P P - - PowerPoint PPT Presentation

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P P - - PowerPoint PPT Presentation

Feb 02, 2024 216 likes •1.15k views

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SLIDE 1

■◆❚❊◆❙❊ ❆❯❚❖▼❖❘P❍■❙▼❙ ❖❋

  • ❘❖❯P❙

▼✐♠❛ ❙t❛♥♦❥❦♦✈s❦✐ ❋❘❆❯❊◆ ■◆ ❉❊❘ ▼❆❚❍❊▼❆❚■❑ ❑♦♥st❛♥③✱ ✷✹ ❏❛♥✉❛r② ✷✵✶✼

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SLIDE 2

❏♦✐♥t ✇♦r❦ ✇✐t❤

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SLIDE 3

❍❡♥❞r✐❦ ▲❡♥str❛ ✭❯▲✮

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SLIDE 4

❆♥❞r❡❛ ▲✉❝❝❤✐♥✐ ✭❯♥✐P❉✮

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SLIDE 5

❏♦♥ ●♦♥③á❧❡③ ❙á♥❝❤❡③ ✭❊❍❯✮

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SLIDE 6
  • r♦✉♣s ❛♥❞ s②♠♠❡tr✐❡s
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SLIDE 7

❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ❳ ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ ❙♥ ❉✷♥

  • ▲ ❱
  • ❛❧

❦ ❆✉t ●

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SLIDE 8

❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠(❳) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳

  • ❙♥

❉✷♥

  • ▲ ❱
  • ❛❧

❦ ❆✉t ●

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SLIDE 9

❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠(❳) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳

  • ❙♥
  • ❉✷♥
  • ▲ ❱
  • ❛❧

❦ ❆✉t ●

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SLIDE 10

❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠(❳) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳

  • ❙♥
  • ❉✷♥
  • ●▲(❱ )
  • ❛❧

❦ ❆✉t ●

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SLIDE 11

❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠(❳) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳

  • ❙♥
  • ❉✷♥
  • ●▲(❱ )
  • ●❛❧(ℓ/❦)

❆✉t ●

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SLIDE 12

❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠(❳) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳

  • ❙♥
  • ❉✷♥
  • ●▲(❱ )
  • ●❛❧(ℓ/❦)
  • ❆✉t(●)
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SLIDE 13

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

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SLIDE 14

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

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SLIDE 15

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

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SLIDE 16

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

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SLIDE 17

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

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SLIDE 18

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

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SLIDE 19

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ❙✻ ✇❤✐❝❤ ❤❛s ✼✷✵ ❡❧❡♠❡♥ts✳

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SLIDE 20

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = ❙✻ ✇❤✐❝❤ ❤❛s ✼✷✵ ❡❧❡♠❡♥ts✳

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SLIDE 21

P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = ❙✻ ✇❤✐❝❤ ❤❛s ✼✷✵ ❡❧❡♠❡♥ts✳

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SLIDE 22

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

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SLIDE 23

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

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SLIDE 24

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

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SLIDE 25

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

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SLIDE 26

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ❉✶✷ ✇❤✐❝❤ ❤❛s ✶✷ ❡❧❡♠❡♥ts✳

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SLIDE 27

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = ❉✶✷ ✇❤✐❝❤ ❤❛s ✶✷ ❡❧❡♠❡♥ts✳

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SLIDE 28

❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = ❉✶✷ ✇❤✐❝❤ ❤❛s ✶✷ ❡❧❡♠❡♥ts✳

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SLIDE 29

❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ✐❞❳ ✳

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SLIDE 30

❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = {✐❞❳, ρ}✳

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SLIDE 31

❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = {✐❞❳, ρ}✳

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SLIDE 32

❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ✐❞❳ ✳

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SLIDE 33

❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠(❳) = {✐❞❳}✳

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SLIDE 34

❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡

▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦✳ ❚❤❡♥ ❙②♠ ❱

  • ▲❦ ❱

✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s ❱ ❱ s✉❝❤ t❤❛t✿ ❢♦r ❛❧❧ ✉ ✈ ❱ ✱ ♦♥❡ ❤❛s ✉ ✈ ✉ ✈ ❀ ❢♦r ❛❧❧ ❦ ❛♥❞ ✈ ❱ ✱ ♦♥❡ ❤❛s ✈ ✈ ❀ t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦✱ t❤❡♥ ●▲❦ ❱

  • ▲♥ ❦ ✳
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SLIDE 35

❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡

▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦✳ ❚❤❡♥ ❙②♠(❱ ) = ●▲❦(❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ✉, ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(✉ + ✈) = α(✉) + α(✈)❀

❢♦r ❛❧❧ ❦ ❛♥❞ ✈ ❱ ✱ ♦♥❡ ❤❛s ✈ ✈ ❀ t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦✱ t❤❡♥ ●▲❦ ❱

  • ▲♥ ❦ ✳
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SLIDE 36

❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡

▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦✳ ❚❤❡♥ ❙②♠(❱ ) = ●▲❦(❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ✉, ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(✉ + ✈) = α(✉) + α(✈)❀
  • ❢♦r ❛❧❧ λ ∈ ❦ ❛♥❞ ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(λ✈) = λα(✈)❀

t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦✱ t❤❡♥ ●▲❦ ❱

  • ▲♥ ❦ ✳
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SLIDE 37

❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡

▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦✳ ❚❤❡♥ ❙②♠(❱ ) = ●▲❦(❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ✉, ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(✉ + ✈) = α(✉) + α(✈)❀
  • ❢♦r ❛❧❧ λ ∈ ❦ ❛♥❞ ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(λ✈) = λα(✈)❀
  • t❤❡ ♠❛♣ α ✐s ❜✐❥❡❝t✐✈❡✳

■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦✱ t❤❡♥ ●▲❦ ❱

  • ▲♥ ❦ ✳
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SLIDE 38

❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡

▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦✳ ❚❤❡♥ ❙②♠(❱ ) = ●▲❦(❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ✉, ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(✉ + ✈) = α(✉) + α(✈)❀
  • ❢♦r ❛❧❧ λ ∈ ❦ ❛♥❞ ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α(λ✈) = λα(✈)❀
  • t❤❡ ♠❛♣ α ✐s ❜✐❥❡❝t✐✈❡✳

■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦✱ t❤❡♥ ●▲❦(❱ ) ∼ = ●▲♥(❦)✳

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SLIDE 39

❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s

▲❡t ℓ/❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳ ❚❤❡♥ ❙②♠ ❦ ❆✉t❦ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s s✉❝❤ t❤❛t✿

  • ▲❦

❀ ❢♦r ❛❧❧ ① ② ✱ ♦♥❡ ❤❛s ①② ① ② ✳ ✭ ❢♦r ❛❧❧ ① ❦✱ ♦♥❡ ❤❛s ① ①✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧ ❦ ❆✉t❦ ✳

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SLIDE 40

❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s

▲❡t ℓ/❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳ ❚❤❡♥ ❙②♠(ℓ/❦) = ❆✉t❦(ℓ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ℓ → ℓ s✉❝❤ t❤❛t✿

  • α ∈ ●▲❦(ℓ)❀

❢♦r ❛❧❧ ① ② ✱ ♦♥❡ ❤❛s ①② ① ② ✳ ✭ ❢♦r ❛❧❧ ① ❦✱ ♦♥❡ ❤❛s ① ①✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧ ❦ ❆✉t❦ ✳

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SLIDE 41

❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s

▲❡t ℓ/❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳ ❚❤❡♥ ❙②♠(ℓ/❦) = ❆✉t❦(ℓ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ℓ → ℓ s✉❝❤ t❤❛t✿

  • α ∈ ●▲❦(ℓ)❀
  • ❢♦r ❛❧❧ ①, ② ∈ ℓ✱ ♦♥❡ ❤❛s

α(①②) = α(①)α(②)✳ ✭ ❢♦r ❛❧❧ ① ❦✱ ♦♥❡ ❤❛s ① ①✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧ ❦ ❆✉t❦ ✳

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SLIDE 42

❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s

▲❡t ℓ/❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳ ❚❤❡♥ ❙②♠(ℓ/❦) = ❆✉t❦(ℓ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ℓ → ℓ s✉❝❤ t❤❛t✿

  • α ∈ ●▲❦(ℓ)❀
  • ❢♦r ❛❧❧ ①, ② ∈ ℓ✱ ♦♥❡ ❤❛s

α(①②) = α(①)α(②)✳ ✭⇒ ❢♦r ❛❧❧ ① ∈ ❦✱ ♦♥❡ ❤❛s α(①) = ①✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧ ❦ ❆✉t❦ ✳

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SLIDE 43

❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s

▲❡t ℓ/❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳ ❚❤❡♥ ❙②♠(ℓ/❦) = ❆✉t❦(ℓ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ℓ → ℓ s✉❝❤ t❤❛t✿

  • α ∈ ●▲❦(ℓ)❀
  • ❢♦r ❛❧❧ ①, ② ∈ ℓ✱ ♦♥❡ ❤❛s

α(①②) = α(①)α(②)✳ ✭⇒ ❢♦r ❛❧❧ ① ∈ ❦✱ ♦♥❡ ❤❛s α(①) = ①✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧ ❦ ❆✉t❦ ✳

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SLIDE 44

❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s

▲❡t ℓ/❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳ ❚❤❡♥ ❙②♠(ℓ/❦) = ❆✉t❦(ℓ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ℓ → ℓ s✉❝❤ t❤❛t✿

  • α ∈ ●▲❦(ℓ)❀
  • ❢♦r ❛❧❧ ①, ② ∈ ℓ✱ ♦♥❡ ❤❛s

α(①②) = α(①)α(②)✳ ✭⇒ ❢♦r ❛❧❧ ① ∈ ❦✱ ♦♥❡ ❤❛s α(①) = ①✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ℓ/❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧(ℓ/❦) = ❆✉t❦(ℓ)✳

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SLIDE 45

❙②♠♠❡tr✐❡s ♦❢ ❣r♦✉♣s

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❙②♠(●) = ❆✉t(●) ❝♦♥s✐sts ♦❢ ❛❧❧ ♠❛♣s α : ● → ● s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ❣, ❤ ∈ ●✱ ♦♥❡ ❤❛s α(❣❤) = α(❣)α(❤)❀

t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ❊①❛♠♣❧❡✿ ❆✉t ✶ ✐❞ ❀ ❆✉t ✐❞ ❀ ❆✉t ♥ ♥ ✳

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SLIDE 46

❙②♠♠❡tr✐❡s ♦❢ ❣r♦✉♣s

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❙②♠(●) = ❆✉t(●) ❝♦♥s✐sts ♦❢ ❛❧❧ ♠❛♣s α : ● → ● s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ❣, ❤ ∈ ●✱ ♦♥❡ ❤❛s α(❣❤) = α(❣)α(❤)❀
  • t❤❡ ♠❛♣ α ✐s ❜✐❥❡❝t✐✈❡✳

❊①❛♠♣❧❡✿ ❆✉t ✶ ✐❞ ❀ ❆✉t ✐❞ ❀ ❆✉t ♥ ♥ ✳

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SLIDE 47

❙②♠♠❡tr✐❡s ♦❢ ❣r♦✉♣s

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❙②♠(●) = ❆✉t(●) ❝♦♥s✐sts ♦❢ ❛❧❧ ♠❛♣s α : ● → ● s✉❝❤ t❤❛t✿

  • ❢♦r ❛❧❧ ❣, ❤ ∈ ●✱ ♦♥❡ ❤❛s α(❣❤) = α(❣)α(❤)❀
  • t❤❡ ♠❛♣ α ✐s ❜✐❥❡❝t✐✈❡✳

❊①❛♠♣❧❡✿

  • ❆✉t({✶}) = {✐❞}❀
  • ❆✉t(Z) = {± ✐❞}❀
  • ❆✉t(Z/♥Z) ∼

= (Z/♥Z)∗✳

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SLIDE 48

❋r♦♠ t❤❡ s②♠♠❡tr✐❡s t♦ t❤❡ ❣r♦✉♣

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SLIDE 49

❊①❛♠♣❧❡ ✶

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❆✉t(●) = ✶ ⇒

  • ✶ ♦r
  • ✷✳
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SLIDE 50

❊①❛♠♣❧❡ ✶

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❆✉t(●) = ✶ ⇒ #● = ✶ ♦r #● = ✷✳

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SLIDE 51

❊①❛♠♣❧❡ ✷

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❆✉t(●) ✐s ❝②❝❧✐❝ ⇒

  • ✐s ❛❜❡❧✐❛♥✳
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SLIDE 52

❊①❛♠♣❧❡ ✷

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❚❤❡♥ ❆✉t(●) ✐s ❝②❝❧✐❝ ⇒

  • ✐s ❛❜❡❧✐❛♥✳
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SLIDE 53

❊①❛♠♣❧❡ ✷

▲❡t ● ❜❡ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣✳ ❚❤❡♥ ❆✉t(●) ✐s ❝②❝❧✐❝ ⇒

  • ✐s ❝②❝❧✐❝✳
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SLIDE 54

❊①❛♠♣❧❡ ✷

▲❡t ● ❜❡ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣✳ ❚❤❡♥ ❆✉t(●) ✐s ❝②❝❧✐❝ ⇒

  • ✐s ♦♥❡ ❜❡t✇❡❡♥
  • Z/✷Z❀
  • Z/✹Z❀
  • Z/♣❦Z✱ ❢♦r ♣ ♦❞❞ ❛♥❞ ❦ ≥ ✵❀
  • Z✳
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SLIDE 55

❊①❛♠♣❧❡ ✸

▲❡t ● ❜❡ ❛ ❣r♦✉♣ ❛♥❞ ❛ss✉♠❡ t❤❛t

  • ● ❤❛s ❝❛r❞✐♥❛❧✐t② ✼✷✾ = ✸✻✳
  • ● ✐s ✷✲❣❡♥❡r❛t❡❞✳
  • ❆✉t(●) ❤❛s ❝❛r❞✐♥❛❧✐t② ✶✵✹✾✼✻✳

❚❤❡♥ ● ✐s ✉♥✐q✉❡ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✳

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SLIDE 56

◆♦♥✲❊①❛♠♣❧❡ ✸

▲❡t ● ❜❡ ❛ ❣r♦✉♣ ❛♥❞ ❛ss✉♠❡ t❤❛t

  • ● ❤❛s ❝❛r❞✐♥❛❧✐t② ✼✷✾ = ✸✻✳
  • ● ✐s ✷✲❣❡♥❡r❛t❡❞✳
  • ❆✉t(●) ❤❛s ❝❛r❞✐♥❛❧✐t② ✶✵✹✾✼✻✳

❚❤❡♥ t❤❡r❡ ❛r❡ ✶✵✵ ♣♦ss✐❜❧❡ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ❢♦r ●✳

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SLIDE 57
  • ❡♥❡r❛❧ ✐❞❡❛

❈♦♥❞✐t✐♦♥s ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ ❆✉t(●) ❘❡str✐❝t✐♦♥s t♦ t❤❡ str✉❝t✉r❡ ♦❢ ● P❇✿ ❉♦❡s ● ❡✈❡♥ ❡①✐st❄

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SLIDE 58
  • ❡♥❡r❛❧ ✐❞❡❛

❈♦♥❞✐t✐♦♥s ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ ❆✉t(●) ↓ ❘❡str✐❝t✐♦♥s t♦ t❤❡ str✉❝t✉r❡ ♦❢ ● P❇✿ ❉♦❡s ● ❡✈❡♥ ❡①✐st❄

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SLIDE 59
  • ❡♥❡r❛❧ ✐❞❡❛

❈♦♥❞✐t✐♦♥s ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ ❆✉t(●) ↓ ❘❡str✐❝t✐♦♥s t♦ t❤❡ str✉❝t✉r❡ ♦❢ ● ↓ P❇✿ ❉♦❡s ● ❡✈❡♥ ❡①✐st❄

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SLIDE 60

■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r♦✉♣s

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SLIDE 61

■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s

▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ α ♦❢ ● ✐s ✐♥t❡♥s❡ ✐❢ ❢♦r ❛❧❧ ❍ ≤ ● t❤❡r❡ ❡①✐sts ❣ ∈ ● s✉❝❤ t❤❛t α(❍) = ❣❍❣−✶✳ ❲r✐t❡ α ∈ ■♥t(●)✳ ▼♦t✐✈❛t✐♦♥✿ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ❛♣♣❡❛r ♥❛t✉r❛❧❧② ❛s s♦❧✉t✐♦♥s t♦ ❛ ❝❡rt❛✐♥ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠✳ ❚❤❡② ✭s✉r♣r✐s✐♥❣❧②✦✮ ❣✐✈❡ r✐s❡ t♦ ❛ ✈❡r② r✐❝❤ t❤❡♦r②✳ ❊①❛♠♣❧❡✿ ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❝②❝❧✐❝ ❣r♦✉♣ ✐s ✐♥t❡♥s❡✳ ■♥♥❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳

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SLIDE 62

■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s

▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❆♥ ❛✉t♦♠♦r♣❤✐s♠ α ♦❢ ● ✐s ✐♥t❡♥s❡ ✐❢ ❢♦r ❛❧❧ ❍ ≤ ● t❤❡r❡ ❡①✐sts ❣ ∈ ● s✉❝❤ t❤❛t α(❍) = ❣❍❣−✶✳ ❲r✐t❡ α ∈ ■♥t(●)✳ ▼♦t✐✈❛t✐♦♥✿ ■♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠s ❛♣♣❡❛r ♥❛t✉r❛❧❧② ❛s s♦❧✉t✐♦♥s t♦ ❛ ❝❡rt❛✐♥ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠✳ ❚❤❡② ✭s✉r♣r✐s✐♥❣❧②✦✮ ❣✐✈❡ r✐s❡ t♦ ❛ ✈❡r② r✐❝❤ t❤❡♦r②✳ ❊①❛♠♣❧❡✿

  • ❊✈❡r② ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❝②❝❧✐❝ ❣r♦✉♣ ✐s ✐♥t❡♥s❡✳
  • ■♥♥❡r ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ✐♥t❡♥s❡✳
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SLIDE 63

■♥t❡♥s✐t②

▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣✲❣r♦✉♣✳ ❚❤❡♥ ■♥t ● P ❈ ✇❤❡r❡ P ✐s ❛ ♣✲❣r♦✉♣✳ ❈ ✐s ❛ s✉❜❣r♦✉♣ ♦❢

♣✳

❚❤❡ ✐♥t❡♥s✐t② ♦❢ ● ✐s ✐♥t ● ❈✳

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SLIDE 64

■♥t❡♥s✐t②

▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣✲❣r♦✉♣✳ ❚❤❡♥ ■♥t(●) ∼ = P ⋊ ❈ ✇❤❡r❡

  • P ✐s ❛ ♣✲❣r♦✉♣✳
  • ❈ ✐s ❛ s✉❜❣r♦✉♣ ♦❢ F∗

♣✳

❚❤❡ ✐♥t❡♥s✐t② ♦❢ ● ✐s ✐♥t ● ❈✳

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SLIDE 65

■♥t❡♥s✐t②

▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t ● ❜❡ ❛ ✜♥✐t❡ ♣✲❣r♦✉♣✳ ❚❤❡♥ ■♥t(●) ∼ = P ⋊ ❈ ✇❤❡r❡

  • P ✐s ❛ ♣✲❣r♦✉♣✳
  • ❈ ✐s ❛ s✉❜❣r♦✉♣ ♦❢ F∗

♣✳

❚❤❡ ✐♥t❡♥s✐t② ♦❢ ● ✐s ✐♥t(●) = #❈✳

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SLIDE 66

❚❤❡ ♣r♦❜❧❡♠

❈❛♥ ✇❡ ❝❧❛ss✐❢② ❛❧❧ ♣✲❣r♦✉♣s ● s❛t✐s❢②✐♥❣ ✐♥t(●) > ✶❄

❨❊❙✦

slide-67
SLIDE 67

❚❤❡ ♣r♦❜❧❡♠

❈❛♥ ✇❡ ❝❧❛ss✐❢② ❛❧❧ ♣✲❣r♦✉♣s ● s❛t✐s❢②✐♥❣ ✐♥t(●) > ✶❄

❨❊❙✦

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SLIDE 68

■♥t❡♥s❡ tr✐♣❧❡s

❆♥ ✐♥t❡♥s❡ tr✐♣❧❡ ✐s ❛ tr✐♣❧❡ (♣, ●, α) s✉❝❤ t❤❛t

  • ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✳
  • ● ✐s ❛ ✜♥✐t❡ ♣✲❣r♦✉♣✳
  • α = ✶ ❜❡❧♦♥❣s t♦ ❈ ✭♦r t♦ ❛ ❝♦♥❥✉❣❛t❡✮✳

■♥t❡♥s❡ tr✐♣❧❡s ❛r❡ q✉✐t❡ r❛r❡✿ ✐❢ ❛ ❣r♦✉♣ ♦❝❝✉rs ✐♥ ❛♥ ✐♥t❡♥s❡ tr✐♣❧❡✱ t❤❡♥ ✐ts str✉❝t✉r❡ ✐s ❛❧♠♦st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ♣ ❛♥❞ ✐ts ❝❧❛ss✳ ❚❤❡r❡ ❛r❡ ♥♦ ✐♥t❡♥s❡ tr✐♣❧❡s ✇✐t❤ ♣ ✷✳

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SLIDE 69

■♥t❡♥s❡ tr✐♣❧❡s

❆♥ ✐♥t❡♥s❡ tr✐♣❧❡ ✐s ❛ tr✐♣❧❡ (♣, ●, α) s✉❝❤ t❤❛t

  • ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✳
  • ● ✐s ❛ ✜♥✐t❡ ♣✲❣r♦✉♣✳
  • α = ✶ ❜❡❧♦♥❣s t♦ ❈ ✭♦r t♦ ❛ ❝♦♥❥✉❣❛t❡✮✳

■♥t❡♥s❡ tr✐♣❧❡s ❛r❡ q✉✐t❡ r❛r❡✿ ✐❢ ❛ ❣r♦✉♣ ♦❝❝✉rs ✐♥ ❛♥ ✐♥t❡♥s❡ tr✐♣❧❡✱ t❤❡♥ ✐ts str✉❝t✉r❡ ✐s ❛❧♠♦st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ♣ ❛♥❞ ✐ts ❝❧❛ss✳ ❚❤❡r❡ ❛r❡ ♥♦ ✐♥t❡♥s❡ tr✐♣❧❡s ✇✐t❤ ♣ ✷✳

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SLIDE 70

■♥t❡♥s❡ tr✐♣❧❡s

❆♥ ✐♥t❡♥s❡ tr✐♣❧❡ ✐s ❛ tr✐♣❧❡ (♣, ●, α) s✉❝❤ t❤❛t

  • ♣ ✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✳
  • ● ✐s ❛ ✜♥✐t❡ ♣✲❣r♦✉♣✳
  • α = ✶ ❜❡❧♦♥❣s t♦ ❈ ✭♦r t♦ ❛ ❝♦♥❥✉❣❛t❡✮✳

■♥t❡♥s❡ tr✐♣❧❡s ❛r❡ q✉✐t❡ r❛r❡✿ ✐❢ ❛ ❣r♦✉♣ ♦❝❝✉rs ✐♥ ❛♥ ✐♥t❡♥s❡ tr✐♣❧❡✱ t❤❡♥ ✐ts str✉❝t✉r❡ ✐s ❛❧♠♦st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ♣ ❛♥❞ ✐ts ❝❧❛ss✳ ❚❤❡r❡ ❛r❡ ♥♦ ✐♥t❡♥s❡ tr✐♣❧❡s ✇✐t❤ ♣ = ✷✳

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SLIDE 71

❊q✉✐✈❛❧❡♥t tr✐♣❧❡s

❊①❛♠♣❧❡✿ ▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ♥ ∈ Z>✵✳ ❋♦r ❛❧❧ α ∈ F∗

♣ \ {✶}✱ t❤❡ tr✐♣❧❡ (♣, F♥ ♣, α) ✐s ✐♥t❡♥s❡✳

❚✇♦ ✐♥t❡♥s❡ tr✐♣❧❡s ♣ ● ❛♥❞ q ● ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ✐s♦♠♦r♣❤✐s♠

  • s✉❝❤ t❤❛t

✶✳ ■t ❢♦❧❧♦✇s

t❤❛t ♣ q✳ ▲❡t ♣ ● ♣ ● ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ♦❢ ✐♥t❡♥s❡ tr✐♣❧❡s✳

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SLIDE 72

❊q✉✐✈❛❧❡♥t tr✐♣❧❡s

❊①❛♠♣❧❡✿ ▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ♥ ∈ Z>✵✳ ❋♦r ❛❧❧ α ∈ F∗

♣ \ {✶}✱ t❤❡ tr✐♣❧❡ (♣, F♥ ♣, α) ✐s ✐♥t❡♥s❡✳

❚✇♦ ✐♥t❡♥s❡ tr✐♣❧❡s (♣, ●, α) ❛♥❞ (q, ● ′, β) ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ✐s♦♠♦r♣❤✐s♠ σ : ● → ● ′ s✉❝❤ t❤❛t β = σασ−✶✳ ■t ❢♦❧❧♦✇s t❤❛t ♣ = q✳ ▲❡t T = {[♣, ●, α] | ♣, ●, α . . .} ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ♦❢ ✐♥t❡♥s❡ tr✐♣❧❡s✳

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SLIDE 73

❆❜❡❧✐❛♥ ❣r♦✉♣s

▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t Z♣ ❞❡♥♦t❡ t❤❡ r✐♥❣ ♦❢ ♣✲❛❞✐❝ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ ω(F∗

♣) = {α ∈ Z∗ ♣ | α♣−✶ = ✶}✳

◆♦t❡ t❤❛t ω(F∗

♣) ∼

= F∗

♣ ❛♥❞ t❤❛t ❡✈❡r② ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣ ❤❛s ❛ ♥❛t✉r❛❧

str✉❝t✉r❡ ♦❢ Z♣✲♠♦❞✉❧❡✳

Pr♦♣♦s✐t✐♦♥

❆ss✉♠❡ t❤❛t✿ ♣ ✐s ♦❞❞✳

  • ✶ ✐s ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣✳

♣

✶ ✳ ✭ ❊①❛♠♣❧❡✿ ✶ ✮ ❚❤❡♥ ♣ ● ❛♥❞ ✐♥t ● ♣ ✶✳

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SLIDE 74

❆❜❡❧✐❛♥ ❣r♦✉♣s

▲❡t ♣ ❜❡ ❛ ♣r✐♠❡ ♥✉♠❜❡r ❛♥❞ ❧❡t Z♣ ❞❡♥♦t❡ t❤❡ r✐♥❣ ♦❢ ♣✲❛❞✐❝ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ ω(F∗

♣) = {α ∈ Z∗ ♣ | α♣−✶ = ✶}✳

◆♦t❡ t❤❛t ω(F∗

♣) ∼

= F∗

♣ ❛♥❞ t❤❛t ❡✈❡r② ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣ ❤❛s ❛ ♥❛t✉r❛❧

str✉❝t✉r❡ ♦❢ Z♣✲♠♦❞✉❧❡✳

Pr♦♣♦s✐t✐♦♥

❆ss✉♠❡ t❤❛t✿

  • ♣ ✐s ♦❞❞✳
  • ● = ✶ ✐s ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ♣✲❣r♦✉♣✳
  • α ∈ ω(F∗

♣) \ {✶}✳ ✭ ❊①❛♠♣❧❡✿ α = −✶ ✮

❚❤❡♥ [♣, ●, α] ∈ T ❛♥❞ ✐♥t(●) = ♣ − ✶✳

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SLIDE 75

❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s

▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳

  • ■❢ ①, ② ∈ ●✱ t❤❡♥ [①, ②] = ①②①−✶②−✶✳
  • ■❢ ❍, ❑ ≤ ●✱ t❤❡♥ [❍, ❑] = [①, ②] | ① ∈ ❍, ② ∈ ❑✳

❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ ● ✐s ❣✐✈❡♥ ❜②

  • ✶
  • ✳
  • ✐

✶

  • ●✐ ✳

■❢ ● ✐s ❛ ♣✲❣r♦✉♣✱ t❤❡r❡ ❡①✐sts ❦ s✉❝❤ t❤❛t ●❦ ✶ ❛♥❞ t❤❡ ✭♥✐❧♣♦t❡♥❝②✮ ❝❧❛ss ♦❢ ● ✐s ❝ ✐

  • ✐
  • ✐

✶

✶ ♠✐♥ ❦

  • ❦

✶

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SLIDE 76

❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s

▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳

  • ■❢ ①, ② ∈ ●✱ t❤❡♥ [①, ②] = ①②①−✶②−✶✳
  • ■❢ ❍, ❑ ≤ ●✱ t❤❡♥ [❍, ❑] = [①, ②] | ① ∈ ❍, ② ∈ ❑✳

❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ ● ✐s ❣✐✈❡♥ ❜②

  • ●✶ = ●✳
  • ●✐+✶ = [●, ●✐]✳

■❢ ● ✐s ❛ ♣✲❣r♦✉♣✱ t❤❡r❡ ❡①✐sts ❦ s✉❝❤ t❤❛t ●❦ ✶ ❛♥❞ t❤❡ ✭♥✐❧♣♦t❡♥❝②✮ ❝❧❛ss ♦❢ ● ✐s ❝ ✐

  • ✐
  • ✐

✶

✶ ♠✐♥ ❦

  • ❦

✶

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SLIDE 77

❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s

▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳

  • ■❢ ①, ② ∈ ●✱ t❤❡♥ [①, ②] = ①②①−✶②−✶✳
  • ■❢ ❍, ❑ ≤ ●✱ t❤❡♥ [❍, ❑] = [①, ②] | ① ∈ ❍, ② ∈ ❑✳

❚❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ ● ✐s ❣✐✈❡♥ ❜②

  • ●✶ = ●✳
  • ●✐+✶ = [●, ●✐]✳

■❢ ● ✐s ❛ ♣✲❣r♦✉♣✱ t❤❡r❡ ❡①✐sts ❦ s✉❝❤ t❤❛t ●❦ = ✶ ❛♥❞ t❤❡ ✭♥✐❧♣♦t❡♥❝②✮ ❝❧❛ss ♦❢ ● ✐s ❝ = #{✐ | ●✐ = ●✐+✶} = −✶ + ♠✐♥{❦ | ●❦ = ✶}.

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SLIDE 78

❙tr❛t❡❣②

❋♦r ❛❧❧ ❝ ∈ Z≥✵✱ ❧❡t T [❝] = {[♣, ●, α] ∈ T | ● ❤❛s ❝❧❛ss ❝}✳ ❚❤❡♥✿

  • T =

❝ T [❝]✳

  • T [✵] = ∅✳
  • T [✶] = {[♣, ●, α] ❛s ✐♥ t❤❡ Pr♦♣♦s✐t✐♦♥}✳
  • T [❝] ❢♦r ❝ = ✷ ❄
  • T [❝] ❢♦r ❝ ≥ ✸ ❄
slide-79
SLIDE 79

❈❧❛ss ✷

▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ♥ ∈ Z>✵✳ ❉❡✜♥❡ (❊❙(♣, ♥), ∗) ❛s

  • ❊❙(♣, ♥) = F♣ × F♥

♣ × F♥ ♣✳

  • (③✶, ②✶, ①✶) ∗ (③✷, ②✷, ①✷) = (③✶ + ③✷ + ①✶ · ②✷, ②✶ + ②✷, ①✶ + ①✷)✳

❊①❡r❝✐s❡✿ ❊❙ ♣ ♥ ❤❛s ♦r❞❡r ♣✷♥

✶ ❛♥❞ ❝❧❛ss ✷✳

▲❡t

♣✳ ❚❤❡♥

③ ② ①

✷③

② ① ✐s ❛♥ ✐♥t❡♥s❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❊❙ ♣ ♥ ✳

Pr♦♣♦s✐t✐♦♥

✷ ♣ ❊❙ ♣ ♥ ♣ ✐s ♦❞❞ ♥

✵ ♣

✶ ✳

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SLIDE 80

❈❧❛ss ✷

▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ♥ ∈ Z>✵✳ ❉❡✜♥❡ (❊❙(♣, ♥), ∗) ❛s

  • ❊❙(♣, ♥) = F♣ × F♥

♣ × F♥ ♣✳

  • (③✶, ②✶, ①✶) ∗ (③✷, ②✷, ①✷) = (③✶ + ③✷ + ①✶ · ②✷, ②✶ + ②✷, ①✶ + ①✷)✳

❊①❡r❝✐s❡✿

  • (❊❙(♣, ♥), ∗) ❤❛s ♦r❞❡r ♣✷♥+✶ ❛♥❞ ❝❧❛ss ✷✳
  • ▲❡t λ ∈ F∗

♣✳ ❚❤❡♥ αλ : (③, ②, ①) → (λ✷③, λ②, λ①) ✐s ❛♥ ✐♥t❡♥s❡

❛✉t♦♠♦r♣❤✐s♠ ♦❢ (❊❙(♣, ♥), ∗)✳

Pr♦♣♦s✐t✐♦♥

✷ ♣ ❊❙ ♣ ♥ ♣ ✐s ♦❞❞ ♥

✵ ♣

✶ ✳

slide-81
SLIDE 81

❈❧❛ss ✷

▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ♥ ∈ Z>✵✳ ❉❡✜♥❡ (❊❙(♣, ♥), ∗) ❛s

  • ❊❙(♣, ♥) = F♣ × F♥

♣ × F♥ ♣✳

  • (③✶, ②✶, ①✶) ∗ (③✷, ②✷, ①✷) = (③✶ + ③✷ + ①✶ · ②✷, ②✶ + ②✷, ①✶ + ①✷)✳

❊①❡r❝✐s❡✿

  • (❊❙(♣, ♥), ∗) ❤❛s ♦r❞❡r ♣✷♥+✶ ❛♥❞ ❝❧❛ss ✷✳
  • ▲❡t λ ∈ F∗

♣✳ ❚❤❡♥ αλ : (③, ②, ①) → (λ✷③, λ②, λ①) ✐s ❛♥ ✐♥t❡♥s❡

❛✉t♦♠♦r♣❤✐s♠ ♦❢ (❊❙(♣, ♥), ∗)✳

Pr♦♣♦s✐t✐♦♥

T [✷] = {[♣, (❊❙(♣, ♥), ∗), αλ] | ♣ ✐s ♦❞❞, ♥ ∈ Z>✵, λ ∈ F∗

♣ \ {✶}}✳

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SLIDE 82

❈❧❛ss ❛t ❧❡❛st ✸

  • ✐✈❡♥ ❛ ♣✲❣r♦✉♣ ●✱ ❧❡t (●✐)✐≥✶ ❜❡ ✐ts ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s✳ ▲❡t

(❢✐)✐≥✶ ❜❡ t❤❡ s❡q✉❡♥❝❡✱ ✇✐t❤ ✈❛❧✉❡s ✐♥ Z≥✵✱ s✉❝❤ t❤❛t t❤❡ ♦r❞❡r ♦❢

  • ✐/●✐+✶ ✐s ❡q✉❛❧ t♦ ♣❢✐✳

Pr♦♣♦s✐t✐♦♥

▲❡t ❝ ✸ ❛♥❞ ❛ss✉♠❡ ♣ ● ❝ ✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳ ❚❤❡ ♦r❞❡r ♦❢ ✐s ❡q✉❛❧ t♦ ✷ ❛♥❞ ✐♥t ● ✷✳ ❋♦r ❛❧❧ ✐✱ t❤❡ q✉♦t✐❡♥t ●✐ ●✐

✶ ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ♣ ❛♥❞

✐♥❞✉❝❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② ✶ ✐ ♦♥ ✐t✳ ❢✐ ✐

✶

✷ ✶ ✷ ✶ ✷ ✶ ❢ ✵ ✵ ✵ ✇✐t❤ ❢ ✵ ✶ ✷ ✳

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SLIDE 83

❈❧❛ss ❛t ❧❡❛st ✸

  • ✐✈❡♥ ❛ ♣✲❣r♦✉♣ ●✱ ❧❡t (●✐)✐≥✶ ❜❡ ✐ts ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s✳ ▲❡t

(❢✐)✐≥✶ ❜❡ t❤❡ s❡q✉❡♥❝❡✱ ✇✐t❤ ✈❛❧✉❡s ✐♥ Z≥✵✱ s✉❝❤ t❤❛t t❤❡ ♦r❞❡r ♦❢

  • ✐/●✐+✶ ✐s ❡q✉❛❧ t♦ ♣❢✐✳

Pr♦♣♦s✐t✐♦♥

▲❡t ❝ ≥ ✸ ❛♥❞ ❛ss✉♠❡ [♣, ●, α] ∈ T [❝]✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

  • ❚❤❡ ♦r❞❡r ♦❢ α ✐s ❡q✉❛❧ t♦ ✷ ❛♥❞ ✐♥t(●) = ✷✳
  • ❋♦r ❛❧❧ ✐✱ t❤❡ q✉♦t✐❡♥t ●✐/●✐+✶ ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r F♣ ❛♥❞ α

✐♥❞✉❝❡s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② (−✶)✐ ♦♥ ✐t✳

  • (❢✐)✐≥✶ = (✷, ✶, ✷, ✶, . . . , ✷, ✶, ❢ , ✵, ✵, ✵, . . .) ✇✐t❤ ❢ ∈ {✵, ✶, ✷}✳
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SLIDE 84

◆♦r♠❛❧ s✉❜❣r♦✉♣s str✉❝t✉r❡

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SLIDE 85

❆♥ ✐♥t❡♥s❡ ❣r❛♣❤

❋✐① ♣ ❛♥❞ ❞❡✜♥❡ T♣ = {[♣, ●, α] | ●, α, . . .}✳ ❚❤❡r❡ ✐s ❛ ✇❡❧❧✲❞❡✜♥❡❞ s❡q✉❡♥❝❡ ♦❢ s❡ts . . . − → T♣[❝ + ✶]

π❝+✶

− − − → T♣[❝] π❝ − → T♣[❝ − ✶] − → . . . ✇❤❡r❡✱ ❢♦r ❛❧❧ ❝✱ t❤❡ ♠❛♣ π❝ ✐s ❞❡✜♥❡❞ ❜② π❝ : [♣, ●, α] → [♣, ●/●❝, α]. ❲❡ ❞❡✜♥❡ ❛ ❣r❛♣❤

♣

❊♣ ❱♣ ✱ ✇❤❡r❡ ❱♣

♣✳

✈ ✇ ❊♣ ✐❢ t❤❡r❡ ❡①✐sts ❝ s✉❝❤ t❤❛t

❝ ✈

✇✳

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SLIDE 86

❆♥ ✐♥t❡♥s❡ ❣r❛♣❤

❋✐① ♣ ❛♥❞ ❞❡✜♥❡ T♣ = {[♣, ●, α] | ●, α, . . .}✳ ❚❤❡r❡ ✐s ❛ ✇❡❧❧✲❞❡✜♥❡❞ s❡q✉❡♥❝❡ ♦❢ s❡ts . . . − → T♣[❝ + ✶]

π❝+✶

− − − → T♣[❝] π❝ − → T♣[❝ − ✶] − → . . . ✇❤❡r❡✱ ❢♦r ❛❧❧ ❝✱ t❤❡ ♠❛♣ π❝ ✐s ❞❡✜♥❡❞ ❜② π❝ : [♣, ●, α] → [♣, ●/●❝, α]. ❲❡ ❞❡✜♥❡ ❛ ❣r❛♣❤ G♣ = (❊♣, ❱♣)✱ ✇❤❡r❡

  • ❱♣ = T♣✳
  • (✈, ✇) ∈ ❊♣ ✐❢ t❤❡r❡ ❡①✐sts ❝ s✉❝❤ t❤❛t π❝(✈) = ✇✳
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SLIDE 87

❚❤❡ ❣r❛♣❤ ❢♦r ♣ = ✸

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SLIDE 88

❊①❛♠♣❧❡ ✸

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❆ss✉♠❡ t❤❛t

  • ● ❤❛s ❝❛r❞✐♥❛❧✐t② ✼✷✾ = ✸✻✳
  • ● ✐s ✷✲❣❡♥❡r❛t❡❞✳
  • ❆✉t(●) ❤❛s ❝❛r❞✐♥❛❧✐t② ✶✵✹✾✼✻✳

❚❤❡♥ ● ✐s ✉♥✐q✉❡ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✳

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SLIDE 89

❚❤❡ ❣r❛♣❤ ❢♦r ♣ > ✸

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SLIDE 90

❚❤❡ ✐♥✜♥✐t❡ ❝❛s❡

❚❤❡♦r❡♠

▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ❝ ∈ Z>✵✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

  • ■❢ ❝ ≥ ✸✱ t❤❡♥ T♣[❝] ✐s ✜♥✐t❡✳
  • T♣[❝] = ∅ ⇐

⇒ ♣ = ✸ ❛♥❞ ❝ ≥ ✺✳

  • ■❢ ♣ > ✸✱ t❤❡♥ # ❧✐♠

← −

❝

T♣[❝] = ✶✳ ■❢ ❧✐♠

❝ ♣ ❝

♣ ● ❝

❝ ❝ ✵✱ ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ t❤❡

♣r♦✲♣✲❣r♦✉♣ ●❧✐♠ ❧✐♠

❝

  • ❝ ❛♥❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠

❧✐♠ ♦❢ ●❧✐♠

t❤❛t ✐s ✐♥❞✉❝❡❞ ❜② t❤❡ ❛✉t♦♠♦r♣❤✐s♠s

❝ ✳

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SLIDE 91

❚❤❡ ✐♥✜♥✐t❡ ❝❛s❡

❚❤❡♦r❡♠

▲❡t ♣ ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t ❝ ∈ Z>✵✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

  • ■❢ ❝ ≥ ✸✱ t❤❡♥ T♣[❝] ✐s ✜♥✐t❡✳
  • T♣[❝] = ∅ ⇐

⇒ ♣ = ✸ ❛♥❞ ❝ ≥ ✺✳

  • ■❢ ♣ > ✸✱ t❤❡♥ # ❧✐♠

← −

❝

T♣[❝] = ✶✳ ■❢ ❧✐♠ ← −

❝

T♣[❝] = {[♣, ● (❝), α(❝)]}❝>✵✱ ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♣r♦✲♣✲❣r♦✉♣ ●❧✐♠ = ❧✐♠ ← −

❝

  • (❝) ❛♥❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ α❧✐♠ ♦❢ ●❧✐♠

t❤❛t ✐s ✐♥❞✉❝❡❞ ❜② t❤❡ ❛✉t♦♠♦r♣❤✐s♠s α(❝)✳

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SLIDE 92

❆ ♣r♦✜♥✐t❡ ❡①❛♠♣❧❡

▲❡t ♣ > ✸ ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t t ∈ Z♣ s❛t✐s❢② ( t

♣) = −✶✳ ❙❡t

❆♣ = Z♣ + Z♣✐ + Z♣❥ + Z♣✐❥ ✇✐t❤ ❞❡✜♥✐♥❣ r❡❧❛t✐♦♥s ✐✷ = t✱ ❥✷ = ♣✱ ❛♥❞ ❥✐ = −✐❥✳ ❚❤❡♥ ❆♣ ✐s ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❧♦❝❛❧ r✐♥❣ s✉❝❤ t❤❛t ❆♣/❥❆♣ ∼ = F♣✷✳ ❚❤❡ ✐♥✈♦❧✉t✐♦♥ · : ❆♣ → ❆♣ ✐s ❞❡✜♥❡❞ ❜② ❛ = s + t✐ + ✉❥ + ✈✐❥ → ❛ = s − t✐ − ✉❥ − ✈✐❥. ▲❡t ● = {❛ ∈ ❆∗

♣ | ❛❛ = ✶ ❛♥❞ ❛ ≡ ✶ ♠♦❞ ❥❆♣} ❛♥❞✱ ❢♦r ❛❧❧ ❛ ∈ ●✱

❞❡✜♥❡ α(❛) = ✐❛✐−✶✳

❚❤❡♦r❡♠

  • ✐s ❛ ♣r♦✲♣✲❣r♦✉♣ ❛♥❞ α ✐s t♦♣♦❧♦❣✐❝❛❧❧② ✐♥t❡♥s❡✱ ✐✳❡✳ ❢♦r ❛♥② ❝❧♦s❡❞

s✉❜❣r♦✉♣ ❍ ♦❢ ● t❤❡r❡ ❡①✐sts ❣ ∈ ● s✉❝❤ t❤❛t α(❍) = ❣❍❣−✶✳ ▼♦r❡♦✈❡r✱ (●, α) ∼ = (●❧✐♠, α❧✐♠)✳

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SLIDE 93