠♠♦♥s Rtunnel ∝ x

Bγ exp (γx) st ♦r♠ s st ♥

①♣rsó♥ s♠♣ RtunnelRtunnel (0)

♥ ♥ó♥ P s ♥ ♠♦♦srr♦♦ t♦s ss ♣r♦①♠♦♥s s ♦♥sr♥ ♥ ♦r♠ ♦♥♥ts♣r♥♦ ♦♥tró♥ RE−CH ② s♣♦♥♥♦ q n ② N ♥♦ ♠♥♣♦r ♣ó♥ P s ♦t♥

R−R(∞)

Ro −R(∞)≈ Rtunnel

Rtunnel (0)= λ exp [γxo (λ− 1)]

♦♥ Rtunnel (0) ≡ Rtunnel (P = 0) Ro ≡ R (P = 0) ② R (∞) ≡ R (P → ∞)P♦r s♠♣ ♥ st ó♥ λ s ♦ ♦♠♦ λN−H(P ; E) ♣♦r ♠♦♦ ♦♦♦ ❬s ❪ ♦♥ CN−H

1 = E/6 ♣sr q ①♣rsó♥ s♣♦♥ ♥ ♣♥♥ s♠♣ x ♦♥ P ést ♦♥t♥ ís s♥ ♣r♦♠ ♥ st♦

st ♦r♠ ró♥ r♥ ♠ás rr ♠str q RtunnelRtunnel (0)

♣♥ ú♥♠♥t t♥só♥ ♣ rt ♠ó♦ ❨♦♥ ♠tr③ ♣♦♠ér P ≡ P/E ② s♣ró♥ ♥ ♥tr s r♦♥s♦♥t♦rs rt γ−1 xoγ r str rs s♠s Rtunnel

Rtunnel (0)♣r r♥ts ♦rs E ♥ ♦r ♦ xoγ ♠♥trs q

♥♥ xoγ ♣r ♥ ♦r ♦ E s ♠str ♥ r r ♠str q t③ó♥ ♣♦í♠r♦s ♦♥ ♦rs ♠

②♦rs E rqr ♣r ♠②♦rs t♥s♦♥s ♣r ♥③r ♥ s♣ró♥ ♠ t♥♦ tró♥♦ x(P ) ② ♣♦r ♦ t♥t♦ ♠s♠♦ ♦r Rtunnel

Rtunnel (0) ♥ ♦♠♣♦rt♠♥t♦ s♣r♦ ♥♦ s♣st♦ ♦r♠ó♥

í♥ ♠tr s ♠♣♦rt♥t ♥♦tr q ♠ó♦ ❨♦♥ ♠tr③P ♣ ♠♥trs ♥r♠♥t♥♦ ♥t ♥t ♥trr③♥t ♥ ♠tr③ s♥ ♠r ♥ ♦r♠ ♣r tr rrr ♣♦t♥ ϕ s r s♥ ♠r xoγ

♥á♦♠♥t r ♠str q ♥♦ xo ② E s rqr♥♠②♦rs t♥s♦♥s ♣r ♥③r ♠s♠ s♠♥ó♥ ♣♦r♥t ♥ Rtunnel

♣r ♦rs ♠♥♦rs γ ♣♦r ♠♣♦ s♥♦ ♥ ♣r r♥♦♠tr③ ♦♥♠♥♦r ♦r ϕ sí ② r r♠r♥ q s♥s ② ♥tr♦ ♠ó♥ ♥ s♥s♦r t♥só♥ ♠á♥ s♦♥ ♥ ♠tr ♣♥ sr ♠♦♦s ♠♥♦ st

♥♥ st ♠tr E rrr t♥♦r♥♦♣♦í♠r♦ ϕ ② st♥ ♠ t♥♦ xo s♦r Rtunnel

0 400 800 12000.0

0.5

1.0

0 1000 2000

xo = 10

E=300 kPa E=700 kPa E=1000 kPa E=1500 kPa E=1800 kPa E=2500 kPa

Rtu

nnel /

Rtu

nnel

(0)

Stress (kPa)

baE=700 kPa

xo = 2.5

xo = 3.5

xo = 5

xo = 6.5

xo = 10 xo = 25

xo = 75

r rs s♠s t♦ ♠ó♦ ❨♦♥ ♠tr③ ♣♦♠ér s♥ r♥♦ E ♣r xoγ = 10 ② xoγ ♦♥ E = 700P s♦r Rtunnel

Rtunnel (0) s rs r♦♥ s s♥♦ ♦♥

λ = λN−H(P ; E)

♠tr③ ♣♦♠ér ♠♥♦ ♥t ♥t ♥trr③♥t s♣ró♥ ♠ ♥tr ♥s ♠♥♦ s ♦♥♦♥s r♦ ♦♠♦ sr ♠♥t ② t♠♣♦ ①♣♦só♥ ♠♣♦ ♠♥ét♦ Hcuring② rrr ♣♦t♥ s♦ t♥♦ tró♥♦ s♥♦ r♥tssst♠s r♥♦♠tr③

♥♠♥t t♦r ♥♦tr q ♠♦♦ srr♦♦ ♥♦ ♣r♣♥t♦s ♥①ó♥ ♥ rs♣st ♣③♦rsst ♦s sst♠s st♦♦♥stt② ♥ r♥ s♥t ♦♥ rs♣st ♦sr ♥ ♠♦s♦♠♣♦st♦s st♦♠ér♦s ♦♥ stró♥ s♦tró♣ r♥♦ ♣r♣r♦s♣♦r ♥str♦ r♣♦ ❬❪ ♦♥ ♥♦ s strtr♥ ♥s r♥♦

•

♣ít♦ s♣st ♣③♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥ st ♣ít♦ s rtr③ó rs♣st ♣③♦rsst ♠tr rr♥ ② s srr♦ó ♥ ♠♦♦ ♦♥sttt♦ q sr ♥tér♠♥♦s rts ♦s rst♦s ①♣r♠♥ts

♦s ♣rá♠tr♦s st ♦t♥♦s ♣♦r ♣ó♥ ♦r♠s♠♦ sr♥ q s ♣s♦♥s r♥♦ ♣rs♥t♥ ♠út♣s s♣r♦♥s r♠♥t♦♥s ♥tr♥s ♥ ♦♥♦r♥ ♦♥ ♦♥♣t♦ ♣s♦♥s♥ r ♥s ♦♠♣ts ♦ s ♠r♦rís ♦♥r♠♥q s ♣s♦♥s ♥♦ s♦♥ ♦♠♣ts s♥♦ q ♣rs♥t♥ rr♦s rrrs ♣rtís r♥♦ q ♥♥ ♣rs♥ s♣r♦♥s♣rs ♥tr ♦s r♣♦s ♣rtís ♥ s s♣r♦♥s sr ♠tr③ ♣♦♠ér s ♣r♦♥s ♠♦♦ srr♦♦ ♥♥q rsst♥ s r♦♥s ♦♥t♦rs r♥♦ s r s r♦♥s r♥♦ ♥tr s s♣r♦♥s ♣ ♦♥srrs ♦♥st♥t ♥♣♥♥t t♥só♥ ♠á♥ ♣ P ♦♥ ♥ ♣r♦①♠ó♥ ♠♦♦♣r ♥ ró♥ ♥♦ ♥ ♥tr rsst♥ étr ♠ R ② t♥só♥ ♠á♥ ♥① ♣ ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ P ♦s ú♥♦s t♦s rqr♦s s♦♥ r t♥só♥♦r♠ó♥ ♠tr③ st♦♠ér s♥ r♥♦ ♥♦ ② ♥ú♠r♦ strtrs ♦♠♥rs ♦♥♥t♦ ♣s♦♥s q trs♥ ♠str ♦♥t♥♦ ♦s tr♦♦s ♦♣st♦s N st ♦r♠ str ♦st♦s ①♣r♠♥ts ♣♦r ♠♦♦ srr♦♦ s r♣r♥ trs ♣rá♠tr♦s st A1 A2 ② A3 q s r♦♥♥ ♥ ♦r♠ rt ♦♥ rs♠r♦só♣s γ xo n ② l

♠♦♦ ♣r q ♣r ♥ sst♠ r♥♦♠tr③ ♦ ♣rá♠tr♦γ ♥♦ t♥ s♥s ♦ s ♠② ♣qñ s ♦♥♦♥s ①♣r♠♥tst③s ♣r ♣r♣rr ② s s♣r q xo st♥ ♠ t♥♦ tró♥♦ P = 0 s ♠② ♣♥♥t t♦rs ①♣r♠♥ts♦♠♦ sr ♦♥♥tró♥ r♥♦ ♠♥t ② t♠♣♦ ①♣♦só♥ Hcuring r♦ ♥trr③♠♥t♦ ♠tr③ t ♠♦♦ ♣rt♠é♥ q s xo s rt♠♥t ♣qñ♦ ♥ ♦r ♦ γ s rqrrá♥ r♥♦ ♥á♠♦ t♥só♥ ♠á♥ ♠②♦r ♣r ♦srr ♥ ♠♦rt♦ ♠②♦r R

P♦r út♠♦ ♦s á♦s ♥♠ér♦s r③♦s s♥♦ ①♣rs♦♥s s♠♣s ♦♥r♠♥ q t③r ♠trs ♦♥ ♠②♦r ♠ó♦ ❨♦♥ s①t♥ r♥♦ ♥á♠♦ t♥s♦♥s ♠á♥s ♣r sst♠ ♦q s ♥tt♠♥t s♣r ♣rtr ♦♥sr♦♥s áss rts ♦♠♣♦rt♠♥t♦ ást♦ ést♦s sst♠s

♣ít♦

s♣st ♠♥t♦rsst①♣r♠♥t♦s ♠♦♦ ②s♠♦♥s

tr ♠② r t s♠ rst ♥♠♥② ②s

♦ s

s♠♥ s r♥ ♦♠♣♦st♦s st♦♠ér♦s strtr♦s ♦♥stró♥ ♥s♦tró♣ r♥♦ étr♠♥t ♦♥t♦r ② ♠♥ét♠♥t t♦ ♣ ①r ♠♥t♦rsst♥ s r♠♦ ♥ s ♦♥t étr ♣♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ s♦ ♥ ♠♦♦ ♣③♦rsst♥ srr♦♦ ♥ ♣ít♦ s ♣r♦♣♦♥ ♥ ♠♦♦ ♦♥sttt♦ ♣r ♠♥t♦rsst♥ ♥s♦tró♣ rrs ♥ st t♣♦ ♠trs ♦s rst♦s①♣r♠♥ts ♣r rsst♥ étr R ♥ ♥ó♥ ♠♣♦♠♥ét♦ ♣♦ ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ H s♦♥ ♠② ♥ st♦s ♣♦r ♠♦♦ srt♦ ♣r s♦ ♠r♦♣rtís ♥ st♦ s♣r♣r♠♥ét♦ 34❬❪ ♥s ♠♥ét♠♥t r♥t r♦ s ③ s ♣rs♥t♥ s♠♦♥s ♣r rs♣st ♠♥t♦rsst s ♦♥ ♦tr♦sr♥♦s ♥ st♦ s♣r♣r♠♥ét♦ ② ♠♥ét♠♥t ♦q♦♠♦str♥♦ ♥ ♣rtr ♥♥ ♠ó♦ ❨♦♥ ♠tr③② ♠♥t③ó♥ stró♥ r♥♦

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥tr♦ó♥

♦s ♦♠♣♦st♦s ♠♥ét♦s ♦r♠♦s ♣♦r s♣rs♦♥s r♥♦s ♠♥ét♠♥t t♦s ♥tr♦ ♥ ♠tr③ ♦rá♥ stá♥ r♥♦ ♥ t ♥ t♥ó♥ r♥t s ♣♦s ♠♦strr r♥st♦s ♠♥t♦rsst♦s ♠♦ ♥ rsst♥ étr ♣♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣r ♥t♥ss ♠♣♦ ♠♥ét♦♠♦rs ❬ ❪ s ♠trs ♠s ♦♠ú♥♠♥t t③s s♦♥ ♦s ♠♦s s♦s♦s ♥tr♦ ♦s q s ♥②♥ ♦s s ②♦s st♦♠ér♦s ♦s r♥♦s ♣♥ ♦♥str str♦♥s s♦tró♣s♦ ♥s♦tró♣s ♣rtís ♠♥éts ♠② r ♥tr③ ís♦qí♠ ② ♠♦r♦ó ♥♦ ♥♦tó♥ ♥ ♣rs♥t tr♦♦s ♦♠♣♦st♦s ♦r♠♦s ♣♦r s♣rs♦♥s ♥s♦tró♣s ♥ ♥ ♠tr③ st♦♠érs s♦♥ rr♦s ♦♠♦ ♣♦r ss ss ♥ ♥és ♦s s ♣♥ sr♣r♣r♦s ♦♥ ♣♦r r♦ ♠tr st♦♠ér♦ ♥ ♣rs♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥ ♣rtr s s ♣rtís ♠♥étss s♣rs♥ ♥ ♣♦♠ts♦①♥♦ P ♥♦ ♣♦í♠r♦ t♦í s♥♥tr ♦ ② ♦ s ♦ r♦ tér♠♦ ♥ ♣rs♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥t♥s Hcuring sst♠♥ r♦ s ♥ ♣st♦ q s ♣rtís ♠♥éts s r♣♥♦r♠♥♦ strtrs t♥rs ♥tr♦ ♠tr③ ♣♦♠ér ♦r♥ts♣rr♥t♠♥t ♥ ró♥ Hcuring r ts ♦r♠ó♥ ♦s sst♠s ♣♦r st í ♥ s rr♥s ❬ ❪ ② ♥♦s ♣ít♦s ♣r♦s ♣rs♥t ss tr í ♣r ♣r♣ró♥ sst♠s s s♦ó♥ ♣♦í♠r♦ ♥ ♥ s♦♥t ♦át ♦♣♦r ♠♣♦ ♥ t♦♥♦ ♥ r♥♦ ♠♥ét♦ s ♥♦r♣♦r ♠③ ② s♦♥t s ♦♠♣t♠♥t ♣♦r♦ ♦ ♥ ♣rs♥ ♥ ♠♣♦ ♠♥ét♦ ❬ ❪ st♦s sst♠s ♦♠♦ sí t♠é♥ q♦ss♦s ♥ s ♦s ♣♥ ♣rs♥tr rs♣st ♠♥t♦rsst ♥♦ ♥ s r ♠♦ ♥ s rsst♥ étr ♣♦r ♣ó♥ ♥ ♠♣♦♠♥ét♦ ①tr♥♦ H Pr ♦srr rs♣st r♥♦ t③r sr ♥♦ só♦ ♠♥ét♠♥t t♦ s♥♦ t♠é♥ ♦♥t♦r ó♠♦ ♣③♦rsst ♠♦ rsst étr ♦♥ t♥só♥ ♠á♥ s♦ r♣♦rt ♣r ♦s sst♠s ❬ ❪ ② ♠♦ ♥ ♣ít♦ ♥ ♣rtr t③r s rs♣st s ♥s♦tró♣

♦♠♦ s ♥♦ ♦♥ ♥tr♦r ♦s sst♠s ♣r♣r♦s ② rtr③♦s ♦rrs♣♦♥♥ s ♦♥ ♣s♦♥s ♦r♠s ♣♦r ♦♠r♦♥s ♣rtís r♥♦ ♠♥ét♦♦♥t♦rs q ♣rs♥t♥♥s♦tr♦♣í étr ♦t ♥ ♣ít♦ s ♦ró ♥ ♦r♠s♠♦ ♦♥sttt♦ ♣r rs♣st ♣③♦rsst ♦s sst♠s ♠♦♦ s s ♥ ♦♣♠♥t♦ ♥tr s ♣r♦♣s ásts ♠tr ② ♥ó♠♥♦ t♥♦ tró♥♦ ♥tr r♦♥s ♦♥t♦rs

♦tr r r♥ó♥ ♥tr H ② Hcuring

♥tr♦ó♥

r♥♦ s♣rs ♣♦r ♥ ♥ ♣ ♠tr③ ♣♦♠ér s♥t st ♥ó♠♥♦ ♦♥stt② t♦r ♠t♥t ♣r ♦♥ó♥ étr trés ♠tr st ♦r♠ ♠♦♦ ♦♥sr s♠♥ó♥ st♥ t♥♦ tró♥♦ ♥♦ s ♦♠♣r♠ ♣♦r ♣ó♥ ♥ t♥só♥ ♠á♥ ♥① ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦

♦♠♦ s strrá ♠s ♥t st♦s sst♠s ♠str♥ ♥♦ só♦ ♣③♦rsst s ♥♦ t♠é♥ ♠♥t♦rsst ♥ ♥str♦ ♦♥♦♠♥t♦ ♥ trtr ♥ s♦ r♣♦rt♦ ♦s ♠♦♦s q t♦♠♥ ♥ ♥t ♦s t♦s ♠♥t♦rsst♦s ♥ ♦♠♣♦st♦s ♦♥ r♥♦s ♠♥t♦t♦s s♣rs♦s♥ ♠trs ♥♦♠♥éts ♦s ♠♦♦s s♦♥ t ② ♦♦r♦rs❬ ❪ ② ❬ ❪ sst♠ ♠♦♦ ♣♦r tt ♥♦ stá ♦r♠♦ ♣♦r ♣s♦♥s ♥ ♠♦ r♥♦ s ♦r♥③♦♠♦ ♥ ♦r ♣rs q r strtr ♥tr♥ ♦♥ ♣r stá ♦r♥ ♥ ♦r♠ ♦♥st ② r♣rs♥t ♥ ♣rtí r♥♦s♣r ss ♥s ♥♠ts ♣♦r ♥ ♣ ♣♦í♠r♦ s♥♦ts r♥s ♥tr ♦s s qí st♦s ② ♦s ♦rs ♣rs② r♦♥ sts ② sts ♥ ♣ít♦ r ó♥

♦tr♦ ♠♦♦ s srr♦♦ ♣♦r q♥ ♦ t③ ♣r str rs♣st ♠♥t♦rsst ♥ ♦♠♣♦st♦ ♦♥ ♠tr③ s ♥♣♦í♠r♦ t♣♦ s♦①♥♦ ♦♥ ♦ ♦♥t♥♦ t s♦♥♦ sí ♠tr③ s ♠♦ ♦♠♦ ♥ ♦ s♦s ♥ r ♥ s♦♦ st♦♠ér♦ ♥♦r♣♦r♥♦ ró♥ ♥tr r♥♦ ② ♠tr③ ♦♠♦♦♥s♥ ♠♦♠♥t♦ r♥♦ sr ♠♥t③♦ ♣♦r ♠♣♦♠♥ét♦ ①tr♥♦ s♣t♦ s ♥tr♦♥s ♥tr ♣rtís r♥♦ t③ ♥ ♠♦♦ s♠♣♦ ♥tr♦♥s t♣♦ ♠♥ét♦ ♣♦r ♣♥t sst♠ s♦s♦ st♦ ♣♦r r ♥♦t♦r♠♥t ♦♠♣♦st♦ st♦♠ér♦ ♥st♦ ♥ ♣rs♥t ss ♣♦r ♦ q s♠♦♦ ♥♦ s ♣ ♣r sr♣ó♥ rs♣st ♠♥t♦rsst ♥str♦ rr♥

♥t♠♥t ♥②♦ ② ♦♦r♦rs ❬ ❪ ♥♦r♠r♦♥ ♥♦r♠s♠♦ ♣r srr ♠♥t♦stró♥ ♦r♠ó♥ ♥ ♣♦r♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥ sst♠s ♣♦♥r♠♦s♣♦r ♦r q ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣ ♥r♥ ♦r♠ó♥ ♠r♦só♣ ♠str ♦♠♣ñ ♥ r♦r♥♠♥t♦ r♥♦ ♦ s♣♦só♥ ♠♦♦ ♥②♦ t ♣♦rísr t♦ ♣r srr rs♣st ♠♥t♦rsst ♦s sst♠s ♥ st♦ ♥ t♦ ♥ ♦ ♠♦♦ sst♠ ♥♦ s ♥♥tr ♦r♠♦ ♣♦r ♣s♦♥s s♥♦ ♣♦r ♥ rr♦ ♣ró♦ ♣rtís ♦♥stró♥ ♥s♦tró♣ Pr r ♣ó♥ ♠♦♦ ♥②♦ ♠♦s r③♦ ♥ tr♥srs♦ ést ss ♥s ♠♦♦♥s ♦♥ ♦t♦ ♣tr♦ ♥str♦ sst♠ ①♣r♠♥t s♥ rst♦s stst♦r♦s sí ♠♦s ♦sr♦ q ♦ ♠♦♦ sst♠ ♥♦t♦r♠♥t

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

rs♣st ♠♥t♦rsst ♥ ♦s s ♦r♠♦s ♣♦r ♣s♦♥sst ♦ ♣♦s♠♥t r ♥ q s strtrs ♦♠♣s r♥♦♣rs♥ts ♥ rr♥ r♥ sst♥♠♥t strtr♦♥t♠♣ ♣♦r ♠♦♦ ♥②♦ ② ♦♦r♦rs ♦♠♦ sí t♠é♥♣♦rq s ♥tr♦♥s ♠♥éts ♥tr s strtrs ♦♠♣s r♥♦♥♦ ♣♥ sr ♦♥t♠♣s ♠♥t ♥ tér♠♥♦s ♥tr♦♥s ♣♦♦s ♠♥ét♦s ♣♥ts ♦♠♦ s sr ♥ ♦s tr♦s ♥②♦t

st ♦r♠ ♣r♥♣ ♦t♦ ♦♥t♥ó♥ s ①t♥r ♦r♠s♠♦ srr♦♦ ♥ ♣ít♦ ♣r ♣rs♥tr ♥ ♠♦♦ ♦♥sttt♦ rs♣st ♠♥t♦rsst rrs ♥ s ♦r♠♦s ♣♦r ♣s♦♥s ♠♥éts♦♥t♦rs ♦ ♦♥♦♥s ♥tr♦ s♠r♦ ♣r♥♣ tó♣♦ sr ♥st♦ s r♦♥ ♦♥ ♦♠♣r♥só♥ ♦r♥ ♦s r♥s t♦s ♠♥t♦rsst♦s ♦sr♦s ♣r st♦ssst♠s ♦♠♣st♦s ♠♦♦ s ♥r ♣r t♦ s ♦ ♦♥ó♥ sst♠ st♦♠ér♦ 34❬❪P 4.2% s♠♥♦♥ ♦♠♦ rr♥ ♥á♦♠♥t ♦ r③♦ ♥ ♣ít♦ ♥tr♦r ② s t♦♠♦ ♦♠♦ t ♣r ♣rs♥tr ♥ ♦♥①ó♥ ♦♥♣t♥tr s rs ② ♣rá♠tr♦s tór♦s ② ss ♠♥ts ♦srs ①♣r♠♥t♠♥t ♦♥tr②♥♦ ó♥ ♣rá♠tr♦s rsts r♥t s♠ó♥ rs♣sts ♠♥t♦rssts rs♣st ♠♥t♦rsst rr♥ st s♥♦ ♠♦♦ srr♦♦

♦♠♣♠♥tr ♠str ♦s sí♠♦♦s t③♦s ♣r ♦r♠s♠♦ ♠♥t♦rsst♥

♥ó♥ ♦s sí♠♦♦s s♦s ♥ ♠♦♦ ♠♥t♦rsst♥ ♥ ♦s sst♠s

í♠♦♦ ♥ó♥H ♠♣♦ ♠♥ét♦ ♣♦P ♥só♥ t♦tPmag ♥só♥ ♠♥étPmec ♥só♥ ♠á♥M ♥t③ó♥ r♥♦Ms ♥t③ó♥ stró♥ r♥♦Mr ♥t③ó♥ r♠♥♥ r♥♦ ♦q♦ ♠♥ét

♠♥tHc ♠♣♦ ♠♥ét♦ ♦rt♦ r♥♦ ♦q♦ ♠♥ét♠♥

tH‡ ♠♣♦ ♠♥ét♦ rtríst♦ r♥♦ ♥ st♦ s♣r♣r

♠♥ét♦

♦♦

♦♦

♦r♠s♠♦ rt

s♣♦♥rá q ♦♥♥tró♥ r♥♦ s t q s ♥r♥ strtrs t♥rs ♥ ♠tr③ ♥s ♥ ♥ ró♥ ♦♥ ♥ ♣qñs♣rsó♥ ♥r s♣♦♥ t♠é♥ q ①st ♣r♦ó♥ ♥tr s♣s♦♥s s s ♦r♠♥ strtrs ♦♠♥rs q ♦♥t♥ srs ♦♣sts ♠str ♣r♦ ♥♦ ② ♣r♦♦♥ ♥ ró♥♣r♣♥r s ♦♠♥s ♥ ♦trs ♣rs s ♦♥♦♥s ①♣r♠♥ts t③s sr♥ ♦♥ó♥

♦♠♦ s t♦ ♥ ♣ít♦ ♥tr♦r ♦♥sr♥♦ strtr ♥tr♥ ♠tr r♥♦ ♥ ♦s sst♠s r ♥♠str ♦ ♠tr ♥tr ♦s tr♦♦s ♦r♦ ♠tá♦ rsst♥ étr ♦ sst♠ t♥ trs ♦♠♣♦♥♥ts ♣r♥♣s rsst♥ étr s♦ ♦♥ t♥♦ tró♥♦ ♥tr ♦s r♦♥s♦♥t♦rs s♣rs ♣♦r ♥ ♣ s♥t ♠tr③ ♣♦♠ér Rtunnel rsst♥ étr ♥ ss r♦♥s ♦♥t♦rs RCR ② rsst♥ ♦♥tt♦ étr♦ ♥tr s r♦♥s ♦♥t♦rs ② ♦str♦♦s ♦r♦ RE−CR

sí ①t♥♥♦ ♠♦♦ srr♦♦ ♥ ♣ít♦ ♥tr♦r ♣r ♥r t♦ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ rsst♥ étr s♠strs s ♥tr ♦s tr♦♦s ♠tá♦s ♣ srrs♥ tér♠♥♦s rts ♠♥t ♥ ①♣rsó♥ s♥ q ♦rrs♣♦♥ ♥ rt♦ q♥t N rsst♥ ♥ ♣r♦ ♥ t♦ ♣rtr s ①♣rs♦♥s ② rst

R (Pmec, H) =n− 1

NRtunnel +

n

NRCR +

2

NRE−CR

♦♥ H s ♠♣♦ ♠♥ét♦ ♣♦ Pmec s t♥só♥ ♠á♥ ♥① ♣ N s ♥ú♠r♦ strtrs ♦♠♥rs r♥♦ q ♦♥t♥ ♦s tr♦♦s ② n s ♥ú♠r♦ r♦♥s ♦♥t♦rs ♥ ♥ s strtrs ♦♠♥rs st ♦r♠ s t♥♥ n − 1 r♦♥s t♥♦ t♥♥ st♣s ♥ ♥ ss strtrs ♦♠♥rs ♥ ♦r♠s♠♦ srr♦♦ ♥ st ♣ít♦ s s♣♦♥rá q r♦r♥③ó♥ r♥♦ ♣♦r ♣ó♥ H ♥♦ ♠♦ N ♥ n

♥ ♣rtr ♣r sst♠ rr♥ 34❬❪P 4.2% s st♠♦ N = 660± 50 ♠♥t ♠r♦s♦♣í ♠♥trs qs r♣r♦ ♦r n = 98 ± 9 ♠♥t ♠♦♦ rs♣st ♣③♦rsst és ♣ít♦ ♣r ♠ás ts P♦r ♦ ♦ ♥ ♣ít♦ qr r♦ q N ♣♥ ár ♠str ♥③ A ♠♥trs q n ♣♥ ♦♥t s♣s♦r ♠str t♥só♥ ♥ Lo ♥ ♣rtr ♣r s ♠♦♥sr③s s♦r rr♥ A = 0.8 ♠2 ② Lo = 2.5 ♠♠

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

qr r♦ t♠é♥ q N ♥♦ s ♥ú♠r♦ ♣s♦♥s s♥♦ ♥ú♠r♦ ♦♠♥s ♣r♦♥ts ♥ s ♦r♠ ♣♦r ♥♠r♦ss♣s♦♥s

♥ st ♣♥t♦ s ♠♣♦rt♥t ♠♥♦♥r q ♥♦ s s♣r q ♥ ♠♥s♠♦ s♦ ♥ r♦tó♥ ② ♥♠♥t♦ s ♣s♦♥s ♥♦♣♦r ♣ó♥ ♠♣♦ ♠♥ét♦ H r♥ ♥ sr♣ó♥ ♣r♦♣♣r rs♣st ♠♥t♦rsst ♠tr ♥ ♣r♠r r ♣rsst♠s ♥ ré♠♥ s♣r♣r♠♥ét♦ ♠♦♠♥t♦ ♠♥ét♦ ♥t♦ ♥♦ ♣♦r H ♥ ♣s♦♥ µ s ♣r♦ H ❯♥ ♥r♠♥t♦ H ♥♦ ♠♦ ró♥ µ q ♣r♠♥ s♠♣r ♣r♦ H P♦r♦ t♥t♦ t♦rq ♥tr H ② µ s ♥♦ ② ♥♦ ①st r③ ♠♣s♦r ♣r r♦tó♥ s strtrs ♣s♦t♥rs ♦ ♣ó♥ H ♥ ré♠♥ s♣r♣r♠♥ét♦ ♥ s♥♦ r ♣r sst♠s ♦♥ r♥♦s ♥ st♦ ♠♥ét♦ ♦q♦ ♦♠♦ sr ♠trs rr♦♠♥ét♦s♦♥ ♣ ♣rs♥trs ♥ t♦rq ♥♦ ♥♦ ♥♦ rst tr q ♥ r♦tó♥ ② ♥ó♥ s ♣s♦♥s ♦ ♣ó♥ H ♣r♦③♥ s♠♥ó♥ st♥ ♥tr ♦s ♣s♦♥s ♦♥sts sú♥ s♣♦♥r ♥ stó♥ ♥ q ♦s ♥tr♦s r♦tó♥ s ♦s r♦♥s ♦♥t♦rs ♦♥sts t♥rs ♥♦ stá♥ ♣rt♠♥t ♥♦ss r q rt q ♥ ♦s ♣♥t♦s ♥♦ s ♣r ró♥ Hcuring ♥ s s♦ ♥ ♣♦str♦r r♥ó♥ ♣♦r ó♥ ♥ H ①tr♥♦♠♥t st♥ t♥♦ tró♥♦ ♥tr s r♦♥s ♦♥t♦rss♠♥②♥♦ ♦♥t ♦ s ♦♥trr♦ ♦ ♦sr♦ ①♣r♠♥t♠♥t st s♠♥ó♥ ♥ ♦♥t t s♦ r♣♦rt♥ ♥♠r♦s♦s st♦s ①♣r♠♥ts ② tór♦s ❬ ❪

♥♥ ♠♣♦ ♠♥ét♦ s♦r ♥♦ ♦s tér♠♥♦s s ú ♦♥t♥ó♥

sst♥ ♦♥tt♦ étr♦ ♥tr s r♦♥s ♦♥

t♦rs ② ♦s tr♦♦s(2NRE−CR

)

♦s tr♦♦s ♠tá♦s t③♦s t♠♥t ♦♠♣st♦s ♣r♥♣♠♥t ♣♦r ♦r♦ ♥♦ s ♠♥t③♥ ♣♦r ♣ó♥ H P♦r ♦ t♥t♦ s♣♦♥r♠♦s ♥ ♣rs♥t ♦r♠s♠♦ q rsst♥ étr ♣♦r ♦♥tt♦ tr♦♦♣s♦♥ ♥♦ ♠ ♣♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦ ♦r RE−CR ♥ s ♦♥♦♥s ♠ó♥ rs♣st ♠♥t♦rsst ♣ st♠rs t③♥♦ ♠♦♦ ♦♠ ❬❪ P♦r ♠♣♦ ♥ s♦ ♣rtr rr♥34❬❪P ♥♦ s♦ rsst étr r♥♦ rsst étr ♦s tr♦♦s á♠tr♦ ♠♦ s ♣s♦♥s ② r③ r♥♦ s st♠ 2

NRE−CR ∼ 10−4Ω á♦t♦ ♠♥t ♣ ♦♥strs ♥ ó♥ st rst♦ s ♥r③ ♣r t♦♦ ♠tr r♥♦ ♦r♠ t q ♠♦♦s♣♦♥ q rsst♥ ♣♦r ♦♥tt♦ ♥♦ t♥ ♦♥tró♥ ♣r

♦♦

rs♣st ♠♥t♦rsst

sst♥ étr s r♦♥s ♦♥t♦rs r

♥♦(nNRCR

)

s♥t ♣s♦ s r ♦♥tró♥ RCR R ② t♠é♥ ♣♦s ♣♥♥ RCR ♦♥ H RCR s rsst♥ étr ró♥ ♦♥t♦r r♦ ♠r♦♣rtís r♥♦ ♥tr♦ ♥♣s♦♥ ♦♠♦ s srr♦♦ ♥ ♣ít♦ ♥♦ s♦sr♦s stá s♣r♦ s♥t ♣♦r ♥ ♣ ♣♦í♠r♦ ♥tr♦ ♥ r♦ ♥ ♣rtí r♥♦ ♣ str ♥ ♦♥tt♦ rt♦ ♦♥♦tr ♣rtí ♦ str s♣r♦ ést ♥ s♦ rr♥ rsst♥ RCR r♣rs♥t rsst♥ ♥ r♦ ♠r♦♣rtís 34❬❪ ♥tr♦ ♥ ♣s♦♥ Pr ♦ sst♠ r♣rs♥tó♥ ♦ étr♦ trés ♥ ú♥♦ r♦ ♦rrs♣♦♥ st♦ ♦t♥♦ tr♦♥s s r♦♥s ♦ ♦♥t♥♦ ♣t ♠tá♥ ♠r♦♣rtí r♦♥s ♥á♦s ♥ ♦tr ♠r♦♣rtí ssr♦♥s ♥ s ♠r♦♣rtís ② s s♣♦♥♥ ♥ ♦♥tt♦ r♥♦ s♥♣ ♣♦♠ér ♥tr s P♦r ♦ rst r③♦♥ s♣rr ♦s s♥t st♥ t ♣r st♦ ♦ t♥♦ tr♦♥s ♥tr ss ♠r♦♣rtís ♥♦ rí ♥♦ ♥ ♠♣♦ ♠♥ét♦ H s ♣♦ st♦r♠ ♥♦ s s♣r q s ♣rs♥t♥ ♦r♠♦♥s ásts ♦ ♣ásts r♦♥s ♦♥tt♦ ♥tr ♠r♦♣rtís r♥♦ ♦♠♦ ♦♥s♥ t♥s♦♥s ♠♥éts ♥s ♣♦r H ♦♥ ♥♥ s♦r RCR ② ♦s t♦s ♠♥t♦rsst♦s ♦ ♣♦r③ó♥ s♣♥s tró♥♦st♠♣♦♦ s♦♥ r♥ts ♣s s ♣rtís ② ♥♦ s♦♥ st♥s

st út♠♦ t♦ ♣♦r③ó♥ s♣♥s tró♥♦s ♠r ♣rtr t♥ó♥ ①st♥ ♣♦♥s rs♣t♦ t♦s ♣♦r③ó♥ s♣♥s ♥ sst♠s ♥ st♦ s♣r♣r♠♥ét♦ ❬❪ ② ♥♦s ♣r ♥♥♦♦♠♣♦st♦s ❬❪ st ♥ó♠♥♦ ♣ ♦rrr ♥tr ♦s ♣rtís s r♦ ♣♦r③ó♥ s♣♥s tró♥♦s s r♥t ♥s♠ tr♦é♥♦ ♦ tr♦♥♦s ♥t♦♥ ♠♥t ♠♥s♠♦ ♥♦r♦s s♥t Pr♠r♦ s ♥ ♦♥♦♦ s♣♥tró♥ q rrr t♥♦ tró♥♦ s ♥ ♣rtí ♦tr ♣rtí s♠♥②♥♦ r r♥ ♣♦r③ó♥ s♣♥s ♥tr ♠s ♦q t♥♦ tró♥♦ s ♦r♦ ♥♦ ♦s tr♦♥s ♥♦ ♠♥ sst♦ s♣í♥ Pst♦ q ♠♣♦ ♠♥ét♦ H t♥ ♣♦r③r ♦ss♣♥s tró♥♦s ♥t♦♥s s ② ♣rs♥t♥ ♥tr③ ís♦qí♠r♥t ♥t r♥ ♣♦r③ó♥ s♣♥s ♥tr ♠s rr ♦♥ H ♦♥ ♦♥s♥t s♠♥ó♥ rrr t♥♦tró♥♦ ② s♠♥ó♥ RCR Pr q ♦ ♠♥s♠♦ t♥ rs ♥sr♦ t♥r r♥t ♣♦r③ó♥ s♣♥s tró♥♦s ♥tr ② q ♥♦ s s♦ ♦ st♦ ♦ q s ♠r♦♣rtís ② s♦♥ ♥st♥s s ♣♥t♦ st s ♦♠♣♦só♥ qí♠ ②

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

rtrísts ♠♦r♦ós ♦ ♥♦ s s♣r♥ t♦s ♣♦r③ó♥ s♣♥s tró♥♦s ♣♥♥ts H s♦r ♦♥ó♥ étr ♣st♦q ♥♦ s trt ♥ ♥s♠ tr♦é♥♦ rí ♣♦s q ② sr♥ st♥ts ♥ s♦ q r r♥s ♥ t♦s s♣rs ♦s s ♥r♥ ♥ s♠trí Pr♦ ♦s t♦s s♦♥ t♦r♦s ②♣r♦♠♥ r♦ ♥♦ s ♦♥sr♥ ♦ r♦ ♠♥♦s ♠ró♥ tr♦♥s trés ♥ ♥ú♠r♦ stíst♠♥t t♦ ♣rtís ♦r♥ 105 ♦♠♦ s ♦ ♥ ♣ít♦

♥ rs♠♥ ♣♦r s ♦♥sr♦♥s rr ♠♥♦♥s ♥♦ s ♥②♥♥ ♣rs♥t ♠♦♦ ♣♦ss t♦s ♠♥t♦rsst♥ s♦r RCR t♦♦s ♠♦♦s ♣ótss s ①♣♦ró ♣r r♥♦ ♠r♦♣rtís 34❬❪ ♣♦r ♠ó♥ ♦♥t étr ♦ ♠tr ♣♦♦ ♦ r♥ts ♦♠♣rs♦♥s ♠á♥s ② ♠♣♦s ♠♥ét♦s ♠♥t s♣♦st♦ srt♦ ♥ ó♥ ② sq♠t③♦ ♥♠♥t ♥ r ♣r tr t♦r ♥tr♣rtó♥ ♦s rst♦s srs ♦rr♥t♣♦t♥ IV rtrísts r ♠str♥q rs♣st étr ♣♦♦ 34❬❪ s ó♠ ♣r t♦s s♦♠♣rs♦♥s ♠á♥s ♣s rsst♥ étr r ♥♦ ♣♦♦ s ♦♠♣r♠♦ ❬r rs ② ❪ st♦ ♣ strs ♥tér♠♥♦s ♦r♠ó♥ ♠♥♦s ♣r♦t♦s trés ♦s s ♦rr♥t étr ♣ r s ♥ tr♦♦ ♦♣st♦ ❬ ❪ r♥t s♦♠♣rsó♥ ♣♦♦ rsst♥ étr ♠♥t ♣r♦ ♣r♠♥ ♦rs ♦s ♣♦s♠♥t ♦ ♦r♠ó♥ ♣ást rátrrrrs r♥t ♦♠♣rsó♥ ❬❪ ♥ s rs ② s ♦t♥♥s ♠s♠s rs ♥♣♥♥t♠♥t ♠♣♦ H ♣♦

r ♠str q ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥♦ ♠♦ rsst étr ♣♦♦ r♥♦ ♣r ♥♥♥ s ♦♠♣rs♦♥s t③s ♦♠♦ s s♣r ♥ s s ♦♥sr♦♥s sts ♠s rr s r ♥♦ s s♣r♥ ♦♥tr♦♥s tér♠♥♦nNRCR t♦ ♠♥t♦rsst♦ ♥ t♠♣♦♦ s ♦s ♥②♥ ♥ ♦r♠s♠♦ qí srt♦ st s♣♦só♥ s ♥r③ ♣r qr t♣♦ r♥♦t③♦ ♦tr t♠é♥ q s ♦♥tró♥ rsst♥ étr t♦tR ♣ ♦t♥rs ♣rtr ♠♦♦ rs♣st ♣③♦rsst r♣ít♦ ♦t♥é♥♦s ♦r n

NRCR ∼ 0.5Ω ♣r s♦ 34❬❪♥ P ♥ ♦trs ♣rs ♥ st ♠♦♦ rsst♥ ró♥♦♥t♦r RCR s ♦♥sr ♥♣♥♥t H ② s ♦♥tró♥ rsst♥ étr t♦t s ♦t♥ ♠♦♦ rs♣st ♣③♦rsst

♦♦

Stress

Res

ista

nce

()

Stress (kPa)

Compression Decompression

0 2 4 60

1

2

3

P=8.7 MPa

P=1.6 MPaR (

)

H (kOe)

P=156 kPa

r rs rtríst IV ♣r r♥♦ 34❬❪ ♣rr♥ts ♦♠♣rs♦♥s ♣♦s sst♥ étr r♥♦ ♥ ♥♦ ♦♠♣rsó♥s♦♠♣rsó♥ sq♠ sst♠ ①♣r♠♥tt③♦ ♣r ♠ó♥ t♦ t♥só♥ ② ♠♣♦ ♠♥ét♦s♦r ♦♥t ♣♦♦ 34❬❪ tr♦♦s ♦r ♣♦♦ ♠str ♠♦ ♦♥t♥♦r rs♥ rí t♦ ♠♣♦♠♥ét♦ s♦r rsst♥ étr r♥♦ r♥ts ♦♠♣rs♦♥s

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♦♥tró♥ ♣r♥♣ t♦ ♠♥t♦rsst♦ (Rtunnel)

s♦s ♥ ♥áss r③♦ ♠♦♦ srr♦♦ s♣♦♥ q ♣r♥♣ ♦♥tró♥ t♦ ♠♥t♦rsst♦ rs ♥ Rtunnel ♥ ♣ít♦ s ♠♦stró q ♣r ♦rrt sr♣ó♥ t♥♦ tró♥♦ trés ♦s s♣rs ♠sérs ♦rrs♣♦♥♥ts ③ ②♦ ♦s r♦♥s ♦♥t♦rs s ♥sr♦ ♣r ♥ ♥r③ó♥ ♦r♠s♠♦ ♠♠♦♥s ❬❪ ♣r t♦♣♦♦í st ♥r③ó♥♣r♠t ♦t♥r ①♣rsó♥ q r s♦ó♥ ♥ít

♥ ♣r♥♣♦ ♦s ♣rá♠tr♦s ϕ ② x ♣♥ sr ♥♥♦s ♣♦r H t♦s ♠♥rs ♥t t♦ H s♦r ϕ strs♦♦ t♦s ♣♦r③ó♥ s♣♥s tró♥♦s q s♦♥ sst♠♦s ♥ s ssó♥ ♥ ó♥ ♣r ♥♦ s s♣r♥ ♦s t♦s♣st♦ q ♥♦ ② tr♦♥t♦♥s ♥ ♦s ♠tr ② ♦ ♥♦ s♦sr ♥♥ H ♥ ♦♥t étr ♣♦♦ r♥♦r s str t♠é♥ q ♥ rr♥ ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♣r♣♥r ♦ étr♦ st♦s ♣r♣♥r ró♥ ♦r♥tó♥ ♣rr♥ s strtrs ♦♠♥rs r♥♦ ♥♦ r ♥♥ú♥ t♦ ♠♥t♦rsst♦ ❬❪ ♥st s♦ ♣ó♥ ♠♣♦ ♠♥ét♦ t♠é♥ ♣♦r③ ♦s s♣♥stró♥♦s ❬❪ ♣r♦ ♥♦ s ♦sr♥ ♠♦s R ♦♥ H st ♦stá ♥ ♦♥♦r♥ ♦♥ ♣ótss s♣rr ♦s t♦s ♣♦r③ó♥ s♣♥s ♥ ♦s sst♠s ♥ s t♦s sts ♦♥sr♦♥s ♣rs♥t ♠♦♦ s♣♦♥ q ϕ s ♥♣♥♥t H

st ♦r♠ t♦ ♠♥t♦rsst♦ s ♥♦r♣♦r trés ♣♥♥ x s♣ró♥ ♠ ♥tr r♦♥s ♦♥t♦rs ♦♥ H ♦srrr♠♦s st ♥ó♠♥♦ ♦♠♦ ♦♣♠♥t♦ ♠♥t♦ást♦ ♦s ♦♥♣t♦s trás st ♥ó♠♥♦ s ♣rs♥t♥ ♦♥t♥ó♥ ♣r ♥♠♣♦ ♠♥ét♦ ①tr♥♦ s r♦♥s ♦♥t♦rs ♥tr♦ ♥ ♣s♦♥ r♥♦ s ♠♥t③♥ ♥r♥♦ ♥ ♥tró♥ ♠♥ét trt ♥tr s st ♥tró♥ r st♥ x st r r③ást s♦ ♦♠♣rsó♥ ♣ s♥t ♣♦í♠r♦ q s♣rs r♦♥s ♦♥t♦rs ♥trt♥ts sí ♣♦r ♣ó♥ H s ♦t♥♥ ♠♦ ♥ s♣ró♥ ♥tr s r♦♥s ♦♥t♦rs t♠át♠♥ts ♣r♦♣♦♥ x = x (M) ♦♥ M ♠♥t③ó♥ s r♦♥s ♦♥t♦rs♥ st s♥t♦ r③ ♠♥ét tú ♥ ♦r♠ ♥á♦ ♥ ♠♣♦ r③ ♠á♥ ♦sss ② ♦♦r♦rs ❬❪ s rr♥ st ♥ó♠♥♦♦♠♦ ♣rsó♥ ♠♥ét ♦ t♥só♥ ♠♥ét trtá♥♦♦ ♦♠♦ ♥ t♥só♥♠á♥ ♦ ♦♥srr♠♦s q t♥só♥ t♦t ♥ ♥ ró♥♥ ♣rtr ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ ♥ P ♥ ♥ ♠str s♦♠t t♥só♥ ①tr♥ ♠á♥ ② ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ H t♥ ♦s ♦♠♣♦♥♥ts t♥só♥ ♠á♥ Pmec ② t♥só♥ ♠♥ét Pmag ♥ st♦s ♠♥t♦rsst♥ sst♠s t♥só♥ ♠á♥ ♥ ró♥ ♦r♥tó♥ r♥♦ Pmec s

♦♦

♠♥t♥ ♦♥st♥t ♥ ♥ ♦r rtrr♦ P ⋆ és ó♥ s rPmec = P ⋆ = constante

P = Pmec + Pmag = P ⋆ + Pmag

♦ ♣ó♥ P ⋆ ② ♥ s♥ Pmag ♦♥t ♠str ♥ ró♥ P ⋆ s rr ♦♠♦ L⋆ ♦ ss♥t♣ó♥ H ♥ ró♥ ♣r♦ ♥ ♠♥t③ó♥ ♥ r♥♦M q ♥r ♥ t♥só♥ ♦♥ q ♦rrs♣♦♥ t♥só♥ ♠♥ét♥ s ró♥ Pmag Pmag r♣rs♥t ♠♦ ♥ t♥só♥ t♦t t♥só♥ s♦r ♠str s P ⋆ ♥ s♥ H ② ♠ P ⋆ + Pmag

♥♦ s ♣ ♥ ♠♣♦ ①tr♥♦ H ♣ró♥ Pmag ♥ ♥♠♦ ♥ ♦♥t ♠r♦só♣ sst♠ ♥ ró♥ L L =L (Pmag) s ♦r♠ó♥ ♠♥t♦ást ♣ ♦ ♥♦ sr ♦sr ♦♦s♥♦ ♣♥♥♦ sst♠ ♦ st♦ ② s ♦♥♦♥s ♦♥t♦r♥♦♠♣sts

yo y*y(P*+Pmag)

P* P* Pmag

H

a b c

r str ♠r♦só♣ ♠tr③ ♣♦♠ér s♥ r♥♦ ♦r♠ ♣rs♠át rt♥r ♦♥t yo ♥tr ♦s ♣③s ♠♥t③s ♣s♦ s♣r ♦♦r r♦s♦ s ♣ s♦r ♦sst♠ ♥ t♥só♥ ♠á♥ rtrr Pmec = P ⋆ ♦q st♦♠ér♦r s s♣s♦r y⋆ s ♠♥t③♥ s ♣③s ♦s ①tr♠♦s s♥rá ♥ t♥só♥ ♠♥ét Pmag ♥ st stó♥ t♥só♥ t♦t s♦r ♦q ♣♦♠ér♦ srá P = P ⋆+Pmag ② s ♦♥t s♣s♦r s rrá y(P ⋆ + Pmag)

Pr r s ♦♥sr♠♦s ♦r ♥ ♠str ♠r♦só♣ ♠tr③♣♦♠ér s♥ r♥♦ ♦r♠ ♣rs♠át rt♥r ♦♥t yo ♥tr ♦s ♣③s ♠♥t③s ♣s♦ s♣r r ♦♠♦s sr ♥ ó♥ ② ♥ ♠②♦r ♣r♦♥ ♥ ♣é♥ ♣♥♥ y ♦♥ t♥só♥ ♣ ♥ ró♥ ♦♥t♥ P ♣sr ♣r♠tr③ ♠♥t r♥ts ♠♦♦s ② ♦♦ ♣♥♥ ♥ y ② P ♦♦♦ ♣♥♥ rát ② ①♣♦♥♥♦♦ r♥ ② ♠♦♦ ♦♦♥②♥ ♥tr ♦tr♦s ♠♦♦

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♦♦ ② ♦♦♦ s ♣♥ ♥ r♥♦ ♦r♠♦♥s rt♠♥t ♣qñs ♠♥♦rs 10% tí♣♠♥t ② ♥♦ st♥ ♥ s rs♦r♠ó♥t♥só♥ ♠á♥ ♥str ♠tr③ st♦♠ér P P♦rs r③ó♥ ② ♦♦ ② ♠♦♦ ♦♦♦ ♥♦ s♦♥ ♦♥sr♦s♥ ♣rs♥t ♠♦♦ ♠♦♦ ♦♦♥②♥ rqr ♥ ♣rá♠tr♦ st ♦♥ ② s t③ ♥ ♥ ♥tr♦ ♠s ①t♥♦ ♦r♠♦♥s r ♣♦r ♠♣♦ ♥ ♣ít♦ st ♦r♠ s ♦♣t ♣♦r ♥sr♣ó♥ ♦♠♣♦rt♠♥t♦ ást♦ ♠tr③ st♦♠ér P♠♥t ② r♥

dy

y= −dP

E

♦♥ E s ♠♦♦ ❨♦♥ ♠tr③ ♣♦♠ér s♥ r♥♦ ② P t♥só♥ t♦t ♣

♣♦♥♠♦s q s ♣ s♦r ♦ sst♠ ♥ t♥só♥ ♠á♥ rtrr Pmec = P ⋆ y⋆ ♦♥t ♦q st♦♠ér♦ ♦ t♥só♥ r s ♠♥t③♥ s ♣③s ♦s ①tr♠♦s ♣♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ rtrr♦ s ♥rá ♥ r③♠♥ét ♥tr s ♣③s ♦♥s♥t♠♥t s rrá ♥ t♥só♥♦♥ rtrr ♥tr③ ♠♥ét s♦r ♦q st♦♠ér♦Pmag st ♦r♠ t♥só♥ t♦t rtrr s♦r ♦q ♣♦♠ér♦srá P = P ⋆ + Pmag ♦♥ y(P ⋆ + Pmag) ♦♥t ♦q ♣♦♠ér♦ s t♥só♥ t♦t r

♥tr♥♦ ①♣rsó♥ ♦ s ♦♥♦♥s ♦♥t♦r♥♦ y⋆ ≡ y(P =P ⋆ = Pmec) s r Pmag = 0 y ≡ y(P ) ≡ y(P ⋆ + Pmag) rst

y

y⋆= exp

(−Pmag

E

)

♥♦ ♠s♠♦ r③♦♥♠♥t♦ q ♥ ó♥ s r♦♥♥Pmag ② x ♠♥t s♣♦só♥ ♦r♠ó♥ í♥ ♥ ss♠♣t♦♥

x

x⋆=

y

y⋆

♦♥ x⋆ s st♥ ♠ ♠r♦só♣ ♥tr r♦♥s ♦♥t♦rs ♠♥t③ó♥ ♥ P♦r ♦ srt♦ ♠s rr ♦ st ♦♥ó♥ t♥só♥ ♠♥ét s ♥ ♥ x⋆ ♦♥ ♦ s ♦t♥ x⋆ ≡ x(Pmec =P ⋆,Pmag = 0)

♥ rs♠♥ x ♥ ♠♦♦ ♠♠♦♥s t♥♦ tró♥♦ ♥r③♦ s r♦♥ ♦♥ Pmag trés s ♦♥s ②

t♦r ♥♦♥trrá ♠s ts ①♣rsó♥ ♥ ♣é♥ ② rr♥sr♦♥s

♦♦

P♦r út♠♦ q ♦♥srr ♣♥♥ Pmag ♦♥ H r③♠♥ét ♥tr ♦s r♦♥s tr♠♥s♦♥s ♣ sr ♣r♦①♠♠♥t♦♥sr ♦♠♦ rt♠♥t ♣r♦♣♦r♦♥ ♣r♦t♦ ♦s ♠♦♠♥t♦s ♣♦rs ♠♥ét♦s s r♦♥s ♠♥éts ♥♣♥♥t s♣ró♥ ♥tr ♠s r♦♥s s s♣ró♥ ♥tr s s ♠② ♣qñ ♦♠♣r♦ ♦♥ t♠ñ♦ ♠♦ rtríst♦ s r♦♥s ❬ ❪s s ♣rs♠♥t s♦ ♦s sst♠s s ♦♥sr♦s qí ♥ t♦ s ♠♦str♦ ♥ ♣ít♦ q s♣ró♥ ♠ ♥tr r♦♥s♦♥t♦rs x stá ♥ ♦r♥ ♥♦s ♣♦♦s ♥♥♦♠tr♦s ♠♥trs q t♠ñ♦ ♠♦ s r♦♥s s stá ♥ ♦r♥ ♦s ♠r♦♥s P♦r ♦t♥t♦ ♦♥srr♠♦s r③ ♠♥ét ♥tr s r♦♥s ♦♥t♦rs ♦♠♦ ♥♣♥♥t x ② ♣r♦♣♦r♦♥ ♣r♦t♦ ♦s ♠♦♠♥t♦s ♣♦rs ♠♥ét♦s s r♦♥s ♦♥sr♥♦ ss r♦♥s♦♥ ♦♠♥ ♠♦ ② ♦ q ♠♦♠♥t♦ ♠♥ét♦ ♣♦r s ♣r♦♣♦r♦♥ ♠♥t③ó♥ ♠tr r♥♦ ♣♦r ♥ ♦♠♥M (H) ♣ srrs

Pmag = M2(H)

(Ms

2

Λ

)

♦♥ Λ s ♥ ♦♥st♥t ♦♠étr♠♦r♦ó Ms s ♠♥t③ó♥ stró♥ ♠tr r♥♦ ② M s ♠♥t③ó♥ ♥♦r♠③ r♥♦ ♣♦r ♥ ♦♠♥ M =M (H)/Ms ♦♥ −1 ≤ M ≤ 1

st ♠♥r s ♦♥s s ♦t♥ ♣♥♥ x ♦♥ H

x(H) = x⋆ exp

(−M

2(H)

K

)

♦♥ K ≡ ΛEM2s

s ♥ ♦♥st♥t st ♣♥♥t sst♠ ♦ st♦ ♦tr q ♣rá♠tr♦ K ♥ Pmag ♥ r♦ ♥á♦♦

strt♠♥t M s ♥ó♥ ♠♣♦ ♠♥ét♦ ♦ ♠♣♦ ♠♥ét♦ ♦s♦r ró♥ ♦♥t♦r iés♠ local,i s ♥ ♥ó♥ ♠♣♦ ♠♥ét♦ ①tr♥♦② ♠♣♦ ♠♥ét♦ ♥♦ ♣♦r s ♦trs r♦♥s ♦♥t♦rs ♠♥t③s ♦ s♣♦só♥ s♣r♣♦só♥ ♥ ♠♥t③ó♥ ró♥ ♦♥t♦r iés♠ ♣①♣rsrs t♦r♠♥t ♦♠♦

♠i = vii (local,i)

♦♥ local,i = +∑

j 6=iij ② ró♥

♠i = s =+

∑

j 6=iij∣

∣

∣+

∑

j 6=iij

∣

∣

∣

♦♥ ij s ♠♣♦ ♠♥ét♦ ♥♦ ♣♦r ró♥ ♦♥t♦r jés♠ s♦r ró♥ ♦♥t♦r iés♠ á♦ t♦ ♠♣♦ ♦ rqr á♦s trt♦s ♦ ♦st♦ ♦♠♣t♦♥ ❬❪ ♥ ♠♦♦ ♣rs♥t♦ qí ♥♦ s ♦♥sr♥ ♦st♦s ♠♣♦ ♠♥ét♦ ♥♦ ♣♦r r♥♦ ♠♥t③♦

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♠ó♦ ❨♦♥ ♥ Pmec ♣r ♥ ♦ Pmag ♠②♦rs ♦rs K♠♣♥ ♠♥♦rs ♦r♠♦♥s rstr qí q x t♦♠ ♦r x⋆

♣r H = 0 ♥ s♦ r♥♦s ♦ ré♠♥ s♣r♣r♠♥ét♦ ❬♣s♣r ♦s r♥♦s M (H = 0) = 0❪ ♥ ♠♦ s s t③ ♥ r♥♦♦q♦ ♠♥ét♠♥t ♦♠♦ s rá ♠s ♥t x t♦♠ ♦rx⋆ ♣r H = ±Hc s♥♦ Hc ♠♣♦ ♦rt♦ ♠tr r♥♦❬♣s ♣r ♦s r♥♦s M (H = ±Hc) = 0❪ ①♣rsó♥ x(H) ♣r♦st♣♦r s s ♥ ♣r ♣rr ♣♥♥ Rtunnel ♦♥ H ♦ q ♦s ♣rá♠tr♦s N n RE−CR ② RCR s ♦♥sr♥♥♣♥♥ts H ♥ ♣rs♥t ♠♦♦ s ♦t♥ s♥t ①♣rsó♥♣r ♠♥t♦rsst♥ ♣♦r♥t

MR% (H) ≡[R (P ⋆, H)−R (P ⋆, H = 0)

R (P ⋆, H = 0)

]× 100

=

[Rtunnel (P

⋆, H)−Rtunnel (P⋆, H = 0)

R⋆o

]× 100

♦♥ R⋆o ≡ R (P ⋆, H = 0)

♥ s ♦ srr♦♦ rst ♥t q ♣rs♥t ♠♦♦ rqr tr♠♥ó♥ r ♠♥t③ó♥ ♠tr r♥♦M (H)② ♦ str ♦s rst♦s ①♣r♠♥ts MR% (H) ♣r ♦t♥r ♦s♦s ♣rá♠tr♦s st (x⋆γ) ② K ♦s s♦s sst♠s ♦♥ r♥♦s♦q♦ ♠♥ét♠♥t q ♣rs♥t♥ stérss ♠♥ét s st♥♥ ♦r♠ s♣r ♥ s♦♥s ss♥ts

st♦s ② ssó♥

♦♠♣♦rt♠♥t♦ ①♣r♠♥t ♠♥ét♦ ② ♠♥t♦rsst♦ ♠tr rr♥ 34❬❪P

Pr ♠trs ♦ st♦ s♣r♣r♠♥ét♦ s rs ♠♥t③ó♥ ♥♦r♠③s M (H) ♥r♠♥t s♦♥ srts ♠♥t♠♥t ♥ó♥ ♥♥ L (H) ❬❪

M (H) = L (H) = coth(H)− 1/H

♦♥ ♠♣♦ ♠♥ét♦ r♦ s ♥ ♦♠♦ H ≡ H/H‡ ② H‡ s ♥♠♣♦ ♠♥ét♦ rtríst♦ s♣í♦ ♣r ♠tr ② t♠♣rtr tr♦

Pr rr♥ s ♣rtís r♥♦ ♠r♦♣rtís 34❬❪ s♥ ♦♠♣♦rt♠♥t♦ ♦ ♣♦r ♦♠♦ s ♠str♥ r ♥ r í♥ ♦♥t♥ ♦rrs♣♦♥ st M (H) = M (H)Ms s♥♦ ♣rtr st st s ♦t♥♥

st♦s ② ssó♥

♦s ♣rá♠tr♦s H‡ = (420±9) ②Ms = (71.0±0.4) ♠−1 ♦tr q ♦♥r♥ ♦s t♦s ①♣r♠♥ts ♥ ♣♥♥ ♥ M♦♥ H ♥ r ♥ ♣t ♣r ♠♣♦s ♠♥ét♦s r♥s ♣♦st♦s♦ ♥t♦s ♥♥ rt♦ ♦♠♣♦♥♥t ♣r♠♥ét♦ s♦♦ ♣t♠tá q r♣rs♥t ♥ ♣qñ ♦♥tró♥ ② ♥♦ s t♦♠ ♥ ♥t♥ ♣rs♥t ♥áss

r r ♠♥t③ó♥ r♥♦ 34❬❪ 25 í♥♦♥t♥ ♦rrs♣♦♥ st r③♦ ♠♥t ①♣rsó♥

♥ ♦s ①♣r♠♥t♦s ♠♥t♦rsst♥ t♥só♥ ♠á♥ ♥①♥ ró♥ ♦rr♥t étr ró♥ ♦r♥tó♥ ♣rr♥ s ♣s♦♥s s ♠♥t♥ s♠♣r ♦ ♥ ♥ ♦r rtrr♦ P ⋆ ≈75 P ♣r rr♥ ♥♦ s ♣ ♥ ♠♣♦ ♠♥ét♦♥ s ró♥ rsst♥ étr sst♠ R ♠ st sst③ó♥ ♦♠♦ s ♠str ♥ r r ♥ s r í♥ ♦♥t♥ ♦rrs♣♦♥ st ♦ ♣♦r ♥ ♣r♦s♦ ró♥♠♦♥♦①♣♦♥♥

MR% (H, t) =

[1− exp

(− t

τR

)]MR%(H, t = ∞)

t♠♣♦ ró♥ ♠♥t♦rsst♦ rtríst♦ ♦sr♦ τR s♣♥♥t ♠♥t ♠♣♦ ♠♥ét♦ ♣♦ t♦♠♥♦ tí♣♠♥t ♦rs r♥♦s s♥♦s ♣r rr♥ ♦s①♣r♠♥t♦s r③♦s ♠str♥ q ♦r τR ♥♦ s ♥♥♦ ♥♦r♠ s♥t ♣♦r t♦rs ♥str♠♥ts r ó♥ s♥♦ q ♥♠♦ ♣r♥ sr ♥ ♣r♦♣ ís ♥trí♥s ás ú♥ τRs s♠r t♠♣♦ ró♥ ást rtríst♦ ♣r ♦ τE

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

s♦♦ ♦t♥ó♥ ♥ t♥só♥ ♠á♥ st ♦ ♣r ♥ ♦r♠ó♥ r r

r ♠♦ t♥só♥ ♠á♥ Pmec ♦ ♣ó♥ ♥ ♦r♠ó♥ s♦r rr♥ ♠♦ rsst♥ étr rr♥ R ♦ ♣ó♥ ♥ ♣♦ ♠♥ét♦ ①tr♥♦ H ①♣rs♦ ♦♠♦ ♠♦ ♥ ♠♥t♦rsst♥ ♣♦r♥t ♥ ♠♦s ♣♥s í♥ ♦♥t♥ ♦rrs♣♦♥ ♥ ♣r♦s♦ ró♥ ♠♦♥♦①♣♦♥♥ S (H, t) =[S(H, t = 0)− S(H, t = ∞)] exp (−t/τS)+S(H, t = ∞) ♦♥ S = t♥só♥ ♠á♥ ♦ MR%

st ♦sró♥ stá ♥ ♦♥♦r♥ ♦♥ sq♠ ís♦ s♠♣♦trás ♠♦♦ qí srt♦ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♥ ♥t♥só♥ ♠á♥ q ♠♦ s ♠♥s♦♥s ♠r♦ ② ♠r♦só♣s ♠str L ② x rs♣t♠♥t rqr♥♦ ♥ t♠♣♦ ♦r♥ τE♣r ♦rr st③ó♥ ♣r♦♣s ts ♦♠♦ rsst♥ étrR r♦♥ ♦♥ x ♦ s rs t♥só♥♦r♠ó♥ r♦♥s ♦♥L sí s♠t ①♣r♠♥t ♥tr τR ② τE stá ♥ ♦♥♦r♥ ♦♥ ssq♠

rstrr♦♥ ♦s ♦rs st③♦s R ♣r ♦r H ♠♥t♦rsst♥ ♣♦r♥t MR% ♥ ♥ s ♠str♦♠♦ ♥ó♥ H ♥ r í♥ ♦♥t♥ ♥ r ♦rrs♣♦♥ st r③♦ ♠♥t s ♦♥s ♦♥sttts ♠♦♦srt♦ st s ♠② ♥ R2 = 0.997 ♦ sts r♣r♥ ♦s ♦rs (x⋆γ) = (9.4±0.7) ② K = 93±9 ♦ q ♠♦♦ s♣♦♥ q rrr ♣♦t♥ s♦ t♥♦ tró♥♦ ϕ ♥♦s ♥♥ ♣♦r H ♦r γ ♣r sst♠s P r♥t♠♥tr♣♦rt♦ γ = 10 ♥♠−1 ❬❪ ♣ sr t③♦ ♦♠♦ ♥ st♠ó♥❯s♥♦ ♦ ♦r s st♠ x⋆ ≈ 0.9 ♥♠

st♦s ② ssó♥

r s♣st ♠♥t♦rsst 34❬❪P 4.2% í♥ ♦♥t♥ ♦rrs♣♦♥ ♠♦♦ ♦♥sttt♦ ♦ ♣♦r s ①♣rs♦♥s

♠♥t ♦r♠ó♥ str♥ ♠♥ét ♠r♦só♣ ♠á①♠♥③ stá ♣♦r (1 − x/x⋆) ♣r M2 = 1 stró♥ ♠♥t③ó♥ ♥ ♥str♦ ♠♦♦ ♦ ♦r ♦rrs♣♦♥ 1 − exp(−1/K)

❬r ♦♥ M2 = 1❪ ❯s♥♦ ♦r r♣r♦ K = 93 s ♣r♥ ♠r♦♦r♠ó♥ ♠♥ét ♠á①♠ 1% ♣r sst♠ rr♥ ❬♣r ♠♣♦s ♠♥ét♦s ♠♦ ♠②♦rs H‡ = (420 ± 9) ❪s♦♦ ést ♠♥t t♠é♥ ♣ st♠rs t♥só♥ ♠♥ét♠á①♠ Pmax

mag ♣r M2 = 1 ♦r ♣r♦ s Pmax

mag = E/K ♥ ♣rtr ♣r ♠trs ♦♠♣st♦s s♦s ♥ P s t♥ E ≈ 700− 800P ♣♥♥♦ r♦ ♥trr③♠♥t♦ ❬ ❪ sí ♣r rr♥ s ♣r Pmax

mag ≈ 7− 9 P s♣t♦ ♦♥r♥♥t ♦r♠ó♥ ♥ rr♥ ♥♦

s ♣ Pmag s ♦♠♥t ♦♥t♥ó♥ ♦ s♣♦só♥ í♥ ♦r♠ó♥ ♠r♦só♣ s ♣s ♣♦♠érs ♥tr r♦♥s ♦♥t♦rss ♦r♠ó♥ ♠r♦só♣ ♠tr③ ❯♥ ♦r♠ó♥ ♠r♦só♣ ♠á①♠ 1% r ♣árr♦ s♣r♦r s r ♥♠tr③ ♣♦♠ér s♥ r♥♦ ♦♠♣r♠rs ♠♥♦s 1% s s ♣ ♥t♥só♥ ♠á♥ ♠♥t q t♥só♥ ♠♥ét ♠á①♠ ♦q ♠ó♦ ❨♦♥ ♥ ró♥ ♦r♥tó♥ ♣rr♥ r♥♦ ró♥ ♣ó♥ H E‖ s ♠②♦r q ♠ó♦ ❨♦♥ ♠tr③ s♥ r♥♦ E s s♣r♥ ♦r♠♦♥s ú♥ ♠♥♦rs♣r ❬ ❪ P♦r ♠♣♦ ♥ rr♥ L⋆ ∼ 1♠♠ ♣♦r ♦ q s ♦♥sr♦♥s rr ♠♥♦♥s ♥ ♦r♠ó♥ ♦r♥ µ♠ ♦rr ♠♣♦s ♠♥ét♦s r♥s ♥♦ s

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

tt ♥♦ s ♥ ♣♦rt ♠str ♦rrs♣♦♥♥t

♠♦♥s rs♣st ♠♥t♦rsst ♣rs ♥ st♦ s♣r♣r♠♥ét♦

♠♦str♦ q ♠♥s♠♦ q ♦r♥ rs♣st ♠♥t♦rsst ♦s sst♠s s t♥♦ tró♥♦ ♥tr r♦♥s ♦♥t♦rs r♥♦ r♥s s♦♦ Rtunnel Pr tr s♠ó♥ rs Rtunnel(H)/R⋆ ♦♥ R⋆ ♥♦ ♦♠♦ R⋆ ≡ Rtunnel(Pmec =P ⋆, Pmag = 0) = Rtunnel(P

⋆,M = 0) s r③ó ♥ ♥áss s♠♣♦ ♠♥t ♠♦♦ t♥♦ tró♥♦ ♠♠♦♥s ♥ ♣s ♦♥t♦rs♣♥s ♣rs ❬❪

Rtunnel(x) ∝ x exp γx

♥ s ♥ ①♣rsó♥ s♠♣ ♣r♠t ♣trr ís ♥♠♥t ♣r♦♠ ♦♥ ♦ ♠♦♦ srr♦♦ ♣r♦ s♥t ①♣rsó♥s♠♣

Rtunnel

R⋆= exp

x⋆γ

[exp

(−M

2 (H)

K

)− 1

]− M2 (H)

K

st ♠♥r s ó t♦ H‡K ② (x⋆γ) s♦r Rtunnel(H)/R⋆

♠♥t s♠ó♥ rs s♥♦ ó♥ r ♦srr ♥ r q ♣♥ ♦rs ♠♦s ♥tr 50− 100%♥ Rtunnel/R

⋆ sú♥ ♦r K ♣r ♠♣♦s ♠♦r♦s H/H‡ ∼ 3s r ♠♦♦ ♣rs♥t♦ ♣r rs♣st ♠♥t♦rsst t ú♥ ♠♣♦s ♠♦r♦s ♥ H = 1000 ② s ♥ r♥s ♠♦s ♥ r

s s♠♦♥s r③s ♠str♥ q s♥s ♥ rs♣st♠♥t♦rsst ♣ sr ♥r♠♥t rs ♠♥rs

s♠♥②♥♦ ♦r K ♣st♦ q s♥s ♠♣♦ ①tr♥♦ H r ♠♥tr ♦r K s♠♥ó♥ K♦rrs♣♦♥ ♠②♦rs ♠♥t③♦♥s stró♥ r♥♦ Ms

r♥♦s ♠s ♠♥t③s ♥♥ ♠②♦rs Pmag ♦ ♥ ♠♥♦rs E♥ ♠tr③ ♠ás ♥ s ♠ás á ♦♠♣r♠r ♦ ♥ ♦ H♦s ♦rs K s♦s ♣r s s♠♦♥s ts ♦rrs♣♦♥♥ Ms ♥ ♥tr♦ 100−150 ♠−1 á♠♥t ♥③ ♣r♠♦s ♠trs ♠♥ét♦s ♠♥trs q ♦s ♦rs E stá♥s♦♦s ♦ s♥tét♦ ② ♥tr ② rs♥s ♣♦①ís ❬❪

♥r♠♥t♥♦ x⋆γ s♥♠♥t ♣rá♠tr♦ x⋆γ tú ♦♠♦ ♠♦♦r s♥s ♠♥t♦rsst ♠♥t ♦s ♠♥s♠♦s

st♦s ② ssó♥

0 3000 6000

0.2

0.4

0.6

0.8

1.0

4.4

10

25

50100

K = 500

H = 500 Oexå = 10

Rtu

nnel

/ R

å

H (Oe)

a b c0 3000 6000

0.6

0.8

1.0

30

20

10

5

xå = 0.5

H = 500 OeK = 50

0 3000 60000.8

0.9

1.0

750

100

200500

1250

H = 2000 Oe

xå = 10K = 50

r rs s♠s Rtunnel/R⋆ ♥ ♥ó♥ H t♦

K ♦♥ H‡ = 500 ② x⋆γ = 10 t♦ x⋆γ ♦♥ K = 50 ② H‡ = 500 t♦ H‡ ♦♥ K = 50 ② x⋆γ = 10

♣r♠r♦ s♦♦ t♦r x⋆ ♦rrs♣♦♥ ♠♦ó♥ ♦♠étr rs♣st ♠♥tr q s♥♦ s♦♦ t♦r γ♦rrs♣♦♥ ♠♦ó♥ ís ♥trí♥s ♥ó♠♥♦ t♥♦tró♥♦ ♠á♥♦á♥t♦ ♥tr r♦♥s ♦♥t♦rs r♥♦ ♦ s x⋆ ≪ γ−1 ♦rs ♣qñ♦s x⋆γ ♦♥tétr s rt♠♥t ♦rs ♣qñ♦s R ② í ♠♦r ♣♦r ♣ó♥ ♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♠tr♣r♠♥ ♦♥ ♦♥t ② s ♣r r♦s t♦s♠♥t♦rsst♦s ♦s ♦rs s♦s ♥ s s♠♦♥s s♦♥ ss ①♣r♠♥t♠♥t ♠♥♦ s ♦♥♦♥s ♣r♣ró♥ ♠♦♥♦ ♥tr③ qí♠ r♥♦ ②♦ ♠tr③ ②♠♦♥♦ ♦r P ⋆

s rs Rtunnel(H)/R⋆ ♦t♥s ♦♥ ♥str♦ ♠♦♦ ♠str♥ ♥♣♥t♦ ♥①ó♥ r r ② ♣♦só♥ ♣ ♥♦♥trrs á♠♥t ♥ ♦r♠ ♥♠ér P ♠♦strrs ♦♥ q ♦ ♣♥t♦ ♥①ó♥ s ♠ ♦rs ♠♥♦rs H ♣r ♦rs ♠②♦rs x⋆ ϕ② Ms ② ♦rs ♠♥♦rs E ② H‡ st♦ út♠♦ s ♦sr r♠♥t ♥ r

♣ó♥ H ♥ ró♥ ♦♣st ♦r♥ ♥ rs♣st♠♥t♦rsst s♠étr Rtunnel (H) = Rtunnel (−H) ♥♦ ♠♦strst♦ s ♥ ♦♥s♥ rt s♠trí ♥ó♠♥♦ ♦♣♠♥t♦ ♠♥t♦ást♦ s ♠♣♦ ①tr♥♦ H s ♣♦ ♦♥♠♥t

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

ró♥ ♦r♥tó♥ ♣rr♥ s strtrs ♦♠♥rs r♥♦

①t♥só♥ ♠♦♦ sst♠s ♦♥ ♠♥t③ó♥♦q

♠♦♦ ♣ sr ①t♥♦ ♣r ♣rr rs♣st ♠♥t♦rsst s ♥s♦tró♣♦s ♦♥ strtrs ♦♠♥rs ♦r♠s ♣♦r r♥♦s♥ st♦ ♠♥ét♦ ♦q♦ s r q♦s q ①♥ stérss♠♥ét rr♦♠♥ét♦s ♠♥trs ♥♦ ①st só♥ ♥tr r♥♦② ♠tr③ s♥ t♦ ♥s ❬❪

♦♠♦ s ♠♦strrá ♠s ♥t s♥♦ r♥♦s ♠♥ét♠♥t ♦q♦s s ♣r stérss ♥ rs♣st ♠♥t♦rsst ♦♥s♥ stérss ♠♥ét r♥♦ trés ♣♥♥ Rtunnel

♦♥ M (H) P♦r st r③ó♥ ♣r♠r ♣s♦ s s♠r s rs ♠♥t③ó♥ ♣r r♥♦s ♦♥ ♠♥t③ó♥ stá ♦q ② ♦ t③rs rs ♣r s♠r rs♣st ♠♥t♦rsst ♦s ♦r♠♦s♦♥ st♦s r♥♦s ♥ ♥ó♥ H P♥ t③rs ♠♦s ♠♦♦s ♣rsrr s rs ♠♥t③ó♥ ♣r st♦s ♦q♦s M (H) ♦♠♦sr ♠♦♦ strt♦♥ ❬❪ tr♦ ♠♦♦ ♠♣♠♥t t③♦♦♠♦ ♥ó♥ st ♣r rs ♠♥t③ó♥ st♦s ♦q♦ss ❬❪

M(H)=

2

πtan−1

[(H + ζ

)tan

(πΘ

2

)]

♦♥ Mr s ♠♥t③ó♥ r♠♥♥t Θ ≡ Mr/Ms s rtr r ♠♥t③ó♥ Hc s ♠♣♦ ♦rt♦ ♠♣♦ ♠♥ét♦r♦ s H ≡ H/Hc ② ♠♥t③ó♥ r s M ≡M/Ms ♦♠♦ s♥ó ♥tr♦r♠♥t s♣♦♥ ♠♣ít♠♥t q rs♣st ♠♥t③ó♥ rtríst r♥♦ ♦rrs♣♦♥ ♥ st♦ st♦♥r♦ ♥ ①♣rsó♥ ζ = −1 s dH

dt > 0 ② ζ = 1 s dHdt < 0 s r

ζ = −1(+1) ♦rrs♣♦♥ r ♠♥t③ó♥ ♠♥t③ó♥rs♣t♠♥t r ♠str rs ♠♥t③ó♥ s♠s s♥♦ ó♥ ♣r r♥ts ♦rs Hc ② ♦rs ♦s Ms ② Θ ♦♥ í♥ ♦♥t♥ ♣♥t ♦rrs♣♦♥ ζ = −1(+1)st ♥♦tó♥ s ♠♥t♥ ♣r s rs s♥ts P ♦srrs ♥♠♣ó♥ s rs ♥ r r ♠str t♦ Hc s♦r rs s♠s Rtunnel/R

⋆ s H ♣r sst♠s ②♦s r♥♦s ♣rs♥t♥ ♦s ♦s ♠♥t③ó♥ r ♦s s rs Rtunnel(H)/R⋆ ♦♥r♥ s♥t♦t♠♥t ♠s♠♦ ♦r R∞

R⋆

♣r H → ±∞ ♦♠♦ ♥ s♦ r♥♦s ♥ st♦ s♣r♣r♠♥ét♦ ♣♥ s♦♠♥t Ms K ② ♣rá♠tr♦ ♥rét♦strtr

st♦s ② ssó♥

x⋆γ ② stá ♦ ♣♦r

Rtunnel

R⋆

H→±∞−−−−−→ R∞R⋆

= exp

x⋆γ

[exp

(− 1

K

)− 1

]− 1

K

-30000 0 30000-100

0

100 Hc = 500 Oe

Hc = 1000 Oe

Hc = 2000 Oe

-25000 0 25000

0.8

0.9

1.0

-2000 0 2000

-30

0

30

c d

-2000 0 2000

0.98

0.99

1.00

a b

Rtu

nnel

/ R

å

Rtu

nnel

/ R

å

H (Oe) H (Oe)

M (

kA.m

-1)

H (Oe)

M (

kA.m

-1)

H (Oe)

r ♦s ♠♥t③ó♥ s♠♦s s♥♦ ♦♥Ms = 100 ♠−1 Θ = 0.3 ② Hc = 500, 1000 ② 2000 ♥ só♦rrs♣♦♥ ♠♥t③ó♥ dH/dt > 0 ② ♣♥t ♦rrs♣♦♥ ♠♥t③ó♥ dH/dt < 0 ♠♣ó♥ ♣♥ q r♠♥t♠str q s rs M(H) ♣rs♥t♥ ♠s♠ r♠♥♥ ♣r♦ r♥t ♦rt t♦ Hc s♦r rs s♠s Rtunnel/R

⋆

♦♠♦ ♥ó♥ H ♦♥ K = 50 x⋆γ = 10 Ms = 100 ♠−1 Θ = 0.3 ②Hc = 500, 1000 ② 2000 ♠♣ó♥ ♣♥ ♦♥ s s♥♥ ♦s ♠á①♠♦s ♥ H = ±Hc

P♦r ♦tr♦ ♦ ♥♦tr q ♠♥trs ♦s r♥♦s ♥ st♦ s♣r♣r♠♥ét♦ ♥♥ ♥ r♠♥t♦ ♠♦♥ót♦♥♦ Rtunnel ♦♥ H s♦ r♥♦s♦♥ ♠♥t③ó♥ ♦q ♥ ♣rs♥ ♥ ♠á①♠♦ s♦t♦ ♥ rs♣st ♥ ♥ó♥ H ♥ ±Hc r str ♥♠♣ó♥ s rs s♠s Rtunnel(H)/R⋆ st ♦♠♣♦rt♠♥t♦ s

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♥ ♦♥s♥ ♦ q ♠♦♦ ♦♥sr q Pmag(H) ♠♥t♦♥ M2 ♣♦r ♦ q Rtunnel s ♥ ♥ó♥ r♥t M2 ♦♠r ♥ ♦♥sró♥ ♣♦r ♠♣♦ s rs ♦♥ ζ = −1 ② H > 0 H s ♥r♠♥t♦♥ st s♦ M2 = (Mr/Ms)

2 ② Rtunnel/R⋆ < 1 ♣r H = 0 ♦ M2

s♠♥② M2 = (Mr/Ms)2 ♥ H = 0 M2 = 0 ♥ H = Hc ♣♦r ♦ q

Rtunnel/R⋆ ≤ 1 ♥ ♥tr♦ 0 ≤ H ≤ Hc ♥♦ H = Hc M2 = 0 ②

Rtunnel/R⋆ = 1 ♠á①♠♦ ♦r ♦r H s ♥r♠♥t ♣♦r ♥♠

Hc s ♦sr ♥ ♥rsó♥ rs♣st ② q M2 ♠♥t ♥♠♥t♦♥ ♦♥s♥t s♠♥ó♥ Rtunnel/R

⋆ sí t♦ ♠♥t♦ Hc s ♦rr♠♥t♦ ♣♦só♥ ♥rsó♥ ♦♠♣♦rt♠♥t♦ rs♣st ♠♥t♦rsst

t♦ ♠♦ó♥ Θ ♥ s rs ♠♥t③ó♥ r♥♦ str ♥ s rs ② ró♥ Rtunnel/R

⋆ s H♣r s♦s s♦s s ♠str ♥ ♦s ♣♥s ❬♣r ♦rs ♦s K x⋆γMs ② Hc❪ ♥r♠♥t♦ Θ ♥ ss rs ♠♣ ♠♥t♦ Mr ②♣♦r ♦ t♥t♦ M(H = 0) sí s ♦sr q Rtunnel/R

⋆ s ♠áss♥s H ♥♦ Θ ♦ Mr ♠♥t s r s ♦sr ♥ ró♥♠ás r♣t rs♣st ♠♥t♦rsst ❬|d(Rtunnel/R

⋆)/dH|♠♥t ♦♥ Θ ①♣t♦ ♥ H = ±Hc ♦♥ s ♥ ♦♠♦ s sró♥tr♦r♠♥t❪ rs♣st ♦♥r ♠s♠♦ ♦r Rtunnel/R

⋆ ≈ 0.8 ♦♥H → ±∞ ♣st♦ q ♦ ♦r ♥♦ ♣♥ Θ

st♦s ② ssó♥

-30000 0 30000-100

0

100

-2500 0 2500

-40

0

40

H (Oe)

M (

kA.m

-1)

H (Oe)

M (

kA.m

-1)

a b

H (Oe)H (Oe)

c d

-40000 0 400000.8

0.9

1.0

-3000 0 3000

0.94

0.96

0.98

1.00

Rtu

nnel

/ R

å

Rtu

nnel

/ R

å

r ♦s ♠♥t③ó♥ s♠♦s s♥♦ ♦♥Ms = 100 ♠−1 Hc = 2000 ② Θ = 0.1, 0.2 ② 0.5 ♠♣ó♥ ♣♥ q r♠♥t ♠str q s rs M(H) ♣rs♥t♥ ♠s♠ ♦rt ♣r♦ st♥t r♠♥♥ t♦ Θ s♦r rss♠s Rtunnel/R

⋆ ♦♠♦ ♥ó♥ H ♦♥ K = 50 x⋆γ = 10 Ms =100 ♠−1 Hc = 2000 ② Θ = 0.1, 0.2 ② 0.5 ♠♣ó♥ ♣♥ ♦♥ s s ♥♥ ♦s ♠á①♠♦s ♥ H = ±Hc

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

r ♠str t♦ K s♦r rs s♠sRtunnel/R⋆

♥ ♥ó♥ H ♣r ♦rs ♦s Ms x⋆γ Θ Mr ② Hc r ♦rrs♣♦♥ ♥ ♠♣ó♥ s rs ♦♥ ♥ ③ ♠s ss ♥♥ ♦s ♠á①♠♦s ♥ H = ±Hc ♥ st s♦ ♦♠♦ s ♣r ♣rtr s rs ♣rs♥t♥ r♥t ♦♥r♥

-30000 0 30000

0.2

0.4

0.6

0.8

1.0

50

5

7

10

30

-3000 0 3000

0.8

0.9

1.050

5

7

10

20

30

100K=400

Rtu

nnel

/ R

å

H (Oe)

a b

r t♦ K s♦r rs s♠s Rtunnel/R⋆ ♥ ♥ó♥

H ♦♥ Ms = 100 ♠−1 x⋆γ = 10 Θ = 0.3 Mr = 30 ♠−1 ②Hc = 2000 ♠♣ó♥ ♣♥ ♦♥ s s ♥♥ ♦s♠á①♠♦s ♥ H = ±Hc

♥♠♥t r ♠str t♦ (x⋆γ) ♥ rs s♠s Rtunnel/R

⋆ ♥ ♥ó♥ H ♦♥ ♦rs ♦ Ms K Θ ② Hc r ♠str ♥ ♠♣ó♥ s rs s♠s ♦♥ ♥♠♥t s s ♥♥ ♦s ♠á①♠♦s ♥ H = ±Hc ♠♥t r ♦♥r ♥ ♦r st♥t♦ ♣r r♥s H

só♦ s ♦♥sr r ♠♥t③ó♥ ♠♣♦s ♠♥ét♦s ♣♦st♦s s ♦sr♥ r♠♥t ♥ ♣♥t♦ ♥①ó♥ ② ♥ ♠á①♠♦ ♥ H = Hc♥t♦ ♣♥t♦ ♥①ó♥ ♦♠♦ ♠á①♠♦ s ♦sr♥ ♥ r♣♦rt r♥t♠♥t ♣♦ P♥ ② ♦♦r♦rs ❬❪ ♣r ♦♠♣♦st♦s ♣r♣r♦s♦♥ ♣rtís rr♦♠♥éts r♦♥ rr♦ ♣♦♦ rt♦ ② ♠tr③ ♣♦rtá♥ ♠♦♦ ♠♣♦ st ♦♥ ♦srr ♥ ♦ tr♦ rs♣st ♠♥t♦rsst ♦s ♦♠♣♦st♦s P ② P ♦♥

st♦s ② ssó♥

-40000 0 40000

0.6

0.7

0.8

0.9

1.0

-3000 0 3000

0.96

0.98

1.00

Rtu

nnel

/ R

å

a b

H (Oe)

28

21

14

7

xå = 0.07

r t♦ x⋆γ ♥ rs s♠s Rtunnel/R⋆ ♥ ♥ó♥

H ♦♥ Ms = 100 ♠−1 K = 50 Θ = 0.3 Mr = 30 ♠−1 ② Hc = 2000 ♠♣ó♥ ♣♥ ♦♥ s s ♥♥ ♦s ♠á①♠♦s ♥H = ±Hc

♦♥t♥♦s r♦♥ rr♦ 60% ② 80% ♥ ♣s♦ rs♣t♠♥ts s♠♦♥s ♠str♥ q ♦s ♠②♦rs ♠♦s rsst♥

♠♣♦s ♠♦r♦s H/Hc ≈ 1.5− 2 s ♦r♥ rr ♣rá♠tr♦ K ♣r♥♣ ♥♥ s♦r Mr stá ♣♦r ♦ ♣rá♠tr♦ ♦tr q♥ ♥str♦ ♠♦♦ K s q♥t ♠♥ét♦ E ♣r s♦ ♠á♥♦á♥t♦ ♠♥♦r s K sst♠ t♥ ♠②♦r rs♣st ♠♥t♦rsst sr ♦s ♠♦s ♦♥t s♦♥ ♠②♦rs ♣r ♥ H ♦

•

♥ st ♣ít♦ s rtr③ó rs♣st ♠♥t♦rsst ♠tr rr♥ ② s ①t♥ó ♦r♠s♠♦ srr♦♦ ♥ ♣ít♦ ♣r ♠♦r ♦s rst♦s ①♣r♠♥ts

♣rs♥t ♠♦♦ t♦♠ ♥ ♥t s ♣r♥♣s rtrísts ♦♥r♥♥ts ♠♥t♦rsst♥ ♥ sst♠s ② st ♠② ♥ ♦srst♦s ①♣r♠♥ts ♣r rr♥ ♠♦♦ ♣ sr

♣ít♦ s♣st ♠♥t♦rsst ①♣r♠♥t♦s ♠♦♦ ②

s♠♦♥s

♣♦ ♣r sst♠s ♦♥ r♥♦s ♥ st♦ s♣r♣r♠♥ét♦ ♦ ♦♥ st♦ ♠♥ét♦ ♦q♦ rr♦♠♥ét♦s Pr s s sst♠s ♥trr♦♥t ♣r♥♣ r ♥ ♦s r♥s t♦s ♠♥t♦rsst♦s ♦sr♦s ①♣ó♥ ♣r♦♣st ♣♦r ♠♦♦ ♦♥sr q t♥♦tró♥♦ ♥tr r♦♥s ♦♥t♦rs r♥♦ s ♠♣♦rt♥t s ♠♥t③ó♥ r♥♦ M s s♥t♠♥t ♣r ♥rr ♥t♥só♥ ♠♥ét ♠♦r Pmag Pmag s ♣③ ♥r ♥ s♠♥ó♥ st♥ ♠r♦só♣ t♥♦ x ss st♥s ♥♦s♦♥ ♠② ♣qñs ♥ ♦♠♣ró♥ ♦♥ st♥ rtríst t♥♦ ♦s s♣t♦s ② s r♦♥♥ ♦♥ ♣rá♠tr♦ K ♠♦♦ s ♦♥t ♦♥ ♣ ♥r ♥ t♥só♥ ♠♥ét trés ♦s ♣rá♠tr♦s Λ ② Ms ♦♠♦ sí t♠é♥ st ♠tr♦r♠ ♣♦r s t♥só♥ ♠ó♦ ❨♦♥ ♠tr③ E s♣t♦ s rtríst♦ t♥♦ tró♥♦ s st♥ t♥♦ s ♠s♦ ♣qñ ♦♥t étr s rt♠♥t ② ♥♦♣ sr á♠♥t ♠♦ ♣♦r r③s ①tr♥s ♠♦♦ ♥♦r♣♦r ís s②♥t ♦♣♠♥t♦ ♠♥t♦ást♦ t③♥♦ s♦♦ ♦s♣rá♠tr♦s γx⋆ ② K ♦♥♥ó♥ ♦s t♦rs ♠♥♦♥♦s ♦♥ q ú♥ ♥ s♠♥ó♥ ♣♦r♥t s♦♦ 1% ♥ x ♣ ♥r r 10% ♠♦ ♥ rs♣st ♠♥t♦rsst ♣♥♥♦ γx⋆♠♥trs q ♣r ♠s♠♦ γx⋆ ♥ ♦r♠ó♥ ♠r♦só♣ r♥ 10% ♥r ♥ 60% ♠♦ ♠♥t♦rsst♦ és r

♠♦♦ s♣♦♥ ♦♥ó♥ ♥s♦tr♦♣í étr ♦ts r ①st♥ ♦♥t ♣r♦ó♥ só♦ ♥ ró♥ ♣ó♥ ♠♣♦ ♠♥ét♦ ①tr♥♦ ♦♥♥tró♥ ♣rtís r♥♦ s ♥♥tr ♥ ♥ ♥tr♦ q sr ♦♥ó♥ ♥♦ s♣r♥ r♥s t♦s ♦♥♥tró♥ r♥♦ ♥ rs♣st ♠♥t♦rsst ♥ ♦trs ♣rs ♦♥♥tró♥ r♥♦ ♥ r♦ s♥t♦ ♥ ♥ró♥ ♦s sst♠s ♦♥ ♣r♦ ♥♦s ♣r♥ r♥s t♦s ♦♥♥tró♥ s ♦♥ó♥ s♦♥sr

♣ít♦

♦♥s♦♥s ♥rs

♥② ♣②s t♦r② s ②s♣r♦s♦♥ ♥ t s♥s tt t s ♦♥②

②♣♦tss ②♦ ♥ ♥r ♣r♦ t ♦♠ttr ♦ ♠♥② t♠s t rsts ♦

①♣r♠♥ts r t s♦♠ t♦r② ②♦♥ ♥r sr tt t ♥①t t♠ t

rst ♥♦t ♦♥trt t t♦r②

t♣♥ ♥ r st♦r② ♦♠

♥ ♣rs♥t ss s str♦♥ rs♦s s♣t♦s r♦♥♦s ♦♥t étr ♥s♦tró♣ ♥ ♦♠♣♦st♦s st♦♠ér♦s strtr♦s ♦♥ r♥♦ s str② ♥ ♦r♠ ♣s♦♥s ♦♥♥ ♦r♥tó♥ ♣rr♥

♥ ♥ ♣r♠r t♣ s ①♣♦ró ♣ó♥ s♦s sst♠s ♥ ♠♣♠♥tó♥ s♣♦st♦s ♥♦♥s ♦ró sñr rr ②rtr③r ♥ ♣r♦t♦t♣♦ rr♦ s♥s♦r t♥só♥ s♦ ♥ P34❬❪ ♠♣♠♥t♥♦ tr♦♦s q ♥♦ tr♥ rs♣st ♠tr ② ♥♦ tr♦r♥ ♠s♠ ♦♥ s♦ ss♦ s rtrísts ♠ás ♠♣♦rt♥ts sst♠ ♠♣♠♥t♦ s♦♥ ♣ s♥srt♥s♦♥s ♠á♥s ② ♣♦t♥♠♥t ♠♣♦s ♠♥ét♦s s♥ ♥s ♠r ♥ ♥ ♥str♠♥tó♥tr♥só♥ rs♣st s rt♠♥t ♥s♦tró♣ ♦ s q ♣♥ ró♥ ♥ q ♦s ♠♣♦s①tr♥♦s stá♥ ♣♦s s♦r rr♦ rs♣st ♦s stí♠♦s s t♦t♠♥t rrs ♠ás t♦♦ s♣♦st♦ ♣rs♥t ① ♦♠♦ sí t♠é♥ rsst♥ ①♣♦só♥ r ② rs♦ss♦♥ts ♦rá♥♦s ② s♦♦♥s ♦ss

s rtrísts ♠♥♦♥s ♥♦s r♦♥ ♥ ♦r♠ ♥tr ①♣♦rr s ss íss q ♦r♥♥ s ♣r♦♣s ♥ ♣r♠r

♣ít♦ ♦♥s♦♥s ♥rs

♣s♦ ♥t♦♥s r t♦ ♦s ♣rá♠tr♦s strtrs rtríst♦s ♦s sst♠s ♠♥♦♥♦s s♦r ♣r♦ ①r♦♥t étr ♣r ú♥♠♥t ♥ ró♥ ♦r♥tó♥♣rr♥ r♥♦ ♥s♦tr♦♣í étr ♦t ♥ st ♠r♦ r♥ r♥ t♥♦ó ♥ ♠♣♠♥tó♥ s♣♦st♦s ♦♠♦ srt♦ ♥ ♣árr♦ ♥tr♦r s t③ó ♥ ♦r ♣r♦t♦ ♦♥sr♥♦ s strtrs r♥♦ ♦♠♦ s♠♥t♦s rts t♥♦♦♠♣t♦♥♠♥t té♥s ♦rít♠s ♦♥t r♦

❯♥ s ♦♥s♦♥s ♣r♥♣s st♦ r③♦ s q rst ♥sr♦ ♥tr♦r ♥s♦tr♦♣í ♥tr♥ ♠♥t ♥ stró♥ ♥r ♥s♦tró♣ ♦s ♦t♦s ♣r♦♥ts ♣r ♥③r ♥ ♦r♠ t ♥ ♦♥♦r♥ ♦♥ s ♦sr♦♥s ①♣r♠♥ts s♠s♠♦ s♥♦♥trr♦♥ ♦♥♦♥s strtrs q sr♥ s ♣r♦s ♥ ♣r♣ró♥ ♦s sst♠s st♦s ♥ st ♦♥t①t♦s ró q ①st ♥ rt ♣♥♥ ♣r♦ ♦t♥r ♦♥ s♣rsó♥ ♥r s ♣s♦♥s ② s ♠♦stró q ♦♥t ♠ ♦s ♦t♦s ♣r♦♥ts t♠é♥ ♥ r♦ ♠♣♦rt♥t♠♥trs q s♣rsó♥ ♥ ♦♥t t♥ ♣♦♦ t♦ s♣t♦ ó♥ t♦ t♠ñ♦ sst♠ ♣r♦t♦ ♠♦♦ srt♦♠str q ♦s sst♠s strtr♦s ♦♠♦ ♦s trt♦s qí ♣♥ ♦♥♦ ♣rs♥tr ♣♥♥♦ t♠ñ♦ ♠str ♠♥♦♥rq rtr♦ ♦♠étr♦♣r♦t♦ ♥♦♥tr♦ ♣r s ♣t♠é♥ sst♠s ♦♠étr♠♥t ♥á♦♦s ♦♥ ♦tr♦s ♦t♦s ♥♠♥s♦♥s ♥♥♦t♦s ♥♥♦rs ♥♥♦s t

P♦str♦r♠♥t s srr♦ó ♥ ♠♦♦ ♦♥sttt♦ q sr ♥ tér♠♥♦s rts rs♣st ♣③♦rsst ♠trs ♦ ♦♥ó♥ ♠♦♦ srr♦♦ s ♣ó♥ ♥r ♣r t♦♦s ♦ssst♠s ♦♠♣st♦s ♦♥ strtr r♥♦ ♥á♦ ♦s ♥③♦s♥ ♣rs♥t ss s t③ó rr♥ ♣r r③r ♥ ♦♥①ó♥♦♥♣t ♥tr s rs ② ♣rá♠tr♦s tór♦s ♠♦♦ ② ss ♠♥ts ♦srs ①♣r♠♥t♠♥t ♦s rst♦s ♦t♥♦s sr♥ qs ♣s♦♥s r♥♦ ♣rs♥t♥ ♠út♣s s♣r♦♥s r♠♥t♦♥s ♥tr♥s ② q t♥♦ tró♥♦ trés ss s♣r♦♥s♦♥stt② ♠♥s♠♦ ís♦ ♠♦♦r rs♣st ♣③♦rsst ♠tr ♠♦♦ srt♦ ♣r♠t r ♥ ♦r♠ ♥ttt t♦ ♦s ♣rá♠tr♦s ♠r♦só♣♦s rtríst♦s sst♠ ♥ r♥♦♥á♠♦ s rs♣st ♣③♦rsst ♥ ♣rtr ♦s á♦s ♥♠ér♦sr③♦s ♦♥r♠♥ q t③r ♠trs ♣♦♠érs ♦♥ ♠②♦r ♠ó♦ ❨♦♥ s ①t♥ r♥♦ ♥á♠♦ t♥s♦♥s ♠á♥s

♥♠♥t ♥♦s ♣r♦♣s♠♦s ①t♥r ♦r♠s♠♦ ♥tr♦r ♣r r♥r♥ ①♣ó♥ t ♠♥s♠♦ s♦♦ rs♣st ♠♥t♦rsst ♦s ♠tr ♥s♦tró♣♦s ♦♥ r♥♦s ♦♥t♦rs ó♠♦s

♠♥ét♠♥t t♦s ♠♦♦ t♦♠ ♥ ♥t s ♣r♥♣s rtrísts ♠r♦só♣s

sst♠ ② st ♠② ♥ ♦s rst♦s ①♣r♠♥ts ♣r rr♥ s♠s♠♦ s ♥♦r♣♦r ♥ tér♠♥♦s s♥♦s ís s②♥t ♦♣♠♥t♦ ♠♥t♦ást♦ ♦♥♥ó♥ ♦s t♦rs ♥♦r♦s♦♥ q ú♥ ♥ s♠♥ó♥ ♠② ♣qñ ♥ s♣ró♥ ♥trr♦♥s ♦♥t♦rs r♥♦ ♣ ♥r rs♣sts ♠♥t♦rssts♦♥srs s ♠♥t③ó♥ r♥♦ s s♥t♠♥t

r♦ ♥t♦♥s q ♠♦♦ s♦ ♥ ♥♥ ♦♣♠♥t♦ ♠♥t♦ást♦ s♦r ♣r♦ t♥♦ tró♥♦ ♣r♠t♣rr rs♣st ♠tr r♥t t♥t♦ t♥s♦♥s ♠á♥s ♦♠♦ ♠♣♦s ♠♥ét♦s ①tr♥♦s st s ♥ rst♦ ♥tr ss ②q ♣r♠t ♥tr♣rtr ② r♦♥③r s rs♣sts ①♣r♠♥ts ♦ss♥s♦rs ♠♣♠♥t♦s

♦s ♦♥♣t♦s qr♦s ♣rt♥♥ sr t t♥ó♦♦s ② ①♣r♠♥tsts ♥ srr♦♦ ② ♦♣t♠③ó♥ ♠trs ♦♠♣st♦sstrtr♦s ② s♣♦st♦s ♥s

♣é♥

♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦

♥③ s r str ♥♥r ♠♦s r ♥

♦s ③♣tr♦s ② sstrs

♦t③♠♥♥

s♠♥ st ♣é♥ sr r♠♥t ♦s ♥♠♥t♦s ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦ ♣ sst♠s ♣r♦t♦s

♥rs

♥ ♣ít♦ ♦③♠♦s ♥str♦ ♥trés ♥ r ♣♥♥ ♦s ♣rá♠tr♦s rtríst♦s ♥ sst♠ ♣r♦t♦ ♦♥ t♠ñ♦ ést ♠r♦ tór♦ ♠ás rt♦ q ♣ t③rs ♣r ést ♥ s s♥ ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦ ♣♦rss ss ♥ ♥és q ♦♥st ①♠♥r s ♥ts íss r ♣♥t♦ rít♦ ② ♣r♦♠♥t s ♥♦ ♦s ♠ét♦♦s ♠s ♠♣♦rt♥ts srr♦♦s ♥ ís tór ♦s út♠♦s tr♥t ② ♥♦ ñ♦s ♥♥t ❲s♦♥é r♦♥♦ ♥ ♦♥ Pr♠♦ ♦ ♥ ís ❬ ❪ ♣♦rs ♦♥tró♥ ♥ srr♦♦ ét♦♦ r♣♦ ♥♦r♠③ó♥ ♥ ♥ ss ♦♠♥③♦s st ♠ét♦♦ ♣♦ ♥ r♠♦♥á♠ s tr♥s♦♥s s s rt♠♥t s♥♦ ♥tr♦r♦ ♥ ♦♥t①t♦ tr♥s♦♥s ♣r♦ó♥

♣é♥ ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦

Pr ♥tr♦r ♦ ♠ét♦♦ ♦rr♠♦s s ♣ó♥ ♥ st♠ó♥ ♦s ♣rá♠tr♦s rít♦s ♦♥sr♠♦s ♥ ♦t♦rí ♥ ♦♥ró♥ ♥ sst♠ ♣r♦ó♥ st♦s ♥r ♣r ♥ ró♥ ♦♣ó♥ p = p0 < p∞ ♦sr♠♦s ♦t♦rí s st♥s ♠②♥s ♥♦ ♣♦r♠♦s st♥r ♦s st♦s ♦♣♦s q♦s ②♥ts ②♣♦r ♦ t♥t♦ ♥♦ ♦srr♠♦s ♦s r♣♦s ♦r♠♦s ♣♦r ♥ ú♥♦ st♦ ♠ás s r♠s q s s♣r♥♥ ♦s strs ② ♦s ♣qñ♦s ♣♥tsq ♦♥t♥ r♥s r♦♥s ♦♣s s ♣r♥ ♥ s③ó♥ st ♦r♠ ♦♥ró♥ ♦t♥ p0 < p∞ ♦sr ♥ ♦r♠st♥t s ♦♠♣♦rt ♦♠♦ ♥ ♦♥ró♥ ♥r ♥ ♦r p1 < p0♥t♠♥t s s♠♦s á♥♦♥♦s ♦t♦rí ♥♦ sr♠♦s ♣s st♥r ♥♥♥♦ ♦s strs ② ♦t♦rí ♣rrá s♦ ♥♦♥ró♥ ♥r ♥ ró♥ ♥ st♦s ♦♣♦s s r p = 0é s ♦srrí s ♥♦s ♠♦s ♥ ♦t♦rí ♥ ♦♥ró♥♥r ♦♥ p0 > p∞ P♦♠♦s sr ♠s♠♦ r③♦♥♠♥t♦ ♣r rq s♦♦ ♦srrí♠♦s ♣qñs r♦♥s st♦s s♦♣♦s ♠q ♥♦s ♠♦s ♦t♦rí s♦s s♣♦s s ♥ ♠♥♦s sr♥s② ♦♥ró♥ ♣r♥trá t♥r ♥ ♠②♦r ♣♦r♥t s ár ♦♣s r ♦t♦rí ♣rrá s♦ ♥ ♦♥ró♥ ♥r ♥♦r p = p1 ♠②♦r q p0 ♥t♠♥t s s♠♦s á♥♦♥♦s ♦t♦rí ♣rrá s♦ ♥ ♦♥ró♥ ♥r ♥ ró♥ st♦s ♦♣♦s ♥tr s r p = 1 ♥ ♠r♦ ♦s ♦rsp = 0 ② p = 1 s ♥♦♠♥♥ ♣♥t♦s ♦s trs ❬ ❪

é s ♦srrí s r♣t♠♦s st ♣r♦♠♥t♦ ♣r p0 = p∞ ♦♠♦s ♠♥♦♥ó ♥ ó♥ ♥ ♠r ♣r♦ó♥ s ♦sr♥ s♠s♠s ♣r♦♣s s♥ ♠♣♦rtr á s s ♦sró♥ t③♦ ♦t♦rí ♣rrá ♠s♠ ♥q ♦♠♥t ♠s ♣qñ s♥♠♣♦rtr á♥t♦ ♥♦s ♠♦s s s s♥t♦ ♦r p∞ s s♣② s ♥♦♠♥♦ ♣♥t♦ ♦ ♥♦ tr ❬ ❪

♦♥sr♠♦s ♦r ♠♦r ♦♥ró♥ ♥ ♥ ♦r♠ q ♠ r♥♦s ♦t♦rí ♥ r r q ♣rt♦♥♠♦s ♥s ♦ ♦qs r♥♦ t♦♦ ♥tr♠♦ ♠r♠♦s r s ♥♣rs♣t t q ♦s st♦s ♥ ♥ s ♦♠♥♥ ♣r r ♥ ♥♦s♣rst♦ ♦ st♦ r♥♦r♠③♦ ♥t♦♥s ♥ r t♥ ♠s♠ s♠trí q r ♦r♥ ② r♠♣③r s s ♣♦r ♦s ♥♦s st♦s s♠ s ♦♥ts t♦s s st♥s s♦♥ ♦r ♠s ♣qñs♥ ♥ t♦r b ♦♥ b s ♠♥só♥ ♥ s r q s ♦♠♣♦♥ b× b st♦s ♥t♦♥s t♦ r♥♦♠③ó♥s r♠♣③r r♣♦ st♦s ♣♦r ♥ st♦ r♥♦r♠③♦ ó♠♦ ♠♦s s st♦ r♥♦r♠③♦ stá ♦♣♦ ♦ ♥♦ ♦ q s♠♦s q st♦ r♥♦r♠③♦ ♣rsr s rtrísts ♣r♥♣s ♥tr♠♦♦r♥ s s♠trí ② s ♦♥t s ♦♥sr q st♦ r♥♦r③♦ stá ♦♣♦ s r♣♦ ♦r♥ st♦s ♣rs♥t ♣r♦♦♥ ♦

♥rs

r ♣r♦ó♥ t♦ tr ♥ tr♥s♦r♠ó♥ r♥♦r♠③ó♥ s♦r ♦♥r♦♥s tí♣s ♣r♦ó♥ ♣r ♦rs p ♣♦r ♦ ② ♥♠ p∞ s ♠str♥ ♥ s rs ② rs♣t♠♥t ♦ r ♣r♦ó♥ rt ♥ ♠♦s s♦s t♦

r ♦♥ró♥ ♣r♦♦♥ st♦ ♥r p = 0.5 < p∞ ♦♥ró♥ ♦r♥ r♥♦r♠③ trs s ♣♦r tr♥s♦r♠ó♥ s tr♦ st♦s ♥ ♥ ♥♦ s♣rst♦ t♦ ♥ tr♥s♦r♠ó♥ ♦♥ ♥rrí ♥ ú♥♦ st♦ s♦♣♦

s r♥♦r♠③♦♥s sss ♣r♦♦♥ ♠♥t♦ rs♣t♦ ♦rp∞ P♦♠♦s r q ♣r p = 0.7 t♦ s trs♦r♠♦♥s ♥r s♣③♠♥t♦ p = 1 Pr p = 0.5 t♦ s trs♦r♠♦♥s♥r s♣③♠♥t♦ p = 0 ♦tr q trtrs ♥ sst♠♣r♦t♦ ♥t♦ ♥♦ s♦♠♦s ♣s ♦♥t♥r tr♥s♦r♠ó♥ r♥♦r♠③ó♥ ♥ ♦r♠ ♥♥ ét♦♦ r♣♦ ♥♦r♠③ó♥♣r♠t ♥♦ s♦♦ st♠r p∞ s♥♦ t♠é♥ ♦tr♦s ♣rá♠tr♦s rít♦s ♦♠♦sr ①♣♦♥♥t rít♦ ν

♦♥t♥ó♥ s ♣rs♥t ♥ ét♦♦ r♣♦ ♥♦r♠③ó♥♣r st♠r p∞ s♥♦ ♠t♦♦♦í srt ♣♦r ②♥♦s ② ♦♦r♦rs ❬❪ Prt♠♦s ♣r♦①♠ó♥ q ♦♥♥t♦ st♦s s ♥♣♥♥t t♦s s ♦trs ② stá rtr③ s♦♠♥t♣♦r ♣r♦ p′ q sté ♦♣ tr♥s♦r♠ó♥ r♥♦r♠③ó♥ ♥tr p ② p′ r ♦ q ís ás ♣r♦ó♥ s ♦♥t ♦ q ♥♠♦s ♦♣ó♥ ♥ s♦♦ s ♦♥t♥ ♥ ♦♥♥t♦ st♦s q ♣r♦♥ ♥ sú♥ r ♣r♦ó♥ ♦s st♦s stá♥ ♦♣♦s ♦♥ ♥ ♣r♦ p♥t♦♥s strá ♦♣ ♦♥ ♥ ♣r♦ p′ ♦♥ p′ s

♣é♥ ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦

r ♦♥ró♥ ♣r♦ó♥ st♦ ♥r p = 0.7 > p∞ ♦♥ró♥ ♦r♥ r♥♦r♠③ trs s ♣♦r tr♥s♦r♠ó♥ s tr♦ st♦s ♥ ♥ ♥♦ s♣rst♦ t♦ ♥ tr♥s♦r♠ó♥ ♦♥ ♥rrí ♥ ú♥♦ st♦ ♦♣♦

♣♦r tr♥s♦r♠ó♥ rrsó♥ ♦ ró♥ rrsó♥ ♦r♠

p′ = R(p).

♥t R(p) s ♣r♦ t♦t q ♦s st♦s ♦r♠♥ ♥ ♠♥♦♣r♦t♦ Pr r s ♦♥sr♠♦s r q ♠str sst ♦♥r♦♥s rt♠♥t ♣r♦♥ts ♥ ♥ ♦♥ b = 2 ♣r♦ q st♦ r♥♦r♠③♦ sté ♦♣♦ stá ♣♦r s♠ s ♣r♦s t♦s s ♦♥r♦♥s ♣r♦♥ts

p′ = R(p) = p4 + 4p3(1− p) + 2p2(1− p)2.

♥ ♥r ♣r♦ p′ ♦s st♦s ♦♣♦s r♥♦r♠③♦s s r♥t ♣r♦ ♦♣ó♥ p ♦s st♦s ♦r♥s P♦r ♠♣♦s♣♦♥♠♦s q ♥♠♦s ♦♥ p = p0 = 0.5 ♦ ♥ s♦ tr♥s♦r♠ó♥ r♥♦r♠③ó♥ ♦r ♦t♥♦ ♣r p′ ♣rtr sp1 = p′ = R(p0) = 0.44 r♣t♠♦s ♥♥t♠♥t s tr♥s♦r♠ó♥ r♠♦s p = 0 ♥♦ ♦s ♣♥t♦s ♦s trs ♣r♠♥t srt♦sPr ♥♦♥trr ♦r ♠r ♣r♦ó♥ p∞ ♠♦s ♥♦♥trr ♦r p∗ = p t q

p∗ = R(p∗)

s ①♣rs♦♥s ② rst p∗ ∈ 0, 1, 0.61804 ♦s ♦s ♣r♠r♦s ♦rs ♦rrs♣♦♥♥ ♦s ♣♥t♦s ♦s trs ♠♥trs q trr♦

s♦ ♣r♦ ♣r♦ó♥

r s st ♦♥r♦♥s rt♠♥t ♣r♦♥ts ♥ ♥ ♦♥ b = 2

♦rrs♣♦♥ ♥str st♠ó♥ ♠r ♣r♦ó♥ q sr♦♠♣r♦ ♦♥ ♦r p∞ = 0.59274621(13) ❬ ❪ ♣rsó♥ st♠ó♥ s st♥t ♥ ♦♥sr♥♦ rt s♣♦só♥ r③ ♣♦♥r q ♦♣ó♥ s ♥♣♥♥t s ♦trss s ♦rrt♦ ♣r ♦s st♦s ♦r♥s ♣r♦ ♦ r♥♦r♠③ó♥ s ♣r♥ ♣rt ♦s ♠♥♦s ♣r♦t♦s ♦r♥s ② s ♥♥♦tr♦s q ♥♦ st♥ ♣rs♥ts ♥ ♥tr♠♦ ♦r♥ st ♣r♦♠áts ♥♦♠♥ ♦♠ú♥♠♥t ♣r♦♠ ♥trs ♦ q st♦s t♦ss♣rs s ♥ ♠♥♦s ♣r♦s ♥r♠♥tr t♠ñ♦ ♥ ♦r♠ ♠♦rr ♥str st♠ó♥ s ♦♥srr s ♠②♦rs ♣♦r♠♣♦ b = 3

♦♥t♥ó♥ s srrá r♠♥t ♦s rst♦s ♥ít♦s rr♦♦s ♣♦r rs♣t♦ s♦ ♥♦ ♣r t♦r ♥trs♦ ♠♣♠♥tó♥ ét♦♦ r♣♦ ♥♦r♠③ó♥ ♣r st♠ó♥ ♦tr♦s ♣rá♠tr♦s rít♦s ❬ ❪

s♦ ♣r♦ ♣r♦ó♥

♥ó♥ ℘L,κ

(p) q ♣r♦ q ♥ sst♠ ♥t♦♠♥s♦♥ t♠ñ♦ L ② ♣rá♠tr♦s strtrs κ ♣r♦ ♥♦♣ó♥ p sr ♥tr♠♥t ♥ ♦♥t①t♦ ♣♥♦ ♣r♦①♠ó♥ ♠ét♦♦ srt♦ ♥ ó♥ ♥tr♦r ②♥♦s ② ♦♦r♦rs ❬❪ ♠♦strr♦♥ q ℘

∞,κ(p∞) = p∞ ② ♣♦r ♦ t♥t♦ ♥t

℘∞,κ

♣rí sr ♥♦♥rs ❬♣s p∞ = p∞ (κ) ♣♥♥t srtrísts strtrs sst♠ ♣r♦t♦ ♥ st♦❪

♦s st♦s r ♦s s♣t♦s ♥rss ♥ó♥ ♣r♦ ♣r♦ó♥ r♦♥ ♥♦s ♣♦r ♥♥s ② ♦♦r♦rs ❬❪♦ tr♦ trt ♣r♦ ♣r♦ó♥ sst♠s rt♥rs② ♠str ♥♠ér♠♥t q ♣r♦ ♣r♦ó♥ ♥ sst♠s ♥♥t♦s ♥s rít s ♥ ♥ó♥ ♥rs ró♥ s♣t♦

♣é♥ ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦

rtá♥♦ r② ❬❪ ♥s♣r♦ ♣♦r s♦s rst♦s ♥♠ér♦s ró♥ ♦r♠ ♥ít ♣r ♥ó♥ s♥♦ ♥r♥③ ♦♥♦r♠ ♣r♦ ♣r♦ó♥ ♥ sst♠s ♣r♦t♦s ♠♥s♦♥s ór♠ s ♥♦♥tr ♥ ♦♥♦r♥ ♦♥ r♦s rst♦s s♦♦s ♣r♦♠s ♣r♦ó♥ ♠♥s♦♥ ❬❪ P♦str♦r♠♥t♥♥s ② ss ♦♦r♦rs ♥r③r♦♥ ór♠ ♥ít r②♣r ♦trs ♦r♠s ♣♦r ♠♣♦ ♣r♦r♠♦s ❬❪ ② ♥♦♥trr♦♥ ♥♦♥♦r♥ ♥tr ♥ ♥♠ér ② s ♣r♦♥s ♥íts ♦♥r♠♥♦ ♥ ♣rtr ♥rs ♣r♦ ♣r♦ó♥♦♠♦ ♥ó♥ ró♥ s♣t♦ t♠é♥ ♣r ss ♦trs ♦r♠s

r♥♦ ♥ ♦r♠ ♥♣♥♥t rssrr ❬❪ ♥♦♥tró q ♥♦s sst♠s ♣r♦ó♥ st♦ ♥ ♥tr♠♦s r♦s ♦ r ♣r♦ó♥ ℜ1 ❬❪ ♣r♦ó♥ ♥ ♥ ró♥♦♥ó♥ ♦♥t♦r♥♦ r ♥ ♦tr ♥ ♥♠ér sst♥t q℘

∞,κ(p∞) = 1/2 6= p∞ ♥ s tr♦ rssrr t♠é♥ sñ q

st ♦r s ♦♥sst♥t ♦♥ ♣r♦♠ ♣r♦ó♥ ♥ ♣r ℘

L,κ(p∞) = 1/2 ♣r t♦♦ L ♥t♦ ♥rs ❬❪

♣r♦ ♣r♦ó♥ ♣r sst♠s ♣r♦♦♥ st♦ ♣♦str♦r♠♥t st ♦♥ ♠♦r ♣rsó♥ ♥♠ér ♣♦r ❩ ❬❪ qé♥♦rr♦♦ró ℘

L,κ(p∞) = 1/2 ② r♠♥tó q ♥ ♥♦ ést ♦r s ♦♥

sst♥t ♦♥ ♥rs rs♣t♦ ♣r♦♠ ♣r♦♦♥ ♥ rst♦ ♦♥tr ♣s tr♥s♦r♠ó♥ ♣r♦①♠ s ♥ s ♥ ❬❪ ♠♣ q ℘

∞,κ(p∞) = p∞ 6= 1/2

♥ ♥ tr♦ ♣♦str♦r ♦ ♦ ♣♦r ♦ ② r♦♥② ❬❪ ♦s t♦rs ♠♦strr♦♥ q st ♣r♥t ♥♦♥rs ℘

∞,κ(p∞) ♥

❬❪ s ♥ rtt♦ ♣ó♥ s♠♣ tr♥s♦r♠ó♥ ♠♦♥♦♣r♠étr r♥♦♠r③ó♥ ② q ♥ ♦rrt ♣ó♥ tr♥s♦r♠ó♥ ♠♣ ♥♦ s♦♦ q ℘

∞,κ(p∞) s ♥rs s♥♦ t♠

é♥ q ℘∞,κ

(p∞) = 1/2 ①t♦ ♣r sst♠s r♦s ♦ r ♣r♦ó♥ ℜ1

♦t♦s ♣♦r r ♥t♦ ♣r♦ ♣r♦ó♥ ♥rs ♦♠♦ ♥ rtríst ♣♥t♦ ♦ tr♥s♦r♠ó♥ p∞ r♦s t♦rs ❬ ❪ str♦♥ ♣♥♥ ♠♥t ♦♥ t♠ñ♦ sst♠ s♦ ② ♦♥ ró♥ s♣t♦ ♦♥t♥ó♥ s sr ♦r♠s♠♦ srr♦♦ ♣r t ♥ ♣♦ sst♠s ♣r♦t♦s s♠♥t♦s rts ♣r♦♥ts

♥ sst♠ ♠♥s♦♥ ró♥ s♣t♦ r = Lp/Ln ♦♥ Lp

s ♦♥t sst♠ ♦♥sr ♥ r ♣r♦ó♥ ② Ln

s ♦♥t sst♠ ♥ ró♥ ♦rt♦♦♥ Lp s ③ ♥♠♦s t♠ñ♦ ♥♦r♠③♦ sst♠ ♦♠♦ L =

√LpLn q ♦rrs♣♦♥

t♠ñ♦ ♥ sst♠ r♦ ár ♥ ♦s ♦t♦s ♣r♦♥tss♠♥t♦s rts ♦♥t ♥tr ♥ s♦ s♠♥t♦s rts ♦♥t ♥♦ ♥tr L ♦rrs♣♦♥ t♠ñ♦ ♥ sst♠ ♠♦

s♦ ♣r♦ ♣r♦ó♥

♦r♠ t q ♦♥t ♦s s♠♥t♦s rts ℓ q ♥♦r♠③♣s ♥ sst♠ t♠ñ♦ ♥ L′ = 1000 ② s♠♥t♦s rts ♦♥t ℓ′ = 10 s ♦♠♣t♠♥t q♥t ♥ tér♠♥♦s ♣r♦t♦s ♥ sst♠ t♠ñ♦ ♥ L = L′/ℓ′ = 100 ② s♠♥t♦s rts ♦♥t ♥tr r ♣♥t♦ rít♦ rtr③♦ ♣♦r Φ∞ ♥ó♥ ♣r♦ ♣r♦ó♥ ℘L,r (Φ) ♣ srrs ♥ tér♠♥♦s ♥♥ó♥ ♥rs s♦ ❬ ❪

℘L,r (Φ) = F (x, yi, z) .

♦s r♠♥t♦s ♥ó♥ ♥rs s♦ F s♦♥ z = C ln r yi =BiωiL

−ϑi ② x = A (Φ− Φ∞)L1/ν ♦s s♦♠rr♦s ♥♥ ♥rs s rs A Bi ② C s♦♥ ♦s t♦rs ♠étr♦s ♥♦♥rss ωis♦♥ s rs rr♥ts s♦ ② ϑi s♦♥ s ♦rr♦♥s ♦s①♣♦♥♥ts s♦ i = 1, 2, . . . ❬ ❪ ❯t③♥♦ ♦♥♦♥s ♦♥t♦r♥♦ rs ② ♦♥sr♥♦ ♦s sst♠s ♦♠♣♠♥tr♦s ♦ss♠♥t♦s rts ② s♣♦ í♦ r♦r s s ♦♥② q ♦ss♠♥t♦s rts ♣r♦♥ ♥ ♥ ró♥ ♦ ♥ s♣♦ í♦ ♣r♦♥ ró♥ ♦rt♦♦♥ ést

F (x, yi, z) + F (−x, −yi,−z) = 1.

r♥♦ rs♣t♦ x yi ② z ② ♥♦ s rs ♥ x = yi = z = 0 sr Φ = Φ∞ L −→ ∞ ② r = 1 s ♦♥② q ∂mF/∂xj∂yk11 · · · ∂zl|0 =0 ♣r m ♣r ①♣♥♥♦ ♥ó♥ ♣r♦ ♣r♦ó♥ r ♣♥t♦ rít♦ ♥♦♥tr♠♦s q

F (x, yi, z) = F (0, 0, 0) + f0 (x, z) +∞∑

i=1

fi (x, z) yi + . . . ,

♦♥ s ♥♦♥s f0 (x, z) ② fi (x, z) stá♥ ♥s ♣♦r

f0 (x, z) =

∞∑

j,l=0

1

j!l!

∂j+lF

∂xj∂zl

∣∣∣∣0

xj zl, ♣r j + l ♠♣r

②

fi (x, z) =

∞∑

j,l=0

1

j!l!

∂j+l+1F

∂xj∂yi∂zl

∣∣∣∣0

xj zl, ♣r j + l ♣r

♦ q ♥ó♥ stró♥ ♣r♦ ♣r♦ó♥ ΓL,r (Φ) =∂℘L,r/∂Φ stró♥ ♣r♦ ♣r♦ó♥ ♣r ♥ sst♠ t♠ñ♦ L ró♥ s♣t♦ r ② ♥s ♦t♦s Φ ♣♦♠♦s ♥r ♠♦♠♥t♦ kés♠♦ stró♥

µk =

∫ ∞

0(Φ− Φ∞)k

∂℘L,r∂Φ

dΦ

= A−kL−k/ν

∫ ∞

−AΦ∞L1/ν

xk∂F

∂xdΦ.

♣é♥ ♦rí r♣♦ ♥♦r♠③ó♥ ♥ s♣♦

stt②♥♦ s ♦♥s ♥ ó♥ ♠♦♠♥t♦kés♠♦ s ♦♠♦

µk (yi, z) = L−k/ν

(g0 (z) + +

∞∑

i=1

gi (z) yi + · · ·)

♦♥ ♠♦s ♥tr♦♦ s ♥♦♥s ♥rs g Pr k ♠♣rs g0 (z)s ♥ ♥ó♥ ♠♣r ② gi (z) s♦♥ ♥♦♥s ♣rs z Pr ♦rs k♣rs g0 (z) s ♣r ② gi (z) s♦♥ ♥♦♥s ♠♣rs P♦r ♦ t♥t♦ ♣r♦sr ♣r ♦s ♣rt♦rs ♦♥ rs♣t♦ z sr ♥♣♥♥t t♣♦ sst♠

♦♠♣♦rt♠♥t♦ s♦ 〈Φ〉L,r ♣sr srt♦ ♠♥t ♥ó♥ s♦ ♠♦♠♥t♦ ♥r③ ♦♥♦♥ts ♣♥♥ts ró♥ s♣t♦ r

〈Φ〉L,r = Φ∞ + L−1/ν∞∑

i=0

ai (r)L−ϑi ,

♦♥ ϑi s♦♥ s ♥♦♠♥s ♦rr♦♥s ♦s ♦♥ts s♦s♥♦ ϑ0 = 0 ❬ ❪ ♥ ♦r♠ s♠r ♣r r♥③ ∆2

L,r ♥tr♦♠♦s ①♣♥só♥ ♥ sr

∆2L,r = L−2/ν

∞∑

i=0

bi (r)L−ϑi .

①♣rsó♥ ② ♣r g0 (z) ② gi (z) ♣♥ ♦t♥rs ss♥ts ①♣rs♦♥s ♣r♦①♠s ♣r ♦s ♣rt♦rs ♦r♥ 0 ② 1 ♣r〈Φ〉L,r ② ∆2

L,r r r = 1 s r z = 0 a0(r) ≈ a0,0 ln(r)+a0,1 ln3(r)

a1(r) ≈ a1,0 + a1,1 ln2(r) b0(r) ≈ b0,0 + b1,0 ln

2(r) ② b1(r) ≈ b1,0 ln(r) +b1,1 ln

3(r) st ♦r♠ ♣r sst♠s r♦s r = 1 s♠♥t♦s rts ♣r♦♥ts s t♥♥ s s♥ts ①♣rs♦♥s s♠♣s s♦

〈Φ〉L,r=1 = Φ∞,r=1 + aL−1/ν−ϑ

②∆L,r=1 = cL−1/ν

♦♥ a ≡ a1,0 ② c ≡√b0,0 sú♥ ♥♦tó♥ s ♥ ó♥ Pr

s♠♥t♦s rts ♦♥t ℓ s ①♣rs♦♥s ♥tr♦rs s ♥r③♥♦♠♦ 〈Φ〉L,r=1,ℓ = Φ∞,r=1,ℓ + aℓL

−1/ν−ϑ ② ∆L,r=1,ℓ = cℓL−1/ν ♦♠♦ s

sr ♥ ó♥ ♣r sst♠s ♠♥s♦♥s s t♥ ν =4/3 ♥ ♣rtr ♣r sst♠s ♠♥s♦♥s s♠♥t♦s rts♣r♦♥ts ♦♥t ♥tr rst ❬❪

ϑ =

0.83± 0.02, r = 1

0, r 6= 1.

♣é♥

♦rí r♦s

♠♥♥t ♥♦ ② ♠♥♦ s ♠♥♦ ♥r

Pr♦r♦s ② ♥trs ♥t♦♥♦ ♦

s♠♥ sr ♥s ♥♦♦♥s ♦rí r♦s ② s♣ó♥ ♥ ♣r♦♠s ♦♥t ② ♣r♦ó♥

♥♦♥s

♥ ♦rí r♦s ♥ r♦ s ♥ r♣rs♥tó♥ ♥ ♦♥♥t♦ ♦t♦s ♦♥ ♥♦s ♦s ♦t♦s stá♥ ♦♥t♦s ♣♦r ♥s ♦s♦t♦s ♥tr♦♥t♦s stá♥ r♣rs♥t♦s ♣♦r str♦♥s ♠t♠áts♠s érts ♦ ♥♦♦s ② ♦s ♥s q ♦♥t♥ ♥♦s ♣rs értss♦♥ ♠♦s rsts í♣♠♥t ♥ r♦ s r♣rs♥t r♠át♠♥t♦♠♦ ♥ ♦♥♥t♦ ♣♥t♦s ♣r ♦s érts ♥♦s í♥s ♦ rs ♣rs rsts ❬❪ s strt♠♥t ♥ r♦ G s ♥ ♣r ♦r♥♦G = (V,E) ♦♥ V s ♥ ♦♥♥t♦ érts ♦ ♥♦♦s ♥♦r♠♠♥t ♥t♦② E s ♥ ♦♥♥t♦ rsts ♦ r♦s q r♦♥♥ st♦s ♥♦♦s ♠♦r♥ r♦ G ② s ♥♦t ♦♥ |V | s ♥ú♠r♦ érts r♦ ♥ ért ♦ ♥♦♦ s ♥ú♠r♦ r♦s E q s ♥♥tr♥ ♥ é

❯♥ ♣r ♥♦ ♦r♥♦ s ♥ ♦♥♥t♦ ♦r♠ a, b ♠♥r qa, b = b, a Pr ♦s r♦s st♦s ♦♥♥t♦s ♣rt♥♥ ♦♥♥t♦ ♣♦t♥ V r♥ 2 s ♥♦t ♣♦r P (V ) = 2 ❯♥ r♦ ♥♦r♦ ♦ r♦ ♣r♦♣♠♥t ♦ s ♥ r♦ G = (V,E) ♦♥ V 6= ∅ ②E ⊆ x ∈ P (V ) : |x| = 2 s ♥ ♦♥♥t♦ ♣rs ♥♦ ♦r♥♦s ♠♥t♦s V

❯♥ r♦ ♥trsó♥ s ♥ r♦ q r♣rs♥t ♥ ♣tró♥ ♥trs♦♥s ♥ ♠ ♦♥♥t♦s ♦r♠♠♥t ♥ r♦ ♥trsó♥

♣é♥ ♦rí r♦s

s ♥ r♦ ♥♦ r♦ ♦r♠♦ ♣♦r ♥ ♠ ♦♥♥t♦s Si i = 1, 2, · · · ♥r♥♦ ♥ ért vi ♣♦r ♦♥♥t♦ Si ② ♦♥t♥♦ ♦s érts vi ②vj ♠♥t ♥ rst s♠♣r q ♦s ♦♥♥t♦s ♣rs♥t♥ ♥trsó♥ ♥♦í s r

E (G) =vi, vj |Si ∩ Sj 6= ∅

❯♥ r♦ ♥trsó♥ s r♣rs♥trs ♣♦r ♥ ♠tr③ ♥trsó♥ Mt q

Mi,j =Mj,i =

0, s ♦s ♠♥t♦s i ② j s ♥trst♥

1, s♦ ♦♥trr♦.

P♦r ♠♣♦ s ♦♥sr♠♦s ♥ ♦♥♥t♦ N s♠♥t♦s rts♣r♦♥ts ♥ tér♠♥♦s r♦s Si = iés♠ ② V = Si, i =1, · · · , N = vi ② E (G) =

vi, vj|Si ∩ Sj 6= ∅

r ♠s

tr r♣rs♥tó♥ ♥ sst♠ 10 s♠♥t♦s rts ♦ sst♠♣ sr ♥tr♣rt♦ ♥ tér♠♥♦s r♦s sú♥ ♠str r ♦♥ s ♠tr③ ♥trsó♥ s♦ r

st♦ ♦♥t ♥ r♦s

♥ sst♠ ♠♥s♦♥ rt♥r s♠♥t♦s rts ♣r♦♥ts ♦♥ ♦ st♦ st ♦r ♣♦♠♦s ♥tr♣rtr s sst♠ ♦♠♦ ♥r♦ ♥ tér♠♥♦s ♣r♦t♦s st♠♦s ♥trs♦s ♥ r s ♦s rsts♦♣sts sst♠ stá♥ ♦♥ts ♣♦r ♥ r♣♦ ♣r♦♥t Pr rs♣♦♥r s ♥trr♦♥t rs ♦♥t r♦ ♣r ♦ s ♥ srr♦♦ ♥♠r♦s♦s ♦rt♠♦s ❯♥♦ ♦s s úsq♥ ♣r♦♥ ♣♦r ss ss ♥ ♥s s ♥ ♦rt♠♦ q♣r♠t r♦rrr t♦♦s ♦s ♥♦♦s ♥ r♦ ♠♥r ♦r♥ ♣r♦ ♥♦♥♦r♠ ♥♦♥♠♥t♦ ♦♥sst ♥ r ①♣♥♥♦ t♦♦s ② ♥♦ ♦s ♥♦♦s q ♦③♥♦ ♦r♠ rrr♥t ♥ ♥ ♠♥♦ ♦♥rt♦ ♥♦ ② ♥♦ q♥ ♠ás ♥♦♦s q str ♥ ♦ ♠♥♦ rrstr♥ ♠♦♦ q r♣t ♠s♠♦ ♣r♦s♦ ♦♥ ♥♦ sr♠s ♥♦♦ ② ♣r♦s♦ ❬❪ r ♠str ♦r♥ ♥ q s st♥ ♦s ♥♦♦s ♥ st♦ ♦♥t ♥ ♥ r♦ ♣qñ♦♠♥t ♦rt♠♦

st♦ ♦♥t ♥ r♦s

r ♥tr♣rtó♥ ♥ sst♠ ♣r♦♥t s♠♥t♦s rts♥ tér♠♥♦s r♦s st♠ ♦♥ 10 s♠♥t♦s rts N = 10 ♦r♠♦ ♣♦r trs strs t♠ñ♦ 5 3 ② 2 ♣rs♥tó♥ sst♠ ♦♠♦ r♦ ♥trsó♥ G = G(V,E) ♦♥ V = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10② E(G) =

1, 2, 2, 3, 2, 4, 3, 4, 3, 7, 5, 6, 8, 9, 9, 10

tr③ ♥trsó♥ s♦ sst♠ ♣r♦♥t

r r♥ ♥ q s st♥ ♦s ♥♦♦s ♥ st♦ ♦♥t♥tr ♥♦♦s ♠♥t ♦rt♠♦ ♣t rst r

♣é♥

♦♦s st

trt s t ♥ ♦ tr s♥ r② t♦♦ ♦♥ ♠ ♦♥② ♦r♦ t r♥ ♥ t ♥② t s ♥♦ t♠ tt t s♦ rtr♥ t♦ t

♣♥♥ss ♥ s♦♥♥ss ♦ srt♦♥s♦♥ ♠tr ♥ ♦♦s t♥s

♦rt ♦♦ r♦r♣

s♠♥ st ♣é♥ sr ♦s ♠♦♦s ♣rstt③♦s ♥ ♣ít♦ ② sr ♠♦♦ ♦♦ ♦♦r♥ ♦♦♦ ② ♠♦♦ ♦♦♥②♥

♦♦ ♦♦♦

❯♥ ♠tr ♣rást♦ ♦ ♠tr ást♦ r♥ ❬❪ s ♥ t♣♦ ♠tr ást♦ ♣r ó♥ ♦♥sttt q r♦♥ t♥s♦♥sstrsss ② ♦r♠♦♥s str♥s ♣ ♦t♥rs ♣rtr ♥ ♣♦t♥ást♦ ♦ ♥rí ást ♦r♠ó♥ q s ♥ó♥ st♦

♣r♠r♦ ♦s ♠♦♦s ♦♥sttt♦s srr s ♠♦♦ ♦♦♦ ♥ r t③r ♦r♠s♠♦ ② ❬❪ r♥ ♥r ♣r♦ ♠②♦r ♦♠♣ ♠t♠át ♦♥t♥ó♥ s ♦rrá ♦♠♦♦ s ♥ ♣rs♣t ♠r♦só♣ ❬❪

♥ ♣♦í♠r♦ s♦♠t♦ ♦r♠ó♥ ♠á♥ ②s ♥s ♣♦♠érs ♥♦ ♣rs♥t♥ ♥r♠♥t♦ s♠ ♦r♠ó♥ í♥ ♠tr ♦r♠ó♥ rt ♥ s rs ♥s ♣♦♠érs q♦r♠♥ ♠str s ♠s♠ q sr r t♦t ♠str ♠r♦só♣❬r r ❪

♣é♥ ♦♦s st

r ♦r♠ó♥ í♥ ♠♣ q ♥ ♣♦♠ér qr ♦r♠ó♥ rt ♠str ♠r♦só♣ ❬❪

♥ Lj,o j = x, y, z s ♠♥s♦♥s ♠str ♥ s♥ ♦r♠ó♥ ♦ ♦r♠ó♥ s ♦♥♦♥s ♠tr ♥ ♦s trs sq♥ ♥s ♦♠♦ λj ≡ Lj

Lj,o R st♥ ♣r♦♠♦ ♥tr ①tr

♠♦s s ♥s ♣♦♠érs ♥ r s ♠str st♥♥ ♦r♠ t♦r s♠♥♦ ♦r♠ó♥ í♥ λj =

RjRj,o

N ♥t ♠ ♠♦♥ó♠r♦s q ♦r♠♥ s ♥s ♣♦

♠érs ② s b ♦♥t ♦s ♠s♠♦s ♦ ♥s♠ ♠r♦♥ó♥♦s♠♥♦ t♦t rt ♦♥♦r♠♦♥ ♥ s ♥♦♥s ♥tr ♠♦♥ó♠r♦s ♥tr♦♣í ♥ ♥ ♦♥ N ♠♦♥ó♠r♦s ② st♥ ♥tr ①tr♠♦s

−→R

♣ rs ♦♠♦ ❬❪

S(N,

−→R)= −3

2kB

−→R 2

Nb2+ S (N, 0)

♦♥ kB s ♦♥st♥t ♦t③♠♥♥ ②

S (N, 0) ≡ 3

2kB ln

(3

2πNb2

)+ kB ln

[∫

C

Ω(N,

−→R)d−→R

]

♦♥ C s s♣♦ ♦♥r♦♥s t♥rs ♦tr q S (N, 0) ♣♥ N ♣r♦ ♥♦

−→R s s ② ♦♥

−→R 2 = R2

x+R2y+R

2z

♦♦ ♦♦♦

♠♦ ♥tró♣♦ ♥ iés♠ ♣♦r t♦ ♦r♠ó♥ s

∆Si ≡ S(N,

−→R)− S

(N,

−→R o

)

= −3

2kBR2

x +R2y +R2

z

Nb2+

3

2kBR2

x,o +R2y,o +R2

z,o

Nb2

= −3

2kB

[(λ2x − 1

)R2

x +(λ2y − 1

)R2

y +(λ2z − 1

)R2

z

Nb2

]

st ♦r♠ ♠♦ ♥tr♦♣í ♠str ♦♠♣st ♣♦r n♥s s

∆S =

n∑

i=1

∆Si = −3

2

kBNb2

n∑

i=1

∑

j

(λ2j − 1

)(Rj,o)

2i

♦♥ j = x, y, z sst♠ stá ♦r♠♦ ♣♦r ♥s s ♣ ♠♦strrs q

❬ ❪⟨R2

j,o

⟩=

1

n

n∑

i=1

(Rj,o)2i =

Nb2

3

s s ②

∆S = −nkB2

(λ2x + λ2y + λ2z − 3

)

❯t③♥♦ r♠♥t♦s ♠② s♥♦s ♣ ♠♦strrs q ♠♦ ♥rí r ♥ r ♣♦♠ér stá s♦♦ ♣r♥♣♠♥t ♠♦s ♥tró♣♦s ♦ rst

∆F ≈ −T∆S =nkBT

2

(λ2x + λ2y + λ2z − 3

)

♦♥ T s t♠♣rtr sst♠♥ ♣rtr s ♠tr s ♥♦♠♣rs ♦♠♥ s ♠♥t♥

♦♥st♥t sí V =∏

j Lj =∏

j λjLj,o =∏

j Lj,o qí∏

j λj = 1♥ tr♦ srt♦ ♥ ♣ít♦ s ♠str♥ ①♣r♠♥t♦s

♦r♠ó♥ ♥① ♦♠♥♦ rtrr♠♥t x ♦♠♦ ró♥ ♦r♠ó♥ ♥ ♥ ♠tr ♥♦♠♣rs s t♥ λx ≡ λ ② λy = λz =

1√λ

sí s ♦ss s t♥

∆F ≈ nkBT

T

(λ2 +

2

λ− 3

)

τ t♥só♥ ♣ s♦r ♠tr Pr ①♣r♠♥t♦ ♥ stó♥

τ = τx =

(∂∆G

∂Lx

)

N,Ly ,Lz

=1

Lx,o

(∂∆G

∂λ

)

♣é♥ ♦♦s st

A = LyLz ár ♠str s♦r s ♣ t♥só♥sí s ♦ss t♥só♥ rr stá ♣♦r

σ =τxA

=nkBT

V

(λ2 − 1

λ

)

♦♥sr♥♦ t♥só♥ ♦♥♥♦♥ ♦ ♥♥r P ♣♦st ♥ ♦♠♣rsó♥ rst

P = 2CN−H1

(1

λ2− λ

)

♦♥ CN−H1 ≡ nkBT

2V q ♦rrs♣♦♥

♦♦ ♦♦

❯♥ ♠♦♦ ♠♦ ♠ás s♥♦ ② ♣ó♥ ♠ás ♠t só♦ ♣r s♦ ♦r♠♦♥s ♣qñs s ♠♦♦ ♦♦

♦♥sr♠♦s q s t♦♠ ♥ ♦q ♣rs♠át♦ rt♥r ♥ ♠tr ást♦ ♦♥t Lx = Lo ♥♦ Ly ② tr Lz s ♦♠♣r♠ ♠tr ♥ ró♥ ♦♥t♥ ♣♥♦ ♥ r③ Fx ♥t♦♥s ♦♥t ♦q s s♠♥rá ♥ ♥ ♥t ∆Lx ♣♦♥r♠♦s q ♠♦ ♥ ♦♥t s ♣qñ♦ rs♣t♦ Lo Pr ♥ r♥ ♥ú♠r♦ ♠trs ♦s ①♣r♠♥t♦s ♠str♥ q ♣r ♦r♠♦♥s s♥t♠♥t♣qñs r③ s ♣r♦♣♦r♦♥ ♦r♠ó♥

Fx ∝ ∆Lx

♦r♠ó♥ ∆Lx ♦q ♣♥ t♠é♥ s ♦♥t P♦♠♦s♦♠♣r♥r ó♠♦ ♠♥t s♥t r♠♥t♦ ♣♠♦s ♦s ♦qsé♥t♦s ♣♦r ss ①tr♠♦s ② ♦s ♦♠♣r♠♠♦s ♠s♠s r③s tú♥ ♥ ♦q ♣♦r ♦ q ♥♦ srrá ♥ ♦r♠ó♥ ∆Lx P♦r ♦ t♥t♦ ♦r♠ó♥ ♥ ♦q ♦♥t 2Lo rá sr ♦s s ♠②♦r q ♥ ♦q ♦♥ ár tr♥srs ♣r♦ ♦♥t Lo Pr ♦t♥r♥ ♥ú♠r♦ ♠s rtríst♦ ♠tr ② ♠♥♦s ♣♥♥t ♦r♠♣rtr ♠s♠♦ ♠♦s t③r ♦♥t ∆Lx/Lo ♣r♦♣♦r♦♥ r③ ♣ ♣r♦ ♥♣♥♥t Lo

Fx ∝ ∆Lx

Lo

r③ Fx t♠é♥ ♣♥ ár tr♥srs ♦q ♣♦♥♠♦sq ♣♠♦s ♦s ♦qs é♥t♦s ♣♦r ♥ s rs trs ♦ st♦♥ró♥ ♣r ♦rr ♥ ♦r♠ó♥ ∆Lx r♠♦s ♣r♥ r③ 2Fx sí r③ ♥sr ♣r ♦r♠r ♥ ♦q ♥ rt♥t ∆Lx sr ♣r♦♣♦r♦♥ ár tr♥srs ♦q A Pr

♦♦ ♦♦ r♥

♦t♥r ♥ ①♣rsó♥ ♥ ♦♥t ♣r♦♣♦r♦♥ ♥tr r③ ♣ s ♥♣♥♥t s ♠♥s♦♥s ♦q s sr

Fx = −EoA∆Lx

Lo

♦ ♥ rsré♥♦ ♥ tér♠♥♦s t♥só♥ ♣ P = FxA

P = −Eo∆Lx

Lo= −Eoǫ

♦♥ Eo s ♥♦♠♥♦ ♠ó♦ ❨♦♥ ♠tr st ①♣rsó♥s ♦♥♦ ♦♠♦ ② ♦♦ ♦ ♠♦♦ ♦♦ ❬❪

t♦♠♥♦ ♠♦♦ ♦♦♦ ♥ ró♥ ♦♥♦♥s r♥s ♥ ♦r♠♦♥s ♣qñs ♦♥sr♥♦ q ♦♥ó♥② str♥ sts♥ λ = 1 + ǫ rst P ≈ −6CN−H

1 ǫ

sí ♣r ♦r♠♦♥s ♣qñs s r♣r ♥ ♦♦♥ ♦♥ ♠ó♦ ❨♦♥ ♠tr ♦ ♣♦r Eo = 6CN−H

1

♦♦ ♦♦ r♥

❯♥ ♠♦♦ tr♥t♦ ♣r srr ♦♠♣♦rt♠♥t♦ ást♦ rt♦s ♠trs s ♦♦ r♥ t③♦ ♣♦r ♥str♦ r♣♦ ♥ ♥♠r♦s♦s tr♦s ♣r♦s ♣r ♠♦♦ stst♦r♦ sst♠s ♦♥r♥♦s r ♥tr③ ♦♠♦ sr ♦tts ❬❪ ♦tts ♦♣s♦♥ s♠r♦ ❬❪ ② ♠♥tt ❬❪ ♦♠♦ sí t♠é♥ ♦♠♣♦st♦s st♦♠ér♦s s♦tró♣♦s ♦♥ r♥♦ rt♦ ❬❪

ás♠♥t ♦ ♠♦♦ ♦♥sst ♥ s♠r q ♦♠♣♦rt♠♥t♦ást♦ ♠tr stá ♦ ♣♦r ② ①♣♦♥♥

dLx

Lx= −dP

Eo

♦♥ Eo s ♥♠♥t ♠ó♦ ❨♦♥ ♠tr ♥ stó♥ ♦ ♦♥ó♥ ♦♥t♦r♥♦ Lx(P = 0) ≡ Lo ♥tr♥♦ rst

Lx

Lo= exp

(− P

Eo

)

♦♠♦ r♦ ♣r t♦r ♥trs♦ ♠♦strr q st ①♣rsó♥ t♠é♥ ♣r♠t r♣rr ② ♦♦ ♣r ♦r♠♦♥s ♣qñs

♥ ♠str q st ♠♦♦ rst t♦ ♣r sr♣ó♥ást ♦s sst♠s st♦s ♥ ♥ ♠♣♦ r♥♦ t♥s♦♥s ♠á♥s ♦♥ ①♣ó♥ ró♥ ♦r♠♦♥s ♣qñs ❬ ❪

♥ t♦ sr♥♦ t♦r f(λ) = 1λ2 − λ ♦♠♦ ♣♦♥♦♠♦ ②♦r ♥ t♦r♥♦

ǫ = 0 s t♥ f(λ) ≈ −6ǫ + 3ǫ2 − ǫ3 + O(ǫ4) Pr ♦r♠♦♥s ♣qñs t♦♠♠♦s♣r♦①♠ó♥ ♣r♠r ♦r♥ r♣rá♥♦s rst♦ ♥♥♦

♣é♥ ♦♦s st

♦♦ ♦♦♥②♥

♦s ♠trs s♦♥ ♠♦♦s ♥ ♦r♠ ♠♥t s sr♣♦♥s ♥tr♦r♠♥t ♠♥♦♥s Pr ♦r♠♦♥s r♥s rrrrs ♠♦♦s ♦r♥ ♠②♦r ❯♥♦ ♦s ♠s t③♦s s ♠♦♦ ♦♦♥②♥ ♣r♦♣st♦ ♥♠♥t ♣♦r ♥ ♦♦♥② ♥ ñ♦ ❬❪ ② r♦r♠♦ ♥ tér♠♥♦s ♥r♥ts r♦s ♣♦r ♦♥ ♥ ♥ ñ♦ ❬❪ s♦s ♥r♥ts t♠é♥ ♠♦s ♥r♥ts ♦r♠ó♥ str♥ ♥r♥ts stá♥ ♥♦s ♦♠♦

I1 ≡∑

j

λ2j

I2 ≡∑

j 6=k

λ2jλ2k

I3 ≡∏

j

λ2j

♦♥ j, k = x, y, z ♦tr q I3 s r♦ ♠♦ rt♦ ♦♠♥

I3 =(

VVo

)2 ② t♦♠ ♦r ♥tr♦ ♣r ♥ s♦♦ ♥♦♠♣rs

♠♥t s r③rá ♥ sr♣ó♥ s♠♣ ♠♦♦ s♥t③r ♦r♠s♠♦ t♥s♦r ②

❱é♥♦♦s ♠♦♦ ♦♦♦ s sr ♥s ♥rí r Υ ≡ ∆F

V ♦♠♦ sr ♣♦t♥ t♦♠♥♦ r♥ ♥tr♦s ♥r♥ts ② ss ♦rs ♥ ♠str ♥♦ ♦r♠

Υ = C0 + C1 (I1 − 3) + C2 (I2 − 3) + C3 (I3 − 1) + · · ·

♦tr q s♥♦ tr♠♥♦ ♦rrs♣♦♥ sr♣ó♥ ♥♦♦♦♥ ♥ rt♦ tér♠♥♦ s ♥ ♥ só♦s ♥♦♠♣rss♠♥trs q trr tér♠♥♦ s ♣r♠r♦ sr s♦♦ s♦♥srs♣t♦ sr♣ó♥ ♥♦♦♦♥ ás

♦♥sr♥♦ ♦r♠♦♥s ♥①s ② tr♥♥♦ sr ♥ rt♦tér♠♥♦ t♥só♥ rr t♦♠ ①♣rsó♥

σ =

(2C1 +

2C2

λ

)(λ2 − 1

λ

)

sí s ♦ss ♦♥sr♥♦ t♥só♥ ♦♥♥♦♥ ♦ ♥♥r P ♣♦st ♥ ♦♠♣rsó♥ rst

P =

(2CM−R

1 +2CM−R

2

λ

)(1

λ2− λ

)

♦♥ C1 ≡ CM−R1 ② C2 ≡ CM−R

2 q ♦rrs♣♦♥ ♥♦♠♥♥ ♥r♥ts ♣s s ♦r s ♥♣♥♥t ó♥ sst♠

♦♦r♥s

♦rí

♦ srs ♦ ♠♣r♦ ♠♦ó♥ é ♠♦rr ② ó ♦ rs ♣③ ♥t♦♥s rr t

s ♦ ♥ rt ♠t ♠♥♦

♦r♦ r♦ á♦♦ ♣rt♦♥tst

❬❪ s♥ ② P Pé ♥ ♥♥rí ♦s ♠trs ♦♠s♦♥

❬❪ ♦r ② ♦r♦ ♥tr♦ó♥ ♥ ♠trs ♣r♥♥r♦s r ♦ó♥ t ♦ srs Prs♦♥ ó♥

❬❪ ♥ r♥♦r P♦②♠r P ♦♠♣♦sts ♦rtrtr ♣♣t♦♥s ❲♦♦ Ps♥ rs ♥ ♥trtr ♥♥r♥ sr ♥

❬❪ ♠rs♥ ②r ❲♥tr♠♥t ② ❲ ♦♥ ♦♠ ♣♣t♦♥s ♦ ♣♦②♠r♦♠♣♦st ♠trs r ♦♠♣♦stss♥ ♥ t♥♦♦② ♦ ♥♦ ♣♣

❬❪ ♦♠♥ ❯ ♥ ❲ ② ❨ ♥♦ ♠ tstr♦♥ r ♦ t ♠♥ ♣r♦♣rts ♦ r♦♥ ♥♥♦t♣♦②♠r ♦♠♣♦sts r♦♥ ♦ ♥♦ ♣♣

❬❪ P♥② t r♥t st ② ♦sst ♥ r♦♥ ♥♥♦ts ♥ r♠♠tr① ♥♥♦♦♠♣♦sts ♣r♣r ②t♠♣rtr ①trs♦♥ ♠ P②ss ttrs ♦ ♥♦ ♣♣

❬❪ ♦s t r ♠ ② ♦rs ♥rr②s ♦ s♥ r♦♥ ♥♥♦ts ♥rt r♦♠ r♥♦♠ ♥t♦rs ② ♦r♥tt♦♥② st sr t♦♥ ♥♦ ttrs ♦ ♥♦ ♣♣

♦rí

❬❪ ♦st♥s♦♥ ❩ ♥ ② ❲ ♦ ♥s ♥ t s♥♥ t♥♦♦② ♦ r♦♥ ♥♥♦ts ♥ tr ♦♠♣♦sts r♦♠♣♦sts s♥ ♥ t♥♦♦② ♦ ♥♦ ♣♣

❬❪ P♥ ② ❳ ♥ ② ♥ r♦♥ ♥♥♦t♣♦②♠r ♦♠♣♦sts t ♠ ♠♣r♦ tr ♦♥tts ♠ P②ssttrs ♦ ♥♦ ♣♣

❬❪ ♥♠r ♦♠♠♥s ♥③r sr ② ❲♥②♥ s♥ r♦♥ ♥♥♦ts ♥ ♦♠♣♦sts ② ♠t ♣r♦ss♥ ♠t♦s ♠ ♣②ss ttrs ♦ ♥♦ ♣♣

❬❪ ♦s♦t③ ② sr ♥s ♥ rr♦ t♥♦♦② ♦r♥ ♦ ♠♥ts♠ ♥ ♠♥t ♠trs ♦ ♥♦ ♣♣

❬❪ ♥r ♠♦ ② r♦ rt♦♥ ♦ s♣r♣r♠♥t ♦♦s ♥ ♠♥t s t qst ♦r t qr♠stt ♦t ttr ♦ ♥♦ ♣♣

❬❪ P t♣♥ ♦ s♦st② ♠♥t ♦t♥ ② t ♦♦ss♣♥s♦♥ ♦ ♠♥t ♣rts ♦ ❯ Pt♥t

❬❪ rr ② r♦ t♦ rr♦s ♣r♦♣rts ♥ ♣♣t♦♥s r③♥ ♦r♥ ♦ P②ss ♦ ♥♦ ♣♣

❬❪ ♥ s ♦r ❳ ② ❩♥ ♥t r♦♦② ♥ ♦s rr♥t ♣♥♦♥ ♥ ♦♦ ♥tr ♥♦ ♥♦ ♣♣

❬❪ ③ r Pr③ ♦r s♦ ♦rs♦ ② rtí♥ r trtr st♦♠r s♠♠tr ♠s s♣②♥ ♠♥t♦ ♥ ♣③♦ rsstt② ♦r♥ ♦ P♦②♠r♥ Prt P♦②♠r P②ss ♦ ♥♦ ♣♣

❬❪ ③ trs ♠♥t♦ást♦s s♦s ♥ ♦♠♣♦st♦s ♥♥♦♦♠♣st♦s ♠♥ét♦s sí♥tss rtr③ó♥ ② ♣ó♥ srr♦♦ s♥s♦rs ss ♦t♦r ❯♥rs ♥♦s rs

❬❪ tt ③ P ♥t♦♥ Pr③ tr ♦r ② r ♥s♦tr♦♣ ♠♥t♦rsst♥ ♥ ♣③♦rsstt②♥ strtr 34sr ♣rts ♥ P st♦♠rs t r♦♦♠ t♠♣rtr ♥♠r ♦ ♥♦ ♣♣

♦rí

❬❪ P ♥t♦♥ ♦r Pr③ tr ② ② r ♥t ♥ st ♣r♦♣rts ♦ ♦24♣♦②♠t②s♦①♥ ♠♥t② ♦r♥t st♦♠r ♥♥♦♦♠♣♦sts ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ❨♥ ❲ ♦ ❩ ♥ ❲ ♥ ❩♥ ② ❲ Pr♣rt♦♥ ♦ r♣♥ ♥♥♦st♣♦②♥♥ ♦♠♣♦st t s♣ ♣t♥ r♦♥ ♦ ♥♦ ♣♣

❬❪ ❲ ❲♥ ❩♦ ❲ ♥ ❩ ♥ ♥ ❩ ❲ ♥t ② ♥ rt♦♥ ♦ r♣♥♣♦②♥♥ ♦♠♣♦st ♣♣r ♥ st ♥♦ tr♦♣♦②♠r③t♦♥ ♦r♣r♦r♠♥ ① tr♦ s ♥♦ ♦ ♥♦ ♣♣

❬❪ P ♣t ♥♠ ♥ str P♦♣ ② ♠r r♦♥ rtrsts ♦ r♦♥ ♥♥♦t t♥♠ tr♥sst♦rs ♥♦t♥♦♦② ♦ ♥♦ ♣

❬❪ ❳ ❩♥ ❳ ❳ P ♥ ♦ ♦t ts② ❨ ❩♦ rtrsts ♦ t tr ♣r♦t♦♥ ♥ r♦♥ ♥♥♦ts♣♦②♠r ♥♥♦♦♠♣♦sts ♦r♥ ♦ P②s ♠str② ♦ ♥♦ ♣♣

❬❪ tt ♦r Pr③ r ② r ♣r♣r♠♥t ♥s♦tr♦♣ st♦♠r ♦♥♥t♦rs ①t♥ rrs♠♥t♦♣③♦rsstt② ♥s♦rs ♥ tt♦rs P②s ♦ ♣♣

❬❪ ♥♦ P ♠♣ ♥♦♥ ② ♦ ♠♦t② r♦♥♥♥♦t t♥♠ tr♥sst♦rs ♦♥ ♣♦②♠r sstrt ♣♣ P②ss ttrs ♦ ♥♦ ♣

❬❪ P ♦♥s r② s♠ ② ❩tt ①tr♠ ♦①②♥ s♥stt② ♦ tr♦♥ ♣r♦♣rts ♦ r♦♥ ♥♥♦ts ♥ ♦ ♥♦ ♣♣

❬❪ ♠♠♦ ♦rt♦s ♦ ② ❩ ♦t ♥♥rsr② rt ♦t♦♥ ♦ tr♦♥ s♥ ♥ r st♦r② s♥ ♦♥srt♦♥s ♥ r♥t ♣r♦rss ♥trs ♦ ♥♦ ♣♣

❬❪ s ♥♠ ② ❯r sstt② ♥ ♣r♦t♦♥ ♥t♦rs ♦♦♥♠♥s♦♥ ♠♥ts t ♥t strt♦♥ P②s ♦ ♥♦ ♣

♦rí

❬❪ Psqr ❯♥♥ ♥ r ② ♦ ♦♥t♥ ♥ tr♥s♣r♥t s♥ r♦♥ ♥♥♦t tr♦s ♦r ♣♦②♠rr♥ s♦r s ♣♣ P②ss ttrs ♦ ♥♦ ♣

❬❪ ③ tt P ♥t♦♥ Pér③ r ② ♦r trtr ♥ ♠♥t ♣r♦♣rts ♦ 2−x♦♠x4 ♥♥♦♣rts ♥ 2−x♦♠x4P ♠♥t♦st♦♠rs s ♥t♦♥ ♦ ♠ ♦♥t♥t ♦r♥ ♦ ♥ts♠ ♥ ♥t trs♦ ♣♣

❬❪ r ♦r③ r♥ ❱ ♦♥ P♦s♦② Pr③ ♠♦ ♦r t ♣♥♥ ♦ t tr ♦♥t♥t t ♣♣ strss ♥ ♥st♥♦♥t♥ ♦♠♣♦sts ♦r♥♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ rt♥ ♥rs♦♥ ♥ ♦ ② ❲♠s♦♥♦♥tr♦♥ ♣r♦t♦♥ ♥ strtr ♣rt ♦♠♣♦sts srt♦♥s ♦ ♥t tr♠♦rsst♥ ♣③♦rsst♥ ♥ ♠rsst♥ P②s ♦ ♥♦ ♣

❬❪ t ② ♦sss P③♦rsstt② ♦ ♠♥t♦r♦♦ st♦♠rs ♦r♥ ♦ P②ss ♦♥♥s ttr ♦ ♥♦ ♣

❬❪ t ② ♦sss tr rsstt② ♠♥s♠ ♥ ♠♥t♦r♦♦ st♦♠r ♦r♥ ♦ P②ss ♣♣ P②ss ♦ ♥♦ ♣

❬❪ t P ♥♦♥ ② ♦sss r♠♦rsst♥ ♥ ♥t ♠♥t♦rsst♥ ♦ ♠♥t♦r♦♦ st♦♠rs ♦r♥ ♦ P②ss ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ♦q ② ♦sss ♥t♦strt♦♥ ♥ ♣③♦rsstt② ♥ st♦♠rs t ♠♥t ♣rts ♦r♥ ♦ ♥ ♥♦ ♥♦ ♣♣

❬❪ ♦③♦s ② t♥ r♦strtr ♥ ♣r♦♣rts ♦ ♠♥t♦r♦♦ st♦♠rs ♣♥ ss Psr

❬❪ ❳ ❩ ❨ ♥ ② ❨ ♥ ♦♥♥r ♣rssr♣♥♥t ♦♥tt② ♦ ♠♥t♦r♦♦ st♦♠rs ♠rt trs ♥ trtrs ♦ ♥♦ ♣

❬❪ r♥á♥③♦♥♦ ③ó♣③ ♦r③ ② ❱♥t ♠t♦♥s ♦ ♣♦②s♣rs ♠♥t♦r♦♦ s strtr ♥ ♥t ♥stt♦♥ ♦r♥ ♦ ♦♦② ♣rs♥t♦ ♥♦ ♣♣

♦rí

❬❪ ❳ ♦♥ ❨ ❳ ② ❳♥ ♥t♦♥ strss ♥♥♥t ♥ ♦♦ ss♣♥s♦♥ ♦ ♣r♠♥t ♥ s♣r♣r♠♥t♣rts s♣rs ♥ rr♦ ♠♠ ♦t ♠ttr ♦ ♥♦ ♣♣

❬❪ r♦ ♥r ② ♠♦ ❯♥rst♥♥ t s♣rs♦♥s ♦ s♣r♣r♠♥t ♣rts ♥r str♦♥ ♠♥t s r ♦ ♦♥♣ts t♦r② ♥ s♠t♦♥s ♦t ttr ♦ ♥♦ ♣♣

❬❪ ❳ ♦♥ ❨ ❳ ❳♥ ② ❲ ♥ ♠t♦♥ ♦ ♠♥t♦♥ rrr♥ ♠r♦strtrs ♦ ♠♥t♦r♦♦ ♣st♦♠rs ♦t ttr ♦ ♥♦ ♣♣

❬❪ ❨ ❳ ❳ ♦♥ ② ❳♥ ♥t♦♥ ♠r♦strtr rtr③t♦♥ ♦ ♠♥t♦r♦♦ ♣st♦♠rs s♥ ♠♣♥s♣tr♦s♦♣② ♦t ttr ♦ ♥♦ ♣♣

❬❪ P♥ ❳♥ ② ❳ ♦♥ ♥t ♣♥♥t tr♦♦♥tt② ♦ t r♣t ♦♣ ♠♥t♦r♦♦ ♣st♦♠rs♦t ♠ttr ♦ ♥♦ ♣♣

❬❪ ❳♥ ❨ ❳ ② ❳ ♦♥ ♥t ♣r♦rss ♦♥ t ♠♥t♦r♦♦ ♣st♦♠rs ♥tr♥t♦♥ ♦r♥ ♦ ♠rt ♥ ♥♦trs ♦ ♥♦ ♣♣

❬❪ ♥ r♦strtr ♥ r♦♦② ♦ ♠♥t ②r ♠trs r ♦ ♣♣ ♥s ♣♣

❬❪ ü♥tr ❨ ♦r♥ ü♥tr ② ♥ ❳r② ♠r♦t♦♠♦r♣ rtr③t♦♥ ♦ strtr ♠♥t♦r♦♦st♦♠rs ♠rt trs ♥ trtrs ♦ ♥♦ ♣

❬❪ ♥r♠♥♥ ② ♥ ♥stt♦♥ ♦ t ♠♦t♦♥ ♦ ♣rts ♥ ♠♥t♦r♦♦ st♦♠rs ② ❳µ ♠rt trs♥ trtrs ♦ ♥♦ ♣

❬❪ ♥r♠♥♥ ü♥tr ♦r♥ ② ♥ ♦♠♣rs♦♥t♥ ♠r♦♥ ♠r♦strtr ♦ ♠♥t♦t ♦♠♣♦sts ♥♦r♥ ♦ P②ss ♦♥r♥ rs ♦ ♣ P Ps♥

❬❪ ♦rát ü♥tr ❨ ♦r♥ ♥r♠♥♥ ② ♥❳µ ♥②ss ♦ ♠♥t ♥ ♣s tr♥st♦♥s ♥ ♠♥t♦r♦♦ st♦♠rs ♠rt trs ♥ trtrs ♦ ♥♦ ♣

♦rí

❬❪ ♦♥r rs rs r ② t♦r rt ① tr♦ rr② ♦r r♦r♥ sr ♣♦t♥ts ♦r♥ ♦ ♥r♦s♥ ♠t♦s ♦ ♥♦ ♣♣

❬❪ ❲ ♦ ♦♥tt ♦r tr② ♦♥♥t♥ ♣r♥t rt♦r t q r②st s♣② ❯ Pt♥t

❬❪ ❨ ❲♥ ❩♥ ❨ ♥ ♥ ② ♥ trss♣♥♥t♣③♦rsstt② ♦ t♥♥♥♣r♦t♦♥ s②st♠s ♦r♥ ♦ ♠trss♥ ♦ ♥♦ ♣♣

❬❪ ♦ rt♦♥ ② ♦②♦ ♦♥ ttr♦♠♥ ♥ str♥ rs♣♦♥s ♦ r♦♥ ♥♥♦ts ♥♥♦♦♠♣♦sts ♥ P ♠rt trtrs ♥ trs ♦♥strtt♦♥ ♥ t ♦♥t♦r♥ ♣♣ ♥tr♥t♦♥♦t② ♦r ♣ts ♥ P♦t♦♥s

❬❪ tt♥ ② P rt P③♦rsst♥ ♥ ♣♦②♠r ♥♥♦♦♠♣♦stst s♣t rt♦ ♣rts ♣♣ ♠trs ♥trs♦ ♥♦ ♣♣

❬❪ ♠♥ ② P rt ts ♦ ♥trt st♥ ♥ ♥♠♥t♦♥ t♥♥♥ rsst♥ ♥ str♥ s♥stt② ♦ ♥♥♦t♣♦②♠r♦♠♣♦st ♠s ♥♦t♥♦♦② ♦ ♥♦ ♣

❬❪ ♠ Pt♣♦♥t ② ② Pr P③♦rsstt② ♦ ❲sP ♥♥♦♦♠♣♦st ♥ r♦ tr♦ ♥ ②st♠s t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♣♣

❬❪ ②♥ ② r♥ ♥t ♣③♦rsst rs♣♦♥s ♥ ③♥♣♦②♠t②s♦①♥ ♦♠♣♦sts ♥r ♥① ♣rssr ♦r♥ ♦P②ss ♣♣ P②ss ♦ ♥♦ ♣

❬❪ tss ♥s ♦s♥s ③ t♦ ♦r♥ Pr♦ ② ♦③③ ♠rt ♣③♦rsst t♥♥♥ ♦♠♣♦st ♦r① r♦♦t s♥s♥ s♥ ♠rt trs ♥ trtrs ♦ ♥♦ ♣

❬❪ ❳ ❨ ♥ ♥ P ♦ ♦ ❩♥ ② ❲ rt♦♥ ♦ tr♠♥s♦♥ r♣♥ ♦♠♣♦② ♠t②s♦①♥ ♦♠♣♦sts ♥ tr ♣♦t♥t ♣♣t♦♥ s str♥ s♥s♦r ♣♣ ♠trs ♥trs ♦ ♥♦ ♣♣

♦rí

❬❪ r ♦ 23 s sss ♦r tr♦♥s ♥ rt♣♣t♦♥s ♥tr♥t♦♥ trs s ♦ ♥♦ ♣♣

❬❪ ❱♦♥♥t♥♦t r♠ r P ②sr ② trässrt② ♦ tr ♣r♦♣rts ♦ ♣③♦rsst ♣sts ♥ tr♠♥t♦♥ ♦ t tr tr♥s♣♦rt ♦r♥ ♦ t r♦♣♥ r♠ ♦t② ♦ ♥♦ ♣♣

❬❪ sr ② ❲♥② t ♦ ♥♥♦t ♥♠♥t♦♥ ♣r♦t♦♥ ♦♥tt② ♥ r♦♥ ♥♥♦t♣♦②♠r ♦♠♣♦stsP②s ♦ ♥♦ ♣

❬❪ ❲t ♦♥♥ ♥s② ② ❲♥②♠t♦♥s ♥ tr ♦♥tt② ♦ ♣r♦t ♥t♦rs ♦ ♥t r♦s t r♦s rs ♦ ① ♥♠♥t P②s ♦ ♥♦ ♣

❬❪ rt♥s ② ♦♦r ♦♥t♥♠ ♣r♦t♦♥ trs♦s ♥ t♦ ♠♥s♦♥s P②s ♦ ♣

❬❪ ♥ts ♥♦ ❱t③ ② ♥r ②r♠♥t♦r♦♦ st♦♠r ♥♥ ♦ ♠♥t ♥ ♦♠♣rss♦♥ ♣rssr ♦♥ ts tr ♦♥tt② ♦r♥ ♦ ♥str ♥♥♥r♥ ♠str② ♦ ♥♦ ♣♣

❬❪ ❩ ❩♥ ❳ ❨ ♥ ❲♥ ② ❲ ②r♦tr♠ s②♥tss ♥ sss♠② ♦ ♠♥tt 34 ♥♥♦♣rts t t♠♥t ♥ tr♦♠ ♣r♦♣rts ♦r♥ ♦ r②st r♦t♦ ♥♦ ♣♣

❬❪ r♥t ♦r P♦rt ♦ ♦ ❱♥r st ② r ♥t r♦♥ ♦① ♥♥♦♣rts s②♥tss st③t♦♥ t♦r③t♦♥ ♣②s♦♠ rtr③t♦♥s ♥ ♦♦♣♣t♦♥s ♠ rs ♦ ♥♦ ♣♣

❬❪ ❩ ❳ ❲♥ ❨♦ ❩ ♥ ♦ ♥❳ ♦ ② ♥ ②♥tss ♦ ♠♥tt ♥♥♦♣rts ♥ ❲ ♠r♦♠s♦♥ ♦r♥ ♦ ♠trs s♥ ♦ ♥♦ ♣♣

❬❪ ❨ ♥ ❨ ❲ ❨ ❲❨ s ❲ ♥ ② ♦ q♦s s♣rs♦♥s♦ ♠♥tt ♥♥♦♣rts t 3

+ srs ♦r ♠♥t ♠♥♣t♦♥s ♦ ♦♠♦s ♥ ♦♥trst ♥ts ♦♠trs ♦ ♥♦ ♣♣

♦rí

❬❪ ♠ ② ♠ ②♥tss ♦ rr♦t ♠♥t ♥♥♦♣rts ② s♦♥♦♠ ♠t♦ ♦r ♦♥trst ♥t ♦r♥ ♦ ♥ts♠ ♥ ♥t trs ♦ ♣♣

❬❪ ❲ P ♦ ♠ ② ❨ ♠ ♥♠ ♠♥tt ♥♥♦♣rts ②♥tss ♠r♦strtr ♥ ♠♥t♣r♦♣rts trs ttrs ♦ ♥♦ ♣♣

❬❪ tr r③ ♦r ③ tt ② r r♦ rs♣♦♥s ♦ ♥s♦tr♦♣ ♠♥t♦r♦♦ st♦♠rs ♦♥ ①♣r♠♥ts P②s ♦ ♥♦ ♣

❬❪ ♦♦② ♦r♥♦ ♦r rrr P ♥t♦♥ tt ③ r Ptt♥r ② ❱ rs r♦♠tr ♣r♦ss♠② ♦ ♠♥t♦tt tr ♥ ♠♥t ♥♥♦♣rts s♥♦ t♣s ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ P ♥t♦♥ r ♦r Pr③ ②② r ②♥tss ♥ rtr③t♦♥ ♦ ♦24 ♠♥t♥♥♦ts ♥♥♦r♦s ♥ ♥♥♦rs ♦r♠t♦♥ ♦ ♠♥t strtrst♦♠rs ② ♠♥t ♥ ♥♠♥t ♦ ♦24 ♥♥♦r♦s♦r♥ ♦ ♥♦♣rt sr ♦ ♥♦ ♣♣

❬❪ P ♥t♦♥ r ② ② ♦r ♥s♦tr♦♣② ♥ r①t♦♥♣r♦sss ♦ ♥①② ♦r♥t ♦24 ♥♥♦♣rts s♣rs ♥P P②s ♦♥♥s ttr ♦ ♥♦ ♣♣

❬❪ ♥ P ♥t♦♥ ③ Pr③ tr ♦r r ② r ♥t ♥ st ♥s♦tr♦♣②♥ ♠♥t♦r♦♦ st♦♠rs s♥ ♥s ♥♥♦♣rts ♥♥♥♦♥s ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ♥ ♦r ② r trtr tr♥ ♠♥t ♣r♦♣rts ♦ 1−x❨x3 ♦t♥ ② s ♦♣r♣tt♦♥ ♦r♥ ♦ ♦②s ♥ ♦♠♣♦♥s ♦ ♣♣

❬❪ r③rr♦ ♦rrsstr♦ ❱ ♦♥③á③ ❯ rt③ ② ♦s ♥tt ♥ ♠♥ttsr ♦rs ♥♥♦♣rts tt ♠♥t ♦r ♦r♥ ♦ ♦ tt ♠str②♦ ♥♦ ♣♣

♦rí

❬❪ ♥ ♥ ♦s P♥r ❨s② P ♥tt ♥♥♦♣rts t t♥ ♦ ♦r sr s♦r♥ ♦ ♦♦ ♥ ♥tr s♥ ♦ ♥♦ ♣♣

❬❪ ss s sr♦ ② ó♣③♥t ②♥tss♦ sr♦t ♠♥tt ♥♥♦♣rts ♦r♥ ♦ ♦♥r②st♥♦s ♦ ♥♦ ♣♣

❬❪ ❨ ❨♥ ❩❨ ❩ ❩♦♥ ts ❨ ❳ ② ❱♥tsr♥②♥tss ♥ rtr③t♦♥ ♦ st q♦s s♣rs♦♥s ♦ sr♥♥♦♣rts tr♦ t t♦♥s ♣r♦ss tr ♠ ♦ ♥♦ ♣♣

❬❪ ♥r ② ❲s r♥♦s♦♥ ♠str② r♦♠ ♦s t♦trs ♦♥ ❲② ♦♥s

❬❪ ♠♥♦ ❲ ❲ ② ♥③r r ♠♦t♦♥ ♦s②r ♣♦② ♠t② s♦①♥ ♥t♦rs ② tr♦t ♥ tr♦t♦③♦♥ trt♠♥t ♦r♥ ♦ ♦♦ ♥ ♥tr s♥♦ ♥♦ ♣♣

❬❪ t ② ② ♠♠ ♣t ♣r♦♣rts ♦ t♥♠s ♦ ③♥ ♦① q♥t♠ ♦ts ♥ ♣♦②♠t②s♦①♥ ❯❱♦♥ ♥ t t ♦ r♦ss♥♥ ♦r♥ ♦ ♦♦ ♥ ♥trs♥ ♦ ♥♦ ♣♣

❬❪ sts r♦♥❩♣ ♥ ♥♥ rs ❱♥ ② ❲t ♥♥ ♦ r♦ss♥r ♦♥♥trt♦♥ ♦♥ tr♦ss♥♥ ♦ P ♥ t ♥t♦r strtrs ♦r♠ P♦②♠r♦ ♥♦ ♣♣

❬❪ ♦♥ts s ♥ts ② P ❯r♥ tr ♦♥tt②♦ ♠t ♣♦rs ♥r ♣rssr ♣♣ P②ss ♦ ♥♦ ♣♣

❬❪ ❲P s♦ ❲ ❨ ♥ ♥ ❨ ❨♥ ② ♥ ① t♠♣rtr s♥s♦r rr② s ♦♥ r♣t♣♦②♠t②s♦①♥ ♦♠♣♦st ♥s♦rs ♦ ♥♦ ♣♣

❬❪ r♠ ② ❨ t② ♦ ♣③♦rsst♥ t♦ r♦♥ ♥♥♦tP ♦♠♣♦st ♠trs ♦r ♥♥♦s♥s♦rs ♥t ♦♥r♥ ♦♥ ♥♦t♥♦♦② ♣♣

♦rí

❬❪ ❳ ② ❲ ♦ tr♥♣♥♥t rsst♥ ♦ ♣♠s ♥r♦♥ ♥♥♦ts ♦♠♣♦st ♠r♦strtrs ♥♦t♥♦♦② r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣

❬❪ r♣rs ♦r③ ♦st♦♥ r♠ s♥❱ P♠ ② ❱ rr② tr♦♥t♥♥♥ ♠♦t♦♥ ♥ ♣r♦t♥♥t♦r ♦ r♣♥ q♥t♠ ♦ts rt♦♥ ♣♥♦♠♥♦♦ ♥rst♥♥ ♥ ♠t②♣rssr s♥s♥ ♣♣t♦♥s ♥♦ ttrs ♦ ♥♦ ♣♣

❬❪ ③ ❲ ❩ ❩ ❩♥ ❨♥ ② ❩ ♥ P③♦rsst ♦r ♦ tr② ♦♥t r♦♥ rstr♠♦♣st st♦♠r ♥♥♦♦♠♣♦sts ♦r♥ ♦ ♥ P②ss ♦ ♥♦ ♣♣

❬❪ ❩s ♥t P♦♥s ♥rts ② r♦sP♦②s♦♣r♥♥♥♦strtr r♦♥ ♦♠♣♦st s♦t tr♥t ♦r♣rssr s♥s♦r ♣♣t♦♥ ♥s♦rs ♥ tt♦rs P②s♦ ♥♦ ♣♣

❬❪ ❨ ② ♦ ♥t ♥t♥st② t ♦♥♣♥ tr ♣t♦r rtrsts ♥ s♦stt② ♦ ♠♥t♦r♦♦ st♦♠r ♦♦ ♥ P♦②♠r ♥ ♦ ♥♦ ♣♣

❬❪ ❨ ② ♦ P②s rtrsts ♦ ♠♥t♦r♦♦ ss♣♥s♦♥s ♥ tr ♣♣t♦♥s ♦r♥ ♦ ♥str♥ ♥♥r♥ ♠str② ♦ ♥♦ ♣♣

❬❪ ♦q ② ♦sss ♥s t ♥ st♦♠rs t ♣rts ♥ ② ♠♥t ♥tr♥t♦♥ ♦r♥ ♦ s♦s ♥strtrs ♦ ♥♦ ♣♣

❬❪ ♥ ②♦ ② P ♦r♠♥ r ♦♥ t ♥s tr♦♣♥ P♦②♠r ♦r♥ ♦ ♥♦ ♣♣

❬❪ ♦q ♦sss ③♦ ② r r♦♠♥♥②ss ♦ ♥ st♦♠r t ♣rts ♦r♥③ ♥ ♥strtr ♦r♥ ♦ ♠trs s♥ ♦ ♥♦ ♣♣

❬❪ t♦ ②r ♦♥♠♥ ② ♠♦♥♥ t♦♥ ♦② ♦♠♣♥t ♠♥t♦t st♦♠rs t ♦♦ss ♠♥t♦r♦♦ rs♣♦♥s ♦r♥ ♦ ♣♣ P♦②♠r ♥ ♦ ♥♦

♦rí

❬❪ ❨ ❳ ② ❨♥ ♥♥ ♦ ♦♠♣♦st♦♥ ♦r♦♥② r♦♥ ♣rts ♦♥ ②♥♠ ♠♥ ♣r♦♣rts ♦ ♠♥t♦r♦♦ st♦♠rs ♦r♥ ♦ ♥ts♠ ♥ ♥t trs♦ ♥♦ ♣♣

❬❪ ❳ ♦♥ ② Prt♥ ♠♥t♦r♦♦ t♦ ♠♥t♦r♦♦ st♦♠rs ♥r ♥♦r♠ ♣rssr ♥ ♦r♥ ♦P②ss ♦♥r♥ rs ♦ ♣ P Ps♥

❬❪ ❳ ♦ ❳ ❲ ♥ ❳ ♦♥ ❨♥ ❲ ♥② ❳ ♥ r♦strtr ♥ ♠♥t♦r♦♦ ♣r♦♣rts ♦t tr♠♦♣st ♠♥t♦r♦♦ st♦♠r ♦♠♣♦sts ♦♥t♥♥ ♠♦ r♦♥② r♦♥ ♣rts ♥ ♣♦② st②r♥t②♥t②♥♣r♦♣②♥st②r♥ ♠tr① ♠rt trs ♥ trtrs♦ ♥♦ ♣

❬❪ ♥r ② ♥qst sstt② ♦ ♦♠♣♦st ♦♥t♥♣♦②♠r s ♥t♦♥ ♦ t♠♣rtr ♣rssr ♥ ♥r♦♥♠♥t ♣♣t♦♥s s ♣rssr ♥ s ♦♥♥trt♦♥ tr♥sr ♦r♥ ♦♣♣ P②ss ♦ ♥♦ ♣♣

❬❪ P ❩ ❲♥ ❨ ② ❨ t② ♦♥ t ♦ ♣♦②♠t②s♦①♥ ♥ ♥t♠s♥t ♠ rtr♥t ♣♦②♣r♦♣②♥ ♦r♥ ♦ ♣♦②♠rrsr ♦ ♥♦ ♣♣

❬❪ ♦r Pr♦t② ♥ ttsts ♦r ♥♥r♥ ♥ t ♥s♥ r♥♥

❬❪ ② P ttsts ♠t♦s ♥ ♣♣t♦♥s ♦♠♣r♥s rr♥ ♦r s♥ ♥str② ♥ t ♠♥♥ tt♦t

❬❪ r♦♥② ② tr ♥tr♦t♦♥ t♦ ♣r♦t♦♥ t♦r② ②♦r r♥s

❬❪ ♥ ② ❲ tr Pr♦t♦♥ ♥ ♦♠♣♦sts ♦r♥ ♦ tr♦r♠s ♦ ♥♦ ♣♣

❬❪ ♠♥ ② ❩ ♥t ♦♥t r♦ ♦rt♠ ♥ ♣rs♦♥ rsts ♦r ♣r♦t♦♥ P②s ttrs ♦ ♥♦ ♣

❬❪ ❩ ② ♠♥ ♦♥r♥ ♦ trs♦ st♠ts ♦r t♦♠♥s♦♥ ♣r♦t♦♥ P②s ♦ ♥♦ ♣

❬❪ ❩ ♣♥♥♥ ♣r♦t② ♥ ♣r♦t♦♥ P②s rttrs ♦ ♥♦ ♣

♦rí

❬❪ ❳ ❩♥ ♥ ❨♥ ❩♥ ❨♥ ② ❨♥ ②♦♥t ♣♦②♠r ♦♠♣♦sts r♦♠ r♦♦♠t♠♣rtr ♦♥ q r ♣♦①② rs♥ t ♦ ♥tr♣s ②r ♦♥ ♣r♦t♦♥ ♦♥t♥ ♥s ♦ ♥♦ ♣♣

❬❪ t♦③r♥♥③ ♥rs ② ♠r③Pst♦r ♦♥♠♦♥♦t♦♥ s③ ♣♥♥ ♦ t rt ♦♥♥trt♦♥ ♥ ♣r♦t♦♥ ♦strt r r♦s ♥r qr♠ ♦♥t♦♥s r♦♣♥ P②s ♦r♥ ♦♥♥s ttr ♥ ♦♠♣① ②st♠s ♦ ♥♦ ♣♣

❬❪ ❨ ♠ ❨ ❨♥ ② ❨♦♦ ①♣♦s ♣r♦t♦♥ ♥ ♥♥♦ts s②st♠ P②s ♦ ♥♦ ♣

❬❪ r ② ♦③♦s Pr♦t♦♥ ♥ ♦♠♣♦st ♦ r♥♦♠ st ♦♥t♥ ♣rts ♦ tt ♦♠♠♥t♦♥s ♦ ♥♦ ♣♣

❬❪ r ♥rs♦♥ ①♥r ② ❲♥r ① ♦♠♥ ts rt♦♥ t♦ t ♦♥st ♦ ♣r♦t♦♥ P②s r ♦ ♥♦ ♣

❬❪ r ♥♥♠ ② ♥rs♦♥ rt ♦r ♦ tt♦♠♥s♦♥ sts s②st♠ P②s ttrs ♦ ♥♦ ♣

❬❪ ❨ ♥ ❩ ❲♥ ❳ ♦ ❨ ❳ ♥ ♥ ❩♦② ❳ ❩♥ ♥ ♣s r♠ ♦ tr ♥r ♥t ♣rssr rs ♦ t ♦st♥st② trt s ♥ ♥s ♦ ♥♦ ♣

❬❪ ♠ ❩♥ ② P♦ ♦♥r♥ ♦rs ①t♥♥ r♦♠♣♦②♠r srs P②s ♦ ♥♦ ♣

❬❪ P♦ ♦rt Ps ♦ ❲tr ②♦♥ ♦ q ♥❱♣♦r ♥r ♦♥s

❬❪ rr t♥♥ ❲♥ r ❯ sr ❲ ♠ ② r♦r qr ♥ r♣♥ ♥♥♦♣rs tr ♦ ♥♦ ♣♣

❬❪ P t♥s ② ärr s♦♥ ♥ ♦♥♥s ♠ttr ♣r♥r

❬❪ P ②♥♦s t♥② ② ❲ ♥ r ♦♥t r♦r♥♦r♠③t♦♥ r♦♣ ♦r ♣r♦t♦♥ P②s ♦ ♣♣

♦rí

❬❪ ❩ ③♥t t♦st r♥♦r♠③t♦♥ r♦♣ ♥ ♣r♦t♦♥ tt♦♥s ♥ r♦ss♦r P②s ttst ♥s ♥ ts ♣♣t♦♥s ♦ ♥♦ ♣♣

❬❪ ➎ t♥♦ ② ♥ts③ s♥ ♥ s②♠♠trs②st♠s ♦ ♣r♦t♥ sts P②s ♦ ♣

❬❪ r② ♥♠r ♦ ♥♣♥t s♣♥♥♥ strs ♥ t♦♠♥s♦♥ ♣r♦t♦♥ ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣

❬❪ ❲tts r♦ss♥ ♣r♦t② ♦r rt ♣r♦t♦♥ ♥ t♦ ♠♥s♦♥s ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣♣

❬❪ ♥♥s ❨ ♥t♥ ② P P♦♦t ♦♥♦r♠ ♥r♥ ♥t♦♠♥s♦♥ ♣r♦t♦♥ ♠ t ♦ ♦ ♥♦ ♠tP ♣♣

❬❪ Prss♥r ② ♦♦♥② ♠r rsts ♦r r♦ss♥ s♣♥♥♥♥ r♣♣♥ ♥ t♦♠♥s♦♥ ♣r♦t♦♥ ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣

❬❪ ② ❲s♦♥ r♠♠s ♣r♦♦ ♦ ❲tts ♦r♠ ♥♥s ♦ Pr♦t② ♦ ♥♦ ♣♣

❬❪ P ♦ ② r♦♥② ♥♦r♠③t♦♥ r♦♣ s♥ ♥ ♥rst② ♥ s♣♥♥♥ ♣r♦t② ♦r ♣r♦t♦♥ P②s ttst♥s ♥ ts ♣♣t♦♥s ♦ ♥♦ ♣♣

❬❪ P ♦ ② r♦♥② ♥ ♥ ♥rst② ♥ t s♣♥♥♥♣r♦t② ♦r ♣r♦t♦♥ P②s ♦ ♥♦ ♣

❬❪ ❲s♦♥ ♥♦r♠③t♦♥ r♦♣ ♥ rt ♣♥♦♠♥ ♥♦r♠③t♦♥ r♦♣ ♥ t ♥♦ s♥ ♣tr P②s ♦ ♥♦ ♣

❬❪ ♥ ♥ ② ❨ ♦♦ ♥ts③ s♥ trs ♦tr ♦♥tt② ♣r♦t♦♥ ♥ ♥♥♦♦♠♣♦sts ♥♦s♥ ♥♥♦♥♥r♥ ♦ ♥♦ ♣♣

❬❪ ❨♦♦ ❲ ♦ ② ❨ ♠ ♦♥tt② ♦ st ♣r♦t♦♥strs t ♥s♦tr♦♣ ♥♠♥ts ♦r♥ ♦ t ♦r♥ P②s♦t② ♦ ♥♦ ♣♣

❬❪ ② ❩♥ ♥ts③ s♥ ♥ st ♣r♦t♦♥ P②s ♦ ♥♦ ♣

♦rí

❬❪ P ♥ ②r ♦♦②♥ ♥ ♦♣③ P② t♥② Pr♦t♦♥ t♦r② ♦♥♦♥ Ptr♦♣②s♦t② sttr ♦ ♥♦

❬❪ r♦♥② ② ♦ ♦♠♠♥t ♦♥ ❵♣♥♥♥ ♣r♦t② ♥ ♣r♦t♦♥ P②s r ttrs ♦ ♥♦ ♣

❬❪ ❩ ♦rrt♦♥t♦s♥ ①♣♦♥♥t ♦r t♦♠♥s♦♥ ♣r♦t♦♥ P②s ♦ ♣

❬❪ tr r ♦r ♦rt♠ t♦rs ♥r t t♦♠♥s♦♥♣r♦t♦♥ trs♦ P②ss ttrs ♦ ♥♦ ♣♣

❬❪ P ② r Pr♦t♦♥ ♥ ♦♥tt② ♦♠♣trst② P②s ♦ ♣♣

❬❪ P ♠s s♦r♥ ② ü qr♠ ♥ ♥♦♥qr♠ ♦r ♦ rr♦s①♣r♠♥ts ♥ t♦r② ❩tsrt ür P②ss ♠ ♦ ♥♦ ♣♣

❬❪ ♠r♦stt r♠ r r ♥♥ ② P ②sr ♦t♦♥ ♦ t t♥♥♥♣r♦t♦♥ ♣r♦♠ ♥ t ♥♥♦♦♠♣♦st r♠ P②s ♦ ♥♦ ♣

❬❪ tt♥ ② P ♥ r ♦♦t ♦♥t♥♠ ♣r♦t♦♥ ♦ ♣♦②s♣rs ♥♥♦rs P②s r ttrs ♦ ♥♦ ♣

❬❪ t s ② st♦tr③s ♥♥♥♣r♦t♦♥ ♦r ♦ ♣♦②s♣rs ♣r♦t ♥ ♦t ♣s♦s♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ P ttr ② r♠ ♥♥♥ ♦♥tt② ♥ ♥s♦tr♦♣♥♥♦r ♦♠♣♦sts ♣r♦t♦♥s ♠♦ ♦r♥ ♦ P②ss♦♥♥s ttr ♦ ♥♦ ♣

❬❪ ❨ ❨ ♦♥ ② ♥ tr♠♥♥t r♦ ♦ t♥♥♥ rsst♥♥ tr ♦♥tt② ♦ ♣♦②♠r ♦♠♣♦sts r♥♦r ② s♣rs r♦♥ ♥♥♦ts ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ♦st♥s♦♥ ② ❲ ♦ ♦♠♥♥t r♦ ♦ t♥♥♥ rsst♥ ♥ t tr ♦♥tt② ♦ r♦♥ ♥♥♦ts♦♠♣♦sts ♣♣ P②ss ttrs ♦ ♥♦ ♣

♦rí

❬❪ s♦② ❯ ♥rr ② P rt♥② ♥♥♥ ♦♥tt②♥ ♣③♦rsstt② ♦ ♦♠♣♦sts ♦♥t♥♥ r♥♦♠② s♣rs ♦♥t ♥♥♦♣tts trs ♦ ♥♦ ♣♣

❬❪ r② ♥♠r ♦ ♥♣♥t s♣♥♥♥ strs ♥ t♦♠♥s♦♥ ♣r♦t♦♥ ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣

❬❪ ♥ ♥♥rs r♥ ② ♦♥ ss t♦r② ♠♦rrsts Pt Ps♥ ♠t

❬❪ ❲ ② ♦tr♦s t♠ts ♦tr ❱rs♦♥ ♦♣♠♥t ♠ tt♣s♠t♦r

❬❪ r ♦♠♣tt♦♥ ♦♠tr② ♦rt♠s ♥ ♣♣t♦♥s♣r♥r

❬❪ ♠♥ ② ❩ st ♦♥t r♦ ♦rt♠ ♦r st♦r ♦♥ ♣r♦t♦♥ P②s ♦ ♣ ♥

❬❪ ♥t♦ ② ♦rqt♦ Prs tr♠♥t♦♥ ♦ t rt trs♦ ♥ ①♣♦♥♥ts ♥ tr♠♥s♦♥ ♦♥t♥♠ ♣r♦t♦♥ ♠♦ ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣

❬❪ ♠r♦stt ♥ t ♥st♦r♦♥t♦r tr♥st♦♥ ♥ ♣♦②♠r ♥♥♦♦♠♣♦sts ss ♦t♦r s♥♥

❬❪ ❱ s♥ ♦♥t♥♠ ♣r♦t♦♥ ♦ ♦r♣♣♥ ss t strt♦♥ ♦ r ♥ ♣♦r t P②s ♦ ♣

❬❪ ③♥ ♥②♦♣ ♦ ♠t♠ts ♣r♥r❱r r♥ ❨♦r

❬❪ s ② ♦rrs rt ♣r♦t st♣st s♥t ♠t♦ ♦r ♥♦♥♥r ♥rs ♣r♦♠s t s♣rst② ♦♥str♥ts ♥rsPr♦♠s ♦ ♥♦ ♣

❬❪ r ② ❲ ♦♥♥r rss♦♥ ❲② rs ♥ Pr♦t②♥ ttsts ❲②

❬❪ ♦ré ♥rrqrt ♦rt♠ ♠♣♠♥tt♦♥ ♥t♦r② ♥ ♠r ♥②ss ❲ts♦♥ ♦ ♦ tr♦ts ♥ t♠ts ♣♣ ♣r♥r r♥ r

❬❪ ♦♥r r♠ r ② P ②sr ♣t♠ ♣r♦t♦♥♦ s♦rr srt ♦♠♣♦sts P②s ♦ ♥♦ ♣

♦rí

❬❪ ♦♥ t ♦ Prssr ♥ ♠♣rtr ♦♥ tr ♦♥tt② ♦ P ♦♠♣♦sts ss ♦t♦r ♦♥♦r ❯♥rst②

❬❪ r P②s ♣r♦♣rts ♦ ♣♦②♠rs ♥♦♦ ♣r♥r

❬❪ ❨ ② ❲ ♥ r ♠♥s♦♥ t♦r② ♦ tr ♦♥ttrsst♥ ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ♦♠③ r♥ ❲ ♥ P ❩♥ ❨ ② ♥ ①♣r♠♥t t♦♥ ♦ r ♠♥s♦♥ t♦r② ♦tr ♦♥tt rsst♥ ♣♣ P②ss ttrs ♦ ♥♦ ♣

❬❪ P ❩♥ ② ❨ ♥ s ♦r tr ♦♥tt rsst♥t ss♠r ♠trs ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ♦♠ ❲♠s♦♥ ② ♦♠ tr ♦♥tts ♦r② ♥♣♣t♦♥ ♣r♥r r♥ r

❬❪ P ❩♥ ❨ ❲ ♥ ♦♠③ r♥ ❩r ② ♥ ♦♥tt rsst♥ t ss♠r ♠trs ♦♥tts♥ t♥ ♠ ♦♥tts ♥ tr ♦♥tts ♦♠ t♦♠ ♦♥r♥ ♦♥ ♣♣

❬❪ P ❩♥ ts ♦ sr r♦♥ss ♦♥ tr ♦♥tt t♥♥ ♥♥♠♥t ss ♦t♦r ❯♥rst② ♦ ♥

❬❪ ♠s♦♥♦ ♥♦♦ ♦ t P②s♦♠ Pr♦♣rts ♦ t ♠♥ts ♣r♥r ❯

❬❪ ❱♥ ② ❱♥ P ♥tr♥ ♠ss♦♥ ♥ r♦♥ ♦ ♥tr rr ♥③ts ♦r♥ ♦ ♣♣ P♦②♠r ♥ ♦ ♥♦ ♣♣

❬❪ t③♥ tr rt ♦♥t♥ ♣♦②♠r ♦♠♣♦sts ♦r♥ ♦ tr♦r♠s ♦ ♥♦ ♣♣

❬❪ tss ❱ ♥s ② Prr ① tt s♥s♥s ♦♥ ♣③♦rsst ♦♠♣♦sts r ♥s♦rs ♦ ♥♦ ♣♣

❬❪ ❳❲ ❩♥ ❨ P♥ ❩♥ ② ❳ ❨ ♠ ♣♥♥ ♦♣③♦rsst♥ ♦r t ♦♥t♦r ♣♦②♠r ♦♠♣♦sts ♦r♥♦ P♦②♠r ♥ Prt P♦②♠r P②ss ♦ ♥♦ ♣♣

♦rí

❬❪ ♥tr r öss rs ② ♦r ♥ ♣♣r♦ ♦r ♠♦♥ ♣③♦rsst ♦r s♥s♦rs s ♦♥ s♠♦♥t ♣♦②♠r ♦♠♣♦sts tr♦♥s r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣

❬❪ ts♦ ② ❲♥② tr ♣r♦♣rts ♦ ♣♦②♠r ♥♥♦♦♠♣♦sts ♦♥t♥♥ r♦ ♥♥♦rs Pr♦rss ♥ P♦②♠r ♥♦ ♣♣

❬❪ ③r ② P♦rt ♦♠♣r♥s st② ♦ tr♥sst♦rs s♦♥ ♦♥t ♣♦②♠r ♠tr① ♦♠♣♦sts tr♦♥ s r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣

❬❪ rs♠r ❨ ♥ ② ♦♥t ♥♥♦♦♠♣♦sts s ♦♥ ♣♦②st②r♥ ♠r♦s♣rs ♥ sr ♥♥♦rs② t① ♥♥ ♣♣ ♠trs ♥trs ♦ ♥♦ ♣♣

❬❪ ♦♥ ② ❩ ❩ ♥ t ♠♥s♠ ♦ ♣③♦rsstt② ♦ r♦♥♥♥♦t ♣♦②♠r ♦♠♣♦sts P♦②♠r ♦ ♥♦ ♣♣

❬❪ Pr ❨ ♦♥ ❨ ♥ ♠ ❨ ♠ ② ♦ ♥t t♥♥♥ ♣③♦rsst♥ ♦ ♦♠♣♦st st♦♠rs t♥tr♦ ♠r♦♦♠ rr②s ♦r trs♥st ♥ ♠t♠♦ tr♦♥ s♥s ♥♥♦ ♦ ♥♦ ♣♣

❬❪ ♠♠♦♥s ♥r③ tr♠ ❱ rtrst ♦r t trt♥♥ t ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣♣

❬❪ ♥ ♥tr♦t♦♥ t♦ ♥♥♥ ♥♥♥ r♦s♦♣② ❯P ①♦r

❬❪ ♥ ② ②s♥ ②♥♠ ♠r t ♥ ♠♦rt♥♥ ♥t♦♥s ②♦♥ ♥r② ♥♠♥t P②s ♦ ♥♦ ♣

❬❪ ❳ ♥ ② ❲ ♠ ♦r ts ♦♥ tr♣③♦rrr ♣r♠trs ♥ ♠t♥st♦r♠t t♥♥ ♥t♦♥s ♥s♦ ♠s ♦ ♥♦ ♣♣

❬❪ tr♥r ♦♥ ♦ tr♦♥ tr♥s♣♦rt ♥ sss♠ ♥♥♦rs ♣♦♠ tss ❩ür

❬❪ ♥②♦ ❱ ♦s♦ ♣♥♥♦ ② ♥r ♥ ♣r♦♣rts ♦ ♠♥t♦s♥st st♦♠rs ♥t♦♥ ♦ t♦♥t♥♠♠♥s ♥ ♠r♦s♦♣ t♦rt ♣♣r♦s ♦t♠ttr ♦ ♥♦ ♣♣

♦rí

❬❪ ♦♠rt ♣♥♥♦ ss ② ♥r ♦♥♦ strss ♥ str♥ ♠♣t♦♥ ts ♥ ♣♦②♠r ♠ts ♦r♥ ♦ ♦♥t♦♥♥ ♥s ♦ ♣♣

❬❪ ❨ ❨ ② ❨P ❩♦ ♦r♠t♦♥ ♦ P ♠♠r♥ ♥ ♠r♦♥tr ② tr r♦♣t ♦♠♣rs♦♥ t♥ ♦♦♥②♥♥ ♥r st ♦♥sttt ♠♦s ♦r♥ ♦ ♦♦ ♥ ♥trs♥ ♦ ♥♦ ♣♣

❬❪ ❲ rss② P♦②♠r qs ♥ t♦rs ②♦r r♥s

❬❪ ②♥♠♥ t♦♥ ② ♥s ②♥♠♥ trs ♦♥ P②ss st♦♣ t♦♥ ❱♦♠ ♥♥♠ t♦♥ ②♥♠♥ trs ♦♥ P②ss s ♦♦s

❬❪ s qrt qt♦♥ ♥r♥ts ♥ rs s♦t♦♥ r t♠t ③tt ♣♣

❬❪ ♠ Pt♣♦♥t ② ② Pr ② strt ♥ s♥st str♥ s♥s♦r s ♦♥ sr ♥♥♦rst♦♠r♥♥♦♦♠♣♦st ♥♥♦ ♦ ♥♦ ♣♣

❬❪ ♥r ♠♥ rt♥ ♣t ② ♦rsrt♦♥ ♦ ♣♦②♠t②s♦①♥ ♦♠♣♦sts t ♥ ♥♥♦♣rt ♥ ♥♥♦r rs ♥ st② ♦ tr ♠♥ ♥ ♠♥t♣r♦♣rts ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

❬❪ ② r♦s ♣♥s qt♦♥s r P♦②♠r♥♥r♥ ♥ ♦ ♥♦ ♣♣

❬❪ ❨P ❲ ❳ ❨ ② ❩♥ ♦♥ ❨♦♥s ♠♦s ♦ rr② ♥♥♦♦♠♣♦sts s♥ ♦♠♣♦st t♦rs P♦②♠r st♥ ♦ ♥♦ ♣♣

❬❪ ♥♦♥ ② ❲♥ t ♦ s♣t rt♦ ♦ ♥s♦♥s ♦♥t st ♣r♦♣rts ♦ ♥rt♦♥② ♥ ♦♠♣♦sts P♦②♠r♦♠♣♦sts ♦ ♥♦ ♣♣

❬❪ ② ♠♣ ♦r♠ ♦r t t ♠♦ ♦ ♥rt♦♥ ♥ ♦♠♣♦sts P♦②♠r ♥♥r♥ ♥ ♦ ♥♦ ♣♣

❬❪ ♥t♦rsst♦r s♥s♦r t ♠♥t♦r♦♦ st♦♠rs♦r♥ ♦ ♥str ♥ ♥♥r♥ ♠str② ♦ ♥♦ ♣♣

❬❪ ♥♥ ♦ t ♠♥t ♦♥ t tr ♦♥tt② ♦♠♥t♦r♦♦ st♦♠rs ♦r♥ ♦ ♥str ♥ ♥♥r♥♠str② ♦ ♥♦ ♣♣

♦rí

❬❪ ❨ ② ♦ ♥t♦rsst♥ rtrsts ♦ ♠♥t♦r♦♦ ♥r ♠♥t ♥str ♥♥r♥ ♠str② sr ♦ ♥♦ ♣♣

❬❪ ♥②♦ ❱ ♦s♦ ♦r♥ ♣♥♥♦ ② ♥r ♥ ♣r♦♣rts ♦ ♥t♦♥st st♦♠rs ♥ ♦♠♦♥♦s ♠♥t ♦r② ♥ ①♣r♠♥t ♥r♦♠♦r②♠♣♦s ♦ ♣♣ ❲② ♥♥ rr②

❬❪ ♥②♦ ❱ P ♦s♦ ♣♥♥♦ ② ♥r♥t♦♥st st♦♠rs ♥ ♦♠♦♥♦s ♠♥t rr rt♥r tt ♠♦ r♦♠♦r ♦r② ♥ ♠t♦♥s ♦ ♥♦ ♣♣

❬❪ ♥ t♦ ❨ ② P③♦rsst str♥s♥s♦rs ♠ r♦♠ r♦♥ ♥♥♦ts s ♣♦②♠r ♥♥♦♦♠♣♦sts♥s♦rs ♦ ♥♦ ♣♣

❬❪ ❨ r ❨♥ ❩ s ② ♥ ♥♥♥t ♥ ♣♦②♠rr♦♥ ♥♥♦t ♥♥♦♦♠♣♦st str♥ s♥s♦rt tr ♦ ♥♦ ♣♣

❬❪ ❳ ♦ ❲ ❲♥ ❳ ♥ ♦ ② ❳♦ t♠♣rtr ♥♥♥ ♥ s♣r♣r♠♥ts♠ ♥ ♦ ♦ t♥♥♥ ♥t♦♥s ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣♣

❬❪ Pr♠♥ t♥t Pt♥ ♥②♦♣②② rr rt ② ② srt♦♥ ♦ ①tr♠② ♦♥ s♣♥ r①t♦♥ t♠s ♥ ♥ ♦r♥ ♥♥♦r s♣♥ tr ♥♦t♥♦♦②♦ ♥♦ ♣♣

❬❪ ❩ ♥ P ♥ ② s♣♥ ♣♦r③t♦♥ ♥ ②t ♥tr ②r③t♦♥ ♥ ♦ ♦♠♣♦st ♠s r♦♥ ♦ ♥♦ ♣♣

❬❪ ❩ ③ ❨ ② ③ ❨♥r ② t ♥♥♥ ts ♦♥ t♥♥ ♠♥t♦rsst♥ ♥ P♦②♠♦ r♥r♠s ♥ts r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣

❬❪ P② ②♥ ♦ r♥ P P♦♥ ♦♥ ② é♥♦♥ ♦t♥♥♥ ♥♥♠♥t ♦ t tr rs♣♦♥s ♦ ♥♥♦♣rt ♥t♦rs ♠ ♦ ♥♦ ♣♣

❬❪ P♠ ♥ ♦③ ♦♥ ♠r ③♦ ♥♦ ② ❨ ③ ♣♥♣♥♥t t♥♥♥ ♥ ♠♥t

♦rí

t♥♥ ♥t♦♥s t ♥♥♦♣rts ♠ ♥ ♥ ♠tr①♦ tt ♦♠♠♥t♦♥s ♦ ♣♣

❬❪ ② P ♥ r ♦♥ r♦① rr② P ③③♥ ③ rt ② s♣ ♦♦♠t♠♣rtrt♥♥ ♠♥t♦rsst♥ ♥ sss♠ ♠② s②♥ts③ ♠t r♦♥ ♥♥♦♣rts ♥♦ ttrs ♦ ♥♦ ♣♣

❬❪ ❳ ❩♥ ❲ ❨ ♥ ❲ ② ❩ ♦ ♥ ♦r♦ t ♠♥t♦rsst♥ ♣♥♦♠♥♦♥ ♥ ♠♦r s②st♠s ♠♦t② s ♦ ♥♦ ♣♣

❬❪ ❨ s②♠ ② ❩t ♥♦♦ ♦ s♣♥ tr♥s♣♦rt ♥ ♠♥ts♠ ♣rss

❬❪ ♣♣rt rt ② ❱♥ ♠r♥ ♦ s♣♥ tr♦♥s ♥ t st♦r tr ♠trs ♦ ♥♦ ♣♣

❬❪ r t♦♦ ② ❨ r ♦♥ ♦ ♣rt ♥trt♦♥s ♥ ♠♥t♦r♦♦ st♦♠rs ♦r♥ ♦ ♣♣ P②ss♦ ♥♦ ♣

❬❪ ❱♦♦♥ r ② P ➆tt♥r ♥t♦stt ♥trt♦♥s ♥ ♦rs t♥ ②♥r ♣r♠♥♥t ♠♥ts ♦r♥ ♦♥ts♠ ♥ ♥t trs ♦ ♥♦ ♣♣

❬❪ ♥♦ ❲ ♥s ♦♦♦s② s ❱rs ② ♥r♥ ♣r♣r♠♥ts♠ ♥ ♦tr ♠♥t trs ♥ r♥r♠trs r ♦♥ ♥ r s②st♠s ♦r♥ ♦ ♥♥♦s♥♥ ♥♥♦t♥♦♦② ♦ ♥♦ ♣♣

❬❪ ❨ ❩♥ q ♣ ②♥♠s ♥ ♣♠♦♥r② r②s Pr♦st

❬❪ r P②s Pr♦♣rts ♦ P♦②♠rs ♥♦♦ ♣r♥r ❨♦r

❬❪ s ② trt♦♥ ♦r② ♦ rr♦♠♥t ②strss ♦r♥♦ ♠♥ts♠ ♥ ♠♥t ♠trs ♦ ♥♦ ♣♣

❬❪ ❨ ❩♥ P♥ ❩ ❲ ❲♥ ❳ ♥ ② ❳♦ ♥♥♠♥t ♦ rr♦♠♥ts♠ ♥ ♥♦♣ ♦♦♣♥ ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣

♦rí

❬❪ ❱♥♦♦ ♦♠r ♦tt ♦rrs rr♦ rr ♥tr ♥③ ② ❲ss♠♥♥ ♣♣r♥ ♦ r♦♦♠t♠♣rtr rr♦♠♥ts♠ ♥ ♦♣ 2 − δ ♠sP②s ♦ ♥♦ ♣

❬❪ tr♥s ② ❨ ♥ tr♠♥t♦♥ ♦ ♣r♥ rr♦♠♥t ♦♠♣♦♥♥ts ♦ ♠♥t③t♦♥ ♥ ♠♥t♦rsst♥ ♦ r♥r♦ ♠s ♦r♥ ♦ ♣♣ P②ss ♦ ♥♦ ♣♣

❬❪ ❲s♦♥ Pr♦♠s ♥ ♣②ss t ♠♥② ss ♦ ♥t ♥t♠ ♦ ♣♣

❬❪ ❲s♦♥ ♥♦r♠③t♦♥ r♦♣ ♥ rt ♣♥♦♠♥ Pss♣ ♥②ss ♦ rt ♦r P②s ♦ ♥♦ ♣

❬❪ ❲s♦♥ r♥♦r♠③t♦♥ r♦♣ ♥ rt ♣♥♦♠♥s ♦ ♦r♥ P②ss ♦ ♥♦ ♣♣

❬❪ ♥♥② ♦r sr ② ♠♥ t♦r② ♦ rt ♣♥♦♠♥ ♥ ♥tr♦t♦♥ t♦ t r♥♦r♠③t♦♥ r♦♣ ①♦r❯♥rst② Prss ♥

❬❪ t♥② ♥ ♥rst② ♥ r♥♦r♠③t♦♥ r ♣rs♦ ♠♦r♥ rt ♣♥♦♠♥ ♦ P②s ♦ ♣♣ r

❬❪ ❨ ♥ ② ❲ öt ♦♥t r♦ st② ♦ t st♣r♦t♦♥♠♦ ♥ t♦ ♥ tr ♠♥s♦♥s P②s ♦ ♣

❬❪ P ♥♥s Pt P P♦♦t ② ❨ ♥t♥ ♥ t♥rst② ♦ r♦ss♥ ♣r♦ts ♥ t♦♠♥s♦♥ ♣r♦t♦♥♦r♥ ♦ sttst ♣②ss ♦ ♥♦ ♣♣

❬❪ r② rt ♣r♦t♦♥ ♥ ♥t ♦♠trs ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣

❬❪ P rssrr ♣r♥ ♥ ♦♥ ♠♥s♦♥s ♦ ♣r♦t♦♥ ♦r♥ ♦ P②ss t♠t ♥ ♥r ♦ ♥♦ ♣

❬❪ r♥s♦♥ s♣ r♥♦r♠③t♦♥ ♦ ♦♥s♦rr ♦♥t♥ tts P②s ♦ ♣♣ ♣

❬❪ ❨ ♥ ② ❯♥rs ♥ts③s♥ ♥t♦♥s ♦r rt s②st♠s t tt ♦♥r② ♦♥t♦♥sP②s ♦ ♥♦ ♣

♦rí

❬❪ ❨ ♥ ② ❯♥rs ♥ts③ s♥ ♥t♦♥s ♦r ♣r♦t♦♥ ♦♥ tr♠♥s♦♥ tts P②s ♦ ♥♦ ♣

❬❪ P s ♥ ② ❯♥rs s♥ ♥t♦♥s ♦r ♦♥♣r♦t♦♥ ♦♥ ♣♥rr♥♦♠ ♥ sqr tts t ♠t♣ ♣r♦t♥ strs P②s ♦ ♥♦ ♣

❬❪ P s ② ♥ Pr♦t♦♥ trs♦s rt ①♣♦♥♥ts♥ s♥ ♥t♦♥s ♦♥ ♣♥r r♥♦♠ tts ♥ tr s P②s ♦ ♥♦ ♣

❬❪ r ♥tr♦t♦♥ t♦ r♣ t♦r② ♦r Pt♦♥s

❬❪ ② r♣ ♦r② t ♦rt♠s ♥ ts ♣♣t♦♥s ♥ ♣♣♥ ♥ ♥♦♦② ♣r♥r

❬❪ r②s strt r♣ ♦rt♠s ♦r ♦♠♣tr t♦rs♦♠♣tr ♦♠♠♥t♦♥s ♥ t♦rs ♣r♥r

❬❪ ♥ ♦♥♥r st ♦r♠t♦♥s ♦r ♥ ♥ ♥♥r♥ ♦r Pt♦♥s

❬❪ ♥st♥ ② ♦② P♦②♠r P②ss ❯P ①♦r

❬❪ tr♦ P②ss ♦ P♦②♠rs ♦♥♣ts ♦r ❯♥rst♥♥ rtrtrs ♥ ♦r ♣r♥r r♥ r

❬❪ ♦♦♥② t♦r② ♦ r st ♦r♠t♦♥ ♦r♥ ♦ ♣♣♣②ss ♦ ♥♦ ♣♣

❬❪ ♥ r st ♦r♠t♦♥s ♦ s♦tr♦♣ ♠trs ❱ rtr ♦♣♠♥ts ♦ t ♥r t♦r② P♦s♦♣ r♥st♦♥s ♦t ♦② ♦t② ♦ ♦♥♦♥ t♠t P②s ♥ ♥♥r♥ ♥s ♦ ♥♦ ♣♣

st ró♥♠♦s

tt♥t ♦t t♥ ♦rr r♥s♦r♠ ♥rr♣tr♦s♦♣② s♣tr♦s♦♣í ♥rrr♦ ♣♦r r♥s♦r♠ ♦rr ♥ t♥ ♦t t♥

r♦♥ ♥♦ts ♥♦t♦s r♦♥♦

❯❱❱s s rt♥ ❯❱❱s ♣tr♦s♦♣② s♣tr♦s♦♣í ❯❱❱s ♣♦r t♥ s

t ♦♥t strs r♣♦s ♦♥t♦rs t♦s

rr♦♠♥t s♦♥♥ s♦♥♥ rr♦♠♥ét

♦ ♦♥t strs r♣♦s ♦♥t♦rs ♦s

♥t♦r♦♦ st♦♠r st♦♠r♦ ♠♥t♦r♦ó♦

❲ t r♦♥ ♥♦ts ♥♦t♦s r♦♥♦ Prs út♣s

♣t r♦s♦♣② r♦s♦♣í Ó♣t

P ♣♦②♥♥ ♣♦♥♥

P ♣♦②♠t②s♦①♥ ♣♦♠ts♦①♥♦

P❳ P♦r ❳② rt♦♥ ró♥ ②♦s ❳ P♦♦s

st②r♥t♥ rr ♦ str♥♦t♥♦

trtr t♦♠r ♦♠♣♦st ♦♠♣♦st♦ st♦♠ér♦strtr♦

♥♥♥ tr♦♥ r♦s♦♣② r♦s♦♣♦ tró♥♦ rr♦

♦rí

❯ ♣r♦♥t♥ ♥t♠ ♥trr♥ s ♥t♦♠tr ♥tó♠tr♦ s♣♦st♦s ♣r♦♥t♦rs ♥trr♥ á♥t

♦t tr ♥s♦tr♦♣② ♥s♦tr♦♣í étr ♦t

r♥s♠ss♦♥ tr♦♥ r♦s♦♣② r♦s♦♣♦ tró♥♦ r♥s♠só♥

❱ ❱rt♥ ♠♣ ♥t♦♠tr ♥t♦♠trí str ❱r♥t

♥ ♥♦ ♥ t ♣ ♣r tr♥♦

s♥♦ ♥ r♦r ♦♥ q ♦ ss

♠ t rt ♦♠♦ t ♥♦

á♥♦ t♦♦ t ♦r③ó♥

♦r♦ r♦ á♦♦ ♣rt♦ ♥tst

♥t♥rs q t♦♦s s♦♠♦s ♦rs

t♦ ♥str ♦t♥ t♥ ♠♣♥s s s♥ts

Pr♦r♠♦s ♥t♦♥s ♥sñr ♦♥ ♠♣♦

r ♥é ró♥♠♦ ♦r♦