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Oct 27, 2023 329 likes •823 views

rs s rss r s r s tr

SLIDE 1

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

◆ór❛ ❙③❛❦á❝s ❝♦✲❛✉t❤♦r✿ ❏♦❤♥ ▼❡❛❦✐♥ ■❚❆❚ ✷✵✶✻ ❆❧❣♦r✐t❤♠✐❝ ❆s♣❡❝ts ♦❢ ❋✐♥❞✐♥❣ ❙❡♠✐❣r♦✉♣s ♦❢ P❛rt✐❛❧ ❆✉t♦♠♦r♣❤✐s♠s ♦❢ ❈♦♠❜✐♥❛t♦r✐❛❧ ❙tr✉❝t✉r❡s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 2

■♠♠❡rs✐♦♥s

❉❡✜♥✐t✐♦♥

❆ ❝♦♥t✐♥♦✉s ♠❛♣ f : Y → X ❜❡t✇❡❡♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✐s ❝❛❧❧❡❞ ❛♥ ✐♠♠❡rs✐♦♥ ✐❢ ❡✈❡r② ♣♦✐♥t y ∈ Y ❤❛s ❛ ♥❡✐❣❤❜♦r❤♦♦❞ U t❤❛t ✐s ♠❛♣♣❡❞ ❤♦♠❡♦♠♦r♣❤✐❝❛❧❧② ♦♥t♦ f (U) ❜② f ✳ ▼♦t✐✈❛t✐♦♥✿ ✈✐s✉❛❧✐③✐♥❣ ♠❛♥✐❢♦❧❞s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 3

■♠♠❡rs✐♦♥s

❉❡✜♥✐t✐♦♥

❆ ❝♦♥t✐♥♦✉s ♠❛♣ f : Y → X ❜❡t✇❡❡♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✐s ❝❛❧❧❡❞ ❛♥ ✐♠♠❡rs✐♦♥ ✐❢ ❡✈❡r② ♣♦✐♥t y ∈ Y ❤❛s ❛ ♥❡✐❣❤❜♦r❤♦♦❞ U t❤❛t ✐s ♠❛♣♣❡❞ ❤♦♠❡♦♠♦r♣❤✐❝❛❧❧② ♦♥t♦ f (U) ❜② f ✳ ▼♦t✐✈❛t✐♦♥✿ ✈✐s✉❛❧✐③✐♥❣ ♠❛♥✐❢♦❧❞s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 4

■♠♠❡rs✐♦♥s

◗✉❡st✐♦♥✿ ❣✐✈❡♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s X, Y ✱ ✐s t❤❡r❡ ❛♥ X → Y ✐♠♠❡rs✐♦♥❄ ❲❡ ♣r♦✈✐❞❡ ✇❛②s t♦ ❛♥s✇❡r ❢♦r t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡✳ ❊①❛♠♣❧❡✿ ✐♠♠❡rs✐♦♥s ❜❡t✇❡❡♥ ✭❞✐r❡❝t❡❞✮ ❣r❛♣❤s ❆❞❞✐t✐♦♥❛❧ r❡str✐❝t✐♦♥✿ r❡s♣❡❝t t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 5

■♠♠❡rs✐♦♥s

◗✉❡st✐♦♥✿ ❣✐✈❡♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s X, Y ✱ ✐s t❤❡r❡ ❛♥ X → Y ✐♠♠❡rs✐♦♥❄ ❲❡ ♣r♦✈✐❞❡ ✇❛②s t♦ ❛♥s✇❡r ❢♦r t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡✳ ❊①❛♠♣❧❡✿ ✐♠♠❡rs✐♦♥s ❜❡t✇❡❡♥ ✭❞✐r❡❝t❡❞✮ ❣r❛♣❤s ❆❞❞✐t✐♦♥❛❧ r❡str✐❝t✐♦♥✿ r❡s♣❡❝t t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 6

■♠♠❡rs✐♦♥s

◗✉❡st✐♦♥✿ ❣✐✈❡♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s X, Y ✱ ✐s t❤❡r❡ ❛♥ X → Y ✐♠♠❡rs✐♦♥❄ ❲❡ ♣r♦✈✐❞❡ ✇❛②s t♦ ❛♥s✇❡r ❢♦r t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡✳ ❊①❛♠♣❧❡✿ ✐♠♠❡rs✐♦♥s ❜❡t✇❡❡♥ ✭❞✐r❡❝t❡❞✮ ❣r❛♣❤s

a c d d

− →

a b c d

❆❞❞✐t✐♦♥❛❧ r❡str✐❝t✐♦♥✿ r❡s♣❡❝t t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 7

■♠♠❡rs✐♦♥s

◗✉❡st✐♦♥✿ ❣✐✈❡♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s X, Y ✱ ✐s t❤❡r❡ ❛♥ X → Y ✐♠♠❡rs✐♦♥❄ ❲❡ ♣r♦✈✐❞❡ ✇❛②s t♦ ❛♥s✇❡r ❢♦r t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡✳ ❊①❛♠♣❧❡✿ ✐♠♠❡rs✐♦♥s ❜❡t✇❡❡♥ ✭❞✐r❡❝t❡❞✮ ❣r❛♣❤s

a c d d

− →

a b c d

❆❞❞✐t✐♦♥❛❧ r❡str✐❝t✐♦♥✿ r❡s♣❡❝t t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 8

❈♦✈❡r✐♥❣ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

❆ ❝♦✈❡r✐♥❣ ✐s ❛ ❝♦♥t✐♥♦✉s ♠❛♣ f : Y → X ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡♥ ❝♦✈❡r Uα ♦❢ X s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ α✱ f −✶(Uα) ✐s ❛ ❞✐s❥♦✐♥t ✉♥✐♦♥ ♦❢ ♦♣❡♥ s❡ts ✐♥ Y ✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s ♠❛♣♣❡❞ ❤♦♠❡♦♠♦r♣❤✐❝❛❧❧② ♦♥t♦ Uα ❜② f ✳ ❊✈❡r② ❝♦✈❡r✐♥❣ ✐s ❛♥ ✐♠♠❡rs✐♦♥✳ ❯♥❞❡r ♠✐❧❞ ❝♦♥❞✐t✐♦♥s ✐♠♣♦s❡❞✱ ❝♦♥♥❡❝t❡❞ ❝♦✈❡rs ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❛r❡ ✐♥ ♦♥❡✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ s✉❜❣r♦✉♣s ♦❢ ✐ts ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✳ ❋✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✿ ❤♦♠♦t♣② ❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ♣❛t❤s ❛r♦✉♥❞ ❛ ❣✐✈❡♥ ♣♦✐♥t✱ ❡q✉✐♣♣❡❞ ✇✐t❤ ❝♦♥❝❛t❡♥❛t✐♦♥

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 9

❈♦✈❡r✐♥❣ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

❆ ❝♦✈❡r✐♥❣ ✐s ❛ ❝♦♥t✐♥♦✉s ♠❛♣ f : Y → X ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡♥ ❝♦✈❡r Uα ♦❢ X s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ α✱ f −✶(Uα) ✐s ❛ ❞✐s❥♦✐♥t ✉♥✐♦♥ ♦❢ ♦♣❡♥ s❡ts ✐♥ Y ✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s ♠❛♣♣❡❞ ❤♦♠❡♦♠♦r♣❤✐❝❛❧❧② ♦♥t♦ Uα ❜② f ✳ ❊✈❡r② ❝♦✈❡r✐♥❣ ✐s ❛♥ ✐♠♠❡rs✐♦♥✳ ❯♥❞❡r ♠✐❧❞ ❝♦♥❞✐t✐♦♥s ✐♠♣♦s❡❞✱ ❝♦♥♥❡❝t❡❞ ❝♦✈❡rs ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❛r❡ ✐♥ ♦♥❡✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ s✉❜❣r♦✉♣s ♦❢ ✐ts ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✳ ❋✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✿ ❤♦♠♦t♣② ❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ♣❛t❤s ❛r♦✉♥❞ ❛ ❣✐✈❡♥ ♣♦✐♥t✱ ❡q✉✐♣♣❡❞ ✇✐t❤ ❝♦♥❝❛t❡♥❛t✐♦♥

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 10

❈♦✈❡rs ♦❢ ❣r❛♣❤s

❆♥ ❡①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣ ♦❢ t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s = t❤❡ ❢r❡❡ ❣r♦✉♣ ♦❢ r❛♥❦ n ❝♦✈❡rs ♦❢ t❤❡ ❜♦✉q✉❡t ♦❢ ❝✐r❝❧❡s ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ s✉❜❣r♦✉♣s ♦❢ t❤❡ ❢r❡❡ ❣r♦✉♣ ♦❢ r❛♥❦ ✭◆✐❡❧s❡♥✲❙❝❤r❡✐❡r t❤❡r♦❡♠ t❤❡ ❢✉♥❞✳ ❣r♦✉♣s ♦❢ ❣r❛♣❤s ❛r❡ ❛❧✇❛②s ❢r❡❡✮

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 11

❈♦✈❡rs ♦❢ ❣r❛♣❤s

❆♥ ❡①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣ ♦❢ t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s = t❤❡ ❢r❡❡ ❣r♦✉♣ ♦❢ r❛♥❦ n = ⇒ ❝♦✈❡rs ♦❢ t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s ← → ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ s✉❜❣r♦✉♣s ♦❢ t❤❡ ❢r❡❡ ❣r♦✉♣ ♦❢ r❛♥❦ n ✭◆✐❡❧s❡♥✲❙❝❤r❡✐❡r t❤❡r♦❡♠ t❤❡ ❢✉♥❞✳ ❣r♦✉♣s ♦❢ ❣r❛♣❤s ❛r❡ ❛❧✇❛②s ❢r❡❡✮

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 12

❈♦✈❡rs ♦❢ ❣r❛♣❤s

❆♥ ❡①❛♠♣❧❡✿ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣ ♦❢ t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s = t❤❡ ❢r❡❡ ❣r♦✉♣ ♦❢ r❛♥❦ n = ⇒ ❝♦✈❡rs ♦❢ t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s ← → ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ s✉❜❣r♦✉♣s ♦❢ t❤❡ ❢r❡❡ ❣r♦✉♣ ♦❢ r❛♥❦ n ✭◆✐❡❧s❡♥✲❙❝❤r❡✐❡r t❤❡r♦❡♠ ⇐ ⇒ t❤❡ ❢✉♥❞✳ ❣r♦✉♣s ♦❢ ❣r❛♣❤s ❛r❡ ❛❧✇❛②s ❢r❡❡✮

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 13

r❛♣❤ ✐♠♠❡rs✐♦♥s

❆ r❡s✉❧t ♦❢ ▼❛r❣♦❧✐s ❛♥❞ ▼❡❛❦✐♥✿ ■♠♠❡rs✐♦♥s ♦✈❡r t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s ← → ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞s ♦❢ t❤❡ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ r❛♥❦ n✳ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞❄

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 14

r❛♣❤ ✐♠♠❡rs✐♦♥s

❆ r❡s✉❧t ♦❢ ▼❛r❣♦❧✐s ❛♥❞ ▼❡❛❦✐♥✿ ■♠♠❡rs✐♦♥s ♦✈❡r t❤❡ ❜♦✉q✉❡t ♦❢ n ❝✐r❝❧❡s ← → ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞s ♦❢ t❤❡ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ r❛♥❦ n✳ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞❄

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 15

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❋❛❝t✿ t❤❡ ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥❝❡ ♦♥ ♣❛t❤s ♦❢ ❛ ❣r❛♣❤ ✐s ✐♥❞✉❝❡❞ ❜② pp−✶ ≡ α(p) ❢♦r ❛♥② ♣❛t❤ p ▲❡t ❞❡♥♦t❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ✐♥❞✉❝❡❞ ❜②

✶ ✶ ✶ ✶ ✶ ✭♥♦ ❣❡♥❡r❛❧ t♦♣♦❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮

▲❡t ❜❡ ❛ ❞✐❣r❛♣❤ ❡❞❣❡✲❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t ✱ ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ ❛♥❞ ❧❡t ❜❡ ❛ ✈❡rt❡①✳ ❚❤❡♥ ♣❛t❤s st❛rt✐♥❣ ❛t ✇♦r❞s ♦✈❡r

✶✳

❉❡✜♥✐t✐♦♥

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ✐s t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❝♦♥s✐st✐♥❣ ♦❢ ✲❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ♣❛t❤s ❛r♦✉♥❞ ✱ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♥❝❛t❡♥❛t✐♦♥✳ ◆♦t❡✿ ✐s ❣❡♥❡r❛t❡❞ ❜② ✱ ✐♥ ❢❛❝t✱ ✐t ✐s ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ t❤❡ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 16

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❋❛❝t✿ t❤❡ ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥❝❡ ♦♥ ♣❛t❤s ♦❢ ❛ ❣r❛♣❤ ✐s ✐♥❞✉❝❡❞ ❜② pp−✶ ≡ α(p) ❢♦r ❛♥② ♣❛t❤ p ▲❡t ≈ ❞❡♥♦t❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ✐♥❞✉❝❡❞ ❜② pp−✶p ≈ p & pp−✶qq−✶ ≈ qq−✶pp−✶ ✭♥♦ ❣❡♥❡r❛❧ t♦♣♦❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮ ▲❡t ❜❡ ❛ ❞✐❣r❛♣❤ ❡❞❣❡✲❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t ✱ ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ ❛♥❞ ❧❡t ❜❡ ❛ ✈❡rt❡①✳ ❚❤❡♥ ♣❛t❤s st❛rt✐♥❣ ❛t ✇♦r❞s ♦✈❡r

✶✳

❉❡✜♥✐t✐♦♥

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ✐s t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❝♦♥s✐st✐♥❣ ♦❢ ✲❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ♣❛t❤s ❛r♦✉♥❞ ✱ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♥❝❛t❡♥❛t✐♦♥✳ ◆♦t❡✿ ✐s ❣❡♥❡r❛t❡❞ ❜② ✱ ✐♥ ❢❛❝t✱ ✐t ✐s ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ t❤❡ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 17

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❋❛❝t✿ t❤❡ ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥❝❡ ♦♥ ♣❛t❤s ♦❢ ❛ ❣r❛♣❤ ✐s ✐♥❞✉❝❡❞ ❜② pp−✶ ≡ α(p) ❢♦r ❛♥② ♣❛t❤ p ▲❡t ≈ ❞❡♥♦t❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ✐♥❞✉❝❡❞ ❜② pp−✶p ≈ p & pp−✶qq−✶ ≈ qq−✶pp−✶ ✭♥♦ ❣❡♥❡r❛❧ t♦♣♦❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮ ▲❡t Γ ❜❡ ❛ ❞✐❣r❛♣❤ ❡❞❣❡✲❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X✱ ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ ❛♥❞ ❧❡t v ❜❡ ❛ ✈❡rt❡①✳ ❚❤❡♥ ♣❛t❤s st❛rt✐♥❣ ❛t v ← → ✇♦r❞s ♦✈❡r X ∪ X −✶✳

❉❡✜♥✐t✐♦♥

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ✐s t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❝♦♥s✐st✐♥❣ ♦❢ ✲❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ♣❛t❤s ❛r♦✉♥❞ ✱ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♥❝❛t❡♥❛t✐♦♥✳ ◆♦t❡✿ ✐s ❣❡♥❡r❛t❡❞ ❜② ✱ ✐♥ ❢❛❝t✱ ✐t ✐s ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ t❤❡ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 18

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❋❛❝t✿ t❤❡ ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥❝❡ ♦♥ ♣❛t❤s ♦❢ ❛ ❣r❛♣❤ ✐s ✐♥❞✉❝❡❞ ❜② pp−✶ ≡ α(p) ❢♦r ❛♥② ♣❛t❤ p ▲❡t ≈ ❞❡♥♦t❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ✐♥❞✉❝❡❞ ❜② pp−✶p ≈ p & pp−✶qq−✶ ≈ qq−✶pp−✶ ✭♥♦ ❣❡♥❡r❛❧ t♦♣♦❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮ ▲❡t Γ ❜❡ ❛ ❞✐❣r❛♣❤ ❡❞❣❡✲❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X✱ ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ ❛♥❞ ❧❡t v ❜❡ ❛ ✈❡rt❡①✳ ❚❤❡♥ ♣❛t❤s st❛rt✐♥❣ ❛t v ← → ✇♦r❞s ♦✈❡r X ∪ X −✶✳

❉❡✜♥✐t✐♦♥

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ L(Γ, v) ✐s t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❝♦♥s✐st✐♥❣ ♦❢ ≈✲❝❧❛ss❡s ♦❢ ❝❧♦s❡❞ ♣❛t❤s ❛r♦✉♥❞ v✱ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♥❝❛t❡♥❛t✐♦♥✳ ◆♦t❡✿ L(Γ, v) ✐s ❣❡♥❡r❛t❡❞ ❜② X✱ ✐♥ ❢❛❝t✱ ✐t ✐s ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ t❤❡ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ FIM(X)✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 19

❚❤❡ t❤❡♦r❡♠ ❝❧❛ss✐❢②✐♥❣ ❣r❛♣❤ ✐♠♠❡rs✐♦♥s

❚❤❡♦r❡♠ ✭▼❛r❣♦❧✐s✱ ▼❡❛❦✐♥✮

▲❡t f : Γ✷ → Γ✶ ❜❡ ❛♥ ✐♠♠❡rs✐♦♥ ♦✈❡r Γ✶✱ ✇❤❡r❡ Γ✶ ❛♥❞ Γ✷ ❛r❡ ❣r❛♣❤s ❧❛❜❡❧❡❞ ♦✈❡r X✱ ❛♥❞ f r❡s♣❡❝ts t❤❡ ❧❛❜❡❧✐♥❣✳ ■❢ vi ∈ V (Γi)✱ i = ✶, ✷✱ s✉❝❤ t❤❛t f (v✷) = v✶✱ t❤❡♥ f ✐♥❞✉❝❡s ❛♥ ❡♠❜❡❞❞✐♥❣ ♦❢ L(Γ✷, v✷) ✐♥t♦ L(Γ✶, v✶)✳ ❈♦♥✈❡rs❡❧②✱ ❧❡t Γ✶ ❜❡ ❛ ❧❛❜❡❧❡❞ ❣r❛♣❤ ❛♥❞ ❧❡t H ❜❡ ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ L(Γ✶, v✶) ❢♦r s♦♠❡ v✶ ∈ V (Γ✶)✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✷✲❝♦♠♣❧❡① Γ✷ ❛♥❞ ❛♥ ✐♠♠❡rs✐♦♥ f : Γ✷ → Γ✶ ❛♥❞ ❛ ✈❡rt❡① v✷ ∈ V (Γ✷) s✉❝❤ t❤❛t f (v✷) = v✶ ❛♥❞ L(Γ✷, v✷) = H✳ ❘❡♠❛r❦s✿

◮ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞s ←

→ ❝❧♦s❡❞ ✉♣✇❛r❞s ✐♥ t❤❡ ♣❛rt✐❛❧ ♦r❞❡r ← → st❛❜✐❧✐③❡rs ♦❢ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❛❝t✐♦♥s

◮ L(Γ, v) ❛♥❞ L(Γ, v′) ❛r❡ ❝♦♥❥✉❣❛t❡✱ ❜✉t ♥♦t ♥❡❝❡ss❛r✐❧②

✐s♠♦r♣❤✐❝ ✭✉♥❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣✮

◮ H, K ⊆ L(Γ✶, v✶) ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s❛♠❡ ✐♠♠❡rs✐♦♥ ✐✛ t❤❡②

❛r❡ ❝♦♥❥✉❣❛t❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 20

■♠♠❡rs✐♦♥s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s

❈❲✲❝♦♠♣❧❡①❡s✿ ❛ ❝❧❛ss ♦❢ ✭✈❡r② ❣❡♥❡r❛❧✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡ ❆ ❈❲✲❝♦♠♣❧❡① ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❜✉✐❧t ✐t❡r❛t✐✈❡❧②✿ ✶✳ ❙t❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ s❡t ♦❢ ♣♦✐♥ts ✭✵✲❝❡❧❧s✮ ✷✳ ❆tt❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧s t♦ t❤❡ ✵✲s❦❡❧❡t♦♥ ✭✶✲❝❡❧❧s✮ ✸✳ ❆tt❛❝❤ ♦♣❡♥ ❞✐s❦s t♦ t❤❡ ✶✲s❦❡❧❡t♦♥ ✭✷✲❝❡❧❧s✮ ✹✳ ✳✳✳ ❉✐♠❡♥s✐♦♥✿ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❈❲✲❝♦♠♣❧❡①❡s ❂ ❣r❛♣❤s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 21

■♠♠❡rs✐♦♥s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s

❈❲✲❝♦♠♣❧❡①❡s✿ ❛ ❝❧❛ss ♦❢ ✭✈❡r② ❣❡♥❡r❛❧✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡ ❆ ❈❲✲❝♦♠♣❧❡① ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❜✉✐❧t ✐t❡r❛t✐✈❡❧②✿ ✶✳ ❙t❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ s❡t ♦❢ ♣♦✐♥ts ✭✵✲❝❡❧❧s✮ ✷✳ ❆tt❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧s t♦ t❤❡ ✵✲s❦❡❧❡t♦♥ ✭✶✲❝❡❧❧s✮ ✸✳ ❆tt❛❝❤ ♦♣❡♥ ❞✐s❦s t♦ t❤❡ ✶✲s❦❡❧❡t♦♥ ✭✷✲❝❡❧❧s✮ ✹✳ ✳✳✳ ❉✐♠❡♥s✐♦♥✿ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❈❲✲❝♦♠♣❧❡①❡s ❂ ❣r❛♣❤s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 22

■♠♠❡rs✐♦♥s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s

❈❲✲❝♦♠♣❧❡①❡s✿ ❛ ❝❧❛ss ♦❢ ✭✈❡r② ❣❡♥❡r❛❧✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡ ❆ ❈❲✲❝♦♠♣❧❡① ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❜✉✐❧t ✐t❡r❛t✐✈❡❧②✿ ✶✳ ❙t❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ s❡t ♦❢ ♣♦✐♥ts ✭✵✲❝❡❧❧s✮ ✷✳ ❆tt❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧s t♦ t❤❡ ✵✲s❦❡❧❡t♦♥ ✭✶✲❝❡❧❧s✮ ✸✳ ❆tt❛❝❤ ♦♣❡♥ ❞✐s❦s t♦ t❤❡ ✶✲s❦❡❧❡t♦♥ ✭✷✲❝❡❧❧s✮ ✹✳ ✳✳✳ ❉✐♠❡♥s✐♦♥✿ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❈❲✲❝♦♠♣❧❡①❡s ❂ ❣r❛♣❤s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 23

■♠♠❡rs✐♦♥s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s

❈❲✲❝♦♠♣❧❡①❡s✿ ❛ ❝❧❛ss ♦❢ ✭✈❡r② ❣❡♥❡r❛❧✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡ ❆ ❈❲✲❝♦♠♣❧❡① ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❜✉✐❧t ✐t❡r❛t✐✈❡❧②✿ ✶✳ ❙t❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ s❡t ♦❢ ♣♦✐♥ts ✭✵✲❝❡❧❧s✮ ✷✳ ❆tt❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧s t♦ t❤❡ ✵✲s❦❡❧❡t♦♥ ✭✶✲❝❡❧❧s✮ ✸✳ ❆tt❛❝❤ ♦♣❡♥ ❞✐s❦s t♦ t❤❡ ✶✲s❦❡❧❡t♦♥ ✭✷✲❝❡❧❧s✮ ✹✳ ✳✳✳ ❉✐♠❡♥s✐♦♥✿ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❈❲✲❝♦♠♣❧❡①❡s ❂ ❣r❛♣❤s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 24

■♠♠❡rs✐♦♥s ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s

❈❲✲❝♦♠♣❧❡①❡s✿ ❛ ❝❧❛ss ♦❢ ✭✈❡r② ❣❡♥❡r❛❧✮ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡ ❆ ❈❲✲❝♦♠♣❧❡① ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❜✉✐❧t ✐t❡r❛t✐✈❡❧②✿ ✶✳ ❙t❛rt ✇✐t❤ ❛ ❞✐s❝r❡t❡ s❡t ♦❢ ♣♦✐♥ts ✭✵✲❝❡❧❧s✮ ✷✳ ❆tt❛❝❤ ♦♣❡♥ ✐♥t❡r✈❛❧s t♦ t❤❡ ✵✲s❦❡❧❡t♦♥ ✭✶✲❝❡❧❧s✮ ✸✳ ❆tt❛❝❤ ♦♣❡♥ ❞✐s❦s t♦ t❤❡ ✶✲s❦❡❧❡t♦♥ ✭✷✲❝❡❧❧s✮ ✹✳ ✳✳✳ ❉✐♠❡♥s✐♦♥✿ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❈❲✲❝♦♠♣❧❡①❡s ❂ ❣r❛♣❤s

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 25

▲❛❜❡❧❡❞ ✷✲❝♦♠♣❧❡①❡s

✶✲s❦❡❧❡t♦♥✿ ❞✐❣r❛♣❤ ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧ ❢♦r ❡❛❝❤ ✷✲❝❡❧❧ ✱ ✇❡ s♣❡❝✐❢② ❛ st❛rt✐♥❣ ♣♦✐♥t ♦♥ ✐ts ❜♦✉♥❞❛r②✿ r♦♦t ♦❢ ✖ t❤❡ ✭❣r❛♣❤✲t❤❡♦r❡t✐❝✮ ♣❛t❤ ❛r♦✉♥❞ t❤❡ ❜♦✉♥❞❛r② st❛rt✐♥❣ ❛t t❤❛t ♣♦✐♥t✿ ❜♦✉♥❞❛r② ✇❛❧❦ ♦❢ ✖ ✷✲❝❡❧❧s✿ ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t ✐♥ ❛ ✇❛② t❤❛t t✇♦ ✷✲❝❡❧❧s ✇✐t❤ t❤❡ s❛♠❡ r♦♦t ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧ ✐❢ t✇♦ ✷✲❝❡❧❧s ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ t❤❡♥ t❤❡ ❧❛❜❡❧ ♦❢ t❤❡✐r ❜♦✉♥❞❛r② ✇❛❧❦ ✐s t❤❡ s❛♠❡✿ ❜♦✉♥❞❛r② ❧❛❜❡❧ ♦❢ ✖

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 26

▲❛❜❡❧❡❞ ✷✲❝♦♠♣❧❡①❡s

✶✲s❦❡❧❡t♦♥✿ ❞✐❣r❛♣❤ ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧

◮ ❢♦r ❡❛❝❤ ✷✲❝❡❧❧ C✱ ✇❡ s♣❡❝✐❢② ❛ st❛rt✐♥❣ ♣♦✐♥t ♦♥ ✐ts ❜♦✉♥❞❛r②✿

r♦♦t ♦❢ C ✖ α(C)

◮ t❤❡ ✭❣r❛♣❤✲t❤❡♦r❡t✐❝✮ ♣❛t❤ ❛r♦✉♥❞ t❤❡ ❜♦✉♥❞❛r② st❛rt✐♥❣ ❛t

t❤❛t ♣♦✐♥t✿ ❜♦✉♥❞❛r② ✇❛❧❦ ♦❢ C ✖ bw(C) ✷✲❝❡❧❧s✿ ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t ✐♥ ❛ ✇❛② t❤❛t t✇♦ ✷✲❝❡❧❧s ✇✐t❤ t❤❡ s❛♠❡ r♦♦t ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧ ✐❢ t✇♦ ✷✲❝❡❧❧s ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ t❤❡♥ t❤❡ ❧❛❜❡❧ ♦❢ t❤❡✐r ❜♦✉♥❞❛r② ✇❛❧❦ ✐s t❤❡ s❛♠❡✿ ❜♦✉♥❞❛r② ❧❛❜❡❧ ♦❢ ✖

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 27

▲❛❜❡❧❡❞ ✷✲❝♦♠♣❧❡①❡s

✶✲s❦❡❧❡t♦♥✿ ❞✐❣r❛♣❤ ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X ✐♥ ❛ ✇❛② t❤❛t t✇♦ ❡❞❣❡s ❣♦✐♥❣ ✐♥t♦ ✭❝♦♠✐♥❣ ♦✉t ❢r♦♠✮ ❛ ✈❡rt❡① ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧

◮ ❢♦r ❡❛❝❤ ✷✲❝❡❧❧ C✱ ✇❡ s♣❡❝✐❢② ❛ st❛rt✐♥❣ ♣♦✐♥t ♦♥ ✐ts ❜♦✉♥❞❛r②✿

r♦♦t ♦❢ C ✖ α(C)

◮ t❤❡ ✭❣r❛♣❤✲t❤❡♦r❡t✐❝✮ ♣❛t❤ ❛r♦✉♥❞ t❤❡ ❜♦✉♥❞❛r② st❛rt✐♥❣ ❛t

t❤❛t ♣♦✐♥t✿ ❜♦✉♥❞❛r② ✇❛❧❦ ♦❢ C ✖ bw(C) ✷✲❝❡❧❧s✿ ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t P ✐♥ ❛ ✇❛② t❤❛t

◮ t✇♦ ✷✲❝❡❧❧s ✇✐t❤ t❤❡ s❛♠❡ r♦♦t ❝❛♥♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧ ◮ ✐❢ t✇♦ ✷✲❝❡❧❧s ❤❛✈❡ t❤❡ s❛♠❡ ❧❛❜❡❧✱ t❤❡♥ t❤❡ ❧❛❜❡❧ ♦❢ t❤❡✐r

❜♦✉♥❞❛r② ✇❛❧❦ ✐s t❤❡ s❛♠❡✿ ❜♦✉♥❞❛r② ❧❛❜❡❧ ♦❢ ρ ∈ P ✖ bl(ρ)

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 28

❊①❛♠♣❧❡s

bl(ρ) = aba−✶b−✶

✷

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 29

❊①❛♠♣❧❡s

bl(ρ) = aba−✶b−✶ bl(ρ) = a✷

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 30

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✖ ✇❤❛t ❛r❡ t❤❡ ❣❡♥❡r❛t♦rs ❛♥❞ t❤❡ ♥❡✇ r❡❧❛t✐♦♥s❄ ❋♦r ❛ ✷✲❝♦♠♣❧❡① ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t ✱ t❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✇✐❧❧ ❜❡ ❛♥ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢

✷

P❛t❤s✿ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❡❞❣❡s ❛♥❞ ✷✲❝❡❧❧s✱ ✇❤❡r❡ ❛ ✷✲❝❡❧❧ ✐s r❡❣❛r❞❡❞ ❛s ❛ ❧♦♦♣ ❜❛s❡❞ ❛t ✐t✬s r♦♦t✳ ❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ❛t ✿ ✐s ❛ ❝❧♦s❡❞ ♣❛t❤ ❜❛s❡❞ ❛t ✿ ❝❧♦s❡ ✉♣✇❛r❞s ✐♥ t❤❡ ♣❛rt✐❛❧ ♦r❞❡r

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 31

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✖ ✇❤❛t ❛r❡ t❤❡ ❣❡♥❡r❛t♦rs ❛♥❞ t❤❡ ♥❡✇ r❡❧❛t✐♦♥s❄ ❋♦r ❛ ✷✲❝♦♠♣❧❡① ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X ∪ ρ✱ t❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✇✐❧❧ ❜❡ ❛♥ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ MX,P = Inv

X ∪ P | ρ✷ = ρ, ρ ≤ bl(ρ) : ρ ∈ P
.

P❛t❤s✿ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❡❞❣❡s ❛♥❞ ✷✲❝❡❧❧s✱ ✇❤❡r❡ ❛ ✷✲❝❡❧❧ ✐s r❡❣❛r❞❡❞ ❛s ❛ ❧♦♦♣ ❜❛s❡❞ ❛t ✐t✬s r♦♦t✳ ❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ❛t ✿ ✐s ❛ ❝❧♦s❡❞ ♣❛t❤ ❜❛s❡❞ ❛t ✿ ❝❧♦s❡ ✉♣✇❛r❞s ✐♥ t❤❡ ♣❛rt✐❛❧ ♦r❞❡r

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 32

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✖ ✇❤❛t ❛r❡ t❤❡ ❣❡♥❡r❛t♦rs ❛♥❞ t❤❡ ♥❡✇ r❡❧❛t✐♦♥s❄ ❋♦r ❛ ✷✲❝♦♠♣❧❡① ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X ∪ ρ✱ t❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✇✐❧❧ ❜❡ ❛♥ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ MX,P = Inv

X ∪ P | ρ✷ = ρ, ρ ≤ bl(ρ) : ρ ∈ P
.

P❛t❤s✿ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❡❞❣❡s ❛♥❞ ✷✲❝❡❧❧s✱ ✇❤❡r❡ ❛ ✷✲❝❡❧❧ ✐s r❡❣❛r❞❡❞ ❛s ❛ ❧♦♦♣ ❜❛s❡❞ ❛t ✐t✬s r♦♦t✳ ❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ❛t ✿ ✐s ❛ ❝❧♦s❡❞ ♣❛t❤ ❜❛s❡❞ ❛t ✿ ❝❧♦s❡ ✉♣✇❛r❞s ✐♥ t❤❡ ♣❛rt✐❛❧ ♦r❞❡r

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 33

❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✖ ✇❤❛t ❛r❡ t❤❡ ❣❡♥❡r❛t♦rs ❛♥❞ t❤❡ ♥❡✇ r❡❧❛t✐♦♥s❄ ❋♦r ❛ ✷✲❝♦♠♣❧❡① ❧❛❜❡❧❡❞ ♦✈❡r t❤❡ s❡t X ∪ ρ✱ t❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ ♣❛t❤s ✇✐❧❧ ❜❡ ❛♥ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ MX,P = Inv

X ∪ P | ρ✷ = ρ, ρ ≤ bl(ρ) : ρ ∈ P
.

P❛t❤s✿ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❡❞❣❡s ❛♥❞ ✷✲❝❡❧❧s✱ ✇❤❡r❡ ❛ ✷✲❝❡❧❧ ✐s r❡❣❛r❞❡❞ ❛s ❛ ❧♦♦♣ ❜❛s❡❞ ❛t ✐t✬s r♦♦t✳ ❚❤❡ ❧♦♦♣ ♠♦♥♦✐❞ ♦❢ C ❛t v✿ L(C, v) = l(p) : p ✐s ❛ ❝❧♦s❡❞ ♣❛t❤ ❜❛s❡❞ ❛t v ω ω✿ ❝❧♦s❡ ✉♣✇❛r❞s ✐♥ t❤❡ ♣❛rt✐❛❧ ♦r❞❡r

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 34

❊①❛♠♣❧❡s

a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

✷ ✷

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 35

❊①❛♠♣❧❡s

a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶ Inva, ρ | ρ✷ = ρ, ρ ≤ a✷ ≥ ρ, aρaω

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 36

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭▼❡❛❦✐♥✱ ❙③✳✮

▲❡t f : C✷ → C✶ ❜❡ ❛♥ ✐♠♠❡rs✐♦♥ ♦✈❡r C✶✱ ✇❤❡r❡ C✶ ❛♥❞ C✷ ❛r❡ ✷✲❝♦♠♣❧❡①❡s ❧❛❜❡❧❡❞ ♦✈❡r X ∪ P✱ ❛♥❞ f r❡s♣❡❝ts t❤❡ ❧❛❜❡❧✐♥❣✳ ■❢ vi ∈ C✵

i ✱ i = ✶, ✷✱ s✉❝❤ t❤❛t f (v✷) = v✶✱ t❤❡♥ f ✐♥❞✉❝❡s ❛♥

❡♠❜❡❞❞✐♥❣ ♦❢ L(C✷, v✷) ✐♥t♦ L(C✶, v✶)✳ ❈♦♥✈❡rs❡❧②✱ ❧❡t C✶ ❜❡ ❛ ❧❛❜❡❧❡❞ ✷✲❝♦♠♣❧❡① ❛♥❞ ❧❡t H ❜❡ ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ MX,P s✉❝❤ t❤❛t H ⊆ L(C✶, v✶) ❢♦r s♦♠❡ v✶ ∈ C✵

✶✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts

❛ ✷✲❝♦♠♣❧❡① C✷ ❛♥❞ ❛♥ ✐♠♠❡rs✐♦♥ f : C✷ → C✶ ❛♥❞ ❛ ✈❡rt❡① v✷ ∈ C✵

✷

s✉❝❤ t❤❛t f (v✷) = v✶ ❛♥❞ L(C✷, v✷) = H✳

✶ ✶ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s❛♠❡ ✐♠♠❡rs✐♦♥ ✐✛ t❤❡② ❛r❡

❝♦♥❥✉❣❛t❡✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 37

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭▼❡❛❦✐♥✱ ❙③✳✮

▲❡t f : C✷ → C✶ ❜❡ ❛♥ ✐♠♠❡rs✐♦♥ ♦✈❡r C✶✱ ✇❤❡r❡ C✶ ❛♥❞ C✷ ❛r❡ ✷✲❝♦♠♣❧❡①❡s ❧❛❜❡❧❡❞ ♦✈❡r X ∪ P✱ ❛♥❞ f r❡s♣❡❝ts t❤❡ ❧❛❜❡❧✐♥❣✳ ■❢ vi ∈ C✵

i ✱ i = ✶, ✷✱ s✉❝❤ t❤❛t f (v✷) = v✶✱ t❤❡♥ f ✐♥❞✉❝❡s ❛♥

❡♠❜❡❞❞✐♥❣ ♦❢ L(C✷, v✷) ✐♥t♦ L(C✶, v✶)✳ ❈♦♥✈❡rs❡❧②✱ ❧❡t C✶ ❜❡ ❛ ❧❛❜❡❧❡❞ ✷✲❝♦♠♣❧❡① ❛♥❞ ❧❡t H ❜❡ ❛ ❝❧♦s❡❞ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞ ♦❢ MX,P s✉❝❤ t❤❛t H ⊆ L(C✶, v✶) ❢♦r s♦♠❡ v✶ ∈ C✵

✶✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts

❛ ✷✲❝♦♠♣❧❡① C✷ ❛♥❞ ❛♥ ✐♠♠❡rs✐♦♥ f : C✷ → C✶ ❛♥❞ ❛ ✈❡rt❡① v✷ ∈ C✵

✷

s✉❝❤ t❤❛t f (v✷) = v✶ ❛♥❞ L(C✷, v✷) = H✳ H, K ⊆ L(C✶, v✶) ❝♦rr❡s♣♦♥❞ t♦ t❤❡ s❛♠❡ ✐♠♠❡rs✐♦♥ ✐✛ t❤❡② ❛r❡ ❝♦♥❥✉❣❛t❡✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 38

❊①❛♠♣❧❡s ♦❢ ✐♠♠❡rs✐♦♥s ♦✈❡r D ∨ S✶ MX,P = Inva, b, ρ | ρ✷ = ρ, ρ ≤ b ak, anρa−n : n ∈ {✶, . . . , k}ω k ∈ N✱ (k = ✹) (ab)nab✷a−✶(ab)−n : n ∈ Nω

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 39

❊①❛♠♣❧❡s ♦❢ ✐♠♠❡rs✐♦♥s ♦✈❡r D ∨ S✶ MX,P = Inva, b, ρ | ρ✷ = ρ, ρ ≤ b a✹, ρ, a✷b✷a−✷, abnb−na−✶ : n ∈ Zω ρ, aρa−✶, a✷ba, a✸ρa✷ω

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 40

❆♥ ✐♠♠❡rs✐♦♥ ♦✈❡r t❤❡ t♦r✉s

MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶ H = a−✶b−✶ab, abρa−✶ω ⊆ MX,P

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 41

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 42

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 43

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 44

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 45

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶✱ ρ ≤ aba−✶b−✶ ⇔ ρ = ρaba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 46

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶✱ ρ ≤ aba−✶b−✶ ⇔ ρ = ρaba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 47

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 48

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s

SLIDE 49

❈♦♥str✉❝t✐♥❣ t❤❡ ❝♦♠♣❧❡①

H = a−✶b−✶ab, abρa−✶ω ≤ MX,P = a, b, ρ | ρ✷ = ρ, ρ ≤ aba−✶b−✶

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ✐♠♠❡rs✐♦♥s ♦✈❡r ✷✲❝♦♠♣❧❡①❡s