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Mar 14, 2023 196 likes •583 views

trt s s s tts ts ts rs

SLIDE 1

▼♦❞❡❧ t❤❡♦r❡t✐❝ ❞②♥❛♠✐❝s ✐♥ ❛ ●❛❧♦✐s ❢❛s❤✐♦♥

♠♦t✐✈❛t✐♦♥s✱ ❞❡✜♥✐t✐♦♥s✱ ❢❛❝ts✱ ♥♦ ♣r♦♦❢s✳✳✳ ❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✶

✶■▼ ❯❲r

❇➛❞❧❡✇♦✱ ✼✳✵✼✳✷✵✶✼

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶ ✴ ✶✼

SLIDE 2

❚❤❡ ❯♥✐✈❡rs❡

SLIDE 3

❚❤❡ ❯♥✐✈❡rs❡ ✭♣❡rs♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮

SLIDE 4

❚❤❡ ❯♥✐✈❡rs❡ ✭♣❡rs♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮

SLIDE 5

❚❤❡ ❯♥✐✈❡rs❡ ✭♣❡rs♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮

SLIDE 6

❚❤❡ ❯♥✐✈❡rs❡ ✭♣❡rs♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮

SLIDE 7

❚❤❡ ❯♥✐✈❡rs❡ ✭♣❡rs♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✮

SLIDE 8

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

▼❛✐♥ ❡①❛♠♣❧❡s ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❆❈❋❆✱ Q❆❈❋❆✱ ✭♣❡r❢❡❝t✴✐♠♣❡r❢❡❝t✮ ❜♦✉♥❞❡❞ P❆❈ ✜❡❧❞s✱ ♣s❡✉❞♦✜♥✐t❡ ✜❡❧❞s r❛♥❞♦♠ ❣r❛♣❤ ✭♦t❤❡r ❣r❛♣❤✲❧✐❦❡ t❤❡♦r✐❡s✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✽ ✴ ✶✼

SLIDE 9

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

▼❛✐♥ ❡①❛♠♣❧❡s ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❆❈❋❆✱ Q❆❈❋❆✱ ✭♣❡r❢❡❝t✴✐♠♣❡r❢❡❝t✮ ❜♦✉♥❞❡❞ P❆❈ ✜❡❧❞s✱ ♣s❡✉❞♦✜♥✐t❡ ✜❡❧❞s r❛♥❞♦♠ ❣r❛♣❤ ✭♦t❤❡r ❣r❛♣❤✲❧✐❦❡ t❤❡♦r✐❡s✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✽ ✴ ✶✼

SLIDE 10

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

▼❛✐♥ ❡①❛♠♣❧❡s ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❆❈❋❆✱ Q❆❈❋❆✱ ✭♣❡r❢❡❝t✴✐♠♣❡r❢❡❝t✮ ❜♦✉♥❞❡❞ P❆❈ ✜❡❧❞s✱ ♣s❡✉❞♦✜♥✐t❡ ✜❡❧❞s r❛♥❞♦♠ ❣r❛♣❤ ✭♦t❤❡r ❣r❛♣❤✲❧✐❦❡ t❤❡♦r✐❡s✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✽ ✴ ✶✼

SLIDE 11

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❉♦✉❜ts✳✳✳

❆r❡ s✐♠♣❧❡ t❤❡♦r✐❡s st✉❞✐❡❞ ❡♥♦✉❣❤❄ P✐♦tr✬s ♣♦✐♥t ♦❢ ✈✐❡✇✳✳✳ ▼❛②❜❡ ❆❈❋❆ ✐s ❛ ✏❣❡♥❡r✐❝✑ s✐♠♣❧❡ t❤❡♦r②❄ ❍♦✇ ♠❛♥② s✐♠♣❧❡ t❤❡♦r✐❡s ♦❝❝✉r ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❆❈❋❆❄ ✭❛r❡ ✏❆❈❋❆✲❧✐❦❡✑❄✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✾ ✴ ✶✼

SLIDE 12

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❉♦✉❜ts✳✳✳

❆r❡ s✐♠♣❧❡ t❤❡♦r✐❡s st✉❞✐❡❞ ❡♥♦✉❣❤❄ P✐♦tr✬s ♣♦✐♥t ♦❢ ✈✐❡✇✳✳✳ ▼❛②❜❡ ❆❈❋❆ ✐s ❛ ✏❣❡♥❡r✐❝✑ s✐♠♣❧❡ t❤❡♦r②❄ ❍♦✇ ♠❛♥② s✐♠♣❧❡ t❤❡♦r✐❡s ♦❝❝✉r ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❆❈❋❆❄ ✭❛r❡ ✏❆❈❋❆✲❧✐❦❡✑❄✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✾ ✴ ✶✼

SLIDE 13

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❉♦✉❜ts✳✳✳

❆r❡ s✐♠♣❧❡ t❤❡♦r✐❡s st✉❞✐❡❞ ❡♥♦✉❣❤❄ P✐♦tr✬s ♣♦✐♥t ♦❢ ✈✐❡✇✳✳✳ ▼❛②❜❡ ❆❈❋❆ ✐s ❛ ✏❣❡♥❡r✐❝✑ s✐♠♣❧❡ t❤❡♦r②❄ ❍♦✇ ♠❛♥② s✐♠♣❧❡ t❤❡♦r✐❡s ♦❝❝✉r ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❆❈❋❆❄ ✭❛r❡ ✏❆❈❋❆✲❧✐❦❡✑❄✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✾ ✴ ✶✼

SLIDE 14

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❉♦✉❜ts✳✳✳

❆r❡ s✐♠♣❧❡ t❤❡♦r✐❡s st✉❞✐❡❞ ❡♥♦✉❣❤❄ P✐♦tr✬s ♣♦✐♥t ♦❢ ✈✐❡✇✳✳✳ ▼❛②❜❡ ❆❈❋❆ ✐s ❛ ✏❣❡♥❡r✐❝✑ s✐♠♣❧❡ t❤❡♦r②❄ ❍♦✇ ♠❛♥② s✐♠♣❧❡ t❤❡♦r✐❡s ♦❝❝✉r ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❆❈❋❆❄ ✭❛r❡ ✏❆❈❋❆✲❧✐❦❡✑❄✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✾ ✴ ✶✼

SLIDE 15

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❆❈❋❆✲❧✐❦❡

❆❈❋❆✲❧✐❦❡ ♠❡❛♥s ✏❛ st❛❜❧❡ t❤❡♦r② ✭❡✳❣✳ ❆❈❋✮ ✇✐t❤ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❛❝t✐♦♥ ❜② ❛✉t♦♠♦r♣❤✐s♠s✑ ❙t❛❣❡ ❞✐r❡❝t✐♦♥s ◆♦✇ ❉❛♥✐❡❧ ❣♦❡s t♦ ✈✐s✉❛❧✐③❡r t♦ ❞r❛✇ ❞✐❛❣r❛♠s ❡①♣❧❛✐♥✐♥❣ ❤♦✇ t♦ ❛❞❞ ❛♥ ❛❝t✐♦♥ ❜② ❛✉t♦♠♦r♣❤✐s♠s t♦ ❛ st❛❜❧❡ t❤❡♦r②

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✵ ✴ ✶✼

SLIDE 16

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

❆❈❋❆✲❧✐❦❡

❆❈❋❆✲❧✐❦❡ ♠❡❛♥s ✏❛ st❛❜❧❡ t❤❡♦r② ✭❡✳❣✳ ❆❈❋✮ ✇✐t❤ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❛❝t✐♦♥ ❜② ❛✉t♦♠♦r♣❤✐s♠s✑ ❙t❛❣❡ ❞✐r❡❝t✐♦♥s ◆♦✇ ❉❛♥✐❡❧ ❣♦❡s t♦ ✈✐s✉❛❧✐③❡r t♦ ❞r❛✇ ❞✐❛❣r❛♠s ❡①♣❧❛✐♥✐♥❣ ❤♦✇ t♦ ❛❞❞ ❛♥ ❛❝t✐♦♥ ❜② ❛✉t♦♠♦r♣❤✐s♠s t♦ ❛ st❛❜❧❡ t❤❡♦r②

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✵ ✴ ✶✼

SLIDE 17

▼♦t✐✈❛t✐♦♥s ❙❡❝t♦r ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s

▲♦♥❣ t❡r♠ ♣❧❛♥

✶ T ♠❝ ✐s st❛❜❧❡ =

⇒ T ♠❝

G ✐s s✐♠♣❧❡ ✭❞♦♥❡✱ s♦♠❡ r❡s✉❧ts ✐♥ t❤✐s t❛❧❦✮

✷ ■❢ T ′ ✐s s✐♠♣❧❡ ✭♣r♦❜❛❜❧② ♠♦r❡ r❡str✐❝t✐♦♥s✮✱ t❤❡♥ t❤❡r❡ ❡①✐st L ✱ T ❛♥❞

G s✉❝❤ t❤❛t T ♠❝

G ❡①✐sts ❛♥❞ ✐s ❜✐✲✐♥t❡r♣r❡t❛❜❧❡ ✇✐t❤ T ′✳

✸ ●❡♦♠❡tr✐❝ ❞❡s❝r✐♣t✐♦♥ ♦❢ s✐♠♣❧❡ t❤❡♦r✐❡s ✐♥ t❤❡ st②❧❡ ♦❢ ❆❈❋❆✳ ❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✶ ✴ ✶✼

SLIDE 18

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

❙❡t✲✉♣

❆ss✉♠❡ t❤❛t T ♠❝ ❛♥❞ T ♠❝

G ❡①✐st ❛♥❞ T ♠❝ ✐s st❛❜❧❡ ❛♥❞ ❤❛s ◗❊ ❛♥❞ ❊■✳

❲❡ ✜① ❛ ♠♦♥st❡r ♠♦❞❡❧ (C,(σg)g∈G) | = T ♠❝

G ❛♥❞ ❛ |C|+✲s❛t✉r❛t❡❞ ♠♦♥st❡r

♠♦❞❡❧ D | = T ♠❝ s✉❝❤ t❤❛t C ⊆ D✳ ❆❢t❡r ✉s❡ ♦❢ ●❛❧♦✐s t❤❡♦r②✱ ✇❡ ❝❛♥ s❛② ♠♦r❡ ❛❜♦✉t ❤♦✇ C ✐s ❝♦♥t❛✐♥❡❞ ✐♥ D✱ ✇❤✐❝❤ ✐s ❡①tr❡♠❧② ✐♠♣♦rt❛♥t ✐♥ ♣r♦♦❢s ✭♥♦t t❤❡ ❝❛s❡ ❢♦r G = Z✱ ❡✳❣✳ ❆❈❋❆✮✳ ❊✳❣✳ ❛❝❧C

L G (A) = ❛❝❧C L (G ·A) = ❛❝❧D L (G ·A)∩C

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✷ ✴ ✶✼

SLIDE 19

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

❙❡t✲✉♣

❆ss✉♠❡ t❤❛t T ♠❝ ❛♥❞ T ♠❝

G ❡①✐st ❛♥❞ T ♠❝ ✐s st❛❜❧❡ ❛♥❞ ❤❛s ◗❊ ❛♥❞ ❊■✳

❲❡ ✜① ❛ ♠♦♥st❡r ♠♦❞❡❧ (C,(σg)g∈G) | = T ♠❝

G ❛♥❞ ❛ |C|+✲s❛t✉r❛t❡❞ ♠♦♥st❡r

♠♦❞❡❧ D | = T ♠❝ s✉❝❤ t❤❛t C ⊆ D✳ ❆❢t❡r ✉s❡ ♦❢ ●❛❧♦✐s t❤❡♦r②✱ ✇❡ ❝❛♥ s❛② ♠♦r❡ ❛❜♦✉t ❤♦✇ C ✐s ❝♦♥t❛✐♥❡❞ ✐♥ D✱ ✇❤✐❝❤ ✐s ❡①tr❡♠❧② ✐♠♣♦rt❛♥t ✐♥ ♣r♦♦❢s ✭♥♦t t❤❡ ❝❛s❡ ❢♦r G = Z✱ ❡✳❣✳ ❆❈❋❆✮✳ ❊✳❣✳ ❛❝❧C

L G (A) = ❛❝❧C L (G ·A) = ❛❝❧D L (G ·A)∩C

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✷ ✴ ✶✼

SLIDE 20

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

❙❡t✲✉♣

❆ss✉♠❡ t❤❛t T ♠❝ ❛♥❞ T ♠❝

G ❡①✐st ❛♥❞ T ♠❝ ✐s st❛❜❧❡ ❛♥❞ ❤❛s ◗❊ ❛♥❞ ❊■✳

❲❡ ✜① ❛ ♠♦♥st❡r ♠♦❞❡❧ (C,(σg)g∈G) | = T ♠❝

G ❛♥❞ ❛ |C|+✲s❛t✉r❛t❡❞ ♠♦♥st❡r

♠♦❞❡❧ D | = T ♠❝ s✉❝❤ t❤❛t C ⊆ D✳ ❆❢t❡r ✉s❡ ♦❢ ●❛❧♦✐s t❤❡♦r②✱ ✇❡ ❝❛♥ s❛② ♠♦r❡ ❛❜♦✉t ❤♦✇ C ✐s ❝♦♥t❛✐♥❡❞ ✐♥ D✱ ✇❤✐❝❤ ✐s ❡①tr❡♠❧② ✐♠♣♦rt❛♥t ✐♥ ♣r♦♦❢s ✭♥♦t t❤❡ ❝❛s❡ ❢♦r G = Z✱ ❡✳❣✳ ❆❈❋❆✮✳ ❊✳❣✳ ❛❝❧C

L G (A) = ❛❝❧C L (G ·A) = ❛❝❧D L (G ·A)∩C

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✷ ✴ ✶✼

SLIDE 21

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

❙❡t✲✉♣

❆ss✉♠❡ t❤❛t T ♠❝ ❛♥❞ T ♠❝

G ❡①✐st ❛♥❞ T ♠❝ ✐s st❛❜❧❡ ❛♥❞ ❤❛s ◗❊ ❛♥❞ ❊■✳

❲❡ ✜① ❛ ♠♦♥st❡r ♠♦❞❡❧ (C,(σg)g∈G) | = T ♠❝

G ❛♥❞ ❛ |C|+✲s❛t✉r❛t❡❞ ♠♦♥st❡r

♠♦❞❡❧ D | = T ♠❝ s✉❝❤ t❤❛t C ⊆ D✳ ❆❢t❡r ✉s❡ ♦❢ ●❛❧♦✐s t❤❡♦r②✱ ✇❡ ❝❛♥ s❛② ♠♦r❡ ❛❜♦✉t ❤♦✇ C ✐s ❝♦♥t❛✐♥❡❞ ✐♥ D✱ ✇❤✐❝❤ ✐s ❡①tr❡♠❧② ✐♠♣♦rt❛♥t ✐♥ ♣r♦♦❢s ✭♥♦t t❤❡ ❝❛s❡ ❢♦r G = Z✱ ❡✳❣✳ ❆❈❋❆✮✳ ❊✳❣✳ ❛❝❧C

L G (A) = ❛❝❧C L (G ·A) = ❛❝❧D L (G ·A)∩C

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✷ ✴ ✶✼

SLIDE 22

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

❲❤② ◗❊ ❛♥❞ ❊■❄

❈♦r♦❧❧❛r② ❆ss✉♠❡ t❤❛t E,A,B ⊆ D✱ A ✐s r❡❣✉❧❛r ♦✈❡r E ✲ ✐✳❡✳ E ⊆ A ❛♥❞ ❞❝❧D

L (A)∩❛❝❧D L (E) = ❞❝❧D L (E)

✲ ❛♥❞ f✶,f✷ ∈ ❆✉tL (D)✱ f✶|E = f✷|E✳ ■❢ A | ⌣

D E B ❛♥❞ f✶(A) |

⌣

D f✶(E) f✷(B)✱ t❤❡♥ t❤❡r❡ ❡①✐sts h ∈ ❆✉tL (D) s✉❝❤

t❤❛t h|A = f✶|A ❛♥❞ h|B = f✷|B✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✸ ✴ ✶✼

SLIDE 23

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

❲❤② ◗❊ ❛♥❞ ❊■❄

❈♦r♦❧❧❛r② ❆ss✉♠❡ t❤❛t E,A,B ⊆ D✱ A ✐s r❡❣✉❧❛r ♦✈❡r E ✲ ✐✳❡✳ E ⊆ A ❛♥❞ ❞❝❧D

L (A)∩❛❝❧D L (E) = ❞❝❧D L (E)

✲ ❛♥❞ f✶,f✷ ∈ ❆✉tL (D)✱ f✶|E = f✷|E✳ ■❢ A | ⌣

D E B ❛♥❞ f✶(A) |

⌣

D f✶(E) f✷(B)✱ t❤❡♥ t❤❡r❡ ❡①✐sts h ∈ ❆✉tL (D) s✉❝❤

t❤❛t h|A = f✶|A ❛♥❞ h|B = f✷|B✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✸ ✴ ✶✼

SLIDE 24

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

■♥❞❡♣❡♥❞❡♥❝❡ ♥♦t✐♦♥

❉❡✜♥✐t✐♦♥ A

⌣

E

B ⇐ ⇒ G ·A

D

| ⌣

G·E

G ·B ❚❤❡ ❛❜♦✈❡ ❞❡✜♥❡s ❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❧❛t✐♦♥✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✹ ✴ ✶✼

SLIDE 25

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼✐♥♦r t❤✐♥❣s

■♥❞❡♣❡♥❞❡♥❝❡ ♥♦t✐♦♥

❉❡✜♥✐t✐♦♥ A

⌣

E

B ⇐ ⇒ G ·A

D

| ⌣

G·E

G ·B ❚❤❡ ❛❜♦✈❡ ❞❡✜♥❡s ❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❧❛t✐♦♥✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✹ ✴ ✶✼

SLIDE 26

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❆❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t

❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t ✐♥ D ♦✈❡r C ✐❢ ❢♦r ❛❧❧ s♠❛❧❧ (M, ¯ σ) (C, ¯ σ) ❛♥❞ ❛❧❧ s♠❛❧❧ A ⊆ C✱ s✉❝❤ t❤❛t M ⊆ A✱ ✐t ❢♦❧❧♦✇s ❛❝❧D

L (A) = ❞❝❧D L

❛❝❧D

L (A)∩C, ❛❝❧D L (M)

❋❛❝t

❖❦ ❢♦r C = D ✭❡✳❣✳ ❛❝t✐♦♥s ♦❢ G = Z✱ ❧✐❦❡ ❆❈❋❆✮ ❖❦ ❢♦r ✜♥✐t❡ G✳ ❖❦ ✐❢ C ✐s ❜♦✉♥❞❡❞ ✭❡✳❣✳ G ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✺ ✴ ✶✼

SLIDE 27

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❆❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t

❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t ✐♥ D ♦✈❡r C ✐❢ ❢♦r ❛❧❧ s♠❛❧❧ (M, ¯ σ) (C, ¯ σ) ❛♥❞ ❛❧❧ s♠❛❧❧ A ⊆ C✱ s✉❝❤ t❤❛t M ⊆ A✱ ✐t ❢♦❧❧♦✇s ❛❝❧D

L (A) = ❞❝❧D L

❛❝❧D

L (A)∩C, ❛❝❧D L (M)

❋❛❝t

❖❦ ❢♦r C = D ✭❡✳❣✳ ❛❝t✐♦♥s ♦❢ G = Z✱ ❧✐❦❡ ❆❈❋❆✮ ❖❦ ❢♦r ✜♥✐t❡ G✳ ❖❦ ✐❢ C ✐s ❜♦✉♥❞❡❞ ✭❡✳❣✳ G ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✺ ✴ ✶✼

SLIDE 28

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❆❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t

❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t ✐♥ D ♦✈❡r C ✐❢ ❢♦r ❛❧❧ s♠❛❧❧ (M, ¯ σ) (C, ¯ σ) ❛♥❞ ❛❧❧ s♠❛❧❧ A ⊆ C✱ s✉❝❤ t❤❛t M ⊆ A✱ ✐t ❢♦❧❧♦✇s ❛❝❧D

L (A) = ❞❝❧D L

❛❝❧D

L (A)∩C, ❛❝❧D L (M)

❋❛❝t

❖❦ ❢♦r C = D ✭❡✳❣✳ ❛❝t✐♦♥s ♦❢ G = Z✱ ❧✐❦❡ ❆❈❋❆✮ ❖❦ ❢♦r ✜♥✐t❡ G✳ ❖❦ ✐❢ C ✐s ❜♦✉♥❞❡❞ ✭❡✳❣✳ G ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✺ ✴ ✶✼

SLIDE 29

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❆❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t

❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t ✐♥ D ♦✈❡r C ✐❢ ❢♦r ❛❧❧ s♠❛❧❧ (M, ¯ σ) (C, ¯ σ) ❛♥❞ ❛❧❧ s♠❛❧❧ A ⊆ C✱ s✉❝❤ t❤❛t M ⊆ A✱ ✐t ❢♦❧❧♦✇s ❛❝❧D

L (A) = ❞❝❧D L

❛❝❧D

L (A)∩C, ❛❝❧D L (M)

❋❛❝t

❖❦ ❢♦r C = D ✭❡✳❣✳ ❛❝t✐♦♥s ♦❢ G = Z✱ ❧✐❦❡ ❆❈❋❆✮ ❖❦ ❢♦r ✜♥✐t❡ G✳ ❖❦ ✐❢ C ✐s ❜♦✉♥❞❡❞ ✭❡✳❣✳ G ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✺ ✴ ✶✼

SLIDE 30

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ■❢ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t✱ t❤❡♥ | ⌣

s❛t✐s✜❡s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦✈❡r ❛ ♠♦❞❡❧

t❤❡♦r❡♠ ✭T ♠❝

G ✐s s✐♠♣❧❡ ❛♥❞ |

⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✮✳

❋❛❝t ✏◆❖ ❤♦♣❡✑ ❢♦r✿ st❛❜✐❧✐t② ♦❢ T ♠❝

G ✐♥ ❣❡♥❡r❛❧ ✭❆❈❋❆✮

✏s✉♣❡rst❛❜❧❡ = ⇒ s✉♣❡rs✐♠♣❧❡✑ ✭Q❆❈❋❆✮ r❡♠♦✈✐♥❣ ✏❛❧❣✳ ❝❧♦s✳ s♣❧✐t✑ ✭❛❧❣✳ ❝❧♦s✳ ❞♦ ♥♦t s♣❧✐t = ⇒ ♥♦t ❜♦✉♥❞❡❞✱ ❡✳❣ D∞✲❚❈❋ ❡①✐sts ❛♥❞ ✐s ♥♦t s✐♠♣❧❡✱ ❜❡❝❛✉s❡ ♠♦❞❡❧s ❛r❡ ♥♦t ❜♦✉♥❞❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✻ ✴ ✶✼

SLIDE 31

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ■❢ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t✱ t❤❡♥ | ⌣

s❛t✐s✜❡s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦✈❡r ❛ ♠♦❞❡❧

t❤❡♦r❡♠ ✭T ♠❝

G ✐s s✐♠♣❧❡ ❛♥❞ |

⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✮✳

❋❛❝t ✏◆❖ ❤♦♣❡✑ ❢♦r✿ st❛❜✐❧✐t② ♦❢ T ♠❝

G ✐♥ ❣❡♥❡r❛❧ ✭❆❈❋❆✮

✏s✉♣❡rst❛❜❧❡ = ⇒ s✉♣❡rs✐♠♣❧❡✑ ✭Q❆❈❋❆✮ r❡♠♦✈✐♥❣ ✏❛❧❣✳ ❝❧♦s✳ s♣❧✐t✑ ✭❛❧❣✳ ❝❧♦s✳ ❞♦ ♥♦t s♣❧✐t = ⇒ ♥♦t ❜♦✉♥❞❡❞✱ ❡✳❣ D∞✲❚❈❋ ❡①✐sts ❛♥❞ ✐s ♥♦t s✐♠♣❧❡✱ ❜❡❝❛✉s❡ ♠♦❞❡❧s ❛r❡ ♥♦t ❜♦✉♥❞❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✻ ✴ ✶✼

SLIDE 32

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ■❢ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t✱ t❤❡♥ | ⌣

s❛t✐s✜❡s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦✈❡r ❛ ♠♦❞❡❧

t❤❡♦r❡♠ ✭T ♠❝

G ✐s s✐♠♣❧❡ ❛♥❞ |

⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✮✳

❋❛❝t ✏◆❖ ❤♦♣❡✑ ❢♦r✿ st❛❜✐❧✐t② ♦❢ T ♠❝

G ✐♥ ❣❡♥❡r❛❧ ✭❆❈❋❆✮

✏s✉♣❡rst❛❜❧❡ = ⇒ s✉♣❡rs✐♠♣❧❡✑ ✭Q❆❈❋❆✮ r❡♠♦✈✐♥❣ ✏❛❧❣✳ ❝❧♦s✳ s♣❧✐t✑ ✭❛❧❣✳ ❝❧♦s✳ ❞♦ ♥♦t s♣❧✐t = ⇒ ♥♦t ❜♦✉♥❞❡❞✱ ❡✳❣ D∞✲❚❈❋ ❡①✐sts ❛♥❞ ✐s ♥♦t s✐♠♣❧❡✱ ❜❡❝❛✉s❡ ♠♦❞❡❧s ❛r❡ ♥♦t ❜♦✉♥❞❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✻ ✴ ✶✼

SLIDE 33

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ■❢ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡s s♣❧✐t✱ t❤❡♥ | ⌣

s❛t✐s✜❡s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦✈❡r ❛ ♠♦❞❡❧

t❤❡♦r❡♠ ✭T ♠❝

G ✐s s✐♠♣❧❡ ❛♥❞ |

⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✮✳

❋❛❝t ✏◆❖ ❤♦♣❡✑ ❢♦r✿ st❛❜✐❧✐t② ♦❢ T ♠❝

G ✐♥ ❣❡♥❡r❛❧ ✭❆❈❋❆✮

✏s✉♣❡rst❛❜❧❡ = ⇒ s✉♣❡rs✐♠♣❧❡✑ ✭Q❆❈❋❆✮ r❡♠♦✈✐♥❣ ✏❛❧❣✳ ❝❧♦s✳ s♣❧✐t✑ ✭❛❧❣✳ ❝❧♦s✳ ❞♦ ♥♦t s♣❧✐t = ⇒ ♥♦t ❜♦✉♥❞❡❞✱ ❡✳❣ D∞✲❚❈❋ ❡①✐sts ❛♥❞ ✐s ♥♦t s✐♠♣❧❡✱ ❜❡❝❛✉s❡ ♠♦❞❡❧s ❛r❡ ♥♦t ❜♦✉♥❞❡❞✮

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✻ ✴ ✶✼

SLIDE 34

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❖t❤❡r r❡s✉❧ts

T ♠❝

G ❤❛s s❡♠✐✲◗❊ ✭s✐♠✐❧❛r ❛s ✐♥ ❆❈❋❆✮✳

T ♠❝

G ❝♦❞❡s ✜♥✐t❡ t✉♣❧❡s✳

❚❤❡♦r❡♠ ■❢ | ⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✱ t❤❡♥ T ♠❝

G ❤❛s ❣❡♦♠❡tr✐❝ ❊■ ✭❡❛❝❤ t✉♣❧❡ ✐s

✐♥t❡r❛❧❣❡❜r❛✐❝ ✇✐t❤ ❛ r❡❛❧ t✉♣❧❡✮✳ T ♠❝

G ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛s ✇❡❛❦ ❊■ ✭❈❈▼❆✮✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✼ ✴ ✶✼

SLIDE 35

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❖t❤❡r r❡s✉❧ts

T ♠❝

G ❤❛s s❡♠✐✲◗❊ ✭s✐♠✐❧❛r ❛s ✐♥ ❆❈❋❆✮✳

T ♠❝

G ❝♦❞❡s ✜♥✐t❡ t✉♣❧❡s✳

❚❤❡♦r❡♠ ■❢ | ⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✱ t❤❡♥ T ♠❝

G ❤❛s ❣❡♦♠❡tr✐❝ ❊■ ✭❡❛❝❤ t✉♣❧❡ ✐s

✐♥t❡r❛❧❣❡❜r❛✐❝ ✇✐t❤ ❛ r❡❛❧ t✉♣❧❡✮✳ T ♠❝

G ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛s ✇❡❛❦ ❊■ ✭❈❈▼❆✮✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✼ ✴ ✶✼

SLIDE 36

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❖t❤❡r r❡s✉❧ts

T ♠❝

G ❤❛s s❡♠✐✲◗❊ ✭s✐♠✐❧❛r ❛s ✐♥ ❆❈❋❆✮✳

T ♠❝

G ❝♦❞❡s ✜♥✐t❡ t✉♣❧❡s✳

❚❤❡♦r❡♠ ■❢ | ⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✱ t❤❡♥ T ♠❝

G ❤❛s ❣❡♦♠❡tr✐❝ ❊■ ✭❡❛❝❤ t✉♣❧❡ ✐s

✐♥t❡r❛❧❣❡❜r❛✐❝ ✇✐t❤ ❛ r❡❛❧ t✉♣❧❡✮✳ T ♠❝

G ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛s ✇❡❛❦ ❊■ ✭❈❈▼❆✮✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✼ ✴ ✶✼

SLIDE 37

❉❡✜♥✐t✐♦♥s ❛♥❞ ❢❛❝ts ▼❛❥♦r t❤✐♥❣s

❖t❤❡r r❡s✉❧ts

T ♠❝

G ❤❛s s❡♠✐✲◗❊ ✭s✐♠✐❧❛r ❛s ✐♥ ❆❈❋❆✮✳

T ♠❝

G ❝♦❞❡s ✜♥✐t❡ t✉♣❧❡s✳

❚❤❡♦r❡♠ ■❢ | ⌣

✐s ❢♦r❦✐♥❣ ✐♥❞❡♣❡♥❞❡♥❝❡✱ t❤❡♥ T ♠❝

G ❤❛s ❣❡♦♠❡tr✐❝ ❊■ ✭❡❛❝❤ t✉♣❧❡ ✐s

✐♥t❡r❛❧❣❡❜r❛✐❝ ✇✐t❤ ❛ r❡❛❧ t✉♣❧❡✮✳ T ♠❝

G ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛s ✇❡❛❦ ❊■ ✭❈❈▼❆✮✳

❉❛♥✐❡❧ ▼❛① ❍♦✛♠❛♥♥ ✭■▼ ❯❲r✮ ▼♦❞❡❧ ❚❤❡♦r② ✷✵✶✼ ✶✼ ✴ ✶✼