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s strs t t sr - - PowerPoint PPT Presentation

Oct 28, 2023 328 likes •623 views

s strs t t sr t trt Pt t

slide-1
SLIDE 1

▼✐s❧❡❛❞✐♥❣ st❛rs✿ ❲❤❛t ❝❛♥♥♦t ❜❡ ♠❡❛s✉r❡❞ ♦♥ t❤❡ ■♥t❡r♥❡t ❄

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-2
SLIDE 2

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-3
SLIDE 3

❍♦✇ ❛❝❝✉r❛t❡ ❛r❡ ♥❡t✇♦r❦ ♠❛♣s ❄

❲❤② ❄ ❚♦ ❞❡✈❡❧♦♣✴❛❞❛♣t ♣r♦t♦❝♦❧s t♦ ■♥t❡r♥❡t P❛❉■❙✱ ❘▼❚P ❚♦ ✉♥❞❡rst❛♥❞ t❤❡ ✐♠♣❛❝t ♦❢ ✉♥❝❡rt❛✐♥t②✿ ♥❡t✇♦r❦s ♠❡tr♦❧♦❣② ❄ ❍♦✇ ❄ ♠✉❧t✐❝❛st [▼❛r❝❤❡tt❛ ❡t ❛❧✳ , ❏❙❆❈ ′✶✶] ♥❡t✇♦r❦ t♦♠♦❣r❛♣❤② ❚r❛❝❡r♦✉t❡ ❚❛❦❡❛✇❛② ❈♦♠♣❛r❡ ✐♥t❡r♥❡t t♦♣♦❧♦❣✐❡s ✐♥st❡❛❞ ♦❢ ❝♦✉♥t✐♥❣ t❤❡♠ ▲♦❝❛❧ ♣r♦♣❡rt✐❡s s✉✛❡r

  • ❧♦❜❛❧ ♣r♦♣❡rt✐❡s s✉✛❡r ❧❡ss

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-4
SLIDE 4

❚r❛❝❡r♦✉t❡ ♣r✐♥❝✐♣❧❡

■❞❡❛❂ s❡♥❞ ✓ ❜✉❣❣② ✔ ❚❚▲ ♣❛❝❦❡ts✱ ❝♦❧❧❡❝t ❡rr♦r ♠❡ss❛❣❡s

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-5
SLIDE 5

❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s

❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s

→

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-6
SLIDE 6

❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s

❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s

→

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-7
SLIDE 7

❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s

❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝ ←

❄

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-8
SLIDE 8

❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s

❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝ ←

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-9
SLIDE 9

❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s

❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝ ←

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-10
SLIDE 10

❚r❛❝❡r♦✉t❡ ▲✐♠✐t❛t✐♦♥s

❙❛♠♣❧✐♥❣ ❜✐❛s✿ ■♠♣❛❝t ♦❢ s♦✉r❝❡s ❧♦❝❛t✐♦♥ ❆❧✐❛s✐♥❣✿ ❍♦✇ t♦ ♠❛♣ ■Ps t♦ t❤❡ ❡q✉✐♣❡♠❡♥t ▲♦❛❞✲❇❛❧❛♥❝✐♥❣✿ ❚r❛❝❡r♦✉t❡ ❛ss✉♠❡s ❛❧❧ ♣❛❝❦❡ts ❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣❛t❤✳✳✳ ❙t❛rs✿ ❉✐s❛❜❧❡❞✴❋✐❧t❡r❡❞ ■❈▼P ♠❡ss❛❣❡s

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝ ←

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-11
SLIDE 11

❘❡❧❛t❡❞ ❲♦r❦s

❲♦r❦s ✐♥✐t✐❛t❡❞ ❜② ❆❝❤❛r②❛ ❛♥❞ ●♦✉❞❛ [❙❙❙✵✾, ■❈❉❈◆✶✵, ■❈❉❈◆✶✶] ▼♦❞❡❧s t♦ ❝❛♣t✉r❡ ✐rr❡❣✉❧❛r ♥♦❞❡s ■rr❡❣✉❧❛r ♥♦❞❡s → ❛♥♦♥②♠♦✉s ♥♦❞❡s ❈❡♥tr❛❧ ❝♦♥❝❡♣t ♦❢ ♠✐♥✐♠❛❧ t♦♣♦❧♦❣✐❡s ❈♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❣❡♥❡r❛❜❧❡ t♦♣♦❧♦❣✐❡s ❖✉r ❛♣♣r♦❛❝❤✿ ▲♦ts ♦❢ ❣❡♥❡r❛❜❧❡ t♦♣♦❧♦❣✐❡s ✐s ♥♦t ❛ ♣r♦❜❧❡♠ ✐❢ t❤❡② ❛r❡ ❛❧❧ s✐♠✐❧❛r

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-12
SLIDE 12

▼♦❞❡❧

  • ✵(❱✵, ❊✵)✿ st❛t✐❝

✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ❂ ❚❛r❣❡t ❚♦♣♦❧♦❣②✳ ✈ ∈ ❱✵ ✐s

❡✐t❤❡r ♥❛♠❡❞✿ ❛❧✇❛②s ❛♥s✇❡rs ✇✐t❤ ✐ts ♦♥❧② ♥❛♠❡ ✭♥♦ ❛❧✐❛s✐♥❣✮ ❡✐t❤❡r ❛♥♦♥②♠♦✉s✿ ❛❧✇❛②s ❛♥s✇❡rs ⋆✳

◆♦t ♥❡❝❡ss❛r✐❧② ♠✐♥✐♠❛❧ ✦ ❞●✵(✉, ✈) ✐s t❤❡ s❤♦rt❡st ♣❛t❤ ❞✐st❛♥❝❡✳

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-13
SLIDE 13

▼♦❞❡❧

  • ✵(❱✵, ❊✵)✿ st❛t✐❝

✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ❂ ❚❛r❣❡t ❚♦♣♦❧♦❣②✳ ✈ ∈ ❱✵ ✐s

❡✐t❤❡r ♥❛♠❡❞✿ ❛❧✇❛②s ❛♥s✇❡rs ✇✐t❤ ✐ts ♦♥❧② ♥❛♠❡ ✭♥♦ ❛❧✐❛s✐♥❣✮ ❡✐t❤❡r ❛♥♦♥②♠♦✉s✿ ❛❧✇❛②s ❛♥s✇❡rs ⋆✳

◆♦t ♥❡❝❡ss❛r✐❧② ♠✐♥✐♠❛❧ ✦ ❞●✵(✉, ✈) ✐s t❤❡ s❤♦rt❡st ♣❛t❤ ❞✐st❛♥❝❡✳

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-14
SLIDE 14

▼♦❞❡❧✴✷

❲❡ ❞♦♥✬t ❦♥♦✇ ●✵✱ ❜✉t ✇❡ ❦♥♦✇ ❛ s❡t ♦❢ tr❛❝❡s T ♦❢ ●✵✳ ❆♥♦♥②♠♦✉s ♥♦❞❡s ❛♣♣❡❛r ❛s st❛rs ✭⋆✮ ✐♥ T ❊❛❝❤ st❛r ❤❛s ❛ ✉♥✐q✉❡ ♥✉♠❜❡r ✭⋆✐, ✐ = ✶..s✮ ✐♥ T ❞❚(✉, ✈) ✐s t❤❡ ♥✉♠❜❡r ♦❢ s②♠❜♦❧s ✐♥ ❚ ∈ T ❜❡t✇❡❡♥ ✉ ❛♥❞ ✈✳ ◆♦ ❆ss✉♠♣t✐♦♥ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ tr❛❝❡s✱ ♦♥ ♣❛t❤ ✉♥✐q✉❡♥❡ss ♥♦r s②♠♠❡tr②✳

↔ ↔

t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝ t✶ = ❜, ⋆✶, ❝ t✷ = ❜, ⋆✷, ❝ t✸ = ❛, ❜, ⋆✸, ❝ t✹ = ❝, ⋆✹, ❜, ❛

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-15
SLIDE 15

❘✉❧❡s

❈♦♠♣❧❡t❡ ❝♦✈❡r✿ ❊❛❝❤ ❡❞❣❡ ♦❢ ●❖ ❛♣♣❡❛rs ❛t ❧❡❛st ♦♥❝❡ ✐♥ s♦♠❡ tr❛❝❡ t♦ T ❘❡❛❧✐t② s❛♠♣❧✐♥❣✿ ❋♦r ❡✈❡r② tr❛❝❡ ❚ ∈ T ✱ ✐❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ s②♠❜♦❧s σ✶, σ✷ ∈ ❚ ✐s ❞❚(σ✶, σ✷) = ❦✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣❛t❤ ✭✐✳❡✳✱ ❛ ✇❛❧❦ ✇✐t❤♦✉t ❝②❝❧❡s✮ ♦❢ ❧❡♥❣t❤ ❦ ❝♦♥♥❡❝t✐♥❣ t✇♦ ✭♥❛♠❡❞ ♦r ❛♥♦♥②♠♦✉s✮ ♥♦❞❡s σ✶ ❛♥❞ σ✷ ✐♥ ●✵✳ ◆♦ ❛ss✉♠♣t✐♦♥ ♦♥ ❝♦✈❡r❛❣❡

❛, ❜, ❝ ❛, ❝, ❜ ❛, ⋆✶, ❝ ❛, ❞, ❝

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-16
SLIDE 16

❘✉❧❡s ✲✷

α✲✭❘♦✉t✐♥❣✮ ❈♦♥s✐st❡♥❝②✿ ❚❤❡r❡ ❡①✐sts ❛♥ α ∈ (✵, ✶] s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② tr❛❝❡ ❚ ∈ T ✱ ✐❢ ❞❚(σ✶, σ✷) = ❦ ❢♦r t✇♦ ❡♥tr✐❡s σ✶, σ✷ ✐♥ tr❛❝❡ ❚✱ t❤❡♥ t❤❡ s❤♦rt❡st ♣❛t❤ ❝♦♥♥❡❝t✐♥❣ t❤❡ t✇♦ ✭♥❛♠❡❞ ♦r ❛♥♦♥②♠♦✉s✮ ♥♦❞❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ σ✶ ❛♥❞ σ✷ ✐♥ ●✵ ❤❛s ❞✐st❛♥❝❡ ❛t ❧❡❛st ⌈α❦⌉✳ ◆♦t❡✿ α > ✵ ⇔ ❧♦♦♣✲❧❡ss r♦✉t✐♥❣ ❛, ❜, ❝ ❛, ✐✶, ✐✸, ✐✹, ❝ ⇒ α ≤ ✷

✺

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-17
SLIDE 17

❙♦♠❡ ♠♦r❡ ❞❡✜♥✐t✐♦♥s

❆ t♦♣♦❧♦❣② ● ✐s α✲❝♦♥s✐st❡♥t❧② ✐♥❢❡rr❛❜❧❡ ❢r♦♠ T ✐❢ ✐t r❡s♣❡❝ts t❤❡ ✸ ♣r❡✈✐♦✉s r✉❧❡s✳ ▲❡t GT = {●, s✳t✳ ● ✐s ✐♥❢❡rr❛❜❧❡ ❢r♦♠ T } ❲❡ st✉❞② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s❡t GT ♦❢ ✐♥❢❡rr❛❜❧❡ t♦♣♦❧♦❣✐❡s✳ ❲❡ ❞❡✜♥❡ t❤❡ ❝❛♥♦♥✐❝ ❣r❛♣❤ ●❝ ❛s t❤❡ str❛✐❣❤t❢♦r✇❛r❞ ❣r❛♣❤ t❤❛t tr❡❛ts ❡❛❝❤ st❛r ❛s ✉♥✐q✉❡✳

→

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-18
SLIDE 18

❍♦✇ t♦ ✐♥❢❡r t♦♣♦❧♦❣✐❡s ❄

✶ ✐♥❢❡rr❛❜❧❡ t♦♣♦❧♦❣②❂ ✶ ♠❛♣♣✐♥❣ ♦❢ st❛rs t♦ ❛♥♦♥②♠♦✉s r♦✉t❡rs✳ ▲❡t ▼❛♣ ❜❡ s✉❝❤ ❢✉♥❝t✐♦♥✳

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-19
SLIDE 19

❍♦✇ t♦ ✐♥❢❡r t♦♣♦❧♦❣✐❡s ✲✷

  • ❝ ✐s ✐♥❢❡rr❛❜❧❡✳ ▼❛♣= ■❞✳

✭✐✮ ✐❢ ⋆✶ ∈ ❚✶ ❛♥❞ ⋆✷ ∈ ❚✷✱ ❛♥❞ ⌈α · ❞❚✶(⋆✶, ✉)⌉ > ❞❈(✉, ⋆✷) ⇒ ▼❛♣(⋆✶) = ▼❛♣(⋆✷)✳ ✭✐✐✮ ⋆✶ ∈ ❚✶ ⋆✷ ∈ ❚✷✱ ❛♥❞ ∃❚ s✳t✳ ⌈α · ❞❚(✉, ✈)⌉ > ❞❈(✉, ⋆✶) + ❞❈(✈, ⋆✷) ⇒ ▼❛♣(⋆✶) = ▼❛♣(⋆✷).

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-20
SLIDE 20

❙t❛r ❣r❛♣❤

❆❧❣♦r✐t❤♠ ❝♦♥str✉❝t✐✈❡ ♣❛rt ❲❡ ❝♦♥str✉❝t t❤❡ ❙t❛r ●r❛♣❤ ●∗(❱∗, ❊∗)✿ ❱❡rt✐❝❡s❂st❛rs ✐♥ t❤❡ tr❛❝❡ ❊❞❣❡s❂✐❢ st❛rs ❝❛♥♥♦t ❜❡ ♠❡r❣❡❞✿ (⋆✶, ⋆✷) ∈ ❊∗ ⇔ ▼❛♣(⋆✶) = ▼❛♣(⋆✷). ✶ ♣r♦♣❡r ❝♦❧♦r✐♥❣ ♦❢ ●∗ ↔ ✶ ▼❛♣ ❢✉♥❝t✐♦♥ ♠✐♥✐♠❛❧ ❝♦❧♦r✐♥❣ → ♠✐♥✐♠❛❧ t♦♣♦❧♦❣② ✬♠❛①✐♠❛❧✬ ❝♦❧♦r✐♥❣ → ●❝

|❱∗|

  • ❦=γ(●∗)

P(●∗, ❦)/❦! ≥ |GT |, γ(●∗) = ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ ●∗ P(●∗, ❦) = ❝❤r♦♠❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ ●∗✳

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-21
SLIDE 21

❙t❛r ●r❛♣❤ ✴✷

→

❚r❛❝❡✿ t✶ = ❛, ⋆✶, ❜ t✷ = ❜, ⋆✷, ✐✶, ✐✷, ✐✸, ⋆✸, ❝ α = ✵.✺

  • ❝ :

→

  • ∗ ❈♦❧♦r✐♥❣s

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-22
SLIDE 22

▲♦♦❦✐♥❣ ❛t ♣r♦♣❡rt✐❡s✴ ●❧♦❜❛❧

❘❡s✉❧ts ❛r❡❜❛❞ ✦ ❈♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ◆♦ ❛ss✉♠♣t✐♦♥ ♦♥ ✓ ❝♦✈❡r❛❣❡ ✔ ⇒ ❙t❛rs ❞✐s❝♦♥♥❡❝t t❤❡ ❣r❛♣❤ ✦ |❝❝(●✶)/❝❝(●✷)| ≤ ♥

✷

❙tr❡t❝❤ ❊✈❡♥ ✐❢ ✇❡ ♦♥❧② ❝♦♥s✐❞❡r ❝♦♥♥❡❝t❡❞ t♦♣♦❧♦❣✐❡s✳ |str❡t❝❤(●)| ≤ ♥+s−✶

✷

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-23
SLIDE 23

▲♦♦❦✐♥❣ ❛t ♣r♦♣❡rt✐❡s✴ ▲♦❝❛❧

■t✬s✇♦rs❡ ✦ ❚r✐❛♥❣❧❡s ❇✐♣❛rt✐t❡ ❝♦♠♣❧❡t❡ ❣r❛♣❤ ✇♦rst ❝❛s❡ |❈✸(●✶)/❈✸(●✷)| ≤ ∞ ❉❡❣r❡❡ ❲♦rst ❝❛s❡ ✐s ♦♥ ❛♥♦♥②♠♦✉s ♥♦❞❡s |❉❊●(●✶) − ❉❊●(●✷)| ≤ ✷(s − γ(●⋆))

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-24
SLIDE 24

❇❡st ❝❛s❡✿ ❋✉❧❧② ❡①♣❧♦r❡❞ t♦♣♦❧♦❣✐❡s

❙tr♦♥❣ ❛ss✉♠♣t✐♦♥s✿ α = ✶ ✿ s❤♦rt❡st ♣❛t❤ r♦✉t✐♥❣ ∀✉, ✈ ∈ ❱✵, ∃❚ ∈ T s✉❝❤ t❤❛t ✉ ∈ ❚ ∨ ✈ ∈ ❚ ❲❡ ❞♦♥✬t s❡❡ ❛♥② str♦♥❣❡r ❝❛s❡ ❘❡s✉❧ts✿

  • ❧♦❜❛❧ ♣r♦♣❡rt✐❡s ❝♦♥s❡r✈❡❞

▲♦❝❛❧ ♣r♦♣❡rt✐❡s✿ st✐❧❧ ❜❛❞ ✦

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-25
SLIDE 25

❖✈❡r❛❧❧ ❘❡s✉❧ts

❆❜s♦❧✉t❡ ❞✐✛❡r❡♥❝❡✿ ●✶ − ●✷ Pr♦♣❡rt② ❆r❜✐tr❛r② ❋✉❧❧② ❊①♣❧♦r❡❞ ✭α = ✶✮ ★ ♦❢ ♥♦❞❡s ≤ s − γ(●∗) ≤ s − γ(●∗) ★ ♦❢ ❧✐♥❦s ≤ ✷(s − γ(●∗)) ≤ ✷(s − γ(●∗)) ★ ♦❢ ❈❈ ≤ ♥/✷ = ✵ ❉✐❛♠❡t❡r ≤ (s − ✶)/s · (◆ − ✶) s/✷ ✭¶✮ ▼❛①✳ ❉❡❣✳ ≤ ✷(s − γ(●∗)) ≤ ✷(s − γ(●∗)) ❚r✐❛♥❣❧❡s ≤ ✷s(s − ✶) ≤ ✷s(s − ✶)/✷ ❘❡❧❛t✐✈❡ ❞✐✛❡r❡♥❝❡✿ ●✶/●✷ ★ ♦❢ ♥♦❞❡s ≤ (♥ + s)/(♥ + γ(●∗)) ≤ (♥ + s)/(♥ + γ(●∗)) ★ ♦❢ ❧✐♥❦s ≤ (ν + ✷s)/(ν + ✷) ≤ (ν + ✷s)/(ν + ✷) ★ ♦❢ ❈❈ ≤ ♥/✷ = ✶ ❙tr❡t❝❤ ≤ (◆ − ✶)/✷ = ✶ ❉✐❛♠❡t❡r ≤ s ✷ ▼❛①✳ ❉❡❣✳ ≤ s − γ(●∗) + ✶ ≤ s − γ(●∗) + ✶ ❚r✐❛♥❣❧❡s ∞ ∞

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-26
SLIDE 26

❈♦♥❝❧✉s✐♦♥

❈♦♥str✉❝t✐✈❡ ♣r♦♦❢s ❈♦♠♣❧❡① ❛❧❣♦r✐t❤♠ ❉♦♥✬t ❝♦✉♥t✱ ❝♦♠♣❛r❡ ✦ ❍✉❣❡ ❞✐ss✐♠✐❧❛r✐t✐❡s ❲♦rst ❝❛s❡ ❛♣♣r♦❛❝❤ ❆ ♣r❛❝t✐❝❛❧ ♣❛rt✿ ❝♦♠♣❛r❡ ✇✐t❤ r❡❛❧✐t② ❞❡✈❡❧♦♣ ♣r♦♣❡rt② ❡st✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs

slide-27
SLIDE 27

❚❤❛♥❦s✦

❨✈♦♥♥❡✲❆♥♥❡ P✐❣♥♦❧❡t✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✳ ❚ré❞❛♥ ▼✐s❧❡❛❞✐♥❣ st❛rs