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rs s r rs r - - PowerPoint PPT Presentation

Feb 15, 2023 216 likes •576 views

rs s r rs r tt r sst rst t

slide-1
SLIDE 1
  • r♦✉♣s ✇❤♦s❡ ♥♦♥✲◆♦r♠❛❧ ❙✉❜❣r♦✉♣s ❛r❡

▼❡t❛❤❛♠✐❧t♦♥✐❛♥

❉❛r✐♦ ❊s♣♦s✐t♦

❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ◆❛♣♦❧✐ ❋❡❞❡r✐❝♦ ■■

❨♦✉♥❣ ❘❡s❡❛r❝❤❡rs ✐♥ ❆❧❣❡❜r❛ ❈♦♥❢❡r❡♥❝❡ ✲ ✷✵✶✾

◆❛♣♦❧✐ ✖ ❙❡♣t❡♠❜❡r✱ ✶✽t❤

❉❛r✐♦ ❊s♣♦s✐t♦

slide-2
SLIDE 2

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

■♥ ✶✾✻✻✱ ●✳▼✳ ❘♦♠❛❧✐s ❛♥❞ ◆✳❋✳ ❙❡s❡❦✐♥ ✐♥tr♦❞✉❝❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❚❤❡② s❛② ❛ ❣r♦✉♣ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ✐❢✿ ∀H ≤ G, H ∈ A ⇒ H ⊳ G ✇❤❡r❡ A ✐s t❤❡ ❝❧❛ss ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ✇✐❧❧ ✉s❡ H t♦ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧❧② SH✲❝❧♦s❡❞✱ ❜✉t ♥♦t ❝❧♦s❡❞ ❜② ❡①t❡♥s✐♦♥s ♦r ❡✈❡♥ ❞✐r❡❝t ♣r♦❞✉❝ts✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 3

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

■♥ ✶✾✻✻✱ ●✳▼✳ ❘♦♠❛❧✐s ❛♥❞ ◆✳❋✳ ❙❡s❡❦✐♥ ✐♥tr♦❞✉❝❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❚❤❡② s❛② ❛ ❣r♦✉♣ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ✐❢✿ ∀H ≤ G, H ∈ A ⇒ H ⊳ G ✇❤❡r❡ A ✐s t❤❡ ❝❧❛ss ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ✇✐❧❧ ✉s❡ H t♦ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧❧② SH✲❝❧♦s❡❞✱ ❜✉t ♥♦t ❝❧♦s❡❞ ❜② ❡①t❡♥s✐♦♥s ♦r ❡✈❡♥ ❞✐r❡❝t ♣r♦❞✉❝ts✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-4
SLIDE 4

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

■♥ ✶✾✻✻✱ ●✳▼✳ ❘♦♠❛❧✐s ❛♥❞ ◆✳❋✳ ❙❡s❡❦✐♥ ✐♥tr♦❞✉❝❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❚❤❡② s❛② ❛ ❣r♦✉♣ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ✐❢✿ ∀H ≤ G, H ∈ A ⇒ H ⊳ G ✇❤❡r❡ A ✐s t❤❡ ❝❧❛ss ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ✇✐❧❧ ✉s❡ H t♦ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧❧② SH✲❝❧♦s❡❞✱ ❜✉t ♥♦t ❝❧♦s❡❞ ❜② ❡①t❡♥s✐♦♥s ♦r ❡✈❡♥ ❞✐r❡❝t ♣r♦❞✉❝ts✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-5
SLIDE 5

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭✷✵✵✻✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-6
SLIDE 6

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭✷✵✵✻✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-7
SLIDE 7

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭✷✵✵✻✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-8
SLIDE 8

▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭✷✵✵✻✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-9
SLIDE 9

❚❤❡ ❝❧❛ss H ❝❛♥ ❛❧s♦ ❜❡ s❤♦✇♥ t♦ ❜❡ L✲❝❧♦s❡❞✱ ✐✳❡✳✿ Pr♦♣♦s✐t✐♦♥ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ✇❤♦s❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣s ❛r❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ ❚❤❡♥ G ✐ts❡❧❢ ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 10

❚❤❡ ❝❧❛ss H ❝❛♥ ❛❧s♦ ❜❡ s❤♦✇♥ t♦ ❜❡ L✲❝❧♦s❡❞✱ ✐✳❡✳✿ Pr♦♣♦s✐t✐♦♥ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ✇❤♦s❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣s ❛r❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ ❚❤❡♥ G ✐ts❡❧❢ ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 11
  • ❡♥❡r❛❧✐③❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❙♦♠❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❛r❡ st✉❞✐❡❞ ✐♥ ❛ ♣❛♣❡r ❜② ▼✳ ❉❡ ❋❛❧❝♦✱ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ❛♥❞ ❈✳ ▼✉s❡❧❧❛ ✐♥ ✷✵✵✾✳ ❚❤❡♦r❡♠ ✷✳ ❬▼✳ ❉❡ ❋❛❧❝♦ ✲ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✲ ❈✳ ▼✉s❡❧❧❛✱ ✷✵✵✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ♠✐♥✐♠❛❧✲♥♦♥✲H ❣r♦✉♣✳ ❚❤❡♥ G ✐s ✜♥✐t❡✳ ❖❢ ❝♦✉rs❡ s✉❝❤ ❛ ❣r♦✉♣ ✐s ❡✐t❤❡r ♣❡r❢❡❝t ✭♦♥❡ ❡①❛♠♣❧❡ ✐s A✺✮ ♦r s♦❧✉❜❧❡ ♦❢ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ❛t ♠♦st ✹✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 12
  • ❡♥❡r❛❧✐③❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❙♦♠❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❛r❡ st✉❞✐❡❞ ✐♥ ❛ ♣❛♣❡r ❜② ▼✳ ❉❡ ❋❛❧❝♦✱ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ❛♥❞ ❈✳ ▼✉s❡❧❧❛ ✐♥ ✷✵✵✾✳ ❚❤❡♦r❡♠ ✷✳ ❬▼✳ ❉❡ ❋❛❧❝♦ ✲ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✲ ❈✳ ▼✉s❡❧❧❛✱ ✷✵✵✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ♠✐♥✐♠❛❧✲♥♦♥✲H ❣r♦✉♣✳ ❚❤❡♥ G ✐s ✜♥✐t❡✳ ❖❢ ❝♦✉rs❡ s✉❝❤ ❛ ❣r♦✉♣ ✐s ❡✐t❤❡r ♣❡r❢❡❝t ✭♦♥❡ ❡①❛♠♣❧❡ ✐s A✺✮ ♦r s♦❧✉❜❧❡ ♦❢ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ❛t ♠♦st ✹✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 13
  • ❡♥❡r❛❧✐③❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡② ❛❧s♦ ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✸✳ ❬▼✳ ❉❡ ❋❛❧❝♦ ✲ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✲ ❈✳ ▼✉s❡❧❧❛✱ ✷✵✵✾❪ ▲❡t G ❜❡ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❤②♣❡r✭❛❜❡❧✐❛♥✲♦r✲✜♥✐t❡✮ ❣r♦✉♣ ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛r❡ ✐♥ H✳ ❚❤❡♥ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ ❛♥❞ ❚❤❡♦r❡♠ ✹✳ ❬▼✳ ❉❡ ❋❛❧❝♦✱ ❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❈✳ ▼✉s❡❧❧❛✱ ✷✵✵✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣ s❛t✐s❢②✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ ♥♦♥✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s✳ ❚❤❡♥ G ✐s ❡✐t❤❡r ❷❡r♥✐❦♦✈ ♦r ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 14
  • ❡♥❡r❛❧✐③❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡② ❛❧s♦ ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✸✳ ❬▼✳ ❉❡ ❋❛❧❝♦ ✲ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✲ ❈✳ ▼✉s❡❧❧❛✱ ✷✵✵✾❪ ▲❡t G ❜❡ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❤②♣❡r✭❛❜❡❧✐❛♥✲♦r✲✜♥✐t❡✮ ❣r♦✉♣ ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛r❡ ✐♥ H✳ ❚❤❡♥ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ ❛♥❞ ❚❤❡♦r❡♠ ✹✳ ❬▼✳ ❉❡ ❋❛❧❝♦✱ ❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❈✳ ▼✉s❡❧❧❛✱ ✷✵✵✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣ s❛t✐s❢②✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ ♥♦♥✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s✳ ❚❤❡♥ G ✐s ❡✐t❤❡r ❷❡r♥✐❦♦✈ ♦r ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 15

k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

▲❡t H✵ = A ❜❡ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ❞❡✜♥❡✱ r❡❝✉rs✐✈❡❧②✱ t❤❡ ❝❧❛ss Hk ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳ ❆ ❣r♦✉♣ G ✐s ✐♥ t❤❡ ❝❧❛ss Hk ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀H ≤ G, H ∈ Hk−✶ ⇒ H ⊳ G ❛♥❞ r❡❢❡r t♦ ❣r♦✉♣s ✐♥ t❤❡ ❝❧❛ss Hk ❛s k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❲✐t❤ k = ✶ ✇❡ ❤❛✈❡ t❤❡ ✉s✉❛❧ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❖❜✈✐♦✉s❧② ✇❡ ❤❛✈❡✿ H✵ ⊆ H✶ ⊆ H✷ ⊆ . . .

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 16

k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

▲❡t H✵ = A ❜❡ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ❞❡✜♥❡✱ r❡❝✉rs✐✈❡❧②✱ t❤❡ ❝❧❛ss Hk ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳ ❆ ❣r♦✉♣ G ✐s ✐♥ t❤❡ ❝❧❛ss Hk ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀H ≤ G, H ∈ Hk−✶ ⇒ H ⊳ G ❛♥❞ r❡❢❡r t♦ ❣r♦✉♣s ✐♥ t❤❡ ❝❧❛ss Hk ❛s k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❲✐t❤ k = ✶ ✇❡ ❤❛✈❡ t❤❡ ✉s✉❛❧ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❖❜✈✐♦✉s❧② ✇❡ ❤❛✈❡✿ H✵ ⊆ H✶ ⊆ H✷ ⊆ . . .

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 17

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡♦r❡♠ ✺✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❉✳❊✳✱ ▼✳ ❚r♦♠❜❡tt✐ ✷✵✶✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ Hk✲❣r♦✉♣✳ ❚❤❡♥ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ❛♥❞ ✐❢ G ✐s s♦❧✉❜❧❡✱ ✐ts ❞❡r✐✈❡❞ ❧❡♥❣t❤ ❞♦❡s ♥♦t ❡①❝❡❡❞ ✸k✳ ▼❛♥② ♦t❤❡r ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❝❛♥ ❜❡ ♣r♦✈❡❞ ❛❧s♦ ❢♦r Hk✲❣r♦✉♣s✿ ❚❤❡r❡ ❡①✐st ♥♦♥ s♦❧✉❜❧❡ H✷✲❣r♦✉♣s✱ ❡✳❣✳ A✺✱ ❛♥❞ H✷✲❣r♦✉♣s ✇❤♦s❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ ✐s ♥♦t ♦❢ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✱ ❡✳❣✳ GL(✷, ✸) ✇❤♦s❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ ✐s SL(✷, ✸) ✇❤✐❝❤ ✐s ❛ ❣r♦✉♣ ♦❢ ♦r❞❡r ✷✹ = ✷✸ · ✸✳ ❊✈❡r② ❧♦❝❛❧❧② ❣r❛❞❡❞ ♠✐♥✐♠❛❧✲♥♦♥✲Hk ❣r♦✉♣ ✐s ✜♥✐t❡✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 18

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡♦r❡♠ ✺✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❉✳❊✳✱ ▼✳ ❚r♦♠❜❡tt✐ ✷✵✶✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ Hk✲❣r♦✉♣✳ ❚❤❡♥ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ❛♥❞ ✐❢ G ✐s s♦❧✉❜❧❡✱ ✐ts ❞❡r✐✈❡❞ ❧❡♥❣t❤ ❞♦❡s ♥♦t ❡①❝❡❡❞ ✸k✳ ▼❛♥② ♦t❤❡r ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❝❛♥ ❜❡ ♣r♦✈❡❞ ❛❧s♦ ❢♦r Hk✲❣r♦✉♣s✿ ❚❤❡r❡ ❡①✐st ♥♦♥ s♦❧✉❜❧❡ H✷✲❣r♦✉♣s✱ ❡✳❣✳ A✺✱ ❛♥❞ H✷✲❣r♦✉♣s ✇❤♦s❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ ✐s ♥♦t ♦❢ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✱ ❡✳❣✳ GL(✷, ✸) ✇❤♦s❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ ✐s SL(✷, ✸) ✇❤✐❝❤ ✐s ❛ ❣r♦✉♣ ♦❢ ♦r❞❡r ✷✹ = ✷✸ · ✸✳ ❊✈❡r② ❧♦❝❛❧❧② ❣r❛❞❡❞ ♠✐♥✐♠❛❧✲♥♦♥✲Hk ❣r♦✉♣ ✐s ✜♥✐t❡✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 19

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡♦r❡♠ ✺✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❉✳❊✳✱ ▼✳ ❚r♦♠❜❡tt✐ ✷✵✶✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ Hk✲❣r♦✉♣✳ ❚❤❡♥ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ❛♥❞ ✐❢ G ✐s s♦❧✉❜❧❡✱ ✐ts ❞❡r✐✈❡❞ ❧❡♥❣t❤ ❞♦❡s ♥♦t ❡①❝❡❡❞ ✸k✳ ▼❛♥② ♦t❤❡r ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❝❛♥ ❜❡ ♣r♦✈❡❞ ❛❧s♦ ❢♦r Hk✲❣r♦✉♣s✿ ❚❤❡r❡ ❡①✐st ♥♦♥ s♦❧✉❜❧❡ H✷✲❣r♦✉♣s✱ ❡✳❣✳ A✺✱ ❛♥❞ H✷✲❣r♦✉♣s ✇❤♦s❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ ✐s ♥♦t ♦❢ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✱ ❡✳❣✳ GL(✷, ✸) ✇❤♦s❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ ✐s SL(✷, ✸) ✇❤✐❝❤ ✐s ❛ ❣r♦✉♣ ♦❢ ♦r❞❡r ✷✹ = ✷✸ · ✸✳ ❊✈❡r② ❧♦❝❛❧❧② ❣r❛❞❡❞ ♠✐♥✐♠❛❧✲♥♦♥✲Hk ❣r♦✉♣ ✐s ✜♥✐t❡✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-20
SLIDE 20

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡ ❝❧❛ss Hk ✐s ❛ ❧♦❝❛❧ ❝❧❛ss✳ ❋✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ Hk✲❣r♦✉♣s ❛r❡ ♣♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡✳ ▲♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s s❛t✐s❢②✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ ♥♦♥✲k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s ❛r❡ ♥❡❝❡ss❛r✐❧② ❷❡r♥✐❦♦✈ ♦r k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛❧❧ ❧✐❡ ✐♥ t❤❡ ❝❧❛ss Hk ❛r❡ Hk✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-21
SLIDE 21

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡ ❝❧❛ss Hk ✐s ❛ ❧♦❝❛❧ ❝❧❛ss✳ ❋✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ Hk✲❣r♦✉♣s ❛r❡ ♣♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡✳ ▲♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s s❛t✐s❢②✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ ♥♦♥✲k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s ❛r❡ ♥❡❝❡ss❛r✐❧② ❷❡r♥✐❦♦✈ ♦r k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛❧❧ ❧✐❡ ✐♥ t❤❡ ❝❧❛ss Hk ❛r❡ Hk✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 22

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡ ❝❧❛ss Hk ✐s ❛ ❧♦❝❛❧ ❝❧❛ss✳ ❋✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ Hk✲❣r♦✉♣s ❛r❡ ♣♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡✳ ▲♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s s❛t✐s❢②✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ ♥♦♥✲k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s ❛r❡ ♥❡❝❡ss❛r✐❧② ❷❡r♥✐❦♦✈ ♦r k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛❧❧ ❧✐❡ ✐♥ t❤❡ ❝❧❛ss Hk ❛r❡ Hk✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 23

Pr♦♣❡rt✐❡s ♦❢ k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❚❤❡ ❝❧❛ss Hk ✐s ❛ ❧♦❝❛❧ ❝❧❛ss✳ ❋✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ Hk✲❣r♦✉♣s ❛r❡ ♣♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡✳ ▲♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s s❛t✐s❢②✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ ♥♦♥✲k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s ❛r❡ ♥❡❝❡ss❛r✐❧② ❷❡r♥✐❦♦✈ ♦r k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛❧❧ ❧✐❡ ✐♥ t❤❡ ❝❧❛ss Hk ❛r❡ Hk✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-24
SLIDE 24
  • r♦✉♣s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs ♦❢

♥♦♥✲k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s

❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ♣r♦✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❝♦♥❝❡r♥✐♥❣ ❣r♦✉♣s ✇❤♦s❡ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s ❤❛✈❡ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs✳ ❚❤❡♦r❡♠ ✻✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✲ ❋✳ ❉❡ ▼❛r✐✱ ✷✵✵✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣ ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❚❤❡♥ G ❤❛s ✜♥✐t❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣✳ ❚❤✐s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❚❤❡♦r❡♠ ✼✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❉✳❊✳✱ ▼✳ ❚r♦♠❜❡tt✐ ✷✵✶✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣ ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs ♦❢ s✉❜❣r♦✉♣s ✇❤✐❝❤ ❛r❡ ♥♦t Hk✳ ❚❤❡♥ G ❤❛s ✜♥✐t❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-25
SLIDE 25
  • r♦✉♣s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs ♦❢

♥♦♥✲k✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ s✉❜❣r♦✉♣s

❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ♣r♦✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❝♦♥❝❡r♥✐♥❣ ❣r♦✉♣s ✇❤♦s❡ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s ❤❛✈❡ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs✳ ❚❤❡♦r❡♠ ✻✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✲ ❋✳ ❉❡ ▼❛r✐✱ ✷✵✵✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣ ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❚❤❡♥ G ❤❛s ✜♥✐t❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣✳ ❚❤✐s ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❚❤❡♦r❡♠ ✼✳ ❬❋✳ ❞❡ ●✐♦✈❛♥♥✐✱ ❉✳❊✳✱ ▼✳ ❚r♦♠❜❡tt✐ ✷✵✶✾❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣ ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♥♦r♠❛❧✐③❡rs ♦❢ s✉❜❣r♦✉♣s ✇❤✐❝❤ ❛r❡ ♥♦t Hk✳ ❚❤❡♥ G ❤❛s ✜♥✐t❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 26

∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❖♥❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ❝❧❛ss H∞ =

  • k∈N

Hk. ◆♦t✐❝❡ ♥♦✇ t❤❛t ✐❢ G ✐s ✐♥ t❤✐s ❝❧❛ss✱ G ❝❛♥♥♦t ❤❛✈❡ ❛♥② ✐♥✜♥✐t❡ ❞❡s❝❡♥❞✐♥❣ ❝❤❛✐♥ ♦❢ s✉❜❣r♦✉♣s✿ G = G✵ > G✶ > G✷ > . . . ✇✐t❤ Gi+✶ ⋪ Gi ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t H∞ ✐s ♣r❡❝✐s❡❧② t❤❡ ❝❧❛ss ♦❢ ❣r♦✉♣s ✇✐t❤ ❛ ❜♦✉♥❞ ♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ s✉❝❤ ✧❜❛❞✧ ❝❤❛✐♥s✳ ◆♦t✐❝❡ t❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t H∞ ⊆ FA ✐✳❡✳ ∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❤❛✈❡ ✜♥✐t❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣✳

❉❛r✐♦ ❊s♣♦s✐t♦

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SLIDE 27

∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❖♥❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ❝❧❛ss H∞ =

  • k∈N

Hk. ◆♦t✐❝❡ ♥♦✇ t❤❛t ✐❢ G ✐s ✐♥ t❤✐s ❝❧❛ss✱ G ❝❛♥♥♦t ❤❛✈❡ ❛♥② ✐♥✜♥✐t❡ ❞❡s❝❡♥❞✐♥❣ ❝❤❛✐♥ ♦❢ s✉❜❣r♦✉♣s✿ G = G✵ > G✶ > G✷ > . . . ✇✐t❤ Gi+✶ ⋪ Gi ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t H∞ ✐s ♣r❡❝✐s❡❧② t❤❡ ❝❧❛ss ♦❢ ❣r♦✉♣s ✇✐t❤ ❛ ❜♦✉♥❞ ♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ s✉❝❤ ✧❜❛❞✧ ❝❤❛✐♥s✳ ◆♦t✐❝❡ t❤❛t ✇❡ ❛❧r❡❛❞② ❦♥♦✇ t❤❛t H∞ ⊆ FA ✐✳❡✳ ∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❤❛✈❡ ✜♥✐t❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-28
SLIDE 28

∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❲❡ ❝❛♥ ❛❧s♦ s❤♦✇ t❤❡ ❝♦♥✈❡rs❡ ✭✐✳❡✳ FA ⊆ H∞✱ ❤❡♥❝❡ t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ❝❧❛ss❡s✮✳ ❋♦r ✐❢ G ✐s ❛ ❣r♦✉♣ s✉❝❤ t❤❛t |G ′| = n = pn✶

✶ . . . pnl l ✱ t❤❡♥ ❛♥②

❞❡s❝❡♥❞✐♥❣ ❝❤❛✐♥ {Gi} ♦❢ s✉❜❣r♦✉♣s ♦❢ G r❡❧❛t❡❞ ❜② Gi+✶ ⋪ Gi ✐s ❜♦✉♥❞❡❞ ✐♥ ❧❡♥❣t❤ ❜② f (n) = n✶ + n✷ + . . . + nl ❛♥❞ s♦ G ✐s Hf (n)✳ ❚❤❡ ❢❛❝t t❤❛t ❛ ❣r♦✉♣ ✇✐t❤ ❜♦✉♥❞ f (n) ♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤✐s t②♣❡ ♦❢ ❝❤❛✐♥s ✐s Hf (n) ❝❛♥ ❜❡ ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n✶ + . . . + nl✳ ❚❤❡ ❜❛s✐s ♦❢ ✐♥❞✉❝t✐♦♥ ✐s t❤❡ ♦❜s❡r✈❛t✐♦♥ t❤❛t ❣r♦✉♣s ✇✐t❤ ❝♦♠♠✉t❛t♦r ♦❢ ♣r✐♠❡ ♦r❞❡r ❛r❡ ♥❡❝❡ss❛r✐❧② ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-29
SLIDE 29

∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❲❡ ❝❛♥ ❛❧s♦ s❤♦✇ t❤❡ ❝♦♥✈❡rs❡ ✭✐✳❡✳ FA ⊆ H∞✱ ❤❡♥❝❡ t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ❝❧❛ss❡s✮✳ ❋♦r ✐❢ G ✐s ❛ ❣r♦✉♣ s✉❝❤ t❤❛t |G ′| = n = pn✶

✶ . . . pnl l ✱ t❤❡♥ ❛♥②

❞❡s❝❡♥❞✐♥❣ ❝❤❛✐♥ {Gi} ♦❢ s✉❜❣r♦✉♣s ♦❢ G r❡❧❛t❡❞ ❜② Gi+✶ ⋪ Gi ✐s ❜♦✉♥❞❡❞ ✐♥ ❧❡♥❣t❤ ❜② f (n) = n✶ + n✷ + . . . + nl ❛♥❞ s♦ G ✐s Hf (n)✳ ❚❤❡ ❢❛❝t t❤❛t ❛ ❣r♦✉♣ ✇✐t❤ ❜♦✉♥❞ f (n) ♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤✐s t②♣❡ ♦❢ ❝❤❛✐♥s ✐s Hf (n) ❝❛♥ ❜❡ ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n✶ + . . . + nl✳ ❚❤❡ ❜❛s✐s ♦❢ ✐♥❞✉❝t✐♦♥ ✐s t❤❡ ♦❜s❡r✈❛t✐♦♥ t❤❛t ❣r♦✉♣s ✇✐t❤ ❝♦♠♠✉t❛t♦r ♦❢ ♣r✐♠❡ ♦r❞❡r ❛r❡ ♥❡❝❡ss❛r✐❧② ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-30
SLIDE 30

∞✲♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s

❲❡ ❝❛♥ ❛❧s♦ s❤♦✇ t❤❡ ❝♦♥✈❡rs❡ ✭✐✳❡✳ FA ⊆ H∞✱ ❤❡♥❝❡ t❤❡ ❡q✉❛❧✐t② ♦❢ t❤❡ ❝❧❛ss❡s✮✳ ❋♦r ✐❢ G ✐s ❛ ❣r♦✉♣ s✉❝❤ t❤❛t |G ′| = n = pn✶

✶ . . . pnl l ✱ t❤❡♥ ❛♥②

❞❡s❝❡♥❞✐♥❣ ❝❤❛✐♥ {Gi} ♦❢ s✉❜❣r♦✉♣s ♦❢ G r❡❧❛t❡❞ ❜② Gi+✶ ⋪ Gi ✐s ❜♦✉♥❞❡❞ ✐♥ ❧❡♥❣t❤ ❜② f (n) = n✶ + n✷ + . . . + nl ❛♥❞ s♦ G ✐s Hf (n)✳ ❚❤❡ ❢❛❝t t❤❛t ❛ ❣r♦✉♣ ✇✐t❤ ❜♦✉♥❞ f (n) ♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤✐s t②♣❡ ♦❢ ❝❤❛✐♥s ✐s Hf (n) ❝❛♥ ❜❡ ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n✶ + . . . + nl✳ ❚❤❡ ❜❛s✐s ♦❢ ✐♥❞✉❝t✐♦♥ ✐s t❤❡ ♦❜s❡r✈❛t✐♦♥ t❤❛t ❣r♦✉♣s ✇✐t❤ ❝♦♠♠✉t❛t♦r ♦❢ ♣r✐♠❡ ♦r❞❡r ❛r❡ ♥❡❝❡ss❛r✐❧② ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳

❉❛r✐♦ ❊s♣♦s✐t♦

slide-31
SLIDE 31
  • r♦✉♣s ✇❤♦s❡ ♥♦♥✲♥♦r♠❛❧ s✉❜❣r♦✉♣s ❜❡❧♦♥❣ t♦ ❛ ❝❧❛ss X

❆❝t✉❛❧❧②✱ ✇❡ ❝❛♥ st❛rt ❢r♦♠ ❛♥② SH✲❝❧♦s❡❞ ❝❧❛ss ♦❢ ❣r♦✉♣s X ❛♥❞ ❞❡✜♥❡✿ X✵ = X ❛♥❞✱ ❢♦r ❛♥② k ∈ N✱ s❛② t❤❛t ❛ ❣r♦✉♣ G ✐s ✐♥ t❤❡ ❝❧❛ss Xk ✐❢ ❛♥❞ ♦♥❧② ✐❢✿ ∀H ≤ G, H ∈ Xk−✶ ⇒ H ⊳ G. ❲❡ ❝❛♥ t❤❡♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ❝❧❛ss X∞ ❛s

k∈N

Xk.

❉❛r✐♦ ❊s♣♦s✐t♦

slide-32
SLIDE 32
  • r♦✉♣s ✇❤♦s❡ ♥♦♥✲♥♦r♠❛❧ s✉❜❣r♦✉♣s ❜❡❧♦♥❣ t♦ ❛ ❝❧❛ss X

■❢ ♦♥❡ ❛ss✉♠❡s t❤❛t X s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s SHL✲❝❧♦s❡❞ X ⊆ FA ▼✐♥✐♠❛❧✲♥♦♥✲X ❣r♦✉♣s ❛r❡ ✜♥✐t❡ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛r❡ X ❛r❡ ❛❧s♦ X ❚❤❡♥ ❛❧❧ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧✐st ❛❧s♦ ❤♦❧❞ ❢♦r Xk ❢♦r ❛♥② k ∈ N

❉❛r✐♦ ❊s♣♦s✐t♦

slide-33
SLIDE 33
  • r♦✉♣s ✇❤♦s❡ ♥♦♥✲♥♦r♠❛❧ s✉❜❣r♦✉♣s ❜❡❧♦♥❣ t♦ ❛ ❝❧❛ss X

■❢ ♦♥❡ ❛ss✉♠❡s t❤❛t X s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s SHL✲❝❧♦s❡❞ X ⊆ FA ▼✐♥✐♠❛❧✲♥♦♥✲X ❣r♦✉♣s ❛r❡ ✜♥✐t❡ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛r❡ X ❛r❡ ❛❧s♦ X ❚❤❡♥ ❛❧❧ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧✐st ❛❧s♦ ❤♦❧❞ ❢♦r Xk ❢♦r ❛♥② k ∈ N

❉❛r✐♦ ❊s♣♦s✐t♦

slide-34
SLIDE 34
  • r♦✉♣s ✇❤♦s❡ ♥♦♥✲♥♦r♠❛❧ s✉❜❣r♦✉♣s ❜❡❧♦♥❣ t♦ ❛ ❝❧❛ss X

■❢ ♦♥❡ ❛ss✉♠❡s t❤❛t X s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s SHL✲❝❧♦s❡❞ X ⊆ FA ▼✐♥✐♠❛❧✲♥♦♥✲X ❣r♦✉♣s ❛r❡ ✜♥✐t❡ P♦❧②❝②❝❧✐❝✲❜②✲✜♥✐t❡ ❣r♦✉♣s ✇❤♦s❡ ✜♥✐t❡ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡s ❛r❡ X ❛r❡ ❛❧s♦ X ❚❤❡♥ ❛❧❧ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧✐st ❛❧s♦ ❤♦❧❞ ❢♦r Xk ❢♦r ❛♥② k ∈ N

❉❛r✐♦ ❊s♣♦s✐t♦

slide-35
SLIDE 35

❚❤❛♥❦ ②♦✉ ❢♦r t❤❡ ❛tt❡♥t✐♦♥✦✦✦

❉❛r✐♦ ❊s♣♦s✐t♦