HAL Id: tel-00552215https://tel.archives-ouvertes.fr/tel-00552215

Submitted on 5 Jan 2011

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Triangulating Point Sets in Orbit SpacesManuel Caroli

To cite this version:Manuel Caroli. Triangulating Point Sets in Orbit Spaces. Computer Science [cs]. Université NiceSophia Antipolis, 2010. English. tel-00552215

❯❱ P P

❯

♣♦r ♦t♥r ttr

♦tr ♥ ♥s

❯♥rsté ♦♣ ♥t♣♦s

♥t♦♥ ♥♦r♠tq

♣rés♥té t s♦t♥ ♣r

♥

r♥t♥ P♦♥t ts ♥ rt ♣s

ès ré ♣r ♦♥q ❯

s♦t♥

♣♣♦rtrs rt P ür ♥♦r♠t rrü♥♦♥ ❯❱ ❯ r♥

①♠♥trs r ❱ ♦ ♥♦r♠ s♣érr Prs♥♦s ❱ ❯♥rst② ♦ rt♥r ❯♥rsté ♦♦s ♦♥q ❯ ♦♣ ♥t♣♦s étrr♥ért ❱ ❯♥rst② ♦ r♦♥♥♥

♥té ♥rs ♦♠tr②t♦r② r

❯❱❨ P P

❯

P

t♦ ♦t♥ t tt ♦

♦t♦r ♦ ♥s

♦ t ❯♥rst② ♦ ♦♣ ♥t♣♦s

♣t② ♦♠♣tr ♥

♣r♣r ♥ ♥ ②

♥ r♦

r♥t♥ P♦♥t ts ♥ rt ♣s

s♦r ♦♥q

♥ ♦♥

rs rt ♦r♥ P ür ♥♦r♠t rrü♥♦♥ ♥ ❯ r♥

①♠♥t♦rs r ♦♥ ❱rèr ♦ ♥♦r♠ s♣érr Prs♥♦s rs ❯♥rst② ♦ rt♥r ♥r ❯♥rsté ♦♦s ♦♥q ♦♣ ♥t♣♦s étrr♥ért ❱tr ❯♥rst② ♦ r♦♥♥♥

♥t ♥rs r ♦♠tr②t♦r② r

és♠é

♥s tt tès ♥♦s ét♦♥s s tr♥t♦♥s é♥s ♣r ♥ ♥s♠ ♣♦♥ts ♥ss s♣s t♦♣♦♦s ér♥ts ♦s ♣r♦♣♦s♦♥s ♥ é♥t♦♥ é♥ér tr♥t♦♥ ♥② ♣♦r ♣srs sss s♣s ♥s q♥ ♦rt♠ ♦♥strt♦♥ ♦s ♦r♥ss♦♥s ♥ ♠♣♥tt♦♥ ♣♦r s ♣rtr t♦r ♣ttr♠♥s♦♥♥

tr st ♠♦té à ♦r♥ ♣r s♦♥ ♦s ♥t s tr♥t♦♥s ♥② ♣ér♦qs ♥s ♥♦♠r① ♦♠♥s ♦♥t str♦♥♦♠ ♥é♥r s♠tér① ♦♠é ②♥♠q s s t s tr♥t♦♥s ♣ér♦qs ♣♥t êtr s ♦♠♠ s tr♥t♦♥s t♦r ♣t ♦s ♦r♥ss♦♥s ♥é♥t♦♥ t ♥♦s é♦♣♣♦♥s ♥ ♦rt♠ ♥ré♠♥t ♣♦r r tr♥t♦♥ ♥② ♥s t♦r ♣t ♦rt♠ st ♣té ♦rt♠ ♥ré♠♥ts ♥s R

d ♦♥trr s tr① ♥térrs sr s tr♥t♦♥s ♣ér♦qs ♥♦sét♦♥s ♠♥t♥r ♣srs ♦♣s ♣ér♦qs s ♣♦♥ts ♦rsq st ♣♦ss réstt ♦r♥ ♣r ♦rt♠ st t♦♦rs ♥ tr♥t♦♥ t♦r ♣t

♦s ♣rés♥t♦♥s ♥ ♠♣♥tt♦♥ ♥♦tr ♦rt♠ à ♣rés♥t s♣♦♥ ♣q♠♥t ♦♠♠ ♥ ♠♦ ♦tèq ♦rt♠s é♦♠étrqs

♦s é♥érs♦♥s s réstts à ♥ ss ♣s é♥ér s♣s q♦t♥ts ♣ts ♥sqà s s♣s q♦t♥ts ♦rr ♦♥st♥t ♣♦st ♥♥ ♥♦s ♦♥sér♦♥s s t♦r ♦ q st ♥ ①♠♣ ss ♦♣ ♣s r s s♣s q♦t♥ts ♦rr ♥ét ♦♥st♥t

♦tsés tr♥t♦♥ ♥② s♣ q♦t♥t ♦♠♣① s♠♣ tr♥t♦♥ ♣ér♦q t♦r ♣t rêt♠♥t ♦rt♠ ♥ré♠♥t rété ♥♥r♠é

♦r

strt

♥ ts ♦r sss tr♥t♦♥s ♦ r♥t t♦♣♦♦ s♣s ♦r ♥ ♣♦♥t sts❲ ♣r♦♣♦s ♦t ♥t♦♥s ♥ ♦rt♠s ♦r r♥t sss ♦ s♣s ♥ ♣r♦ ♥♠♣♠♥tt♦♥ ♦r t s♣ s ♦ t tr♠♥s♦♥ t t♦rs

♦r s ♦r♥② ♠♦tt ② t ♥ ♦r s♦tr ♦♠♣t♥ tr♠♥s♦♥♣r♦ ♥② tr♥t♦♥s ♥ ♥♠r♦s ♦♠♥s ♥♥ str♦♥♦♠② ♠tr ♥♥r♥ ♦♠ ♦♠♣t♥ ②♥♠s t Pr♦ tr♥t♦♥s ♥ ♥rst♦♦ s tr♥t♦♥s ♦ t t t♦rs ❲ ♣r♦ ♥t♦♥ ♥ ♦♣ ♥ ♥t♥r♠♥t ♦rt♠ t♦ ♦♠♣t ♥② tr♥t♦♥s ♦ t t t♦rs ♦rt♠s ♠♦t♦♥ ♦ t ♥r♠♥t ♦rt♠ ♦r ♦♠♣t♥ ♥② tr♥t♦♥s ♥E

d ❯♥ ♣r♦s ♦r ♦♥ ♣r♦ tr♥t♦♥s ♦ ♠♥t♥♥ sr ♣r♦♦♣s ♦ t ♥♣t ♣♦♥t st ♥r ♣♦ss s♦ t ♦t♣t ♦ ♦r ♦rt♠ s r♥t t♦ ②s tr♥t♦♥ ♦ t t t♦rs ❲ ♣r♦ ♥ ♠♣♠♥tt♦♥ ♦ ♦r♦rt♠ tt s ♥ ♠ t♦ r♦ ♣ s ♣rt ♦ t ♦♠♣tt♦♥♦♠tr② ♦rt♠s rr② ❲ ♥r③ t ♦r ♦♥ t t t♦rs ♦♥t♦ ♠♦r ♥r ss ♦ t ♦rt s♣s s s ♦rt s♣s ♦ ♦♥st♥t ♣♦st rtr ❲ rtr♠♦r ♦♥sr t ♠ rr ss ♦ ♦rt s♣s ♦ ♦♥st♥t ♥trtr

②♦rs ♥② tr♥t♦♥ ♦rt s♣ s♠♣ ♦♠♣① ♣r♦ tr♥t♦♥ t t♦rs ♦r♥ s♣ ♥r♠♥t ♦rt♠ ♦s ♥ ♠♥♦

♦r

♥♦♠♥ts

♦r ♦♠♣s ♥ t ♣rs♥t tss ♦ ♥♦t ♥ ♣♦ss t♦t ts♣♣♦rt ♦ ♠♥② ♣♦♣ tt ♦♠♣♥ ♠ r♥ t st tr ②rs t s ♠♣♦sst♦ ♠♥t♦♥ t♠ t ♥rtss ♦ t♦ s ts ♥s t♦ ♠♥t♦♥ t♦s♦ ♥♥ ♠ ♠♦st

rst ♦ ♦ t♦ t♥ ♠② s♦r ♦♥q ♦r ♥ ♣r♦♣♦s ts♥trst♥ t♦♣ ♥ ♦r ♣t♥ ♠ s st♥t s ②s ♥ ♠ ♦t ♦ s♣♣♦rt t s r ♣sr t♦ t♦ ♦r t r

t s rt ♦♥♦r t♦ ♠ tt rt ♦r♥ ♥ ♦♥ ♥ ♣t t♦ r♠② tss ♠ r② rt t♦ t ①♠♥t♦rs r ♦♥ ❱rér ♥♦s rs ♥r ♥r ♥ rt ❱tr ♦r ♦♠♥ t♦ ♦♣ ♥t♣♦s t♦ tt♥ ♠②♥s rtr t♥ ♥rs r ♦r ♣t♥ t♦ st♥ ①♠♥t♦r

♦ t♦ ♠♥t♦♥ ♦ rt♦ ♦r ♥ strt s♦♠ ♥t rsr s♦r ♣ ♠ s♥♥t② t♦ t strt t t t♦♣ s♦ ♥t t♦ t♥ r♥♥② ♦r rt sss♦♥s ♦♥ ♦r♠③♥ t ♣r♦♠ rt ♦s s ②s ♦r ♠t qst♦♥s ♦r sss♥ ♥trst♥ ♣r♦♠s ♦ ♥② ♥

♥t t♦ t♥ ❱ssr♦♥ s♦♣♦♦s ♥ ♦♥♦ ♦r tr ♦rts t♦ sts ♦♠♣tt② t♥ t Pr♦ tr♥t♦♥s ♥ t r♠sr ♥ t ❱♦♠ ♠sr rs♣t②

r♥ ♠② ♦r t ♦s♦♥ t♦ ♦♦♣rt t rsr t♠s r♦♠ ❯ r③str ♥ r♦♠ t ♥rst② ♦ r♦♥♥♥ tr♥s

♠ rt t♦ t ♣♦♣ r♦♠ r③ s♥ ♦③r ♥ s ♣r♦t t♠ ♣rtr② ❲♦♥ ♥r ♦♠s r♥r ♦r♥rr rt ❱♦t♥r s s r♥③ r♥♠♠r ❲ ♦ts ♦ ♥trst♥ sss♦♥s r♥ ♦r ♠t sts

♥ t s♦♣ ♦ t ss♦t t♠ r sr ♠t sts t ♥ ♥ ❲rt ♥ rt ❱tr r♦♠ r♦♥♥♥ r② ♣♣rt t rt sss♦♥s♦t ♦♥ ♣♣t♦♥s ♦ ♠② ♦ s s ♦♥ ♠② rsr t♦♣ ts

♠ r② t♥ ♦r t s♣♣♦rt ♦t r♦♠ t ♣r♦t r♥ t ♦ ♣r♦ss ♦ ♠♣♠♥tt♦♥ ♥ s♠ss♦♥ ♦ ♠② ♣ ♣♣rt t ♦♣♥ t♠♦s♣r♥ t trst tt s ♣t ♥ ♠ r♦♠ t r② ♥♥♥

r♦ t ♣r♦t r♥s t ♣♦sst② t♦ t ♥ t♦ ♥ ①♥s t r♥ ♦♠♣tr s♥tsts ♥ ♠t♠t♥s ♦♥ ♦♠tr②rt t♦♣s rr♦s ♠t♥s r ②s ♣♦st ①♣r♥ ♥ ♦t♥ ♣r♦ ♠ t ♥ s♦♥ s♣ ♣r♦♠s

♥② ♥t t♦ t♥ t rr♥t ♥ ♦r♠r ♠♠rs ♦ t ♣r♦t t♠sé♦♠étr ♥ ♦r tr s♣♣♦rt ♥ t r♥② t♠♦s♣r ♥ t♣sr t♦ ♣ss t tr ②rs ♦ ♠② P sts

♦♥ r② ♣♦st ♥ ♦♣♥ ♥tr♥t♦♥ t♠♦s♣r ♦t t s s♦ts ♦ ♠ t♥ t♦ r②♦② ♠t ♥ ♦ ♥r ♠② st② ♦ ♠♦st♦r ②rs ♥ r♥ ②s ♣ r② ♦♦ ♠♠♦r② ♦ ♠② t♠ ♥ ♦♣♥t♣♦s

ÿ ♠öt ♠ ♠♥r ♠ ♥♥ ♠ stts ♥trstüt③t♥ ♠r r♠öt t ♠♥♥ sr♥ ❲ ③ srt♥

♦♥t♥ts

♥tr♦t♦♥

♦tt♦♥ r♥t♦♥s ♥ E

d ♠♣ ♦♠♣①s ♥② tr♥t♦♥ ♦rt♠s t♦ ♦♠♣t ♥② tr♥t♦♥s ♦ E

d ♥r♠♥t ♦rt♠

rt s♣s Pr♦♠ stt♠♥t tt ♦ t rt ♦♥trt♦♥s

Pr♦ tr♥t♦♥s

t t♦rs ♥② tr♥t♦♥ ♦ T

3c

♥t♦♥ P♦♥t sts tt ♦ ♥♦t ♥ ♥② tr♥t♦♥ ♦ T

3c

♦rt♠ ♦♠♥ ♦♥ ♦♠♥ ❲t ♥② tr♥t♦♥

♥②ss ♦♠♣①t② ♥②ss ♠r ♦ sts

♠♣♠♥tt♦♥

♥tr♦t♦♥ t♦ tr♥t♦♥s

tr♥t♦♥ trts tr♥t♦♥ t strtr

♣r♦ tr♥t♦♥s s♥ sts rts ♦r♥ s♣s P♦♥t ♦t♦♥ P♦♥t ♥srt♦♥

❱rt① r♠♦ ss ♣t♠③t♦♥s t♦♥ ♥t♦♥t②

♦♠♣①t② t② ♦ ♥ tr♥t s♥ ①♣r♠♥ts

♥♣t ♣♦♥t sts ♦♥strt♦♥ ♦ t ♥② tr♥t♦♥ P♦♥t ♥srt♦♥ ♥ T

3c

tr♥t♦♥ rr② ❱rt① r♠♦ ♣ ♦r♥ ♦♠♥ ♦♠♣rs♦♥ ♦ t rtr ♦ t♦♥

♣♣t♦♥s Pr♦ ♣ s♣s Pr♦ sr ♠sr Pr♦ ♦♠ ♠sr Pr♦ ♦② ♦rt♠

♦♥s♦♥

♥② tr♥t♦♥s ♦ ♦tr s♣s

Pr♠♥rs t s♣s

♦s ♥ ♠♥♦s r♥t♦♥s ♦ ♦s ♥ ♥♦s ♦rt♠ t ♦r♦s

♣r s♣s r♥t♦♥s ♦ t s♣r ♣r ♦rt s♣s r♥t♦♥s ♦ s♣r ♦rt s♣s

②♣r♦ s♣ ②♣r♦ ♣♥ H

2 ♦ t♦rs r♥t♦♥s ♦ t ♦ t♦rs sss♦♥

♦♥s♦♥ ♥ tr ♦r

strt♦♥ t♦ s♠♣ ♦♠♣①s strt♦♥s ♦♥ s♣s ②♣r♦ ♦rt s♣s

♣tr

♥tr♦t♦♥

r♥t♦♥s r ♦♥sr t♦ ♦♥ ♦ t ♠♦st ♠♣♦rt♥t strtrs ♥ ♦♠♣tt♦♥♦♠tr② ♥② tr♥t♦♥ s s♣ t②♣ ♦ tr♥t♦♥ tt ①tssr r② ♥trst♥ ♥ s ♣r♦♣rts ♦tr t ts t ❱♦r♦♥♦ r♠t s ♦ rt ♠♣♦rt♥ t♦ ♠♥② ♣♣t♦♥s ♥② tr♥t♦♥ ♥ t ❱♦r♦♥♦r♠ r st ♠♥② ♥t ♦rt♠s r ♥♦♥ ♥ ♠♥② ♠♣♠♥tt♦♥s①st ♦r ♠♦st ♦ ts rsts r rstrt t♦ ♥② tr♥t♦♥s ♥ td♠♥s♦♥ ♥ s♣ E

d♥ ts ♦r ♦♥sr t ♣r♦♠ ♦ ♥♥ ♥ ♦♠♣t♥ ♥② tr♥t♦♥s

♦ s♣s ♦tr t♥ Ed ♥tr♦t♦r② st♦♥ s ♦r♥③ s ♦♦s ❲ rst sss

t ♠♦tt♦♥ ♦r ♦r ♦r ♥ t rqr r♦♥ ♥♦ ♦♥ ♥②tr♥t♦♥s ♥ ♦♥ ♦rt s♣s tr ts ♥tr♦t♦♥s stt t ♣r♦♠ ♦♥sr ♥ ts ♦r ♥② r ♣r♦s ♦r ♦♥ ♥② tr♥t♦♥s ♥❱♦r♦♥♦ r♠s ♦ s♣s ♦tr t♥ E

d ♥ ♣rs♥t ♦r ♦♥trt♦♥s t♦ t t♦♣

♦tt♦♥

s ♦r s ♠♦tt ② ♥s ♦r ♦♠♣t♥ ♣r♦ tr♥t♦♥s ♥ r♥t ♦♠♥s♦ s♥ r r sr rs♦♥s ♦r ♦r♥ ♥ ♣r♦ s♣ rst② t ♥♣t♥ ♣♦♥t st t ♥ ♥r♥t ♣r♦t② ♥♦tr rs♦♥ s tt ♦t♥ t s③ ♦t ♠♦ s t♦♦ t♦ r♥ ♦♠♣tt♦♥s ♦♥ t s♦ t ①♣r♠♥ts r r♥ ♦♥ s♠s♠♣ r♣t ♣r♦② t♦ ♦ ♦♥r② ts ♥ ♥② tr r s♦♠♥trst♥ ♠t♠t qst♦♥s ♦♥ tr♥t♦♥s ♥ ♠ss r t s ♦ ♣r♦tr♥t♦♥s ♥ ♣ t♦ ♥ rtr ♥st ♥ ss t s sr t♦ ♦♠♣t ttr♥t♦♥ ♦ s♠r ♣♦♥t st ♦♥t♥♥ ♦♥② ♦♥ ♣r♦ ♦♣② ♦ ♥♣t ♣♦♥tt♦ ♦ r♥♥② ♥ t ♦♠♣tt♦♥

❲ ♥♦ ♦♥rt ①♠♣s ♦ ♣r♦♠s ♥ r♥t s ♦ s♥ tts ♣r♦ tr♥t♦♥s ♦♠ ♦ t♠ ♥ ♣rs♥t t t Pr♦s♣t ❲♦rs♦♣ ♦♥ ♦♠tr ♦♠♣t♥ ♥ Pr♦ ♣s t ♦♣♥t♣♦sétrr♥é r♥ t♦r ♥ t t ♦rs♦♣ ♥ r♥t♥ s♣s ♦r ♥rst♥♥ t ♦r t t ♦r♥t③ ♥tr ♥ ♥ tr♥s♦♠r

tt♣♦r♥tsPr♦♣s❲♦rs♦♣tt♣♦r♥t③♥tr♥♥♦♣♣s

♥tr♦t♦♥

❲ rst ♦♥sr ♥ ①♠♣ ♦ ♥ ♥r♥t② ♣r♦ ♣r♦♠ ♥ ♠tr s♥srsrrs ♦♣ ♥ ♠trs tt s♦ s♦♠ s♣ ♣r♦♣rts ♦r ①♠♣♦s♥ t ❬♦❪ ♦♣ ♦♥ s♦s ♠trs tt ♥ s t♦ r♣r♦♥ ♦♥s ♥ ♦rr t♦ r② t ♠tr ♣r♦♣rts t② s ♥t ♠♥t s♠t♦♥s♦r s♣② t② ♥t t♦ ♦♠♣t ♥②s ♦♠ ♠s ♦ t s♦s s♦s r ♣r♦ ♥ ♦rr t♦ ♠s t ♦ s♦ t ss t♦ ♠s ♦♥② ♦♥♣r♦ ♦♠♥ s tt t ♠s ♥ r♣t ♥ t♦tr s r ❲♥ ♦♠♣t♥ ♥ ♣r♦ stt♥ t♦ ♦♣s ♦ t ♠s t t♦tr ② ♦♥strt♦♥

♥ ♦s♠♦♦② t s ♦ str♦♥♦♠② ♥ t rs strtr ♥ t ♥rs t s♦ ♦s♠♦♦ ♣r♥♣ stts ❱ ♦♥ s♥t② r s t♣r♦♣rts ♦ t ❯♥rs r t s♠ ♦r ♦srrs ❬❪ s ♦r s♠t♦♥s ♦♥t rs strtr ♥ t ♥rs s♥t② r s♠♣ ♦ t strtr ♥ s r♣t ♣r♦② s r ❲ ♥ ♥ ♦♥tt t str♦♥♦♠rs♦ ♦♣ ♥② tsst♦♥ st♠t♦r ❬❲❪ ♥ ♦ r ♥trst♥ s♥ ♣r♦ ♥② tr♥t♦♥s ♥ ❱♦r♦♥♦ r♠s ② r s♦ ♥trst♥ tt ♥♠rs ♦ ♣r♦ ♣ s♣s ❬❲❱P+❪

r♥r ♠trs s♦ ♣rt ♣r♦♣rts t ♠r♦s ♥ ♦♥t♥♠♠♥♣r♦♣rts t ♠r♦s ♦ ♥rst♥ t rt♦♥ t♥ ♦t ♥s ♦ ♣r♦♣rts ♠t♦ s ♦♥ rr② ♦ ♥② tr♥t♦♥s s ♥ ♦♣ ❬rr❪ ♦ ♦ ♦♥r② ts t t♦rs s ♣r♦ stt♥

str♦♣②s s♠t♦♥s ♦t♥ ♦r s♥ ♠t♦ ♣t ♠s r♥♠♥ttt s s ♦♥ rr rs s s t♦ ♣r♦♠s ♥ t s♥ r♥ s♦ tr s♦♠ ♥tr♠t r s③ s s tt s ♥♦t r② ♣t ♦r ①tr♠ss ♦r t r s r♥ r s♠r s♥ s rqr ♠s t ♥♥ ♦t r ♦♥sr② ♠♦r ♦♠♣t ❯s♥ ❱♦r♦♥♦ r♠s ♥st ♦ rr rss♠s ♣t ♦♣t♦♥ ♥ ts s r r ♣♣t♦♥s ♦ ts ♥ ♣r♦ stt♥s ❬❪

♠r ♣♣r♦s r s ♥ ♦♠♣tt♦♥ strtr ♦♦② ❬r❪ ②♥♠s ❬♠ q❪ ♣rt ②♥♠s ❬❪ s♦ ♠♥s ❬❪ ♦♠ ♦♠♣t♥ ❬❲❪ t ts st ♥ r r♦♠ ①st ♦♠ ♦ ts ♣♣t♦♥s s♦ s♦r r♥t t②♣s ♦ ♣r♦ts s s ♣r♦ts ♥ ss t♥ tr rt♦♥s ♥ E

3

♦r ♣♦♥t sts ♥rt ② ♠♦r ♦♠♣t s♦♠trs s s rt♦♥s ♦r r♦tt♦♥sr r s♦ ♠t♠t qst♦♥s r ♣r♦ tr♥t♦♥s ♥ s

♦r ①♠♣ t ♥ Pr♦♠ ❬♦❪ ♦ ♥ s♣ ♣rtt♦♥ ♥t♦ s ♦ q♦♠ t t st r ♦ sr t♥ t♠ r ♦r t tr♥t ♦♥②♦♠ ♣rtt♦♥ t tr♥t ♦tr ♣r♦♣♦s ② ♥ ♠s rs rr♥t② ♣r♦rss ♠ ♦♥ ts t♦♣ s♥ ♣r♦ ❱♦r♦♥♦ r♠s ❬❪ rr ♠♦r t♦♣s ❬♦❪ ♦rs ♦♥ tt ♥♠r s♥trs ♦ ♣♦♥t sts ♥ ♣r♦stt♥

r♥t♦♥s ♥ Ed

♥ ts st♦♥ r♣tt t ♥t♦♥ ♥ t ♣r♦♣rts ♦ t ♥② tr♥t♦♥ ❲ strt t ♥r②♥ ♦♥♣ts s s s♠♣ ♦♠♣①s ♥ tr♥t♦♥s♥ ♥tr♦ ♥② tr♥t♦♥s s♦♠ ♠♣♦rt♥t ♣r♦♣rts ♥ ①♣♥ ♥t t ♥r♠♥t ♦rt♠ tt s s ssq♥t②

r♥t♦♥s ♥ Ed

r rs strtr ♦ t ♦s♠ ♦rts② ❲rt ♦♥ s♦ ♦rts② ♦s♥ tr♥t ♦♥②♦♠ ♠ ❬♣❪

♠♣ ♦♠♣①s

❲ ♥♦ ♥tr♦ t ♥♦t♦♥s ♦ s♠♣s ♥ s♠♣ ♦♠♣①s ♦r rtr r♥s ❬❩♦♠❪

❲ ♥♦t t d♠♥s♦♥ ♥ s♣ ② Ed st♥ t♥ p ♥ q s

♥♦t ② st(p, q) t ♥t ♦ t t♦r v := q − p r♦♠ p t♦ q s ♥♦t ② ‖v‖ st A ⊆ E

d s s t♦ ♦♥① ♦r ♣♦♥ts p, q ∈ A t ♥ s♠♥t t♥ p♥ q s ♦♥t♥ ♥ A

t p0, p1, . . . pk ♣♦♥ts ♥ Ed k ≤ d ♥r ♦♠♥t♦♥ λ0p0+λ1p1+ . . .+λkpk

t∑k

i=0 λi = 1 λi ∈ R s♣♥s ♥ ♥ rt② ♣♦♥ts r s t♦ ♥②♥♣♥♥t t ♥ rt② s s♣ ♦ ♠♥s♦♥ k

♥t♦♥ ks♠♣① ks♠♣① σ ♥ Ed k ≤ d s t ♦♥① ♦ k + 1

♥② ♥♣♥♥t ♣♦♥ts Pσ = p0, p1, . . . , pk

s♠♣① τ ♥ ② Pτ ⊆ Pσ s ♦ σ ♥ s σ s ♦ s s ♥♦t② σ ≥ τ ♥ τ ≤ σ ♦t tt σ ≥ σ ♥ σ ≤ σ

r ①st sr ♥t♦♥s ♦ s♠♣ ♦♠♣①s ♥ t trtr t♥ t②rstrt t♦ ♥t ♥♠r ♦ s♠♣s ❬❩♦♠ ❱❪ ♥ t sq t ♥♥ts♠♣ ♦♠♣①s s♦ s t ♥t♦♥ ♥ ♥ ❬❪

♥t♦♥ ♠♣ ♦♠♣① s♠♣ ♦♠♣① s st K ♦ s♠♣s stt

σ ∈ K, τ ≤ σ ⇒ τ ∈ K

σ, σ′ ∈ K ⇒ σ ∩ σ′ ≤ σ ♥ σ ∩ σ′ ≤ σ′

r② ♣♦♥t ♥ s♠♣① ♦ K s ♥♦r♦♦ tt ♥trsts t ♠♦st ♥t② ♠♥②s♠♣s ♥ K ♦ ♥t♥ss

♦rs K s ♥t t♥ ♦♥t♦♥ s ②s ❲ s♦♠ ♠♦r ♥t♦♥s t K s♠♣ ♦♠♣① sst ♦ K s

s♠♣ ♦♠♣① s t s♦♠♣① ♦ K str ♦ sst L ⊆ K ♦♥ssts

♥tr♦t♦♥

♦ t ♦s ♦ s♠♣s ♥ L

t(L) := σ ∈ K | ∃τ ∈ L, σ ≥ τ.

♦sr L ♦ L s t s♠st s♦♠♣① ♦♥t♥♥ L

L := τ ∈ K | ∃σ ∈ L, τ ≤ σ.

♥ ♦ L s ♥ s ♦♦s

(L) := t(L) − t(L).

r ♦r ♥ strt♦♥ ♦ str ♦sr ♥ ♥ ♥ ssq♥t sss♦♥s

r t t(v) s tr♥s ♥ s s♠♥ts ♥tr t(v) str♥s s s♠♥ts ♥ ♥♥t rts t (v) s ♥s ♥ ♥♥trts

♥trst ♥ t ♥♦♥ ♦ t s♠♣s r♦♠ st L ♥♦t ②⋃L ❲

♥♦t t ♥tr♦r ♦ s♠♣① σ ② σ ♦ s♠♣s σ ♥ τ r s t♦ ♥tr♥②s♦♥t tr ♥tr♦rs r s♦♥t σ ∩ τ = ∅ s r

r σ ♥ τ r ♥tr♥② s♦♥t

♦t tt t ♥t♦♥ ♦ s♠♣ ♦♠♣①s ♥ t ssq♥t ♥t♦♥s r♣r② ♦♠♥t♦r ♥ ♦ ♥♦t ♣♥ ♦♥ t ♠♥ s♣ s ♥ ♦rr t♦ sts ♥t♦♥s ♥ s♣s ♦tr t♥ E

d ♦♥② t ♥ ♦s t s♠♣s rrqr t♦ ♥

❲ ♥ ♥♦ ♦♥sr tr♥t♦♥s r r② s strtrs ♥ ♦♠♣tt♦♥♦♠tr② r♥t♦♥s r ② s ♥ t②♣s ♦ ♣♣t♦♥s s s ♥t ♠♥t♠t♦s ♥ ♠s♥ ❲ rst t ♦r♠ ♥t♦♥

♥t♦♥ r♥t♦♥ t X t♦♣♦♦ s♣ ♥ s♠♣s r♥ tr♥t♦♥ ♦ X s s♠♣ ♦♠♣① K s tt

⋃K s ♦♠♦♠♦r♣ t♦X tr♥t♦♥ ♦ X ♥ ② ♣♦♥t st S s tr♥t♦♥ ♦ X s tt t st♦ rts ♦ t tr♥t♦♥ s ♥t t♦ S

r♥t♦♥s ♥ Ed

♥ Ed t ♥♦♥ ♦ ♥t ♥♠r ♦ s♠♣s s ♦♠♣t rs E

d ts s ♥♦t ♦♠♣t ♦ ♦r♥ t♦ t ♦ ♥t♦♥ tr s ♥♦ ♥t tr♥t♦♥ ♦ E

d ♦r ♥ ♥ tr♥t♦♥s ♦ t ♦♥♣♦♥t♦♠♣tt♦♥ E

d∪∞ ♦ Ed r ♦♠

♥t♦r s♠♣s ♦r♠ ② ∞ ♥ d ♥t rts r ♦♥sr s ♥♥t ds♠♣sr♦♠ ♥♦ ♦♥ ♥ rr t♦ ♥t tr♥t♦♥ ♦ t ♣♦♥t st S ♥ E

d t②♠♥ tr♥t♦♥ ♦ t ♣♦♥t st S ∪ ∞ ♥ E

d ∪ ∞

♥② tr♥t♦♥

♥② tr♥t♦♥ s ♥♠ tr ♦rs ♥② ♦ s ♦♥ ♦ t rst t♦①♣♦r t r② ♥trst♥ ♥ s ♣r♦♣rts ♦ ts ② ♥♦ st strtr ♥♦♠♣tt♦♥ ♦♠tr② ❬❪

♥ ♦rr t♦ ♥ t ♥② tr♥t♦♥ ♦ Ed ♥ ② ♣♦♥t st rst ♥

s♦♠ ♥♦tt♦♥ t S srt ♣♦♥t st ♥ Ed t σ ds♠♣① t d+1 rts

♦ σ ♦♥ t ♦♥r② ♦ ♥q② ♥ d t r♠sr♥ ♦ σ ♦♥ ♦ t rts ♦ σ s ∞ t♥ t r♠sr♥ ♦ σ s s♣ ♦♥② t ②♣r♣♥ ♥ ② t r♠♥♥ d rts

♥t♦♥ ♥② tr♥t♦♥ ds♠♣① ♥ d♠♥s♦♥ tr♥t♦♥♦s r♠sr♥ ♦s ♥♦t ♦♥t♥ ♥② rt① ♦ t tr♥t♦♥ ♥ ts ♥tr♦r ss t♦ t ♥② ♣r♦♣rt② tr♥t♦♥ ♦ E

d t rt① st S s t♥② tr♥t♦♥ ♦ E

d ♥ ② S ds♠♣① ♥ t tr♥t♦♥ s t♥② ♣r♦♣rt②

❲ rt DT (S) t♦ ♥♦t t ♥② tr♥t♦♥ ♦ Ed ♥ ② S

♦ ①♣♦r ♠♦r ♣r♦♣rts ♦ t ♥② tr♥t♦♥ ♥♦ ♦♥sr ts strtr t ❱♦r♦♥♦ r♠

♥t♦♥ ❱♦r♦♥♦ r♠ ❱♦r♦♥♦ V or(p,S) ♦ ♣♦♥t p ∈ S ♦♥ssts♦ ♣♦♥ts q ♥ E

d s tt st(q, p) < st(q, pi) ♦r pi ∈ S − p ❱♦r♦♥♦ r♠ V D(S) ♦ t ♣♦♥t st S s t ♣rtt♦♥ ♦ E

d ♥t♦ t ❱♦r♦♥♦s ♦ t ♣♦♥ts ♥ S

r ♦r ♥ strt♦♥ ♦ ♥② tr♥t♦♥ ♥ ❱♦r♦♥♦ r♠♥ t ♣♥ ♦r rtr r♥ ♦♥ ♥② tr♥t♦♥s ♥ ❱♦r♦♥♦ r♠ss ❬ ❨ r ❪

r♣ ♦ t ❱♦r♦♥♦ r♠ s t ♦♦♥ ♣r♦♣rts sts ♦rrs♣♦♥ t♦ t rts ♦ t r♣ t ❱♦r♦♥♦ rts ♦rrs♣♦♥ t♦ s ♥ t r♣ t trs ♦ ♥tr♠t ♠♥s♦♥ ♦♦ ♠♣t② ♦ t ❱♦r♦♥♦r♠ ♦ S s t ♥② r♣ ♦ S

❲ s② tt S s ♥ ♥r ♣♦st♦♥ tr s ♥♦ sst ♦ ♠♦r t♥ k +1 ♣♦♥ts tt ♦♥ ♦♥ ks♣r ♦r ♥ ♦♥ k♣♥ ♦r k < d t ♣♦♥ts ♥ S r ♥ ♥r ♣♦st♦♥t♥ t ♥② r♣ s t ♥② tr♥t♦♥

t ♣♦♥ts r ♥♦t ♥ ♥r ♣♦st♦♥ t♥ t ssts ♦ ♦s♣r ♣♦♥ts ♦r♠♦♥① ♣♦②r ♥ t ♥② r♣ ♥ ♦rr t♦ tr♥s♦r♠ t ♥② r♣ t♦ ♥② tr♥t♦♥ ts ♣♦②r ♠st tr♥t s t ♣♦♥ts r ♦s♣r ♥② tr♥t♦♥ ♦ ts ♣♦②r s t ♥② ♣r♦♣rt② ♦r ♣rt ♣r♣♦sst s ♦t♥ s t♦ ②s ♥q② ♥ ♥② tr♥t♦♥ ♦r ♣♦♥t ststt r ♥♦t ♥ ♥r ♣♦st♦♥ ts ♥ s♥ s②♠♦ ♣rtrt♦♥ ❬❪

♥tr♦t♦♥

r ♥② tr♥t♦♥ s♦ ♥s ♥ ❱♦r♦♥♦ r♠ s ♥s ♦ ♣♦♥ts ♥ t ♣♥

❱♦r♦♥♦ r♠ ♥ ♥r③ t♦ t ❱♦r♦♥♦ r♠ ♦r ♣♦r r♠ r r t s♦ t s tt t♦ ♣♦♥t ♥ t st S ♥t ♥ st♥ s r♣ ② t ♣♦r st♥ t (p, wp) ♥ (q, wq) ♥♦t♣♦♥ts p ♥ q ♥ E

d t♦tr t tr ts wp ♥ wq rs♣t② ♥ t ♣♦rst♥ s ♥ ②

Π((p, wp), (q, wq)) := st(p, q)2 − wp − wq.

t ♥② tr♥t♦♥ ♦r rr tr♥t♦♥ ♥ ♥ s t ♦ t ♣♦r r♠ ♥ t s♠ ② s t ♥② tr♥t♦♥ s t ♦t ❱♦r♦♥♦ r♠ ❲t ♥② tr♥t♦♥s r ♥r③t♦♥ ♦ ♥② tr♥t♦♥s ♦♠tr② t② ♥ ♥rst♦♦ s ♦♦s t ♣♦♥ts(p, wp) ♥ (q, wq) ♥ ♦♥sr s s♣rs ♦ rs

√wp ♥

√wq ♥tr t p ♥

q rs♣t② ♣♦r ♣r♦t ♦ t♦ t ♣♦♥ts s 0 ♥ ♦♥② t ♦rrs♣♦♥♥ s♣rs ♥trst ♦rt♦♦♥② s t♦ s ♣♦♥ts r s t♦ ♦rt♦♦♥♥♦♦s② t ♣♦r ♣r♦t ♦ t♦ t ♣♦♥ts s strt② ♥t ♦r ♣♦st tt ♣♦♥ts r s t♦ s♣r♦rt♦♦♥ ♦r s♦rt♦♦♥ rs♣t② ♥ ♥♦♥♥rt s d + 1 t ♣♦♥ts ♠t ①t② ♦♥ ♦♠♠♦♥ ♦rt♦♦♥ t♣♦♥t (o, wo) (o, wo) s ♦rt♦♦♥ t♦ ♦ t d + 1 t ♣♦♥ts d + 1t ♣♦♥ts r s t♦ t t ♥② ♣r♦♣rt② ♦r ♥② ♦tr t♣♦♥t ♥ S t ♣♦r ♣r♦t t (o, wo) s ♥♦♥♥t t ♥♣t ♣♦♥t st s ♥♥r ♣♦st♦♥ ♥♦t ♠♦r t♥ d + 1 t ♣♦♥ts t s♠ ♦♠♠♦♥ ♦rt♦♦♥ ♣♦♥t t t ♥② tr♥t♦♥ s t st ♦ ds♠♣s ♦r♠② (d + 1)t♣s ♦ ♣♦♥ts ♥ ts ♣r♦♣rt② ❬❪ ♦r trt♠♥t ♦ ♥rt ss ♦t tt t s♠ ♠♦♥t t♦ t t ♦ ♣♦♥t ♥ S tt ♥② tr♥t♦♥ ♦s ♥♦t ♥ r ♦r t ♥②tr♥t♦♥ ♦r ♠♦r ♦♥ t ♥② tr♥t♦♥s s ❬❪

♦st ♦ t sr ♥ ♣tr ①t♥s t♦ t ♥② tr♥t♦♥s❲r ts s ♥♦t t s trt t ♥② tr♥t♦♥s s♣rt②

Pr♦♣rts ♦ tr♥t♦♥s

s③ ♦ ♥② ♣♥r tr♥t♦♥ s ♥r ♥ t ♥♠r ♦ rts ♥ s♥ s② r♦♠ t rs ♦r♠ t n, e, f ♥♦t t ♥♠r ♦ rts s ♥

r♥t♦♥s ♥ Ed

r t ♥② tr♥t♦♥ ♦ E2

s ♥ t tr♥t♦♥ rs♣t② ♥ rs ♦r♠ stts tt n − e + f = 2rtr♠♦r s tr s ♥ ♦rrs♣♦♥s t♦ t♦ s ts 2e = 3f P♥ ts ♥t♦ rs ♦r♠ s f = 2n − 4 ∈ Θ(n)

s③ ♦ tr♠♥s♦♥ tr♥t♦♥ ♥ qrt ♥ t ♥♠r ♦ rtst c ♥♦t t ♥♠r ♦ s ♥ e t ♥♠r ♦ s s ♦ rs♦♥ ♦rs ♦r♠ stts n − e + f − c = 0 ♠♣s c ∈ O(e) ♥ s ♥ ② ♣r ♦ rts s♦ t ♥♠r ♦ s s ♥ O(n2) r r ①♠♣s ♦ qrts③ ♥② tr♥t♦♥s s♦ t ♦♥ s tt s r s③ ♦ ♥②

tr♥t♦♥s ♦ Ed s ♦♥ ② O(n⌈ d

2⌉) s tt ♦♥ ❬❪

r qrt s③ ♥② tr♥t♦♥ ♦ E3

♦rt♠s t♦ ♦♠♣t ♥② tr♥t♦♥s ♦ Ed

r r r♦s ♦rt♠s ♥♦♥ ♦r ♦♠♣t♥ t ♥② tr♥t♦♥ ❬♦♦r ❲t❪ rst ♦rt♠s ♥ ♦♣ ♦r ♣♥r ♥② tr♥t♦♥s ② t t② ♦ t ♥② tr♥t♦♥ ♥ t ❱♦r♦♥♦ r♠ ♦rt♠s ♦r ♦♠♣t♥ ❱♦r♦♥♦ r♠s ♦ ♣♦♥t sts ♥ s t♦ ♦♠♣t ♥②tr♥t♦♥s s

♥ E2 t s③ ♦ t ♥② tr♥t♦♥ s ♥ O(n) s s♥ ♦ ♦r ♦♥

♦♥ t ♦♠♣①t② ♦ ♦♠♣t♥ t s ♥ Ω(n log n) s t s♦rt♥ ♣r♦♠ ♥ r t♦ ♥② tr♥t♦♥ ♦♠♣tt♦♥ ♠♥ ♣♦♥ts ♥ ♦♥ ♦♥ ♦t ♥t ♣r♦ ♥♣t ♣♦♥ts r ♣rt ♦ t ♦♥① ♦ t tr♥t♦♥ sr ② ♥ ①trt r♦♠ t tr♥t♦♥ ♥ ♥rs♥ ♦rr ♦ tr x♦♦r♥t ♥ ♥r t♠ s t ♦♥strt♦♥ ♦ t ♥② tr♥t♦♥ ♠st ♥Ω(n log n)

♥tr♦t♦♥

r ❯s♥ t ♥② tr♥t♦♥ ♦rt♠ t♦ s♦rt ♣♦♥ts ♦♥ x

❲ r② ♥tr♦ t ♠♦st ♦♠♠♦♥② s ♣♣r♦s t♦ ♦♠♣t ♥② tr♥t♦♥s rst ♦r t ♣♥r s ♥ t♥ ♦r r ♠♥s♦♥s

♥r♠♥t s ♣♣r♦ s ♥t ♥ ♣rt ♥ s② t♦ ♠♣♠♥t t s rst♥ sr ② ❬♦❪ ♥ ❬❲t❪

♥ ♣♦♥t st S t ♦rt♠ strts t tr♥t♦♥ ♦ d + 1 ♣♦♥ts r♦♠ S♥ ♥r ♣♦st♦♥ s ♥ssr② ♥② ♥ ♥srts t r♠♥♥ ♣♦♥ts ♦♥② ♦♥ rst♦r♥ t ♥② ♣r♦♣rt② tr ♣♦♥t ♥srt♦♥ ♣♦♥t ♥srt♦♥♦♥ssts ♦ ♣♦♥t ♦t♦♥ st♣ ♥ ♣♦♥t ♥srt♦♥ st♣

♥ t ♥ ♣♣r♦ t ♣♦♥t ♦t♦♥ st♣ s ♥ O(n⌈ d2⌉) ♥ t ♦rst s

♣♦♥t ♥srt♦♥ st♣ ♥srts t ♣♦♥t ♥t♦ t s♠♣① tt ♦♥t♥s t ♥ rst♦rs t♥② ♣r♦♣rt② ♦rsts ♦♠♣①t② ♦ t ♣♦♥t ♥srt♦♥ st♣ s ts ♦♥② O(n⌈ d

2⌉) r r n ♣♦♥t♦t♦♥ ♥ ♥srt♦♥ st♣s s♦ t ♦r ♦♠♣①t② s

O(n⌈ d2⌉+1)

s rst ♥ ♠♣r♦ ♦♥sr② s♥ r♥♦♠③t♦♥ ♥ ♣♦♥t♦t♦♥t strtr ♥♦♠③t♦♥ ♠♥s r t♦ ♥srt t ♣♦♥ts ♦ S ♥ r♥♦♠ ♦rr ❲♥♦ t rsts ♦ r♥♦♠③ ♦rsts ♥②ss ♦♥sr ♦rsts♣♦♥t st S ♥ r♥♦♠③ ♥srt♦♥ ♦rr

rst rsts ♥ ♦r ♥② tr♥t♦♥s ♦ E2 ♦♥② ❯s♥

♣♦♥t♦t♦♥ tstrtr t ♣♦♥t ♦t♦♥ st♣ ♥ ♠♣r♦ t♦ ♥ ①♣t♦♠♣①t② ♦ O(log n) r s ♣♦♥t♦t♦♥ tstrtrs ♥ ♣r♦♣♦s rqr♥ ①♣t O(n) s♣ ❬❪ rst ♣r♦♣♦s ♥ ♦♥ rs♦♥ t ♣♦♥t st♠st ♥♦♥ ♥ ♥ ❬❪ ♥tr♦ ♥ ♦♥♥ rs♦♥ ♥ ❬ ❪sr ② ②♥♠ ♣♣r♦ ♦♥ ♦r rt① r♠♦ ❲♥ s♥ s ♣♦♥t♦t♦♥ tstrtr ♥ r♥♦♠③ ♥srt♦♥ ♦rr t ①♣t ♦♠♣①t②♦ t ♣♦♥t ♥srt♦♥ st♣ s O(1) s ②s ♥ ♦r ♦♣t♠ r♥♦♠③ ♦rsts♦♠♣①t② ♦ O(n log n)

r♥♦♠③ ♥r♠♥t ♥②ss ♥ s② ①t♥ t♦ E3 rst♥ ♥ ①♣t

t♠ ♥ s♣ ♦♠♣①t② O(n2) s ♦♣t♠ ❬❪ t t② ♥ ①t♥s t♦

d ♠♥s♦♥s ♥ ts s t ①♣t t♠ ♥ s♣ ♦♠♣①t② s O(n⌈ d2⌉) ❬

❪ ❲ sr t ts ♦ t ♥r♠♥t ♦rt♠ ♥ t♦♥ ♦

r♥t♦♥s ♥ Ed

♦♥qr ♦♥qr ♣♣r♦ t♦ ♦♠♣t ♣♥r ♥② tr♥t♦♥s s ♥ sr ② ❬❪ ♥ ♠♣r♦ ② ❬②❪ t t② ♦♠♣tst ❱♦r♦♥♦ r♠ t ♥② tr♥t♦♥ ♥ t♥ ♦♥strt ♥ ♥r t♠

♣♦♥t st S s ♣rtt♦♥ ♦♥ t x①s ♦r t y①s ♥t♦ t♦ ♦t q②r ssts S1 ♥ S2 ♥ t ❱♦r♦♥♦ r♠s ♦ S1 ♥ S2 r ♦♠♣t rrs②♥② tr s s♥ st♣ tt ♦♠♣ts t ❱♦r♦♥♦ r♠ ♦ S r♦♠ t ❱♦r♦♥♦r♠s ♦ S1 ♥ S2 s ♦rt♠ s t ♦♣t♠ ♦rsts ♦♠♣①t② O(n log n)t ♦s ♥♦t rt② ①t♥ t♦ r ♠♥s♦♥s

♣♥ ♦rt♠ s s ♥ ♥ ♦rt♠ t♦ ♦♠♣t ♣♥r ❱♦r♦♥♦ r♠s❬♦r❪ t ♠♥t♥s ♥ tt s♣s ♦r t ♣♥ ♥ t ❱♦r♦♥♦ r♠ ♦ tr② ♥♦♥tr ♣♦♥ts ♥ t ♥ ts s t ♥ s♣s t ♣♥ t ❱♦r♦♥♦r♠ s ♣t ♦♥ t ② ♥ ♥ ♣♦♥ts ♥♦♥tr ② t s♣♥ r s ♦rt♠ s ♦♣t♠ ♦rsts ♦♠♣①t② O(n log n)

♥r♠♥t ♦rt♠

♠♣♠♥tt♦♥ ♦ ♥② tr♥t♦♥s ♥ t ♦♠♣tt♦♥ ♦♠tr② ♦rt♠s rr② ❬❪ ss ♠♦ rs♦♥ ♦ t ♥r♠♥t ♦rt♠ sr② ♦②r ❬♦❪ ♥ ❲ts♦♥ ❬❲t❪ ❲ ♥♦ ♣rs♥t ♥ ♠♦r t t ♦rt♠s ② t ♠♣♠♥tt♦♥ ❬❨ P❪ ♦♥ ♦r ♦rt♠ t♦ ♦♠♣tt ♥② tr♥t♦♥ ♦ r♥t ♦rt s♣s s s

t rst ①♣♥ t ♠♣♦② ♣♦♥t ♦t♦♥ strt② t♦tr t t♦ ♦♣t♦♥♠♣r♦♠♥ts ♦♥ s♥ ♣♦♥t♦t♦♥ tstrtr ♥ t ♦tr ♦♥ ♣r♦r♠♥s♦♠ ♣r♣r♦ss♥ ♦♥ t ♣♦♥t st ♦r ♦♥strt♥ t tr♥t♦♥ ♥ ♣rs♥tt♦ ♦♠♠♦♥ s♦t♦♥s t♦ rst♦r♥ t ♥② ♣r♦♣rt② ♦ t tr♥t♦♥ tr ♥♥srt ♥ ♣♦♥t ♥ ♥② s♦ ♦ t♦ r♠♦ rt① r♦♠ tr♥t♦♥

P♦♥t ♦t♦♥ ♥ ♦rr t♦ ♥ t tt ♦♥t♥s ♥ ♣♦♥t ♥ ♣♣r♦ r♠♠r♥ st♦st s s ❬P❪ ♣♦♥t ♦t♦♥ ♣r♦r strts t ♥rtrr② c t ♦♦ss t f ♦ t t r♥♦♠ t v ♥♦t t rt① ♦ ctt ♦s ♥♦t ♦♥ t♦ f t qr② ♣♦♥t ♥ v ♦♥ r♥t ss ♦ t ②♣r♣♥♦♥t♥♥ f t ♣♦♥t ♦t♦♥ ♦rt♠ ♣r♦s t♦ t ♥♦r♥ ♦♥ t ♦♣♣♦sts ♦ f ♥ ♦s t s♠ ♥ t r♠♠rs t tt s ♥ st ♦r ♥♦rr t♦ s ♦♥ ♦♠tr tst s t ♦♦ss f t r♥♦♠ ♠♦♥st t s ♦ c trs t ♦♥t♥♥ t ♣♦♥t tr ♥t ♥♠r ♦ st♣s t ♣r♦t②r ♦r ♥ strt♦♥ ♥

s ♥tr♦ ♥ t♦♥ sr ♣♦♥t♦t♦♥ tstrtrs ♥ ♣r♦♣♦s ♥ t ♥② rr② ❬❪ s ♠♣♠♥t ♦rt♠ ♥♠r ♦tr♥t♦♥s ♦ r② s♠r ssts ♦ t ♥♣t ♣♦♥t st r ♠♥t♥ ♦r ♦ ts tr♥t♦♥s tr r ♥s t♥ ♦rrs♣♦♥♥ rts ♥ t ♥①t ♥r ♥t ♥①t ♦rsr tr♥t♦♥ ♥ t rr② ♣♦♥t ♦t♦♥ strts ♥ t ♦rssttr♥t♦♥ t♥ ♦s ♦♥ t♦ t ♥①t ♥r tr♥t♦♥ ♥ strts t r♠♠r♥ st♦st sr ♦ t rt① ♦rrs♣♦♥♥ t♦ t ♦t♣t ♦ t ♣♦♥t♦t♦♥ ♦♥ t ♦rsr tr♥t♦♥

t ♦ ♣♦♥t st s ♥♦♥ t ♥ t ♥ ♦rr ♥ ② tt t♦ sss♣♦♥ts ♥ t ♦rr♥ r s♣t② ♦s ♥ t ♣♦♥t ♦t♦♥ ♥ ♥t③ t

♥tr♦t♦♥

r t ❲ ♦♦ ♦♥ ♦ t rr♦s ♦s♥ t r♥♦♠ t♦ t ♦sr t♦ pt ❲t♦t t r♥♦♠ ♦ ♦ t ♥♦r t♦ st t ♦rt♠ ♦ ♥ ♣♥ ♥ ♥♥t ♦♦♣ ♥ ts stt♦♥

t st ♥srt ♣♦♥t ♥ ♣rt ts ♣♣r♦ ②s r② ♦♦ rsts s ♣♣r♦s ♣t s♦rt♥ ❬❪ ♠♣♠♥tt♦♥ ♦r s♦rt♥ ♣♦♥ts s♣t② s ❬❪ t♦rt ♥②ss ♥ ♦♥ ♥ ❬❪

P♦♥t ♥srt♦♥ ♦ ♦ ts st♣ s t♦ rst♦r t ♥② ♣r♦♣rt② ♥ ♥srt♥ t ♥ ♣♦♥t ♥t♦ t tr♥t♦♥ ♥ t t♦♠♥s♦♥ s ♥ ♣♣r♦ s♥s♦ ♣s s ♦t♥ s t t s t t♦ ①t♥ ts ♣♣r♦ t♦ r ♠♥s♦♥s ❬❪ ❲ s ♥ tr♥t ♣♣r♦ ♣r♦♣♦s ♥♣♥♥t② ② ♦②r ❬♦❪♥ ❲ts♦♥ ❬❲t❪ tt ①t♥s s② t♦ d ♠♥s♦♥s ♥ s st ♦r ♠♣♠♥tt♦♥ t♦ ts s♠♣t②

♣♦♥t ♥srt♦♥ s ♦♥ ♥ tr st♣s

♥t② t s tt ♠st ♠♦ t ♦♥t r♦♥

t s r♦♠ t ♦♥t r♦♥

t ♦ rt ♥ st♣ t s s♣♥♥ ② t ♥ ♣♦♥t ♥ t ♦ ts

♦♥t r♦♥ ♦♥ssts ♦ ds♠♣s ♦s r♠sr♥ s ♦♥t♥ t♥ ♣♦♥t s r

♦ ♥ t ♦♥t r♦♥ ♣trst sr ♦rt♠ s s t strts t t ♦♥t♥♥ t ♥ ♣♦♥t ♦r ts ♥♦rs t s tr t② ♦♥ t♦ t♦♥t r♦♥ ♥♦r s ♥ ♦♥t ts ♥♦rs t t r② st ♦♥sr tst rrs② ♥♦r s ♥♦t ♥ ♦♥t t sr s st♦♣♣

♦♥t r♦♥ s strs♣ ♣♦②t♦♣ ♥ t ♥ ♣♦♥t s ♦t ♥ ts r♥ t ♥ ♣♦♥t ♥ s t ♦ ♦♥r② ♦ t ♣♦②t♦♣ rtr♠♦r t ♦♥r②♦ t ♣♦②t♦♣ ♦♥ssts ♦ (d − 1)s♠♣s ② ♦♥strt♦♥ s tr t♥ t ds♠♣s ♥ ♦♥t t s ♣♦ss t♦ ♣ t ♦ ② ♦♥strt♥ ♥ ds♠♣s r♦♠t (d − 1)s♠♣s tt r ts ♦ t ♣♦②t♦♣ ♥ t ♥ ♣♦♥t s s tsr tr♥t♦♥ s r

s ♣♣r♦ rt② ①t♥s t♦ t t ♥② tr♥t♦♥ ♥ ts st ♦♥t r♦♥ ♦♥ssts ♦ ds♠♣s ♦s ♦rt♦♦♥ s♣rs r s♦rt♦♦♥t♦ t s♣r ♥tr t t ♥ ♣♦♥t ♥ ♦s sqr rs s t t ♦ t

rt s♣s

r t tr♥s ♥ ♦♥t t p t str ♦ p ♥ t ♣t♥② tr♥t♦♥

r ♦♥t r♦♥s ♦r ♥② tr♥t♦♥ t ♥ t ♥②tr♥t♦♥ rt

♥ ♣♦♥t s r s♠ ♣trst sr ♦rt♠ s ♦r t ♥②tr♥t♦♥ ♥ s ♦♥t♦♥ ♦r ts ♦rt♠ t♦ ♦r s tt t ♦♥tr♦♥ ♠st ♦♥trt s tr ♦r t t ♥② tr♥t♦♥

❱rt① r♠♦ ♦r ♥r tr♥t♦♥s rt① r♠♦ s ♥♦t ♣♦ss s r♠♦♥ rt① ♠t ♦ tt ♥♥♦t tr♥t t ö♥rt ♣♦②r♦♥❬❪

♦r t ♥② tr♥t♦♥ ♥ t t ♥② tr♥t♦♥ rt① ♥ r♠♦ ♥ t ♦♦♥ ② rst t rt① ♥ ♥♥t s r r♠♦ r♦♠t tr♥t♦♥ s s ♦ ♦s ♦♥r② s ♣♦②t♦♣ t (d−1)s♠♣s sts ♥ t rts ♦ tt ♣♦②t♦♣ r tr♥t s♥ t ♦rt♠ sr♦ t♦ ♦♠♣t t ♥② tr♥t♦♥ ♦r t t ♥② tr♥t♦♥ ♦ ♣♦♥t st ♦♥r② ♦ t ♦ ♦r♠s s♦♠♣① ♦ ts tr♥t♦♥ ♥ tst ♥tr♦r ♦ ts ♣♦②t♦♣ ♥ s ♥t♦ t ♦

rt s♣s

❲ ♥♦ s♦rt ♥tr♦t♦♥ t♦ ♦rt s♣s ♥ ♦tr ♠t♠t ♦♥♣ts tt s tr♦♦t t tss ❬r♠❪ ♦r rtr r♥

♥tr♦t♦♥

t♦♣♦♦ s♣ s st X t♦tr t ♦t♦♥ Ω ♦ ssts ♦ X t t♣r♦♣rts

• ∅ ♥ X r ♥ Ω

• ♥♦♥ ♦ ♥② ♦t♦♥ ♦ sts ♥ Ω s ♥ ♥ Ω

• ♥trst♦♥ ♦ ♥② ♥t ♦t♦♥ ♦ sts ♥ Ω s ♥ ♥ Ω

♦t♦♥ Ω s s♦ t♦♣♦♦② ♦♥ t st X ♥ ♠♥t ω ♦ Ω s ♥♦r♦♦ ♦ ♥ ♠♥t x ∈ X x ∈ ω sst O ♦ X s ♦♣♥ t s ♥♦r♦♦ ♦ ♦ ts ♣♦♥ts sst C ♦ X s ♦s ts ♦♠♣♠♥t s ♦♣♥ t♦♣♦♦s♣ s srt ♠♥ts ♦ X ♣♣r s s♥t♦♥ sts ♥ Ω ♦t tt ♥ srtt♦♣♦♦ s♣ t s♥t♦♥ sts r ♦t ♦♣♥ ♥ ♦s ♥t♦♥ f r♦♠ t♦♣♦♦ s♣ X t♦ t♦♣♦♦ s♣ Y s ♦♥t♥♦s ♦r ♣♦♥t x ♦ X ♥ ♥♦r♦♦ N ♦ f(x) ♥ Y t ♣r♠ f−1(N) s ♥♦r♦♦ ♦ x ♥ X ♦♥t♥♦s ♥t♦♥s r s♦ ♠♣s ♦♠♦♠♦r♣s♠ r♦♠ t♦♣♦♦ s♣ X t♦ t♦♣♦♦ s♣ Y s ♥t♦♥ tt s t ♦♥t♥♦s ♥ s ♦♥t♥♦s ♥rs s ♥t♦♥ ①sts X ♥ Y r ♦♠♦♠♦r♣ ♦r t♦♣♦♦② q♥t

❲ s t ♥♦tt♦♥ G :=< g1, . . . , gk > t♦ ♥♦t t r♦♣ ♥rt ② ♠♥tsg1, . . . , gk t g, h ♠♥ts ♦ G ♦r t♦ r♦♣ ♠♥ts g, h ∈ G t r♦♣ ♦♣rt♦♥tt ♦♠♥s g ♥ h s rtt♥ s gh t♦♣♦♦ r♦♣ s r♦♣ t♦tr t t♦♣♦♦② t t ♣r♦♣rt② tt ♦t t r♦♣ ♦♣rt♦♥ ♥ t ♥t♦♥ ♠♣♣♥ r♦♣ ♠♥t t♦ ts ♥rs r ♦♥t♥♦s r♦♣ s srt t s t♦♣♦♦ r♦♣t t srt t♦♣♦♦② ♥ ts ♦r ♦♥② ♦♥sr srt r♦♣s r♦♣ G ss t♦ t♦rs♦♥r t ♦♥② ♠♥t ♦ ♥t ♦rr s t ♥tt②

t♦♣♦♦ r♦♣ G s s t♦ t s r♦♣ ♦ ♦♠♦♠♦r♣s♠s ♦♥ s♣ X ts ♠♥ts r ♦♠♦♠♦r♣s♠s r♦♠ X t♦ X t r♦♣ ♦♣rt♦♥ s ♦♠♣♦st♦♥ ♦ t♦♦♠♦♠♦r♣s♠s ♥ t ♦♦♥ ♣r♦♣rts r

• (hg)(x) = h(g(x)) ♦r g, h ∈ G ♦r x ∈ X

• e(x) = x ♦r x ∈ X r e s t ♥t ♠♥t ♦ G

• t ♥t♦♥ G × X → X, (g, x) 7→ g(x) s ♦♥t♥♦s

t x ♣♦♥t ♦ X t♦♥ ♦ G s s♦♥t♥♦s t x tr s ♥♦r♦♦U ♦ x s tt g ∈ G | gU ∩ U 6= ∅ s ♥t t♦♥ s s♦♥t♥♦s t s s♦♥t♥♦s t ♥② x ∈ X st ♦ ♠s g(x) ♦r ♠♥ts g ♦ G s t ♦rt ♦ x r♦♠ t t tt G s r♦♣ ♦♦s tt t♦ ♥trst♥ ♦rtsr q ♥ ts ♦rts ♦r♠ ♣rtt♦♥ ♦ X s ♥s t ♦♦♥ q♥rt♦♥ x ∼ y x = g(y) ♦r s♦♠ g ∈ G q♦t♥t s♣ X/G s t st ♦ ♦rts♦ X ♥r t t♦♥ ♦ G r♦♠ ♥♦ ♦♥ s t tr♠ ♦rt s♣ ♦r X/G s ts ♠♦r ♦♠♠♦♥② s ♥ ♦♠tr ♦♥t①ts ♥ ♠♣s③s t t tt ♦♥sr s♣ ♦ ♦rts ❲ X/G t ♦rt s♣ ♦ X ♥r t s♦♥t♥♦s t♦♥ ♦ G ❲s t ♥♦♠♥t♦♥ ② s♥ t ♦♦♥ s♦rt ①♣rss♦♥s t♦ ♥♦t X ♥ G ♥tr r ♥♦ ♠ts X s t ♥r②♥ s♣ ♦ X/G ♥ G s t r♦♣ ♦ t♦♥♥♥ X/G t g ∈ G ♥ p ∈ X ❲ s t s♦rt ♥♦tt♦♥ gp t♦ ♥♦t g(p)

♥♠♥t ♦♠♥ ♦ G s sst ♦ X tt ♦♥t♥s t st ♦♥ ♣♦♥t ♦ ♦rt ♥ ♦t♥ rqrs t ♥♠♥t ♦♠♥ t♦ ♦♥♥t ♥ ♦s ♥ t♦♦♥t♥ ♥♦t ♠♦r t♥ ♦♥ ♣♦♥t ♦ t s♠ ♦rt ♥ ts ♥tr♦r

Pr♦♠ stt♠♥t

❲ rtr♠♦r ♥ t ♦♥♣t ♦ ♦r♥ s♣s ♥ s ♦♦s

♥t♦♥ ♦r♥ s♣ t X t♦♣♦♦ s♣ ♠♣ ρ : X → X s ♦r♥ ♠♣ ♥ X s s t♦ ♦r♥ s♣ ♦ X t ♦♦♥ ♦♥t♦♥♦s ♦r ♣♦♥t x ∈ X tr s ♥ ♦♣♥ ♥♦r♦♦ V ♥ ♦♠♣♦st♦♥ ♦ρ−1(V ) s ♠② Uα ♦ ♣rs s♦♥t ♦♣♥ ssts ♦ X ♥ s ② tt ρ|Uα

s ♦♠♦♠♦r♣s♠ ♦r α t hx ♥♦t t r♥t② ♦ t ♠② Uα ♦rrs♣♦♥♥t♦ s♦♠ x ∈ X t ♠①♠♠ h := maxx∈X hx s ♥t t♥ X s ♥ hst♦r♥ s♣ ♦ X

♦r♥ s♣ X ♦ X s s t♦ ♥rs ♦r♥ s♣ t s ♦r♥ s♣♦ ♦r♥ s♣s ♦ X ♦r ①♠♣ st ♦r♥ s♣ ♦ r s ♥ r s r r ♥ s ♥rs ♦r♥ s♣ ♦ r

r t r t st ♦r♥ s♣

s♣ s ♠♥♦ ♦ ♠♥s♦♥ d ♦r d♠♥♦ ♦ ts ♣♦♥ts s ♥♦r♦♦ ♦♠♦♠♦r♣ t♦ E

d ♦t tt t♦rs♦♥r r♦♣ ♦s ♥♦t ① ♣♦♥ts gp 6= hp ♦s ♦r p ∈ E

d g, h ∈ G ♥ g 6= h X s ♠♥♦ ♥ G s srt♥ t♦rs♦♥r t♥ X/G s ♠♥♦ ♥ ts s t t♦♥ ♦ G s s♦ ♣r♦♣r②s♦♥t♥♦s G s ♥♦t t♦rs♦♥r t♥ X/G ♥ ♣♦♥ts ♦s ♥♦r♦♦ s♥♦t ♦♠♦♠♦r♣ t♦ E

d s♦ s♥r ♣♦♥ts ♥ ts s X/G s ♥ ♦r♦ s ♥r③t♦♥ ♦ ♠♥♦s ❬ P❪

❲♥ t ♦t rtr ♦ ♠♥♦s ♠♥ ♠♥♥♥ rtr s ♥r③t♦♥ ♦ ss♥ rtr t♦ d♠♥s♦♥ ♠♥♦s ❬❪ ❲ ♦♥sr♦♥② ♦rt s♣s ♦s ♥r②♥ s♣s ♦♥st♥t rtr

Pr♦♠ stt♠♥t

♦ ♦ ts ♦r s t♦ ♥t♦♥ ♦ t ♥② tr♥t♦♥ ♦ r♥t ♦rts♣s ♥ t♦ s ♥t ♦rt♠s t♦ ♦♠♣t t

♥ ♦rt s♣ ♦♥sr s t s♦ t t♦rs t t♦rs ♥ ♥rt② ♥t②♥ ♦♣♣♦st s ♦ sqr s r

r ♥t②♥ ♦♣♣♦st s ♦ sqr ②s t♦rs

♥tr♦t♦♥

tr♥t♦♥ ♦ t t t♦rs s ♣r♦ ② ♦♥strt♦♥ t ♥ r♣t ♣r♦② ♥ x ♥ yrt♦♥ ♥ E

2 s r ♥ ♣tr sr ♥ ♠♦r

r ♣r♦ tr♥t♦♥

t ♦ t♦ ♦♥sr t t t♦rs s ♥ ♦rt s♣ t ♥♣t ♣♦♥t st s t♦♦ s♠ ♦r ♥♦t strt t ♦s ♥♦t ♥ tr♥

t♦♥ s ♠♥s tt t ♣rtt♦♥ ♦ t t t♦rs ♥t♦ rts s♠♥ts ♥ tr♥ss ♥♦t s♠♣ ♦♠♣① ♥ ♦♦♥② ♥♦t tr♥t♦♥ s r

r s s ♥♦t s♠♣ ♦♠♣① s t ♥trst♦♥ ♦ σ ♥ τ s p ∪ e s ♥♦t s♠♣①

t s ♥sst r ♦♥ t t tt ♦♠♣t♥ tr♥t♦♥ s♠♣ ♦♠♣①s ♠♣♦rt♥t ♦r sr rs♦♥s rst tr♥t♦♥ s ♥ s s♠♣ ♦♠♣① ♥t trtr ❬r♠ ♥ ♣ ❩♦♠❪ ♦r♦r s♥♥ t strtr t♦ ♥t② st♦r tsst♦♥s tt r ♥♦♥s♠♣ ♦♠♣①s ∆♦♠♣①s ❬t❪ ♦ qt ♥♦ tr♥t♦♥ t strtrtt rs ♥ ♦r ♠♣♠♥tt♦♥ ss♠s t strtr t♦ s♠♣ ♦♠♣①❬P❪ ♥ ♠♦r ♠♣♦rt♥t② ♦rt♠s s♥ tr♥t♦♥ s ♥♣t r② ♦♥ tt tt t tr♥t♦♥ s s♠♣ ♦♠♣① ts s t s ♦r ♥st♥ ♦r ♠s♥♦rt♠s ❬❨ ❨❪ s s ♦rt♠s t♦ ♦♠♣t αs♣s r t②♥ ♥ t ♣r♦ s ② sr ♣♣t♦♥s ♠♥t♦♥ t t ♥♥♥ ♦ ts♥tr♦t♦♥

♥ ♣♣r♦ t♦ rs♦ sss rs♥ ♥ ♦♠♣t♥ t ♥② tr♥t♦♥ ♦

tt ♦ t rt

♥ ♦rt s♣ s t♦ s sr ①♣t ♦♣s ♦ t ♥♣t ♣♦♥t st r ①♠♣sr ts ♣♣r♦ s s r ♣rs♥t ♥ t♦♥ ❲ ♥t t♦ ♥♠r ♦♦♣s tt s s♥t② r t♦ ♦t♥ s♠♣ ♦♠♣① t s♠ ♥♦ s tt♥ ♠♣♠♥tt♦♥ ♦ ♥ ♦rt♠ s st s ♥ ♣rt ♥♠r ♣♥s ♦♥♦t t ♦rt s♣ ♥ t ♥♣t ♣♦♥t st ❲ s♦ ♠ t ♦r♥ t♦t t♦♥♦♣s ♥r ♣♦ss

tt ♦ t rt

♥ ts st♦♥ sr ①st♥ ♦r ♦♥ tr♥t♦♥s ♥ s♣s ♦tr t♥ Ed s

♣② ♦r ♥ rt♦♥ t♦ ♦rt s♣s t rst ♦♥sr ♦r sss♥ ♥②tr♥t♦♥s ♦ r♥t s♣s ♦ ♦♥st♥t rtr ♥ tr♥ t♦ sss♥ s♦tr ♣s ♥ ♥② ♦♥sr ♣♣rs tt trt t ♣r♦♠ ♦r ♠♦r ♥rs♣s ♥ ♥ ♠♦r ♥r stt♥s

r♠ ♥ árq③ sr ♦ t♦ ♦♠♣t tr♥t♦♥s ♦ ♣♦♥t sts ♦♥ srs ❬❪ ② ①♣t② trt t ss ♦ t ②♥r t s♣r ♥ t t♦rs② ♦ ♥♦t sss t ♥② tr♥t♦♥

♥ st ♦ ♣♦♥ts S ♦♥ sr t② ♥ ♣s♦tr♥t♦♥ s ♠①♠♠st ♦ s♠♥ts r t s♦rtst ♦ss ♦♥♥t♥ t♦ ♣♦♥ts r♦♠ S s tt♥♦ t♦ s♠♥ts ♥trst ② ♦ ♥♦t ♦♥sr ♥rt ss t ♠♣t② ss♠♦ss t♦ ♥q② ♥ s♠♥ts ♦ t ♣s♦tr♥t♦♥ ♣rtt♦♥ tsr ♥t♦ ♣♦②♦♥ r♦♥s t♦rs ♣r♦ t ♦♦♥ rsts s s♦ r

②♥r t ♥♣t ♣♦♥ts r ♥♦t ♦♥t♥ ♥ ♦♥ ②♥r t♥ t ♣s♦tr♥t♦♥ ♣rtt♦♥s t ②♥r ♥t♦ t♦ ♥♦♥ r♦♥s ♥ ♠♥② ♦♥r♦♥s tt r tr♥s

♣r t ♥♣t ♣♦♥ts r ♥♦t ♦♥t♥ ♥ ♦♥ s♣r t♥ t ♣s♦tr♥t♦♥ ♣rtt♦♥s t s♣r ♥t♦ tr♥s ts ♣rtt♦♥ s tr♥t♦♥

♦rs t st ♦♥ qr♥t ♦ t ♥♠♥t sqr ♦ t t t♦rs ♦s ♥♦t ♦♥t♥♥② ♥♣t ♣♦♥t t♥ t ♣s♦tr♥t♦♥ s ♥♦t tr♥t♦♥

② ♦ ♥♦t ♥ ♦rt♠ ♦r ♦♠♣t♥ tr♥t♦♥s ♦ t t♦rs

r t r♥t♦♥ ♦ t ②♥r t Ps♦tr♥t♦♥ ♦ t t♦rs♥♦ ♠♦r s♦rtst ♦ss ♥ t♦t ♥tr♦♥ ♥trst♦♥s ❬❪

♥tr♦t♦♥

③ó♥ ♥ ♦ ❬❪ sss t ♦♠♣tt♦♥ ♦ ❱♦r♦♥♦ r♠s ♦♥ ♦r♦s strtr ♦ t ❱♦r♦♥♦ r♠ s t ♥② ♣r♦♣rt② t s ♥♦t ♥ssr② s♠♣ ♦♠♣① ② ♦♥② ♦♥sr qt rstrt ss ♦ ♦r♦s rt s♣s♦ E

2 ♥ S2 ♥r t t♦♥ ♦ srt r♦♣s ♦ s♦♠trs ♥ t ♥ s

ts r t ♣♣r r♦♣s t ♣♦♥t r♦♣s ♥ t r③ r♦♣s ♥ t s♣rs t ♣♦♥t r♦♣s ♦♥② ❬❪ t G ♦♥ s r♦♣ ♥ S t ♥♣t ♣♦♥tst G s ♥♥t t♥ GS t st ♦ ♣♦♥ts ♥ t ♦rts ♦ t ♣♦♥ts ♦ S s♥♥t t V D(S) ♥♦t t ❱♦r♦♥♦ r♠ ♦ S ♥ E

2 ♦r S2 t♦rs s♦ tt

♦r ♥② G tr s ♥t ♣♦♥t st S∗ ⊂ GS ♥ ♥♠♥t ♦♠♥ F s ttV D(S∗) ∩ F = V D(GS) ∩ F s ♠♥s tt V D(S∗) ∩ F ♥r t t♦♥ ♦ G s t♥♥t ❱♦r♦♥♦ r♠ ♦ GS s r ♦ r s t♦ ♦♠♣t s s

r t ❱♦r♦♥♦ r♠ ♦ tr ♣♦♥ts ♦♥ ②♥r t ♥ s ♥♦ ♦r ❬❪

♥ ♦ t♦rs ♣r♦ tt ♥ t ♦rst s ♦r t ♦rst r♦♣s S∗

♦♥ssts ♦ ♣ t♦ ♦♣s ♦ t ♣♦♥t st S♥ ts ♣♣r♦ t ♦♠♣tt♦♥ ♦ t ❱♦r♦♥♦ r♠ s ♦♥ ♥ E

2 ♦r S2 rs♣

t② s ♥② ♦ t ♥♦♥ ♦rt♠s ♦ ♦♠♣①t② O(n log n) ♥ s st ♥♠r ♦ ♣♦♥ts ♥ S∗ s t ♠♦st t♠s t ♥♠r ♦ ♣♦♥ts ♥ S ts ♦s ♥♦t♥ t s②♠♣t♦t ♦r ♦ t ♦rt♠ t t ♦s ♥♦t ♣♣r r② ♣rttr ♥ tt t ♦rt♠ s s♦ ♦♥ ② t♦r ♦ t st 37 ♥ t ♦rst s

♥♥② ♥ ❬❪ ①♠♥ tr♥t♦♥s ♦ t ♣r♦t ♣♥ s♥ ♦rt s♣ ♦ S

2 ♥r t t♦♥ ♦ t r♦♣ ♥t②♥ t♦ ♥t♣♦ ♣♦♥ts ②s ♥ ♥r♠♥t ♦rt♠ ♥ strt t ♥ ♥t tr♥t♦♥ ♦ s① ♦ t ♥♣t♣♦♥ts ② r♣rs♥t t tr♥t♦♥ ♥ ♣r♦r♠ t ♦♠♣tt♦♥s ♥ t ♣r♦t♣♥ ts ② s♦ t ♣r♦♠s rs♥ t♦ t t tt t ♣r♦t ♣♥ s♥♦t ♦r♥t tr♥t♦♥s t② ♦♠♣t ♦ ♥♦t t ♥② ♣r♦♣rt②

♦♥ ♥ s♦♥ ❬❪ s♦ tt ♥ t s ♦ t tr♠♥s♦♥ t t♦rs ♣r♦ ♥② tr♥t♦♥ ♥ ①trt r♦♠ ♥② tr♥t♦♥ ♦ ♦♣s ♦ t ♣♦♥t st ♥ t♦♥ ♣rs♥t ♣rt ♦ tr ♣r♦♦ ♥ ♠♦r t② ♦ ♥♦t sss tr t ♥♠r ♦ rqr ♦♣s ♥ r ♣♥♥♦♥ t ♥♣t ♣♦♥t st ② ♦ ♥♦t ♦r ♥ t t t♦rs t ♦♥sr ♥♠♥t♦♠♥ ♥ ② t ❱♦r♦♥♦ ♦ s♥ ♣♦♥t ♥ t t♦rs ♥ t② ♦♠♣t t♥② tr♥t♦♥ ♦ t ♦♥① ♦ t ♥♣t ♣♦♥t st ♥s t ♥♠♥t

tt ♦ t rt

♦♠♥ t♦tr t ts ♦♣s ♥ t ♥t ♥♠♥t ♦♠♥s ♥ E3 r

♦rt♠ ♦rs ♥ tr st♣s

t S t ♥♣t st ♦ ♣♦♥ts ♥ t ♥♠♥t ♦♠♣t t st S∗ ♦ ♦♣s ♦ ♣♦♥t ♥ S

♦♠♣t t ♥② tr♥t♦♥ ♦ S∗ ♥ E3

①trt t s♠♣s r♦♠ t ♥② tr♥t♦♥ ♦ S∗ tt r ♣rt ♦ t♥♥t ♣r♦ ♥② tr♥t♦♥

♦t♣t ♦ ts ♦rt♠ s st ♦ ttrr ♦s ♥♦♥ s ♥♠♥t ♦♠♥♦ t tr♠♥s♦♥ t t♦rs

♥ s ts ♣♣r♦ t② ♦♠♣ts ♥② tr♥t♦♥ ♥ E3 t ♥ s

♥② ♦ t ♥♦♥ ♦rt♠s ♦r t ♥♠r ♦ ♥♣t ♣♦♥ts t♦ t ♦rt♠ s t♠s rr t♥ t ♥♠r ♦ ♣♦♥ts ♥ S s ♦s ♥♦t ♥ t t♦rt♦♠♣①t② ♦ t ♦rt♠ t ♦r ♣rt s t ♦ ♥ t♦ r t ♥♠r♦ rqr ♦♣s ♦ t ♥♣t ♣♦♥ts

♦tr

♥ t ♥♥r♥ ♦♠♠♥t② ♦♠♣s♦♥ ♣r♦♣♦s ♥ ♦rt♠ ♦r ♦♠♣t♥ ♥②tr♥t♦♥s ♦ t tr♠♥s♦♥ t t♦rs ♦♥ ♣t♦♥s ♦ ♣♦♥ts ❬♦❪♦r t ♦rt♠ ② rs ♦♥ t ss♠♣t♦♥ tt t ♥♣t ♣♦♥t st s s♥t② r ♥ strt t strts t ♥ ♥t tr♥t♦♥ ♦ t rts♦s t♦ t ♦r♥rs ♦ t ♥♠♥t s ♥t strtr s ♥♦t s♠♣♦♠♣① ♥ t s ♥♦t rqr t♦ t ♥② ♣r♦♣rt② P♦♥ts r trt② s♥ sr t♥qs ♥ ♦rr t♦ ♥ ♥♦♥sst♥s t♦ ♦t♥♣♦♥t rr♦rs♥ ♦r r♦♥ t♠ ❬❪ ♠ s tt t ♥♣t ♣♦♥t st s s♥t②r ♥ strt t♥ t ♥ ♦t♣t ♥② tr♥t♦♥ ♦ ttr♠♥s♦♥ t t♦rs ♠ s s♣♣♦rt ② ①♣r♠♥ts ♦♥ strtt ♦ ♣r♦♦s r ♣r♦ ①♣r♠♥ts ①t r♥♥♥ t♠ ♦ ♦t O(n1.11)❯♥♦rt♥t② t ♦ s ♥♦t s♦ ♦ ♥♦t ♦♠♣r t t♦ ♦r ♠♣♠♥tt♦♥

♦♥ ♥ r♦t t s♦tr ♣ s ♦r ♦♠♣t♥ ♦t ♣r♦ ♥♥♦♥♣r♦ ❱♦r♦♥♦ r♠s ❬❪ ♦r t ts t♦ ♥ s ♥♦t ♠♥t♥② t t♦r ♥②♠♦r ♥ s ♦ ♣r♦ ❱♦r♦♥♦ r♠s t ♦rs t ♦♣s ♦ ♥♣t ♣♦♥t

rtr♠♦r tr r ♦tr ♠♣♠♥tt♦♥s r♦♠ rsrrs ♣t t♦ tr s♣ s ♦r ♥st♥ ❱♦r ♣r♥ ♠♣♠♥t ♣r③ rs♦♥ ♦ t♦♠♥s♦♥ ♣r♦ ♠s ♥rt♦r t♦ r♥ ①♣r♠♥ts ♦♥ ②♥♠s ❬♣r ♣r❪t s ② ♣t t♦ t s rr ♥ t s ♥♦t ♥t♥ t♦ ♦r ♣s

♦r ♥r s♣s

♦♠ ♦r ♦♥ tr♥t♦♥s ♥ ♠♦r ♥r s♣s ♥ ♣ssr♥♥r ♥ sss tr♥t♦♥s ♥ ♥r t♦♣♦♦ s♣s ❬❪

♦r t② ss♠ t s♣ t♦ ♠ ♥ Ed ♥ t② s♣♣♦s tt t② r

♥ s♥t② s♠♣ ♣♦♥t st ♥ t② ♦♠♣t t ♥② tr♥t♦♥

♥tr♦t♦♥

♦ t ♣♦♥t st ♥ Ed ♥ ♥ t tr♥t♦♥ ♦ t ♥ t♦♣♦♦ s♣ t♦

rstrt♦♥ ♦ t tr♥t♦♥ ♥ Ed ♥ s ② t rstrt ♥② tr♥t♦♥

s s♦♠♣① ② ♦♥strt♦♥ ♥ ts tr♥t♦♥ t sss♦♥ ♠st tr♠♥♥ tr t rstrt tr♥t♦♥ s ♦♠♦t♦♣② q♥t t♦ t ♥t♦♣♦♦ s♣ s t♦♣ s ♥ st t♦ ①t♥s rsr s ♦r ♥st♥ ❬ ♣ ❪

♦ rrs ♥ s♦♥ sss ♦r♦ tr♥t♦♥s ♥ r②st♦r♣ r♦♣s❬❪ r ♣r♦♠ s ♥ s♦♠ s♥s ♥rs t♦ t ♣r♦♠ ♥t t♦ sss ♥ tr♥t♦♥ ♦ s♣ ♥ t ♦rrs♣♦♥♥ s♣ r♦♣ t r♦♣ G s ttt ♥ tr♥t♦♥ s tr♥t♦♥ ♦ t ♦rt s♣ E

3/G

♦♥trt♦♥s

❲ ♠t♠t② s♦♥ ♥t♦♥ ♦ t ♥② tr♥t♦♥ ♦ t tr♠♥s♦♥ t t♦rs ♥ ② st ♦ ♣♦♥ts ❲ ♣r♦♣♦s ♦♥sst♥t ♣♣r♦ ♦trt♥ ♣♦♥t sts tt ♦ ♥♦t ♥ ♣r♦ ♥② tr♥t♦♥ s♥ ♦r♥s♣s ❲ ♣r♦ ♦♥t♦♥s t♦ tr ♣r♦ ♥② tr♥t♦♥ ♦ ♦r♥ s♣ ♦ t t t♦rs ♥ ♦♥rt t♦ ♥② tr♥t♦♥ ♦ t tt♦rs ts ♥ s♠♣ ♦♠tr rtr t♦ r② ts ♦♥t♦♥s

❲ ♣rs♥t ♥ ♣tt♦♥ ♦ t ♥♦♥ ♥r♠♥t ♦rt♠ ♥ E3 ❬♦

❲t❪ tt ♦s t♦ ♦♠♣t ♥② tr♥t♦♥s ♦ t tr♠♥s♦♥ t t♦rs❲ ♦s ♦♥ t ♥r♠♥t ♦rt♠ ♦r sr rs♦♥s ts ♣rt ♥② s ♥♣r♦ ♥ ♣rtr ② t ② ②♥♠ ♠♣♠♥tt♦♥ ♥ ❬P❪ ♠♦r♦r ②♥♠ ♦rt♠ ♦♥ t♦ r② ♥srt ♥ r♠♦ ♣♦♥ts s ♥ssr② ♥r♥t ♦r ♠s♥ ♦rt♠s ♥ s♦tr s ♦♥ ♥② r♥♠♥t ♠t♦s s♦r ♥st♥ ❬ ❨ ❪ ❲ ①t♥ t ♦ ♦r t♦ r♥t sss ♦tr♥t♦♥s s s t ♥② tr♥t♦♥s

❲ s♦ ♣r♦ ♥ ♠♣♠♥tt♦♥ ♦ t ♦rt♠ s♦tr s ♥ r♥ ♣t ② t t♦r ♦r ♦r ♥s♦♥ ♥ rs♦♥ ♦ t rr②❬❪ s s♦tr s ♠♦♥strt ② ♦ ❬❪ ❲ r ♥ ♦♥tt t srs ♦♦r ♠♣♠♥tt♦♥ ♥ tr s t st ♦♥ ♣t♦♥ rrr♥ t ❬♦❪

❲ sss ①t♥s♦♥s tt ♦ ♦r s♥ ♦r ♦ ♥ ♦♠♥t♦♥ t ♦tr ♦♠♣♦♥♥ts s s t ♣ s♣ ♣ ❬❨❪ ♦r t sr ♠s ♥ ♦♠ ♠s♥rt♦rs ❬❨ ❨❪

❲ ①t♥ ♣rts ♦ ♦r ♦r ♦♥ ♥② tr♥t♦♥s ♦ t t t♦rs t♦ r♥tsss ♦ s♣s s s ♦rt s♣s ♦ t d♠♥s♦♥ ♥ s♣ E

d ♥ td♠♥s♦♥ s♣r S

d ❲ ♣rs♥t s♦♠ ♣r♠♥r② s ♦♥ ♦ t♦ ①t♥ t s♦ t♦♦rt s♣s ♦ t ②♣r♦ s♣ H

d

tss s ♦r♥③ s ♦♦s ♥ ♣tr sss ♣r♦ tr♥t♦♥s ♥t ❲ ♣rs ♥t♦♥ s s ♥ ♦rt♠ ♥ ♣r♦ sr ♠♣♦rt♥t♣r♦♣rts ♥ ♣tr sr t ♠♣♠♥tt♦♥ ♥ s♦ s♦♠ ①♣r♠♥ts ♥ ♣♣t♦♥s ♣tr s t ①t♥♥ t t♦r② ♦♣ ♥ ♣tr ♦♥t♦ rtr t ♦rt s♣s s s s♣r ♥ ②♣r♦ ♦rt s♣s

♣tr

Pr♦ tr♥t♦♥s

♥ ts ♣tr rst s♦ ♦ t♦ ♥ t t t♦rs s ♥ ♦rt s♣ ♥ t ♠♥♣rt ♦ t ♣tr s♦ ♦ t♦ ①t♥ t ♥t♦♥ ♦ ♥② tr♥t♦♥ t♦t tr♠♥s♦♥ t t♦rs ♥ ♣rs♥t ♥ ♦rt♠ ❲ ①t♥ t ♦rt♠ t♦t ♥② tr♥t♦♥s ❲ ♥② ♣r♦ t ♦rrt♥ss ♦ t ♦rt♠ ♥s♦ tt t s r♥♦♠③ ♦rsts ♦♣t♠ ♦♠♣①t② ♦ O(n2)

t t♦rs

t rst ♣rs ♥t♦♥ ♦ t t t♦rs tt ♦♥sr tr♦♦tt rst ♦ t ♣tr ♥ r s♦♠ ♦ ts ♥♦♥ ♣r♦♣rts ♥ sts♥♦tt♦♥ ♥② ♥t♦♥ ♦ s♠♣s ♥ t t t♦rs

t tx ty ♥ tz ♥♦t t ♥t tr♥st♦♥s ♦♥ t x y ♥ z①s ♥ E3

rs♣t②

♥t♦♥ T3c t c := (cx, cy, cz) ∈ (R\0)3 ♥ G t r♦♣ < cxtx, cyty, cztz >

♦rt s♣ E3/G s t t♦rs ♥ ♥♦t t ② T

3c ❲ rtr♠♦r ♥♦t t

♣r♦t♦♥ ♠♣ ② π : E3 → T

3c

♥♦tr ♦♠♠♦♥② s ♥♦♠♥t♦♥ s ♥ t♦rs ❬❪ r t ♦r ♥ ♠♥s tt t s♣ s ♦♥st♥t ♠♥♥♥ rtr ③r♦ t s t♦rss t s ♦♠♦♠♦r♣ t♦ t tr♠♥s♦♥ t♦rs ♠ ♥ E

4 t t♦rs T

3c tt ♦♥sr r s d♦ s ♥♠♥t ♦♠♥ ♦r

♥r t t♦r t ♥♠♥t ♦♠♥ s ♥♦t rstrt t♦ ♦s t ♥ d♠♥s♦♥ ♣r♣♣ ♥r s s trt ♥ t♦♥

♦t tt t ♦rts ♦ G r s♦♠♦r♣ t♦ Z3 ♥ T

3 × Z3 s s♦♠♦r♣ t♦ E

3 T3 s

♠tr s♣ t stT(π(p), π(q)) := min st(p′, q′) ♦r p′ ∼ p, q′ ∼ q ♦t tt π s♦♥t♥♦s

♦♥sr t ♦s ♦ [u, u + c0]× [v, v + c1]× [w,w + c2] s ♥♠♥t

♦♠♥ ♦r G ♦♣♥ ♦ Dc = [0, cx) × [0, cy) × [0, cz) ♦♥t♥s ①t② ♦♥

r♣rs♥tt ♦ ♠♥t ♦ T3c ❲ t ♦r♥ ♦♠♥ ❲ ♥♦t ♦♦r♥t

s t♦r ♠t♣t♦♥ ② ∗ (ax, ay, az) ∗ (bx, by, bz) := (axbx, ayby, azbz) ♦t ttts ♥ s♦ ♥rst♦♦ s ♠tr① ♠t♣t♦♥ r t a ♥ b r t ♥trs

Pr♦ tr♥t♦♥s

♦♥ t ♦♥s ♠♣

ϕc : Dc × Z3 → E

3

(p, ζ) 7→ p + c ∗ ζ

s t ♦♥st ♦♥ ♦ Dc s ♥t ‖c‖ ❲ s② tt t♦ ♣♦♥ts p1, p2 ∈ E3

r ♣r♦ ♦♣s ♦ ♦tr t② ♦t ♥ t s♠ ♦rt ♦r q♥t② trs ♣♦♥t p ∈ Dc s tt p1, p2 ∈ ϕc(p × Z

3)♦ tr♥ t♦rs t ♥t♦♥ ♦ s♠♣s ♥ T

3c r s ♥♦ ♠♥♥ ♥t♦♥

♦ ♦♥① ♥ T3c ♥ ttrr♦♥ s ♥♦t ♥q② ♥ ② ♦r ♣♦♥ts ❲ tt

t rt① ♥ ♥tr t♦r ♥♠ ♦st tt s♣s ♦♥ r♣rs♥tt ♦t ♦♥ ♦rt s r ♥tt② t ♦sts tr♠♥ ② s♠♣① r♣sr♦♥ t t♦rs ♥ t ♦ ♥t♦♥ ♦ ϕc t ♦sts r t ♥♠rs ζ ∈ Z

3

r strt♦♥ tr ♣♦♥ts p1, p2, ♥ p3 ♦ ♥♦t ♥q② ♥ tr♥ r♥ tr♥ s t tr♥ (p1,

(00

)) (p2,

(10

)) (p3,

(11

))

s s♥ ♥ t♦♥ t s♠♣ ♦♠♣① ♥t♦♥ s ♣r② ♦♠♥t♦r ♥♦s ♥♦t ♣♥ ♦♥ t s♣ ♥② t s♠♣① ♥t♦♥ ♠st ♣t t♦ T

3c s

♥ ♦♥ ♥ s♠r ② s ♥ ❬❲❪

♥t♦♥ ks♠♣① ♥ T3c t P st ♦ k+1 k ≤ 3 ♣♦♥t♦st ♣rs (pi, ζi)

♥ Dc×Z3 0 ≤ i ≤ k t (P) ♥♦t t ♦♥① ♦ ϕc(P) = pi+c∗ζi | 0 ≤ i ≤ k

♥ E3 t rstrt♦♥ π|(P) ♦ π t♦ t ♦♥① ♦ P s ♥t t ♠ ♦ (P)

② π s ks♠♣① ♥ T3c

♥ ♦tr ♦rs t ♠ ♥r π ♦ s♠♣① ♥ E3 s s♠♣① ♥ T

3c ♦♥② t

♦s ♥♦t s♥trst ♦r t♦ r s♦s t ♦♥① s A B ♥ C ♦ tr♣♦♥t♦st ♣rs ♥ [0, 1)2 ×Z

2 rs♣t②(p1,

(02

))s ♣♦♥t ♥ t ♦rt ♦ rt① ♦

A tt s ♥s A tr rts ♦ B r ♥ t s♠ ♦rtr r ♥♥t② ♠♥② sts ♦ ♣♦♥t♦st ♣rs s♣②♥ t s♠ s♠♣①

♥t♦♥ ♦ ♥ ♦ s ♣t ♦r♥② t σ ks♠♣① ♥ ② stPσ ⊆ Dc × Z

3 s♠♣① τ ♥ ② st Pτ ⊆ Dc × Z3 s ♦ σ ♥ s σ s

♦ ♥ ♦♥② tr s s♦♠ ζ ∈ Z3 s tt (pi, ζi + ζ) | (pi, ζi) ∈ Pτ ⊆ Pσ

♥② tr♥t♦♥ ♦ T3c

s st♦♥ s ♦r♥③ s ♦♦s t rst ♥t♦♥ ♦ t ♥② tr♥t♦♥ ♦ T

3c ❲ ♦sr tt tr r ♣♦♥t sts ♥ T

3c tt ♦ ♥♦t ♥ ♥②

♥② tr♥t♦♥ ♦ T3c

A

C

(p0,

(2

2

))(p1,

(2

2

))

(p2,

(2

2

))

(p0,

(2

1

))(p1,

(2

1

))

(p2,

(2

1

))

(p0,

(0

0

))

(p1,

(0

0

))(p2,

(0

0

))

(p0,

(1

1

))(p1,

(1

1

))

(p2,

(1

1

))

(p0,

(2

0

))(p1,

(2

0

))

(p2,

(2

0

))

(p0,

(1

2

))(p1,

(1

2

))

(p2,

(1

2

))

(p0,

(0

2

))

(p1,

(0

2

))(p2,

(0

2

))

(p0,

(0

1

))(p1,

(0

1

))

(p2,

(0

1

))

(p0,

(1

0

))(p1,

(1

0

))

(p2,

(1

0

))

B

r strt♦♥ π(A) ♥ π(B) r ♥♦t s♠♣s ♦r π(C) s s♠♣①

tr♥t♦♥ ♦ T3c ♥ ♥ssr② ♥ s♥t ♦♥t♦♥ ♦r ♣♦♥t st t♦

♥ ♥② tr♥t♦♥ ♦ T3c ♦r♠ ♥ t s♦♥ ♣rt sss ♦

t♦ t ♣♦♥t sts tt ♦ ♥♦t ♥ ♥② tr♥t♦♥ ♦ T3c

♥t♦♥

t s r tt tr♥t♦♥ ♦ ♣♦♥t st S ♥ E3 s s♠♣ ♦♠♣① t rt①

st S t s ♥② tr♥t♦♥ ♥ ♦♥② ttrr♦♥ stss t ♥②♣r♦♣rt② ts r♠sr♥ ♦s ♥♦t ♦♥t♥ ♥② ♣♦♥t ♦ S ♥ ts ♥tr♦r r♦♠♥♦ ♦♥ ②s ss♠ ♥② tr♥t♦♥s ♦ E

3 t♦ ♥q② ♥ ②s♥ s②♠♦ ♣rtrt♦♥ s sr ♥ t♦♥

❲ ♥t t♦ ♥ t ♥② tr♥t♦♥ ♦ T3c ♦r ♥ ♣♦♥t st π(S)

s t♦ s t ♣r♦t♦♥ ♥r π ♦ ♥② tr♥t♦♥ ♦ E3 ♥ ② t

♥♥t ♣r♦ ♣♦♥t st GS := ϕc(S × Z3) ❲t♦t ♦ss ♦ ♥rt② ♥ ss♠

tt t ♣♦♥ts ♦ S ♥ Dcr♦♠ ♥♦ ♦♥ ♥ ②s ♥♦t t tr ♦♥t♦♥s ♦ ♥t♦♥

♠♣ ♦♠♣①

♠♠ ♦r ♥② ♥t ♣♦♥t st S ⊂ Dc st ♦ s♠♣s K ♥ E3 tt s

♥ t ♥② ♣r♦♣rt② t rs♣t t♦ GS s s♠♣ ♦♠♣① ♥ E3

Pr♦♦ ❲ ♥ t♦ s♦ tt K s t ♦ ♥t♥ss ♣r♦♣rt② ss♠ tt tr s rt① v t ♥ ♥♥t ♥♠r ♦ ♥♥t s♠♣s ♥ ts

♥ ♥♥t ♥♠r ♦ ♥♥t s ♥ S ♦♥t♥s ♦♥② ♥t ♥♠r ♦ ♣♦♥ts tr♠st t st ♦♥ ♣♦♥t q ♥ S ♦ ♥♥t② ♠♥② ♣r♦ ♦♣s r ♥t t♦v ♣r♦ ♦♣s ♦ q ♦r♠ r ♥ t ♠tr ♦ t rst ♠♣t② s♦♥ ② ‖c‖ ♦ r♠sr♥ s ♦ ttrr tt r ♦s ♦ s t♥v ♥ ♣r♦ ♦♣s ♦ q tt r rtr ② t♥ ‖c‖ ♥♥♦t ♠♣t② s s ♦♥trt♦♥ ♥ ♥ rts ♥ K r ♥♥t t♦ ♦♥② ♥t ♥♠r ♦ s♠♣s

t s ♥♦ ♦♥sr ♣♦♥t p ♥ E3 tt s ♥♦t rt① ♥ K t s ♥ t ♥tr♦r

♦ ttrr♦♥ t♥ t s ♥♦r♦♦ tt ♥trsts ♦♥② ♦♥ s♠♣① t s ♥t ♥tr♦r ♦ tr♥ t♥ t s ♥♦r♦♦ tt ♥trsts tr s♠♣s ttr♥ ♥ t t♦ ♥♥t ttrr ss♠ ♥♦ tt p s ♥ t ♥tr♦r ♦ ♥

Pr♦ tr♥t♦♥s

t♥ t s ♥♦r♦♦ ♥trst♥ ♦♥② t tr♥s ♥ ttrr tt r ♥♥tt♦ t t♦ ♥♣♦♥ts ♦ t ♦r♥ t♦ t ♦ sss♦♥ ts r ♦♥② ♥t②♠♥②

♥ GS ♦♥t♥s ♣♦♥ts ♦♥ ♥ ♥♥t r ♥② ♣♦♥t p ∈ E3 s ♦♥t♥ ♥ s♦♠

s♠♣① ♥ ② ♣♦♥ts ♥ GS ♦tr t ♠♠ ts ♠♣s tt t st♦ s♠♣s t ♣♦♥ts ♦ GS s rts ♥ rs♣t♥ t ♥② ♣r♦♣rt② s ♥② tr♥t♦♥ ♦ E

3 ♥ ♥♦t t ② DT (GS) ❯s♥ π ♥ ♥♦ ♥t ♥② tr♥t♦♥ ♦ T

3c

♥t♦♥ ♥② tr♥t♦♥ ♦ T3c t DT (GS) ♥② tr♥t♦♥

♦ GS ♥ E3 π(DT (GS)) s s♠♣ ♦♠♣① ♥ T

3c t♥ t ♥②

tr♥t♦♥ ♦ T3c ♥ ② S ♥ ♥♦t t ② DTT(S)

r r ♣♦♥t sts ♦r π(DT (GS)) s ♥♦t s♠♣ ♦♠♣① s r ♦♥ ♣

rst ♦ ts st♦♥ s ♦t t♦ st②♥ ♥ S t② ♥s ♥②tr♥t♦♥ ♦ T

3c ❲ s♦ tt ♥t♦♥ t② ♠s s♥s ❲ r② tt

t s♠♣s ♠t ♥r π tt ♣r♦ ♦♣s ♦ s♠♣① ♥ DT (GS) r♠♣♣ ♦♥t♦ t s♠ s♠♣① ♥ T

3c ♥r π ❲ s♦ ♣r♦ tt π(DT (GS)) s

st ♦ s♠♣s t♥ t s ♦♥t♦♥s ♥ ♥② sss ♥r r♠st♥s ♦♥t♦♥ s ②s t ♥ssr② ♥ s♥t ♦♥t♦♥♦♥ π(DT (GS)) t♦ tr♥t♦♥ ② ♥t♦♥ t s s♠♣ ♦♠♣①♦s ♥♦♥ s ♦♠♦♠♦r♣ t♦ T

3c

t s strt t t rst ♠♠

♠♠ t rstrt♦♥ ♦ π t♦ ♥② s♠♣① ♥ DT (GS) s ♥t t♥ π(DT (GS))s st ♦ ♥tr♥② s♦♥t s♠♣s ♥ T

3c tt ♦ ♥♦t ♦♥t♥ ♥② ♣♦♥t ♦ π(S) ♥ tr

♥tr♦r

Pr♦♦ ♦♥sr ttrr♦♥ σ ♦ DT (GS) ♦s rts r ♦rt♣ ♦ ♣♦♥tsPσ ⊂ GS σ stss t ♥② ♣r♦♣rt② s♦ ♣r♦ ♦♣s ϕc(Pσ × Z

3) s♦ ♥ ♠♣t② r♠sr♥ s s♦s tt ts ♣r♦ ♦♣s ♦r♠ ttrr ♦DT (GS)

♦t tt ts s ♥ tr ♥ ♥rt ss ♥ ♥rs s ♥ ❬❪t♥ t ♥② tr♥t♦♥ ♦ st ♦ ♦s♣r ♣♦♥ts ♦♥② ♣♥s ♦♥ tr ①♦r♣ ♦rr s tr♥st♥ t st ♦ ♣♦♥ts ♦s ♥♦t ♥ tr ①♦r♣ ♦rr ♣r♦ ♦♣s ♦ tt ♣♦♥t st r tr♥t ♥ t s♠ ②

♦♦♥② π ♦♣ss ♣rs② t ♣r♦ ♦♣s ♦ σ ♦♥t♦ ts q♥ ss♥ T

3c s ♥② ♦r♠♥s♦♥ s♠♣① ♥ DT (GS) s ♥♥t t♦ s♦♠ ttrr♦♥ ♥

ts s ♥ ② sst ♦ ts rts t s♠ ♦s ♦r s♠♣s ♦ ♥② ♠♥s♦♥♦ t ♣r♦t♦♥s ♥r π ♦ t♦ ♥tr♥② s♦♥t k♠♥s♦♥ s♠♣s σ ♥

τ ♥ DT (GS) r tr q ♦r ♥tr♥② s♦♥t ♦r k ≥ 1 t♦ t tt② ♦ πt♥ ♦t s♠♣s ♥ tr rs♣t ♠s s♠ r♠♥t ♠♣s tt t♥tr♦r ♦ s♠♣① ♥♥♦t ♦♥t♥ ♥② rt①

❲ ♦sr tt π(DT (GS)) s ♥t DT (GS) s ♦② ♥t ♠♠ t str ♦ ♥② rt① s ♥t s S s srt s♦ GS s srt ♥ ttrr rt♥ ♦♠ rr t♥ s♦♠ ♦♥st♥t ♦♦♥② tr r ♦♥② ♥t② ♠♥②

♥② tr♥t♦♥ ♦ T3c

ttrr ♥ssr② t♦ t ♦r♥ ♦♠♥ Dc ♥ ts T3c ♥t② ♠♥② ttrr

♦♥② ♥t② ♠♥② s s♦ t ♦r ♥♠r ♦ s♠♣s ♥ π(DT (GS)) s ♥tt♦♦

♦ r ♥♦ tt s♠♣s ♥ DT (GS) r ♠♣♣ s s♠♣s ♦♥t♦ T3c t♥

t ♦ tr♥t♦♥ s ♠♣♣ ♦♥t♦ st ♦ s♠♣s ♥ T3c ❲ ♥♦ ♦♥sr t

♥♥ rt♦♥

srt♦♥ ss♠ tt t rstrt♦♥ ♦ π t♦ ♥② s♠♣① ♥ DT (GS) s ♥t τ s s♠♣① ♥ π(DT (GS)) ♥ τ ′ ≤ τ t♥ τ ′ s s♠♣① ♥ π(DT (GS)) s♦♦s ♠♠t② r♦♠ t t tt ♥♥ rt♦♥s r ♠♥t♥ ② π ♥ r♦♠♠♠

t ♦♥② r♠♥s t♦ s♦ ♦♥t♦♥ t ♥trst♦♥ ♦ t♦ s♠♣s σ ♥ τ ♥π(DT (GS)) s ♥♦tr s♠♣① χ tt s ♥♥t t♦ ♦t σ ♥ τ

♠♠ ss♠ tt t rstrt♦♥ ♦ π t♦ ♥② s♠♣① ♥ DT (GS) s ♥tt σ, τ ∈ π(DT (GS)) ♥② t♦ s♠♣s ♥ T

3c t♥ σ ∩ τ s st ♦ s♠♣s ♥

π(DT (GS))

Pr♦♦ ❲t♦t ♦ss ♦ ♥rt② ss♠ tt σ ∩ τ 6= ∅ ❲ s♦ tt σ ∩ τ =⋃p∈σ∩τ χp r χp s s♠♣① ♥ π(DT (GS)) ♥♦♥ s ♥t s tr r

♦♥② ♥t② ♠♥② s♠♣s ♥ π(DT (GS)) ♦♥sr ♣♦♥t p ∈ σ ∩ τ p s rt①♦ π(DT (GS)) t♥ t s ♥♦t ♦♥t♥ ♥ t ♥tr♦r ♦ ♥② ♦tr s♠♣① ♦r♥ t♦♠♠ ♥ st χp = p p s ♥♦t rt① ♥ π(DT (GS)) t♥ p ∈ σ′ ♥p ∈ τ ′ ♦r s♦♠ ♣r♦♣r s σ′ ≤ σ ♥ τ ′ ≤ τ s σ ♥ τ r ♥tr♥② s♦♥t♠♠ ♥ σ′ ♥ τ ′ r ♥ tr ♥tr♥② s♦♥t ♦r ♥t t ♦♦stt t② r t s♠ ♥ st χp := σ′ = τ ′ ② ♦♥t♦♥ t s♠♣① χp s♦♥t♥ ♥ π(DT (GS))

♠♠r tt⋃t(v) ♥♦ts t ♥♦♥ ♦ t s♠♣s ♥ t str ♦ v ❲ ♥

♥♦ ♦r♠t t ♦♦♥ s♥t ♦♥t♦♥ ♦r π(DT (GS)) t♦ s♠♣ ♦♠♣①

♠♠ ♦r rts v ♦ DT (GS) t rstrt♦♥ ♦ t ♣r♦t♦♥ ♠♣ π|S t(v)

s ♥t t♥ π(DT (GS)) ♦r♠s s♠♣ ♦♠♣①

Pr♦♦ ❲ st K = π(DT (GS)) t σ s♠♣① ♦ DT (GS) ♥ v ♥ ♥♥t rt①♥ σ ⊆ ⋃

t(v) ts t rstrt♦♥ ♦ π|S t(v) t♦ σ s ♥t s ♥ K s st♦ s♠♣s ♠♠

♦♥t♦♥s ♥ ♦♦ r♦♠ t ♦ sss♦♥ t r♠♥s t♦ s♦ ♦♥t♦♥ ♦♥sr t♦ s♠♣s σ, τ ∈ K t σ ∩ τ 6= ∅ ② ♥t♦♥ ♦ s♠♣① tr①st sts Pσ,Pτ ♥ Dc × Z

3 s tt σ = π((Pσ)) ♥ τ = π((Pτ )) r♦♠ ♠♠ ♥♦ tt σ ∩ τ s st ♦ s♠♣s ♥ K ♦ tr ①sts rt① v ∈ σ ∩ τ♥ σ, τ ∈ t(v) ② ss♠♣t♦♥ π|S t(v) s ♥t s♦ π s ♥t ♦♥ σ ♥ τ ♥σ ∩ τ = π((Pσ)) ∩ π((Pτ )) = π((Pσ ∩ Pτ )) s♦ t rstrt♦♥ ♦ π|S t(v) t♦(Pσ ∩Pτ ) s ♥t ♦ r♦♠ ♥t♦♥ t ♦♦s tt σ ∩ τ s s♠♣① ♥σ ∩ τ ⊆ σ, τ σ ∩ τ ≤ σ, τ

s t st ♥r♥t ♦r t t♦r♠ ♥ t ♦♦♥ ♠♠ tt s♦s ttt ♥② tr♥t♦♥ s ♥ ♥ ♥t♦♥ s t② tr♥t♦♥ ♦ T

3c

♥ t s♥s ♦ ♥t♦♥

Pr♦ tr♥t♦♥s

♠♠ |π(DT (GS))| s ♦♠♦♠♦r♣ t♦ T3c

Pr♦♦ ② ts ♦♥strt♦♥ |DT (GS)| = E3 ♥ π s srt ♦♦♥② π(|DT (GS)|)

s q t♦ T3c ♥ t ♥ ♦ qts

π(|DT (GS)|) = π

⋃

σ∈DT (GS)

σ

(1)

= π

⋃

τ∈π(DT (GS))

π−1(τ)

(2)=

⋃

τ∈π(DT (GS))

τ =⋃

σ∈DT (GS)

π(σ) = |π(DT (GS))|

♦s t t ♦♦♥ r♠♥ts

s st♣ st rr♦♣s t s♠♣s ♥ r♥t ♦rr t ♦s ♥♦t ♥ t st ♠♠

r s ♦♥② ♥t ♥♠r ♦ ♠♥ts ♥ π(DT (GS))

❲ r ♥♦ r② t♦ ♣r♦ t ♠♥ t♦r♠ ♦ ts st♦♥ s ♥ssr②♥ s♥t ♦♥t♦♥ ♦r π(DT (GS)) t♦ tr♥t♦♥ ♦ T

3c t s r tt t

st♦♥ ♦ s♠♣ ♦♠♣① s t s♦♠♣① tt ♦♥ssts ♦ s ♥ rts

♦r♠ ss♠ tt t rstrt♦♥ ♦ π t♦ ♥② s♠♣① ♥ DT (GS) s ♥tπ(DT (GS)) s tr♥t♦♥ ♦ T

3c ♥ ♦♥② ts st♦♥ ♦s ♥♦t ♦♥t♥ ♥② ②

♦ ♥t ss t♥ ♦r q t♦ t♦

Pr♦♦ ❲ rst s♦ t ♣rt ❲ st K = π(DT (GS)) r♦♠ ♠♠ ♥srt♦♥ ♥♦ tt K s ♥t st ♦ s♠♣s tt s ♦♥t♦♥s ♥ ss♠ tt K s ♥♦t s♠♣ ♦♠♣① r♦♠ ♠♠ tr s rt①v ∈ K ♦r π|S t(v) s ♥♦t ♥t s ♠♣s t ①st♥ ♦ t♦ r♥t ♣♦♥tsp, q ∈ ⋃

t(v) t π(p) = π(q) t σ ♥♦t t s♠♣① ♦ K tt ♦♥t♥s π(p) = π(q)♥ ts ♥tr♦r ♥ tr r t♦ r♥t s♠♣s σ′

E∈ π−1(σ) ♥ σ′′

E∈ π−1(σ)

♦♥t♥♥ p ♥ q rs♣t② s σ′E♥ σ′′

Er ♦t ♠♥ts ♦ t(v) t u, w

rts r♥t r♦♠ v t u ≤ σ′E♥ w ≤ σ′′

Es tt π(u) = π(w) rts

u ♥ w ②s ①st s π(σ′E) = π(σ′′

E) rts u, w r s♦ ♠♥ts ♦ t(v)

♥ ts tr r s (u, v) ♥ (v, w) ♥ DT (GS) r♦♠ π(u) = π(w) ♦♦s tt t♣r♦t♦♥ ♦ (u, v) ♥ (v, w) ♥r π ♦r♠s ② ♦ ♥t t♦ ♥ T

3c ♦♥trts

t ss♠♣t♦♥ tt π|S t(v) s ♥t ♦ K ♠st s♠♣ ♦♠♣① ♦♦♥♥t♦♥ ♥ ♥♦ ♥♦t DTT(S) = π(DT (GS)) ♠♠ s♦s tt DTT(S)s t② tr♥t♦♥ ♦ T

3c ♦♥s t rst ♣rt ♦ t ♣r♦♦

♦ ♦♥sr t ♦♥② ♣rt r ♥♥♦t ♥② ②s ♦ ♥t ♦♥ s♦ t ss♠♣t♦♥ tt t rstrt♦♥ ♦ π t♦ ♥② s♠♣① ♥ DT (GS) s ♥t ss♠π(DT (GS)) s s♠♣ ♦♠♣① ♦♥t♥♥ t♦ s σ ♥ τ σ 6= τ tt ♦r♠ ②♦ ♥t t♦ ♥ σ∩ τ ♦♥ssts ♦ t t♦ ♥♣♦♥ts ♦ t s♠♥ts σ ♥ τ ♦♥♦t ♦r♠ s♠♣① ♥ π(DT (GS)) ♦♥trts ♦♥t♦♥ ♦ ♥t♦♥

r ♦r ♥ strt♦♥ ♦ ♠♠ ♥ ♦r♠

♥② tr♥t♦♥ ♦ T3c

r strt♦♥ s r♦♥ s ϕc(t(p) × Z3) ∩ Dc ♠♠

r r sr ②s ♦ ♥t t♦ ♦r♥t♥ r♦♠ p ♦r♠

♥ t ♣r♦♦ ♦ ♦r♠ s♦♥ tt π|S t(v) s ♥♦t ♥t t♥tr r ②s ♦ ♥t t♦ ♥ π(DT (GS)) s q♥t t♦ π(DT (GS)) ♥♦t ♥ s♠♣ ♦♠♣① ♦♦♥② ts ♦♥t♦♥ s ♥♦t ♦♥② s♥t t s♦ ♥ssr②♥ ♠♠

P♦♥t sts tt ♦ ♥♦t ♥ ♥② tr♥t♦♥ ♦ T3c

♥ ts st♦♥ ①♣♥ ♦ ♥ ♥t r♣rs♥tt♦♥ ♦ t ♣r♦ tr♥t♦♥ DT (GS) tt s s♠♣ ♦♠♣① ♥ π(DT (GS)) s ♥♦t s♠♣ ♦♠♣① S ♦s ♥♦t ♥ ♥② tr♥t♦♥ ♦ T

3c ♦♠♣t ♥ ♥t② st ♦r♥

s♣ s t♦♥ ♦t tt E

3 ts t t ♣r♦t♦♥ ♠♣ π s ♦r♥ ♠♣ s ♥rs ♦r♥s♣ ♦ T

3c ♠♥s tt t s ♦r♥ s♣ ♦r ♦r♥ s♣s ♦ T

3c ❬r♠❪

♦r ♥♥♦t s t t♦ ♦♠♣t t ♥② tr♥t♦♥ s t s ♥ ♥♥t♥♠r ♦ sts ❲ ♥♦ ♦♥strt ♥t② st ♦r♥ s♣ tt s s♥t②r s♦ tt ♥② ♣♦♥t st P ♥s ♥② tr♥t♦♥ ♦ t

t h = (hx, hy, hz) ∈ N3 T

3h∗c s ♦r♥ s♣ ♦ T

3c t♦tr t t ♦r♥

♠♣ ρh := π π−1h

r πh : E3 → T

3h∗c ♥♦ts t ♣r♦t♦♥ ♠♣ ♦ T

3h∗c s ρ−1

h(p)

♦r ♥② p ∈ T3c ♦♥ssts ♦ hx ·hy ·hz r♥t ♣♦♥ts T

3h∗c s hx ·hy ·hzst ♦r♥

s♣ ♦r♥ ♦♠♥ s Dh∗c = [0, hxcx) × [0, hycy) × [0, hzcz) hx = hy = hz s t ♥♦tt♦♥ πh := πh t h := hx ·hy ·hz ♦r π27 ♥ ♦r♠ ♦ ♦ttt T

3h∗c s t t♦rs ♥

♦♥ ♥ s♦♥ ❬❪ s♦ tt ♦♥② t ♣♦♥ts ♦ GS ♦♥t♥ ♥ Dc ♥t ♦♣s tt srr♦♥ t ♥ ♥ ♥♥ ♦♥ t s♠♣s tt r ♦♠♣t②♦♥t♥ ♥ Dc ❲ s t s ♦ tr ♣r♦♦ t♦ ♣r♦ ♦r♠ s♦ rst st ♦ tr ♣♣r♦ ♣r♦♦ s s ♦♥ ♦♥sr♥ ♣rts ♦ ttr♥t♦♥ ♥ ♥♥ ② ♣♦♥t s s ♦♥ ♥ tr st♣s s♥ ♥♦ss♠s ❬❪ t F (1) ♥♦t t ❱♦r♦♥♦ ♦ t ♦r♥ ♥ t ❱♦r♦♥♦ r♠ ♦t ♦rt ♦ t ♦r♥ ♥r G ♥ F (1) s ♥♠♥t ♦♠♥ ♦ G t rtr♠♦rQ ♣♦♥t st ♥ F (1) A ⊕ B ♥♦ts t ♥♦s s♠ ♦ A ♥ B ♥ ♥F (i) := F (1) ⊕ F (i−1)

Pr♦ tr♥t♦♥s

♠♠ ❬❪ t σ ♥ τ s♠♣s ♥ t ♥♥t ♣r♦ ♥② tr♥t♦♥ DT (GQ) s tt ♦♥ ♦ t rts ♦ σ s ♥ t ♥tr ♦ F (1) ♥ τ∩F (1) 6= ∅♦s ♥

t ♥tr ♦ t r♠sr♥ ♦ σ s ♥ F (1)

σ s ♦♠♣t② ♦♥t♥ ♥s F (2)

τ s ♦♠♣t② ♦♥t♥ ♥s F (3)

Pr♦♦ ❲ r② st t ♣r♦♦s ♦ t tr ♣r♦♣rts

s ♦♦s rt② r♦♠ t t tt F (1) s ❱♦r♦♥♦ ♦ t ❱♦r♦♥♦ r♠♦ t ♦rt ♦ t ♦r♥ ♥r t t♦♥ ♦ G t ♥tr ♦ t r♠sr♥ ♦ σ s ♦ts ♦ F (1) t♥ t ♦ ♦sr t♦ s♦♠ ♦tr ♣♦♥t ♦ GQ♦♥trt♥ t t tt t s t ♥tr ♦ t r♠sr♥ ♦ σ

s ♦♦s rt② r♦♠ ♥ r♦♠ t t tt F (2) s t ♥♦s s♠ ♦F (1) t ts

s ♦♦s rt② r♦♠ t ♥t♦♥ ♦ F (3)

r ♦r ♥ strt♦♥ ♦ ♠♠

r strt♦♥ t s s♥t t♦ ♦♥sr F (3) ♥ ♦rr t♦ ♦♠♣t ♥②tr♥t♦♥ ♦ t t t♦rs

r♦♠ ♠♠ ♦♦s tt tr♥s ♦ t ♣r♦ ♥② tr♥t♦♥ ♦ Qr ♦♥t♥ ♥s F (3) ♥ s♦ t s s♥t t♦ ♦♠♣t t ♥t ♥② tr♥t♦♥DT (GQ ∩ F (3)) ♥ E

3 ♦r♥ t♦ ♠♠ s♠♣s ♦ DT (GQ ∩ F (3)) tt t st ♦♥ rt① ♥ F (1) r s♠♣s ♦ t ♥♥t ♣r♦ ♥② tr♥t♦♥DT (GQ) ♥ t st ♦ ts s♠♣s ♦rs F (1) ♣♣②♥ t t♦♥ ♦ G ♦♥ t ②s♥ ♥♥t ♣r♦ ♣rtt♦♥ ♦ E

3 ♦t tt ts rst rt② ①t♥s t♦ Ed

❯s♥ t ♣♣r♦ ♦ t ♣r♦♦ ♦ ♠♠ ♥ s♦ t ♦♦♥

♦r♠ π27(DT (GS)) s s♠♣ ♦♠♣①

Pr♦♦ ❲ s♦ tt tr r ♥♦ ②s ♦ ♥t ♥ π27(DT (GS)) t Dc(i, j, k) ♥♦tt tr♥st♦♥ ♦ Dc ② (i · cx, j · cy, k · cz)

Dc(i, j, k) := [i · cx, (i + 1) · cx) × [j · cy, (j + 1) · cy) × [k · cz, (k + 1) · cz).

♦rt♠

ss♠ tt tr s ② ♦ ♥t t♦ ♥ π27(DT (GS)) ♥ tr r rtsv, v′, v′′ ∈ DT (GS) s tt t s (v, v′) ♥ (v′, v′′) (v, v′) 6= (v′, v′′) r ♦♥t♥ ♥DT (GS) ♥ tt π27(v) = π27(v

′′) t Dc(i, j, k),Dc(i′, j′, k′) ♥ Dc(i

′′, j′′, k′′) ♥♦tt tr♥st♦♥s ♦ Dc tt ♦♥t♥ v, v′ ♥ v′′ rs♣t② ♦r♥ t♦ ♠♠ s♠♣① ♥trsts ♦t Dc(i, j, k) ♥ Dc(i

′, j′, k′) t♥ |i − i′| ≤ 1 |j − j′| ≤ 1 ♥|k−k′| ≤ 1 ❲t♦t ♦ss ♦ ♥rt② ♥ ♦♦s v ♥ v′ s tt (i, j, k), (i′, j′, k′) ∈0, 13 ♦t tt π27(Dc(i, j, k)) = π27(Dc(i mod 3, j mod 3, k mod 3)) s♦ t st ♦♥♦ t i′′, j′′ ♥ k′′ ♠st ♥ 0 − 3, 1 − 3, 0 + 3, 1 + 3 ♦r π27(v) = π27(v

′′) t♦ ♦s s ♥♦t ♣♦ss ♦r♥ t♦ ♠♠

❲t t s♠ r♠♥tt♦♥ s ♥ t ♣r♦♦ ♦ ♠♠ |π27(DT (GS))| s ♦♠♦♠♦r♣ t♦ t st ♦r♥ s♣ T

3h∗c t h = (3, 3, 3) s ♦♠♦♠♦r♣

t♦ T3c❲ ♣rr t♦ s t r♠♦r ♦ ♦r♥ s♣s rtr t♥ st t ♦t ♦♣s

♦ t ♣♦♥ts s ♥ ❬❪ s t ♦s rt♥ rt ♦♥rs ♥ t tstrtr t ♥② rt♦♥s r ♦♠♣t ♦r s♠♣s

s s♥ ♥ t ♥①t st♦♥ t ♥r♠♥t ♦rt♠ tt ♣rs♥t rqrs st② str♦♥r rst t♥ ♦r♠

♦rt♠

s ♠♥t♦♥ ♥ t ♥tr♦t♦♥ tr s str♦♥ ♠♦tt♦♥ ♦r rs♥ t st♥r♥r♠♥t ♦rt♠ ❬♦❪ t♦ ♦♠♣t ♣r♦ ♥② tr♥t♦♥ t s r♦ st ♦ t ♦rt♠ tt ♣r♦♣♦s ♦r ♣rs♥t t ♥ ♠♦r t tr

• ❲ strt ♦♠♣t♥ ♥ ♥t②st ♦r♥ s♣ T3h∗c ♦ T

3c t h ♦s♥

s tt πh(DT (GS)) s r♥t t♦ tr♥t♦♥♦r♠ s♦s tt s ♦r♥ s♣ ②s ①sts h = (3, 3, 3) s♦♥ ♣♦ss ♦ ♥ t t ♦rt♠ rqrs st② str♦♥r rst t♥♦r♠ ♥ ts ♥s t♦ s ♦tr ♦r♥ s♣s Dc s ♥♦t

• ♥ ♣rt t ♣♦♥t st s r ♥ rs♦♥② strt t s ② tttr ♥ ♥srt t ♣♦♥ts ♦ sst S ′ ⊂ S t ssq♥t π(DT (GS ′′))♦r S ′ ⊂ S ′′ ⊆ S r s♠♣ ♦♠♣①s ♥ T

3c

♥ ts s sr ♣r♦ ♦♣s ♦ s♠♣s ♦ πh(DT (GS ′)) ♥ stt♦ ♦♠♣t♥ π(DT (GS)) ♥ T

3c ② ♥ t ♣♦♥ts t ♥ S \ S ′

♥ ts ② ♥ ❬❪ ♦ ♣t♥ ♣♦♥ts s s♦♦♥ s ts s ♣♦ss ♥ss ♥ S s s♠ ♦r ② strt ♣♦♥t st t ♦rt♠ ♥r ♥trs t s♦♥♣s ♥ rtr♥s πh(DT (GS)) s tr♥t♦♥ ♦ T

3h∗c st ♦♠♦♠♦r♣ t♦

T3ct♦♥ s ♦r♥③ s ♦♦s ❲ sr t ♦rt♠ ♦r t rstrt s

♥ Dc s ♥ ♣r♦ ts ♦rrt♥ss s ♦rrs♣♦♥s t♦ t t rr♥t ♠♣♠♥tt♦♥ ♣r♦s ♥ t s♦♥ ♣rt r① t ♦♥t♦♥ ♦♥ Dc ♦♥t t♦ ♦ ♥ sr ♦ t ♦rt♠ ♥ ♣t t♦ ts s ♥②♥ t♦♥ s♦ ♦ t♦ ①t♥ t ♦rt♠ t♦ ♦♠♣t t ♥②tr♥t♦♥s s

Pr♦ tr♥t♦♥s

♦t tt ♦r st♥ t♦ ♦♠♣t♥ ♥ T3c t s ♥♦t s♥t t♦ tst tr

π(DT (S ′c)) s s♠♣ ♦♠♣① ♥ ♥ ♣♦♥t ♦ rt ② ♦ ♥tt♦ s r ♦ str♦♥r ♦♥t♦♥ s ♥ ♦r t st

r strt♦♥ ♥ ♣♦♥t ♥ s♠♣ ♦♠♣① ♥ rt ② ♦♥t t♦

♦♦♥ ♦srt♦♥ s ♥ t ssq♥t ♣r♦♦s

srt♦♥ t ∆ ♥♦t ttrr♦♥ ♥ DT (GS) ♥ B∆ ts r♠sr♥ t ♠tr ♦ B∆ s s♠r t♥ cmin := mincx, cy, cz t♥ π|B∆

s ♥t♥ π|∆ s ♥t s s♥ ∆ ⊂ B∆ s ∆ s s♠♣① ♥ T

3c

♦♠♥

t ♦r♥ ♦♠♥ Dc s t ♥t c t ♥r♠♥t ♦rt♠ sst st ♦r♥ s♣ T

33c r 3c = (3c, 3c, 3c) ♦r♥ ♦♠♥ ♦r ts

♦r♥ s♣ s D3c❲ ♣r♦ ♦ tt ♦♥ s ♥ t tr♥t♦♥ ♦ T

33c

r s♦rtr t♥ 1√6c

♦♠♣t♥ s♠♣ ♦♠♣① ♥ T3c s ♣♦ss

♦rt♠ ♦r ♣s♦♦ st♥ ♦ t ♦rt♠ t ♦♠♣ts tr ♥② tr♥t♦♥ ♦ T

3c s ♥ ♥ ♥t♦♥ ♦r ♥② tr♥t♦♥

♦ T33c s st ♦♠♦♠♦r♣ t♦ T

3c

♦ s♦ t ♦rrt♥ss ♦ t ♦rt♠ t r♠♥s t♦ sts t ♦♦♥ t♦♣r♦♣rts

tr ♥srt♦♥ 27 s ♥② tr♥t♦♥ ♦ T33c t s ♠♣s③ ♦♥

t t tt ♦r♠ ♥♥♦t s r s ♥ t ♥♥r ♦♦♣ st♣ t st ♦ ♣♦♥ts ♣rs♥t ♥ 27 ♦s ♥♦t ♦♥t♥ t ♣r♦ ♦♣s ♦ pt p ♣♦♥t ♥ Dc ♥ Tp ⊆ ϕc(p×Z

3)∩D3c Tp s sst ♦ t r ♦ ♦♣s ♦ p tt t♥ D3c ♥ 27 s ②s ♦ t ♦r♠ π27(DT (GS ∪T 3c

p ))t T 3c

p = ϕ3c(Tp × Z3) ♠♠ s♦s tt ts s tr♥t♦♥

s ♥ π27(DT (GS)) r s♦rtr t♥ 1√6c t♥ ♥ st t♦ ♦♠♣t♥

♥ T3c rtr♦♥ ♦r ♣r♦♦

♦rt♠

♦rt♠ ♦♠♣t ♥② tr♥t♦♥ ♦ T3c r♦♠ ♣♦♥t st

♥♣t t S ♦ ♣♦♥ts ♥ Dc ♦ ♥t c ∈ E3 \ 0

t♣t DTT(S) ♣♦ss ♦trs π27(DT (GS)) S ′ ⇐ S P♦♣ p r♦♠ S ′

S ⇐ p 27 ⇐ π27(DT (ϕc(p × Z

3))) ♥ ♣r♦♠♣t

t ♦♥st ♥ 27 s ♦♥r t♥ 1√6c ♦

P♦♣ p r♦♠ S ′ S ⇐ S ∪ p ♦r p′ ∈ p + c ∗ ζ | ζ ∈ 0, 1, 23 ♦ ♥srt p′ ♥t♦ 27

♥ ♦r 27 = π27(DT (GS)) S ′ = ∅ t♥ rtr♥ 27 = π27(DT (GS)) ♥♦♥tr♥

♣♦♥t st

♥

♦♠♣t DTT(S) r♦♠ 27 st t♦ T3c

♥srt ♣♦♥ts r♠♥♥ ♥ S ′ ♥t♦ DTT(S) ♦♥ ② ♦♥ rtr♥ DTT(S)

♠♠ t S ⊂ Dc ♥t ♣♦♥t st p ∈ Dc ♣♦♥t ♥ Tp sst ♦ϕc(p × Z

3) ∩ D3c Dc s t♥ π27(DT (GS ∪ T 3cp )) s tr♥t♦♥ ♦ T

33c

t T 3cp := ϕ3c(Tp × Z

3)

Pr♦♦ ❲ rst ♦♥sr tr♥t♦♥ ♥ ② ♦♥② ♦♥ ♣♦♥t S = q ♦r s♦♠q ∈ Dc ♥ π27(GS) ♦♥ssts ♦ 27 ♣♦♥ts rr♥ s rr r ❲t♦t ♦ss ♦♥rt② ♥ ss♠ q = (0, 0, 0) s tr♥t♦♥s ♦ ♣♦♥t sts r ♥r♥tt rs♣t t♦ tr♥st♦♥s ♣♦♥t st π27(GS) s ② ♥rt r r r♥t ♠♣t② s tt t ♣♦♥ts ♦♥ tr ♦♥r② ② r ♥tr t((

12 , 1

2 , 12

)+ ζ

)∗ c t ζ ∈ 0, 1, 23 ♥ rs r =

√3

2 c ≈ 0.866 c ❲t♦t ♦ss ♦♥rt② ♥♦ ♦♥sr t B ♥tr t

(12 , 1

2 , 12

) t s s② t♦ tt t

♥trst♦♥ ♦ t ♠ ♦ ♥② ♣r ♦ ts s ♥r π27 ♦♥ssts ♦ ♦♥ ♦♥♥t st tt π27 rstrt t♦ t ♥♦♥ ♦ ♥② ♣r ♦ ts s s ♥t

s t ♥t ♦ ♥ ♦ t D3c s 3c ♥ t ♠tr ♦ B s√

3 c ♥ ♦♥t (3 −

√3) c ♦ ♥♦ t♦ ♥tr♦ ② ♦ ♥t t♦ t B′ rst

♠♣t② ♥ ② t ♣♦♥ts (2, 0, 0) (2, 0, 1) (2, 1, 0) (2, 1, 1) ♥(

1+√

32 , 1

2 , 12

) t

tr♥s ♦t tt t rs ♦ B′ s s♠r t♥ 0.712 c r ①st ♦tr s s ♦♥t y ♥ z①s t s♥ Dc s tr r r t s♠ t ♦ t ♦r♣t♥ B ♥ B′ s ss t♥ (2 ·0.712+

√3−3)c < 0.155 c ♥ π27|B∪B′ s ♥♦t ♥t

♥ π27(B ∩ B′) ♦♥ssts ♦ t♦ s♦♥t ♦♥♥t ♦♠♣♦♥♥ts♦ t♦ ①♣♦t s♣ ♣r♦♣rt② ♦ ♦rt♠ π27(GS) ♦♥ssts ♦

rr ♦rt♦♦♥ r ♦ ♣♦♥ts ♥ π27(T 3cp ) s sst ♦ t r ♦ ♦♣s ♦ p

r♦♠ ts ♥♦ tt tr ♥ ♣♦♥t p1 ∈ T 3cp t ♥①t ♣♦♥t r ② s♦♠

t♦r ♥ c∗Z3 ♦r t♦ t♦ ♦r♠ ② ♦ ♥t t♦ r♦ss♥ t s B ♥ B′

ts ♥t ♦♥ ♦♥ ①s ♦ t♦ t♥ 1.266c ♥ 1.424c s s ♥♦t ♣♦sss [1.266, 1.424] ∩ Z = ∅ s r ♦t tt ♠♦ p1 ♥s B t♥ trs ♦ B′ ♦♠s s♠r ♥ ts t ♦r♣ r♥ s ♦♥② sst ♦ [1.266, 1.424]

Pr♦ tr♥t♦♥s

r ♦ ♣r♦ ♦♣② ♦ p1 ♥ ♥s B ∩ B′

r s ♥♦ rtr ♣♦sst② ♦r ② ♦ ♥t t♦ t♦ ♦r s t s♦rtst♦♥s tr♦ D3c ♥t

√18 c s rr t♥ 2

√3c

❯s♥ ♦r♠ ts ♣r♦s tt π27(DT (GS ∪ T 3cp )) s tr♥t♦♥ ♥ S

♦♥ssts ♦ ♦♥② ♦♥ ♣♦♥t S ♦♥t♥s ♠♦r ♣♦♥ts t♥ t ♠♣t② s ♥ ♦♥② s♠r ♥ ♠♦r ♦s ②s ♦ ♥t t♦

♦ ♦♠tr rtr♦♥ t♦ tr π(DT (GS)) s s♠♣ ♦♠♣①♥ ts tr♥t♦♥ ♦ T

3c

rtr♦♥ t ♠tr ♦ t r♠sr♥ ♦ ♥② ttrr♦♥ ♥ DT (GS)s s♠r t♥ 1

2 c t♥ π(DT (GT )) s s♠♣ ♦♠♣① ♦r ♥② ♥t T ⊂ Dc tS ⊆ T

Pr♦♦ s ♦ ttrr♦♥ r ♦♠♣t② ♦♥t♥ ♥ ts r♠sr♥ ♥r ts ♦♥ ② t s ♠tr t ♠tr ♦ ♥② r♠sr♥ ss♠r t♥ 1

2 c t♥ s ♥ t tr♥t♦♥ r s♦rtr t♥ 12 c ♥ ♦rr t♦ rt

② ♦ ♥t t♦ t s♠ ♦ t ♥ts ♦ t t♦ s ♥s t♦ t st c s♥♦t ♣♦ss ♦t s r s♦rtr t♥ 1

2 c r♦♠ srt♦♥ ♥ ♦r♠ t ♦♦s tt π(DT (GS)) s s♠♣ ♦♠♣① ♠♦r ♣♦♥ts t ♠tr ♦t rst ♠♣t② ♥♥♦t ♦♠ rr ♠ ♦♦s

❲ ♥♦ ♣r♦ t ♦♠tr rtr♦♥ tt s s ♥ ♣rt

rtr♦♥ t st♦♥ ♦ DT (GS) ♦♥t♥s ♦♥② s s♦rtr t♥ 1√6c r

c s t ♥t ♦ Dc t♥ π(DT (T c)) s s♠♣ ♦♠♣① ♦r ♥② ♥t T ⊂ Dc

t S ⊆ T s♦ t s tr♥t♦♥ ♦ T3c

Pr♦♦ ss♠ tt tr s B ♦ ♠tr d tt ♦s ♥♦t ♦♥t♥ ♥② ♣♦♥t ♦ GS♥ ts ♥tr♦r ♦♥sr t ttrr♦♥ ∆ ♥ DT (GS) tt ♦♥t♥s t ♥tr ♦ B ♥t ♦ t rst ♦ ∆ s ♦♥ r♦♠ ♦ ② t ♥t ♦ t rrttrr♦♥ t r♠sr♥ B s 2d√

6 ♦ s ♥ DT (GS) r s♦rtr

t♥ 1√6c t♥ t ♠tr ♦ ♥② ♠♣t② s s♠r t♥ 1

2 c ♠ ♦♦s r♦♠rtr♦♥ ♥ ♠♠

♦t tt rtr ♥ r ♦♥② s♥t r rtr♦♥ s rt♥ rtr♦♥ r r ♣♦♥t sts t ♠①♠♠ ♠♣t② ♠tr s♦rtr t♥12 c t s ♦♥r t♥ 1√

6c ♠♦r t sss♦♥ ♦♥ t t♦ rtr ♥ ♦♥

♥ t♦♥ ♠♠ ♥ rtr♦♥ ♣r♦

♦rt♠

♦r♠ ♦rt♠ s ♦rrt t ②s ♦♠♣ts ♥② tr♥t♦♥ ♦♠♦♠♦r♣ t♦ T

3c

r rs♦♥ ♦ ♠♠ s ♥ t ♥①t st♦♥ ♦♦s♠♠t② r♦♠ rtr♦♥

♦r♦r② t S ⊂ Dc S 6= ∅ ♥ T ⊂ D4c ♥t ♣♦♥t sts Dc s t♥ π64(DT (GS ∪ T 4c)) s tr♥t♦♥ ♦ T

34c

Pr♦♦ rst ♠♣t② ♥ GS s ♠tr√

3 c ♦♠♥ D4c s ♦ ♥t 4c s ♠♦r t♥ t

√3 c s rtr♦♥ ♣♣s

♦t tt ♥ ♠♠ ♦r♦r② ♦s ♥♦t rqr T t♦ sst ♦ ♣♦♥t r ϕc(p ×Z

3) ♦r s♦♠ ♣♦♥t p ∈ Dc ♥ t♦ ts rst s r ♥ tt t②s rr ♥♠r ♦ ♣r♦ ♦♣s t♦ ♦♥sr t s ♠ sr t♦ ♣r♦ ♥ ts t♥r③s s② t♦ ♦tr stt♥s s s♥ ♦ ♦r sr ♥ ♣♣r♦tt ♦s ♦♠♣t♥ ♥ T

33c

♠♠② ♣♦♥ts ♥ ♦rr t♦ ♦ ♦♠♣t♥ ♥ T33c

t t ♥♥♥ t ♦rt♠strts t ♥ ♥t tr♥t♦♥ ♦ ♠♠② ♣♦♥t st P ♦s♥ s tt ♥② s♣rst♦ P s tr♥t♦♥ ♥ T

3c ♣♦ss ♠♠② ♣♦♥t st ♦ ♣♦♥ts s ♥ s

♦♦s

P :=c · (x, y, z) | x, y ∈

0, 1

3 , 23

, z ∈

0, 1

2

∪c ·

(16 + x, 1

6 + y, 14 + z

)| x, y ∈

0, 1

3 , 23

, z ∈

0, 1

2

r ♦r ♥ strt♦♥ ♦ t ♣r♦ ♥② tr♥t♦♥ ♦ t ♠♠②♣♦♥t st

r tr♥t♦♥ ♦ t ♠♠② ♣♦♥t st

♠tr ♦ t rst ♠♣t② ♥ t st P s s♠r t♥ 12c s rqr ②

rtr♦♥ ♣♦♥t st P s s② t♦ sr ♥ s♥t② s♠ ♦r ♣rt st ♦ ♥trst♥ t♦ ♥stt rtr t♦ ♥ t s♠st ♣♦ss ♣♦♥t st ♥ t♦♠♥s♦♥s t s ♥♦♥ tt t s♠st tr♥t♦♥ ♦ t♦rs s rts ❬ö❪♦r t tr♠♥s♦♥ t♦rs tr♥t♦♥ ♦ rts s ♥♦♥ ❬❪ t t s ♥♦t♥♦♥ tr ts s t s♠st ♣♦ss ♥♠r ♦ rts ♦r ♦t ss t s♠st♣♦♥t sts ♥♥ ♥② tr♥t♦♥ t t ♠tr ♦ t rst r♠sr♥ ♦♥ ② 1

2c r ♥♥♦♥

Pr♦ tr♥t♦♥s

♥ t ♥♣t ♣♦♥ts ♥ S r ♥srt ♥t♦ t ♥t tr♥t♦♥ ♥ ② Pt ♣♦♥ts ♦ P r r♠♦ r♦♠ t tr♥t♦♥ ② ♦♥strt♦♥ t ♣♦♥t st S ∪ P♥s ♥② tr♥t♦♥ ♦ T

3c t ♥♣t ♣♦♥t st t♦t P ♦s ♥♦t ♥

tr♥t♦♥ ♦ T3c ♥②♠♦r t♥ t tr♥t♦♥ ♦♥rt t♦ T

33c

r♥ tr♠♦ ♦ ♦♥ ♦ t ♠♠② ♣♦♥ts

♦♥② ♦r ♦ ts ♣♣r♦ s t r♠♦ ♦ t ♣♦♥ts ♥ P r s t♦♥② t ♣♦t♥t ♦♥rs♦♥ t♦ T

33c

t t♦t s♥ t ♠♠② ♣♦♥t st t ♦tr♥t♦♥ ♦ ♦♠♣t ♥ T

33c

♥ ts s s ② t♦ ♥ ♠♦r ①♣♥s t ♥♣t ♣♦♥t st s r ♥♦ t♥ t ♦r ♦ r♠♦♥ ♣♦♥tss ♥ s t ♦♠♣t② ♦s ♦♠♣t♥ ♥ T

33c t s str t♥ t st♥r

♣♣r♦

♦♥ ♦♠♥

♦ sss♦♥ st r♠♥s t ♦r♥ ♦♠♥ Dc s ♥r ♦ c = (cx, cy, cz) ♥② t ♦♥st♥ts t ♥♠r ♦ sts ♦ t ♦r♥ s♣ t♦ strtt ♥ t ♥t trs♦ ♥ t♦ ♣t

t Th∗c t h = (hx, hy, hz) ♦r♥ s♣ ♦ T3c t ♣r♦t♦♥ ♠♣ πh∗c ♥

♦r♥ ♦♠♥ Dh∗c❲ rst rtr♦♥ t♦ ♥ t♦ st t♦ T

3c ♥ ♦rt♠ ♥ t

s ♦ ♥♦♥ ♦♠♥

rtr♦♥ t ♠tr ♦ t r♠sr♥ ♦ ♥② ttrr♦♥ ♥ DT (GS)s s♠r t♥ 1

2 cmin r cmin = mincx, cy, cz t♥ π(DT (GT )) s s♠♣ ♦♠♣① ♦r ♥② ♥t T ⊂ Dc t S ⊆ T

Pr♦♦ ♣r♦♦ ♦ ts rtr♦♥ s ss♥t② t s♠ s ♦r rtr♦♥ t♠tr ♦ ♥② r♠sr♥ s s♠r t♥ 1

2 cmin t♥ s ♥ DT (GS) rs♦rtr t♥ 1

2 cmin s♦rtst ♣♦ss ♥♦♥tr ② ♥ T3c s ♥t cmin ts

t♦ s t st♦♥ ♦ DT (GS) ♥♥♦t ♦r♠ s ② ♠ ♦♦s r♦♠♦r♠

❯s♥ rtr♦♥ ♥ t ♦r♥ s♣ rqr ♥ ♦rt♠

♠♠ t S ⊂ Dc S 6= ∅ ♥ T ⊂ Dh∗c ♥t ♣♦♥t sts t hi :=⌈2‖c‖

ci

⌉

♦r i = x, y, z ♥ πh∗c(DT (GS ∪ T h∗cp )) t T h∗c := ϕh∗c(T × Z

3) s tr♥t♦♥♦ T

3h∗c

Pr♦♦ rst ♠♣t② ♥ GS s ♠tr s♠r t♥ ‖c‖ ♦♠♥ Dh∗cs ♦ ♦ ♥ts hici ♦r i = x, y, z ♦♥t♦♥ hi =

⌈2‖c‖

ci

⌉♠♣s tt

hici ≥ 2‖c‖ ♣♣②♥ rtr♦♥ t♦ t t t♦rs t ♦r♥ ♦♠♥ Dh∗c ♣r♦st ♠

❲t ♥② tr♥t♦♥

t ♥② tr♥t♦♥ ♦r rr tr♥t♦♥ ♥r③s t ♥②tr♥t♦♥ ♥ t sts r s♣rs s♦ t ♣♦♥ts ❲ s♦rt♥tr♦t♦♥ ♦♥ t ♥② tr♥t♦♥s ♥ t♦♥

♦rt♠

t ♥② tr♥t♦♥ t t ♥② tr♥t♦♥ ♥ ♥♥q② ♥ ♥ ♥rt ss ❬❪ r s♦ tt t ♦ sss♦♥ ♥t♦♥ ♦rs ♥ t s♠ ② ♦r t ♥② tr♥t♦♥s ❲ s♦ ♣r♦ ♦♠tr rtr♦♥ s♥ t ♥t s♠r t♦ rtr♦♥ ♦s s t♦♥r③ ♦rt♠ t♦ ♦♠♣t ♣r♦ t ♥② tr♥t♦♥ s

t S st ♦ t ♣♦♥ts ♥ E3 ♣rs (p, wp) ∈ E

3 × R t Dc ♦ ♥t c ♥ S st ♦ t ♣♦♥ts ♥ Dc t W ♥ w rs♣t②♥♦t t rst ♥ s♠st t ♥ S t ❲(GS) ♥♦t t t ♥②tr♥t♦♥ ♦ E

3 ♥ ② GS

rtr♦♥ t ♦♥st ♦ ❲(GS) s s♦rtr t♥√

16 c2 − 8

3(W − w)

t♥ π(❲(T c)) s tr♥t♦♥ ♦ T3c ♦r ♥② ♥t T ⊂ Dc t S ⊆ T

♦t tt W = w ts ♦♥ rs t♦ 1√6c s t ♥t trs♦ ♦r

♥② tr♥t♦♥s ♣r♦♦ ♦ ts rst ss t ♦♦♥ ①r② ♠♠

♠♠ ♦♥sr ❲(GS) t ♦rt♦♦♥ t ♣♦♥t ♦ ttrr♦♥s t wo t♥ t ♥t ♦ t ♦♥st ♦ t ttrr♦♥ s ♥♦t rr t♥2√

wo + W − w

Pr♦♦ t (o, wo) ♥ ♦rt♦♦♥ t ♣♦♥t ♥ (p, wp), (q, wq) t♦ ♦ tt ♣♦♥ts ♦rt♦♦♥ t♦ (o, wo) ♥ st(o, p) =

√wp + wo ♥t

st(p, q) tt♥s ts ♠①♠♠ t s♠♥t [p, q] ♦♥t♥s o ♥ st(p, q) =√

wo + wp+√wo + wq s ①♣rss♦♥ s ♠①♠③ t t s ♠①♠③ ❲ ♥♦ tt ♥

♥ t ts ♦ ♣♦♥ts ♥ t tr♥t♦♥ ② t s♠ ♠♦♥t ♦s ♥♦t ♥ ttr♥t♦♥ ♦ ♥ ss♠ W − w t♦ t ♠①♠♠ ♣♦ss t ♠♦♦s r ♦r ♥ strt♦♥ ♥

o√

wp

q

p

√wo

√wo

√wq

r strt♦♥ ♦♥st ♣♦ss ♦ ttrr♦♥ t ♦rt♦♦♥s♣r ♦ rs

√wo

Pr♦♦ ♦ rtr♦♥ t λ t ♥t ♦ t ♦♥st ♥ ❲(GS) ♥

t r♠rs ♦ ♥② ttrr♦♥ ♥ ❲(GS) ♥♥♦t rr t♥√

38λ r♦♠

♠♠ ♦♦s tt t ♦♥st ♣♦ss ♦ ttrr♦♥ s ♦♥ ② λmax :=

2

√(√38λ

)2

+ W − w ♥♦ ♦♦s λ t♦ s♠r t♥√

16 c2 − 8

3(W − w) t♥

λmax s s♠r t♥ 12 c s ♥♦ ② ♦ ♥t t♦ ♥ ♦r ♥ π(❲(GS)) s

tr♥t♦♥ ♦r♥ t♦ ♦r♠ t♦tr t t q♥t ♦ srt♦♥

Pr♦ tr♥t♦♥s

♦r t ♥② tr♥t♦♥s ♥ rtr ♣♦♥ts ♥♥♦t ♥rs t s③ ♦t rst r♠sr♥ ♥ s♦ π(❲(T c)) s tr♥t♦♥ s

♦r♦r② t η ♥ s η :=⌈8·(3 + 4

c (W − w))3/2⌉♥ h := η3 t Tp

sst ♦ t h ♦♣s ♦ p ♥ Dηc ♥ T ηcp := ϕηc(Tp × Z

3) ♥ πh(❲(GS ∪ T ηcp ))

s tr♥t♦♥ ♦ T 3ηc

Pr♦♦ S ♦♥ssts ♦ ♦♥② ♦♥ ♣♦♥t t♥ t t ♥② tr♥t♦♥ ♥② GS t② s ♥② tr♥t♦♥ s ♣r♦ ♦♣s ♦ t ♣♦♥t t s♠ t ♦ t rst ♦rt♦♦♥ s♣r ♦♥s t t rst r♠s♣r♥ s rs

√3

2 c ❲t t s♠ r♠♥t s ♥ t ♣r♦♦ ♦ rtr♦♥ t♠tr ♦ t rst ♦rt♦♦♥ s♣r tr ♥ ♥② ♦tr ♣♦♥t s ♦♥ ②

2·√

(√

32 c)2 + W − w ♦ ♠ sr tt ts ①♣rss♦♥ s s♠r t♥ 2·η ·c ♦♦s

η = ⌈8·(3 + 4c (W − w))3/2⌉

♥②ss

♦♠♣①t② ♥②ss

♥ ts t♦♥ s♦ tt s♥ t ♥② rr② t r♥♦♠③ ♦rsts♦♠♣①t② ♦ ♦rt♠ s s②♠♣t♦t② q t♦ t ♦♥ ♦ t ♦rt♠ ♦r♦♠♣t♥ t ♥② tr♥t♦♥ ♦ E

3t s rst sss t ♦♦♥ t♦ ♣♦♥ts ♦r ♦♥sr t ♥② rr②

♥ ♠♦r t ♦ t♦ tst ♦r t ♥t ♦ t ♦♥st ♥ ♦ t♦ ♦♥rtr♦♠ t tr♥t♦♥ ♦ T

3h∗c t♦ t tr♥t♦♥ ♦ T

3c

❲ ♠♥t♥ ♥ ♥s♦rt t strtr E tt rr♥s s tt r ♦♥rt♥ t trs♦ 1√

6cmin s s♦♦♥ s E s ♠♣t② ♥♦ tt t ♦♥st s s♠r

t♥ t trs♦ t♦t ♥♠r ♦ s tt r ♥srt t♦ ♥ r♠♦ r♦♠ Es t ♠♦st ♣r♦♣♦rt♦♥ t♦ t t♦t ♥♠r ♦ s♠♣s tt r rt ♥ str♦②r♥ t ♦rt♠ ❲ ♥ rt ss r♦♠ t s♠♣s t♦ tr s ♥ E ♥ t ♠♥t♥♥ ♦ E ♦s ♥♦t ♥ t ♦rt♠ ♦♠♣①t②

♦ ♦♥rt t tr♥t♦♥ ♦ T3h∗c t♦ DTT(S) ♥ ♦♥rt t♦ T

3c ♥

t♦ trt ♦r s ♥ rts t♦ t ♣r♦ ♦♣s ♣♥ ♦♥② ♦♥rtr♠♦r ♥ t♦ ♣t t ♥♥ rt♦♥s ♦ t♦s ttrr ♦s ♥♦rs ♥ t s s ♥r ♥ t s③ ♦ t tr♥t♦♥ ♥ ts ♦♠♥t ②t ♠♥ ♦♦♣

♥② rr② ♦r ♦rt♠ s ♥r♠♥t ♥ ♥ ♦♠♥t t ♥② rr② ❬❪ s s♦ t♦♥ ♥ ♥ts t strtrs s♥ ♦r ♥t ♦♠♣tt♦♥ ♦ ♥② tr♥t♦♥ ♦ E

d t s t ♥r♠♥t② ♥ s sr s t ♥tr♠t s st♦r t ♥② tr♥t♦♥s♦ ♥ ♥rs♥ sq♥ ♦ ssts ♦ t st ♦ ♥♣t ♣♦♥ts t st st♦rst ♦♠♣t tr♥t♦♥ r r ♣♦♥trs t♥ s♦♠ rts ♥ r♥t s♦rrs♣♦♥♥ t♦ t s♠ ♥♣t ♣♦♥t strtr ♦s ♦r st ♣♦♥t ♦t♦♥ ♥t ♦♠♣t tr♥t♦♥ ❲ rr t rr t♦ t ♦r♥ ♣♣r ♦r ♠♦r ♦♠♣tsr♣t♦♥ ♦ ts t strtr

♥②ss

s strtr ♥ ♣t t♦ ♦r ♦rt♠ t T3h∗c t ♦r♥ s♣ ♦s♥

s ♣rs♥t t t ♥♥♥ ♦ t♦♥ ♥ ♦ t rr② st♦rs tr♥t♦♥ ♦ T

3h∗c ♥ t s rt ♥ ts tr♥t♦♥ s ♦♥rt ♥ ♣♦ss

t♦ tr♥t♦♥ ♦ T3c ♦t tt ♥ l st♦rs tr♥t♦♥ ♦ T

3c t♥

t ♥①t l + 1 s s♦ ♥ T3c s♥ t ♦♥t♥s ♠♦r ♣♦♥ts ♥ ts s♦ st♦rs

tr♥t♦♥ ♦ T3c ② rtr♦♥ ♦r s♦♠ l ♥ st♦r tr♥t♦♥

♦ T3h∗c t ♥①t l + 1 s ♦♥rt ♥t♦ T

3c ♥ ts s ♦r t rts

♦rrs♣♦♥♥ t♦ ♣r♦ ♦♣s ♦ ♥ ♥♣t ♣♦♥t ♥ tt l tr ♣♦♥tr t♦ t l + 1 st t♦ t s♠ rt① ♦rrs♣♦♥♥ t♦ ts ♥♦♥♣t ♣♦♥t ♥T

3c

r♥♦♠③ ♥②ss ♦ ❬❪ ss♠s t ♥srt♦♥ ♦ ♣♦♥ts ♦ S t♦ ♣r♦r♠♥ r♥♦♠ ♦rr ♥s t♦ ts ♥②ss ♥ ♦♠♣t♥ ♥ T

3h∗c r ♠♥♦r

♣♦♥ts r ♥srt ♥ sts ♦ ♦♥st♥t s③ t ♥♠r ♦ ♣r♦ ♦♣s ♥ ts stsr ♥srt ♥ r♥♦♠ ♦rr rst r♠♥s t s♠

♦r♠ ♦rt♠ s ♦♣t♠ r♥♦♠③ ♦rsts t♠ ♥ s♣ ♦♠♣①t② O(n2)

Pr♦♦ t S t ♥♣t ♣♦♥t st rt① st ♦ t ♥② tr♥t♦♥ ♥ i s ♥♦t ② Si ♥ t s ♦ t rr② r ♥♠r r♦♠ ♦tt♦♠ t♦ t♦♣ tts S0 = S t 1/α ♥♦t t ♣r♦t② tt ♣♦♥t s ♥ Si+1 ♥ tt t s ♥ Si♥ ♦rt♠ t ♣♦♥ts ♦ S ♥ ♥srt ♥ r♥♦♠ ♦rr t ♥ ♦♠♣t♥♥ T

33c ♦♣s ♦ ♣♦♥t r ♥srt ♦♥st② t S ′

i t st ♦♥t♥♥ ♦♣s ♦ ♣♦♥t ♦ Si

r♥♦♠③ ♦rsts ♥②ss ♥ ❬❪ s♦s tt t ①♣t ♦st ♦ t ♥ i s ♥r ♥ t rt① r ♥ α ♥ O(αn) t ♥♣t ♣♦♥ts r♥srt ♥ r♥♦♠ ♦rr ♣r♦♦ s s ♦♥ t t tt t ♥♠r ♦ ♣♦♥ts ♥Si tt r ♦sr t♦ qr② ♣♦♥t q t♥ t♦ ♥② ♦tr ♣♦♥t ♥ Si+1 s ♥ O(α) s♣r♦♣rt② ①t♥s t♦ t s ♦ ♦♠♣t♥ ♥ st ♦r♥ s♣ rt① st ♦t ♥② tr♥t♦♥ ♥ i s S ′

i s ♦r ♣♦♥t ♦ Si tr r ♦♣s ♥S ′

i t ♦ ♦♥ ♥ t ♠♦st t♠s r s st ♥ O(α) s t ♦st♦ t ♥ i s st ♥ O(αn) ♥ s♠♠♥ ♣ ♦r s ②s ♥ ①♣tO(n) ♦♠♣①t② ♦r ♦♥ ♣♦♥t ♥srt♦♥

♥ t ♥♥♥ ♦ ts st♦♥ s♦ tt t ♠♥t♥♥ ♦ t strtr E♦♥② rqrs t♦t ♥♠r ♦ O(n2) s t♦ ♥srt ♥ r♠♦ rtr♠♦r s♦ tt t ♦♥rs♦♥ r♦♠ T

33c

t♦ T3c s O(n2) t♦♦ ♥ t s ♣♣ ♦♥② ♦♥

r♥ t ♦rt♠ r♥

s♦ t ♠①♠♠ s③ ♦ t strtr E s ♥ O(n2) ❲ ♥srt t ♠♦st 27n ♣♦♥ts♥t♦ t ♥② tr♥t♦♥ s♦ t s②♠♣t♦t s③ ♦ t rr② ♦s ♥♦t ♥t rs♣t t♦ ❬❪

n ♥ t ♦♥ ♦r t ♣♦♥t ♥srt♦♥ ♦♠s r♦♠ t ♦rsts rt① r ♥t tr♥t♦♥ s ♦r tr♥t♦♥s t ♠①♠♠ rt① r s♠r t♥ O(n)t rsts ♥ st ♠♣r♦ s ♦♠♣①t② rst s♦ ♦s ♦r t ①t♥s♦♥s t♦♥♦♥ ♦♠♥s t♦♥ ♥ t♦ t ♥② tr♥t♦♥s t♦♥

Pr♦ tr♥t♦♥s

♠r ♦ sts

♥ ts st♦♥ ♥ st♠t♦♥ ♦ t ♥♠r ♦ ♣♦♥ts rqr t♦ st t♦st ♦r♥ s♣ t ♥♣t ♣♦♥t st s ♥♦r♠② strt

❲ rst ♥ t ♥♦tt♦♥s s ♥ t ♦♦♥ sss♦♥ t E ♥ ♥t ♥X r♥♦♠ r ♦r E ❲ ♥♦t t ♣r♦t② ♦ E ② p(E) ♥ t ①♣t ♥ t r♥ ♦ X ② E[X] ♥ V [X] rs♣t② ❲♥ tr s ♥♦ ♠t②♦r t r♥♦♠ r s♦ s µ = E[X] ♥ σ =

√V [X] t♦ ♥♦t t ①♣t

♥ t st♥r t♦♥ rs♣t②❲ ♥t t♦ ♣rt t ♥♠r ♦ ♣♦♥ts rqr ♥ ♦rr t♦ rtr

♥ ♦r ♣rs② ♥ st ♦ ♣♦♥ts ♥♦r♠② strt ♥ t ♥t t s t ①♣t ♦r t ♥♠r ♦ ♣♦♥ts s tt t rst ♠♣t② s♠tr s s♠r t♥ 1

2 s ♣r♦♠ ♥ ♠♦ s P♦ss♦♥ ♣r♦ss ❬ ♥❪ ♦♥sr ♥ ♥♥t

r♥♦♠ ♣♦♥t st Π ♥ ② ♥t♥t♥st② P♦ss♦♥ ♣r♦ss ♦♥ E3 ♥ t C :=

[0, 3√

n]3 P♦ss♦♥ strt♦♥ s t ♦♦♥ ♣r♦♣rts t A ♥♦t sst ♦ E

3 ♦♦♠ ‖A‖ ♣r♦t② tt A ♦♥t♥s ①t② k ♣♦♥ts ♦ Π s ♥ ② p(|A ∩Π| = k) = e−‖A‖ ‖A‖k

k! ♥ t ♣r♦t② tt A ♦s ♥♦t ♦♥t♥ ♥② ♣♦♥t ♦ Π sp(|A ∩ Π| = 0) = e−‖A‖

①♣t ♥♠r ♦ ♣♦♥ts ♥ ♦♠ ♦ ♠sr ‖A‖ s t♥ ♥ s

E[|A ∩ Π|] =

∞∑

k=0

k · p(|A ∩ Π| = k) =

∞∑

k=0

k · e−‖A‖ ‖A‖k

k!= ‖A‖

② t ♣r♦♣rts ♦ t P♦ss♦♥ strt♦♥ ♥ r♦♠ t ♦♥strt♦♥ ♦ C ♦♦stt ①♣t ♥♠r ♦ ♣♦♥ts ♥ C s n t ΠC ♥♦t t st ♦ ♣♦♥ts ♦ Π tt ♥s C ΠC := Π ∩ C ❲ ♥♦t t ♥♠r ♦ ♣♦♥ts ♥ ΠC ② nC

t GC ♥♦t t r♦♣ ♦ tr ♦rt♦♦♥ ①s♥ tr♥st♦♥s t ♥♠♥t♦♠♥ C ♥ DT (GCΠC) s t ♥♥t ♣r♦ ♥② tr♥t♦♥ ♥ ② ΠC

♥ DT (GCΠC) ∩ C t st ♦ ttrr ♦ DT (Π) tt r ♦♠♣t② ♦♥t♥ ♥sC t♦tr t tr s ♥ t ♦♦♥ ♦♥sr DT (GCΠC) ∩ C ts s ♠♥s s♠♣♥ ♦t tt ♦♥sr t ♣♦♥t st GCΠC ♥st ♦ Π ♦ ♥♦t♥ t st ♦ ♣♦♥ts ♥s C ♦ t ♣♦♥t st GCΠC ∩ C st ♦♦s t P♦ss♦♥strt♦♥ ♣r♦♣rt② tt s ② s ♥ t ssq♥t sss♦♥

♥ s♠r ② s ♦♥ ② ❬❨❪ ♥ s♦ t ♦♦♥ t♦r♠

♦r♠ t C ♥ ΠC ♥ s ♦ t µO ♥♦t t ①♣t ♦ t♦♠ ♦ t rst ♠♣t② ♥ t ♥♥t ♣r♦ ♥② tr♥t♦♥ DT (GCΠC)♥ t ♦♦♥ ♦s

0.5 lnn

n≤ µO ≤ 5 lnn + 2

n.

Pr♦♦ ❲ ♥♦t ② Bx ♦ ♦♠ x Bx s s t♦ ♠♣t② Bx ∩ Π = ∅ ♣r♦t② tt B5 ln n s ♠♣t② s ♥ ② p(B5 ln n ♠♣t②) = e−5 ln n = 1

n5 ♥p(Bx ♠♣t②) ≤ 1

n5 ♦r x ≥ 5 lnn ❲ ♦♠♣t t ♣r♦t② ♦ t ①st♥ ♦ ♥♠♣t② r♠sr♥ ♦ ♦♠ t st 5 lnn ♥ DT (Π)∩C r♠sr♥ ♥DT (Π) s ♥ ② ♣♦♥ts ts tr r

(nC

4

)♣♦ssts t♦ ♥ ttrr♦♥ t

♣♦♥ts ♥ Π ∩ C s rts ❲ ♥ ♣♣r ♦♥ ♦r t ♣r♦t② tt(nC

4

)≥ n4

♥②ss

t X t r♥♦♠ r tt srs t ♥♠r ♦ ♣♦♥ts r♦♠ Π tt ♥ Cr♦♠ t t tt X s P♦ss♦♥ strt♦♥ ♥♦ tt µ = n ♥ σ =

√n ❲

♥♦ ♣♣② t ♥t ♥qt② P (X − µ ≥ k) ≤ σ2

σ2+k2 ♦♦s♥ k = n ts ②s

P (X − n ≥ n) ≤ 1n+1 ♠♣s tt t ♣r♦t② ♦r nC ≥ 2n s t ♠♦st 1

n+1

nC = 2n t♥ tr r(2n4

)< n4 ♣♦ssts t♦ ♦♦s r♦♠ t BC ♥♦t t st ♦

r♠sr♥ s ♥ ② ♣♦ss t♣s ♦ ♣♦♥ts ♥ Π ∩ C ♣r♦t②tt ♦♥ ♦ t s ♥ BC s ♦♠ rr t♥ 5 lnn ♥ s ♠♣t② ♥ nC ≤ 2n st♥ p(∃ ♠♣t② B5 ln n ∈ BC | nC ≤ 2n) ≤

(2n4

)· 1

n5 < 1n+1

♦ ♥② t ♦♦♥ ♣r♦ts

p(∃ ♠♣t② B5 ln n ∈ BC | nC ≤ 2n) ≤ 1

np(nC ≤ 2n) ≤ 1

p(∃ ♠♣t② B5 ln n ∈ BC | nC ≥ 2n) ≤ 1 p(nC ≥ 2n) ≤ 1n

♠♠♥ ♣ t ♦♥t♦♥ ♣r♦ts s p(∃ ♠♣t② B5 ln n ∈ BC) ≤ 2n

t Y ♥♦t t r♥♦♠ r tt srs t ♠tr ♦ t rst ♠♣t②r♠sr♥ ♥ (Π)∩C ♥ r♦♠ t ♦ sss♦♥ t ♣r♦t②p(Y ≥ 5 lnn) ≤ 2

n ♦♠ ♦ t s ♦♥ r♦♠ ♦ ② t ♦♠ ♦ C s n ♦ ♥ ♥ ♣♣r ♦♥ ♦r t ①♣t ♦ Y t f(x) ♥♦tt ♣r♦t② ♥st② ♥t♦♥ ♦ t strt♦♥ ♦ Y ♥

E[Y ] =

∫ ∞

0xf(x)dx =

∫ 5 ln n

0xf(x)dx +

∫ n

5 ln nxf(x)dx +

∫ ∞

nxf(x)dx

≤ 5 lnn · p(Y ≤ 5 lnn) + n · p(Y ≥ 5 lnn) + 0 ≤ 5 lnn + 2

r♦♠ ❬❨❪ ♥♦ tt E[Y ] ≥ 0.5 lnn P♦ss♦♥ ♣r♦ss sr s♦ r s ♠♦ s tt t r♦t ♦ t

tt ♦♥t♥s t ♣♦♥ts ♦ t tr♥t♦♥ s t ♥♠r ♦ ♣♦♥ts ♥ t ❲♥♦♥sr♥ ♣r♦ tr♥t♦♥s ♥t t♦ t s③ ♦ t ♦♥st♥t ♥ ♠♦ ② rs♥ Π s tt t C s rs t♦ [0, 1]3 ♦sss♦♥ r♠♥s s t ♥② tr♥t♦♥ s ♥r♥t t♦ rs♥

t Π′ ♥♦t t rs rs♦♥ ♦ t ♣♦♥t st Π rstrt t♦ [0, 1]3 ♥ ♦♣♣r♦② ♦♥t♦ E

3 Π′ := 13√

n· p | p ∈ Π t Y ′ ♥♦t t r♥♦♠ r tt

♦rrs♣♦♥s t♦ Y t t rs♣t t♦ t rs rs♥ s t ♦♠♦ t ♠♣t② s ② n s♦ ts ♣r♦s t t♦r♠ ♦r µ = E[Y ′]

t ♥♦ Y ′d ♥♦t t r♥♦♠ r tt srs t ♠tr ♦ t rst ♠♣t②

♥ DT (Π′) ❲ t t ♦♥s ♦r ts ①♣t rt② r♦♠ t ♦♥s ♦♥ t♠♣t② ♦♠s ♥ ♦r♠

3

√3

π

lnn

n≤ E[Y ′

d] ≤ 3

√30 lnn + 12

π · n

♦ ♥ st♠t t ♥♠r ♦ ♥♦r♠② strt ♣♦♥ts rqr ♦r t ①♣t ♦ t rst ♠tr t♦ s♠r t♥ 1

2 rr♥♥ t t ♣rtqt♦♥ ②s 24 ≤ n rrr♥♥ t rt ♣rt s n ≤ 507 ♦ ♦r ♥♦r♠②strt r♥♦♠ ♣♦♥t st t ①♣t ♦ t s③ ♦ t rst ♠♣t② r♠sr♥ s s♠r t♥ 1

2 t ♣♦♥t st s ♠♦r t♥ ♣♦♥ts

t ♦♥s r♥t ♦ t ♥♦♥ ②s ♥qt②

Pr♦ tr♥t♦♥s

♦t tt t ♦♥st ♥ ♥② tr♥t♦♥ s ♦♥ ② t ♠tr ♦t rst ♠♣t② s t s♠ ♥②ss ♦rs ♦r st♠t♥ t ♥♠r ♦ ♣♦♥tsrqr s tt t ♦♥st ♥ t tr♥t♦♥ s s♦rtr t♥ 1√

6s rqr ②

rtr♦♥ ♥ ts s t♦ r♣ t ♦♥st♥t ♦ 12 ♦r t ♠tr ②

1√6 r♦♠ qt♦♥ t♥ ♦♦s 56 ≤ n ♥ n ≤ 1030♥ t♦♥ ♣rs♥t ①♣r♠♥t rsts ♦♥ t ♥♠r ♦ ♣♦♥ts rqr ②

rtr ♥

♣tr

♠♣♠♥tt♦♥

♥ ts ♣tr sr ♦r ♠♣♠♥tt♦♥ ♦ ♣r♦ tr♥t♦♥s ♥ ♠♣♠♥tt♦♥ s s ♦♥ t tr♥t♦♥s ♠♣♠♥tt♦♥ tt r② ①stt rst r ♥tr♦t♦♥ t♦ ♥ ♠♦r t ♦♦ ♦♥ t♠♣♠♥tt♦♥ ♦ t tr♥t♦♥s trrs ♣rs♥t t ♠♣♠♥tt♦♥ ♦t ♣r♦ tr♥t♦♥s ♥ ♠♦r t ♥②ss ♦ t ♦♠♣①t② ♦t ♠♥ ♥t♦♥s ♥ s♦ t ♣rt ♥② ♦ ♦r ♠♣♠♥tt♦♥ ♥ ①♣r♠♥ts ♦♥ ♦t ♥rt ♥ r♦r t ♥② ♦rt ♦♥ ①t♥s♦♥s tt ♠♣♠♥t t♦♥②

♥tr♦t♦♥ t♦

♦♠♣tt♦♥ ♦♠tr② ♦rt♠s rr② s ♦t♦♥ ♦ ♦♣♥ s♦r♠♣♠♥tt♦♥s ♦ ♦♠tr t strtrs ♥ ♦rt♠s ❬❪ ts s♥ ♦♦s t♥r ♣r♦r♠♠♥ ♣r♠ ❬s ❪ s ♥s t sr t♦ s② ♣ r♥t♦♠♣♦♥♥ts t♦tr

♥ ♥r ♣r♦r♠♠♥ ♦♥♣t srs t t②♣s ♥ ♦♣rt♦♥s tt ♠st ♠ ss s ♠♦ ♦ s♣ ♦♥♣t t ♠♣♠♥ts t rqr♦♣rt♦♥s ♥ t②♣s

♠♣♠♥tt♦♥s ♦ ♦♠tr ♦rt♠s ♥ strt② s♣rt t ♦♠♥t♦r♣rts ♦ t ♦rt♠s r♦♠ t ♦♠tr ♦♠♣tt♦♥s ♥ ts ② r♥t ♠♣♠♥tt♦♥s ♦ t ♦♠tr ♦♠♣tt♦♥s ♥ ♣ ♥t♦ ♥ ♦rt♠ ♠♣♠♥tt♦♥♦♠ s ♦♠tr ♥t♦♥t② s ② t ♥ t ♦♠tr② r♥ tt sr ♦

♥ ♠♦r ♣r♦♠ ♥ ♠♣♠♥t♥ ♦♠♣tt♦♥ ♦♠tr② ♦rt♠s s tt ♦♠♣trs ♦ ♥♦t s♣♦s ♦ r t ♣♦sst② t♦ ♣r♦r♠ ①t ♦♠♣tt♦♥s♦♥ r ♥♠rs ♦♥♦ rr♦rs ♥ ♦♠♣tt♦♥s ♥ t♦ ♥♥ ♦r ♥ t♠♣♠♥tt♦♥ ♦ ♦♠tr ♦rt♠s tt♥r t ❬P+❪ s♦♥ tt ts s♥♦t ♣r② t♦rt ♣r♦♠ ♥r② tr s tr♦ t♥ ♥② ♥ ①t♥ss ♥r ♣r♦r♠♠♥ ♣♣r♦ ♥s t sr t♦ ♦♦s st ♥♠rt②♣ ♥ ♣ t ♥t♦ t ♦♠tr② r♥

♠r t②♣s

♥ ♣r♥♣ t sr ♥ ♣r♦ s ♦♥ ♥♠r t②♣ t♦ ♣r♦r♠ t rt♠t ♦♣rt♦♥s

♠♣♠♥tt♦♥

rqr ② ♥ ♦rt♠ ♥ tr r sr ♥♠r t②♣s

• t♥ ♦t♥♣♦♥t ♥♠r t②♣ ♦ ♦ ♥♠r t②♣ sr② st t ♦s ♥♦t ♣r♦r♠ ①t ♦♠♣tt♦♥s t ♦♥♦r♠s t♦ t st♥r ❬♦ ❪

• t♣rs♦♥ ♦t♥♣♦♥t ♥♠r t②♣s ② r ♣rs♦♥ t♥♦ t t② r s♦r ♥ st ♥♦t t♦ ②s ♣r♦ ①t ♦♠♣tt♦♥s

• ♥tr rt♠t s ♥♠r t②♣ ♠♥t♥s t♦ ♥♠rs tt r ♥r② ♦ st ♥ ♥①t ♥♠r t②♣ s ♦r ♥ ♣♣r ♦♥ ♦ t ♥♠r t②t② s♦ r♣rs♥t

• t♦♥ ♥♠rs s ♥♠r t②♣ ♠♥t♥s t♦ ♥tr ♥♠rs ♥♠rt♦r♥ ♥♦♠♥t♦r s ♦♥ s t ♥♠rs t♦ r♣rs♥t r rt♦♥ ts ♥♠rt②♣s r ①t t t ①♣♥s ♦ s♦r ♣r♦r♠♥

• r ♥♠rs ♥♠r t②♣s ♥ r♣rs♥t r r♦♦ts ♦ ♣♦②♥♦♠st ♥tr ♦♥ts ❯♥♦rt♥t② ♦♠♣tt♦♥s ♦♥ r ♥♠rs r qt①♣♥s

♦♠tr② r♥s

♥ ♦♠tr ♦rt♠s tr r t♦ t②♣s ♦ ♦♠tr ♦♠♣tt♦♥s ♣rtt♦♥s ♥ ♦♠tr ♦♥strt♦♥s

♦♠tr ♣rts ♥ st ♦ ♦♠tr ♦ts ♥t t♦ ♥♦ tr ♦r♥♦t t② rt♥ ♣r♦♣rt② st ♦ ♣♦ss ♥srs ♦ ♦♠tr ♣rts s♠ ♥ srt ①♠♣ ♥ tr ♣♦♥ts ♥ E

2 tr t tr♣♦♥t s t♦ t t ♦r t♦ t rt ♦ t ♥ ♥ ② t rst t♦ ♣♦♥ts rr tr ♣♦ss ♥srs t rt ①t② ♦♥ t ♥

♦♠tr ♦♥strt♦♥ ♥ st ♦ ♦♠tr ♦ts ♦♠♣t ♥ ♦♠tr♦t ♦t ♦ t♠ ①♠♣ ♦♠♣t t ♥tr ♦ t r♠sr♥ r ♦tr ♣♦♥ts ♥ E

2

♦♠tr② r♥ ♠st ♣r♦ ♦t♦♥ ♦ ♦♥st♥t s③ ♦♠tr ♦ts♥ E

2 ♥ E3 ♥ ♦♠tr ♣rts ♥ ♦♥strt♦♥s ♦ ♦♥st♥t t♦♥ t♠

♥♦♥①st st ♦♥ssts ♦

• ♦♠tr ♦ts P♦♥t❴ P♦♥t❴ ♠♥t❴ ♠♥t❴

r♥❴ r♥❴ trr♦♥❴ r❴ ♣r❴

• Prts ♦♥r❴ ♦♥r❴ q❴ q❴

r♥tt♦♥❴ r♥tt♦♥❴

r♥tr❴ r♥t♣r❴

• ♦♥strt♦♥s ♦♠♣tr❴ ♦♠♣tr❴

♦♥strtr♠♥tr❴ ♦♥strtr♠♥tr❴

♦♠♣tqrst♥❴ ♦♠♣tqrst♥❴

tr♥t♦♥s

♦t tt tr r ♦♠tr ♦♥strt♦♥s tt ♦ ♥♦t ♣r♦r♠ ♥② ♦♠♣tt♦♥s ♦♥strtr♥❴ s ♦♥strt♦♥s ♦ ♥♦t ♠♣♦s ♥② rqr♠♥ts ♦♥ t♥♠r t②♣s ♥ t ♦♦♥ ts t②♣ ♦ ♦♥strt♦♥s tr ♦♥strt♦♥s

r r r♥t ♣♣r♦s t♦ ♠♣r♦ t ♣r♦r♠♥ ♦ t r♥ ♥ ♣♣r♦t♦ rt t ♣rt t♦♥ s sr ② ❬❱❪ ♥ rst st♣ t ♣rts t s♥ st t ♣♣r♦①♠t ♥♠r t②♣ t ♥♦♥ rr♦r ♦♥ trst ♥ rt s♥ t rr♦r ♦♥ t ♣rt t♦♥ s ♦♥ trs trs♦rts t♦ s♦r t ①t ♥♠r t②♣ rstrt♥ t ♦♠♣tt♦♥ r♦♠ srt s♣♣r♦ s s ♦♥ t tt ♥ ♠♦st ss t s s♥t t♦ s ♥ ♣♣r♦①♠t♥♠r t②♣ t♦ t t ♣rt ♥② ♥ ss t ①t t♦♥ ♠st s ♦ t ♦r ♦ t♥ ♣rts t s ♦t ② t s♣♣r♦♠ s♥ ♥ ♣♣r♦①♠t ♥♠r t②♣ sr ♣♣r♦ s s♦ rt♠ttr♥ ♥ t tr r♥ ♣r♦s tr ♣rts ♦r t ♦♠tr② r♥❬P P❪ ♥♦tr ♣♣r♦ tt ♦♥♥trts ♦♥ ♦♠tr ♦♥strt♦♥s s ③② t♦♥ ❬P❪ ♥ ts s t rt♠t ①♣rss♦♥ tt s t♦ ts ♥♦t t ♠♠t② t ♦♥② ♥ rqr

♦t tt ①t t♦♥ ♦ ♣rts s sr t♥ ①t ♦♠♣tt♦♥ ♦ ♦♥strt♦♥s s♦ ♦r ♦rt♠s tt ♦ ♥♦t ♣r♦r♠ ♥② ♦♠tr ♦♥strt♦♥s ♥♠r t②♣♣r♦♥ ♦♥② ①t ♣rts ♥ s t♦ ♠♣r♦ t ♣r♦r♠♥

♦♠tr ♦rt♠s

♥ t ♣tr ♥ ♦rt♠ ♠♣♠♥t ♥ ♥s st ♦ ♦♠tr ♦ts♣rts ♥ ♦♥strt♦♥s t rqrs s r ♣r♦ tr♦ t♠♣t ♣r♠tr t s♦ ♦♠tr trts ss t♥ t r♥ ♥ sr s trts sst s♦♠t♠s s♣③ ♦ts ♣rts ♦r ♦♥strt♦♥s r ♥ s♦ s♣trts ss s rqr trts ss ts ts t ♥♠r t②♣ s t♠♣t r♠♥t sr ♥ r② ♦♦s t ♥♠r t②♣ t♦ s ♥ s ♥ ♥①♥ t ♣r♦ trts ss t sr ♦♥ ♠♣♠♥tt♦♥ ♦♥ t ts ♦st rs♣♦♥st② t♦ ♦♦s ♥ ♣♣r♦♣rt ♥♠r t②♣ ♣♥♥ ♦♥ tr ♦r♥♦t ♦♠tr ♦♥strt♦♥s r s ♥ t ♦rt♠ r ♦r s♠t ♦♥ t t♠♣t ♣r♠tr rr②

CGAL Algorithm

Geometric traits ...NT ...

r s ♦ t♠♣ts ♥ t②♣ ♦rt♠ ♥♠r t②♣

tr♥t♦♥s

s ♦r s s ♦♥ t tr♥t♦♥s ❬P❪ tt ♠♥② ♥ ♦♣ ② ②♥ P♦♥ ♥ ♦♥q t ♦♠♣ts sr t②♣s ♦ tr♥t♦♥s ♦♣♦♥t sts ♥ E

3 tr♥t♦♥s r st♦r ♥ t tr♥t♦♥ t strtr ❬P❪ ❲ r② ♥tr♦ t ♠♣♠♥tt♦♥ t strss ♦♥ t ♣rts tt rrqr ♥ t sss♦♥ ♦♥ t ♠♣♠♥tt♦♥ ♦ ♣r♦ tr♥t♦♥s tr ♦♥

♠♣♠♥tt♦♥

♣rt r♦♠ ♦♠♣t♥ tr♥t♦♥s ♦ E3 tr s s♦ ♥t♦♥t② ♦r ♦♠♣t♥

♥② tr♥t♦♥s ♥ rr tr♥t♦♥s ♦rt♠ s s ♥r♠♥t ♣♦♥ts r ♥srt ♦♥ ② ♦♥ ❲ ♥♦ r② r t s♦tr s♥

ss r♥t♦♥❴ ♠♣♠♥ts t ♥t♦♥t② t♦ ♦♠♣t ♥ ss tr♥t♦♥s ♦ ♠♥s♦♥ ≤ 3 t s t♠♣t ② t ♦♠tr trts ♥ t t strtrr r t♦ r sss ♥②❴r♥t♦♥❴ ♥ r❴tr♥t♦♥❴ s♦ r

Triangulation_3

GT:class

TDS:class

Delaunay_triangulation_3

GT:class

TDS:class

Regular_triangulation_3

GT:class

TDS:class

r s♥ ♦ t tr♥t♦♥s rrs t♦ t ♦♠tr trts ss♥ t♦ t tr♥t♦♥ t strtr

tr♥t♦♥ trts

♦ts ♣rts ♥ ♦♥strt♦♥s rqr ② t ♥② tr♥t♦♥ ♦♠♣tt♦♥ r ♦♥t♥ ♥ t r♥ s t s ♥♦t rqr t♦ ♣r♦ s♣♦♠tr trts ss ♦ ♦♠♣t ♥② tr♥t♦♥s

♦♠tr trts ♦r ♦♠♣t♥ tr♥t♦♥s ♠st ♦♥t♥

• ♦♠tr ♦ts P♦♥t❴ ♠♥t❴ r♥❴ trr♦♥❴

• ♦♠tr ♣rts ♦♠♣r❳❨❩❴ r♥tt♦♥❴ ♥ ♠♦r ♦r trt♥♥rt ♠♥s♦♥s

• ♦♠tr ♦♥strt♦♥s ♦♠♣tt♦♥ ♦ tr♥t♦♥s ♦♥② rqrs tr♦♥strt♦♥s t♦ ♦♥strt t ♦♠tr ♦ts

♦r ♥② tr♥t♦♥s t ♣rt r♥t♣r❴ s rqr t♦♥② t tsts ♦r ♦r ♥ ♣♦♥ts tt r ♥♦t ♦♣♥r tr t ♣♦♥t s ♥s♦ts ♦r ♦♥ t s♣r ♥ ② t rst ♦r ♣♦♥ts ♦r t ❱♦r♦♥♦ ♦t♣t t ♦♥strt♦♥ ♦♥strtr♠♥tr❴ s rqr t♦♥② ♥ ♥② ♦r rr tr♥t♦♥s t P♦rst❴ ♣rt s rqr ♥st ♦ t r♥t♣r❴tst ♦t tt ♦♥strtr♠♥tr❴ s t ♦♥② ♥♦♥tr ♦♥strt♦♥ ♥ t s♦♥② rqr ♦r t ♦♠♣tt♦♥ ♦ t ❱♦r♦♥♦ r♠ s ♦r ♦♠♣t♥ t ♥② tr♥t♦♥ t ss t♦ s ♥♠r t②♣ tt ♦♥② ♣r♦s ①t ♣rtt♦♥ ♥ ♥♦ ①t ♦♥strt♦♥s

tr♥t♦♥ t strtr

tr♥t♦♥ t strtr ♠♣♠♥ts t t strtr t♦ st♦r tr♥t♦♥s♦ ♠♥s♦♥ ≤ 3 ♦r♠② t st♦rs s♠♣ ♦♠♣① ♦♠♦♠♦r♣ t♦ tr♠♥s♦♥ ♦♠♣t s♣ t♦t ♦♥rs s E

3 s ♥♦t ♦♠♣t rt① t ♥♥t②v∞ s t♦ t ♣♦♥t st ♥ t st♦r ♦♠♣① s tr♥t♦♥ ♦ E

3 ∪ ∞ r

♣r♦ tr♥t♦♥s

♦♥② sr t ♠♥s♦♥ s t s ♦ ♠♥s♦♥ tr s ttr♥t♦♥s ♦ T

3c tt ♥t t♦ st♦r ②s ♠♥s♦♥ tr

tr♥t♦♥ t strtr st♦rs tr♥t♦♥s ♥ t ♦♦♥ ② t st♦rst rts ♥ t s r rts ♦rrs♣♦♥ t♦ ♣♦♥ts ♥ t ♦♠tr ②r ♥s ♦rrs♣♦♥ t♦ ttrr rt① ♦♥t♥s ♣♦♥tr t♦ ♦♥ ♦ t s t s♥♥t t♦ t♦♥② t ♦♥t♥s t ♦♦r♥ts ♦ t ♣♦♥t t ♦rrs♣♦♥s t♦ s st♦r t ♦♦♥ ♥♦r♠t♦♥

• ♦r ♣♦♥trs t♦ rts ♥① r♦♠ 0 t♦ 3

• ♦r ♣♦♥trs t♦ ♥t s ♥① r♦♠ 0 t♦ 3 r ♥① i ♦rrs♣♦♥s t♦ t♥t ♦♣♣♦st ♦ rt① i

♦r ♠♥s♦♥ s♠♣s s ♥ ts r ♠♣t② r♣rs♥t s♦♦s ♥ ♥ ss tr♦ ♦♥ ♦ t s t s ♥♥t t♦ ♥t♥ tt♦ ♥s tt tr♠♥ t rts tt ♦♥ t♦ t t ♥ sstr♦ ♦♥ ♦ t t♦ s t s ♥♥t t♦ ♥t♥ t ♥① ♦ t rt① tt s♦♣♣♦st t♦ t t t ♦♥② rt① ♦ t tt ♦s ♥♦t ♦♥ t♦ t t r ♦r ♥ strt♦♥

vextex 0 vertex 1

vertex 2

vertex 3

facet 0

neighbor 0

edge (1, 3)

r ♣rs♥tt♦♥ ♦ tr♥t♦♥ ♥ t t strtr ❬P❪

ss r♣rs♥t♥ t tr♥t♦♥ t strtr s r♥t♦♥❴t❴strtr❴t s t♠♣t ② rt① ss ♥ ss t sss r ♣r♦ t ♥② ssstt ♠t t rqr♠♥ts ♥ ♥ t rs♣t ♦♥♣ts ♥ s r♦♠ t tstrtr t rts ♥ s r t②♣② ss s♥ ♥s ♥ ♥

srs ♦♥♣t tt r♣rs♥ts ♣♦♥trs t②♣s t♦ ss rts ♥ s ♥ ttr♥t♦♥ t strtr r ♥♠ ❱rt①❴♥ ♥ ❴♥

♣r♦ tr♥t♦♥s

♥ ts st♦♥ sr t ♠♣♠♥tt♦♥ ♦ t ♦rt♠ ♦r ♦♠♣t♥ t ♣r♦♥② tr♥t♦♥ ♦ ♣♦♥t st s sr ♥ ♣tr

❲ rst ♥♠♥ ♦♥♥t♦♥s ♦r t ♦♦♥ sss♦♥s ❲ s t ♥♦tt♦♥sT

3c ♥ T

33c

s ♥ ♥ ♣tr t ss♠ tt c r♣rs♥ts ♦ ♥t c♠♠r tt rts r♣rs♥t ♦rts ♦ t ♥♣t ♣♦♥ts ♥r t t♦♥ ♦ t r♦♣(Z3,+) ♦r (3Z

3,+) ♥ ♦♠♣t♥ ♥ T3c ♦r T

33c rs♣t② ❲♥ ♦♠♣t♥ ♥ T

33c

s t ♦♦♥ ♥♠♥ ♦♥♥t♦♥s ② ♥ ♦r♥ rt① ♠♥ rt① ♦sr♣rs♥tt ♥ D3c s ♥ Dc ② ♣r♦ ♦♣② ♠♥ rt① ♦s r♣rs♥tt♥ D3c ♦s ♥♦t ♥ Dc r ♦r ♥ strt♦♥

♠♣♠♥tt♦♥

r strt♦♥ ♥ ♦r♥ rt① t ts t ♣r♦ ♦♣s ♥ st♦r♥ s♣ ♦ T

2c

r♦♠ ♣tr ♥ t♦♥ ♥ t ♦♦♥ ♠♥ r♥s t♥E

3 ♥ T3c t rs♣t t♦ t ♠♣♠♥tt♦♥

• s ♦ ♦sts

• s ♦ ♦r♥ s♣s

• ♥♦ ♥ ♦r rt① t ♥♥t②

• ♥♦ ♥ ♦r trt♥ ♠♥s♦♥ t♦ ♥ s♠r

❲ ♥♦ t ♦r s♥ ♦ t ♠♣♠♥tt♦♥ ♥ ♣rs♥t ♥ ♠♦r t♦ t ♦sts r ♠♣♠♥t ♦ t ♦♠tr trts ss ♠st ♣t ♦ t♦r♥ s♣s r ♠♥ ♥ ♦ t r♥t st♣s ♦ t ♦rt♠ ♠st ♣ts s ♣♦♥t ♦t♦♥ ♣♦♥t ♥srt♦♥ ♥ rt① r♠♦ ♥② ♣rs♥t t♦♥♥t♦♥ts s s ss ♥t♦♥s s♣ t♦ T

3c ♣tt♦♥ ♦ t ♣♦♥t ♦t♦♥

strts ♥ ♥t♦♥s ♦r ❱♦r♦♥♦ ♦t♣t

s♥

♣r① Pr♦❴ t♦ ss ♥♠s tr♠♥s t t②♣ ♦ ♣r♦t② t tr♠♥stt ♦♠♣t ♥ s♣ ♦♠♦♠♦r♣ t♦ T

3c s ♦♣♣♦s t♦ ♦tr ♣♦ss ♣r♦

s♣s s s ♦rt s♣s ♦ E3 ♥r t t♦♥ ♦ r♦♣ s♣♥♥ ② t♦ ♦r ♦♥

tr♥st♦♥s ♦♥② s s♦ t♦♥ s♥ ♦ t ♣r♦ tr♥t♦♥s s r② s♠r t♦ t s♥ ♦ t

tr♥t♦♥s ss Pr♦❴❴tr♥t♦♥❴ ♦♥t♥s t ♥t♦♥t② tt s ♥♦t s♣ t♦ ♥② tr♥t♦♥s t s ♠♥② ss ♥t♦♥s ♥t ♣♦♥t ♦t♦♥ ♥t♦♥t② s♣③t♦♥ Pr♦❴❴♥②❴tr♥t♦♥❴♦♥t♥s t ♥② s♣ ♥ ❱♦r♦♥♦ s♣ ♥t♦♥t② t s ♣♦ss t♦ ♥♦tr ss Pr♦❴❴rr❴tr♥t♦♥❴ s s♣③t♦♥ ♦ Pr♦❴❴tr♥t♦♥❴ ♥ t s♠ ② s ♦r t rr tr♥t♦♥s ♥ E

3 s r ♥ t♦ tr s ♥♦ ♦♥♣t ♣r♦♠ t ss Pr♦❴❴rr❴tr♥t♦♥❴s ♥♦t ♥ ♠♣♠♥t

s ♥ t tr♥t♦♥s t ♣r♦ tr♥t♦♥ sss t♦ t♠♣t ♣r♠trs t ♦♠tr trts ♥ t tr♥t♦♥ t strtr rst♦♥ s sr ♥ t♦♥ ♦r t s♦♥ ♦♥ t ①st♥ ♠♣♠♥tt♦♥ ♦ ttr♥t♦♥ t strtr ♥ s s sr ♥ t♦♥ t tr♥t♦♥t strtr st♦rs s♠♣ ♦♠♣① t♦t ♦♥r② tr♥t♦♥ ♥ T

3c s

s♠♣ ♦♠♣① ♦♠♦♠♦r♣ t♦ T3c ♥ ts t♦t ♦♥r② ♦ ♥ rs t

♣r♦ tr♥t♦♥s

Periodic_3_triangulation_3

GT:class

TDS:class

Periodic_3_Delaunay_triangulation_3

GT:class

TDS:class

Periodic_3_regular_triangulation_3

GT:class

TDS:class

r s♥ ♦ t ♣r♦ tr♥t♦♥s rrs t♦ t ♦♠trtrts ss ♥ t♦ t tr♥t♦♥ t strtr

ss r♥t♦♥❴t❴strtr❴ rt② ♥② t rt① ss ♥ t ss♥ t♦ ♥ s sr ♥ t♦♥

sts

s sr ♥ t♦♥ s♠♣s r r♣rs♥t ② ♣♦♥t♦st ♣rs r t♣♦♥ts ♥ t ♦r♥ ♦♠♥ ♥ t ♦sts r ♠♥ts ♦ Z

3 ♠♠r tt ♣♦♥t p t♦tr t ♥ ♦st o ♦rrs♣♦♥s t♦ t ♣♦♥t ϕc(p, o) = p + c ∗ o ♥ E

3 ϕc(p, o) | o ∈ Z

3 srs t ♦rt ♦ p ♥r t t♦♥ ♦ t r♦♣ (Z3,+) ks♠♣① ♥ T

3c s ♥ s t ♣r♦t♦♥ ♦♥t♦ T

3c ♦ t ♦♥① ♦ t ♠s ♦ k + 1

♣♦♥t♦st ♣rs ♥r ϕc ♥t♦♥ ♦r♥② ks♠♣① ♥ T33c

s ♥ ♥t s♠ ② ② t ♠s ♦ k + 1 ♣♦♥t♦st ♣rs ♥r ϕ3c tr♥t♦♥ tstrtr ♦s ♥♦t ♥♦ t ♦r♥ ♦♠♥ ♦ t s♣ ♦ tr♥t♦♥ t st♦rs sts r ♠♣♠♥t s tr♠♥s♦♥ ♥tr t♦rs t ♦♠♣r♥s ♥tr♦r ♦♥♥♥ ❲ ♥♦ ♣rs♥t t r♥t ss ♦ ♦sts ♥ t ♣r♦tr♥t♦♥s

s sr ♥ t♦♥ t tr♥t♦♥ t strtr st♦rs ♦♥② t s ♥t rts ♦ tr♥t♦♥ s ♥ t t t♦rs s♠♣s r ♥ ② ♣♦♥t♦st♣rs ♠st ♣t t r♣rs♥tt♦♥ ♦ t tr♥t♦♥ t strtr s ♦♦s♦ sr st♦r ♦r rts ♥ ♦r ♦sts t♦ r♣rs♥t ttrr♦♥ rt① r♣rs♥ts t ♦rt ♦ ♣♦♥t ♦ t ♥♣t ♣♦♥t st rt① ss ts r♠♥s♥♥ t ♦♥② st♦rs t ♣♦♥t ♦ t ♦rt tt s ♥ t ♦r♥ ♦♠♥ ♦ttt t ♦sts ♠st tt t♦ t s ♥ ♥♥♦t tt t♦ t rts s rt① ♠t r♥t ♦sts ♥ r♥t s s r

r strt♦♥ ♥ σ t rt① v s ♦st(00

) ♥ τ t rt① v s ♦st

(10

)

♦t tt ♦r ♥② ♦st o ∈ Z3 t ♦r ♣♦♥t♦st ♣rs (pi, oi) ♥ (pi, oi + o)

i = 0 . . . 3 ♥ t s♠ ♥r t rs♣t ♣r♦t♦♥ ♠♣ ♦r t ♠♣♠♥tt♦♥

♠♣♠♥tt♦♥

♥ t ♦♦♥ ♦♥♥t♦♥ ♥ ♦rr t♦ ♥♦♥ ② ♦ st♦r♥ s

♦♥♥t♦♥ t C t rts v0, . . . , v3 ♥ ♦rrs♣♦♥♥ ♦stso0, . . . , o3 t oi = (oix, oiy, oiz) ♥

mino0x, o1x, o2x, o3x = 0

mino0y, o1y, o2y, o3y = 0

mino0z, o1z, o2z, o3z = 0

♥tt② ts ♠♥s tt t r♣rs♥tt♦♥ ♦ t ♥ E3 s s ♦s s ♣♦ss

t♦rs t ♦r♥ t♦t ♦♥ ♦r ♥t ♦sts ♠ ♥r ϕc ♦ t s②s ♥tr② ♦♥t♥ ♥ t qr♥t x ≥ 0 y ≥ 0 z ≥ 0 s r

r strt♦♥ ♠s ♦ s ♦ ♣r♦ tr♥t♦♥ ♦r♥ t♦♦♥♥t♦♥

♦t tt st♦r♥ t♦♥ ♥♦r♠t♦♥ ♥ t s s rt t rs♣t t♦ ♠♠♦r②s s♦ r ♥trst ♥ ♥♥ r♣rs♥tt♦♥ tt s s ♦♠♣t s ♣♦ss ♦♦♥ ♠♠ s♦s tt ♦♥ t ♣r ♠♥s♦♥ ♦ t s♣ s s♥t

♠♠ ♦r ♥② tr♥t♦♥s ♦ ♦t T3c ♥ T

33c t s s♥t t♦ ♥♦

t ♦sts s♥ tr ts

Pr♦♦ ♠ ♦♦s r♦♠ stt♠♥t ♦ ♠♠ F (2) ♦♥ssts ♦ t ♦♣s♦ t ♥♠♥t tt ♥ rss ② ♦sts s♥ tr ts

♦t tt ♠♠ t② ♦s ♦r ♥② t♦rs t rt♥r ♥♠♥t♦♠♥

s ♦♥t♥s ♦r ♦sts rqr tr ts t♦ st♦r ♦r♥ t♦♠♠ ♥ t♦ st♦r ①tr ts ♣r ❲ rr♥t② s ♦♥ ♥s♥

♥t t♦ st♦r t ♦r ♦sts ♦ t s s♣♥t r♣rs♥tt♦♥ s ♦♥② st♦ st♦r ♦sts ♥ t tr♥t♦♥ t strtr ♦r ♦♠♣t♥ t ♦sts s tr♠♥s♦♥ ♥tr t♦r tt ♥ r♣rs♥t ♠ r r♥ ♦ ♦sts

♦r s♦♠ ♦ t ssq♥t② ♣rs♥t ♦rt♠s ♥ t♦ ♥ ♥♦r ♦sts♦♥sr ♥② tr♥t♦♥ ♦ T

3c ♦r T

33c ♦♥♥t♦♥ tr♠♥s ttrr♦♥

τ ♥ E3 t♦ t ♥♦♥ r♣rs♥tt♦♥ ♦ c ♥ t ♥② tr♥t♦♥ ♦ t

σ t ♥♦♥ r♣rs♥tt♦♥ ♦ ♥t t♦ c ♥ σ ♥ τ r ♥♦t ♥ssr②♥t ♥ E

3 s r

♣r♦ tr♥t♦♥s

♥t♦♥ ♦r ♦st r s ♥ ♦st ζ s tt ϕc(σ, ζ) s ♥t t♦τ ❲ ts ♦st ζ ♥♦r ♦st r♦♠ τ t♦ σ

♥♦r ♦st ♥ tr♠♥ s ♦♦s t v ♦♥ ♦ t ♦♠♠♦♥ rts♦ τ ♥ σ ♥♦r ♦st s t r♥ t♥ t ♦st ♦ v ♥ τ ♥ t ♦st♦ v ♥ σ s r

r strt♦♥ ♥♦r ♦st r♦♠ τ t♦ σ s(10

)

♦r ♦sts r s ♥ t ♠♣♠♥tt♦♥ ♦ t r♠♠r♥ st♦st ♦r t ♣♦♥t ♦t♦♥ ♥ ♥ t ♠♣♠♥tt♦♥ ♦ ♥❴♦♥ts tt tr♠♥s t♦♥t r♦♥ ♦ ♣♦♥t

♦ t ♣♦♥ts tt r ♥♦t rts ♦ t tr♥t♦♥ s t ♦♦♥♥♦tt♦♥ t pv ♥♦t t ♣♦♥t ♦ t rt① v ♦ t tr♥t♦♥ ② ♦st o ♦ rt① v t rs♣t t♦ ♣♦♥t p ♠♥ t ♦st ♦ t ♥♣♦♥t pv ♥ t ♥② t ♥♣♦♥ts ϕc(p, 0) ♥ ϕc(pv, o)

rts

♦♠trrts ♦♥♣t ♦r ♣r♦ tr♥t♦♥s s ss♥t② t s♠ s ♦r tr♥t♦♥s ♠♥ ♥ s tt ♣rt ♥ ♦♥strt♦♥ ♥t ♣♦♥ts ♥ ♦rrs♣♦♥♥ ♦sts s r♠♥ts t♦♥② t♦ t rs♦♥ tt ts♦♥② ♣♦♥ts ♥ ♦rr t♦ ♣r♦ t ♥ ♣rts ♥ ♦♥strt♦♥s ♥ ♣trtt ♦rs s sr ② t ♦♦♥ ♣s♦♦ st♥

♦♦ ♣r♦❴♣rtP♦♥t ♣ P♦♥t ♣

st ♦ st ♦ ④

rtr♥ ♣rt♣ ♦ ♣ ♦

⑥

♣r♦❴♦♥strt♦♥ s ♥ ♥ t s♠ ② ♦t tt t ♥t ♦ t♦r♥ ♦♠♥ s rqr t♦ ♣r♦r♠ t ♦♠♣tt♦♥ ♦ t ♣rt ♦r ♦♥strt♦♥ ♦r♥ ♦♠♥ s ♦② ss♦t t t tr♥t♦♥ s♦ t s ♠♠r ♦ tss Pr♦❴❴tr♥t♦♥❴ ♥ ♥♦t ♦ t trts ss ❲ ♥♦ t trts sst ♣♦♥tr t♦ t ♦r♥ ♦♠♥ st♦r ♥ t ss Pr♦❴❴tr♥t♦♥❴

♣rts ♥ ♦♥strt♦♥s t♥ ♦♥② ♣♦♥ts ♦ ♥♦t rqr ② t♦rt♠ t ♣r♦ t♠ ♦r ♥② rs♦♥s ② ♥ ♦sts r③r♦ ♥ ts s ♥♦ tr♥st♦♥s t♦ ♦sts ♥ t♦ ♣r♦r♠ ♥ ts ♣rts♥ ♦♥strt♦♥s ♦ ♦♥sr② ss rt♠t ♦♣rt♦♥s

①t t♦♥ s sr ♥ t♦♥ t tr♥t♦♥s ♠♣♠♥tt♦♥ rqrs ①t ♣rts ♥ ♥♦ ①t ♦♥strt♦♥s ♦t♣t s ♦♠♥t♦r

♠♣♠♥tt♦♥

strtr tt ♦♥t♥s ♦♥② ♥♠♦ ♥♣t ♣♦♥ts ♦♠♣t♥ t ❱♦r♦♥♦ r♠s t ♦♥② s r ♥♦♥tr ♦♥strt♦♥s r ♥ ♥♠② ♦r ♦♠♣t♥ tr♠♥tr ♦ ttrr

♦r ♣r♦ tr♥t♦♥s t stt♦♥ s st② r♥t r♥ ♥ ♦rt♠ r♥s♦♠ ♣♦♥ts t♦ tr♥st ♥ ♦rr t♦ ♠ s♠♣s ♥ E

3 ♦r t♦♥ ♦♦♠tr ♣rts s r r♥st♥ ♣♦♥ts s ♦♠tr ♦♥strt♦♥ ♥

r strt♦♥ ♦ tst ♦♥ s ♦ t r♠r ♦ t ♣♦♥t q s p♠st tr♥st rst

t s ♥♦t ①t t ♦rrt ♦t♣t ♦ ♥ ♦rt♠ r♥ ♥ ♥ ts tr♠♥t♦♥ ♥♥♦t r♥t ♥ t ♦tr ♥ ♦ ♥♦t ♥t t♦ rqr ①t ♦♥strt♦♥s ♦r ♥②rs♦♥s ♦ ♦ ♦♥strt♦♥s rr♦♣ t tr♥st♦♥s ♥ ♣rt t♦♥s ♥♥ ♣rts t s ② ♥tr♦ t ♣rts t♥ ♣♦♥t♦st ♣rs sr♠♥ts ♦ tr♥st♦♥ ♦ t ♣♦♥ts s ♥♦ ♣rt ♦ t ♣rt t♦♥s ♠♥s t ♦♦r♥ts ♦ t tr♥st ♣♦♥ts r ♥r r♣rs♥t ♥ ♠♠♦r② t♦♥② trt s②♠♦② ♥ ts ② t ♥♦♥ ♥ ② ♥t t♥qs ♦r①t ♣rt t♦♥ ♥ ♣♣ ♦r t s♠ rs♦♥ ♥ ♦♠♣t♥ ♥ T

33c

t ♣♦♥ts ♦ts Dc st♦r t ♦♦r♥ts ♦ t ♦rrs♣♦♥♥ ♣♦♥ts ♥ Dc ♥ rss♥ ♥ ♦st ♦♦r♥ts ♦ ♣♦♥ts ♦ts ♦ Dc r ♥r ♦♠♣t ①♣t②

♦r♥ s♣s

r sr ♦ ♦r ♠♣♠♥tt♦♥ ♥s ♦r♥ s♣s ♥ ♦ ♥♦♥rs♦♥ t♥ r♥t ♦r♥ s♣s ♦t tt ♥ ♥r tr♥t♦♥s ♦ s♠♣♦♥t sts rqr ♦♠♣tt♦♥ ♥ T

33c rs ♦r r ♣♦♥t sts ♥ ♦♠♣t ♥ T

3c

♦ ♦♥ r② ♠♣♦rt♥t ♠♣♠♥tt♦♥ ss s tt tr s♦ ♥♦t ♥② ♦r ♦r♥♥ ♦r♥ s♣s ♥ ♦♠♣t♥ ♥ T

3c t♦t t♦♥ ♦♣s

♥ ♦rr t♦ ♦♠♣t ♥ T33c ♦♣s ♦ ♣♦♥t r ♥srt ♠♠r tt ♥ T

33c

rts t r♣rs♥tt ♥ Dc ♦r♥ rts ♥ ♦tr rts ♣r♦♦♣s

♦♣② st ❲♥ ♦♠♣t♥ ♥ T33c rts r♣rs♥t ♦rts ♥r t t♦♥ ♦ t

r♦♣ (3Z3,+) ♦ ♣♦♥t ♥ D3c ♦ ♥ ♦rt r♣rs♥t ② rt① ♦s ♥♦t ♥ssr②

♥ Dc ♦r t ♥♣t ♣♦♥t st ♦ t ♦rt♠ s ♦♥t♥ ♥ Dc ♦r♥ t♦t♦♥ t tr♥t♦♥ t strtr st♦rs ♦r rt① t ♦♦r♥ts ♦ ♣♦♥t ♥ D3c ♦r♥ t♦ t♦♥ t ♦♠♣tt♦♥ ♦ t ♣♦♥t ♦♦r♥ts ♥ D3c

♦ ♣r♦ ♦♣② ♦ ♥ ♥♣t ♣♦♥t s ♦♠tr ♦♥strt♦♥ ♥ ♦rr t♦ ♦ ts♦♠tr ♦♥strt♦♥ st♦r ♦r ♣r♦ ♦♣② t ♦rrs♣♦♥♥ ♥♣t ♣♦♥t p♥ Dc ♥ ♦♣② st s s tt ϕc(p, s) s t ♣r♦ ♦♣② ♥ D3c ♦♣② sts r

♣r♦ tr♥t♦♥s

♦♥② rqr ♥ ♦♠♣t♥ ♥ T33c s♦ ♦ ♥♦t st♦r t♠ ♥ t tr♥t♦♥ t

strtr t♦ ♦ ♠♠♦r② ♦r ② r st♦r ♥ s♣rt t strtr ♥ ttr♥t♦♥ ss ts s sr ♦ ♦sts ♦♣② sts r ♠♥ts ♦ Z

3♥ t ♠♣♠♥tt♦♥ rs t ♦st ss t♦ r♣rs♥t ♦♣② sts

t strtrs r sr t strtr t♦ r♦r t ♦♣② st ♦ ♥rt① s t strtr t♦♥② st♦rs t ♥♦r♠t♦♥ rts r ♣r♦ ♦♣s ♦ ♦tr ♥ ♥♦r♠t♦♥ tt s ♥♦t ♦♥t♥ ♥ t tr♥t♦♥ tstrtr tr ❲ t♦ s st♠♣ t ❱rt①❴♥ s ② t②♣ ♥st♣r❱rt①❴♥♦♣②❴st s t②♣ t♦ rtr ♦r rt① tt s♥♦t ♥ ♦r♥ rt① ts ♦rrs♣♦♥♥ ♦r♥ rt① ♥ t ♦♣② st ❲ ts♠♣ ♦r♥❴♠♣ s t t② ♦rrs♣♦♥s t♦ t ♠t♠t ♥t♦♥ ♦ t♦r♥ ♠♣ s ♥t♦♥ t s st♦r ♥ t ss Pr♦❴❴tr♥t♦♥❴ r ♦r ♥ strt♦♥ ♦ t s ♦ t ♦r♥❴♠♣

r strt♦♥ ♦r♥❴♠♣ ♠♣♠♥ts t rr♦ rt♦♥

♦r s♦♠ ♦rt♠s ♦ ♥♦t ♦♥② ♥ t♦ ♥ t ♦r♥ ♦♣② r♦♠ ♥ rt①t ♠st s♦ ♥ ♣r♦ ♦♣s ♦ ♥ ♦r♥ rt① r s ♥♦tr ♠♣tt st♦rs ♦r ♦r♥ rt① ts ♣r♦ ♦♣s ② t②♣ ♦ ts ♠♣ s ♥❱rt①❴♥ t t②♣ s stt♦r❱rt①❴♥ r t ♦♣② sts r♠♣t② ♥♦ s tr♥r② ♥♠rs ♥ t ♥① ♦ t ♣r♦ ♦♣② ♥ t t♦r t ♦♣② t ♦♣② st (i, j, k) s st♦r s t t♦r ♠♥t ♦ ♥① i ·32 + j ·31 +k ·30❲ ts ♠♣ rrs❴♦r♥❴♠♣

♣ tr ♦ t ♥t ❲ ♠♣♠♥t rtr♦♥ ♥ ♦rr t♦ ♥ t♦ ♦♥rt t tr♥t♦♥ t♦ T

3c ♦ ♦ ♦r ♥ ♦♠♣t♥ ♥ T

3c

♠♥t♥ ♥ ①r② t strtr tt ♦♥t♥s s tt r t♦♦ ♦♥ ♦r♥ t♦rtr♦♥ s t strtr ♦♥② ♦♥t♥s t ♥ ♦♠♣t♥ ♥ T

33c ♦trs

t s ♠♣t② t ♦rs ♥ t ♦♦♥ ② s t strtr s st♠♣ t ②t②♣ ❱rt①❴♥ ♥ t②♣ stst❱rt①❴♥ ♥ s t♦♦ ♦♥♦♥ ♦ ts rts s st♦r s ② ♥ t ♦tr ♦♥ s ♦♥t♥ ♥ t ♦rrs♣♦♥♥st ❲ ts t strtr ♦♥❴s❴st s r

♥ ♦rr ♥♦t t♦ st♦r s t rqr ♦♠♣rs♦♥ ♥t♦♥ ♦r ❱rt①❴♥s♥ ♥ t ♦♥♥t♦♥ tt ❱rt①❴♥ ♥ ② s ②s s♠r t♥ ♥② ♦ t❱rt①❴♥s ♥ t ♦rrs♣♦♥♥ st ♦r t ♦♠♣rs♦♥ ♥t♦♥ ♦♠♣r t♠♠♦r② rsss ♦ t ❱rt①❴♥s

❲♥ r ♦♠♣t♥ ♥ T33c t♠ ♥srt ♥ ♣♦♥t ♦r r♠♦ rt①

♣t t ♦♥❴s❴st ♥ s ♦ ♣♦♥t ♥srt♦♥ r♠♦ s ♦ t

♠♣♠♥tt♦♥

r ♦♥❴s❴st

♦♥t r♦♥ tt r ♦♥t♥ ♥ t ♦♥❴s❴st ♥ ♦r ♦ t ♥② ♥srt s tr t s ♦♥r t♥ t trs♦ ♥ rtr♦♥ ♥ s♦ ♥srt t ♥t♦ t ♦♥❴s❴st ♦t tt ts ♣♣r♦ r♠♦s ♥ r♥srtst s♠ s r♦♠ t ♦♥r② ♦ t ♦ s r♥♥t ♦r t s qtt t♦ tst ♦r ♥ tr t s ♦♥ t ♦♥r② ♦ t ♦ t tr♥s ♦tt♦ str t♦ r♠♦ ♥ r♥srt s♦♠ s r♥♥t② rtr t♥ tst♥ ♦r ♦t♠ tr t② ♦♥ t ♦♥r② ♦ t ♦

♥ s ♦ rt① r♠♦ ①t② t s♠ ♣♣r♦ s s

♦♥rs♦♥ r♦♠ T33c

t♦ T3c ♥ t ♦♥❴s❴st s ♠♣t② t rr♥t tr♥

t♦♥ ♥ ♦♥rt t♦ T3c ♦ ♦ s♦ trt ♦r t♠s ♦r s ♥ ♦♥ ♦r

rts s s♦ r

r s tt t s r t t② r ♥♦t ♥♦♥t rs♣t t♦ t ♦r♥ ♦♠♥ ♦ T

3c ♥ t s♥s ♦ ♦♥♥t♦♥

♦r ♥♠r s rrt t ♥♦r ♣♦♥trs t♦ t ♥ ♥♦rs ss♦♥ ♥ r t② ♣♦♥t t♦ ♠r ♥ ♥♦rs ♥ ♦♥ s♥ t ♦r♥❴♠♣

♦t tt rts t t② r ♥♦t ♦r♥ rts t ♣r♦♦♣s ♦r ♥♠r s rrt t rt① ♥s tt ♣♦♥t t♦ rtstt r t♦ t t♦ t ♦rrs♣♦♥♥ ♦r♥ rts

t ♠r s

❱rt① t rts tt r ♥♦t ♦r♥ rts

♦♥rs♦♥ r♦♠ T3c t♦ T

33c

♦r♥ t♦ rtr♦♥ ♥ ♦♠♣t♥ ♥ T3c

♥r ♥ t♦ st t♦ T33c

st ♣ ♥srt♥ ♣♦♥ts ♦r rt①r♠♦ ♥ ♠ t ♥ssr② t♦ st t♦ T

33c ♦ r♥ rt① r♠♦ ♥

♥ ♦♠♣t♥ ♥ T3c ♥ t♦ tt t ♥② rt s ♦ ♥♦t s

tt r ♦♥r t♥ t trs♦ ♥ rtr♦♥

♣r♦ tr♥t♦♥s

r strt♦♥ ♦♦r s t t rts tt t r ♥♦t r♥ rr♦s ①♠♣r② ♥t t ♥ ♥② rt♦♥s

♦♥rs♦♥ r♦♠ T3c t♦ T

33c

tr♥s ♦t t♦ ♠♦r ♦♠♣t t♥ t ♦♥rs♦♥r♦♠ T

33c

t♦ T3c sr ♦ t s ♦♥ ♥ t ♦♦♥ ② rst rt ♦♣s

♦ rt① ♥ ♦♥strt t ♦r♥❴♠♣ ♥ t rrs❴♦r♥❴♠♣ trt♦r s ♥ st♦r t rt① ♦sts ♥ ♥♦r ♦sts ♥ s♦♠ t♠♣♦rr② tstrtrs ♦ rt ♣r♦ ♦♣s ♦ ♦t♥♥ tr rts r♦♠ trrs❴♦r♥❴♠♣ s♥ t rt① ♦sts r♦♠ t t♠♣♦rr② t strtr ♦ttt t ♦r♥ s r ♦♠♣t② ♦♥t♥ ♥s t ♦r♥ ♦♠♥ ♦ T

33c s

t② ♥♦ tr♥ t♦ s t ♦sts ③r♦ ♥ t ♦♥s tt ♥♦♥③r♦ ♦sts♦r rts ♦ ♥♦♥③r♦ ♦sts ♠st r♣ ② tr ♣r♦ ♦♣s tt rs♣t ♦♣② st ♦r♥ t♦ t rrs❴♦r♥❴♠♣ t r♠♥s t♦ st t♥♦r rt♦♥s ♦ ♦t t ♥② rt s ♥ t ♦r♥ s t ♣♦♥trs♦ t rts t rt① ♦sts ♥ t s ♥ ♥② t♦ st ♣ t ♦♥❴s❴st

P♦♥t ♦t♦♥

♣♦♥t ♦t♦♥ s r♥t t♦ ♦r ♦♥② ♦r ♥♣t ♣♦♥ts ♥ ♥ t ♦r♥ ♦♠♥

r♠♠r♥ st♦st sr ♥ t♦♥ ♦rs ♥ ♣r♦ tr♥t♦♥s s s t r♣rs♥tt♦♥s ♥ E

3 ♦ t s♠♣s ♦r t ♦♠tr tsts t♠ r♦♠ ♦♥ t♦ t ♥①t ♥ t♦ ♣ tr ♦ t ♥♦r ♦sts♥ ♦rr t♦ ♠♥t♥ t ♦st ♦ t ♥♣t ♣♦♥t t rs♣t t♦ t rts ♦ t rr♥t s r

s strt t r♥♦♠ ♦ t tr♥t♦♥ ts strt♥ s ♥♦♥♠♣t② ♥trst♦♥ t t ♦♥r② ♦ t ♥♠♥t ♦♠♥ t ♥ ♣♣♥ ttt s r② ♦s t♦ t ♣♦♥t t♦ ♦t ♦r ♥ ♦♥t♥s t t ts ♥♦♥ r♣rs♥tt♦♥ ♥E

3 ♦r♥ t♦ ♦♥♥t♦♥ ♣♣rs ♦♥ t ♦♣♣♦st s ♦ t ♥♠♥t ♦♠♥s ♥ tt s② ② tst♥ tr t strt♥ s t st ♦♥ ♥♦♥③r♦ ♦st ♥ ts s ♣♥♥ ♦♥ t ♦♦r♥ts ♦ t ♣♦♥t strt t ♣♦♥t♦t♦♥ t t ♣r♦ ♦♣② ♦ t t t ♦♣♣♦st s ♦ t ♥♠♥t ♦♠♥ sr s ♣r♥ts t r♦♠ trrs♥ t ♦ ♦♠♥ t strt♥ ♥ t ♦♥t♥♥ t ♣♦♥t r t② ♦s s s ♦rs s♣② ♥s♥ t s♣t s♦rt♥ s t♦♥

♠♣♠♥tt♦♥

r strt♦♥ ❲♥ trrs♥ t s t ♦st ♦ t ♣♦♥tt rs♣t t♦ t rr♥t ♥s r♥ tr♥ strt♥ r tr♥ ♦♥t♥♥ t qr② ♣♦♥t q

r strt♦♥ s t ♣♦♥t s ♥ t t ♦ t ♥♠♥t ♦♠♥ strt t t ♠♦r ♣♣r♦♣rt ♣r♦ ♦♣② ♦ t ♥t r♥ tr♥strt♥ r tr♥ ♦♥t♥♥ t qr② ♣♦♥t q

P♦♥t ♥srt♦♥

r ♦♠♣t♥ ♥ T3c t♥ t ♣♦♥t ♥srt♦♥ ♥ ♠♣♠♥t ♥ ①t② t s♠

② s sr ♥ t♦♥ ♦r ♥ T33c t♥ ♣♦♥t ♠st ♥srt

t♠s ② t ♥srt♦♥ ♦rt♠ s sr ♥ ♦rt♠ ♥ t♦♥② ♠st♠♥t♥ t ♠♣s ♦r t ♣r♦ ♦♣s ♦ rts s s t ♦♥❴s❴st

s sr ♥ t♦♥ t ♥srt♦♥ st♣ rst ♥ts t s tt r ♥♦♥t t t ♥② ♥srt ♣♦♥t ♥ t ts ts s ♥ ♦ ♥ t tstrtr tt s r ② ♥ tr♥s t t ♥ ♣♦♥t s rt① t ♦♥r② ♦t ♦ ♥ t ♦♥r② ♦ t ♥♠♥t ♦♠♥ ♥♦♥♠♣t② ♥trst♦♥ s♦♠♦ t ♥ s ♥ t ♦ rts t ♥♦♥③r♦ ♦sts s r t♦ ♣♦ss t♦ ♦♠♣t ts ♦sts ② ♥ tr♦ t s ♦ts t ♦ ♣♥ tr ♦ t ♥♦r ♦sts t ts ♣♣r♦ ♦ qt ♦st② ♥tr♠s ♦ r♥♥♥ t♠ ❲ s t ♦♦♥ ♠♦r t♠♥t t ss s♣♥t♣♣r♦

• r♥ t st♣ ♦ ♥t②♥ s ♥ ♦♥t t t ♥ ♣♦♥t ♦♠♣t t♦sts ♦ rt① ♦ ♥ ♦♥t t rs♣t t♦ t ♥ ♣♦♥t s

♣r♦ tr♥t♦♥s

♦sts r t♠♣♦rr② st♦r ♥ t rts

• tr ♥ rt t ♥ s ♦♥② ♥ t♦ t t st♦r ♦sts r♦♠ trts t♦ ♦rrt② st t ♦sts ♥ t s

• tr ♣♦♥t ♥srt♦♥ t ♦sts tt r st♦r ♥ t rts t♦ r ♦♥ ♦rr t♦ ♥♦t ♥trr t tr ♣♦♥t ♥srt♦♥s

s ♣♣r♦ rqrs t♦ st♦r ♦♥ ♦st ♥ rt① ♥ ts rqrs t s ♦ s♦♠①tr ♠♠♦r② ♥ tt t ♥♠r ♦ rts ♥ tr♥t♦♥ s ♥r② s♠♦♠♣r t♦ t ♥♠r ♦ s s ♠♠ ts ♠♠♦r② ♦r s ♣ts s t ♦♥② s r ♦sts r tt t♦ rts

r strt♦♥ ♦ ♦♦r ♥s s♦ t ♦♥r② ♦ t ♦r♥ ♥srt♦♥ ♦ t ♣♦♥t ♦sts tt t♦ t ♦s rts r s♦♥ t②♥t ♦ t♦ tr♥st t rts ♥ ♦rr t♦ t t s ♣♦②♦♥

♦t tt ts ♣♣r♦ ♦♥② ♦rs t ♦ s ♦♠♦♠♦r♣ t♦ ♦s t ♥♦♥ ♦ s♠♣s ♦ t str ♦ t ♥② ♥srt ♣♦♥t s t strtrtr t ♣♦♥t ♥srt♦♥ s s♠♣ ♦♠♣① t♥ t ♦ s ♦♠♦♠♦r♣ t♦ ♠♠ ♥ ♦r♠

❱rt① r♠♦

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

♠♠r tt t rst st♣ ♦ t rt① r♠♦ ♦♥ssts ♥ t♥ ttrr tt ♦♥t♥ t rt① r♦♠ t tr♥t♦♥ ②♥ ♦ ② ♦r♠ ♣ ts ♦ s ②s ♦♠♦♠♦r♣ t♦ ♥tr♦r ♦ t ♦ ♠st tr♥t ♥ ts tr♥t♦♥ ♠st s ♥ t rr♥t t strtr s t♦ s ♦♠♦♠♦r♣ t♦ t tr♥t♦♥ ♦ t ♦ s ♦♠♣t s t ♥②tr♥t♦♥ ♦ E

3 ♥ ② t rts ♦ t ♦ s s ♦♥ s♥ t tr♥t♦♥ ♠♣♠♥tt♦♥

r st s ♥ ♠♣♠♥tt♦♥ ss tr♥t♦♥ ♦ t ♦ ♠st s♥t♦ t ♦ rt ② t♥ t rt① r♦♠ t ♣r♦ tr♥t♦♥ ♦r ttr♥t♦♥ t strtrs ♦ tr♥t♦♥ ♥ ♣r♦ tr♥t♦♥r ♥♦t t②♣ ♦♠♣t t s s ♦r t ♣r♦ tr♥t♦♥s r♥t rt①♥ sss ♦r st♦r♥ t ♦sts r rqr ♦ t s ♥♦t s♥t t♦ ♦♣② t

♠♣♠♥tt♦♥

♣r♠ts r♥ t s♥ ♣r♦ss t t② ♠st s♦ ♦♥rt t♦ ♦ts ♦ trt t②♣

t♦♥② ♥ ♦♠♣t♥ ♥ T33c

t ♦♥❴s❴st ♥s t♦ ♣t s s t ♦r♥❴♠♣ ♥ t rrs❴♦r♥❴♠♣

❲♥ ♦♠♣t♥ ♥ T3c t♦ r② tr r♠♦ tt tr r ♥♦ s tt

r t♦♦ ♦♥ ♦r♥ t♦ rtr♦♥ trs t tr♥t♦♥ s t♦ ♦♥rtt♦ T

33c

ss

s sr ♥ t♦♥ t rts st♦r t ♦rrs♣♦♥♥ ♣♦♥t ♦♦r♥ts t ♦♦r♥ts ♦ t rt① ♦ ♥① ♥ ♥ ss ② rt①♣♦♥t♦r ∈ 0, 1, 2, 3 ♦r ♥ ♦rr t♦ ♣r♦t s♠♣① ♥t♦ E

3 t ♦sts ♥ t ♥t ♦ t ♦r♥ ♦♠♥ ♠st t♥ ♥t♦ ♦♥t r r t♦ t♥s t♦ ♦♥sr ♦r♥ ♦♠♥ ♦ T

3c s r♥t r♦♠ t ♦r♥ ♦♠♥ ♦ T

33c

❲♥ ♦♠♣t♥ ♥ T33c t ♣♦♥t ♦♦r♥ts ♦ts ♦ Dc ♠st ♦♠♣t s♥ t

♦r♥❴♠♣ ♦ ts t♥ts r♦♠ t sr ♥tr♦ ♥ ♠♠r♥t♦♥ ♣♦♥t t♦ t ♣r♦ tr♥t♦♥s t♦ rt①♣♦♥t ♥♥♦ r♣ ② t♦ ts♣♦♥t ♥st ♦ t♥ t ♣♦♥t ♦♦r♥ts r♦♠ t tr♥t♦♥ t strtr rt② t ♥t♦♥ ♣♦♥t s♦ tst ♦sts r♦♠ t t strtr ♥ t ♣♦♥t ♦♦r♥ts ♥ ♦♣② sts r♦♠ t♦r♥❴♠♣ t tr♥t♦♥ s r♣rs♥t ♥ T

33c

♦ ss t s♠♣s ♦ t tr♥t♦♥ t trt♦rs ♥ rt♦rs ♦ t tr♥t♦♥ t strtr r ♦r ♥♦t tt t tr♥t♦♥ t strtr♦s ♥♦t t ♥♦r♠t♦♥ tr t st♦rs tr♥t♦♥ ♦ T

3c ♦r ♦ T

33c t

tr♥t♦♥ s r♣rs♥t ♥ T33c t trt♦r rtr♥s ♦♣s ♦ s♠♣① t s

♥♦t ♣♦ss t♦ tr ♦t ♦♣s t t t strtr s s♦ s ♠t♠trs♦♥ ❲ ♦♠♣t ♥ T

33c

s t tr♥t♦♥ ♦s ♥♦t ①st ♥ T3c trt♦rs

♦ rtr♥ ♦♥② ♦♥ ♦♣② ♦ s♠♣① ts ♦ t♦ ♥♦♥sst♥s ♥ t♥② ♥ ♥♥ rt♦♥s s rtr♥ s♠♣s ♦ ♥♦t ♦r♠ tr♥t♦♥s r ♦r ♥ ①♠♣ ♦r s♦♠ ♣♣t♦♥s t s s t♦ ♦♥② ♦♥ ♦♣② ♦ s♠♣① ♥ t rtr♥ st ♦ s♠♣s ♦s ♥♦t ♦r♠ tr♥t♦♥

r strt♦♥ ♥ trt♦r rtr♥♥ ♦♥② ♦♥ ♣r♦ ♦♣② ♦ s♠♣①rtr♥s ♦r ♥st♥ t pqr t ♦♥② ♦♥ ♦ ts tr ♥t s

♦ ♣r♦ s ♥t♦♥t② t♦t ♥♦♥sst♥s s♦♠ ♦♠tr trt♦rs

♣r♦ tr♥t♦♥s

tt ♦♥② ♦t♣t s♠♣s t♦t ♥② ♥ ♥♥ rt♦♥s ♥ ♣r♦♠s t♦s s♦♥ ♥ r ♥♥♦t ♦r s ♦♠tr trt♦rs ①st ♦r ♦r t②♣s♦ s♠♣s ♣♦♥ts s♠♥ts tr♥s ♥ ttrr r r ♦r r♥t ♦♣t♦♥st♦ s♣② s♠♣s rtr♥ s s♦ r

• trts ♦r t s♠♣s tt r st♦r t s♦s t s♠ ♦rs t trt♦rs r♦♠ t tr♥t♦♥ t strtr

• ❴❱❴ trts ♦r s♠♣s tt ♥♦♥♠♣t② ♥trst♦♥t t ♥♠♥t ♦♠♥ ♦ t t t♦rs t tr♥t♦♥ s st♦r ♥ ♦♣s ♦ t ♦r♥ ♦♠♥ ♥ s ♦ T

33c s ♠♥s tt t s♠♣s tt

♥trst t ♦♥r② ♦ t rr♥t ♥♠♥t ♦♠♥ rtr♥ srt♠s

• ❯❯ trts ♦♥② ♦r t ♦r♥ ♦♣② ♦ s♠♣①

• ❯❯❴❱❴ trts ♦♥② ♦r t♦s s♠♣s tt ♥♦♥♠♣t② ♥trst♦♥ t t ♦r♥ ♦♠♥ s ♦r ❴❱❴ ♦♥r② s♠♣s rtr♥ sr t♠s

r ♦r ♠♦s ♦r t ♦♠tr trt♦rs ♦♣ t ♦♣rt ❴❱❴ ♦tt♦♠ t ❯❯ ♦tt♦♠ rt ❯❯❴❱❴r t tr♥t♦♥ s r♣rs♥t ♥ T

33c

♥ t ttrr ♦t♣t ② t trt♦rs rs♦♥

♣t♠③t♦♥s

r r sr ♦♣t♠③t♦♥s ♦r tr♥t♦♥ ♦♠♣tt♦♥ s sr ♥ t♦♥ s ♦♣t♠③t♦♥s ♥ ♣t t♦ ♣r♦ tr♥t♦♥s

♠♣♠♥tt♦♥

rr② s sr ♥ t♦♥ t tr♥t♦♥ rr② ♠♥t♥s ♥t♥♠r ♦ ②rs ♦ tr♥t♦♥s r t ♣♣r ②rs r ♦rsr t♥ t ♦r ②rs♦ ♦♥♥t t ②rs rt① s ♣♦♥tr t♦ ts ♦rrs♣♦♥♥ rt① ♥ t ②r♦ ♦♥ ♥ ♥ t ②r ♦ ♣ tr s ♦♥ ♥ t ②r ♦ ♦t tt rt① s ♦rrs♣♦♥♥ rt① ♥ t ②r ♦ s♦ t ♦♥♣♦♥tr ②s ♣♦♥ts t♦s♦♠ rt① ①♣t ♦r rts ♥ t ♦r♠♦st ②r

♦r ♣r♦ tr♥t♦♥s t s ② ♣r♦ tt t ♦rsr ♣♣r ②rs rr♣rs♥t ♥ T

33c

t ♥r ♦r ②rs r r♣rs♥t ♥ T3c ❲ ♥① t ②rs

r♦♠ ♦tt♦♠ t♦ t♦♣ t ♦r♠♦st ②r s ♥① t ②r i t ♣♣r♠♦st②r r♣rs♥t ♥ T

3c s♦ ②r i + 1 s r♣rs♥t ♥ T

33c ♥ t ♣r♦ ♦♣s

♦ ♦r♥ rts ♥ ②r i + 1 ♦ ♥♦t ♦rrs♣♦♥♥ rts ♥ t ♦r ②r i♥ ts tr ♦♥♣♦♥tr s ♥♦t st ♦ ♦ ♥♥ ♦♥♣♦♥trs t t♦♥♣♦♥trs ♦ ♣r♦ ♦♣s ♣♦♥t t♦ t ♦r♥ rt① ♥st Pr♦ ♦♣s ♦♥♦t ♣♣♦♥trs s t② r ♥♦t rqr s r

r strt♦♥ ♣ ♥ ♦♥♣♦♥trs ♦ ♣r♦ ♦♣s ♣♦♥t t♦ t ♦r♥rt① ♦♥ ②r ♦r

♣t s♦rt♥ s♣t s♦rt♥ s sr ♥ t♦♥ ♥ rs t♦t♥② ♠♦t♦♥ ♦t♣t sq♥ ♦ t s♣t s♦rt♥ s t ♣r♦♣rt② tt t♦♣♦♥ts tt r ♦s ♥ t sq♥ s♠ ♥ st♥ ♥ E

d s t ♥♣t♣♦♥t st ♦r t ♣r♦ tr♥t♦♥ s ♥ Dc s sst ♦ E

d ts ♣r♦♣rt②r♠♥s tr

t♦♥ ♥t♦♥t②

❲ ♣r♦ s♦♠ t♦♥ ♥t♦♥t② ♥t♦♥t② tt s ♥♦t rqr t♦ ♦♠♣tt ♥② tr♥t♦♥ ts t s s ♦r t sr ♥tr

♦r ♦♥rs♦♥ t♥ ♦r♥ s♣s s s♦♥ ♥ t♦♥ t rtr♦♥ s s♥t t ♥♦t ♥ssr② t♦ tr t rr♥t ♣♦♥t st ♥s ♥② tr♥t♦♥ ♦ T

3c t s ♣♦♥t st ♥ ♥ tr♥t♦♥ ♥ T

3c ♥ rtr♦♥ s

♥♦t ♥ssr② ♦♥t♦♥ s ♥ ② ♦r♠ ♦r t s ♣♦ss tttr ♥ ♣♦♥t t♦ t ♣♦♥t st t ♦s ♥♦t ♥ tr♥t♦♥ ♥ T

3c ♥②♠♦r

rtss ♦r tr♥t♦♥ tt s ♥♦t ♦♥ t♦ ♥ ♥②♠♦r t ♠t s t♦♦♥rt t r♦♠ T

33c

t♦ T3c ♥t♦♥ s❴tr♥t♦♥❴♥❴❴st ♠♣♠♥ts t

♣r♦ tr♥t♦♥s

tst ♦r t ♦♥t♦♥ ♥ ♦r♠ ♥t♦♥s ♦♥rt❴t♦❴❴st❴♦r♥♥ ♦♥rt❴t♦❴❴st❴♦r♥ ♥ s t♦ ♦♥rt t♥ T

3c ♥ T

33c

❱♦r♦♥♦ r♠ s ♥ t ♥② tr♥t♦♥ ♣r♦ ♥t♦♥t②♦r ♦♠♣t♥ ❱♦r♦♥♦ r♠s ♦t tt t ♦♠♣tt♦♥ ♦ ❱♦r♦♥♦ r♠s rqrs♦♠tr ♦♥strt♦♥s ♥ ts s ①t ♦♥② t r♥ ♣r♦s ①t ♦♥strt♦♥s

♦ rtr♥ t ❱♦r♦♥♦ r♠ ♦♥② ♥ ♥t♦♥s tt ♦r ♥ s♠♣① rtr♥ts s ②s t ♦♦♥ t②♣s ♦ ♦♠tr ♦ts

♥② ❱♦r♦♥♦

P♦♥tt ♠♥t P♦②♦♥❱rt① P♦②r♦♥

♦t tt ♣♦②♦♥s ♥ ♣♦②r r ♥♦t s♠♣s ♥ ♦ ♥♦t ♦♥st♥t s③ s♦tr r ♥♦ r♥ ♦ts ♥ st t♦ r♣rs♥t t♠ ♥ t ♥②tr♥t♦♥s ♦♥② ♥t♦♥s ♦r ts ♥ s r s s♥t t♦sr t ❱♦r♦♥♦ r♠ s ♦ rts ♥ s ♥ ♦♥strt s②r♦♠ ts ♦t♣t ♥ t ♣r♦ s t ♥t♦♥s ♦r ts ♥ s r s♥tt♦♦ ♦r sr♥ t ♣r♦ ❱♦r♦♥♦ r♠ ♦r ♦r t ♣♦②♦♥s ♥ ♣♦②r s♦♠ rts ♠t ♥ t♦ tr♥st ♥ ♦rr t♦ ♠♣ t ♣♦②r♥ ♣♦②♦♥s ♥t♦ E

3 t s t ♠♦r ♥♦ t♦ ♦♠♣t t ♦sts tt tr♠♥ts tr♥st♦♥s s♥ t ♥ t ♥t♦♥s ♦♥② r♦r s♦ ♣r♦ ♥t♦♥s ♦r rts ♥ s tt s t ♥♥ rt♦♥s ♥ t ♥②tr♥t♦♥ t♦ r♦♥strt t rt① ♦sts ♦ t ♣♦②r ♥ ♣♦②♦♥s s s ♠♦r♥t t♥ ♥ ♦ ♣♣r♦ s♥ t ♥t♦♥s ♦r ts ♥ s ♦♥② ♥t♦♥s ♦r rts ♥ s ♦t♣t t rs♣t ♦ts s ♣♦♥t sts tr♦ ♥♦t♣t trt♦r s t ♦♠♣tt♦♥ ♦ ❱♦r♦♥♦ rt① rqrs ♦♠tr ♦♥strt♦♥tr s ♥♦ rs♦♥ t♦ ♦ rtr ♦♠tr ♦♥strt♦♥s ♦r tr♥st♥ ♣♦♥ts s rtr♥ t tr♥st ♣♦♥ts ♥st ♦ ♣rs ♦ ♣♦♥ts ♥ ♦sts

❲ ♠♣♠♥t ♣r ♥t♦♥ ♣r♦❴r♠♥tr tt ♦♠♣ts t r♠♥tr ♦r ♥ ♥ rtr♥s t ♣♦♥t ♥s t ♦♠♥ ♥ ♥ ♦st tt s♣st ♣r♦ ♦♣② ♦ ts ♣♦♥t tt ♦rrs♣♦♥s ♦♠tr② t♦ t r♠♥tr ♦ t♥♦♥ r♣rs♥tt♦♥ ♦ t ♥♣t ♥ E

3 s ♦st ♥♥♦t ♥rr rt②r♦♠ t ♦sts ♦ t s tr s ♥♦ rt♦♥ t♥ t ♦st ♦ r♠♥tr♦ ♥ t ♦sts ♦ t ts s r s ♦♠tr tsts ♠st ♣♣ t♦ ♥ t rt ♦st ♥ −1, 0, 13

t② ♥t♦♥s t♥ ♦r s ♦♦s

tr♥s t r♠♥tr ♦ t ♥sDc s ♦♠♣t ② ♣r♦❴r♠♥tr

t tr♥s t s ♣r ♦ ts ♥♣♦♥ts r♣rs♥t ② ♣♦♥t♦st ♣rs♦♦♥ ♦♥♥t♦♥

♦ ♥ s ♣♦②♦♥ ❲ rtr♥ ♥ ♦rr st ♦ ♣♦♥ts tt ♥ts ♣♦②♦♥ ♦t tt ts ♣♦♥ts ♦ ♥♦t ♥ssr② ♥s t ♦♠♥ ❲t t ♣♦②♦♥ s ♦♦s rt ♦r s ♥♥t t♦ t ♦r ♦♠♣t t r♠♥tr ♥ t rqr ♦st t rs♣t t♦ ♦♥ ♦ trts ♦ t ♥♣t s♥ t ♣r ♥t♦♥ ♣r♦❴r♠♥tr

♠♣♠♥tt♦♥

r strt♦♥ ♦sts ♦ tr♥ ♦ ♥♦t tr♠♥ t ♦st ♦ tsr♠♥tr

❱rt① ♦ rt① s ♣♦②r♦♥ ❲ rtr♥ st ♦ ♣♦♥ts tt ♥ ts♣♦②r♦♥ s tr ♦♥① s ♦ ts ♣♦♥ts ♦ ♥♦t ♥ssr② ♥s t ♦♠♥ t r ♦s♥ s tt tr ♦♥① s ♣r♦t♦♥ ♦ t ♦ t rt① ♥t♦ E

3 ♥ ♦rr t♦ ♦t♥ t ♣♦②r♦♥ ts t ♥②rt♦♥s ♥ ①trt r♦♠ t ♥② tr♥t♦♥

♦♠♣①t②

s s♥ ♥ t♦♥ ♦rt♠ s t s♠ r♥♦♠③ ♦rsts ♦♠♣①t②s t ♦rt♠ ♦r ♦♠♣t♥ ♥② tr♥t♦♥s ♥ E

3 ♦r ts s ♥♦t tr♦r t ♠♣♠♥tt♦♥ s ①♣♥ ♥ t♦♥ ♥ ♦♠♣t♥ ♥ T

33c

t ♦♦r♥ts ♦ ♣♦♥ts tt ♦ ♥♦t ♥ Dc r ♥♦t st♦r ♥ t tr♥t♦♥ t strtr♦r ♠♠♦r② ♥② rs♦♥s ② r tr♦ t t♦♥ t strtr♦r♥❴♠♣ tt ♦s ♥♦t ♦ ♦r ♦♥st♥t t♠ ss ♥ t♦ ♦ ♠♠♦r② ♦r ♥ ♦♠♣t♥ ♥ T

33c

♦ ♥♦t ♠r t t♦♦ ♦♥ s ♥ t t strtrrt② t ♣ tr ♦ t♠ ♥ t ♦♥❴s❴st s sr ♥ t♦♥ s t ♦r♥❴♠♣ ts t strtr ♦s ♥♦t ♦ ♦r ♦♥st♥t t♠ ss ♥ tsst♦♥ ①♠♥ t ♠♣t ♦ t ♦r♥❴♠♣ ♥ t ♦♥❴s❴st ♦♥ t t♠♦♠♣①t② ♦ t ♠♣♠♥tt♦♥

s ♦ ts t♦ t♦♥ t strtrs s ♦ ♠♦tt ② ♣rt♦♥srt♦♥s ♦r ♠♦st ♣♣t♦♥s t s♠s ♠ ♠♦r ♠♣♦rt♥t t♦ ② t♠♥ s♣ ♥t ♥ ♦♠♣t♥ ♥ T

3c rs t ♥② ♦ ♦♠♣tt♦♥ ♥ T

33c

sss ♠♣♦rt♥t ♥ ♦rr t♦ t ♦♥ ♦ O(n2) ♦♥ t ♦♠♣①t② s ♣r♦ ♥t♦♥ ♦ ♥ t♦ st♦r t ♥♦r♠t♦♥ ♦♥t♥ ♥ t ♦r♥❴♠♣ ♥t ♦♥❴s❴st rt② ♥ t tr♥t♦♥ t strtr t♦ ♦t♥ ♦♥st♥t t♠ss ♦r t ♠♠♦r② rsr ♦r t t ♠♠rs rqr t♦ st♦r ts t ♥t s ♥ rts ♦ s♦ rsr ♥ t tr♥t♦♥ s r♣rs♥t ♥ T

3c

s♥ ♥♦♥♥② r ♠♠♦r② ♦♥s♠♣t♦♥ ♦ t tr♥t♦♥ t strtrs ♥ t♦♥ t ①♣t ♦♠♣①t② ♦ t r♥♦♠③ ♦rt♠ ♦r

♦rsts ♥♣t ①♣t ♦♠♣①t② s t ①♣tt♦♥ ♦r t ♦♠♣①t②♦r ♣♦ss ♥srt♦♥ ♦rrs

t S st ♦ n rts ♥ t e ♥ m ♥♦t t ♥♠r ♦ s ♥ s ♦ t♥② tr♥t♦♥ ♥ ② S s ♦♠♣t ② ♦rt♠ t N ♥♦t t♥♠r ♦ rts ♥ M t ♥♠r ♦ s ♦ t tr♥t♦♥ ♦r rtr♦♥ ♣♣s ♦r ♦rt♠ sts t♦ t T

3c t tr♥t♦♥ ♦ n ♣♦♥ts s

♦♠♣①t②

st ♦♠♣t ♥ T33c t♥ N = n ♥ M = m ♥s♦♥s O(1) ⊆ O(N) ⊆ O(n) ♥

O(1) ⊆ O(M) ⊆ O(m) ♦ ss t♦ rts ♥ t ♦r♥❴♠♣ s ♥ O(log N)t ss t♦ s ♥ t ♦♥❴s❴st ♦♥ ♦ t rts ♠st ♦♥ rst s ♥ O(log N) ♥ t♥ t s♦♥ rt① ♠st sr ♦r ♥ t ♦rrs♣♦♥♥ st s ♥ O(N) s t ♦r ♦♠♣①t② ♦r ss t♦ t ♦♥❴s❴st s O(N)

♠♠ t T tr♥t♦♥ ♦ ♦♠♣t 3♠♥♦ t♦t ♦♥r② tts n rts ♥ m s ♥ t ♦♦♥ t♦ stt♠♥ts ♦

♥♠r ♦ s ♦ T s ♥ ② e := n + m

♥♠r ♦ rts s ♥♦t rr t♥ t ♥♠r ♦ s n ≤ m

Pr♦♦ t f ♥♦t t ♥♠r ♦ ts r rtrst χ ♦ ♥② 3♠♥♦t♦t ♦♥r② s ❬♥❪ ♥ s ♥ ② t ♦r♠

χ = n − e + f − m.

rtr♠♦r ♦r tr♥t♦♥ ♥♦ tt 2f = 4m ♣♣②♥ ts t♦ t ♦♦r♠ ②s 0 = n − e + 2m − m ♥ ts e = n + m ♣r♦s t rst stt♠♥t t s tr s ♥ s ♥♥t t♦ t st tr ts s♦ 3f ≥ 3eP♥ ts ♥t♦ t ♦r♠ ♦ ②s 0 = n − e + f − m ≥ n − e + e − m = n − m♥ ts n ≤ m

r♦♠ ♠♠ ♥ ♥♦♥ tr♥t♦♥ ♣r♦♣rts t♦♥ ♥♦tt

O(1) ⊆ O(n) ⊆ O(m) = O(e) ⊆ O(n2)

♥ O(N) ⊆ O(M)

♥t♥♥ ♦ t ♦♥❴s❴st ♥ t♦♥ s♦ tt t t♦t ♥♠r♦ s tt r ♥srt ♥t♦ ♥ r♠♦ r♦♠ t ♦♥❴s❴st s ♣r♦♣♦rt♦♥ t♦t s♠♣s tt r rt ♥ str♦② ② t ♦rt♠ tr t ♦♥rs♦♥ r♦♠T

33c

t♦ T3c t ♦♥❴s❴st s ♥♦t s ♥②♠♦r s♦ t ♥♠r ♦ ♥s t♦ t

♦♥❴s❴st s ♣r♦♣♦rt♦♥ t♦ N2 s ♥ ♦r ♠♣♠♥tt♦♥ ss t♦ rt①s ♥ O(N) t ♠♥t♥♥ ♦ t ♦♥❴s❴st s ♥ O(N3)

❲ ♥♦ sss t ♥t♦♥ts ♣rs♥t ♥ t♦♥

♦♥rs♦♥ r♦♠ T33c

t♦ T3c ♠♦st ①♣♥s st♣s r t ♦r trt♦♥s ♦r

M s r ♦ t♠ rqr t♦ r ♦♣② sts r♦♠ t ♦r♥❴♠♣ s ♥O(log N) ♦r ♦♠♣①t② s O(M log N)

♦♥rs♦♥ r♦♠ T3c t♦ T

33c

♥ t ♠♦st ①♣♥s st♣s r t trt♦♥s ♦r s ❯♥ ♦ ♥ s♦♠ ♦ ts trt♦♥s ♠♥ts ♥ t♦ ♥srt ♦r ♦♥ ♥t ♦ ♦r♥ ♠♣ ♦♥ s s ♥ O(log M)

♦♥strt♦♥ ♦ t ♦r♥❴♠♣ s ♥ O(N log N)t♦♥② ♥ t ♥ ♥ t♦ trt ♦r s ♥ ♦rr t♦ ♥ t s tt

r t♦♦ ♦♥ ♦r♥ t♦ ♠♠ tr r O(M) ♠♥② s ♥srt♦♥ ♦s ♥t♦ t ♦♥❴s❴st s ♥ O(N) ♥ ts t ♦r ♦♠♣①t② ♦ t ♥t♦♥r♠♥s O(MN)

♠♣♠♥tt♦♥

P♦♥t ♦t♦♥ ❲♥ ♦♠♣t♥ ♥ T33c ♦r t rts ♦ st t ♦♣②

sts ♠st ♦♠♣t O(log N) ♥ t ♦rsts O(M) s ♠st st ②s O(M log N) ❲♥ ♦♠♣t♥ ♥ T

3c t s ♦♥ ♦ O(m) ♣♣s

P♦♥t ♥srt♦♥ ❲♥ ♦♠♣t♥ ♥ T33c t sr♦t♥ tt ♥s t s ♥ ♦♥t

s t♦ st O(N) s ♥ t ♦rst s ♥ t s t♦ t t ♦♣② sts ♦r t rts♦ ♦ ts s s♦ ts ts O(N log N) t♦♥② t ♦♥❴s❴st ♠st ♣t ♦r ♣ t♦ O(N) s s♦ t ♦r ♦♥ s O(N2) ❲♥ ♦♠♣t♥ ♥ T

3c

t ♦♠♣①t② r♠♥s O(n)

❱rt① r♠♦ ♦♠♣①t② ♦ t rt① r♠♦ s ♥ O(n2) ♥ t ♦rst s❲♥ ♦♠♣t♥ ♥ T

33c

t ♦♣② sts ♦r t ♦ rts ♠st ①trt r♦♠ t♦r♥❴♠♣ s ♥ O(N log N) ♥ ♣ t♦ O(N) s ♠st r♠♦ r♦♠ t♦♥❴s❴st s ♥ O(N2) ②♥ ♥ ♦r ♦rsts ♦♠♣①t② ♦ O(N2)

rr② ♥ t♦♥ sss tt t tr♥t♦♥ rr② ♦ ❬❪ ♥ s ♦r ♣r♦ tr♥t♦♥s t♦♦ r ♥②③ t ♠♣t ♦ ♦♠♣t♥ ♥ T

33c

♦♥ t ♦♠♣①t② ♦ ♦t ♣♦♥t ♦t♦♥ ♥ ♣♦♥t ♥srt♦♥ ♥ s♥ t ♥②rr②

t α ♥♦t t rt♦ t♥ t ①♣t ♥♠r ♦ ♣♦♥ts ♥ t♦ ♥t ②rs♦r♥ t♦ ❬❪ t ①♣t ♦♠♣①t② ♦ t ♣♦♥t ♦t♦♥ ♥ s♦♠ ②r i sO(αi+1n) ♦r♥ t♦ t sss♦♥ ♦♥ t ♦♠♣①t② ♦ t ♣♦♥t ♦t♦♥ ②r i sr♣rs♥t ♥ T

33c t ①♣t ♦♠♣①t② ♦ t ♣♦♥t ♦t♦♥ O(αi+1n log N)

t ②r I t ♦r♠♦st ②r tt s r♣rs♥t ♥ T33c ♥ t t♦t ♦♠♣①t②

♦ t ♣♦♥t ♦t♦♥ s

i<I∑

i=0

αi+1n +

i=∞∑

i=I

αi+1n log N = n ·(

α(1 − αI)

1 − α+

αI+1

1 − αlog N

)∈ O(n log N)

♦t tt t ♦♥t αI+1

1−α ♦ log N t r② s♠ t r♦♥ Ir♥ ♣♦♥t ♥srt♦♥ t♦♥② t ♣ ♥ ♦♥ ♣♦♥trs ♠st st ♦r ②rs

r♣rs♥t ♥ T3c ts t♦ ♦♣rt♦♥s r ♥ O(1) ♦r ②rs r♣rs♥t ♥ T

33c t

♦♥ ♣♦♥trs ♦ t ♣r♦ ♦♣s t♦ st t♦ t rs♣t ♦r♥ rt① ♥ t②r ♦ s rqrs qr② ♣r ♣♦♥tr ♥ t ♦r♥❴♠♣ s ♥ O(log N) ♣r♦t② ♦ ♥srt♥ ♣♦♥t ♥t♦ ②r i s αi s♦ t ①♣t ♥♠r ♦ ♥srt♦♥s♣r ♣♦♥t s

∑∞i=0 αi ∈ O(1)

s s♥ ♦ ♣♦♥t ♥srt♦♥ ♥t♦ t ♣r♦ ♥② tr♥t♦♥ s ♥ O(n +N2) ⊆ O(nN) ♥srt♥ ♥t♦ t rr② ♦s ♥♦t ♥ ts s♦ t r♥♦♠③ ♦rsts ♦♠♣①t② s O(n2N)

♠♠r② ♦r ♥ ♦r ♦ t rsts ♦ ts st♦♥ N s ♦♥st♥t t s②♠♣t♦t r♥♥♥ t♠ s t s♠ s ♦r ♦♠♣t♥ ♥②

tr♥t♦♥s ♥ E3 ♥♠② ♥ t ♦rst s O(n3) ♥ ♠♣r♦ t♦ r♥♦♠③

♦rsts O(n2) ② s♥ t rr② ♥ s t ♣♦♥t st ♦s ♥♦t tr♥t♦♥t♥ N = n ♥ log n t♦r s ②s ♦rsts ♦♠♣①t② ♥ O(n3 log n)r♥♦♠③ ♦rsts O(n2 log n) ♥ s♥ t rr②

②r t t s♠st ♥① ♥ rst ♥♠r ♦ ♣♦♥ts

t② ♦ ♥ tr♥t s♥

E3

T3

♦♥❴s❴st ♠♥t♥♥ O(N3)T

33c

→ T3c O(M log N)

T3c → T

33c

O(MN)P♦♥t ♦t♦♥ O(m) O(m + M log N)P♦♥t ♥srt♦♥ O(n) O(n + N2)❱rt① r♠♦ O(n2) O(n2)P♦♥t ♦t♦♥ t rr② O(n) O(n log N)

r♥t♦♥ O(n3) O(n3N) t rr② O(n2) O(n2N)

r ♦ t rsts ♦ t♦♥

t② ♦ ♥ tr♥t s♥

♥ ts t♦♥ sr ♥ ♦r ♥ tr♥t s♥ tt ♥trts t tr♥t♦♥s ♥ t ♣r♦ tr♥t♦♥s s tr♥t s♥ s♥ sr ♥ ❬❪ ♠♦tt♦♥ ♦r s ♥ ♣♣r♦ s tt t ♦rt♠s♦r ♦♠♣t♥ tr♥t♦♥s ♥ E

3 ♥ T3c r ♥♦t sst♥t② r♥t ♦ s t♦

♣t s tt ♥t♦♥s s ♣♦ss ♥ t ♦ ♦r t tr♥s ♦t tt ♠♦st ♦t ② ♦♣t♠③ ♥t♦♥s ♦ss ♥② ② ♥r③♥ t♠ s♥t② t♦ ♦r ♦r♦t s♣s

tr♥t s♥ s ♦♣ t♥ t ♦♦♥ ♣rtrts ♦ E3 ♥ T

3

♥t♦ ♦♥t

E3 rt① t ♥♥t② ♥rt ♠♥s♦♥s

T3 ♦r♥ s♣s ♦sts

♦ ♦ t tr♥t s♥ s t♦ ♠ t ♣♦ss t♦ s② ①t♥ t rr♥t♠♣♠♥tt♦♥ t♦ r♥t s♣s t t st ♣♦ss r♥♥② ♥ ♦ s ♦ t tr♥t s♥ s t♦ s♣t t ss r♥t♦♥❴ ♥t♦ tr sss rt② ♥rt♥ r ♠♥ s♣ ss ♣r♦s ♥t♦♥t② tt♣♥s ♦♥ t s♣ t s ♥♦ t♠♣t ♣r♠tr ♦ t tr♥t♦♥ t♦tr tt tr♥t♦♥ t strtr ♥ t ♦♠tr trts

r tr♥t s♥

♠♣♠♥tt♦♥

tr♥t♦♥ t strtr ♥ t ♦♠tr trts r t s♠ s sr ♥t♦♥ t s ♥♦ st t t sss ♥ ♠♦r t

♥r s s ss ♦♥t♥s t ♥t♦♥t② tt ♦s ♥tr ♣♥ ♦♥ ts♣ ♥♦r ♦♥ t tr♥t♦♥ t②♣ s r ♠♦st② r♣♣r ♥t♦♥s r♦♠ t tr♥t♦♥ t strtr ♥ ♦tr ♥r ♣r ♥t♦♥s

♠♥ s♣ s ss s ♥ ♥ t s♥ ♥ s s t♦ ♥ r②t♥tt ♣♥s ♦♥ t ♠♥ s♣ ♦ t tr♥t♦♥r s ♥t♦♥t② tt ♣♥s ♦♥ ♦t s♣ ♥ tr♥t♦♥ t②♣ ♥ ts ss s st♦rs ❬❱❪ tt ♠♦② t ♥t♦♥t② r♥t t♦ t tr♥t♦♥ t②♣♥ t s♣ ss

r♥t♦♥ ss tr♥t♦♥ ss ♣r♦s t s♠ ♥tr s ♥ t s♥ sr ♥ t♦♥ ♥♣♥♥t ♦ t ♠♥ s♣ t ♣r♦s ♥r♦rt♠s ♦r ♣♦♥t ♦t♦♥ ♣♦♥t ♥srt♦♥s ♥ ♣s s ss s ♥② s♣③ t♦♥② tr♥t♦♥ ♥ rr tr♥t♦♥

♥ rtr s♣s s qt s♠♣ t ts s♥ ♥② ♥ s♣ ss ♥ ♣♦ss② r♥t ♦♠tr trts ss s ♥ s♥ t ♦rt♠s ♣r♦ ② ttr♥t♦♥ sss r ② ♥r

rst ♦ tt ♠♥ ♥t♦♥s ♦ ♥ t s ss ♥ t rs♣ts♣ ♥ tr♥t♦♥ sss ♦♥t♥ ♦♥② st♦rs tt ♠♣♠♥t s♣ s♣ ♥tr♥t♦♥ s♣ ♥t♦♥t② ♦r ♥ ts ② ♦ ♦♥② ♠r ♦t ♦t ♦ ♥s ♦ ♦ t♦t ♦s♥ ♥② ♥② s t tr♥t♦♥sr ② ♦♣t♠③ ♥ ② s t s ♥♦t ♣t t♦ ♠♣♠♥t ♠♦t♦♥stt tr ♥rt② ♥st ♥② s ♠♥s tt ♠♦st t ♥t♦♥t② ♥s♣ ♥ t s♣ ss ♠♣s tt ♠♦st ♦ t ♦ ♣t ♥ts rsts ♣rrr ♥ ♦♠♣t② s♣rt ♠♣♠♥tt♦♥ ♦r t ♣r♦tr♥t♦♥s

①♣r♠♥ts

♥ t♦♥ ♥tr♦ tr ♠♣♠♥tt♦♥s ♦ ♦rt♠s t♦ ♦♠♣t t ♣r♦ ♥② tr♥t♦♥ ❯♥♦rt♥t② ♠♦st ♦ t♠ r tr ♥♦t ♣② ♦r ♥♦t ♠♥t♥ ♥②♠♦r ❲ ♦♠♣r ♦r ♠♣♠♥tt♦♥ ♦ t ♦rt♠ t♦♦♠♣t t ♣r♦ ♥② tr♥t♦♥ t♦ t ♠♣♠♥tt♦♥ ♦r t ♥② tr♥t♦♥ ♦ E

3 ❲ ♣rs♥t s♦♠ ①♣r♠♥ts ♦♠♣r♥ t ♦♠♣tt♦♥♦ t ♥② tr♥t♦♥s ♦ E

3 ♥ T3 ♥ ② t s♠ st ♦ ♣♦♥ts ♥ D3

c♦r ♥♦r♠② strt ♥♣t ♣♦♥ts t ♦♠♣tt♦♥ ♦ t ♣r♦ ♥② tr

♥t♦♥ s ①♣t t♦ st② s♦r t♥ t ♦♠♣tt♦♥ ♦ t ♥② tr♥t♦♥ ♦ E

3 t♦ ♦r s ② ♠♥♥ t ♦sts ♥ t ♦♠♣tt♦♥ ♥♦r♥ s♣s

t rst s♦ ①♣r♠♥ts ♦♥ t ♦♥strt♦♥ ♦ t ♥② tr♥t♦♥ s♥r♥t ♦♥rt♦♥s ♦ t ♦ sr ♦♣t♠③t♦♥s t ♦r t♦t s ♦ ts♣t s♦rt♥ ♥ t ♠♠② ♣♦♥t st ♥♦tr tst srs rstrts ♦♥ t ♣♦♥t ♥srt♦♥♥ T

3c s t ♠♦st rt ♣rt ♥ t ♥② tr♥t♦♥ ♦♥strt♦♥ ❲ ♥srt

t ♣♦♥ts ♥t♦ ♣r♦♠♣t tr♥t♦♥ ♥ T3c ♥ ts ② ♦ ♦♠♣tt♦♥ ♥

T33c

t ♥ t s ♦ t ♠♠② ♣♦♥ts ts ♣♦♥ts r ♦s♥ t r♥♦♠ ♥ ♦♥♦t r♠♦ t♠ ♥ ♥ r② ♣rs♥t rsts ♦ t s♠ ①♣r♠♥ts s♥ t

①♣r♠♥ts

tr♥t♦♥ rr② s ♣♦♥t♦t♦♥ tstrtr ♦♦♥ ①♣r♠♥t tstst rt① r♠♦ ♥t♦♥t② ♥ r♥ t ♥② tr♥t♦♥ ♦♥strt♦♥tsts ♥ t ♣♦♥t ♥srt♦♥ tsts s♥ ♦♠tr trts ss tt s s♣③ t♦ ts tt Dc s t ♥t ♥ ♥② ♦♠♣r t t♦ rtr ♦ t♦♥ s ♦ t ♦♣t♠③t♦♥s s s r♥t ♣r♦♣rts ts ①♣r♠♥ts ♣ t♦♥t② s♣ ♥ts ♥ s♦rt♦♠♥s ♥ ♦r ♠♣♠♥tt♦♥

♥♣t ♣♦♥t sts

❲ r♥ ①♣r♠♥ts ♦♥ ♦t ♥rt t ♥ r t r♦♠ str♦♥♦♠② ② ♦rts②♦ Pr♦ ♥ ♥ ❲rt ♥ ♥♠rs t ♦r♥ ♦♠♥ s t ♦♣♥♥t [0, 1)3

❲ ♥♦t t t sts s ♦♦s

♣♦♥ts ♥♦r♠② strt ♥ t ♦♣♥ ♥t

s♣ ♣♦♥ts ♥♦r♠② strt ♦♥ s♣r ♦ rs 0.5 ♥tr t (0.5, 0.5, 0.5)

str ♥♣t t r♦♠ rsr ♦♥ str♦♥♦♠②

❲ ♥♦ sss t r♥t ①♣r♠♥ts ♥ t r♦s ♦ t ts tr♥♥♥ t♠s ♦r ♥♣t ♣♦♥t sts t ♥♠r ♦ ♥♣t ♣♦♥ts ♥ ♥ t rst ♦♠♥ r♥♥♥ t♠ ♦r ♦ ♥② tr♥t♦♥ ♦♠♣tt♦♥ ♦ E

3 ♥ T3c s s♦♥

♥ t s♦♥ ♥ tr ♦♠♥ rs♣t② rtr♠♦r ♥ t st ♦♠♥ t t♦r ② t ♦♠♣tt♦♥ ♦ t ♥② tr♥t♦♥ ♦ T

3c s s♦r s ♥ r♥♥♥

t♠s r ♥ ♥ s♦♥s ①♣r♠♥ts r r♥ ♦♥ ♥ ♥t ❳♦♥ ♦r t ③r♥♥♥ ♦r ♥ ❲ ♦rt tsts tt t♦♦ ♦♥r t♥ s♦♥s♦rt tsts r ♠r s ♥ t ts

♦♥strt♦♥ ♦ t ♥② tr♥t♦♥

❲ r♥ tr srs ♦ ①♣r♠♥ts ♥ t t t ♦♥rt♦♥ ♦♥ s♥ ts♣t s♦rt♥ ♥ ♦♥ s♥ ♦t s♣t s♦rt♥ ♥ t ♠♠② ♣♦♥t st

♦r ①♣r♠♥t st t t ♦r t ♣♦♥t sts str ♥ s♣ s s ♣♦t ♦ ts t ♦t tt t ♣♦t ①s r ♦t s ♦rt♠②

t ♦♥rt♦♥

E3

T3c t♦r

102

103

104

105

106

str E3

T3c t♦r

1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

♣t②♥ ♥sttt r♦♥♥♥ tr♥s

♠♣♠♥tt♦♥

s♣ E3

T3c t♦r

102

103

104

105

106

107

♥ ts ①♣r♠♥t s tt ♦r t ♣♦♥t st ♦ 102 ♣♦♥ts t s♦♦♥ t♦ t ♦r ♦ ♦♠♣t♥ ♥ T

33c

s qt r ♦r rr ♣♦♥t sts t s♦♦♥♦♠s s♠r s t ♥t ♦♠♣tt♦♥ ♥ T

33c

s ss ♠♣t ♦♥ t ♦rr♥♥♥ t♠ ♥② ♦r ♣♦♥t sts ♦ ♦t 106 ♣♦♥ts t ♦♠♣tt♦♥ ♦ ♣r♦tr♥t♦♥s ts ♦t t s♠ t♠ s t tr♥t♦♥ ♦ ♣♦♥t sts ♥ E

3 s♦♦♥ ♦♠s ♥

①♣r♠♥ts str s♦ tt t r♦r ♣♦♥t st s s♥t② strtt♦ ♦r s♠r t♦

♦r t s♣ ①♣r♠♥t t r s♦♦♥ ♣rssts ♥ ♥ r♦s r t ♣r♦♥② tr♥t♦♥ s ♥r r♣rs♥t ♥ T

33c

s ♦r ts ♣♦♥t st tr r②s s ♦♥r t♥ t trs♦ ♦ rtr♦♥ s t ♣r♦t♦♥ ♦ t st♦♥ ♦ t ♣r♦ ♥② tr♥t♦♥ ♦ t s♣r ♦♥t♦ T

3c s ♥ ♠♦st ss

②s ♦ ♥t ♥♥♦t ♥ ♦♣ t♦ ♦ ♠ ttr ♦r♦r r ♥♦t r♦ ♥② s ♥ r♦r ♣♣t♦♥s r s stt♦♥ ♦ ♦r r♦♥s♦♦♥ ♦r rr ♣♦♥t sts s t♦ ♥♥ t tr♥t♦♥ ♦ t ♦r♥ s♣r sr st♠♣s ♥ t♦♥ ♥t♦♥ s r ♥♦ s s t ♣r t♦ ♣② ♥ ♦rr t♦ ♦ s ♠ ♦r s ♣♦ss ♥ ♦♠♣t♥ ♥ T

3c

s r② t ♣r♥♣ ♦

❲t s♣t s♦rt♥

E3

T3c t♦r

102

103

104

105

106

107

str E3

T3c t♦r

1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

①♣r♠♥ts

s♣ E3

T3c t♦r

102

103

104

105

106

107

❲♥ s♥ t s♣t s♦rt♥ t ♦r ♦ t r♥♥♥ t♠s ♦♠s ♠♦st♥r ♥ t ♥♠r ♦ ♣♦♥ts ♣r♥♣ ♥ ♦♠s r♦♠ t t tt t ♣♦♥t♦t♦♥ st♣ ts rt ♦♥sr② ♣♦♥t ♦t♦♥ ♥t♦♥ s r② s♠rr♥♥♥ t♠s tr ♦♠♣t♥ ♥ E

3 ♦r T3c ♦ ♥ t tsts ♦ t t ♦♥rt♦♥

t ♣♦♥t ♦t♦♥ s ♥ tt♥t♥ t ♦♥ t ♦r r♥♥♥ t♠ r♥ ♥t s ♦ s♥ s♣t s♦rt♥ ♦♠♣t♥ ♣r♦ tr♥t♦♥ ♦ r ♥♠r ♦strt ♣♦♥ts s ♦t t♠s s♦r t♥ ♦♠♣t♥ tr♥t♦♥ ♦ E

3 rsts ♦ t str ①♣r♠♥t ♥ r t t ①♣r♠♥t s♣t s♦rt♥ ♦s ♥♦t ♦r r② ♥t② ♦♥ t st ♦ ♣♦♥ts ♦♥ s♣r ♥

t ♣r♦ s ts t s ♦t ② t ①♣♥s ♥♥ ♦ t ♦r♥ s♣

❲t ♠♠② ♣♦♥t st ♥ t s♣t s♦rt♥

E3

T3c t♦r

102

103

104

105

106

str E3

T3c t♦r

1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

s♣ E3

T3c t♦r

102

103

104

105

106

♠♣♠♥tt♦♥

♦r t ①♣r♠♥t t s♦♦♥ ♦r t st ♦ 102 ♣♦♥ts s ①♣♥ ②t t tt ts ♣♦♥t st ♦s ♥♦t ♥ ♥② tr♥t♦♥ ♦ T

3c ♥ ts t

♦rt♠ ♠st st t♦ ♦♠♣t♥ ♥ T33c

r♠♦♥ t ♠♠② ♣♦♥ts ♦r rr♣♦♥t sts ♥♦ ♦♠♣tt♦♥ ♥ T

33c

s rqr t ♥ s♦ t s♦♦♥ s ♠ ss t♦r s t♥ ♥ ♦t tt t rr t ♣♦♥t st t ss s ts♦♦♥ ♥ ①♣♥ ♥ t ♦♦♥ ② ♥ ♦r ♠♣♠♥tt♦♥ ♣r♦s♣③ ♥t♦♥s ♦r ♥ ♠♦r ♥t② t s tt ③r♦ ♦sts ♣rt♦ t s♦♦♥ ♥ T

3c s t♦ ♦st ♠♥♣t♦♥s r♥ t ♣♦♥t ♥srt♦♥ ♥ s

t ♥♦♥③r♦ ♦sts r ♥♦ s t ♣♦♥t st s ♥♦r♠② strt t ♣r♥t♦ s t ♥♦♥③r♦ ♦sts t ♥♠r ♦ s tt r ♦t ♦♥ t ♦♥r② ♦t ♦r♥ ♦♠♥ s s♠r ♦r r ♣♦♥t sts t♥ ♦r s♠ ♣♦♥t sts s♦ t rts♦♦♥ rss ♥ t ♥♠r ♦ ♣♦♥ts r♦s

♦r t s♣ ①♣r♠♥t t 104 ♣♦♥ts t s♦♦♥ s ss t♥ ♦r t s s ts ♣♣r♦ ♦s ♥srt♥ ♣♦♥ts ♥ T

33c ♥② t ♣♦st♣r♦ss♥ ♦ r♠♦♥

t ♠♠② ♣♦♥ts ♥ ♦♥rt♥ t tr♥t♦♥ t♦ T33c

s ①♣♥s s rst ssts tt ♦r ♣♦♥t sts ♦ ♠♦r t♥ ♦t ♣♦♥ts tt ♦ ♥♦t ♥ tr♥t♦♥♥ T

3c t s s t♦ s t ♠♠② ♣♦♥t st

P♦♥t ♥srt♦♥ ♥ T3c

r strt t ♣r♦♠♣t tr♥t♦♥ ♦ T3c r♦♠ 1000 ♥♦r♠② strt

♥♣t ♣♦♥ts ♥ ♥srt ♠♦r ♣♦♥ts ❲ ♦ ts t ♥ t♦t s♣t s♦rt♥ ♦ t♥♣t ♣♦♥ts ♦ ♦ ts ①♣r♠♥ts s t♦ ♦♠♣r t ♥srt♦♥ ♦ ♣♦♥ts ♥ T

3c

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

❲t♦t s♣t s♦rt♥

E3

T3c t♦r

102

103

104

105

106

str E3

T3c t♦r

1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

s♣ E3

T3c t♦r

102

103

104

105

106

①♣r♠♥ts

r s ♥ ♠♦r r② ♥ tr ①♣r♠♥ts tt t r t ♥♠r♦ ♣♦♥ts t ss s t s♦♦♥ ♦t tt s♦ t s ♦ ♣♦♥ts ♦♥ s♣r ♥♥♦ ♦♠♣t s② t s s tr r t♦♥ ♣♦♥ts ♥sr♥ tt ♦♠♣tt♦♥s ♥ ♦♥ ♥ T

3c rtr t♥ T

33c s t♦♥ ♣♦♥ts r ♦♥t♥

♥ t ♥ rst t♦

❲t s♣t s♦rt♥

E3

T3c t♦r

102

103

104

105

106

107

str E3

T3c t♦r

1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

s♣ E3

T3c t♦r

102

103

104

105

106

107

s ①♣r♠♥t ♦♥r♠s t ♦ ①♣♥t♦♥s t♦t ♥ ♥ rsts

tr♥t♦♥ rr②

❲ r♥ t s♠ ①♣r♠♥ts s ♦ s♥ t tr♥t♦♥ rr② ♥ t ♣r♦ s t rr② srs r♦♠ t t tt tr r ②s ②rs r♣rs♥t♥ tr♥t♦♥ ♦ T

33c ❲ ♦♥② s♠♣ ♦ t rsts ♦r t ♣♦♥t st ♦ 106

♣♦♥ts

♠♣♠♥tt♦♥

E3

T3c t♦r

t s♣t s♦rt♥ ♠♠② ♣♦♥t st s♣t s♦rt♥ ♣♦♥t ♥srt♦♥ ♥ T

3c

♣♦♥t ♥srt♦♥ ♥ T3c s♣t s♦rt♥

♥ ♣rt ♦♥② t tr♥t♦♥ ♦ t ♣♣r♠♦st s r♣rs♥t ♥ T33c s♦ t

s♦♦♥ s t t♦r t♥ ♥ ♦♠♣rt② s♠ t tt ts♦♦♥ s s♠r ♦r ①♣r♠♥ts s♦s tt ♥ T

3c t tr♥t♦♥ rr②

s♦s s♠r ♦r s ♥ E3

❱rt① r♠♦

r r♠♦ rts r♦♠ tr♥t♦♥ ♦ t ♥ s③

E3

T3c t♦r

104

105

106

str E3

T3c t♦r

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

❲ s tt t r♥♥♥ t♠ ♦ r♠♦♥ rts ♦s ♥♦t ♠ ♣♥ ♦♥t s③ ♦ t tr♥t♦♥ ♦r ♦r t ♣r♦ tr♥t♦♥ s ♦♠♣rt②s♠ t ss s♦♦♥ ♦ t♦r t♥ ♥ ♥② t tr♥t♦♥♠st ♦♥rt t♦ T

33c

r♥ t r♠♦ t s♦♦♥ t♦r ♦s ♣ t♦ ♦t

♣ ♦r♥ ♦♠♥

♥ t ♣r♦ tr♥t♦♥s t ♦r♥ ♦♠♥ ♦ t ♣r♦ s♣ ♥ ♦s♥ ② t sr ❲♥ ♦♠♣t♥ ♣rts ♦ ♣♦♥t♦st ♣rs ♣♦♥ts ♠st tr♥st ② ♠t♣s ♦ t ♥t ♦ t ♦r♥ ♦♠♥ s s ♥♠r② ♠♦r♥♦ t♥ s♠♣② ①♥ t ♦r♥ ♦♠♥ t♦ t ♥t

❲ ♦ rt s♣ trts ss tt ♣r♠ts ♦♥② t ♥t s ♦r♥ ♦♠♥ ①♣r♠♥ts ♦ s♦ tt ♥ ts s ♥ ♦t ♦♥ t r♥♥♥ t♠ ♦r♦t 106 ♥♦r♠② strt ♣♦♥ts ♥ t ♥ t ♦s♠♦♦ t st ♦ 106

♣♦♥ts

T3c T

3c, c = [0, 1)3 ♣♣ ♥

st♥r

s♣t

♠♠② s♣t

♥s T3c

♥s T3c s♣t

①♣r♠♥ts

str T3c T

3c, c = [0, 1]3 ♣♣ ♥

st♥r

s♣t

♠♠② s♣t

♥s T3c

♥s T3c s♣t

❲ s tt t s♣♣ s r ♥ t ss ♥ s♣t s♦rt♥ s s s♥ ①♣♥ ② t t tt ♦r s♣③t♦♥ t♦ c = [0, 1)3 rts t ♣rtt♦♥ ♥ t ♣rt ♦ t r♥♥♥ t♠ s ♣ ♥ ♣rt t♦♥ s r ♥s♥ s♣t s♦rt♥ ❲ tt t rst♥ s♣♣ s ♥♦t s♥t t♦ ♥ts tr ♥ t ♣ rs

♦♠♣rs♦♥ ♦ t rtr ♦ t♦♥

♥ t ①♣r♠♥ts ♦ s♥ tt tr ♦♠♣t♥ ♥ T3c ♦r T

33c

s ♠♣t ♦♥ t r♥♥♥ t♠s r tst ♦r ♥ ♣♦♥t sts tr ♥srt♦♥ ♦ ♦ ♠♥②♣♦♥ts t tr♥t♦♥ s s♦♠ ♦ t ♦♥t♦♥s ♣rs♥t ♥ ♣tr ❲ s t♦♦♥ ♥♦tt♦♥ ♦r t r♥t ♦♥t♦♥s

② s s t ♥♠r ♦ ♣♦♥ts ♦r t ♣♦♥t st ♥s tr♥t♦♥♦ T

3c ♦r t rst t♠ ♥ssr② ♥ s♥t ♦♥t♦♥ ♦ ♦r♠ s

s t♦ ♦♠♣t ts ♣r♦♣rt② ♦t tt ② ♥ rtr ♣♦♥ts t ♣♦♥t st ♥♦s ts ♣r♦♣rt② ♥

♠tr s s t ♥♠r ♦ ♣♦♥ts ♦r t ♣♦♥t st s t♦♥t♦♥ ♦ rtr♦♥ ♦r t rst t♠ t s ♥ tr♥t♦♥ ♦ ts ♣♦♥tst t ♠tr ♦ t rst r♠sr♥ s s♠r t♥ t ♥t♦ t ♦r♥ ♦♠♥

♥t s s t ♥♠r ♦ ♣♦♥ts ♦r t ♣♦♥t st s t ♦♥t♦♥ ♦ rtr♦♥ t ♥t rtr♦♥

❲ r♥ t tsts ♦♥ t ♣♦♥t sts ♥ str ♦r t ♣♦♥t st s♣ ♦r ♥st♥ts rtr r ♥r ❲ r♥ ①♣r♠♥t t ♥ ♦♥ r♥♦♠ ♥srt♦♥♦rr ♥ ♦♥ s♥ s♣t s♦rt♥ s sr ♥ t♦♥

rsts t♦t t s ♦ s♣t s♦rt♥ r s♦♥ ♥ t t♦ ♦♦♥ ts

102

103

104

105

106

107

str 1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

♥ t t♦ ♦♦♥ ts t rst t t s ♦ s♣t s♦rt♥ r ♥

♠♣♠♥tt♦♥

102

103

104

105

106

107

str 1 · 103

1 · 104

1 · 105

2 · 105

5 · 105

1 · 106

2 · 106

t ♣♦t s♦s t ①♣r♠♥ts t♦t s♣t s♦rt♥ t rt ♣♦t s♦s t①♣r♠♥ts t s♣t s♦rt♥

tsts r r♥ t♠s ♥ t ♣rs♥t ♥♠rs r t rt♠t ♠♥s ♦ rsts s s ♦♥ s t rsts ♦ ♦♥ r♥ ♥ r② t♦ t r♥♦♠③t♦♥♥ t ♦rt♠

s ♦r♠ s ♥ssr② ♥ s♥t ♦♥t♦♥ ♦♥ tr t ♣♦♥t st♥s ♥② tr♥t♦♥ ♦ T

3c t ♥♠rs ♥ ♦♠♥ ♥ ♦♥sr s t

st ♣♦ss rst ♦t tt t ♦♥t♦♥ ♦ ♦r♠ s ♥♦t s♥t ♦r ♦r t♣r♣♦s s t♦ tr ♥② s♣rst ♦ t ♣♦♥t st s tr♥t♦♥ ♥T

3c ❲ ♥ ♦♥sr t rsts ♦ ♦♠♥ s ♦r ♦♥s ♦♥ t ♠♥♠♠ ♥♠r ♦

♣♦♥ts rqr t♦ st t♦ T3c ♥ s t t♦ ♠sr t qt② ♦ ♦r ♦♠tr rtr

s ♦ s♣t s♦rt♥ ♥rss t rqr ♥♠r ♦ ♣♦♥ts t s s♥♦ t ♣♦♥ts r ♥srt ♥ s♣ ♦rr ts t ts ♦♥r ♥t t ♣♦♥t st ss♥t② strt ♥st rsts ♦r t ♣♦♥t st str s♦ tt t♥♠r ♦ rqr ♣♦♥ts s r② s♥st t♦ t ♥♣t ♣♦♥t st

ts s♦ tt ♥r② ♣♦♥t sts ♦ ♦t r♥♦♠ ♣♦♥ts r② ♥ ♥② tr♥t♦♥ ♦ T

3c ♦r t ♥ ♣♣♥ tt ♥srt♥ ♣♦♥t ♥ s

tr♥t♦♥ s t♦ ♣♦♥t st tt ♦s ♥♦t ♥ ♥② tr♥t♦♥ ♦ T3c

♥②♠♦r ♥ ♥r ♦t ♣♦♥ts r ♥ssr② ♦r t rst ♠♣t② ♠tr t♦ s♠r t♥ ♥ ♥t ♦ t ♦r♥ ♦♠♥ ♦r♥ t♦ rtr♦♥ s ♣♦♥t sts ♥ ♥② tr♥t♦♥ ♦ T

3c ♥ rtr ♣♦♥ts ♦

r ♦♠♣t♥ t r♠♥tr ♦ ttrr♦♥ s ♠♦r ①♣♥s t♥ ♦♠♣t♥ tst♥ ♦ t♦ ♣♦♥ts t s ② ♦rt♠ tsts ♦r t ♦♥st ♥t t♦ s♠r t♥ 1√

6c s rtr♦♥ s ts s str♦♥r rtr♦♥ ♦t t♦

♣♦♥ts r rqr ♦r t ♦rt♠ t♦ st t♦ ♦♠♣t♥ ♥ t st ♦r♥s♣ ❲♥ t t st ♦♠s r ts ♥♠r ♥ ♦♥sr ♥

♣♣t♦♥s

♣♣t♦♥s

t♦ t ♠♦rt② ♦ t s s② ♣♦ss t♦ ♣ r♥t ♥t♦♥ts t♦tr ♦r ♥st♥ tr r sr ♦rt♠s tt s tr♥t♦♥s t ♣s♣ ♦♠♣tt♦♥ ❬❨❪ ♦r t ♠s♥ ♦rt♠s ❬❨ ❨❪

♦r t ♥tr ♦ t ♠♣♠♥tt♦♥ ♦ ♣r♦ tr♥t♦♥s s s♦♠♠t♠t② ♠♦tt r♥s t♦ t ♥tr ♦ t ♥♦♥♣r♦ tr♥t♦♥ss ♥ t♦ ♥tr♦ ♣trs t♦ ♠ t ♣r♦ tr♥t♦♥s ♦♠♣t t♦♦tr ♦rt♠s

Pr♦ ♣ s♣s

♣ s♣s ♥ sr ♥ ❬ ❪ s ①t♥s♦♥ s ♥ ♠♦tt ②t ♥ ♦ str♦♥♦♠rs ♦r ♦♠♣t♥ ♣r♦ ♣ s♣s ♦ s♠t♦♥s ♦♥ t ♦s♠ ❬❲❱P+❪ s ♦♥ t ♦s♠♦♦ ♣r♥♣ t♦♥ ts s♠t♦♥s rr♥ ♦♥ ♦♠♣rt② s♠ s♠♣ ♦ t ♦s♠ ♦r t ♦♥rs ♦ ts♠♣ ♥ sst♥t t ♦♥ t t♦♣♦♦② ♦ ts ♣ s♣s ② ♦♠♣t♥♣r♦ ♣ s♣s ts ts ♥ ♦

♥t♦♥ ❬❪ ♥ ♣♦♥t st S ♥ ♣r♠tr α t 0 < α < ∞ t♣ s♣ ♦ S s t sst ♦ s♠♣s ♦ DT (S) tt r♠sr♥ ♦rs α ♦r s♠r tt ♦s ♥♦t ♥② ♣♦♥t ♦ S ♥ ts ♥tr♦r

r♣ t ♥② tr♥t♦♥ ② t ♥② tr♥t♦♥ t♥ t t ♣ s♣s ♥t♦♥ rt② ①t♥s t♦ ♣r♦ tr♥t♦♥s

♠♣♠♥tt♦♥ ♦ ♣ s♣s ts ♥ ♠♣♠♥tt♦♥ ♦ tr♥t♦♥♦rt♠ s t♠♣t ♣r♠tr r♥ t ♦♥strt♦♥ ♦ t ♣ s♣ rst t t ♥② tr♥t♦♥ s ♦♠♣t ♥ t♥ t ♣ s♣ ♦♥ t♦♣ ♦ tt tts t♦ s♠♣① t s♠st α ♦r t ♣♣rs ♥ t ♣ s♣ ♥♣r♦s trt♦rs t♦ ♦t♣t ♣ s♣s ♦r ♥ α

r② s♠ ♣♣r♦ ♦rs ♦r ♣r♦ tr♥t♦♥s ♣r♦tr♥t♦♥ ♠♣♠♥tt♦♥ ♦t♣ts tr♥t♦♥ ♦♥ t♦♣ ♦ t ♣ s♣♥ ♦♠♣t ♥ ♦rr t♦ ♦ s♦ t ss ♣❴s♣❴ ♠st ♥st♥tt tt ss Pr♦❴❴♥②❴tr♥t♦♥❴ s t♠♣t ♣r♠tr t s♦♠ts rs ♥ t ♠♣♠♥tt♦♥ tt r s♦ s ♦♦s ♦t tt ♣r♦t ♣ s♣s r ♥♦t ②t s tr s ♥♦ ♠♣♠♥tt♦♥ ♦ t♣r♦ ♥② tr♥t♦♥s ②t

♥ ♦rr t♦ sts ♦♠♣tt② t♥ t ♣ s♣ ♠♣♠♥tt♦♥ ♥t ♣r♦ tr♥t♦♥s t♦ r♥ s♦♠ t②♣s ♥ ♥t♦♥s tt①st ♥ t tr♥t♦♥s t ♥♦t ♥ t ♣r♦ tr♥t♦♥ss r ♠♥② ♥t♦♥s ♥ t t rt① t ♥♥t② ♥ ♥rt ♠♥s♦♥s♥ tr♥ ♦t t♦ tr ♦r t ♣r♦ s

• ②♣ ♥t❴❬s♠♣①❪❴trt♦r ♥ t ♣r♦ tr♥t♦♥ s♠♣s r♥t ts ts t②♣ ♥ ♥ t♦ t s♠ s t ❬♠♣①❪❴trt♦r

• ♥t♦♥ ♠♥s♦♥ ♥ t ♣r♦ tr♥t♦♥ r ②s ♦♠♣t♥ ♥ tr♠♥s♦♥s s ts ♥t♦♥ rtr♥s ②s

♠♣♠♥tt♦♥

• ♥t♦♥ s❴♥♥t ♦r ♥ s♠♣① tr t s t rt① t♥♥t② s rt① s tr s ♥♦ rt① t ♥♥t② ♥ ♣r♦ tr♥t♦♥ ts♥t♦♥ ②s rtr♥s s

• ♥t♦♥ ♥♠r❴♦❴♥t❴❬s♠♣s❪ r r ♥♦ ♥♥t s♠♣s ♥ t ♣r♦ tr♥t♦♥ s♦ ts ♥t♦♥ st rtr♥s t ♦t♣t ♦ ♥♠r❴♦❴❬s♠♣s❪

rtr♠♦r s sr ♥ t♦♥ ♥♥♦t rt② ss ♣♦♥t ♦♦r♥ts♦ rts tr♦ t tr♥t♦♥ t strtr ♦ ♥ t ♣ s♣ ♠♣♠♥tt♦♥r② ♣♦♥t ♦♦r♥t ss ♦ t ♦r♠ rt①♣♦♥t s t♦ r♣② ♥ ①♣rss♦♥ ♦ t ♦r♠ ts♣♦♥t t♦♥

♥ t ♣r♦ ♣ s♣s r t s ♥trst♥ t♦ ♦♠♣t tr tt♥♠rs s ♥ ♦♠♣s s♥ t ♦rt♠ ♦ ❬❪ ♥ t ♦r ♦ ❬s❪r s♦s sr♥s♦t ♦ ♣r♦t♦t②♣ ♠♣♠♥tt♦♥

r r♥s♦t ♦ ♠♦ ♣r♦t♦t②♣ ♦r ♦♠♣t♥ ♣r♦ ♣ s♣s ♥ trtt ♥♠rs

Pr♦ sr ♠sr

rsts ♦ ts st♦♥ r ♦♥t ♦r t ❱ssr♦♥ s♦♣♦♦s ❬❪ sr ♠sr ♣r♦s ♥rt♦♥ ♦ tr♥r ♠ss t♦ ♣♣r♦①♠t

s♠♦♦t srs t ♠♣♠♥ts t ♠s♥ ♦rt♠ sr ♥ ❬❪ ♦r sr♠s ♦♠♣t ② s r

♦rt♠ ♦rs s ♦♦s t ♠♥t♥s ♥② tr♥t♦♥ ♥ q ♦s♦ sr ts t s sr t ts ❱♦r♦♥♦ ♥trsts tsr r s st ♦ s♦ r♥♠♥t rtr tt tr t s ♦♦♦r ♦rt♠ t♥ strts t ♥② tr♥t♦♥ ♦ E

3 ♥ ② st ♦♥t ♣♦♥ts ♦♥ t sr t ♠♥t♥s q ♦ sr ts ♥ ♥ st♣ ♥ ♣♦♥t ♦♥ t sr s t♦ r♥ t rr♥t t sr ♥② ♦ sr t s r♠sr♥ ♦ ts t tt s ♥tr t t ♥trst♦♥♦ t sr ts ❱♦r♦♥♦ t t sr s r ♥ ② ♦♥ t sr ts r♦♠ t q r r♥ tt s t ♥tr ♦ tr sr ♥② s ♥srt ♥t♦ t ♥② tr♥t♦♥ ♥ t ♥ sr ts r t♦ t q

♠♣♠♥tt♦♥ ♦ ts ♦rt♠ ♥ s sr ♥ ❬❨❪ t ♣r♦s t♦♦♥ tr r♥♠♥t rtr s t

♣♣t♦♥s

r sr ♠s ♦♠♣t ② ♠♦ ② ❱s ♦♠♣t♥ Pst②

• s♣t rtr♦♥ sr t s ♦♦ ts ♠♥♠♠ ♥ s rr t♥ s♦♠sr♥ trs♦

• ❯♥♦r♠ s③ rtr♦♥ sr t s ♦♦ t rs ♦ ts sr ♥② s s♠r t♥ s♦♠ sr♥ trs♦

• rtr rtr♦♥ sr t s ♦♦ t st♥ ♦ ts r♠♥tr t♦ t♥tr ♦ t r ♥② s s♠r t♥ s♦♠ sr♥ trs♦

t s ♦r t trs♦s r ♣r♦

r sr ♥② ♦ sr t

t rtr ♥ r♣ ② t srs ♦♥ rtr ♥ ♥ t♦t s t♠♣t ♥ t s♠ ② t ♥r②♥ ♥② tr♥t♦♥ ♠♣♠♥tt♦♥♥ ①♥

♥ ♦rr t♦ ♦♠♣t ♣r♦ sr ♠ss ♣ t ♣r♦ ♥② tr♥t♦♥s ♥ t sr ♠sr ❲♥ ♦♥ s♦ s♠r ♥tr ♣r♦♠s ♦rs ♦r t ♣r♦ ♣ s♣s s t♦♥ ♦st ♦ t♠ ♥ rs♦ ♥ ts♠ ② ♣r♦♠ ♣♣rs s ♥st ♦ s♥ t ♣♦♥t ♥srt♦♥ ♠t♦ ♦t ♣r♦ ♥② tr♥t♦♥ t sr ♠sr rst ♦♠♣ts t s ♥ ♦♥tt ts ♣♦♥t s♥ t ♠t♦ ♥❴♦♥ts ♥ t r♠♦s ts s ♥ s♥ s s♥ t ♠t♦ ♥srt❴♥❴♦ s sr ♥ t♦♥ ♥ ♦♠♣t♥ ♣r♦ tr♥t♦♥s t ♦sts ♦ t rts ②♥ ♦♥ t ♦♥r② ♦ t ♦

♠♣♠♥tt♦♥

t rs♣t t♦ t ♥ ♣♦♥t ♠st st♦r s ♥♦r♠t♦♥ s rt② st♦r ♥ trts ♥ ♠st ♥ ♣ tr r♥ ♦ ♥❴♦♥ts ♥ ♦rr t♦ ♥♦t ss ts tr ♦♥ s ♥♣ s ♦♥ ② ♥srt❴♥❴♦ s ♥ ♦♠♣t♥ ♣r♦ ♥② tr♥t♦♥ ♦r t♦ ♥❴♦♥ts tr ♠st ♦rrs♣♦♥♥ t♦ ♥srt❴♥❴♦ ♥ tr ♠st ♥♦t ♥② s t♦ ♥❴♦♥ts ♥t♥ sr ♠sr ♦r ♦s ♥♦t ②s ♥srt❴♥❴♦ tr ♦ ♥❴♦♥ts ♦ s ♠♦ rs♦♥ ♦ ♥❴♦♥ts tt ♥s ♣ trt① ♦st ♦r strt♥ t♦ tt s ♥ ♦♥t

♥② t t♦♥ ♦ t r♥♠♥t rtr rqr ♦♠tr ♦♠♣tt♦♥s ♦ tr♥♠♥t rtr ♠st ♣t ♥ ♦rr t♦ trt t ♦sts ♦rrt② ② r ♠ t♦ t ♦rt♠ tr♦ t♠♣t r♠♥ts s♦ t② ♥ r♣ s② ❲rt ♥ st ♦ r♥♠♥t rtr♦♥ sss ♦r t ♣r♦ s ♦t tt ♥ ♦rr t♦♦rrt② ♦♠♣t tr♥st ♣♦♥ts r♦♠ ♣♦♥t♦st ♣rs t r♥♠♥t rtr♦♥ sss♠st ss t♦ t ♥t ♦ t ♦r♥ ♦♠♥ ♦ t ♣r♦ tr♥t♦♥s ♣♦♥tr t♦ t ♣r♦ tr♥t♦♥ ♠st ♣r♦ t♦ t ♦♥strt♦r ♦ trtr♦♥ sss ♦♦♥ ♠♦t♦♥s ♠st ♠♣♠♥t ♦r t s♣ rtr

s♣t rtr♦♥

❯s t ♦sts t♦ ♦rrt② ♠ t ♥ t ♥t♦ R3 ♥ ♦rr t♦ ♦♠♣t ts ♠♥♠♠

♥ s r

r strt♦♥ ♣r♦ s♣t rt♦ rtr♦♥

❯♥♦r♠ s③ rtr♦♥

s rtr♦♥ s ♠♦r ♦♠♣t s t♦ ♦♠♣t t ♥tr ♦ t sr♥② ♦ ♥ t t s ♥♦t ♣♦ss t♦ ♥r t ♦st ♦ t ♥tr ♦ tsr ♥② r♦♠ t ♦sts st♦r ♥ t tr♥t♦♥ s ♥ s♥ ② s♠r ①♠♣ s ♥ r ♦♥ ♣ ♦r r♠sr♥ s ♦ s ❲ srt ♦st o tt ♠♥♠③s t st♥ t♥ ♦♥ rt① ♦ t t ♥ t ♦♠♣t♥tr t ♦st o ♦rrt♥ss ♦ ts ♣♣r♦ ♦♦s r♦♠ t t tt r♦♠♣t♥ tr♥t♦♥ s♥ ♦rt♠ ♥♦ tt t st ♠♣t② s ♠tr s♠r t♥ t ♦♠♥ ♥t ♦ ♦♥ ♦ t ♣r♦ ♦♣s ♦ t♥tr s ♦sr t♦ t t rts t♥ t ♦♠♥ ♥t ♥ t ♦trs rrtr ♥ tr♠s ♦ ♣rts ts ♣♣r♦ rqrs sr ♦♠♣rs♦♥s r ♦r ♥ strt♦♥ ♦ t ♣r♦ rs♦♥ ♦ ts rtr♦♥

rtr rtr♦♥

r t s♠ ♣r♦♠ s ♦r t ♥♦r♠ s③ rtr♦♥ ❲ ♥ ♣♣② t s♠

♣♣t♦♥s

r strt♦♥ ♣r♦ ♥♦r♠ s③ rtr♦♥

s♦t♦♥ s s r

r strt♦♥ ♣r♦ rtr rtr♦♥

♥ r s♦ s♦♠ ♣r♦ ♠ss ♦♠♣t t t sr ♠srs♥ t ♣r♦ ♥② tr♥t♦♥s rs s♦ t ♦♣s ♦ ♠s ❲ s tt t ♦♣s ♣rt② r t t ♦♥rs s ①♣t r♦♠t ② ♦ ♦♥strt♥ t ♠ss

r s♦s s♦♠ ♣r♦ sr ♠ss r③ ♣ ♥ t ②r♦ ♥t♦♥s r tr♣② ♣r♦ ♠♥♠ srs ❬❪ ②♥r s ♥♦t tr♣② ♣r♦ t♦♥② ♣r♦ ♥ ♦♥ rt♦♥ rtss t ♥ ♠s ❲ ♦t♥ t ♣s♥ ♥t♦♥ r♦♠ ♦s♥ t ❯ ♥ ♠ ♦ ♦rs ♦♥ ♦♥ s♦♥

♠♣♠♥tt♦♥ sr ♥ ts st♦♥ s ♥♦t ②t ♣s ♥

Pr♦ ♦♠ ♠sr

s ♦♥t♥t♦♥ ♦ t ♦r ♦♥ ♣r♦ sr ♠s♥ sr ♥ t♦♥ s♦♦r ♦♥ ♦♠♣t♥ ♣r♦ ♦♠ ♠ss s s ♦♥t ♦r t ♦♥♦

♣r♦♠s ♥ s♦t♦♥s ♥ ts s r r② s♠r t♦ t s sr ♥t♦♥ ❲ t♦ ♣t t s♠ tr r♥♠♥t rtr ♦♥ t sr tr♥s♥ t♦♥② t♦ rtr ♦♥ t s ♥ t ♦♠

• s rtr♦♥ ♣♣r ♦♥ ♦♥ t rt♦ t♥ t r♠rs ♥ ts♦rtst ♦ ttrr♦♥

• s rtr♦♥ ♣♣r ♦♥ ♦♥ t rs ♦ t r♠sr♥

♦♠♣tt♦♥ ♦ t t♦ rtr ♦r ♣r♦ ♠s♥ ♥ ♦♥ ♥ t s♠② s t ♦♠♣tt♦♥ ♦ t t rtr sr ♦ r ♦r ①♠♣s♦ ♣r♦ ♦♠ ♠ss ♦♠♣t t t ♦♠ ♠sr ♥ t ♣r♦tr♥t♦♥s

♠♣♠♥tt♦♥ sr ♥ ts st♦♥ s ♥♦t ②t ♣s ♥

♠♣♠♥tt♦♥

r ♦♠ ♣r♦ sr ♠ss r③ ♣ ♥t♦♥ ②r♦ ♥t♦♥ ②♥r ♣s ♥ ♥t♦♥ ♦rts② ♦s♥

Pr♦ ♦② ♦rt♠

♥ ❱♦r♦♥♦ r♠ ♦② trt♦♥ r♣s t sts ② t ♥tr♦s ♦ tr❱♦r♦♥♦ s ❬♦❪ ♦② ♦rt♠ ♦s r♣t ♦② trt♦♥s ❱♦r♦♥♦r♠ s ♥♦♥ t♦ ♦♥r t♦rs ♥tr♦ ❱♦r♦♥♦ r♠ ② r♣t② ♣♣②♥ ♦② trt♦♥s ❬❪ ♦② trt♦♥s r ♦r ♥st♥ s ♥ ♠s ♦♣t♠③t♦♥ ❬♦❪

r ♠♣♠♥tt♦♥ ♦rs s ♦♦s ♥ rst st♣ t ❱♦r♦♥♦ r♠ ♦ t ♥♣t♣♦♥ts s ♦♠♣t ♥ trt♦♥ t ♥tr♦ ♦ ❱♦r♦♥♦ s ♦♠♣t ♥ ♥♥ st ♦ s ♠♦ t♦ t ♥tr♦ ♦ t s♠ ♥ t ♠♣♠♥tt♦♥ t② r♦♠♣t t ❱♦r♦♥♦ r♠ ♦ t ♥tr♦s r♦♠ srt ❯s♥

♥t♦♥s ♦ t ♣r♦ ♥② tr♥t♦♥ t s strt♦rr t♦ ♠♣♠♥t ♣r♦ rs♦♥ ♦ t ♦② ♦rt♠ ♥t♦♥s rtr♥ t s s ♦♥① st♦ ♣♦♥ts ♥ E

3 s♦ t ♥tr♦ ♦♠♣tt♦♥ ♥ E3 ♥ rs ♥② ♥tr♦ s

♦ts ♦ Dc t ♠st tr♥st ♥s t ♦r♥ ♦♠♥ r ♦r ♥strt♦♥ ♦ ♦r ♣r♦ ♦② s♦tr tt s ♣② s ♠♦ ♥

♥ ♥ ♦♠♣t ♦② trt♦♥s ♦♥ ♣r♦ ❱♦r♦♥♦ r♠s r ♥trst♥ tr♠♥♥ tr t ♥♠r ♦ ♣♦♥ts s ♥ ♥♥ ♦♥ t ♠♥♠♠ ♥ ♠①

♣♣t♦♥s

r Pr♦ ♦♠ ♠ss r③ ♣ ♥t♦♥ ♣s ♥ ♥t♦♥♦rts② ♦s♥

r t ♥t ♣r♦ ❱♦r♦♥♦ r♠ ♦ r♥♦♠ ♣♦♥ts t ♦♥r♥tr♦ ❱♦r♦♥♦ r♠ tr ♦t ♦② trt♦♥s

♠♠ r ♥s ♦ t ♣r♦ ♥② tr♥t♦♥ ❲ ①♣t ♥ ♥sr t♦ts qst♦♥ t♦ s ♥ ♠s ♦♣t♠③t♦♥ ❬❱❪ ❲ t ♦♦♥ ①♣r♠♥t ❲♦♠♣t t ♣r♦ ♥② tr♥t♦♥ ♦ st ♦ n ♥♦r♠② strt r♥♦♠♣♦♥ts ♥ Dc ♥ r♥ ♦② trt♦♥s ♥t t ♠♥♠♠ ♥ ♠①♠♠ r♥ ♦ t♦ ♦♥st ♦② trt♦♥s ♥♦t r ② ♠♦r t♥ 0.01 ♦r 10 ♦♥st trt♦♥s ❲ r♥ ♥♣♥♥t ①♣r♠♥ts ♦r n t♥ 1 ♥ 100 sr

❯♥♦rt♥t② t rsts ♦ ts ①♣r♠♥t r ♥♦t r② ♦♥s ♦r s♠ ♣♦♥tsts r② ♦♦ ①tr♠ r ♥s ♥ tt♥ ♦r ♦r r ♣♦♥t stst rsts r ♠ ss r s♣② s♥ t rsts ♣♥ ② ♦♥ t strt♥♦♥rt♦♥ ♦r ♣♦♥t sts ♦ ♦t ♣♦♥ts ♥ ♦t ♣♦♥ts s tt s♦♠ ♦t strt♥ ♦♥rt♦♥s ♥ ② r② ♦♦ rsts

♠♣♠♥tt♦♥

020406080

100120140160180

0 10 20 30 40 50 60 70 80 90 100

max anglemin angle

angle

n

r ♥♠♠ ♥ ♠①♠♠ r ♥ ♦ ♣r♦ ♥② tr♥t♦♥♦ n ♣♦♥ts tr ♦② trt♦♥s

♦♥s♦♥

♥ ts ♣tr ♣rs♥t ♦r ♠♣♠♥tt♦♥ ♦ ♣r♦ tr♥t♦♥s tt s♣② tr♦ t ♦♣♥ s♦r rr② ❲ sr ♦r ♠♣♠♥tt♦♥s s t s♠rts ♥ t r♥s t♦ t tr♥t♦♥s ❲ ♥♦r ♦♥ t ♦♠♣①t② ♦ t ♠♥ ♥t♦♥s ♥ s♦♠ ♥trst♥ ♥t♦♥s tt rs♣ t♦ T

3 ♥ ①♣r♠♥ts ♦♥ ♦t ♥rt ♥ r♦r t r② t ♥②♦ ♦r ♠♣♠♥tt♦♥ ♥ t ♦r r♥♦♠ ♣♦♥t sts ♦r ♠♣♠♥tt♦♥ tr♥s ♦t t♦ ♦♥② ♦t t♠s s♦r t♥ t ♦♠♣tt♦♥ ♦ t ♥② tr♥t♦♥ ♦ E

3s s rt ♠♣r♦♠♥t ♦♠♣r t♦ t t♦r ♦ t st ♥ ♦♠♣t♥ t ♦♣s ♦ ♣♦♥t ♥② s♦ s♦♠ ①♠♣r② ♣♣t♦♥s r t ♣r♦ tr♥t♦♥s ♥ s ♥st ♦ t tr♥t♦♥s rqr♥ ♦♥② ♠♥♦r♣tt♦♥s

t♦r ♥ ♠♣r♦ t♦ s♥ trts ss s♣③ t♦ t ♥t

♣tr

♥② tr♥t♦♥s ♦ ♦tr

s♣s

♥ ts ♣tr ♥r③ t rsts ♦ ♣tr t♦ ♦tr s♣s ♦r s♣② ♦♥sr tr sss ♦ ♦rt s♣s

t s♣s s♣s ♦ ♦♥st♥t ♠♥♥♥ rtr ③r♦ s r ♦rt s♣s♦ E

d

s♣s ♦ ♦♥st♥t ♣♦st rtr ♦rt s♣s ♦ t d♠♥s♦♥ s♣rS

d

s♣s ♦ ♦♥st♥t ♥t rtr ♦rt s♣s ♦ t d♠♥s♦♥ ②♣r♦s♣ H

d

st ss s ♠ rr t♥ t t♦ ♦ ❲ ♦♥② ♣r♠♥r② sss♦♥♦♥ t s♣ s ♦ t ♦ t♦rs

Pr♠♥rs

♥♠r ♦ ♥t♦♥s ♥ ♣tr ♥ ♥r③ t♦ ♦r ♦♥ rtr r♦ ss♦ ♦rt s♣s ♥♥ s♣s ♦ r♥t rtr r ♥r③ rs♦♥s♦ ts ♥t♦♥s tt r rs ♥ t ssq♥t st♦♥s

t M d♠♥♦ t t ♦♦♥ ♣r♦♣rts

r s ♥t♦♥ ♦ t ♥② tr♥t♦♥ ♦ M ♥ ② st ♦ ♣♦♥ts ♥M

♥ ♦rt♠ ♦r ♦♠♣t♥ t ♥② tr♥t♦♥ ♦ M r♦♠ ♥ ♣♦♥t sts ♥♦♥

♥ ts ♣tr sss t ss ♦ M = Ed M = S

d ♥ M = Hd

t G srt r♦♣ ♦ s♦♠trs t♥ ♦♥ M ♥ X := M/G t ♦rt s♣ ♦ M

♥r t t♦♥ ♦ G t ♣r♦t♦♥ ♠♣ π : M → X ♦ t♦ ♦♥sr tr♥t♦♥s♦ X ♦r♥ t♦ ♥t♦♥ rst ♥t♦♥ ♦r s♠♣① ♥ X s s ♥♠♠t ①t♥s♦♥ ♦ t ♥t♦♥ ♦ s♠♣s ♥ t t t♦rs ♥t♦♥

♥② tr♥t♦♥s ♦ ♦tr s♣s

♥t♦♥ ♠♣① ♥ M t σ ks♠♣① ♥ M t rstrt♦♥ π|σ ♦ πt♦ σ s ♥t t ♠ ♦ σ ② π s ks♠♣① ♥ X

♥tt② ts ♥t♦♥ rqrs s♠♣s ♥♦t t♦ s♥trst ♥ t ♦rt s♣ ❲rt DT (Q) t♦ ♥♦t t ♥② tr♥t♦♥ ♦ M ♥ ② ♣♦♥t st Q t P ♣♦♥t st ♥ M ❲ ♥ ♥♦ ♣t ♥t♦♥ t♦ t ♥② tr♥t♦♥ ♦X ♥ ② π(P)

♥t♦♥ ♥② tr♥t♦♥ ♦ X π(DT (GP)) s tr♥t♦♥ ♦ X

ss♠s tt t s s♠♣ ♦♠♣① ♦r♥ t♦ ♥t♦♥ t♥ t t ♥② tr♥t♦♥ ♦ X ♥ ② π(P)

❲ ♥tr♦ s♦♠ s ♥♦t♦♥s r♦♠ r♦♣ t♦r② tt r s tr ♦♥ t G r♦♣ ♥ H ♥♦t sr♦♣ ♦ G H s ♥♦r♠ ♥ G t s ♥r♥t ♥r♦♥t♦♥ ♦r h ∈ H ♥ g ∈ G ghg−1 ∈ H ♦r r♦♣ ♠♥t g ∈ G tst gh | h ∈ H s ♦st ♦ H ♥ G ♥① ♦ sr♦♣ H ♥ G s ♥ st ♥♠r ♦ ♦sts ♦ H ♥ G

♥ t ssq♥t st♦♥s ♣r♦ ♣t rs♦♥s ♦ ♦r♠ ♥ ♣r♦ s♦♠♦♠tr tsts t♦ tr ♥ ♣♦♥t st ♥s tr♥t♦♥ ♥ X = M/G♦r r♥t sss ♦ s♣s M ♥ r♦♣s G

t s♣s

♥ ts st♦♥ ♦♥sr d♠♥s♦♥ t s♣s t t♦rs sss ♥ ♣tr s s♣ s ♦ t s♣s ♦♥sr ♥ ts st♦♥ ❲ t ♥② tr♥t♦♥ ♦t t t♦rs s t ♥s ♦ ♠♥② ♣♣t♦♥ s s♦♠ ♦ t♠ ♦♠♣tt♦♥♦♦② ❬r❪ rqr ♠♦r ♥r ♠♥♦s tt r ♦rt s♣s ♦ E

3 ♥r t t♦♥♦ ♦tr r②st♦r♣ r♦♣s

❲ rst ♥tr♦ ♦s ♥ d♠♥♦s ♥ tr ♣r♦♣rts t♦♥ t♥ sts ♥② tr♥t♦♥s ♦ ♦s ♥ d♠♥♦s ♥ s♦s s♥ tr t♦r♠ tt tr s ②s ♥t②st ♦r♥ s♣ ♦ t ♠♥♦♥ t ♥② tr♥t♦♥ s ♥ ♦r ♥② st ♦ ♣♦♥ts t♦♥ sssst ♥r③t♦♥ ♦ ♦rt♠ s ♣ t♦ ♦s ♥ ♠♥♦s ♦st♦♥♣ts ♠♥t♦♥ ♦ r t♥ r♦♠ ❬❪

♦s ♥ ♠♥♦s

♦s ♠♥♦ s ♦♠♣t ♠♥♦ t♦t ♦♥r② d♠♥♦ s ♥ ♦r t r② ♣♦♥t s ♥♦r♦♦ s♦♠tr t♦ ♥♦r♦♦ ♥ E

d d♠♥s♦♥ r r♦♣ GB s srt r♦♣ ♦ s♦♠trs ♦ E

d s ttt ♦rt s♣ E

d/GB s ♦♠♣t r♦♣s r s♦ r②st♦r♣ r♦♣s ♦rs♣ r♦♣s ❬❪

♦r♠ r ❬❪

• t GB d♠♥s♦♥ r r♦♣ r s r♦♣ GT ♦ d ♥r②♥♣♥♥t tr♥st♦♥s tt s ♥♦r♠ sr♦♣ ♦ GB ♦ ♥t ♥① r♦♣GT s tr♥st♦♥ sr♦♣ ♦ GB

t s♣s

• ♦r ♥② d tr s ♦♥② ♥t ♥♠r ♦ d♠♥s♦♥ r r♦♣s ♣ t♦s♦♠♦r♣s♠

♦t tt t ♦rt s♣ Ed/GB s ♥♦t ♥ssr② ♠♥♦ GB s ♣♦♥ts

① ts ♣♦♥ts ♦ ♥♦t ♥♦r♦♦ ♥ Ed/GB tt s ♦♠♦♠♦r♣ t♦

♥♦r♦♦ ♥ Ed ♦rt s♣ E

d/GB ♥ ②s sr ② t ♠♦r ♥r♦♥♣t ♦ ♥ ♦r♦ ❬P ❪ ♦r t ♦rt s♣ t♦ ♠♥♦ t r♦♣ ♦t♦♥ ♥♥ t ♠st ♥♦t ① ♣♦♥ts ♥ ♦tr ♦rs t r♦♣ ♦ t♦♥ ♠st t♦rs♦♥r t ♥tt② ♠st t ♦♥② ♠♥t ♦ ♥t ♦rr GT s sr♦♣♦ d ♥♣♥♥t tr♥st♦♥s ♦ GB t♥ E

d/GT s dt♦rs t s s♥t t♦ ♦♥srt♦rs♦♥r r r♦♣s t♦ ♦♠♣t② ss② ♦s ♥ ♠♥♦s

♦r♠ ❬❪ ♥② ♦s ♥ d♠♥♦ s q ♣ t♦ ♦♠♦r♣s♠t♦ ①t② ♦♥ ♦rt s♣ E

d/GB r GB s t♦rs♦♥r d♠♥s♦♥ rr♦♣

♦r♥ t♦ ♦r♠ tr r ♦♥② ♥t② ♠♥② d♠♥s♦♥ rr♦♣s ♣ t♦ s♦♠♦r♣s♠ ♥ ♠♥s♦♥ 2 tr r 17 ♥ ♠♥s♦♥ 3 tr r 230 ♥♠r ♦ r r♦♣s ② ♠♥s♦♥ s ss♥ t ♥ t ♥♥♥②♦♣ ♦ ♥tr q♥s ❬♦❪ t② r ♥♦♥ ♣ t♦ ♠♥s♦♥ ♥♠r ♦t♦rs♦♥r r r♦♣s s ss♥ t t② r ♥♦♥ ♣ t♦ ♠♥s♦♥

♥ t♦ ♠♥s♦♥s tr r ♦♥② t♦ t♦rs♦♥r r r♦♣s ♥ ts t♦♦s ♥ ♠♥♦s ♣ t♦ s♦♠♦r♣s♠ t t♦rs ♥ t ♥ ♦tt ♥ tr♠♥s♦♥s tr r ♦s ♥ ♠♥♦s ♦r ♦ r ♥♦♥♦r♥t sst♦♥ s ♥ ♥ ❬❲❪ ♥ ❬❪

• tr ♥r② ♥♣♥♥t tr♥st♦♥s t♦rs

• ♦♥ sr ♠♦t♦♥ ♥ t♦ ♥r② ♥♣♥♥t tr♥st♦♥s ♦rt♦♦♥ t♦ t sr ♠♦t♦♥ ♥ r♦tt ② π 2π/3 π/2 π/3

• tr ♦rt♦♦♥ sr ♠♦t♦♥s tt r♦tt ② π

• t♦ ♥r② ♥♣♥♥t rt♦♥s ♥ ♣♥ r r t♦ r♥t r♦♣s♥rt ♥ ts ②

• ♦♥ rt♦♥ ♥ ♦♥ sr ♠♦t♦♥ tt r♦tts ② π ♦t ♥ ①s ♣r t♦t rt♦♥ ♣♥ ♥ ♦rt♦♦♥ t♦ t tr♥st♦♥ ❲tr t sr ①s s♥s t rt♦♥ ♣♥ ♦r ♥♦t ②s t♦ r♥t ♠♥♦s

rst s① ♠♥♦s r ♦r♥t ♥ t ♦r st ♦♥s r ♥♦t s tr♥rt♦rs ♦♥t♥ rt♦♥s

r♥t♦♥s ♦ ♦s ♥ ♥♦s

t GF t♦rs♦♥r d♠♥s♦♥ r r♦♣ P ♥t ♣♦♥t st ♥ Ed X :=

Ed/GF ♦s ♥ ♠♥♦ t ♣r♦t♦♥ ♠♣ π : E

d → X ♥ DT (GFP) t♥② tr♥t♦♥ ♦ E

d ♥ ② t ♥♥t ♣♦♥t st GFP♦r t sss♦♥s ♦ ♥ t ♦♦♥ t♦ s

♥② tr♥t♦♥s ♦ ♦tr s♣s

♠♥♠♠ st♥ δ(G) ② r♦♣ G ♠♦s ♣♦♥t

δ(G) = minp∈Ed,g∈G,g 6=1G

st(p, gp),

r 1G ♥♦ts t ♥t ♠♥t ♦ G ♦t tt G s t♦rs♦♥r ♥ srtt♥ δ(G) > 0 ♦s

♠tr ∆(S) ♦ t rst d B ♥ Ed tt ♦s ♥♦t ♦♥t♥ ♥② ♣♦♥t ♦

st S ♥ ts ♥tr♦r

❲ ♥♦ ♥r③ ♦r♠ ♣

♦r♠ t st♦♥ ♦ π(DT (GFP)) ♦s ♥♦t ♦♥t♥ ②s ♦ ♥t ≤ 2t♥ π(DT (GFP)) s tr♥t♦♥ ♦ X

♦st ♣rts ♦ t ♣r♦♦ ♦ ♦r♠ r ♦♠♣t② ♦♠♥t♦r ♥ ♦ ♥♦t♣♥ ♦♥ t s♣ ♣r♦♦ ♦ ♠♠ s s ♦♥ t t tt t rstrt♦♥ ♦π t♦ s♠♣① s ♦♠♦♠♦r♣s♠ srt♦♥ ♦♦s rt② r♦♠ ♠♠ ♥ t s♠♣ ♦♠♣① ♥t♦♥ ♣r♦♦s ♦ ♠♠s ♥ r♥ s ♦♥ ♠♠ ♥ t ♣r♦♣rts ♦ π s ♣rt r♦♠ ♠♠ t♦ sss♦♥ ♦ t♦♥ rt② ♥r③s t♦ ♦s ♥ d♠♥♦s ❲♥♦ ♣r♦ t ♥r③ rs♦♥ ♦ ♠♠

♠♠ t K st ♦ s♠♣s ♥ Ed ♦s rts r ①t② t ♠♥ts ♦

GFP ♥ tt s ♦♥t♦♥s ♥ ♦ ♥t♦♥ ♥ t ♥② ♣r♦♣rt②t rs♣t t♦ GFP ♥ K stss t ♦ ♥t♥ss ♣r♦♣rt② s s Ks s♠♣ ♦♠♣①

Pr♦♦ ❲ rst ♦♥sr ♣♦♥t p tt s rt① ♥ K ♦t tt δ(GF ) > 0 ♥ ∆(GFP) <∞ ♦ s GF s t♦rs♦♥r r r♦♣ ♦♥st ♦ K s ♦♥② ∆(GFP) ♥ ts ♥② ♣♦♥t ♥♥t t♦ p ♠st ♥ ♦ rs ∆(GFP) ♥trt p s GF s srt t ♥♠r ♦ ♣♦♥ts ♦ GFP tt ♥s s s ♥t♦♦♥② p s ♥♥t t♦ ♦♥② ♥t② ♠♥② s♠♣s

t s ♥♦ ♦♥sr ♣♦♥t p ♥ Ed tt s ♥♦t rt① ♥ K t σ ♥♦t t s♠♣①

tt ♦♥t♥s p ♥ ts ♥tr♦r ♥ t vσ ♥♦t rt① ♦ σ t St(vσ) ♥♦t t st ♦s♠♣s tt vσ s ♥♥t t♦ ♦ s♦♥ tt St(vσ) ♦♥t♥s ♦♥② ♥t②♠♥② ♠♥ts st St(σ) ♦ s♠♣s tt σ s ♥♥t t♦ s sst ♦ St(vσ) tst s ♥t r s ♥♦r♦♦ U(p) tt s ♥♦♥♠♣t② ♥trst♦♥ t ①t② t♠♥ts St(σ)

rtr♦♥ ♠♥t♦♥s t trs♦ 12 ♣♥s ♦♥ t r♦♣ G ♥r

③ rs♦♥ ♦ ts ♦r♦r② ♦♦s ② s♠♣ ♦♠tr rs♦♥♥ r♦♠ ♦r♠

♦r♦r② ∆(GFP) < δ(GF )2 t♥ π(DT (GFP ′)) s tr♥t♦♥ ♦ X ♦r ♥②

♥t P ′ ⊇ P

♦r ♥② t♦rs♦♥r r r♦♣ tr r ♣♦♥t sts s tt t ♦♥t♦♥ ♦♦r♦r② s s δ s strt② ♣♦st ♥ ∆ ♥ ♠ rtrr②s♠ ② t ♦ ♦ t ♣♦♥t st

♥② ♥r③ rs♦♥ ♦ ♠♠

t s♣s

♠♠ r s ♥♦r♠ sr♦♣ GC ♦ GF ♦ ♥t ♥① s tt t ♣r♦t♦♥♦ t ♥② tr♥t♦♥ ♦ GFP ∪ GCQ ♥ E

d ♦♥t♦ XC = Ed/GC s tr♥t♦♥

♦r ♥② ♥t ♣♦♥t st P ♥ Ed ♥ ♥② Q ⊆ GF q t ♥② q ∈ E

d

Pr♦♦ ♦r♥ t♦ ♦r♠ tr s r♦♣ GT ♦ d ♥r② ♥♣♥♥t tr♥st♦♥s tt s ♥♦r♠ sr♦♣ ♦ GF t ♥t ♥① h′ ❲ ♦♦s ♥rt♦rs g1, . . . , gd

♦ GT ♥ t ♦♦♥ ② t g1 t s♦rtst tr♥st♦♥ ♥ GT t gi+1 t s♦rtst tr♥st♦♥ ♥ GT tt s ♥r② ♥♣♥♥t ♦ t tr♥st♦♥s g1, . . . , gi ♦t tt∆(GT p) ♦s ♥♦t ♣♥ ♦♥ s♣ ♦ ♦ p ♥ ts ♥ ♦♥sr ♦♥st♥t ❲♥ ♥ ♥ ♥tr ♦♥t c s tt ♦r gi t ♥qt② st(p, gc

i p) > 2∆(GT p)♦s ♦r ♥② p ∈ E

d r♦♣ GC ♥rt ② gc1, . . . , g

cd s sr♦♣ ♦ GT ♦ ♥①

cd t t ♣r♦♣rt② δ(GC) > 2∆(GT p) ♦r ♥② p ∈ Ed s GT s ♥♦r♠ ♥ GF

ggT g−1 ∈ GT ♦r g ∈ GF , gT ∈ GT ② ♦♥strt♦♥ ♦ GC tr s t♦♥ t♥t gT ∈ GT ♥ t gC ∈ GC ♥ ② gC = gc

T ♦ t s s② t♦ s tt GC s ♥♦r♠sr♦♣ ♦ GF t ♥① h = h′ · cd ♦t tt ∆(GCGFP) = ∆(GFP) ≤ ∆(GT p) ♦r ♥②p ∈ E

d s ∆(GCGFP) < δ(GC)2 ♦s ♥ ♦r♥ t♦ ♦r♦r② t ♣r♦t♦♥

♦ t ♥② tr♥t♦♥ ♦ GCGFP = GFP ♦♥t♦ XC ♦r♠s tr♥t♦♥ r♠♥s tr ♥ ♥ ♥ rtr ♣♦♥ts

♦t tt t ♣r♦♦ s ♦♥strt t srs ♦ t♦ ♦♥strt GC r♦♠ GT r♦♣ GT ♥ ♦♥strt r♦♠ GF s♥ t ♠strrr ♦rt♠ ❬♠❪ ♥ ♠♣♠♥tt♦♥ ♦ t ♠strrr ♦rt♠ s ♦r♥st♥ ♥ P ❬♣❪ ♠♠ ♠♥s tt tr ①sts s♣ XC ♥ t♣♦♥t st π(P) ♥s ♥② tr♥t♦♥ s♣ XC s ♦r♥ s♣ ♦ X t ♥t ♥♠r ♦ sts ❬r♠❪ ♠♠ ♥ s♦ ♥rst♦♦ ② ♦♥strt♥XC r♦♠ X rt② s ♦♦s

♦s ♥ d♠♥♦ s dt♦rs s ♦r♥ s♣ t ♥t ♥♠r ♦sts s ♦♦s r♦♠ ♦r♠ s sss ♦ ♥♠♥t ♦♠♥ ♦ tdt♦rs s d♠♥s♦♥ ②♣r♣r♣♣ ② ♥ t♦ ♦ ts ②♣r♣r♣♣st♦tr t ♥ ♦r♥ s♣ tt s ♥ dt♦rs ❲ ♥ ♦♥strt XC ②♥ s ♠♥② ♦♣s ♦ t ♥♠♥t ♦♠♥ s ♥ssr② t♦ t ♦♥t♦♥ ♥♦r♦r② ∆(GCGFP) = ∆(GFP) < δ(GC)

2 r ♦r ♥ strt♦♥ ♥t♦ ♠♥s♦♥s

r ♥t ♥♠r ♦ ♦♣s ♦ t ♥♠♥t ♦♠♥

s ♥ ①♠♣ ♦♥sr t t ♥ ♦tt E2/GK r GK s t r♦♣ ♥rt

② tr♥st♦♥ gt ♥ rt♦♥ gg tt s rt♦♥ t♦tr t tr♥st♦♥♣r t♦ t rt♦♥ ①s s r r♦♣ ♥rt ② gt ♥ g2

g s

♥② tr♥t♦♥s ♦ ♦tr s♣s

tr♥st♦♥ sr♦♣ ♦ GK ♦ ♥① ♦ ♥ ♦♦s sr♦♣ ♦ ts tr♥st♦♥sr♦♣ t ♥t ♥① tt s t ♦♥t♦♥ ♦ ♠♠ s ♥ r

r ♣rt ♦ t ♥♥t ♣♦♥t r GKp

♦t tt ♥ ♦t ♦r♦r② ♥ ♠♠ t t ♦♥t♦♥ ♦ t♦r♠ ∆ < δ

2 ♥ ♦r♦r② ♣t t ♣♦♥t st P t♦ rs ∆ ♥ ♠♠ ♣t t r♦♣ GC t♦ ♥rs δ

♦rt♠

♦rt♠ ♥r③s t♦ X = Ed/GF s♥ t rsts ♥ ♥ t ♣r♦s st♦♥

♦rt♠ s ♥r♠♥t t ♣♦♥ts ♦ P r ♦♥ ② ♦♥

• ♦rt♠ tr♠♥s GC r♦♠ GF s sr ♥ t♦♥

• ♥ t ♦rt♠ strts ♦♠♣t♥ ♥ t hst ♦r♥ s♣ XC = Ed/GC

s ♥ ♠♠ ♥srt♥ h ♦♣s ♣r ♥♣t ♣♦♥t r ♦♣② ♦ ♣♦♥tp ♥ ♠♥t ♦ ts ♦rt ♥r t t♦♥ ♦ t q♦t♥t r♦♣ GF /GC ♣♦♥tgp ♦r g ∈ GF /GC

• ♥ t ♦♥t♦♥ ♦ ♦r♦r② s ♠t ♦r t rr♥t ♣♦♥t st t ♦rt♠sts t♦ ♦♠♣t♥ ♥ X ♥ ♦♥t♥s t♦ ♥srt ♦ t r♠♥♥ ♣♦♥ts ♦♥②♦♥

P s s tt t ♦♥t♦♥ ♦ ♦r♦r② s ♥r t♥ t ♦rt♠rtr♥s t tr♥t♦♥ ♦ t ♦r♥ s♣ XC

♦ sss ♣♣r ♥♠② ♦ t♦ st♦r t rr♥t tr♥t♦♥ ♥ ♦ t♦ ♥srt ♣♦♥t

♣♥t t strtr

tr♥t♦♥ ♥ st♦r s r♣ ♥ t ♦♦♥ ② ♠♥s♦♥ s♠♣sr st♦r t st ♦ tr rts ♥ ♥♦rs rt① ♦♥t♥s t ♦♦r♥ts♦ t ♣♦♥t t ♦rrs♣♦♥s t♦ t♦♥② ds♠♣① st♦rs t ♥♦r♠t♦♥ ♦♥ ♦t♦ ♠♣ t s♦♠tr② ♥t♦ E

d ♥ ♣♣r♦♣rt ♠♥t ♦ t s♠♣① ♣r♠ ♥rt ♣r♦t♦♥ ♠♣ π ♦r ts ♣♣r♦ s ♥♦t r② s♣ ♥t s♥ ♦r r♠♥s♦♥s t ♥♠r ♦ ds♠♣s ♥ tr♥t♦♥ ♥ r♦ r② r ♠♦r s♣♥t ♣♣r♦ s t♦ st♦r t st♦♥ ❬❪ ♥ ts s ♠st sr ② ts t♦ rts t♦tr t tr ♦sts r t ♦sts r ♠♥ts ♦GC ♦r GF rs♣t② ds♠♣① ♥ t♥ ♦♥strt ② tr♥st♥ s stt tr ♦sts t ♦♠♠♦♥ rts r s r

♣r s♣s

r t r s t (10

)t♦ ♦t ♦sts ♦ e3 t♦ ♦r♠ t tr♥

pqr

P♦♥t ♥srt♦♥

♦r t ♣♦♥t ♥srt♦♥ t ♣♣r♦ ② ♦②r ❬♦❪ ♥ ❲ts♦♥ ❬❲t❪ ♥ s X s ♦r♥t t ♥srt♦♥ r♦t♥ ♥srs tt t strtr ♦♥② st♦rs♣♦st② ♦r♥t s♠♣s X s ♥♦♥♦r♥t ts s ♥♦t ♣♦ss ♥ ts s ♠st ♣♣② ♥ ♦r♥tt♦♥ tst ♦♥ t ♣r♠ ♥r π ♦ t s♠♣① rst ♦r tst♥tr t ♣♦♥t s ♥s ♦r ♦ts t rs♣t d

t ♦r♦s

♥tr qst♦♥ s ♦ t ①t♥ t rsts t♦ ♥r ♦r♦s rsts ♦t♦♥ ① r r♦♣s t ① ♣♦♥ts ♦r r♦♠ t rt♦r♠ ♥♦ tt ♥② ♦r♦ s ♥t② st ♦r♥ s♣ tt s ♦s♥ ♠♥♦ ♥ ♦♥ ♦r ♣♣r♦ ♦rs ♦ t ♣♣r♦ ♥♥♦t♦♠♣t tr♥t♦♥ ♥ t ♦r♦ t ♥ ②s ♦♠♣t ♥② tr♥t♦♥♥ tt ♦r♥ s♣

♣r s♣s

♥ ts st♦♥ ①t♥ t ♣♣r♦ sr ♥ ♣tr ♦♥t♦ ♦rt s♣s ♦ s♣r♥r t t♦♥ ♦ srt r♦♣ ♦ s♦♠trs ❲ rst sss ♥② tr♥t♦♥s♦ t s♣r ♥ ♥t② t ♦rt s♣s tt ♦r ♣♣r♦ ♥ ♥ ♥ ♥②sss t ♥② tr♥t♦♥ ♦ s♣r ♦r♦

r♥t♦♥s ♦ t s♣r

t Sd ♥♦t d♠♥s♦♥ ♥t s♣r ♥ ♦rr t♦ s ♥t♦♥s ♥

♠st ♥t♦♥ ♦r s♠♣s ♥ t ♥② tr♥t♦♥ ♦ t d♠♥s♦♥s♣r ♦♦♥ ♥t♦♥s r strt♦rr ①t♥s♦♥s ♦ t t s

t ρ : Ed+1 −0 → S

d ♥♦t t r ♣r♦t♦♥ ρ ♣r♦ts ♣♦♥ts ♦ r②R strt♥ t t ♦r♥ ♦♥t♦ t ♥trst♦♥ ♣♦♥t ♦ R ♥ S

d

♥t♦♥ ♣r s♠♣① t Q st ♦ k +1 ♣♦♥ts ♥ Sd t k ≤ d t

(Q) ♥♦t t ♦♥① ♦ Q ♥ Ed+1 (Q) ♦s ♥♦t ♦♥t♥ t ♦r♥

t ♠ ♥r ρ ♦ (Q) s♣r ks♠♣①

r♠sr♥ ♦ s♣r ds♠♣① ♥ ♥ ♥ t s♠ ② s ♦rE

d s♥ t s♣r ♠tr t ♥ ♦♥strt s ♦♦s ♦♥sr t (d − 1)

♥② tr♥t♦♥s ♦ ♦tr s♣s

♥ ② t d rts ♦ t s♠♣① ②♥ ♥ (d−1)②♣r♣♥ ♥ Ed ♣r♦t♦♥

♥r ρ ♦ ts (d − 1) s t r♠sr♥ ♦ t ds♠♣①

♥t♦♥ ♣r ♥② tr♥t♦♥ ♥② tr♥t♦♥ ♦ P♥ S

d s tr♥t♦♥ ♦ Sd s tt t r♠sr♥ ♦ ds♠♣① ♦ t

tr♥t♦♥ ♦s ♥♦t ♥② ♣♦♥t ♦ P ♥ ts ♥tr♦r

♥② tr♥t♦♥ ♦ t s♣r Sd ♥ ② P s t ♦♥① ♦ P ♥

Ed+1 s P s ♥ t d♠♥s♦♥ s♣r ♠ ♥ E

d+1 ♣♦♥ts ♦ P ♣♣r ♥ts ♦♥① ♥ E

d+1 t s ♠♦r ♥t t♦ rt② ♦♠♣t t ♥② tr♥t♦♥♥ d♠♥s♦♥ s♣ ♥st ♦ rs♦rt♥ t♦ E

d+1 ❬❪ sr ♦ t♦ ♦♠♣t s♣r ❱♦r♦♥♦ r♠ ♥ t♦ ♠♥s♦♥s ② ♦♠♣t♥ t♦ ♣♥r ❱♦r♦♥♦ r♠s♥ ❬+❪ sr t♦ ♣♣r♦s ♥ ♥ ♠♣♠♥tt♦♥ ♦r t s ♦ d = 2s ♣♣r♦s s t ♦rt♠ ♦r ♦♠♣t♥ t ♥② tr♥t♦♥ ♥ E

2 ♥♣t t ♣rts ♦r♥② t♦ t ♦rrt rsts ♦r t ♥② tr♥t♦♥♦ t s♣r

♣♣r♦s sss ♥ ❬+❪ ①t♥ t♦ d ♠♥s♦♥s ♣rt t♦ tsttr t r♠sr♥ ♦ ds♠♣① ♦♥t♥s ♥ ♣♦♥t ♥ ♦♠♣ts ♦♦s t σ s♣r ds♠♣① rts ♦ σ ♥ d♠♥s♦♥②♣r♣♥ ♥ E

d+1 t p ♥♦t t qr② ♣♦♥t p ♥ t ♦r♥ ♦♥ r♥t ss♦ t ②♣r♣♥ t♥ p s ♥s t r♠sr♥ ♦ σ

t ♣♦♥ts ♦ P ♥ s♣r P ♦s ♥♦t ♥ ♥② tr♥t♦♥ ♦ S

d ♥ ♦rr t♦ ♥② tr♥t♦♥ tt s ♦♠♦♠♦r♣ t♦ Sd

♥ ♦r ♥st♥ ♥tr♦ rt rt① s♠r t♦ t rt① t ♥♥t② ♦r ♥②tr♥t♦♥s ♦ E

d s t♦♥

♣r ♦rt s♣s

♥ s♠r ♠♥♥r s ♥ t♦♥ ♦r ♣♣r♦ ♦rs ♦r d♠♥s♦♥ srt r♦♣s♦ s♦♠trs ♦ t s♣r S

d r♦♣ ♦ s♦♠trs ♦ t d♠♥s♦♥ s♣r s t♦rt♦♦♥ r♦♣ O(d + 1) srt sr♦♣s ♦ O(d + 1) r ♥t s ts♣r s ♦♠♣t trs t ♦rt ♦ ♣♦♥t ♦ ♥♥t ♥ ♥♥t ssts♦ ♦♠♣t s♣ ♥ ♠t♦♥ ♣♦♥t ♥ r ts ♥♦t srt srtsr♦♣s ♦ O(d + 1) r ♣♦♥t r♦♣s ♥ ♦r ♠♥s♦♥ tr r ♥♥t②♠♥② ♦ t♠ ❲ r② sss t t♦ ♥ tr♠♥s♦♥ ♣♦♥t r♦♣s

r r t♦ t②♣s ♦ ♥t ♣♦♥t r♦♣s ♥ t♦ ♠♥s♦♥s ② r♦♣s Cn♥rt ② r♦tt♦♥ ♦ 360/n r r♦♣s ♥rt ② r♦tt♦♥ ♦ 360/n♥ rt♦♥ s r ♦r ♥ strt♦♥

♥ tr ♠♥s♦♥s tr r s♥ srs ♦ ♣♦♥t r♦♣s ♦ ♦♥t♥s ♥♥♥t ♥♠r ♦ r♦♣s ♥ s♥ ♠♦r ♣♦♥t r♦♣s s♥ ♥♥t srs r♦tt♦♥ ♦ 360/n ♦r ♥② n > 0 s ♦♥ ♥rt♦r r♠♥♥ s♥ ♣♦♥t r♦♣sr t s②♠♠tr② r♦♣ ♦ t ttrr♦♥ t♦tr t t♦ sr♦♣s s s ts②♠♠tr② r♦♣s ♦ t ♦tr♦♥ ♥ t ♦sr♦♥ t ♦♥ sr♦♣ ♦rrtr r♥ s ❬♦①❪

♦t tt ♠♦♥ t tr♠♥s♦♥ ♣♦♥t r♦♣s t ♦♥② r♦♣ t♦t ①♣♦♥ts s t r♦♣ tt ♠♣s ♣♦♥t t♦ ts ♥t♣♦ ♣♦♥t ♦♥ t s♣r ♦rts♣ ♥r t t♦♥ ♦ ts r♦♣ s t ♣r♦t ♣♥ s ts t ♦♥② t♦♠♥s♦♥ s♣r ♠♥♦ ss t s♣r ts ♦tr ♦rt s♣s r t②♦r♦s ❬t❪

♣r s♣s

r t C3 r♦tt♦♥ ♦ 360/3 t D5 r♦tt♦♥ ♦ 360/5 ♥ rt♦♥ s rs D r t rs♣t ♥♠♥t ♦♠♥s

r♥t♦♥s ♦ s♣r ♦rt s♣s

♥r③t♦♥ ♦ ♦r♠ ♥ ♦rt♠ s s♠r t♦ t♦♥ ♠♥ r♥ s tt t ♥♠r ♦ r♦♣s t♦ ♦♥sr s ♥♥t t r♦♣s t♠ss r ♥t

t GP ♥♦t d♠♥s♦♥ ♣♦♥t r♦♣ X := Sd/GP ♥ ♦rt s♣ t ♣r♦t♦♥

♠♣ π : Sd → X ♥ DT (GPP) t ♥② tr♥t♦♥ ♦ S

d ♥ ② t ♥t♣♦♥t st GPP

❲ ♥♦ ♥r③ ♦r♠

♦r♠ t st♦♥ ♦ π(DT (GPP)) ♦s ♥♦t ♦♥t♥ ②s ♦ ♥t ≤ 2t♥ π(DT (GPP)) s tr♥t♦♥ ♦ X

♦r t♦♥ t ♥t♦♥s ♥ ♠♠s ♦ t♦♥ ♥r③ rt②t♦ X ①♣t ♠♠ ♦r t ♣r♦♦ ♦ ♠♠ tr♥s ♦t t♦ tr ♥t s♣r s s GPP s ♥t ♥ s♦ DT (GPP) s ♥t t♦♦

s GPP s ♥t ♥ ②s ♦♦s t s♣r Sd s ♥t ♦r♥ s♣ ♦ X ♥

t ♣♦♥t st GPP ♥s tr♥t♦♥ t ♦♥t♥s ♠♦r t♥ d + 2 ♣♦♥ts ♥♦t t♥ ♦♥ s♣r

❲t t ♠♦t♦♥s sr ♥ t♦♥ t ♥r♠♥t ♦rt♠ ② ♦②r♥ ❲ts♦♥ ♦rs ♦r ♦♠♣t♥ t ♥② tr♥t♦♥ ♦ S

d ♦rt♠ ♥ ♠♦ t♦ ♦r ♥ X = S

d/GP s♥ t rsts r♦♠ t♦♥

• ♦rt♠ strts ♦♠♣t♥ ♥ t |GP |st ♦r♥ s♣ Sd ♥srt♥ |GP |

♦♣s ♣r ♥♣t ♣♦♥t

• ♥ t ♣r♦t♦♥ ♦ t rr♥t tr♥t♦♥ ♥r π s ♥♦ ②s ♦ ♥t t♦♥ t ♥ s♦♥ tt ♥ rtr ♣♦♥ts ♥♥♦t ♥tr♦ ②s ♦ ♥tt♦ t ♦rt♠ sts t♦ ♦♠♣t♥ ♥ X

♦t tt GP s ♥♦t t♦rs♦♥r t♥ t ♦rt♠ rtr♥s DT (GPP) ❲ ♥♦ ♦♠tr rtr♦♥ s♠r t♦ rtr♦♥ t♦ ♥ t♦ st t♦ ♦♠♣t♥♥ X ♦r t s ♦ t r ♣r♦t ♣♥ ❲ ♦ ♥♦t ♥r rtr♦♥

♥② tr♥t♦♥s ♦ ♦tr s♣s

r ♣r♦t ♣♥

t G ♥♦t t r♦♣ ♥rt ② t s♦♠tr② tt ♠♣s ♣♦♥t ♦ S2 ♦♥t♦ ts ♥t♣♦

♣♦♥t ♦rt s♣ S2/G s t r ♣r♦t ♣♥ ❲ ♥ ♦r♠t t ♦♦♥

rtr♦♥

rtr♦♥ t rst s♣r s ♥ S2 tt ♦s ♥♦t ♦♥t♥ ♥② ♣♦♥t ♦

GP s ♠tr s♠r t♥ π/2 t♥ P ♥s t ♥② tr♥t♦♥ ♦ t r♣r♦t ♣♥ X/GPr♦♦ s ♣r♦♦ ♦♥t♥s t s♠ s s t ♣r♦♦ ♦ rtr♦♥ ♣t t♦ ts♣r s t rst ♠♣t② s♣r s s ♠tr s♠r t♥ π

2 t♥ t♦♥st ♥ t ♥② tr♥t♦♥ ♦ GP s s♦rtr t♥ π

2 ♥ ts ♣t ♦ t♦s s s♦rtr t♥ π ♥ ♥♥♦t ♦r♠ t♦② ♥ ♣r♦t ♦♥t♦ S

2/G

rtr♦♥ ♥ s ② t ♦rt♠ sr ♦ t♦ ♥ t♦ ♦♥rtt ♦♠♣t ♥② tr♥t♦♥ r♦♠ t st ♦r♥ s♣ S

2 t♦ S2/G

②♣r♦ s♣

r sss ♦ t♦ ①t♥ t ♣♣r♦ ♦ ♣tr t♦ ♦rt s♣s ♦ t ②♣r♦♣♥ ♥r t t♦♥ ♦ srt r♦♣ ♦ ②♣r♦ s♦♠trs ❲ rst s♦rt♥tr♦t♦♥ ♦♥ t ②♣r♦ ♣♥ H

2 ♥ ♦♥ ②♣r♦ ♥② tr♥t♦♥s ♥ sss r♦♣s ♦ ②♣r♦ s♦♠trs ♥ ①t♥ ♦r♠ ♥ ♦rt♠ t♦ ts stt♥ r♦♣s ♦ ②♣r♦ s♦♠trs tr♥ ♦t t♦ ♥♥t ♥ trt♦r② s ♠ rr t♥ ♦r r♦♣s ♦ ♥ s♦♠trs ❲ sss tr r♥t♣♣r♦s ♦r t s ♦ t r♦♣ ♦ t ♦ t♦rs

②♣r♦ ♣♥ H2

②♣r♦ ♣♥ s t ♣r♦♣rt② tt ♥ ♥ l ♥ ♣♦♥t p ♥♦t ♥ l tr r♥♥t② ♠♥② ♥s tr♦ p tt ♦ ♥♦t ♥trst l s r

r Pr ♥s ♥ t P♦♥ré s ♥s tr♦ t ♣♦♥t p r ♣r t♦l ❬♣❪

r r ♦r ♦♠♠♦♥ ♠♦s t♦ r♣rs♥t t ②♣r♦ ♣♥ t tr♠♥♦ t P♦♥ré s ♠♦ t P♦♥ré ♣♣r ♣♥ ♠♦ ♥ t ②♣r♦♦♠♦ ❬t❪ ❲ ♦♥② s t P♦♥ré s ♠♦ ♥ t ssq♥t sss♦♥s

②♣r♦ s♣

♥ t P♦♥ré s ♠♦ ♣♦♥ts ♦ t ②♣r♦ ♣♥ r ♠♣♣ ♥s t♥ ♥t s ♣♦♥ts ♦♥ t ♥t r r ♣♦♥ts t ♥♥t② ②♣r♦ ♥s♦rrs♣♦♥ t♦ ♠trs ♦ t ♥t s ♦r rr rs tt ♥trst t ♥t r ♦rt♦♦♥② ②♣r♦ rs ♦rrs♣♦♥ t♦ ♥ rs t♦ tr ♥trs r

t② r ♥♦t ♥tr ♥ t ♦r♥ ♦ st♥ ♥t♦♥ s ♥ ② ds =2√

dx2+dy2

1−(x2+y2)

s t st♥ ♦ ♣♦♥t(xy

)t♦ t ♦r♥ s 2 tanh−1(

√x2 + y2)

r r ♦r t②♣s ♦ ②♣r♦ s♦♠trs t② ♥ rtr③ ② tr ①♣♦♥ts t ♥♥t② ②♣r♦ s♦♠tr② tt s ♥♦ ① ♣♦♥ts s r♦tt♦♥ ts ①t② ♦♥ ① ♣♦♥t t ♥♥t② t s ♠t r♦tt♦♥ ②♣r♦ tr♥st♦♥sr s♦♠trs t ①t② t♦ ① ♣♦♥ts t ♥♥t② s r t tr ♦r♥tt♦♥♣rsr♥ ②♣r♦ s♦♠trs ♦rt t②♣ ♦ ②♣r♦ s♦♠trs r t ♥♦♥♦r♥tt♦♥♣rsr♥ rt♦♥s ❬t❪ ❲ ♥♦ ♥tr♦ ②♣r♦ tr♥st♦♥s♥ ♠♦r t

t♦ ♣♦♥ts t ♥r♥t ② tr♥st♦♥ t ♥ ♥q ♦s ♥ t ②♣r♦♣♥ tr♥st♦♥ t tr♥sts ♣♦♥t ♦♥ ts ♥r♥t ♦s tr♥st♦♥t ♠♣s ♣♦♥t ♦ st♥ d t♦ t ♦s ♦♥t♦ ♥♦tr ♣♦♥t ♦♥ t qst♥t ♥ ♦st♥ d t♦ t ♦s s s t♦ t t tt t s ♥ s♦♠tr② st(p, q) =st(t(p), t(q)) ♦t tt p ♥ q ♦ ♥♦t t s♠ st♥ t♦ t ♥r♥t ♦st♥ t st♥ ② t② r tr♥st rs st(p, t(p)) 6= st(q, t(q)) ts t ♦sr ♣♦♥t p s t♦ t ♥r♥t ♦s t s♦rtr st♥ st(p, t(p)) ② t② r tr♥st s s ♥♠♥t r♥ t♦ t ♥ ♣♥ ♦r♥ strt♦♥ ♦ ②♣r♦ tr♥st♦♥s s r ♦r ♥st♥ ♥ s t t♦① ♣♦♥ts r ♥♦t ♥t♣♦ ♥♦t tt t st♥ ♦ ②♣r♦ tr♥st♦♥ ♦♥ t♥ ♦r s rr t♥ ♦♥ t ②♣r♦ ♦s

r t r♥st♦♥ ♦♥ ♠tr ♦ t P♦♥ré s t r♥st♦♥♦♥ ♥r ♦s

s ♦ss r ♥q ♥ t ②♣r♦ ♣♥ t ②♣r♦ ♦♥① ♦ ♣♦♥tst P ♥ ♥ ♥ t ss ② s t s♠st st S s tt t ♦st♥ t♦ ♣♦♥ts p ♥ q ♦ S s ♥tr② ♦♥t♥ ♥ S ♥ ②♣r♦ ♦r②♣r♦ tr♥ s t ♦♥① ♦ t♦ ♦r tr ♣♦♥ts ♥ H

2 rs♣t②♥ ❬❪ t t♦rs ♦♥sr ②♣r♦ ❱♦r♦♥♦ r♠s s♥ t ♣r♦♣rt② tt

♥ t P♦♥ré ♠♦s ②♣r♦ rs r ♥ rs ♥ ts ② t ♦rt♠♦r ♦♠♣t♥ t ♥ ♥② tr♥t♦♥ ♥ s rt② t♦ ♦♠♣t t

♥② tr♥t♦♥s ♦ ♦tr s♣s

②♣r♦ ♥② tr♥t♦♥ s t ♠♣t② rs r ①t② t s♠ ♥ ♦tss

s♥ ♥ ♦ ❬❪ sr ♦ t♦ ♦♠♣t ②♣r♦ ❱♦r♦♥♦ r♠s s♥ t tr♠♥ ♠♦ r ♣♣r♦ s s t ♦r ♦ ❬❪ ♥ ①t♥ t♦ H

d ♦r♥tt♦♥♣rsr♥ srt r♦♣s ♦ s♦♠trs ♦♥ H

2 r s♥r♦♣s ❬t❪ ♥ ❲♥ ❬❲❪ sr s♦♠trs ♥ srt s♦♠tr② sr♦♣s♦ d♠♥s♦♥ ②♣r♦ s♣s

♦ t♦rs

r ♦♥② ♦♥sr t ♦ t♦rs s ♦♥ ♦ t ♠♦st s♠♣ ♦rt s♣s ♦♦♥st♥t ♥t rtr

❲ rst rqr s♦♠ ♠♦r ♥♦t♦♥s r♦♠ r♦♣ t♦r② t H r♦♣ ♥ H′ ♥♦t sr♦♣ ♦ H q♦t♥t r♦♣ H/H′ s t st ♦ ♦sts ♦ H′ ♥ H t t♣r♦t ♦ ssts s r♦♣ ♦♣rt♦♥ r♦♣ s s t♦ r r♦♣ ♠♥t♥ rtt♥ ♥ ♥q ② s ♣r♦t ♦ ♥rt♦rs ❲ ♥♦t t r r♦♣ t♥rt♦rs G ② < G > t R ♥♦t sst ♦ < G > st ♦ rt♦♥s q♦t♥t r♦♣ ♦ t r r♦♣ < G > ♥ t ♥♦r♠ sr♦♣ ♦ < G > ♥rt ②t rt♦♥s ♥ R s ♥♦t ② < G | R > s ♥♦tt♦♥ s r♦♣ ♣rs♥tt♦♥

♦ t♦rs ♥ ♥♦ ♦♥strt s ♥ ♦rt s♣ ♥r t t♦♥ ♦ r♦♣ ♥rt ② ♦r ②♣r♦ tr♥st♦♥s t a, b, c, ♥ d ♥♦t ♦r ②♣r♦tr♥st♦♥s ♥ a, b, c, ♥ d tr rs♣t ♥rs tr♥st♦♥s r r t st t♦r♦♣s t♥ ♦♥ H

2 tt ♥ ♦ t♦rs

G := < a, b, c, d | abcdabcd >

G′ := < a, b, c, d | ababcdcd >

r♦♣s G ♥ G′ r s♦♠♦r♣ t ♥♦t q ❬❪ s♦♠♦r♣s♠ ♥ ♦♥strt ♦♠tr② s♥ t ♦rt♠ sr ♥ ❬❱❨❪ ♦r ♠♦r sss♦♥s♦♥ ts s s♦ ❬ r❪

r ♦♥sr t r♦♣s ♦r t ♥♠♥t ♦♠♥ s t rr ♦t♦♥♥tr t t ♦r♥ ♥rt♦rs ♦ G ♥ G′ rs♣t② ♠♣ ♦t♦♥ s ♦♥t♦♦t♦♥ s s ♥ ♥tr♣rt s ♥t②♥ t s s s♦♥ ♥ r

♥rt♦rs ♦r G ♥ ♦s♥ s ♦♦s t a t ②♣r♦ tr♥st♦♥ tt

tr♥sts ♣♦♥ts ♦♥ t ♥ x①s ② 2 tanh−1(√

21+

√2

) t b, c, d t s♠

tr♥st♦♥s r♦tt ② π/4 ♥♠♥t ♦♠♥ ♦ G s t rr ♦t♦♥ t♥s ♦ π/4 ♦♥sr t t♥ ♦ t ②♣r♦ ♣♥ ② s ♦t♦♥s t rt① ♦ t t♥ t ♥♠♥t ♦♠♥s ♠t t♦ ♦♠♣t t ♥ ♦ 2π

❲ ♦ ♥♦t ♦♥sr t r♦♣ G′ r♦♣ G′ s t ♥t tt t s sr t♦s tt t ♥tt♦♥s t ♥s ♦♥ t ♦t♦♥ ② ♦ t♦rs ♦r ts♥rt♦rs r ♠♦r ♦♠♣t t♦ t

r♥t♦♥s ♦ t ♦ t♦rs

❲ rst ①t♥ ♦r♠ t♦ t s ♦ t ♦ t♦rs t P ♥t ♣♦♥t st ♥H

2 G t r♦♣ ♦ t ♦ t♦rs s ♥ ♥ t♦♥ H2/G t ♦rt s♣ t

♦r S, T ssts ♦ H ST := st | s ∈ S ♥ t ∈ T

②♣r♦ s♣

r t ♥tt♦♥ s♠ ♦r r♦♣ G t ♥tt♦♥ s♠ ♦rr♦♣ G′

♣r♦t♦♥ ♠♣ π : H2 → H

2/G ♥ DT (GP) t ♥② tr♥t♦♥ ♦ H2 ♥

② t ♥♥t ♣r♦ ♣♦♥t st GP ♥ ♦r♠ ♥ ♦r♠t s ♦♦s♦r t ②♣r♦ s

♦r♠ t st♦♥ ♦ π(DT (GP)) ♦s ♥♦t ♦♥t♥ ②s ♦ ♥t ≤ 2t♥ π(DT (GP)) s tr♥t♦♥ ♦ X

♦♥② ♠ss♥ ♣rt ♥ t ♣r♦♦ ♦ ♦r♠ s t ♦ ♥t♥ss ♠♠ t ♦tr ♠♠s r ♣r② ♦♠♥t♦r ♥ ♦♥② r s♥ ♣r♦♣rts ♦ s♠♣s♥♣♥♥t ♦ tr ♠♥

❲ ♥♦ ♥r③ ♠♠

♠♠ st ♦ s♠♣s K tt s ♣ ♥ t ♥②♣r♦♣rt② t rs♣t t♦ GP s s♠♣ ♦♠♣① ♥ H

2

Pr♦♦ ♥ ♦rr t♦ s♦ tt K s s♠♣ ♦♠♣① t r♠♥s t♦ ♣r♦ tt ♦♥t♦♥ ♦ t s♠♣ ♦♠♣① ♥t♦♥ ♦ ♥t♥ss s ❲ s s♠rr♠♥tt♦♥ s ♥ t ♣r♦♦ ♦ ♠♠ s ♣

ss♠ tr s rt① v t ♥ ♥♥t ♥♠r ♦ ♥♥t s♠♣s ♥ ts ♥♥♥t ♥♠r ♦ ♥♥t s t tv ♥♦t t ②♣r♦ tr♥st♦♥ tt ♠♦s vt♦ t ♦r♥ ♥② ♣r♦♣rt② s ♥r♥t ♥r t t♦♥ ♦ s♦♠trs s♦ ♥♣♣② tv ♦♥t♦ K t♦t ♦s♥ t ♥② ♣r♦♣rt② s t♦t ♦ss ♦ ♥rt② ♥ ss♠ tt v s ♥ t ♦r♥ ♦rt ♦ v s ♣♦♥t st ♥ t ♠tr♦ t rst ♠♣t② s s ♦♥ ② t ♠tr ♦ t r♠sr♥ r ♦ t

♥♠♥t ♦t♦♥ s ∆ := 2 sinh−1

(√2(1+

√2)

sin(π/8)

)≈ 4.90 ♦ r♠sr♥

rs ♦ ②♣r♦ tr♥s tt r ♥♥t t♦ v ♥ ♣♦♥t ♥ t ♦rt ♦ ♣♦♥t ♥P ♦ st♥ rr t♥ ∆ r♦♠ v ♥♥♦t ♠♣t② s ♦ rs ∆ ♥tr t t♦r♥ s ♦♠♣t r♦♠ t t tt G s srt t ♦♦s tt t ♥♠r ♦ ♠♥ts♦ t ♦rt ♦ ♣♦♥t ♥ P tt ♥s ts s s ♥t ♥ P ts s ♥t t♦♦

❲ s♦♥ tt tr r ♦♥② ♥t② ♠♥② tr♥s ♥♥t t♦ v ♠♣srt② tt t ♥♠r ♦ s ♥♥t t♦ v s ♥t t♦♦

♥ t s♣s sss s♦ r ②s s ♦r♥ s♣s t♦ rs♦ tsss ♦ ♥♣t ♣♦♥t sts tt ♦ ♥♦t ♥ ♥② tr♥t♦♥ ♦ sss

♥② tr♥t♦♥s ♦ ♦tr s♣s

② t s ♠ ♠♦r t ♥ ♦ ss ♥trst t♦ ♦♥strt ♦r♥ s♣s ♦ ②♣r♦♦rt s♣s ❲ ♣rs♥t t♦ tr♥t ♣♣r♦s

♦♠♣t t ♥t ♥♠r ♦ ♦♣s ♦ t ♥♠♥t ♦♠♥ ♦ t r♦♣ ♥♦♠♣t t ♥② tr♥t♦♥ ♦ H

2 ♥ ② ts ♥t st ♦ ♣♦♥ts

trt t ① ♥t ♣♦♥t st tt s r♥t t♦ ♥ ♥② tr♥t♦♥ ♦ t ♦rt s♣ ♥ r♠♦ ts ♣♦♥ts ♦♥② tr s♦♠ rtr♦♥ s♠r t♦rtr♦♥ s

♦r♥ s♣s

♦r♥ ♠♦tt♦♥ ♦r s♥ ♦r♥ s♣s ♦r ♦♠♣t♥ tr♥t♦♥s ♥ t tt♦rs s tt t ♦r♥ s♣s ♦ t t t♦rs r ♦♠♦♠♦r♣ t♦ t t t♦rs♥ s ♦♦s r♦♠ t ♠♥♥rt③ ♦r♠ ❬r❪ t X t♦♣♦♦s♣ ♥ X kst ♦r♥ s♣ χ(X) ♥♦ts t r rtrst ♦ s♣X t♥ t qt♦♥ χ(X) = k · χ(X) ♦s

tt t ♥s ♦ sr t ♥♠r ♦ ♥s s ♥ ② 1 − 12χ s

t r rtrst ♦ t t t♦rs s ③r♦ ♥② ♦r♥ s♣ ♦ t t t♦rs s♥ t t♦rs ♦ t♦rs s r rtrst −2 ♦♦♥② t rrtrst rss ♥ t ♥♠r ♦ sts ♥ t ♦r ♥rss s ♦rrs♣♦♥st♦ rr ♥♠r ♦ ♥s

♦ r♠♥t ♦♥ t ♥♠r ♦ ♥s ♥ s♦ ♥rst♦♦ ♥ ♠♦r♥tt ② t s ♦♥sr t♦♠♥s♦♥ t♦rs ♠ ♥ E

3 t♦rs st♦ ♥♦♥tr ♦♦♣s s ♦ ♦r♥ s♣ ♦rrs♣♦♥s t♦ ♦♥sr♥ rt♦rs t ♠♦r ♣♦♥ts ♦♥ t ♦ t ♥♦♥tr ♦♦♣s r ♦♥r ♥ t♦②s ♥ ttr♥t♦♥ ♥ ♦ ♣r♦ss ♦ ♥rt♥ t♦st ♦r♥ s♣ ♦t ♦ t♦rs ♦rrs♣♦♥s t♦ tt♥ ♦♥ ♥♦♥tr ♦♦♣ ♥ ♦t ♦ t♦r ♥ st ♦♣♥ s t♦ t ♦tr ♦ t♦rs s r ♥ ts ② ♦♥ ♥ s♥ ①♣♥ ♦♥ ♥♦♥tr ♦♦♣ s ♦♠ ♦♥r ♥ t ♦tr t♦ ♥sr ♥♥ ♥② rt s♣ s t♦ ♥s ♦ ♦r♥ ♦♦♣ ♥t ♥ ♦♥♥ ♦ ♦ ♦♦♣ ♥t ♦ ♥♦t t② ♥ t ♥♠r ♦ ♥s ts♠ ♦♦♣ ♥t s t ♦r♥ ♦ t t ♠t ♣♦ss t♦ t ♥ t ♦♦♣s ♥ ♠♦r ♦♠♣t ② ♥ ♦rr t♦ ♦ ♥tr♦♥ ♥ s♦rt ♦♦♣s

r ♦♥strt♥ t♦st ♦r♥ s♣ ♦ t ♦ t♦rs ❬❪

s t ♣♣rs t t♦ ♦♥strt ♦r♥ s♣ ♥ ②s ♦ ♥t t♦ ♥ ♦ t♦♥② t ♦r♥ ♠♦tt♦♥ ♦ s♥ ♦r♥ s♣s ♥♠② ttt② r t♦♣♦♦② q♥t t♦ t ♦r♥ s♣ s ♥♦t ♥ ♥②♠♦r

②♣r♦ s♣

♦♣s ♦ t ♥♠♥t ♦♠♥ ♥ H2

t F ♥♦t t ♥♠♥t ♦♠♥ ♦ G ♥tr t t ♦r♥ ♥ P ⊂ F ♥t

♣♦♥t st ♠tr ♦ F s ∆ := 2 sinh−1

(√2(1+

√2)

sin(π/8)

)≈ 4.90 s t ♠tr

♦ t rst ♣♦ss ♠♣t② r t rs♣t t♦ GP ♦ tr ♥♥♦t ♥② s♦♥r t♥ ∆ ♥ t ♥② tr♥t♦♥ ♦ GP s ♣♦♥ts tt r rtr ②t♥ 3

2∆ r♦♠ t ♦r♥ ♦ ♥♦t ♥② ♥♥ ♦♥ t s♠♣s ♦ DT (GP) tt♥trst F ♦ t♦ ♦♥sr t sst G ♦ tr♥st♦♥s g ∈ G s tt gF♥ t s ♦ rs 3

2∆ ♥tr t t ♦r♥ ♥♦♥♠♣t② ♥trst♦♥ ❲ ♥ s♥s ♦rt♠ ♥ ♦rr t♦ ♥t② t st G ❬②♥❪ ❲ t♥ ♦♠♣t t ♥②tr♥t♦♥ ♦ H

2 ♥ ② t ♣♦♥t st GP ♥ ①trt s♠♣s tt ♥trstF Pr♠♥r② st♠t♦♥s s♦ tt t ♥♠r ♦ ♥♠♥t ♦♠♥s t♦ ♦♥srs r ♦ s ts ♣♣r♦ tr♥s ♦t t♦ ♥♦t r② ♣rt

♥♦tr s t♦ ♥r③ t rst ♦ ❬❪ tt sr ♥ t♦♥ t♦ t ②♣r♦ ♣♥ ❯♥♦rt♥t② ♥♦s s♠s ♦ ♥♦t ①t♥ t♦ t ②♣r♦s rtss ♥ ♥ st tt s ♥♦♦s t♦ F (n) ♥ ②♣r♦ s♣ tF

(1)p ♥♦t t ❱♦r♦♥♦ ♦ p ♥ t ②♣r♦ ❱♦r♦♥♦ r♠ ♦ Gp ♥ F

(1)p s

♥♠♥t ♦♠♥ ♦ G ♦ ♥ F(n)p :=

⋃q∈Fp

F(n−1)q

♦r♠ t τ s♠♣① ♥ DT (GP) t t st ♦♥ rt① ♥ F(1)p ♥ τ

s ♦♠♣t② ♦♥t♥ ♥s F(3)p

Pr♦♦ ♣r♦♦ ♦rs ♥ t s♠ ② s t ♣r♦♦ ♦ ♠♠ ♥ ❬❪ t σ♥ τ ♥♦t s♠♣s ♥ DT (GP) s tt σ s p s ♦♥ ♦ ts rts ♥ τ s t

st ♦♥ rt① ♥ F(1)p

F(1)p ♦♥t♥s t ♥tr cσ ♦ t r♠sr♥ r ♦ σ s r s s

t♦ t ♦♥strt♦♥ ♦ F(1)p s ❱♦r♦♥♦ cσ s ♦ts ♦ F

(1)p t♥

♣♦♥t ♥ t ♦rt ♦ p ♦ ♦sr t♦ cσ t♥ p s ♦♥trt♦♥ t♦ tt tt cσ s t r♠♥tr ♦ s♠♣① t p s rt①

σ ⊂ F(2)p ♦t tt r♦♠ t ♥t♦♥ ♦ F

(1)p ♦♦s tt cσ ∈ F

(1)p t♥

p ∈ F(1)cσ ♥ s♦ σ ⊂ F

(1)cσ ⊂ F

(2)p ② ♥t♦♥ ♦ F

(2)p

t q ∈ F(1)p rt① ♦ τ r♦♠ t ♦ r♠♥tt♦♥ ♥♦ tt τ ⊂ F

(2)q

♥ ② ♥t♦♥ ♦ F(3)p t ♥s♦♥ F

(2)q ⊂ F

(3)p ♥ ts τ ⊂ F

(3)p ♦s

♥ ♦rr t♦ ♦r♠t rtr♦♥ t♦ tr ♣♦♥t st ♥s ♥②tr♥t♦♥ ♦ t ♦ t♦rs ♥ s♠r ♣♣r♦ s ♦r t t t♦rs

rtr♦♥ t ♠tr ♦ t rst r♠sr♥ r ♦ t tr♥s ♦ t

♥② tr♥t♦♥ ♦ GP∩F(3)p tt ♥trst F

(1)p s s♦rtr t♥ tanh−1

(√2

1+√

2

)≈

1.53 t♥ ♥② ♣♦♥t st Q t P ⊆ Q ♥s ♥② tr♥t♦♥ ♦ t ♦ t♦rs

♥② tr♥t♦♥s ♦ ♦tr s♣s

r cσ s ♥ F(1)p

Pr♦♦ r♦♠ t♦♥ ♥♦ tt t s♦rtst st♥ ② ♣♦♥t ♥

tr♥st ♥r G s t st 2 tanh−1(√

21+

√2

)≈ 3.06 ♦ r♠sr♥ rs ♦

tr♥s tt ♥trst F(1)p r s♦rtr t♥ ts ♦♥st♥t t♥ tr ♥♥♦t ②s

♦ ♥t t♦ ♥ tr ♥ ♥ ♣♦♥ts ♥ ts Q ♥s ♥② tr♥t♦♥♦ t ♦ t♦rs ♦r♥ t♦ ♦r♠

♦rt♠ srs ♦ t♦ ♦♠♣t ♥② tr♥t♦♥s ♦ t ♦ t♦rs ♣♦ss

♦rt♠ ♦♠♣t ♥② tr♥t♦♥ ♦ H2/G ♥ ② t ♣♦♥t st P

♥♣t t P ♦ ♣♦♥ts ♥ F(1)0

t♣t π(DT (GP)) t s tr♥t♦♥ ∅ ♦trs P ′ ⇐ P P♦♣ p r♦♠ P ′

P ⇐ p H ⇐ DT (GP ∩ F

(3)0 )

t ♦♥st ♥ H ∩ F(1)0 s ♦♥r t♥ tanh−1

(√2

1+√

2

)♦

P♦♣ p r♦♠ P ′ P ⇐ P ∪ p ♦r p′ ∈ Gp ∩ F

(3)0 ♦

♥srt p′ ♥t♦ H

♥ ♦r H = DT (GP ∩ F(3)0 )

P ′ = ∅ π(DT (GQ)) s tr♥t♦♥ ♦r ♥② Q ⊇ P t♥ ♦t♦ s ①t

♥

♦♠♣t π(DT (GP)) r♦♠ H st t♦ ♦ t♦rs

♥srt ♣♦♥ts r♠♥♥ ♥ P ′ ♥t♦ π(DT (GP)) ♦♥ ② ♦♥ rtr♥ π(DT (GP))

♦rrt♥ss ♦ t ♦rt♠ ♦♦s r♦♠ ♦r♠s ♥ s srtr♦♥ ♥ s t ♥♣t st P ♦s ♥♦t ♥ ♥② tr♥t♦♥ ♦ H

2/G

②♣r♦ s♣

tr r t② sr ♦♣t♦♥s ♣♦ss ♦r s♠♣t② ♦s t♦ rtr♥ ♥♦t♥ ♥♥ t ♠♣♠♥tt♦♥ t ♦ ♦ ♥trst t♦ rtr♥ t rr♥t strtr H t♦ s♦ ♣♦ss t♦ rtr♥ t s♠♣s r♦♠ H tt ♥trst F

(1)0 s tr ♠s

♥r π ♦r♠ ♣rtt♦♥ ♦ H2/G ♦r♥ t♦ ♦r♠

♠♥ t② s t♦ t② ♦♠♣t t ♣♦♥t st PG ∩ F(3)0 ♥ s

♣♣r♦ s t♦ ♦♠♣t t st G ♦ s♦♠trs ♥ G s tt GF(1)0 ⊇ F

(3)0 ❲

♥♦t ♦♥ ts ♦♠♣tt♦♥ t ♥ s♠♣ ♦r ♦♥ ♦♥ t r♥t② ♦ G❲ ♦♥sr t ♣rtt♦♥ ♦ t ②♣r♦ ♣♥ ♥t♦ ②♣r♦ ♦t♦♥s ♥ ② GF

(1)0

♦♥sr t ♠♥ts ♦ G tt ♠♣ F(1)p t♦ ♦♥ ♦ ts ♥♦rs ♥ t ♦t♦♥ ♣rtt♦♥

♦ H2 sr ♦ ♥♦♥ ♦ F

(1)0 ♥ ts ♥♦rs r② ♦r♠s sst ♦

F(3)p r s ♦♥ ♥♦r ♣r t ♥ ♠♦r ♥♦rs ♣r rt①

♦tr t t ♦r♥ ♦♣② F(1)p ts ♠s 1 + 8 + 40 = 49 ♦♣s t♦ ♦♥sr ♥

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

♣♣r♦ sr ♦ s ♥rtss ♥ s t s ①t♥s t♦ ♦trsrt r♦♣s ♦r GP ∩ F

(3)0 ♥ ♦♠♣t

♠♠② ♣♦♥t st

♥ ts ♣♣r♦ strt t ♥ ♥t tr♥t♦♥ ♦ ♣♦♥t st PD tt s ♦s♥s tt ♥② ♣♦♥t st tt ♦♥t♥s PD ♥s ♥② tr♥t♦♥ ♦ H

2/G ss t s♠ ♣♣r♦ s sr ♥ t♦♥ ♦r t s ♦ t t t♦rs t♥♣t ♣♦♥t st ts ♥s ♥② tr♥t♦♥ t ♣♦♥ts ♦ PD ♥ r♠♦r♦♠ t tr♥t♦♥

t P ♣♦♥t st s tt PD ⊆ P ❲ ♥t t♦ ♣r♦ tt π(DT (GPD)) s ♥② tr♥t♦♥ ♦ H

2/G t s s♥t t♦ s♦ tt t rstrt♦♥ ♦ π ♦♥ t♦♥♦♥s♦♥t ss tt r ♠♣t② t rs♣t t♦ GPD s ♥t s ♠♣s tt trstrt♦♥ ♦ π ♦♥ ♥② t♦ ♠♣t② ss ♥ GP s ♥t t♦♦ ♥ tr ♥♥♦t ②s ♦ ♥t t♦ ♥ ts π(DT (GP)) s ♥② tr♥t♦♥ ♦ H

2/G ♦r♥t♦ ♦r♠

❲ ♣r♦♣♦s ♣♦♥t st PD ♦ ♣♦♥ts ♥ r ♣♦♥ts ♥s t ♥♠♥t♦♠♥ ♦♥ t ♦rr ♥ ♥ t ♦r♥r s ②♣r♦ r♠rs r ♥r♠rs t ♥ s♦♥ tt π(DT (P ∪ PD)) s ♥② tr♥t♦♥ ♦ H

2/Gs ♣♣r♦ ♣♣rs t♦ t ♠♦st ♣rt ♠♦♥ t ♣rs♥t ♦♥s r

s tt t ♥♣t ♣♦♥t st ♦s ♥♦t ♥ ♥② tr♥t♦♥ ♦ H2/G t

♥② tr♥t♦♥ ♥ ② ♠♦ ♥♣t ♣♦♥t st s rtr♥ s ♣♣r♦♥r③s t♦ ♦tr r♦♣s ♥ ♣♣r♦♣rt ♠♠② ♣♦♥t st ♥ ♥

sss♦♥

❲ ♣rs♥t tr ♣♣r♦s t♦ ♥ ss ♥ t ♥♣t ♣♦♥t st ♦s ♥♦t♥ ♥② tr♥t♦♥ ♦ t ♦ t♦rs ♥ t ②♣r♦ s t ♠♦tt♦♥♦r s♥ ♦r♥ s♣s s ♦♥ ♥ t s ♦ ♥ ♦r s♣r s♣s s ♥♦t ♥♥②♠♦r ♦r♥ s♣s r ♥♦t ♦♠♦♠♦r♣ t♦ t ♦r♥ s♣ s ♥ t t t♦rs♥ t ♥rs ♦r♥ s♣ s ♥♦t ♥t s ♥ s♣r ♦rt s♣s ♦ t ♥tr♣♣r♦ ♦ t♦ ♦♠♣t ♥ H

2 s♥ s ♠♥② ♦♣s ♦ t ♥♣t ♣♦♥t st s

♥② tr♥t♦♥s ♦ ♦tr s♣s

r Pr♦♣♦s ♠♠② ♣♦♥t st PD ♦ ♣♦♥ts ♦r t ♦ t♦rs

♥ssr② t♦ ①trt sst ♦ t tr♥t♦♥ ♦♥t♦ t ♥tt♦♥s ♥ ②t r♦♣ G ♥ ♣♣ s s ♣♦ss ♥ t♦ t ♥♠r ♦ rqr ♦♣str♥s ♦t t♦ r② r ♥ ♥② ♥st ♦ ♥♥ t s♣ ♣rs♥t ♥♣♣r♦ tt ♠♦s t ♣♦♥t st s ♣♣r♦ s r② ♣rt ♥ s t ♥t♥♣♣t♦♥ ♥ ♦♣ t ①tr ♣♦♥ts ♥ s ♦ s♠ ♦r ② strt ♥♣t ♣♦♥tsts

♣tr

♦♥s♦♥ ♥ tr ♦r

❲ ♣rs♥t ♥ ♣♣r♦ t♦ ♦♠♣t t ♥② tr♥t♦♥ ♦ T3c ♥ ②

♥ ♣♦♥t st P ♦♣ ♦rt♠ s ♠♣♠♥t ♥ s tr♦t ♦♣♥ s♦r rr② ①t♥s♦♥s ♥ ♣tt♦♥s ♦ ts ♣♣r♦ t♦ ♦tr t♥ s♣r ♦rt s♣s ♥ t♦ t ♦ t♦rs ♦ ♦♥st♥t ♥t rtr rsss r♥ t ♦rs ♦ ts ♦r s♦♠ ♠♦r ♥trst♥ qst♦♥s r♦s ♣rs♥t ♥ ts ♣tr

strt♦♥ t♦ s♠♣ ♦♠♣①s

♥ ts ♦r ♦♥♥trt ♦♥ ♦♠♣t♥ s♠♣ ♦♠♣①s ♦t ♦r t ♣r♣♦s ♦♠t♠t s♦♥♥ss ♦ t ♥t♦♥ ♦ t ♥② tr♥t♦♥s s s ♦r ♣rt ♦♥srt♦♥s ♥ t ♠♣♠♥tt♦♥ s s♦ t♦♥ rtss t♠t ♥trst♥ t♦ ♦♥sr tsst♦♥s tt t ♥② ♣r♦♣rt② t r ♥♦ts♠♣ ♦♠♣①s ♥♦t tr♥t♦♥s r r② sss t ♣r♦♠s rs♥♥ ♦♠♣t♥ t strtrs ♦tr t♥ s♠♣ ♦♠♣①s

① s♠♣ ♦♠♣①s

❲ ♦ r① ♦♥t♦♥ ♦ t s♠♣ ♦♠♣① ♥t♦♥ t♦ t ♦♦♥ t σ ♥ σ′ s♠♣s ♥ K ♥ t ♥trst♦♥ σ ∩σ′ s st ♦ s♠♣s ♥ K st♦♥ ♦ ts r① s♠♣ ♦♠♣① ♦ ♥♦♥tr ②s ♦ ♥tt♦ t ♥♦t ♦ ♥t ♦♥ ♦ ♦♥trt t s♠♣① ♥t♦♥ ♦t ♦♥st♥ts ♦♥ t ♥♠r ♦ rqr ♦♣s ♥ t ♥t ♦ ♠♣r♦s r t s♠s s t♦ s♥ ♥ ♥t t strtr ♦r r① s♠♣ ♦♠♣①s s♦ ts ♣♣r♦ ♠t ♥ ♥trst♥ rt♦♥ ♦r rtr ①♠♥t♦♥♥ sr r s tt ♦rt♠s tt rqr t t strtr t♦ s♠♣♦♠♣① ♠st ♠♦ ♦r♥② ♦r ♥st♥ ♥srt♥ ♥ ♣♦♥t t♦ t tr♥t♦♥ s♥ t str♦ ♣♣r♦ s sr ♥ t♦♥ rqrs t ♦ t♦ ♦♠♦♠♦r♣ t♦ d ① s♠♣ ♦♠♣①s ♦ ♥♦t ts ♣r♦♣rt②

∆♦♠♣①s

tr ♣rs♥ts ❲♦♠♣①s rst ♥tr♦ ② ❲t ♥ ♥ srs ♠♦r rstrt t②♣ ♦ ♦♠♣①s t s♦ ∆♦♠♣①s ❬t❪ ♥ t ∆♦♠♣①t s♠♣s r r♣rs♥t ② ♠♣s tt r ♥♦t ♥ssr② ♥t ♦♥ t s♠♣①

♦♥s♦♥ ♥ tr ♦r

r r① s♠♣ ♦♠♣① ♥ T2 tt s t ♥② ♣r♦♣rt②

♦♥rs tr ♥ s♦♦♣s ♥ ts ② T2 ♥ tsst ② t♦ tr♥s

tr s ♥ ♦♥ rt① ♦r t s ♥♦t sr ♦ t strtr st♦r♥ ∆♦♠♣①s ♦ ♠♣♠♥t r s ♦♥♦♥ ♦r ♦♥ ts t♦♣ ② ♦♥ ❱rèr♥ ❬❱❪

strt♦♥s ♦♥ s♣s

r rr♥t ♣♣r♦ s s♦♠ r ♠tt♦♥s ♦♥ t t②♣s ♦ s♣s t ♥ ♥ r ①♣♦r t rs♦♥s ♦ ts rstrt♦♥s ♥ tr② t♦ ♦♣ s ♦ t♦ ♥ sss

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

♥ ①♠♣ ♦ r♦♣ t ♥ ♥♦♥ ♥♠♥t ♦♠♥ s t r♦♣ GC ♥rt ② ♦♥ tr♥st♦♥ t♥ ♦♥ E

2 q♦t♥t s♣ E2/GC s t ②♥r t s

ss♠ tt GC s ♥rt ② t ♥t tr♥st♦♥ ♦♥ t x①s t P ♣♦♥t st♥ E

2 t♥ DT (GCP) t ♥② tr♥t♦♥ ♦ E2 ♥ ② GCP s t ♦♦♥

♣r♦♣rts

t pmax t ♣♦♥t ♥ P t rst y♦♦r♥t t t ∈ GC ♦♥ ♦ t♥rt♥ ♥t tr♥st♦♥s ♥ tr s ♥ t♥ pmax ♥ tpmax ♥DT (GCP)

♥♦♥ ♦ s♠♣s ♥ DT (GCP) s t ♦♥① ♦ GCP s ♥♦t qt♦ E

2 ♣♦♥ts ♦ rr y t♥ pmax ♦ ♥♦t ♥ t ♦♥①

❲♥ ♣r♦t♥ DT (GCP) ♦♥t♦ E2/GC t rst ♣r♦♣rt② ♠♣s tt tr s s

♦♦♣ ♥ pmax ♦♦♥② t st trst ♦r♥ s♣ s ♥ssr② t♦ ♦②s ♦ ♥t t♦ ♥♣♥♥t ♦ t ♣♦♥t st P s r

s♦♥ ♣r♦♣rt② ♠♣s tt t ♣r♦t tr♥t♦♥ s ♥♦t tr♥t♦♥ ♦ ②♥r t ♦♥② ♦ ♥ ♥♥s s ②s ♣r♦♠ ♦r t ♠♣♠♥tt♦♥ s trr♥t ♠♣♠♥tt♦♥ ♥♥♦t ♥ s♣s t ♦♥rs ♥ t s ♦ E

2

♥ E3 rt① t ♥♥t② s ♥ ♥tr♦ ♥ ♦rr t♦ r♣rs♥t tr♥t♦♥ ♦ t

♦♥ ♣♦♥t ♦♠♣tt♦♥s E2 ∪ ∞ ♥ E

2 ∪ ∞ rs♣t② s t♦♥ s

②♣r♦ ♦rt s♣s

r ♥② tr♥t♦♥ ♦ GCP

s ♥♦t ♣♦ss ♦r t ②♥r s t s t♦ ♦♥rs ♥ rt① t ♥♥t② ♦ ♣rt ♦ ♣ ♦♥ ♦t ♦♥rs s ts ♥♦r♦♦ ♦ ♥♦t ♦♠♦♠♦r♣t♦ E

2 ♥ s♦ E2/GC ∪ ∞ ♦ ♥♦t ♠♥♦ ♦♣t♦♥ ♦ ♥ t♦ r♥trts t ♥♥t② ♦♥ ♦r t ♣♣r ♣ ♥ ♦♥ ♦r t ♦r ♣ s ♠♦r t t♦♥ ♥ tr♠s ♦ ♠♣♠♥tt♦♥

♥ tr ♠♥s♦♥s tr r t♦ r♥t t②♣s ♦ ②♥rs ♦rt s♣s E3/GC

♥ E3/G2C r G2C ♥♦ts t r♦♣ ♥rt ② t♦ r♥t ♥t tr♥st♦♥s

E3/GC s s♣ tt s ♣r♦ ♦♥ ♦♥ ①s E

3/G2C s ♣r♦ ♦♥ t♦ ①s ♦r♦t s♣s ♣rt ♣♣t♦♥s ①st s ❬r❪ s♦ ts s♦ ♥ ♥trst♥rt♦♥ ♦ rsr t♦ ♣rs

♠♦r ♥r ♦r♠t♦♥ ♦ t ♣r♦♠ ♦ ♦♥sr ♦rt s♣s ♦ Ed ♥r

t t♦♥ ♦ r r♦♣ ♦ ♠♥s♦♥ s♠r t♥ d

②♣r♦ ♦rt s♣s

❯♥ ♦r t s ♦ t ♥ s♣r ♦rt s♣s ♦ ♥♦t ②t ♥r ♣♣r♦♦r ②♣r♦ ♦rt s♣s ♥ t♦♥ s♦♠ ♣♣r♦s ♦r t ♦ t♦rstt r ①♣t t♦ ①t♥ t♦ ♦tr ♦rt s♣s ♦ H

2 ♥ ♥ ♦ Hd ♦r s t

ss ♦ srt r♦♣s ♦ ②♣r♦ s♦♠trs s ♠ rr t♥ ♦r t t ♦r ts♣r s tr r st ♠♥② ♦♣♥ qst♦♥s ♦♥ ts t♦♣ ♥ ♥①t st♣ t ♦ s t♦ ①t♥ ♦r ♥♥s t♦ srs ♦ r ♥s ♥♦tr t♦♣ t♦ ①♠♥ s♦rt s♣s ♦ r♠♥s♦♥ ②♣r♦ s♣s

♦♥s♦♥ ♥ tr ♦r

♦r♣②

❬❪ ♥ ♠♥t ♦♠♥q tt ♥ r rs ♦♠♣①t② ♦ ♥② tr♥t♦♥ ♦r ♣♦♥ts ♦♥ ♦r♠♥s♦♥ ♣♦②r ♥ Pr♦♥s♦ t t ②♠♣♦s♠ ♦♥ srt ♦rt♠s ♣s

❬❪ r ♥ ♦♦♠♦♥② Pr♦ ♦rts ♦♥ t rr ②♣r♦♦t♦♥ P②s

❬s❪ r s♣♦ ♦♠♣t♥ ♦♠♦♦② ♦ s♦♠♣①s ♦ t t♦rstt♣sr♣ts♥♦r♥sr♣ts❲s♥

♦rs♣♦r❴s♣♦❴❲❴♣

❬❪ ♥ ♠♥t ♥ ♦ ♥ ü♥tr ♦t ♥r♠♥t ♦♥strt♦♥s♦♥ ♥ Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥♦♠tr② ♣s

❬❪ r♥③ r♥♠♠r ♥ ♦ ♥ ❱♦r♦♥♦ r♠s ♥ örür ♥ ♦r ❯rrt t♦rs ♥♦♦ ♦ ♦♠♣tt♦♥ ♦♠tr②♣s sr ♥ Psrs ❱ ♦rt♦♥ ♠str♠

❬❪ ①♥r♦ ♦ t♦rs strt♦♥ tt♣♥♣♦r

♦❴t♦rs❴strt♦♥♣♥

❬r♠❪ r r♠str♦♥ s ♦♣♦♦② ♣r♥r❱r

❬❪ r ♥♥② ♥ ♦♥q r♥t♥ t r ♣r♦t♣♥ ♥t♠t s♣ts ♦ ♦♠♣tr ♥ ♥♦r♠t♦♥ ♥s

❬r❪ r♥③ r♥♠♠r ❱♦r♦♥♦ r♠s sr② ♦ ♥♠♥t ♦♠tr t strtr ♦♠♣t♥ r②s ♣t♠r

❬s❪ tt str♥ ♥r Pr♦r♠♠♥ ♥ t ❯s♥ ♥ ①t♥♥ t t♥r ♠♣t rr② s♦♥❲s② ♦♥♠♥Ps♥ ♦ ♥ ♦st♦♥ ❯

❬P❪ ré rö♥♥♠♥♥ rst♦♣ r♥ ♥ ②♥ P♦♥ ♥tr rt♠t ②s ♥t ②♥♠ trs ♦r ♦♠♣tt♦♥ ♦♠tr② srt♣♣ t♠ts

P❨

❬❪ ♥♥ ♦ss♦♥♥t r rs ♥ ♠ ♦r♥s ♥r♠♥t♦♥strt♦♥ ♦ t ♥② r♣ ♥ ♠♠ ♠♥s♦♥ ♥ Pr♦♥s♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s

❬r❪ r♥r ♦♠♣tt♦♥ strtr ♦♦② Pr♦ tr♥t♦♥s ♦r ♠♦r ②♥♠s tt♣s♦♣♥rr♦♠tr

♦♦rt♦♥sr♣r♦r♠t♠

❬❨❪ rs r♥ ♣♣st♥ ♥ r♥s ❨♦ ①♣t ①tr♠s ♥ ♥② tr♥t♦♥ ♥tr♥t♦♥ ♦r♥ ♦ ♦♠♣tt♦♥ ♦♠tr②♥ ♣♣t♦♥s

❬❪ r Ür ♥sr♣♣♥ s n♠♥s♦♥♥ s♥ ♠s ♠t ♥♠ ♥♥ ♥♠♥tr ött♥rrt♥

❬❪ ♦r♦ ♥ ♦ st ♥② tr♥t♦♥s ♥ tr ♠♥s♦♥s ♦♠♣tr t♦s ♥ ♣♣ ♥s ♥ ♥♥r♥ ♠r

❬P❪ ♦ ②♥ ♦t ♥ ♦♥ P♦rt r♠♥s♦♥ ♦r♦s♥ tr ♦♠tr strtrs ♦été ♠té♠tq r♥ Prs

❬❪ ♥♥ ♦ss♦♥♥t ♥ t ♦t Pr♦② ♦♦ s♠♣♥ ♥♠s♥ ♦ srs r♣ ♦s

❬♦❪ r♥ ♦②r ♦♠♣t♥ rt tsst♦♥s ♦♠♣tr ♦r♥

❬❪ ♥ ♥ ♦♥strt♥ ♥② tr♥t♦♥s ♦♥ s♣♥rs ♥ r♦♣♥ ②♠♣♦s♠ ♦♥ ♦rt♠s ♦♠ ♦ tr♦ts ♥ ♦♠♣tr ♥ ♣s

❬❨❪ ♥♥ ♦ss♦♥♥t ♥ rtt ❨♥ ♦rt♠ ♦♠tr② ♠r ❯♥rst② Prss ❯ r♥st ② ré rö♥♥♠♥♥

❬♠❪ ♥ q ♠♣②♦ ❱♦r♦♥♦ ♣rt ②♥♠s tt♣s♦♣♥rr♦♠tr♦♦rt♦♥sr

♣r♦r♠t♠

❬+❪ ♥ r♦ Pr♦ str♦ ést♥ ♦r♦t r ♦r♦♥q ♥ ♠ ❲♦r♠sr ♦st ♥ ♥t ♥② tr♥t♦♥s ♦ ♣♦♥ts ♦♥ ♦r ♦s t♦ s♣r ♥ t ♥tr♥t♦♥ ②♠♣♦s♠♦♥ ①♣r♠♥t ♦rt♠s ♦♠ ♦ tr ♦ts ♥ ♦♠♣tr ♥ ♣s

❬❪ ❲♥ ♥ ♠ ② ♥ ♦s ♥ ♣rt ♥②♠s♥ ♦rt♠ ♦r r ss ♦ ♦♠♥s ♥ Pr♦♥s ♦ t t♥tr♥t♦♥ s♥ ♦♥t ♣s

P❨

❬❪ ♥ r♦ ❱ssr♦♥ s♦♣♦♦s ♥ ♦♥q s♥ ♦tr♣②♣r♦ srs ♥ P♦str ♣rs♥tt♦♥ t ♥tr♥t♦♥ ♦♥r♥ ♦♥ rs ♥ rs

❬❪ ♦♠♣tt♦♥ ♦♠tr② ♦rt♠s rr② tt♣♦r

❬❪ P r♥tqt② ♠s ♥rt♦♥ ♦r r srs♥ Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr②♣s

❬❪ ♥ r♦ ♦ rt♦ ♥ ♦♥q ♦♣♥ t tr♥t♦♥s r♦♠ t ♥r②♥ s♣ ♥ ❲♦rs♦♣ ♦♥ ♦rt♠♥♥r♥ ♥ ①♣r♠♥ts ♣s

❬❪ ♦①tr ♥ ❲ ♦sr ♥rt♦rs ♥ t♦♥s ♦r srtr♦♣s ♣r♥r❱r r♥ r ❨♦r ♦②♦

❬♦①❪ ♦①tr r ♦♠♣① P♦②t♦♣s ♠r ❯♥rst② Prss♠r ♥♥ ♥ t♦♥

❬❪ ♥♥t rs♦♥ ♥ Ptr ❲ ♦r ♦rt♠s ♦r ♠tr ♣rs ♥♦♥① s tt r ♦♣t♠ r♥♦♠③ ♥ ♥r♠♥t ♥ Pr♦♥s♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s

❬❪ ♥ r♦ ♥ ♦♥q ❱♦ ♥ t ♦♠♣tt♦♥ ♦ ♣r♦ tr♥t♦♥s ♥ Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥♦♠♣tt♦♥ ♦♠tr② ♣s

❬❪ r r r ♥ r r r♠rs ♥ tr r③♦♣♦♠♣tt♦♥ ♦♠tr② ♦rt♠s ♥ ♣♣t♦♥s ♣r♥r❱rr♥ r♠♥② ♥ t♦♥

❬❪ ♥r♦ Prã♦ ♦ r♠♦ ♠♥♥♥ ♦♠tr② räsr ♦st♦♥

❬❪ ♦s ♥♦ ♥ rrt sr♥♥r ♥ ♥r♠♥t ♦rt♠♦r tt ♥♠rs ♦ s♠♣ ♦♠♣①s ♦♥ t s♣r ♦♠♣tr ♦♠tr s♥

❬❪ ♥ r ♠♥♥♦ ♥ ♦♥r♥ ♦ t ♦② ♦rt♠ ♦r ♦♠♣t♥ ♥tr♦ ❱♦r♦♥♦ tsst♦♥s ♦r♥ ♦♥♠r ♥②ss

❬❪ ♦rs ♥② r s♣èr ③st ♠ t♥t♠ts stst♥♥②

❬❪ rst♦♣ ♣t s♦rt♥ ♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥

tst♦❴t♠❴♠♥♣st♠P♣t♦rt♥

P❨

❬❪ r rs ♥② rr② ♥tr♥t♦♥ ♦r♥ ♦ ♦♥t♦♥s ♦ ♦♠♣tr ♥

❬❪ ♥♥ rts ♥ Ptr ❱ ♦♥② ②♥♠ ♦♠tr② ♦r ♠ts ss♣t ♣rt ②♥♠s tt♣①①①♥♦s♦♥♠t

❬❪ ♦ rrs ♥ ♥ s♦♥ r♦ tr♥t♦♥s ♥r②st♦r♣ r♦♣s Pr♦ t♠t ♥r

❬❪ ♦ P ♦♥ ♥ ♥ s♦♥ Pr♦ ♦♥ t♥s Pr♦t♠t ♥r

❬❪ rs sr ♥ ② ②♥♠ ♥② tr♥t♦♥ ♥ ♦rt♠ ①♣t t♠ ♣r ♦♣rt♦♥ ♦♠♣tt♦♥ ♦♠tr②♦r② ♥ ♣♣t♦♥s

❬❪ r rs t♥ sr ♥ ♦♥q s♣ ♦ s♣rs ♦♠tr t♦♦ t♦ ♥② t② rsts ♦♥ ❱♦r♦♥♦ r♠s ♥ Pr♦♥s♦ t t ♥♥ ♦♥r♥ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s rs♦♥ s sr ♣♦rt tt♣♥rr♥r

❬P❪ r rs ♥ ②♥ P♦♥ ♥t ①t ♦♠tr ♣rts ♦r♥② tr♥t♦♥s ♥ Pr♦♥s ♦ t t ❲♦rs♦♣ ♦♥ ♦rt♠♥♥r♥ ♥ ①♣r♠♥ts ♣s

❬P❪ r rs ②♥ P♦♥ ♥ ♦♥q ❲♥ ♥ tr♥t♦♥ ♥tr♥t♦♥ ♦r♥ ♦ ♦♥t♦♥s ♦ ♦♠♣tr ♥

❬❪ ♦rt r♠♥ ♥ r r t♦rs ♥♦♦ ♦ ♦♠tr♦♣♦♦② sr ♠str♠ ♦♥♦♥ Prs

❬❪ r rs ♥ ♦♥q Prtrt♦♥s ♥ rt① r♠♦♥ ♥② tr♥t♦♥ ♥ Pr♦♥s ♦ t t ②♠♣♦s♠ ♦♥ srt ♦rt♠s ♣s

❬❪ r rs ♥ ♦♥q Prtrt♦♥s ♥ rt① r♠♦♥ ♥② ♥ rr tr♥t♦♥s sr ♣♦rt

❬❪ r rs ♥ ♦♥q Prtrt♦♥s ♥ rt① r♠♦♥ ♥② ♥ rr tr♥t♦♥s ♦ ♣♣r ♥ sr♣♦rt

❬❪ ♦r♥s ♠♦♥ t tr♥sr s♥ ♥strtr rstt♣♦r♥tsPr♦♣s❲♦rs♦♣

❬q❪ ♥ q ♠♣②♦ ♦ ♦♥r② ♦♥t♦♥s tt♣

s♦♦r♦❲♥①♣♣♦♥r②❴♦♥t♦♥s

P❨

❬r❪ r♥♦ r♥ ts ♠♥s ♦ r♥r ♠trs tt♣

♦r♥tsPr♦♣s❲♦rs♦♣

❬❱❪ r ♦♥ ❱rèr ♥ ♦♥q ♥t r♣rs♥tt♦♥s ♦tr♥t♦♥s ♦ rtrr② ♠♥s♦♥ Prs♦♥ ♦♠♠♥t♦♥

❬②❪ ① ②r str ♥♦♥qr ♦rt♠ ♦r ♦♥strt♥ ♥② tr♥t♦♥s ♦rt♠

❬❨❪ r♥ r♥ ♥ rtt ❨♥ ♣ s♣s ♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥tst♦❴t♠❴♠♥

♣st♠P♣♣s

❬❪ rrt sr♥♥r r♣tr ♥ ♠♥ ♥ ts♣ ♦ st ♦ ♣♦♥ts ♥ t ♣♥ r♥st♦♥s ♦♥ ♥♦r♠t♦♥♦r②

❬❪ rrt sr♥♥r ♥ r♥st P ü r♠♥s♦♥ ♣ s♣s r♥st♦♥s ♦♥ r♣s ♥r②

❬❪ rrt sr♥♥r ♥ ♠s ♥r♠♥t t♦♣♦♦ ♣♣♥♦rs ♦r rr tr♥t♦♥s ♦rt♠

❬❪ rrt sr♥♥r ♥ ♠s r♥t♥ t♦♣♦♦ s♣s♥tr♥t♦♥ ♦r♥ ♦ ♦♠♣tt♦♥ ♦♠tr② ♥ ♣♣t♦♥s

❬♦r❪ t♥ ♦rt♥ s♣♥ ♦rt♠ ♦r ❱♦r♦♥♦ r♠s ♦rt♠

❬P❪ ♥rs r ♥ ②♥ P♦♥ ♥r ③② t♦♥ s♠ ♦r ①t♦♠tr ♦♠♣tt♦♥s ♥ Pr♦♥s ♦ t ♥ rr②♥tr ♦trs♥

❬❪ ♦ ♥ ♦♥q ♥r ♣r♦r♠♠♥ ♥ t rr② ♥ ♥♥ ♦ss♦♥♥t ♥ ♦♥q t♦rs t♦♠♣tt♦♥ ♦♠tr② ♦r rs ♥ rs ♣r♥r❱r t♠ts ♥ ❱s③t♦♥

❬❱❪ t♥ ♦rt♥ ♥ rst♦♣r ❱♥ ❲② tt ♥②ss ②s ♥t ①t ♥tr rt♠t ♦r ♦♠♣tt♦♥ ♦♠tr② r♥st♦♥s ♦♥ r♣s ②

❬❲❪ ❳ ♥ ❳♥t♦ ❲♥ s♦♠trs ♥ srt s♦♠tr② sr♦♣s ♦②♣r♦ s♣s s♦ t♠t ♦r♥

❬❪ r♦ r ♥ ♦♥tr①♠♣ t♦ ♥s ♦♥tr ♦♥ ♠♥♠ srs P♦s♦♣ ③♥ ttrs

❬♣❪ ♣ r♦♣s ♦rt♠s ♣r♦r♠♠♥ s②st♠ ♦r ♦♠♣tt♦♥ srt r tt♣♣s②st♠♦r

P❨

❬❪ ♦♥ r♠ rt♥ röts ♥ ás③ó ♦ás③ t♦rs ♥♦♦♦ ♦♠♥t♦rs sr ♠str♠ s♥♥ ❨♦r

❬❱❪ r ♠♠ r ♠ ♣ ♦♥s♦♥ ♥ ♦♥ ❱sss s♥Pttr♥s s♦♥❲s② ♥

❬❪ ♦♥s s ♦♥ ♥t ♥ rr ♥♦♠③ ♥r♠♥t ♦♥strt♦♥ ♦ ♥② ♥ ❱♦r♦♥♦ r♠s ♦rt♠

❬❪ r r♠ ♥ rt♦ árq③ ♦♠♣tt♦♥ ♦♠tr② ♦♥ rsr ♠ Psrs

❬♦❪ ♦r ❲t r② ♦♠♣tr s♥tst s♦ ♥♦ ♦t ♦t♥♣♦♥t rt♠t ♦♠♣t♥ r②s r

❬r❪ ♦♥ rts♦r♥ r ♦♠tr② ♦♠ ♦ rt t①ts ♥t♠ts ♣r♥r❱r ❨♦r

❬t❪ ♥ tr r ♦♣♦♦② ♠r ❯♥rst② Prss

❬♥❪ ♥ ♦♠♥t♦r ♥tr♦t♦♥ t♦ ♦♣♦♦② ❲ r♠♥♥ r♥s♦

❬❲❪ ❲ ♥t③s ♥ ❲♥t r♠♥s♦♥ s ♠♦r♠♥t♠ts ♥♥♥

❬❪ t♥r ♦r ♥r② ♦t♥ ♣♦♥t rt♠t t 754 −1985 ❨♦r ❨ ♣r♥t ♥ P ♦ts

❬❪ s P t♥r ♦r ♦t♥P♦♥t rt♠t ❨♦r ❨ ❯ st

❬t❪ t♥ t♦ s♥ r♦♣s ❯♥rst② ♦ ♦ Prss ♦♥ ♦♥♦♥

❬❪ ❲♠ ♦ t♦ ①② ♦r♠t♦♥ ♣r♥r Pr①s Ps♥ str ❯

❬♣❪ ♥r ♣rt tr♥t ♦♥②♦♠ tt♣♥♣

♦rr♥t❴♦tr♣

❬❪ ❲♦♥ ü♥ ♥ ♥tr ss♠♥♥ r♦♠♦r tsst♦♥ ♦ s♣ ♥ ♣rtr rt① tr♥t♦♥ ♦ t ♠♥s♦♥t♦rs ♠♥sr♣t ♠t♠t

❬P+❪ t③ tt♥r rt ♦r♥ ②♥ P♦♥ t♥ rr ♥ ❨♣ssr♦♦♠ ①♠♣s ♦ r♦st♥ss ♣r♦♠s ♥ ♦♠tr ♦♠♣tt♦♥s ♥Pr♦♥s ♦ t t r♦♣♥ ②♠♣♦s♠ ♦♥ ♦rt♠s ♦♠ ♦tr ♦ts ♥ ♦♠♣tr ♥ ♣s ♣r♥r❱r

P❨

❬r❪ s P r②t sst♦♥s ♥ r♥r ♠trs tt♣

s♦♣♥rr♦♠tr♦♦rt♦♥sr♣r♦r♠t♠

❬❪ ♦♥ ♥tr♦t♦♥ t♦ ♦♣♦♦ ♥♦s ♣r♥r❱r ❨♦r

❬♦❪ trt P ♦② st sqr q♥t③t♦♥ ♥ P r♥st♦♥s ♦♥♥♦r♠t♦♥ ♦r②

❬❪ ♥ tr ♦ ♦rt♠s ♦r ♦♥strt♥ ♥② tr♥t♦♥s ♥tr♥t♦♥ ♦r♥ ♦ ♦♠♣tr ♥ ♥♦r♠t♦♥ ♥s

❬②♥❪ ♦r ②♥♦♥ ♥ ♥s ♦rt♠ t♠ts ♥♥♥

❬❪ s ♥ t ♦♠♦♥♦s ♣♥r P♦ss♦♥ ♣♦♥t♣r♦ss t♠t♦s♥s

❬ö❪ st ös tt♥♥ s ös ss ❩r ♦r rP♦②ër ♥ r ♠♥trr♥tst s♠♠t ❲r ♣s

❬♦❪ rt♥ ♦s♥ Pr♦t② ♥ t s♥ ♦ ♦♥ s♦s tt♣♦r♥tsPr♦♣s❲♦rs♦♣

❬❪ rs ③ó♥ ♥ ♦♠ás ♦ ❱♦r♦♥♦ r♠s ♦♥ ♦r♦s ♦♠♣tt♦♥ ♦♠tr② ♦r② ♥ ♣♣t♦♥s

❬❪ ②års ♥ Ptr ♠♥s♦♥ r♠♥s♦♥ ♣r♦❱♦r♦♥♦ r♥ ♠♦s ♥ ♠r♦♠♥ s♠t♦♥s ♦ t♦♣sst ♦♠♣tt♦♥ trs ♥

❬❪ ②♦♥ ♥♠ ♥ tr ♦♥ ❱♦r♦♥♦ r♠s ♦♥t s♣r ♦♠♣tt♦♥ ♦♠tr② ♦r② ♥ ♣♣t♦♥s

❬❪ r♥ s♥ ♥ r ♦ ②♣r♦ ❱♦r♦♥♦ r♠s ♠ s②♦♠♣t♥ sr ♣♦st♦r② ♦ s

❬♣❪ P♦♥r s ②♣r♦ ♣r ♥s tt♣♥♣♦r

P♦♥r❴s❴②♣r♦❴♣r❴♥ss

❬P❪ ②♥ P♦♥ ♥ ♦♥q tr♥t♦♥ t strtr ♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥tst♦❴t♠❴♠♥♣st♠P

❬P❪ ②♥ P♦♥ ♥ ♦♥q tr♥t♦♥s ♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥tst♦❴t♠❴♠♥

♣st♠Pr♥t♦♥

P❨

❬♦❪ ❱♥ss ♦♥s tt ♥♠r s♥trs ♦ ♦♠♦♥♦s P♦ss♦♥ ♣♦♥t♣r♦sss P②s

❬❨❪ r♥t ♥ té♣♥ ② ♥ rtt ❨♥ ♠s ♥rt♦♥ ♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥tst♦❴t♠❴♠♥♣st♠Ps❴

❬♣❪ ♣♣rt ♥② r♥♠♥t ♦rt♠ ♦r qt② ♠♥s♦♥ ♠s♥rt♦♥ ♦r♥ ♦ ♦rt♠s

❬❱❪ ü♥tr ♦t ♥ rt ❱tr ♦♠♣tt♦♥ t♦♣♦♦② ♥ ♥tr♦t♦♥♥ ♥♥ ♦ss♦♥♥t ♥ ♦♥q t♦rs t ♦♠♣tt♦♥ ♦♠tr② ♦r rs ♥ rs ♣r♥r❱r t♠ts♥ ❱s③t♦♥

❬❨❪ r♥t ♥ ♥ rtt ❨♥ s♥ ♦♠♥s ♦♥ ②♣s s♠♦♦t srs ♥ Pr♦♥s ♦ t t ♥tr♥t♦♥ s♥♦♥t ♣s

❬❨❪ r♥t ♥ ♥ rtt ❨♥ sr ♠s ♥rt♦♥♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥tst♦❴t♠❴

♠♥♣st♠Prsr

❬♥❪ s ♥tó ♥tr ♦♠tr② ♥ ♦♠tr Pr♦t② s♦♥❲s② ♥

❬❪ ♥ ♦♥ ♥♥t ♣r♦ ♠♥♠ srs t♦t s♥trst♦♥♥ ♦t

❬❪ ♦♥t♥ ♦♥t♦♥ r♥t♥ t ①st♥ ♦ r♠♥s♦♥ ♦♥str♥ ♥② tr♥t♦♥s ♥ Pr♦♥s ♦ t t♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s

❬❪ ♦♥t♥ trr ♠s ♥rt♦♥ ② ♥② r♥♠♥t ♥ Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s

❬❪ ♦♥t♥ ♣ ♦rt♠s ♦r ♦♥strt♥ r♠♥s♦♥ ♦♥str♥ ♥② tr♥t♦♥s ♥ Pr♦♥s ♦ t t♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥ ♦♠tr② ♣s

❬♠❪ rs ♠s ♦♠♣t♥ t ♥t② Prs♥t r♦♣s ♠r❯♥rst② Prss ♠r ❯

❬♦❪ ♦♥ ♦♥♥ ♥②♦♣ ♦ ♥tr sq♥s tt♣

rsrtt♦♠⑦♥ssq♥s

❬♦❪ rr② ♦s ♣rsst♥t ♦s♠ ♥ ts ♠♥t strtr ♦r② ♥ ♠♣♠♥tt♦♥ r❳

P❨

❬♣❪ ♥ ♣♥r r ♦♣♦♦② r ♦♦ ♦♠♣♥② ❨♦r

❬♣r❪ ❱♦r ♣r♥ ♦♠♣tt♦♥ ②♥♠s ♥ ❱♦r♦♥♦ tsst♦♥s tt♣s♦♣♥rr♦♠tr♦♦rt♦♥sr

♣r♦r♠t♠

❬♣r❪ ❱♦r ♣r♥ ♣r s ♠♦ ♥♥r♥t ♦s♠♦♦ ②r♦②♥♠ s♠t♦♥s ♦♥ ♠♦♥ ♠s ♦♥t② ♦ts ♦ t ♦②str♦♥♦♠ ♦t②

❬t❪ ♦♥ t ♦♠tr② ♦ rs ♣r♥r❱r ❨♦r

❬❪ ♦♥ ♥ s tt♣t♦rs♠t♠ss♦tr

❬♦❪ ❲♠ ♦♠s♦♥ ♥ t s♦♥ ♦ s♣ t ♠♥♠♠ ♣rtt♦♥ rP♦s♦♣ ③♥

❬♦❪ rst♥ ♦♠♣s♦♥ st ♥ r♦st ♥② tsst♦♥ ♥ ♣r♦♦♠♥s ♥tr♥t♦♥ ♦r♥ ♦r ♠r t♦s ♥ ♥♥r♥

❬r❪ ❲♠ r r♣♣♥r ♥♥♥ r äss♥ ♠r ❲ss♥st♥ t♠tsP②ss ss

❬❪ ❲♠ rst♦♥ r♠♥s♦♥ ♦♠tr② ♥ ♦♣♦♦② ❱♦♠ Pr♥t♦♥ ❯♥rst② Prss rs②

❬❪ ❲♠ P rst♦♥ ♦♠tr② ♥ ♦♣♦♦② ♦ r♥♦s tt♣♠sr♦r♣t♦♥s♦♦st♠

❬♦❪ ♥ ♦r♥♦s ♣t♠st♦♥ ♠s ès ♦t♦rt ♥ s♥s❯♥rsté ♦♣♥t♣♦s r♥

❬❱❪ ést♥ ❱tt Prs♦♥ ♦♠♠♥t♦♥ tt♣rts

♥s②♦♥r⑦tt

❬❲❪ ♥ ♥ ❲rt ♥ ❲♠ ♣ ♦s♠ ♦♠tr ♥②ss tt♣str♦r♥⑦②rtt♠♣t♦♥

②♣

❬❲❱P+❪ ♥ ♥ ❲rt rt ❱tr r♥ Pt♥ ♦ r♥ ♥ ♦rt♦ ♣ s♣ t♦♣♦♦② ♦ t ♦s♠ r❳❱

❬❱❨❪ rt ❱tr ♥ ❨♣ ♦♠♣tt♦♥ ♦♠♣①t② ♦ ♦♠♥t♦rsrs ♥ Pr♦♥s ♦ t t ♥♥ ②♠♣♦s♠ ♦♥ ♦♠♣tt♦♥♦♠tr② ♣s

❬❲t❪ ❲ts♦♥ ♦♠♣t♥ t n♠♥s♦♥ ♥② tsst♦♥ t♣♣t♦♥s t♦ ❱♦r♦♥♦ ♣♦②t♦♣s ♦♠♣tr ♦r♥

P❨

❬❲❪ ❲ss ♦ ②r♦♣♦ ♠♥strr♥ ts srr♦♥♥ tr strtr ♦♠tr ♠♥r tt♣s♦♣♥rr♦♠tr r

❬❲❪ P ❲s♦♥ r ♣s ♠r ❯♥rst② Prss ♠r

❬❨❪ rtt ❨♥ tr♥t♦♥s ♥ ❯sr ♥ r♥ ♥ t♦r ♦r t♦♥ tt♣♦r♥

tst♦❴t♠❴♠♥♣st♠Pr♥t♦♥

❬❩♦♠❪ r ❩♦♠♦r♦♥ ♦♣♦♦② ♦r ♦♠♣t♥ ♠r ❯♥rst② Prss♠r

P❨