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srt sts tt - - PowerPoint PPT Presentation

Mar 13, 2023 291 likes •904 views

tt s ssts srt sts tt t t sr

slide-1
SLIDE 1

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s

❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

▲✐✈❛t ❚②❛♣❛❡✈

◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❙❛r❛t♦✈ ❙t❛t❡ ❯♥✐✈❡rs✐t②

■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ p✲❆❉■❈ ▼❆❚❍❊▼❆❚■❈❆▲ P❍❨❙■❈❙ ❆◆❉ ■❚❙ ❆PP▲■❈❆❚■❖◆❙ ✭p✲❆❉■❈❙✳✷✵✶✺✮✱ ❇❡❧❣r❛❞❡✱ ❙❡♣t❡♠❜❡r ✽✱ ✷✵✶✺

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-2
SLIDE 2

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❚❤❡ ❛✉t♦♠❛t♦♥ tr❛♥s❢♦r♠❛t✐♦♥s ♦✈❡r ❛❧♣❤❛❜❡t Fp = {0, 1, . . . , p − 1}✱ ✇❤❡r❡ p ♣r✐♠❡ ♥✉♠❜❡r ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ r✐♥❣ ♦❢ p✲❛❞✐❝ ✐♥t❡❣❡rs Zp ✐♥ p✲❛❞✐❝ ♠❡tr✐❝ t❤❛t s❛t✐s❢② t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ ✇✐t❤ ❛ ❝♦♥st❛♥t ✶✳ ❆♥ ✭s②♥❝❤r♦♥♦✉s✮ ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ✻✲t✉♣❧❡ ✇❤❡r❡ ✐s ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t✱ ✐s ❛ s❡t ♦❢ st❛t❡s✱ ✐s ❛♥ ♦✉t♣✉t ❛❧♣❤❛❜❡t✱ ✐s ❛ st❛t❡ ✉♣❞❛t❡ ♠❛♣✱ ✐s ❛♥ ♦✉t♣✉t ♠❛♣✱ ✐s ❛♥ ✐♥✐t✐❛❧ st❛t❡✳ ◆♦t❡ t❤❛t ❛r❡ ✜♥✐t❡ ❛❧♣❤❛❜❡ts✱ ❤♦✇❡✈❡r ❝♦✉❧❞ ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ♦❢ st❛t❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-3
SLIDE 3

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❚❤❡ ❛✉t♦♠❛t♦♥ tr❛♥s❢♦r♠❛t✐♦♥s ♦✈❡r ❛❧♣❤❛❜❡t Fp = {0, 1, . . . , p − 1}✱ ✇❤❡r❡ p ♣r✐♠❡ ♥✉♠❜❡r ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ r✐♥❣ ♦❢ p✲❛❞✐❝ ✐♥t❡❣❡rs Zp ✐♥ p✲❛❞✐❝ ♠❡tr✐❝ t❤❛t s❛t✐s❢② t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ ✇✐t❤ ❛ ❝♦♥st❛♥t ✶✳ ❆♥ ✭s②♥❝❤r♦♥♦✉s✮ ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ✻✲t✉♣❧❡ A = (I, S, O, S, O, s0) ✇❤❡r❡ I ✐s ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t✱ S ✐s ❛ s❡t ♦❢ st❛t❡s✱ O ✐s ❛♥ ♦✉t♣✉t ❛❧♣❤❛❜❡t✱ S : I × S → S ✐s ❛ st❛t❡ ✉♣❞❛t❡ ♠❛♣✱ O : I × S → O ✐s ❛♥ ♦✉t♣✉t ♠❛♣✱s0 ∈ S ✐s ❛♥ ✐♥✐t✐❛❧ st❛t❡✳ ◆♦t❡ t❤❛t ❛r❡ ✜♥✐t❡ ❛❧♣❤❛❜❡ts✱ ❤♦✇❡✈❡r ❝♦✉❧❞ ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ♦❢ st❛t❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-4
SLIDE 4

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❚❤❡ ❛✉t♦♠❛t♦♥ tr❛♥s❢♦r♠❛t✐♦♥s ♦✈❡r ❛❧♣❤❛❜❡t Fp = {0, 1, . . . , p − 1}✱ ✇❤❡r❡ p ♣r✐♠❡ ♥✉♠❜❡r ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ r✐♥❣ ♦❢ p✲❛❞✐❝ ✐♥t❡❣❡rs Zp ✐♥ p✲❛❞✐❝ ♠❡tr✐❝ t❤❛t s❛t✐s❢② t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ ✇✐t❤ ❛ ❝♦♥st❛♥t ✶✳ ❆♥ ✭s②♥❝❤r♦♥♦✉s✮ ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ✻✲t✉♣❧❡ A = (I, S, O, S, O, s0) ✇❤❡r❡ I ✐s ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t✱ S ✐s ❛ s❡t ♦❢ st❛t❡s✱ O ✐s ❛♥ ♦✉t♣✉t ❛❧♣❤❛❜❡t✱ S : I × S → S ✐s ❛ st❛t❡ ✉♣❞❛t❡ ♠❛♣✱ O : I × S → O ✐s ❛♥ ♦✉t♣✉t ♠❛♣✱s0 ∈ S ✐s ❛♥ ✐♥✐t✐❛❧ st❛t❡✳ ◆♦t❡ t❤❛t I, O ❛r❡ ✜♥✐t❡ ❛❧♣❤❛❜❡ts✱ ❤♦✇❡✈❡r S ❝♦✉❧❞ ❜❡ ❛♥ ✐♥✜♥✐t❡ s❡t ♦❢ st❛t❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-5
SLIDE 5

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❆♥ ✭s②♥❝❤r♦♥♦✉s✮ ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ✻✲t✉♣❧❡ A = (I, S, O, S, O, s0) ✇❤❡r❡ I ✐s ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t✱ S ✐s ❛ s❡t ♦❢ st❛t❡s✱ O ✐s ❛♥ ♦✉t♣✉t ❛❧♣❤❛❜❡t✱ S : I × S → S ✐s ❛ st❛t❡ ✉♣❞❛t❡ ♠❛♣✱ O : I × S → O ✐s ❛♥ ♦✉t♣✉t ♠❛♣✱s0 ∈ S ✐s ❛♥ ✐♥✐t✐❛❧ st❛t❡✳ ▲❡t✬s ❝♦♥s✐❞❡r ♦♥❧② ❛❝❝❡ss✐❜❧❡ ❛✉t♦♠❛t❛✿ ✇❤❡r❡ ❛♥② st❛t❡ s ∈ S ♦❢ ❛✉t♦♠❛t♦♥ A ✐s r❡❛❝❤❛❜❧❡ ❢r♦♠ ✐♥✐t✐❛❧ st❛t❡ s0 ❛❢t❡r ❛ ✜♥✐t❡ ✐♥♣✉t ✇♦r❞ u ✇❛s ❢❡❞ t♦ t❤❡ ❛✉t♦♠❛t♦♥✳ ❲❡ ❛ss✉♠❡ ❢✉rt❤❡r t❤❛t ✳ ❆s ❡✈❡r② ❛✉t♦♠❛t♦♥ tr❛♥s❢♦r♠s t❤❡ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥t♦ ♦✉t♣✉t s❡q✉❡♥❝❡✱ ✇❡ ♠❛② s❛② t❤❛t ❛✉t♦♠❛t♦♥ ♣❡r❢♦r♠s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ♦♥ ✳ ❆✉t♦♠❛t❛ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ✭♥♦♥✲❛✉t♦♥♦♠♦✉s✮ ❞②♥❛♠✐❝❛❧ s②st❡♠s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✲❛❞✐❝ ✐♥t❡❣❡rs ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 6

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❆♥ ✭s②♥❝❤r♦♥♦✉s✮ ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ✻✲t✉♣❧❡ A = (I, S, O, S, O, s0) ✇❤❡r❡ I ✐s ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t✱ S ✐s ❛ s❡t ♦❢ st❛t❡s✱ O ✐s ❛♥ ♦✉t♣✉t ❛❧♣❤❛❜❡t✱ S : I × S → S ✐s ❛ st❛t❡ ✉♣❞❛t❡ ♠❛♣✱ O : I × S → O ✐s ❛♥ ♦✉t♣✉t ♠❛♣✱s0 ∈ S ✐s ❛♥ ✐♥✐t✐❛❧ st❛t❡✳ ▲❡t✬s ❝♦♥s✐❞❡r ♦♥❧② ❛❝❝❡ss✐❜❧❡ ❛✉t♦♠❛t❛✿ ✇❤❡r❡ ❛♥② st❛t❡ s ∈ S ♦❢ ❛✉t♦♠❛t♦♥ A ✐s r❡❛❝❤❛❜❧❡ ❢r♦♠ ✐♥✐t✐❛❧ st❛t❡ s0 ❛❢t❡r ❛ ✜♥✐t❡ ✐♥♣✉t ✇♦r❞ u ✇❛s ❢❡❞ t♦ t❤❡ ❛✉t♦♠❛t♦♥✳ ❲❡ ❛ss✉♠❡ ❢✉rt❤❡r t❤❛t I = O = Fp✳ ❆s ❡✈❡r② ❛✉t♦♠❛t♦♥ A tr❛♥s❢♦r♠s t❤❡ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥t♦ ♦✉t♣✉t s❡q✉❡♥❝❡✱ ✇❡ ♠❛② s❛② t❤❛t ❛✉t♦♠❛t♦♥ A ♣❡r❢♦r♠s ❛ tr❛♥s❢♦r♠❛t✐♦♥ fs0 ♦♥ Zp✳ ❆✉t♦♠❛t❛ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ✭♥♦♥✲❛✉t♦♥♦♠♦✉s✮ ❞②♥❛♠✐❝❛❧ s②st❡♠s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✲❛❞✐❝ ✐♥t❡❣❡rs ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-7
SLIDE 7

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❆♥ ✭s②♥❝❤r♦♥♦✉s✮ ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ✻✲t✉♣❧❡ A = (I, S, O, S, O, s0) ✇❤❡r❡ I ✐s ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t✱ S ✐s ❛ s❡t ♦❢ st❛t❡s✱ O ✐s ❛♥ ♦✉t♣✉t ❛❧♣❤❛❜❡t✱ S : I × S → S ✐s ❛ st❛t❡ ✉♣❞❛t❡ ♠❛♣✱ O : I × S → O ✐s ❛♥ ♦✉t♣✉t ♠❛♣✱s0 ∈ S ✐s ❛♥ ✐♥✐t✐❛❧ st❛t❡✳ ▲❡t✬s ❝♦♥s✐❞❡r ♦♥❧② ❛❝❝❡ss✐❜❧❡ ❛✉t♦♠❛t❛✿ ✇❤❡r❡ ❛♥② st❛t❡ s ∈ S ♦❢ ❛✉t♦♠❛t♦♥ A ✐s r❡❛❝❤❛❜❧❡ ❢r♦♠ ✐♥✐t✐❛❧ st❛t❡ s0 ❛❢t❡r ❛ ✜♥✐t❡ ✐♥♣✉t ✇♦r❞ u ✇❛s ❢❡❞ t♦ t❤❡ ❛✉t♦♠❛t♦♥✳ ❲❡ ❛ss✉♠❡ ❢✉rt❤❡r t❤❛t I = O = Fp✳ ❆s ❡✈❡r② ❛✉t♦♠❛t♦♥ A tr❛♥s❢♦r♠s t❤❡ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥t♦ ♦✉t♣✉t s❡q✉❡♥❝❡✱ ✇❡ ♠❛② s❛② t❤❛t ❛✉t♦♠❛t♦♥ A ♣❡r❢♦r♠s ❛ tr❛♥s❢♦r♠❛t✐♦♥ fs0 ♦♥ Zp✳ ❆✉t♦♠❛t❛ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ✭♥♦♥✲❛✉t♦♥♦♠♦✉s✮ ❞②♥❛♠✐❝❛❧ s②st❡♠s ♦♥ t❤❡ s♣❛❝❡ ♦❢ p✲❛❞✐❝ ✐♥t❡❣❡rs Zp✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-8
SLIDE 8

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❋♦r ❡✈❡r② ❛❝❝❡ss✐❜❧❡ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t❛ As = (I, S, O, S, O, s)✱ ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛♠✐❧② F = {fs : s ∈ S} ♦❢ ✶✲▲✐♣s❝❤✐t③ tr❛♥s❢♦r♠❛t✐♦♥s fAs✱ s ∈ S ♦♥ Zp✳ ❘❡♠✐♥❞❡r✿ tr❛♥s✐t✐✈✐t② ♦❢ ❢❛♠✐❧✐❡s ♦❢ ♠❛♣♣✐♥❣s ❆ ❢❛♠✐❧② ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ♦♥ t❤❡ s❡t ✐s ❝❛❧❧❡❞ tr❛♥s✐t✐✈❡ ✐❢ ❢♦r ❛♥② ♣❛✐r t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✳ ◆♦t❡ t❤❛t ❛ ❜✐❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s s❛✐❞ t♦ ❜❡ tr❛♥s✐t✐✈❡✱ ✇❤❡♥❡✈❡r t❤❡ ❢❛♠✐❧② ✐s tr❛♥s✐t✐✈❡✳ ❈♦♠♣❧❡t❡ tr❛♥s✐t✐✈❡ ❆✉t♦♠❛t♦♥ ✐s s❛✐❞ t♦ ❜❡ ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡✱ ✐❢ ❢❛♠✐❧② ✱ ✐s tr❛♥s✐t✐✈❡ ♦✈❡r ✱ ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-9
SLIDE 9

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❋♦r ❡✈❡r② ❛❝❝❡ss✐❜❧❡ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t❛ As = (I, S, O, S, O, s)✱ ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛♠✐❧② F = {fs : s ∈ S} ♦❢ ✶✲▲✐♣s❝❤✐t③ tr❛♥s❢♦r♠❛t✐♦♥s fAs✱ s ∈ S ♦♥ Zp✳ ❘❡♠✐♥❞❡r✿ tr❛♥s✐t✐✈✐t② ♦❢ ❢❛♠✐❧✐❡s ♦❢ ♠❛♣♣✐♥❣s ❆ ❢❛♠✐❧② F ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ♦♥ t❤❡ s❡t M ✐s ❝❛❧❧❡❞ tr❛♥s✐t✐✈❡ ✐❢ ❢♦r ❛♥② ♣❛✐r (a, b) ∈ M × M t❤❡r❡ ❡①✐sts f ∈ F s✉❝❤ t❤❛t f(a) = b✳ ◆♦t❡ t❤❛t ❛ ❜✐❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥ f : M → M ✐s s❛✐❞ t♦ ❜❡ tr❛♥s✐t✐✈❡✱ ✇❤❡♥❡✈❡r t❤❡ ❢❛♠✐❧② {e, f±1, f±2, . . .} ✐s tr❛♥s✐t✐✈❡✳ ❈♦♠♣❧❡t❡ tr❛♥s✐t✐✈❡ ❆✉t♦♠❛t♦♥ ✐s s❛✐❞ t♦ ❜❡ ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡✱ ✐❢ ❢❛♠✐❧② ✱ ✐s tr❛♥s✐t✐✈❡ ♦✈❡r ✱ ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-10
SLIDE 10

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❋♦r ❡✈❡r② ❛❝❝❡ss✐❜❧❡ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t❛ As = (I, S, O, S, O, s)✱ ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛♠✐❧② F = {fs : s ∈ S} ♦❢ ✶✲▲✐♣s❝❤✐t③ tr❛♥s❢♦r♠❛t✐♦♥s fAs✱ s ∈ S ♦♥ Zp✳ ❘❡♠✐♥❞❡r✿ tr❛♥s✐t✐✈✐t② ♦❢ ❢❛♠✐❧✐❡s ♦❢ ♠❛♣♣✐♥❣s ❆ ❢❛♠✐❧② F ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ♦♥ t❤❡ s❡t M ✐s ❝❛❧❧❡❞ tr❛♥s✐t✐✈❡ ✐❢ ❢♦r ❛♥② ♣❛✐r (a, b) ∈ M × M t❤❡r❡ ❡①✐sts f ∈ F s✉❝❤ t❤❛t f(a) = b✳ ◆♦t❡ t❤❛t ❛ ❜✐❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥ f : M → M ✐s s❛✐❞ t♦ ❜❡ tr❛♥s✐t✐✈❡✱ ✇❤❡♥❡✈❡r t❤❡ ❢❛♠✐❧② {e, f±1, f±2, . . .} ✐s tr❛♥s✐t✐✈❡✳ ❈♦♠♣❧❡t❡ tr❛♥s✐t✐✈❡ ❆✉t♦♠❛t♦♥ A ✐s s❛✐❞ t♦ ❜❡ ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡✱ ✐❢ ❢❛♠✐❧② fs mod pn : Z/pnZ → Z/pnZ✱ s ∈ S ✐s tr❛♥s✐t✐✈❡ ♦✈❡r Z/pnZ✱ n = 1, 2, 3, . . .✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 11

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆✉t♦♠❛t❛ ❛♥❞ ❞②♥❛♠✐❝s

❱✳❙✳ ❆♥❛s❤✐♥ s❤♦✇❡❞ t❤❛t ❛♥ ❛✉t♦♠❛t♦♥ A ✐s ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ α(fs0) = 1 ❢♦r ❛ ❝❧♦s✉r❡ ♦❢ ❛❧❧ t❤❡ ♣♦✐♥ts ( x mod pk

pk

, fs0(x) mod pk

pk

)✱ x ∈ Zp✱ k = 1, 2, 3, . . . ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✉♥✐t sq✉❛r❡ E2 = [0, 1] × [0, 1] ⊂ R2✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-12
SLIDE 12

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

▲❡t✬s ❡♥✉♠❡r❛t❡ s②♠❜♦❧s ♦❢ t❤❡ ❛❧♣❤❛❜❡t Fp = {0, 1, . . . , p − 1} ✇✐t❤ ♥❛t✉r❛❧ ♥✉♠❜❡rs αi ∈ F = {1, . . . , p}✳ ◆❡①t ❧❡t✬s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✇♦r❞ u = αk−1 . . . α1α0 ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t F t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r u = α0 + α1

p+1 + . . . + αk−1 (p+1)k−1 ✳

❲❡ ❝♦♥s✐❞❡r ❛❧❧ ♣♦✐♥ts ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ sq✉❛r❡ ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ r✉♥s t❤r♦✉❣❤ ❛❧❧ ✜♥✐t❡ ✇♦r❞s✳ ❚❤❡ s❡t ♦❢ t❤❡s❡ ♣♦✐♥ts ✐s ❝❛❧❧❡❞ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ ✳ ❚❤❡ ❝❡rt❛✐♥ ✐♥t❡r❡st ❢♦r ✉s ♣r❡s❡♥ts t❤❡ t❛s❦ t♦ ❞❡s❝r✐❜❡ ❛ tr❛♥s✐t✐✈❡ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t♦♥ ♠❛♣s ❜② ♠❡❛♥s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-13
SLIDE 13

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

▲❡t✬s ❡♥✉♠❡r❛t❡ s②♠❜♦❧s ♦❢ t❤❡ ❛❧♣❤❛❜❡t Fp = {0, 1, . . . , p − 1} ✇✐t❤ ♥❛t✉r❛❧ ♥✉♠❜❡rs αi ∈ F = {1, . . . , p}✳ ◆❡①t ❧❡t✬s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✇♦r❞ u = αk−1 . . . α1α0 ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t F t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r u = α0 + α1

p+1 + . . . + αk−1 (p+1)k−1 ✳ ❲❡ ❝♦♥s✐❞❡r ❛❧❧ ♣♦✐♥ts ♦❢

t❤❡ ❊✉❝❧✐❞❡❛♥ sq✉❛r❡ Γ = [1, p + 1) × [1, p + 1) ⊂ R2 ♦❢ t❤❡ ❢♦r♠ ( u, fs0( u)) ✇❤❡r❡ u r✉♥s t❤r♦✉❣❤ ❛❧❧ ✜♥✐t❡ ✇♦r❞s✳ ❚❤❡ s❡t ♦❢ t❤❡s❡ ♣♦✐♥ts Ω(fs0) ⊂ Γ ✐s ❝❛❧❧❡❞ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ A✳ ❚❤❡ ❝❡rt❛✐♥ ✐♥t❡r❡st ❢♦r ✉s ♣r❡s❡♥ts t❤❡ t❛s❦ t♦ ❞❡s❝r✐❜❡ ❛ tr❛♥s✐t✐✈❡ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t♦♥ ♠❛♣s ❜② ♠❡❛♥s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-14
SLIDE 14

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

▲❡t✬s ❡♥✉♠❡r❛t❡ s②♠❜♦❧s ♦❢ t❤❡ ❛❧♣❤❛❜❡t Fp = {0, 1, . . . , p − 1} ✇✐t❤ ♥❛t✉r❛❧ ♥✉♠❜❡rs αi ∈ F = {1, . . . , p}✳ ◆❡①t ❧❡t✬s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✇♦r❞ u = αk−1 . . . α1α0 ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t F t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡r u = α0 + α1

p+1 + . . . + αk−1 (p+1)k−1 ✳ ❲❡ ❝♦♥s✐❞❡r ❛❧❧ ♣♦✐♥ts ♦❢

t❤❡ ❊✉❝❧✐❞❡❛♥ sq✉❛r❡ Γ = [1, p + 1) × [1, p + 1) ⊂ R2 ♦❢ t❤❡ ❢♦r♠ ( u, fs0( u)) ✇❤❡r❡ u r✉♥s t❤r♦✉❣❤ ❛❧❧ ✜♥✐t❡ ✇♦r❞s✳ ❚❤❡ s❡t ♦❢ t❤❡s❡ ♣♦✐♥ts Ω(fs0) ⊂ Γ ✐s ❝❛❧❧❡❞ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ A✳ ❚❤❡ ❝❡rt❛✐♥ ✐♥t❡r❡st ❢♦r ✉s ♣r❡s❡♥ts t❤❡ t❛s❦ t♦ ❞❡s❝r✐❜❡ ❛ tr❛♥s✐t✐✈❡ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t♦♥ ♠❛♣s ❜② ♠❡❛♥s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-15
SLIDE 15

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❊①❛♠♣❧❡ ♦❢ ❣❡♦♠❡♥t✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥✿

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-16
SLIDE 16

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❋♦r ❡✈❡r② st❛t❡ s ∈ S ♦❢ ❛♥ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ♠❛♣ Rs : Fp → Fp t❤❛t tr❛♥s❢♦r♠s ✐♥♣✉t s②♠❜♦❧ x ∈ Fp ✐♥t♦ ♦✉t♣✉t s②♠❜♦❧ y ∈ Fp✳ ◆♦t❡✱ t❤❛t Rs ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❛ s✉r❥❡❝t✐✈❡ ♠❛♣✳ ❈♦♥s✐❞❡r ❛♥ ❛✉t♦♠❛t♦♥ ❛♥❞ t❤❡ ❢❛♠✐❧② ✳ ❆ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ❡❛❝❤ st❛t❡ ♦❢ ❛ ♠❛♣ ❝r❡❛t❡s ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ ✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛✉t♦♠❛t❛ ❝❧❛ss t❤❛t ✐s ❝♦♥str✉❝t❡❞ t❤✐s ✇❛②✳ ❉❡♥♦t❡ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ ✳ ❚❤❡♦r❡♠ ✶ ✭▲✳❚✳✱ ✷✵✶✸✮ ❆♥ ❛✉t♦♠❛t♦♥ ✐s ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t❤❡ ❛✉t♦♠❛t♦♥ ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ✱ s✉❝❤ t❤❛t ❛♥❞ ❛r❡ ❛✣♥❡ ❡q✉✐✈❛❧❡♥ts✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-17
SLIDE 17

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❋♦r ❡✈❡r② st❛t❡ s ∈ S ♦❢ ❛♥ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ♠❛♣ Rs : Fp → Fp t❤❛t tr❛♥s❢♦r♠s ✐♥♣✉t s②♠❜♦❧ x ∈ Fp ✐♥t♦ ♦✉t♣✉t s②♠❜♦❧ y ∈ Fp✳ ◆♦t❡✱ t❤❛t Rs ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❛ s✉r❥❡❝t✐✈❡ ♠❛♣✳ ❈♦♥s✐❞❡r ❛♥ ❛✉t♦♠❛t♦♥ A ❛♥❞ t❤❡ ❢❛♠✐❧② {Rs : s ∈ S}✳ ❆ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ❡❛❝❤ st❛t❡ ♦❢ ❛ ♠❛♣ ❝r❡❛t❡s ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ ✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛✉t♦♠❛t❛ ❝❧❛ss t❤❛t ✐s ❝♦♥str✉❝t❡❞ t❤✐s ✇❛②✳ ❉❡♥♦t❡ ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ ✳ ❚❤❡♦r❡♠ ✶ ✭▲✳❚✳✱ ✷✵✶✸✮ ❆♥ ❛✉t♦♠❛t♦♥ ✐s ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t❤❡ ❛✉t♦♠❛t♦♥ ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ✱ s✉❝❤ t❤❛t ❛♥❞ ❛r❡ ❛✣♥❡ ❡q✉✐✈❛❧❡♥ts✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-18
SLIDE 18

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❋♦r ❡✈❡r② st❛t❡ s ∈ S ♦❢ ❛♥ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ♠❛♣ Rs : Fp → Fp t❤❛t tr❛♥s❢♦r♠s ✐♥♣✉t s②♠❜♦❧ x ∈ Fp ✐♥t♦ ♦✉t♣✉t s②♠❜♦❧ y ∈ Fp✳ ◆♦t❡✱ t❤❛t Rs ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❛ s✉r❥❡❝t✐✈❡ ♠❛♣✳ ❈♦♥s✐❞❡r ❛♥ ❛✉t♦♠❛t♦♥ A ❛♥❞ t❤❡ ❢❛♠✐❧② {Rs : s ∈ S}✳ ❆ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ❡❛❝❤ st❛t❡ s ∈ S ♦❢ ❛ ♠❛♣ Rs ❝r❡❛t❡s ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ B✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛✉t♦♠❛t❛ ❝❧❛ss K(A) t❤❛t ✐s ❝♦♥str✉❝t❡❞ t❤✐s ✇❛②✳ ❉❡♥♦t❡ ΩB(fs) ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ B✳ ❚❤❡♦r❡♠ ✶ ✭▲✳❚✳✱ ✷✵✶✸✮ ❆♥ ❛✉t♦♠❛t♦♥ ✐s ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t❤❡ ❛✉t♦♠❛t♦♥ ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ✱ s✉❝❤ t❤❛t ❛♥❞ ❛r❡ ❛✣♥❡ ❡q✉✐✈❛❧❡♥ts✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 19

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❋♦r ❡✈❡r② st❛t❡ s ∈ S ♦❢ ❛♥ ❛✉t♦♠❛t♦♥ A ✇❡ ❛ss♦❝✐❛t❡ ❛ ♠❛♣ Rs : Fp → Fp t❤❛t tr❛♥s❢♦r♠s ✐♥♣✉t s②♠❜♦❧ x ∈ Fp ✐♥t♦ ♦✉t♣✉t s②♠❜♦❧ y ∈ Fp✳ ◆♦t❡✱ t❤❛t Rs ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❛ s✉r❥❡❝t✐✈❡ ♠❛♣✳ ❈♦♥s✐❞❡r ❛♥ ❛✉t♦♠❛t♦♥ A ❛♥❞ t❤❡ ❢❛♠✐❧② {Rs : s ∈ S}✳ ❆ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♦❢ ❡❛❝❤ st❛t❡ s ∈ S ♦❢ ❛ ♠❛♣ Rs ❝r❡❛t❡s ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ B✳ ▲❡t✬s ❝♦♥s✐❞❡r ❛✉t♦♠❛t❛ ❝❧❛ss K(A) t❤❛t ✐s ❝♦♥str✉❝t❡❞ t❤✐s ✇❛②✳ ❉❡♥♦t❡ ΩB(fs) ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ B✳ ❚❤❡♦r❡♠ ✶ ✭▲✳❚✳✱ ✷✵✶✸✮ ❆♥ ❛✉t♦♠❛t♦♥ A ✐s ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t❤❡ ❛✉t♦♠❛t♦♥ B ∈ K(A) ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ω(fs0) ⊂ Ω(fs0)✱ ω(fs) ⊂ ΩB(fs) s✉❝❤ t❤❛t ω(fs0) ❛♥❞ ω(fs) ❛r❡ ❛✣♥❡ ❡q✉✐✈❛❧❡♥ts✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 20

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❆♥ ❛✉t♦♠❛t♦♥ A ✐s ❝♦♠♣❧❡t❡❧② tr❛♥s✐t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t❤❡ ❛✉t♦♠❛t♦♥ B ∈ K(A) ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ω(fs0) ⊂ Ω(fs0)✱ ω(fs) ⊂ ΩB(fs) s✉❝❤ t❤❛t ω(fs0) ❛♥❞ ω(fs) ❛r❡ ❛✣♥❡ ❡q✉✐✈❛❧❡♥ts✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳ ■❢ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✇♦r❞s ♦❢ ✜♥✐t❡ ❧❡♥❣t❤ k ❛s ♣r❡✜①❡s ♦❢ ✐♥✜♥✐t❡ ✇♦r❞s ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t Fp✱ t❤❛t ✐♥ t❤❡✐r t✉r♥ ❛r❡ ❡❧❡♠❡♥ts ♦❢ r✐♥❣ ♦❢ p✲❛❞✐❝ ✐♥t❡❣❡rs Zp✱ t❤❡ ❝♦✐♥❝✐❞❡♥❝❡ ♦❢ t❤❡s❡ ♣r❡✜①❡s ♠❡❛♥s t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✐♥✜♥✐t❡ ✇♦r❞s ♦♥ ❛ ✉❧tr❛♠❡tr✐❝ s♣❛❝❡ Zp ✐s ❡q✉❛❧ t♦ p−k✳ ❚❤❛t ♠❡❛♥s✱ t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ fs✲ ✐♠❛❣❡s ♥♦t ♠♦r❡ t❤❛♥ p−k✱ ❜❡❝❛✉s❡ t❤❡ ♠❛♣♣✐♥❣s fs : Zp → Zp ❛r❡ ✶✲▲✐♣s❝❤✐t③✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 21

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

■♥ ❛❞❞✐t✐♦♥✱ ❛❧❧ s✉❝❤ ✐♥✜♥✐t❡ ✇♦r❞s ❤❛✈❡ ❛ ❝♦♠♠♦♥ ♣r❡✜① ✇✐t❤ ❧❡♥❣t❤ k ≥ 0 ❛s p✲❛❞✐❝ ✐♥t❡❣❡rs ❢❛❧❧ ✐♥t♦ t❤❡ ❜❛❧❧ Bp−k = {z ∈ Zp : |z − a|p ≤ p−k} ✇✐t❤ r❛❞✐✉s p−k ❛♥❞ ❝❡♥t❡r ❛t a ∈ Zp✳ ▼♦r❡♦✈❡r✱ t❤❡ ✇♦r❞ u ♦❢ ❧❡♥❣t❤ k ❛s ♣r❡✜① ♦❢ z ∈ Zp ✐s t❤❡ r❡❞✉❝t✐♦♥ ♦❢ z ♠♦❞✉❧♦ pk✱ ✐✳❡✳ u ∈ Z/pkZ✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ ✱ t❤❡♥ ❛♥❞ ❢❛❧❧s ✐♥t♦ ♦♥❡ ❜❛❧❧✱ ❛♥❞ ♣♦✐♥ts ♦❢ ✱ t❤❛t ❝♦rr❡s♣♦♥❞ t♦ ✐♥♣✉t✴♦✉t♣✉t ✇♦r❞s ♦❢ ❧❡♥❣t❤ ❝♦✐♥❝✐❞❡ ❛♥❞ ❢♦r♠ ❡❧❡♠❡♥ts ♦❢ ❛♥❞ s❡ts✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 22

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

■♥ ❛❞❞✐t✐♦♥✱ ❛❧❧ s✉❝❤ ✐♥✜♥✐t❡ ✇♦r❞s ❤❛✈❡ ❛ ❝♦♠♠♦♥ ♣r❡✜① ✇✐t❤ ❧❡♥❣t❤ k ≥ 0 ❛s p✲❛❞✐❝ ✐♥t❡❣❡rs ❢❛❧❧ ✐♥t♦ t❤❡ ❜❛❧❧ Bp−k = {z ∈ Zp : |z − a|p ≤ p−k} ✇✐t❤ r❛❞✐✉s p−k ❛♥❞ ❝❡♥t❡r ❛t a ∈ Zp✳ ▼♦r❡♦✈❡r✱ t❤❡ ✇♦r❞ u ♦❢ ❧❡♥❣t❤ k ❛s ♣r❡✜① ♦❢ z ∈ Zp ✐s t❤❡ r❡❞✉❝t✐♦♥ ♦❢ z ♠♦❞✉❧♦ pk✱ ✐✳❡✳ u ∈ Z/pkZ✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ fs0(z) ≡ fs(z) (mod pk)✱ t❤❡♥ fs0(z) ❛♥❞ fs(z) ❢❛❧❧s ✐♥t♦ ♦♥❡ ❜❛❧❧✱ ❛♥❞ ♣♦✐♥ts ♦❢ Ω(fs0)✱ Ω(fs) t❤❛t ❝♦rr❡s♣♦♥❞ t♦ ✐♥♣✉t✴♦✉t♣✉t ✇♦r❞s ♦❢ ❧❡♥❣t❤ k ❝♦✐♥❝✐❞❡ ❛♥❞ ❢♦r♠ ❡❧❡♠❡♥ts ♦❢ ω(fs0) ❛♥❞ ω(fs) s❡ts✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-23
SLIDE 23

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❖❜✈✐♦✉s❧② fs0(z) ≡ fs(z) (mod pk) ✐♠♣❧✐❡s t❤❛t fs0(z) ≡ fs(z) (mod pℓ) ❢♦r ❛❧❧ ℓ = 1, . . . , k − 1✱ t❤❡r❡❢♦r❡ t❤❡ ♣♦✐♥ts ♦❢ ✐♠❛❣❡s ♦❢ ✇♦r❞s ♦❢ ❧❡♥❣t❤ ℓ ❝♦✐♥❝✐❞❡✳ ❚❤❡ ❝❛s❡ ♦❢ ♠❡❛♥s t❤❛t ✱ ❢❛❧❧ ✐♥t♦ ❞✐✛❡r❡♥t ❜❛❧❧s ♦❢ r❛❞✐✉s ❛♥❞ t❤❡ ♣♦✐♥ts ✐♥ ❛♥❞ ❜② ❛♣♣r♦♣r✐❛t❡ ✐♥♣✉t✴♦✉t♣✉t ✇♦r❞s ♦❢ ❧❡♥❣t❤ ✶ ✐s ❝♦♠♣❛t✐❜❧❡ ❜② ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥ ✱ ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 24

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❖❜✈✐♦✉s❧② fs0(z) ≡ fs(z) (mod pk) ✐♠♣❧✐❡s t❤❛t fs0(z) ≡ fs(z) (mod pℓ) ❢♦r ❛❧❧ ℓ = 1, . . . , k − 1✱ t❤❡r❡❢♦r❡ t❤❡ ♣♦✐♥ts ♦❢ ✐♠❛❣❡s ♦❢ ✇♦r❞s ♦❢ ❧❡♥❣t❤ ℓ ❝♦✐♥❝✐❞❡✳ ❚❤❡ ❝❛s❡ ♦❢ fs0(z) ≡ fs(z) (mod p) ♠❡❛♥s t❤❛t fs0(z)✱ fs(z) ❢❛❧❧ ✐♥t♦ ❞✐✛❡r❡♥t ❜❛❧❧s ♦❢ r❛❞✐✉s p−1 ❛♥❞ t❤❡ ♣♦✐♥ts ✐♥ ω(fs0) ❛♥❞ ω(fs) ❜② ❛♣♣r♦♣r✐❛t❡ ✐♥♣✉t✴♦✉t♣✉t ✇♦r❞s ♦❢ ❧❡♥❣t❤ ✶ ✐s ❝♦♠♣❛t✐❜❧❡ ❜② ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥ φ1 : ( u, fs( u)) → ( u, fs0( u) + d)✱ d ∈ Z✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 25

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

■❢ fs0(z) ≡ fs(z) (mod pk) ❛♥❞ fs0(z) ≡ fs(z) (mod pk−1)✱ t❤❡♥ fs0(z), fs(z) ∈ Bpk−1 ❛♥❞ ♣♦✐♥ts ♦❢ ✐♠❛❣❡s ω(fs0) ❛♥❞ ω(fs) ❢♦r ✇♦r❞s ✇✐t❤ ❧❡♥❣t❤ k ❛r❡ ❝♦♠♣❛t✐❜❧❡ ❜② tr❛♥s❢♦r♠❛t✐♦♥ φk : ( u, fs( u)) → ( u, fs0( u) + d)✱ d ∈ Q✳ ❚❤❡ tr❛♥s✐t✐✈✐t② ♦❢ ❢❛♠✐❧② ❢♦r ❛♥ ❛✉t♦♠❛t♦♥ ♠❡❛♥s t❤❛t ❢♦r ❛ ❣✐✈❡♥ ♣❛✐r ♦❢ str✐♥❣s ✇✐t❤ ❧❡♥❣t❤ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ✇♦r❞ s✉❝❤ t❤❛t s❡♥❞s t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t ♦❢ ✐♥t♦ t❤❡ st❛t❡ s✉❝❤ t❤❛t ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 26

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

■❢ fs0(z) ≡ fs(z) (mod pk) ❛♥❞ fs0(z) ≡ fs(z) (mod pk−1)✱ t❤❡♥ fs0(z), fs(z) ∈ Bpk−1 ❛♥❞ ♣♦✐♥ts ♦❢ ✐♠❛❣❡s ω(fs0) ❛♥❞ ω(fs) ❢♦r ✇♦r❞s ✇✐t❤ ❧❡♥❣t❤ k ❛r❡ ❝♦♠♣❛t✐❜❧❡ ❜② tr❛♥s❢♦r♠❛t✐♦♥ φk : ( u, fs( u)) → ( u, fs0( u) + d)✱ d ∈ Q✳ ❚❤❡ tr❛♥s✐t✐✈✐t② ♦❢ ❢❛♠✐❧② F ❢♦r ❛♥ ❛✉t♦♠❛t♦♥ A ♠❡❛♥s t❤❛t ❢♦r ❛ ❣✐✈❡♥ ♣❛✐r u, w ♦❢ str✐♥❣s ✇✐t❤ ❧❡♥❣t❤ k t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ✇♦r❞ v s✉❝❤ t❤❛t s❡♥❞s t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t ♦❢ s0 ✐♥t♦ t❤❡ st❛t❡ s s✉❝❤ t❤❛t fs( u) = w✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 27

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❚❤❡ s❡t ♦❢ ♠❛♣s Rs ❣❡♥❡r❛t❡s ❛✉t♦♠❛t♦♥ B ∈ K(A) s✉❝❤ t❤❛t t❤❡ ♣♦✐♥t ( u, fs0( u)) ∈ ω(fso) ✐s tr❛♥s❢❡rr❡❞ ✐♥t♦ ❛ ♣♦✐♥t ( u, w) ∈ ω(fs) ⊂ ΩB(fs) ❜② tr❛♥s❢♦r♠❛t✐♦♥ φ : ( u, fs0( u)) → ( u, cfs0( u) + d)✱ c, d ∈ Q✳ ❙✉✣❝✐❡♥❝②✳ ■❢ t❤❡ ✐♠❛❣❡s ❛♥❞ ❝♦✐♥❝✐❞❡✱ t❤❡♥ ❢♦r s♦♠❡ ♦❢ ❧❡♥❣t❤ ✱ ❛♥❞ ✳ ■❢ ✱ ❛r❡ ✲❡q✉✐✈❛❧❡♥t✱ t❤❡♥ ✳ q✳❡✳❞✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 28

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❚❤❡ s❡t ♦❢ ♠❛♣s Rs ❣❡♥❡r❛t❡s ❛✉t♦♠❛t♦♥ B ∈ K(A) s✉❝❤ t❤❛t t❤❡ ♣♦✐♥t ( u, fs0( u)) ∈ ω(fso) ✐s tr❛♥s❢❡rr❡❞ ✐♥t♦ ❛ ♣♦✐♥t ( u, w) ∈ ω(fs) ⊂ ΩB(fs) ❜② tr❛♥s❢♦r♠❛t✐♦♥ φ : ( u, fs0( u)) → ( u, cfs0( u) + d)✱ c, d ∈ Q✳ ❙✉✣❝✐❡♥❝②✳ ■❢ t❤❡ ✐♠❛❣❡s ❛♥❞ ❝♦✐♥❝✐❞❡✱ t❤❡♥ ❢♦r s♦♠❡ ♦❢ ❧❡♥❣t❤ ✱ ❛♥❞ ✳ ■❢ ✱ ❛r❡ ✲❡q✉✐✈❛❧❡♥t✱ t❤❡♥ ✳ q✳❡✳❞✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 29

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❚❤❡ s❡t ♦❢ ♠❛♣s Rs ❣❡♥❡r❛t❡s ❛✉t♦♠❛t♦♥ B ∈ K(A) s✉❝❤ t❤❛t t❤❡ ♣♦✐♥t ( u, fs0( u)) ∈ ω(fso) ✐s tr❛♥s❢❡rr❡❞ ✐♥t♦ ❛ ♣♦✐♥t ( u, w) ∈ ω(fs) ⊂ ΩB(fs) ❜② tr❛♥s❢♦r♠❛t✐♦♥ φ : ( u, fs0( u)) → ( u, cfs0( u) + d)✱ c, d ∈ Q✳ ❙✉✣❝✐❡♥❝②✳ ■❢ t❤❡ ✐♠❛❣❡s ω(fs0) ❛♥❞ ω(fs) ❝♦✐♥❝✐❞❡✱ t❤❡♥ fs0( u) ≡ fs( u) (mod pk) ❢♦r s♦♠❡ u ♦❢ ❧❡♥❣t❤ k✱ ❛♥❞ fs ∈ F✳ ■❢ ω(fs0)✱ ω(fs) ❛r❡ φ✲❡q✉✐✈❛❧❡♥t✱ t❤❡♥ fs ∈ F✳ q✳❡✳❞✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 30

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❱✳❙✳ ❆♥❛s❤✐♥ ♣r♦♣♦s❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❛✉t♦♠❛t♦♥ ♠❛♣♣✐♥❣s ❜② ♠❡❛♥s ♦❢ β✲❡①♣❛♥s✐♦♥✳❚❤❡ β✲❡①♣❛♥s✐♦♥s ❛r❡ r❛❞✐① ❡①♣❛♥s✐♦♥s ✐♥ ♥♦♥✲✐♥t❡❣❡r ❜❛s❡s❀ ●✐✈❡♥ x ∈ R✱ x ≥ 0 ❛♥❞ β ∈ R✱ β > 1 ✇❡ ❝❛❧❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ x = ∞

n=1 χnβ−n ❛ β✲❡①♣❛♥s✐♦♥✱

✇❤❡r❡ χi ∈ {0, 1, . . . , ⌊β⌋}✳ ❋♦r ✇♦r❞ x = xn−1 . . . x1x0✱ xi ∈ Fp ✇❡ ♣✉t ✐♥t♦ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❛ ♣♦✐♥t ← − x = (β−n(xn−1βn−1 + . . . + x1β + x0)) mod 1 ∈ [0, 1)✳ ❈♦♥s✐❞❡r✐♥❣ ❛❧❧ ♣♦✐♥ts ✱ ✇❤❡r❡ r❛♥❣❡s ♦✈❡r ❛❧❧ ✜♥✐t❡ ✇♦r❞s✱ ✇❡ ❣❡t ❛ s❡t ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✉♥✐t sq✉❛r❡ ✳ ❆ ❝❧♦s✉r❡ ♦❢ ❛❧❧ s✉❝❤ ♣♦✐♥ts ✐s t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ ✭ ✲♣❧♦t ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✮✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ ♦❢ t❤❡ tr❛♥s✐t✐✈❡ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t♦♥ ♠❛♣♣✐♥❣s ✐♥ t❤❡ ✲♣❧♦ts r❡♠❛✐♥s ♦♣❡♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 31

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❱✳❙✳ ❆♥❛s❤✐♥ ♣r♦♣♦s❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❛✉t♦♠❛t♦♥ ♠❛♣♣✐♥❣s ❜② ♠❡❛♥s ♦❢ β✲❡①♣❛♥s✐♦♥✳❚❤❡ β✲❡①♣❛♥s✐♦♥s ❛r❡ r❛❞✐① ❡①♣❛♥s✐♦♥s ✐♥ ♥♦♥✲✐♥t❡❣❡r ❜❛s❡s❀ ●✐✈❡♥ x ∈ R✱ x ≥ 0 ❛♥❞ β ∈ R✱ β > 1 ✇❡ ❝❛❧❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ x = ∞

n=1 χnβ−n ❛ β✲❡①♣❛♥s✐♦♥✱

✇❤❡r❡ χi ∈ {0, 1, . . . , ⌊β⌋}✳ ❋♦r ✇♦r❞ x = xn−1 . . . x1x0✱ xi ∈ Fp ✇❡ ♣✉t ✐♥t♦ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❛ ♣♦✐♥t ← − x = (β−n(xn−1βn−1 + . . . + x1β + x0)) mod 1 ∈ [0, 1)✳ ❈♦♥s✐❞❡r✐♥❣ ❛❧❧ ♣♦✐♥ts (← − x , fs0(← − x ))✱ ✇❤❡r❡ x r❛♥❣❡s ♦✈❡r ❛❧❧ ✜♥✐t❡ ✇♦r❞s✱ ✇❡ ❣❡t ❛ s❡t ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✉♥✐t sq✉❛r❡ E2 = [0, 1] × [0, 1]✳ ❆ ❝❧♦s✉r❡ ♦❢ ❛❧❧ s✉❝❤ ♣♦✐♥ts ✐s t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ A ✭β✲♣❧♦t ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✮✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ ♦❢ t❤❡ tr❛♥s✐t✐✈❡ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t♦♥ ♠❛♣♣✐♥❣s ✐♥ t❤❡ ✲♣❧♦ts r❡♠❛✐♥s ♦♣❡♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 32

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s ♦❢ ❛✉t♦♠❛t❛

❱✳❙✳ ❆♥❛s❤✐♥ ♣r♦♣♦s❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❛✉t♦♠❛t♦♥ ♠❛♣♣✐♥❣s ❜② ♠❡❛♥s ♦❢ β✲❡①♣❛♥s✐♦♥✳❚❤❡ β✲❡①♣❛♥s✐♦♥s ❛r❡ r❛❞✐① ❡①♣❛♥s✐♦♥s ✐♥ ♥♦♥✲✐♥t❡❣❡r ❜❛s❡s❀ ●✐✈❡♥ x ∈ R✱ x ≥ 0 ❛♥❞ β ∈ R✱ β > 1 ✇❡ ❝❛❧❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ x = ∞

n=1 χnβ−n ❛ β✲❡①♣❛♥s✐♦♥✱

✇❤❡r❡ χi ∈ {0, 1, . . . , ⌊β⌋}✳ ❋♦r ✇♦r❞ x = xn−1 . . . x1x0✱ xi ∈ Fp ✇❡ ♣✉t ✐♥t♦ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❛ ♣♦✐♥t ← − x = (β−n(xn−1βn−1 + . . . + x1β + x0)) mod 1 ∈ [0, 1)✳ ❈♦♥s✐❞❡r✐♥❣ ❛❧❧ ♣♦✐♥ts (← − x , fs0(← − x ))✱ ✇❤❡r❡ x r❛♥❣❡s ♦✈❡r ❛❧❧ ✜♥✐t❡ ✇♦r❞s✱ ✇❡ ❣❡t ❛ s❡t ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ✉♥✐t sq✉❛r❡ E2 = [0, 1] × [0, 1]✳ ❆ ❝❧♦s✉r❡ ♦❢ ❛❧❧ s✉❝❤ ♣♦✐♥ts ✐s t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡ ♦❢ ❛✉t♦♠❛t♦♥ A ✭β✲♣❧♦t ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✮✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ ♦❢ t❤❡ tr❛♥s✐t✐✈❡ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t♦♥ ♠❛♣♣✐♥❣s ✐♥ t❤❡ β✲♣❧♦ts r❡♠❛✐♥s ♦♣❡♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 33

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❲♦r❞s

▲❡t A = (I, S, O, S, O, s0) ❜❡ ❛♥ ❛✉t♦♠❛t♦♥✱ ✇❤❡r❡ I = O = Fp✳ ❙t❛t❡ ✉♣❞❛t❡ ❢✉♥❝t✐♦♥ S ❛♥❞ ♦✉t♣✉t ❢✉♥❝t✐♦♥ O ❝❛♥ ❜❡ ❝♦♥t✐♥✉❡❞ t♦ t❤❡ s❡t I∗ × S ✭I∗ ✐s ❛ s❡t ♦❢ ❛❧❧ ✜♥✐t❡ ✇♦r❞s ♦✈❡r ❛❧♣❤❛❜❡t I✮ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rr❡♥t r✉❧❡s✿ S(e, s) = s, O(e, s) = e, S(x · w, s) = S(w, S(x, s)), O(x · w, s) = O(x, s) · O(w, S(x.s)), ✇❤❡r❡ x ∈ I✱ s ∈ S✱ ❛♥❞ w ∈ I∗ ❛r❡ ❛r❜✐tr❛r② ❡❧❡♠❡♥ts✳ ❚❤❡ ❛✉t♦♠❛t♦♥ A ❞❡✜♥❡s ❛ ❢✉♥❝t✐♦♥ O(·, s0) : I∗ → O∗ t❤❛t s♣❡❝✐✜❡s t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥ ♦♥ ✜♥✐t❡ ✇♦r❞s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 34

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❲♦r❞s

▼♦r❡♦✈❡r✱ st❛t❡ ✉♣❞❛t❡ ❢✉♥❝t✐♦♥ S ❛♥❞ ♦✉t♣✉t ❢✉♥❝t✐♦♥ O ❝❛♥ ❜❡ ❝♦♥t✐♥✉❡❞ t♦ t❤❡ s❡t I∞ × S ✭I∞ ✐s ❛ s❡t ♦❢ ❛❧❧ ✐♥✜♥✐t❡ ✇♦r❞s ♦✈❡r ❛❧♣❤❛❜❡t I✮✳ ❆ ♠❛♣♣✐♥❣ ✐s s❛✐❞ t♦ ❜❡ ❞❡✜♥❡❞ ❜② ❛✉t♦♠❛t♦♥ ✐❢ ❢♦r ❛♥② ✳ ❚❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ❜② ❛♥ ❛✉t♦♠❛t♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❛❝t✐♦♥ ♦♥ ✐♥✜♥✐t❡ ✇♦r❞s ♦❢ t❤✐s ❛✉t♦♠❛t♦♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 35

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❲♦r❞s

▼♦r❡♦✈❡r✱ st❛t❡ ✉♣❞❛t❡ ❢✉♥❝t✐♦♥ S ❛♥❞ ♦✉t♣✉t ❢✉♥❝t✐♦♥ O ❝❛♥ ❜❡ ❝♦♥t✐♥✉❡❞ t♦ t❤❡ s❡t I∞ × S ✭I∞ ✐s ❛ s❡t ♦❢ ❛❧❧ ✐♥✜♥✐t❡ ✇♦r❞s ♦✈❡r ❛❧♣❤❛❜❡t I✮✳ ❆ ♠❛♣♣✐♥❣ fs0 : I∞ → O∞ ✐s s❛✐❞ t♦ ❜❡ ❞❡✜♥❡❞ ❜② ❛✉t♦♠❛t♦♥ A ✐❢ fs0(w) = O(w, s0) ❢♦r ❛♥② w ∈ I∞✳ ❚❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ❜② ❛♥ ❛✉t♦♠❛t♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❛❝t✐♦♥ ♦♥ ✐♥✜♥✐t❡ ✇♦r❞s ♦❢ t❤✐s ❛✉t♦♠❛t♦♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 36

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❲♦r❞s

▼♦r❡♦✈❡r✱ st❛t❡ ✉♣❞❛t❡ ❢✉♥❝t✐♦♥ S ❛♥❞ ♦✉t♣✉t ❢✉♥❝t✐♦♥ O ❝❛♥ ❜❡ ❝♦♥t✐♥✉❡❞ t♦ t❤❡ s❡t I∞ × S ✭I∞ ✐s ❛ s❡t ♦❢ ❛❧❧ ✐♥✜♥✐t❡ ✇♦r❞s ♦✈❡r ❛❧♣❤❛❜❡t I✮✳ ❆ ♠❛♣♣✐♥❣ fs0 : I∞ → O∞ ✐s s❛✐❞ t♦ ❜❡ ❞❡✜♥❡❞ ❜② ❛✉t♦♠❛t♦♥ A ✐❢ fs0(w) = O(w, s0) ❢♦r ❛♥② w ∈ I∞✳ ❚❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ❜② ❛♥ ❛✉t♦♠❛t♦♥ ✐s ❝❛❧❧❡❞ t❤❡ ❛❝t✐♦♥ ♦♥ ✐♥✜♥✐t❡ ✇♦r❞s ♦❢ t❤✐s ❛✉t♦♠❛t♦♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 37

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❲♦r❞s

❘❡♠✐♥❞ t❤❛t ✭✐♥✐t✐❛❧✮ ❛✉t♦♠❛t♦♥ ❛❝t✐♦♥ ♦♥ ❛ ✐♥♣✉t s❡q✉❡♥❝❡s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❧❛❜❡❧❡❞ ✐♥✜♥✐t❡ ❤♦♠♦❣❡♥❡♦✉s tr❡❡ T(A)✳ ❚❤❡ ✈❡rt✐❝❡s ♦❢ s✉❝❤ tr❡❡ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ st❛t❡s ♦❢ ❛✉t♦♠❛t♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② s②♠❜♦❧ x ∈ I ♦❢ t❤❡ ✐♥♣✉t ❛❧♣❤❛❜❡t ❛♥ ❛rr♦✇ ❧❛❜❡❧❡❞ ❜② x|O(x, s) st❛rts ❢r♦♠ st❛t❡ s t♦ st❛t❡ S(x, s)✳ ❚❤❡ r♦♦t ♦❢ t❤✐s tr❡❡ T(A) ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✐♥✐t✐❛❧ st❛t❡ s0 ♦❢ ❛✉t♦♠❛t♦♥ A✳ ❋r♦♠ ❡❛❝❤ ✈❡rt❡① ❣♦❡s ❡①❛❝t❧② #I = p ❛rr♦✇s✳ ❚♦ ✜♥❞ ♦✉t ❛❝t✐♦♥ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥ A ♦♥ t❤❡ ✇♦r❞ w ∈ I∞✱ ✇❡ s❤♦✉❧❞ ♠♦✈❡✱ st❛rt✐♥❣ ❢r♦♠ r♦♦t ♦❢ t❤❡ tr❡❡ T(A)✱ ❛❧♦♥❣ t❤❡ ❛rr♦✇s ♦❢ t❤❡ tr❡❡ s♦ t❤❛t t❤❡ ✇♦r❞ w r❡❛❞s ♦♥ t❤❡ ❧❡❢t ♣❛rts ♦❢ t❤❡ ❧❛❜❡❧s ❛❧♦♥❣ t❤❡ ❛rr♦✇s❀ t❤❡♥ t❤❡ ♣r♦❞✉❝t ♦❢ ❛❧❧ r✐❣❤t ♣❛rts ♦❢ t❤❡ ❧❛❜❡❧s ✇✐❧❧ ❜❡ ❡q✉❛❧ t♦ O(w, s0) ∈ O∞✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-38
SLIDE 38

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❋♦r ✐♥st❛♥❝❡✱ A = ({0, 1}, S, {0, 1}, S, O, s0)✱ ✇❤❡r❡ S = {s0, s1, s2, . . .} ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❧❛❜❡❧❡❞ ❞✐r❡❝t❡❞ tr❡❡ T(A)✳ ❚❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ tr❡❡ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ st❛t❡s ♦❢ ❛✉t♦♠❛t♦♥✱ ❛♥❞ ❢♦r ❡✈❡r② s②♠❜♦❧ x ∈ {0, 1} ♦❢ t❤❡ ✐♥♣✉t ❛❧♣❤❛❜❡t ❛♥ ❛rr♦✇ ❧❛❜❡❧❡❞ ❜② x|O(x, s) st❛rts ❢r♦♠ st❛t❡ s t♦ st❛t❡ S(x, s)✿ S(0, s0) = s4✱ O(0, s0) = 1✱ S(1, s4) = s5✱ O(1, s4) = 0, . . .

❋✐❣✉r❡ ✿ ❋✐rst ❧❡✈❡❧s ♦❢ t❤❡ tr❡❡ ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-39
SLIDE 39

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❉②♥❛♠✐❝s❀ ▼❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ♠❛♣s

❉②♥❛♠✐❝❛❧ s②st❡♠ ♦♥ ❛ ♠❡❛s✉❛r❛❜❧❡ s♣❛s❡ S ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ tr✐♣❧❡ (S, µ, f)✱ ✇❤❡r❡ S ✐s ❛ s❡t ❡♥❞♦✇❡❞ ✇✐t❤ ❛ ♠❡❛s✉r❡ µ✱ ❛♥❞ f : S → S ✐s ❛ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥✳ ❆ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✐s ❛ s❡q✉❡♥s❡ ♦❢ ♣♦✐♥ts ♦❢ t❤❡ s♣❛❝❡ ✱ ✐s ❝❛❧❧❡❞ ❛♥ ✐♥✐t✐❛❧ ♣♦✐♥t ♦❢ t❤❡ tr❛❥❡❝t♦r②✳ ❘❡♠✐♥❞❡r✿ ♠❡❛✉s✉r❡✲♣r❡s❡r❣✐♥❣ ♠❛♣s ❆ ♠❛♣♣✐♥❣ ♦❢ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ ✐♥t♦ ❛ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ ❡♥❞♦✇❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐st✐❝ ♠❡❛s✉r❡ ❛♥❞ ✱ r❡s♣❡❝t✐✈❡❧②✱ ✐s s❛✐❞ t♦ ❜❡ ♠❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ✇❤❡♥❡✈❡r ❢♦r ❡❛❝❤ ♠❡❛s✉r❛❜❧❡ s✉❜s❡t ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-40
SLIDE 40

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❉②♥❛♠✐❝s❀ ▼❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ♠❛♣s

❉②♥❛♠✐❝❛❧ s②st❡♠ ♦♥ ❛ ♠❡❛s✉❛r❛❜❧❡ s♣❛s❡ S ✐s ✉♥❞❡rst♦♦❞ ❛s ❛ tr✐♣❧❡ (S, µ, f)✱ ✇❤❡r❡ S ✐s ❛ s❡t ❡♥❞♦✇❡❞ ✇✐t❤ ❛ ♠❡❛s✉r❡ µ✱ ❛♥❞ f : S → S ✐s ❛ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥✳ ❆ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✐s ❛ s❡q✉❡♥s❡ x0, x1 = f(x0), . . . , xi = f(xi−1) = fi(x0), . . . ♦❢ ♣♦✐♥ts ♦❢ t❤❡ s♣❛❝❡ S✱ x0 ✐s ❝❛❧❧❡❞ ❛♥ ✐♥✐t✐❛❧ ♣♦✐♥t ♦❢ t❤❡ tr❛❥❡❝t♦r②✳ ❘❡♠✐♥❞❡r✿ ♠❡❛✉s✉r❡✲♣r❡s❡r❣✐♥❣ ♠❛♣s ❆ ♠❛♣♣✐♥❣ F : S → S ♦❢ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ S ✐♥t♦ ❛ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ Y ❡♥❞♦✇❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐st✐❝ ♠❡❛s✉r❡ µ ❛♥❞ ν✱ r❡s♣❡❝t✐✈❡❧②✱ ✐s s❛✐❞ t♦ ❜❡ ♠❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ✇❤❡♥❡✈❡r µ(F −1(S)) = ν(S) ❢♦r ❡❛❝❤ ♠❡❛s✉r❛❜❧❡ s✉❜s❡t S ⊆ S✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-41
SLIDE 41

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❉②♥❛♠✐❝s❀ ▼❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ♠❛♣s

❆ ♠❛♣♣✐♥❣ F : S → S ♦❢ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ S ♦♥t♦ S ❡♥❞♦✇❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐st✐❝ ♠❡❛s✉r❡ µ✱ ✐s s❛✐❞ t♦ ❜❡ ♠❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ✇❤❡♥❡✈❡r µ(F −1(S)) = µ(S) ❢♦r ❡❛❝❤ ♠❡❛s✉r❛❜❧❡ s✉❜s❡t S ⊆ S✳ ❈♦♥s✐❞❡r ❞②♥❛♠✐❝❛❧ s②st❡♠ ♦♥ ✱ ✇❤❡r❡ ♠❛♣ ❞❡✜♥❡❞ ❜② s♦♠❡ ❛✉t♦♠❛t♦♥ ✳ ❚❤❡ r✐♥❣ ❝❛♥ ❜❡ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ✱ t❤✉s ❜❡❝♦♠✐♥❣ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❚❤❡ ❧❛tt❡r ♠❡❛s✉r❡ ✐s ❛ ♥♦r♠❛❧✐③❡❞ ❍❛❛r ♠❡❛s✉r❡✳ ❚❤❡ ❜❛s❡ ♦❢ ❡❧❡♠❡♥t❛r② ♠❡❛s✉r❛❜❧❡ s✉❜s❡ts ❛r❡ ❛❧❧ ❜❛❧❧s ♦❢ ♥♦♥✲③❡r♦ r❛❞✐✐ ❀ ❛♥❞ ✇❡ ♣✉t ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-42
SLIDE 42

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❉②♥❛♠✐❝s❀ ▼❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ♠❛♣s

❆ ♠❛♣♣✐♥❣ F : S → S ♦❢ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ S ♦♥t♦ S ❡♥❞♦✇❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐st✐❝ ♠❡❛s✉r❡ µ✱ ✐s s❛✐❞ t♦ ❜❡ ♠❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ✇❤❡♥❡✈❡r µ(F −1(S)) = µ(S) ❢♦r ❡❛❝❤ ♠❡❛s✉r❛❜❧❡ s✉❜s❡t S ⊆ S✳ ❈♦♥s✐❞❡r ❞②♥❛♠✐❝❛❧ s②st❡♠ (Zp, µ, fA) ♦♥ Zp✱ ✇❤❡r❡ ♠❛♣ fs0 : Zp → Zp ❞❡✜♥❡❞ ❜② s♦♠❡ ❛✉t♦♠❛t♦♥ A = (Fp, S, Fp, S, O, s0)✳ ❚❤❡ r✐♥❣ Zp ❝❛♥ ❜❡ ❡♥❞♦✇❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ✱ t❤✉s ❜❡❝♦♠✐♥❣ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❚❤❡ ❧❛tt❡r ♠❡❛s✉r❡ ✐s ❛ ♥♦r♠❛❧✐③❡❞ ❍❛❛r ♠❡❛s✉r❡✳ ❚❤❡ ❜❛s❡ ♦❢ ❡❧❡♠❡♥t❛r② ♠❡❛s✉r❛❜❧❡ s✉❜s❡ts ❛r❡ ❛❧❧ ❜❛❧❧s Bp−k(a) ♦❢ ♥♦♥✲③❡r♦ r❛❞✐✐ p−k❀ ❛♥❞ ✇❡ ♣✉t µ(Bp−k(a)) = p−k✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-43
SLIDE 43

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

◆♦✇ ❝♦♥s✐❞❡r t❤❡ ❛❝②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛✳ ❆♥ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ ✭tr❛♥s❞✉❝❡r✮ ✐s ❛ ✻✲t✉♣❧❡ B = (I, S, O, S, O, so)✱ ✇❤❡r❡ I✱ O ❛r❡ ✜♥✐t❡ ❛❧♣❤❛❜❡ts✱ S ✐s ❛ s❡t ♦❢ st❛t❡s✱ S : I × S → S ✐s t❤❡ st❛t❡ ✉♣❞❛t❡ ❢✉♥❝t✐♦♥✱ O : I × S → O∗✱ ✇❤❡r❡ O∗ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ✜♥✐t❡ str✐♥❣s ✭✇♦r❞s✮ ♦✈❡r O✱ ❛♥❞ s0 ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-44
SLIDE 44

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

  • ✐✈❡♥ ❛ tr❛♥s❞✉❝❡r B ❛♥❞ ❛♥ ✐♥✜♥✐t❡ ✐♥♣✉t str✐♥❣

. . . α2α1α0 ∈ I∞✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦✉t♣✉t s❡q✉❡♥❝❡ {βi} ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❜② βi = O(αi, si) ❚❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ . . . β2β1β0 ♦❢ t❤❡ ♦✉t♣✉t s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ t❤❡ ♦✉t♣✉t str✐♥❣✳ ❚❤✐s str✐♥❣ ✐s ✉s✉❛❧❧② ✐♥✜♥✐t❡✱ ❜✉t ✇✐❧❧ ❜❡ ✜♥✐t❡ ✐❢ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② βi✬s ❛r❡ ♥♦♥❡♠♣t②✳ ❲❡ s❛② t❤❛t t❤❡ tr❛♥s❞✉❝❡r ✐s ♥♦♥❞❡❣❡♥❡r❛t❡ ✐❢ ❡✈❡r② ✐♥✜♥✐t❡ ✐♥♣✉t str✐♥❣ r❡s✉❧ts ✐♥ ❛♥ ✐♥✜♥✐t❡ ♦✉t♣✉t str✐♥❣✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❢✉♥❝t✐♦♥ fs0 : I∞ → O∞ ♠❛♣♣✐♥❣ ❡❛❝❤ ✐♥♣✉t str✐♥❣ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦✉t♣✉t str✐♥❣ ✐s ❝❛❧❧❡❞ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ ❣✐✈❡♥ tr❛♥s❞✉❝❡r✳ ❘❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ❛❧✇❛②s ❝♦♥t✐♥✉♦✉s✱ ❛♥❞ ❛♥② ✐♥✈❡rt✐❜❧❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✐s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-45
SLIDE 45

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

  • ✐✈❡♥ ❛ tr❛♥s❞✉❝❡r B ❛♥❞ ❛♥ ✐♥✜♥✐t❡ ✐♥♣✉t str✐♥❣

. . . α2α1α0 ∈ I∞✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦✉t♣✉t s❡q✉❡♥❝❡ {βi} ✐s ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② ❜② βi = O(αi, si) ❚❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ . . . β2β1β0 ♦❢ t❤❡ ♦✉t♣✉t s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ t❤❡ ♦✉t♣✉t str✐♥❣✳ ❚❤✐s str✐♥❣ ✐s ✉s✉❛❧❧② ✐♥✜♥✐t❡✱ ❜✉t ✇✐❧❧ ❜❡ ✜♥✐t❡ ✐❢ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② βi✬s ❛r❡ ♥♦♥❡♠♣t②✳ ❲❡ s❛② t❤❛t t❤❡ tr❛♥s❞✉❝❡r ✐s ♥♦♥❞❡❣❡♥❡r❛t❡ ✐❢ ❡✈❡r② ✐♥✜♥✐t❡ ✐♥♣✉t str✐♥❣ r❡s✉❧ts ✐♥ ❛♥ ✐♥✜♥✐t❡ ♦✉t♣✉t str✐♥❣✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❢✉♥❝t✐♦♥ fs0 : I∞ → O∞ ♠❛♣♣✐♥❣ ❡❛❝❤ ✐♥♣✉t str✐♥❣ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦✉t♣✉t str✐♥❣ ✐s ❝❛❧❧❡❞ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❜② t❤❡ ❣✐✈❡♥ tr❛♥s❞✉❝❡r✳ ❘❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ❛❧✇❛②s ❝♦♥t✐♥✉♦✉s✱ ❛♥❞ ❛♥② ✐♥✈❡rt✐❜❧❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✐s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-46
SLIDE 46

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❘♦✉❣❤❧② s♣❡❛❦✐♥❣✱ ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✐s ❛ ❛✉t♦♠❛t♦♥ t❤❛t ❝♦♥✈❡rts ❛♥ ✐♥♣✉t str✐♥❣ ♦❢ ❛r❜✐tr❛r② ❧❡♥❣t❤ t♦ ❛♥ ♦✉t♣✉t str✐♥❣✳ ❚❤❡ tr❛♥s❞✉❝❡r r❡❛❞s ♦♥❡ s②♠❜♦❧ ❛t ❛ t✐♠❡✱ ❝❤❛♥❣✐♥❣ ✐ts ✐♥t❡r♥❛❧ st❛t❡ ❛♥❞ ♦✉t♣✉tt✐♥❣ ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ s②♠❜♦❧s ❛t ❡❛❝❤ st❡♣✳ ❆s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡rs ❛r❡ ❛ ♥❛t✉r❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡rs✱ ✇❤✐❝❤ ❛r❡ r❡q✉✐r❡❞ t♦ ♦✉t♣✉t ❡①❛❝t❧② ♦♥❡ s②♠❜♦❧ ❢♦r ❡✈❡r② s②♠❜♦❧ r❡❛❞✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ r❡♣r❡s❡♥t❡❞ ❜② ▼♦♦r ❞✐❛❣r❛♠✿

❋✐❣✉r❡ ✿ ❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-47
SLIDE 47

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❘♦✉❣❤❧② s♣❡❛❦✐♥❣✱ ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✐s ❛ ❛✉t♦♠❛t♦♥ t❤❛t ❝♦♥✈❡rts ❛♥ ✐♥♣✉t str✐♥❣ ♦❢ ❛r❜✐tr❛r② ❧❡♥❣t❤ t♦ ❛♥ ♦✉t♣✉t str✐♥❣✳ ❚❤❡ tr❛♥s❞✉❝❡r r❡❛❞s ♦♥❡ s②♠❜♦❧ ❛t ❛ t✐♠❡✱ ❝❤❛♥❣✐♥❣ ✐ts ✐♥t❡r♥❛❧ st❛t❡ ❛♥❞ ♦✉t♣✉tt✐♥❣ ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ s②♠❜♦❧s ❛t ❡❛❝❤ st❡♣✳ ❆s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡rs ❛r❡ ❛ ♥❛t✉r❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡rs✱ ✇❤✐❝❤ ❛r❡ r❡q✉✐r❡❞ t♦ ♦✉t♣✉t ❡①❛❝t❧② ♦♥❡ s②♠❜♦❧ ❢♦r ❡✈❡r② s②♠❜♦❧ r❡❛❞✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ r❡♣r❡s❡♥t❡❞ ❜② ▼♦♦r ❞✐❛❣r❛♠✿

❋✐❣✉r❡ ✿ ❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 48

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

▲❡t✬s ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s♣❡❝✐❛❧ t②♣❡ ❢♦r ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✇❤❡r❡ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❛❧♣❤❛❜❡ts ❛r❡ s❛♠❡✳ ▼♦r❡ ♦✈❡r✱ ❧❡t✬s I = O = Fp = {0, 1, . . . p − 1}✳ ❆ ♠❛♣♣✐♥❣ ✐s ❝❛❧❧❡❞ ✲✉♥✐t ❞❡❧❛② ✇❤❡♥❡✈❡r ❣✐✈❡♥ ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r tr❛s❧❛t❡❞ ✐♥♣✉t str✐♥❣ ✭✈✐❡✇❡❞ ❛s ❛ ✐♥✜♥✐t❡ str❡❛♠ ♦❢ s②♠❜♦❧s✮ ✐♥t♦ ♦✉t♣✉t str✐♥❣ ♦❢ ❢♦r♠ ✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 49

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

▲❡t✬s ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ s♣❡❝✐❛❧ t②♣❡ ❢♦r ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✇❤❡r❡ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❛❧♣❤❛❜❡ts ❛r❡ s❛♠❡✳ ▼♦r❡ ♦✈❡r✱ ❧❡t✬s I = O = Fp = {0, 1, . . . p − 1}✳ ❆ ♠❛♣♣✐♥❣ fs0 : Zp → Zp ✐s ❝❛❧❧❡❞ n✲✉♥✐t ❞❡❧❛② ✇❤❡♥❡✈❡r ❣✐✈❡♥ ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r B = (Fp, S, Fp, S, O, s0) tr❛s❧❛t❡❞ ✐♥♣✉t str✐♥❣ α = . . . αn . . . α1α0 ✭✈✐❡✇❡❞ ❛s ❛ ✐♥✜♥✐t❡ str❡❛♠ ♦❢ s②♠❜♦❧s✮ ✐♥t♦ ♦✉t♣✉t str✐♥❣ ♦❢ ❢♦r♠ β = . . . βn+1βn✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 50

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❲❡ ❛ss✉♠❡ t❤❛t ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✇♦r❦s ✐♥ ❛ ❢r❛♠❡✇♦r❦ ♦❢ ❞✐s❝r❡t❡ t✐♠❡ st❡♣s✳ ❆ ✉♥✐t✲❞❡❧❛② ♠❛♣♣✐♥❣ ✐s ♦♥❡ t❤❛t s✐♠♣❧② ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t ♦♥❡ t✐♠❡ ✉♥✐t ❧❛t❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✐s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ ✇✐t❤ ✉♥✐t✲❞❡❧❛②✱ t❤❛t ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✐♥❝♦♠♠✐♥❣ ❧❡tt❡r✱ ♦✉t♣✉ts ❛♥ ❡♠♣t② ✇♦r❞❀ ❛❢t❡r t❤❛t✱ t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t♣✉ts t❤❡ ✐♥❝♦♠✐♥❣ ✇♦r❞s ✇✐t❤♦✉t ❝❤❛♥❣❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-51
SLIDE 51

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❲❡ ❛ss✉♠❡ t❤❛t ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✇♦r❦s ✐♥ ❛ ❢r❛♠❡✇♦r❦ ♦❢ ❞✐s❝r❡t❡ t✐♠❡ st❡♣s✳ ❆ ✉♥✐t✲❞❡❧❛② ♠❛♣♣✐♥❣ ✐s ♦♥❡ t❤❛t s✐♠♣❧② ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t ♦♥❡ t✐♠❡ ✉♥✐t ❧❛t❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✐s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ ✇✐t❤ ✉♥✐t✲❞❡❧❛②✱ t❤❛t ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✐♥❝♦♠♠✐♥❣ ❧❡tt❡r✱ ♦✉t♣✉ts ❛♥ ❡♠♣t② ✇♦r❞❀ ❛❢t❡r t❤❛t✱ t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t♣✉ts t❤❡ ✐♥❝♦♠✐♥❣ ✇♦r❞s ✇✐t❤♦✉t ❝❤❛♥❣❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

slide-52
SLIDE 52

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❲❡ ❛ss✉♠❡ t❤❛t ❛♥ ❛s②♥❝❤r♦♥♦✉s tr❛♥s❞✉❝❡r ✇♦r❦s ✐♥ ❛ ❢r❛♠❡✇♦r❦ ♦❢ ❞✐s❝r❡t❡ t✐♠❡ st❡♣s✳ ❆ ✉♥✐t✲❞❡❧❛② ♠❛♣♣✐♥❣ ✐s ♦♥❡ t❤❛t s✐♠♣❧② ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t ♦♥❡ t✐♠❡ ✉♥✐t ❧❛t❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✐s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ ✇✐t❤ ✉♥✐t✲❞❡❧❛②✱ t❤❛t ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✐♥❝♦♠♠✐♥❣ ❧❡tt❡r✱ ♦✉t♣✉ts ❛♥ ❡♠♣t② ✇♦r❞❀ ❛❢t❡r t❤❛t✱ t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t♣✉ts t❤❡ ✐♥❝♦♠✐♥❣ ✇♦r❞s ✇✐t❤♦✉t ❝❤❛♥❣❡s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 53

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❆ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✐s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ ✇✐t❤ ✉♥✐t✲❞❡❧❛②✱ t❤❛t ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✐♥❝♦♠♠✐♥❣ ❧❡tt❡r✱ ♦✉t♣✉ts ❛♥ ❡♠♣t② ✇♦r❞❀ ❛❢t❡r t❤❛t✱ t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t♣✉ts t❤❡ ✐♥❝♦♠✐♥❣ ✇♦r❞s ✇✐t❤♦✉t ❝❤❛♥❣❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✐❢ t❤❡ tr❛♥s❞✉❝❡r r❡❛❞s ❛s ✐♥♣✉t ❛ s②♠❜♦❧ αt ❛t t✐♠❡ t✱ ✐t ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s②♠❜♦❧ ❛s ♦✉t♣✉t ❛t t✐♠❡ t + 1✳ ❆t t✐♠❡ t = 0✱ t❤❡ tr❛♥s❞✉❝❡r ♦✉t♣✉ts ♥♦t❤✐♥❣✳ ❲❡ ✐♥❞✐❝❛t❡ t❤✐s ❜② s❛②✐♥❣ t❤❛t t❤❡ tr❛♥s❞✉❝❡r tr❛♥s❧❛t❡s ✐♥♣✉t ✐♥t♦ ♦✉t♣✉t ✱ ✇❤❡r❡ ✐s ❡♠♣t② ✇♦r❞✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 54

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❆ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✐s ❞❡✜♥❡❞ ❜② ✜♥✐t❡ ❛s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥ ✇✐t❤ ✉♥✐t✲❞❡❧❛②✱ t❤❛t ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✐♥❝♦♠♠✐♥❣ ❧❡tt❡r✱ ♦✉t♣✉ts ❛♥ ❡♠♣t② ✇♦r❞❀ ❛❢t❡r t❤❛t✱ t❤❡ ❛✉t♦♠❛t♦♥ ♦✉t♣✉ts t❤❡ ✐♥❝♦♠✐♥❣ ✇♦r❞s ✇✐t❤♦✉t ❝❤❛♥❣❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✐❢ t❤❡ tr❛♥s❞✉❝❡r r❡❛❞s ❛s ✐♥♣✉t ❛ s②♠❜♦❧ αt ❛t t✐♠❡ t✱ ✐t ✇✐❧❧ ♣r♦❞✉❝❡ t❤❡ s②♠❜♦❧ ❛s ♦✉t♣✉t ❛t t✐♠❡ t + 1✳ ❆t t✐♠❡ t = 0✱ t❤❡ tr❛♥s❞✉❝❡r ♦✉t♣✉ts ♥♦t❤✐♥❣✳ ❲❡ ✐♥❞✐❝❛t❡ t❤✐s ❜② s❛②✐♥❣ t❤❛t t❤❡ tr❛♥s❞✉❝❡r tr❛♥s❧❛t❡s ✐♥♣✉t . . . α2α1α0 ✐♥t♦ ♦✉t♣✉t . . . β2β1e✱ ✇❤❡r❡ e ✐s ❡♠♣t② ✇♦r❞✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 55

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❆ n✲✉♥✐t ❞❡❧❛② tr❛♥s❞✉❝❡r ✐s ♦♥❡ t❤❛t ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t n t✐♠❡s ✉♥✐t ❧❛t❡r❀ t❤❛t ✐s✱ t❤❡ ✐♥♣✉t . . . αn . . . α1α0 tr❛♥s❧❛t❡❞ ✐♥t♦ ♦✉t♣✉t . . . βn+1βnen✱ ♠❡❛♥✐♥❣ ❛❣❛✐♥ t❤❛t tr❛♥s❞✉❝❡r ♣r♦❞✉❝❡s ♥♦ ♦✉t♣✉t ❢♦r t❤❡ ✜rst t✐♠❡s s❧♦ts✳ ❚❤❡♦r❡♠ ✷ ❆ ✲✉♥✐t ❞❡❧❛② ♠❛♣♣✐♥❣ ✐s ❝♦♥t✐♥✉♦✉s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 56

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t❛

❆ n✲✉♥✐t ❞❡❧❛② tr❛♥s❞✉❝❡r ✐s ♦♥❡ t❤❛t ♣r♦❞✉❝❡s t❤❡ ♦✉t♣✉t n t✐♠❡s ✉♥✐t ❧❛t❡r❀ t❤❛t ✐s✱ t❤❡ ✐♥♣✉t . . . αn . . . α1α0 tr❛♥s❧❛t❡❞ ✐♥t♦ ♦✉t♣✉t . . . βn+1βnen✱ ♠❡❛♥✐♥❣ ❛❣❛✐♥ t❤❛t tr❛♥s❞✉❝❡r ♣r♦❞✉❝❡s ♥♦ ♦✉t♣✉t ❢♦r t❤❡ ✜rst n t✐♠❡s s❧♦ts✳ ❚❤❡♦r❡♠ ✷ ❆ n✲✉♥✐t ❞❡❧❛② ♠❛♣♣✐♥❣ fs0 : Zp → Zp ✐s ❝♦♥t✐♥✉♦✉s✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 57

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

  • ✐✈❡♥ n✲✉♥✐t ❞❡❧❛② tr❛♥s❞✉❝❡r B = (Fp, S, Fp, S, O, s0)✱ ❛♥❞ tr❡❡

T(B) ✇❡ ❝❛❧❧ ❛ r❡❛❝❤❛❜❧❡ s❡t ♦❢ st❛t❡s V ♦❢ ❛✉t♦♠❛t♦♥ B✱ ✐❢ ❡❛❝❤ ✈❡rt❡① ♦❢ tr❡❡ T(B) ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❧❡♠❡♥t ♦❢ V ✐s ❛❝❝❡ss✐❜❧❡ ❢♦r n st❡♣s ❢r♦♠ t❤❡ r♦♦t s0✳ ❊❧❡♠❡♥t s ∈ V ✐s ❝❛❧❧❡❞ r❡❛❝❤❛❜❧❡ st❛t❡✳ ❚❤❡♦r❡♠ ✸ ❆ n✲✉♥✐t ❞❡❧❛② ♠❛♣♣✐♥❣ fs0 : Zp → Zp ✐s ♠❡❛s✉r❡✲♣r❡s❡r✈✐♥❣✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❡①❛❝t❧② pn r❡❛❝❤❛❜❧❡ st❛t❡s s ∈ V s✉❝❤ t❤❛t ❢♦r ❛♥② ✜♥✐t❡ ♦✉t♣✉t ✇♦r❞ β t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥ B ♦♥ s✉✐t❛❜❧❡ ✐♥♣✉t ✇♦r❞s α(β) ❝♦✐♥❝✐❞❡ ✇✐t❤ ♦✉t♣✉t ✇♦r❞ O(s, α(β)) = β✳

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 58

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❚❤❛♥❦ ❨♦✉✦

▲✐✈❛t ❚②❛♣❛❡✈ ❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 59

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❆s②♥❝❤r♦♥♦✉s ❛✉t♦♠❛t♦♥

  • ▲✐✈❛t ❚②❛♣❛❡✈

❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛

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SLIDE 60

❆✉t♦♠❛t❛ ❛s ❞②♥❛♠✐❝❛❧ s②st❡♠s P❛rt ■✿ ❆✉t♦♠❛t❛ P❛rt ■■✿ ●❡♦♠❡tr✐❝❛❧ ✐♠❛❣❡s P❛rt ■■■✿ ▼❡❛s✉r❡✲♣r❡s❡✈✐♥❣ ❢♦r ❛✉t♦♠❛t❛ P❛rt ■❱✿ ❈r✐t❡r✐❛ ♦❢ ♣r❡s❡r✈❡ t❤❡ ♠❡❛✉s✉r❡

❯♥✐❧❛t❡r❛❧ s❤✐❢t

  • ▲✐✈❛t ❚②❛♣❛❡✈

❉✐s❝r❡t❡ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❆✉t♦♠❛t❛