t trts t - PowerPoint PPT Presentation

❚✐❣❤t tr✐❛♥❣✉❧❛t✐♦♥s✿ ❛ ❧✐♥❦ ❜❡t✇❡❡♥ ❝♦♠❜✐♥❛t♦r✐❝s ❛♥❞ t♦♣♦❧♦❣② ❏♦♥❛t❤❛♥ ❙♣r❡❡r ▼❡❧❜♦✉r♥❡✱ ❆✉❣✉st ✶✺✱ ✷✵✶✻

❚♦♣♦❧♦❣✐❝❛❧ ♠❛♥✐❢♦❧❞s ▸ ✭●❡♦♠❡tr✐❝✮ ❚♦♣♦❧♦❣② ✐s st✉❞② ♦❢ ♠❛♥✐❢♦❧❞s ✭s✉r❢❛❝❡s✮ ✉♣ t♦ ❝♦♥t✐♥✉♦✉s ❞❡❢♦r♠❛t✐♦♥ ▸ ❈♦♠♣❧✐❝❛t❡❞ ❞❡❢♦r♠❛t✐♦♥s ♠❛❦❡ r❡❝♦❣♥✐t✐♦♥ ♦❢ ❛ ♠❛♥✐❢♦❧❞ ❞✐✣❝✉❧t ✭✐❢ ♥♦t ✐♠♣♦ss✐❜❧❡✮ ▸ ❘❡❝♦❣♥✐t✐♦♥ ♠✐❣❤t st✐❧❧ ❜❡ ❡❛s② ❢♦r ✏s✉✣❝✐❡♥t❧② ♥✐❝❡✑ ❞❡❢♦r♠❛t✐♦♥s

❈♦♥✈❡①✐t② ⇒ ❝♦♥✈❡①✐t② ▸ ❚♦♣♦❧♦❣✐❝❛❧ ❜❛❧❧s ✴ s♣❤❡r❡s ▸ ▲✐♠✐t❡❞ ✉s❡ ✐♥ ❣❡♦♠❡tr② ❛♥❞ t♦♣♦❧♦❣②✿ ♠❛♥② ♦❜❥❡❝ts ❛r❡ ♥♦t ❜❛❧❧s ✴ ❝❛♥♥♦t ❜❡ ❝♦♥✈❡① ▸ ■♥t✉✐t✐✈❡ ♥♦t✐♦♥✿ ♠❛♥② ♦❜❥❡❝ts ❧♦♦❦ ♠♦r❡ ❝♦♥✈❡① t❤❛♥ ♦t❤❡rs

❚✐❣❤t♥❡ss ▸ ❆♥ ❡♠❜❡❞❞✐♥❣ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ❉ ✐♥t♦ s♦♠❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ E ❞ ✐s s❛✐❞ t♦ ❜❡ t✐❣❤t✱ ✐❢ ✐t ✐s ✏❛s ❝♦♥✈❡① ❛s ♣♦ss✐❜❧❡✑ ❣✐✈❡♥ ✐ts t♦♣♦❧♦❣✐❝❛❧ ❝♦♥str❛✐♥ts ▸ ❉ ✐s t✐❣❤t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐♥t❡rs❡❝t✐♥❣ ❉ ✇✐t❤ ❛ ❤❛❧❢ s♣❛❝❡ ❤ ❞♦❡s ♥♦t ✐♥tr♦❞✉❝❡ ♥❡✇ t♦♣♦❧♦❣✐❝❛❧ ❢❡❛t✉r❡s ✭❡❣✳ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦r ❤♦❧❡s✮ H 0 ( D ) = F h H 0 ( D ∩ h ) = F 2 → � ֒ D D ▸ ❚✐❣❤t♥❡ss ❣❡♥❡r❛❧✐s❡s ❝♦♥✈❡①✐t②✱ ❛♥❞ ♠✐♥✐♠✐s❡s t♦t❛❧ ❛❜s♦❧✉t❡ ❝✉r✈❛t✉r❡✳ ✶ ✶ ❉❡✈❡❧♦♣❡❞ ❜② ❆❧❡①❛♥❞r♦✈✱ ✶✾✸✽❀ ▼✐❧♥♦r✱ ❈❤❡r♥ ❛♥❞ ▲❛s❤♦❢✱ ❑✉✐♣❡r✱ ✶✾✺✵✬s✳

❚✐❣❤t♥❡ss

▼♦rs❡ t❤❡♦r② ▼ö❜✐✉s✳ ❚❤❡♦r✐❡ ❞❡r ❡❧❡♠❡♥t❛r❡♥ ❱❡r✇❛♥❞ts❝❤❛❢t✱ ✶✽✻✸

▼♦rs❡ t❤❡♦r② ▸ ●✐✈❡♥ ❛ ♠❛♥✐❢♦❧❞ ▼ ✱ ❧♦♦❦ ❛t ❛ ❢✉♥❝t✐♦♥ ❢ ∶ ▼ → R ▸ ❢ ❤❛s ✜♥✐t❡❧② ♠❛♥② ❝r✐t✐❝❛❧ ❧❡✈❡❧s ❢ − ✶ ( α ) ▸ ❢ → ✏❤❡✐❣❤t ❢✉♥❝t✐♦♥✑✱ ❢ − ✶ ( ▼ ) → ✏❝♦♥t♦✉r ❧✐♥❡s✑ ▸ ❉❡✜♥❡s ❤❛♥❞❧❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ▼ ⇒ ✏✜♥✐t❡✑ ❞❡s❝r✐♣t✐♦♥ ♦❢ ▼ ▸ P♦✇❡r❢✉❧✿ ✉s❡❞ t♦ ♣r♦✈❡ ❤ ✲❝♦❜♦r❞✐s♠ t❤❡♦r❡♠

▼♦rs❡ t❤❡♦r② ❛♥❞ t✐❣❤t♥❡ss ▸ ❙♦♠❡ ❤❡✐❣❤t ❢✉♥❝t✐♦♥s ❛r❡ ❜❡tt❡r t❤❛♥ ♦t❤❡rs✿ ❢❡✇❡r ❝r✐t✐❝❛❧ ❧❡✈❡❧s ♠❡❛♥s ♠♦r❡ ❡✣❝✐❡♥t ❤❛♥❞❧❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ▸ ❍♦✇ ❣♦♦❞ ❝❛♥ ❛ ▼♦rs❡ ❢✉♥❝t✐♦♥ ❜❡❄ ❚❤❡♦r❡♠ ✭▼♦rs❡ r❡❧❛t✐♦♥s✮ ▲❡t ▼ ❜❡ ❛ ♠❛♥✐❢♦❧❞✱ ❛♥❞ ❧❡t ❢ ∶ ▼ → R ❜❡ ❛ ▼♦rs❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ ❞ µ ( ❢ ) ≥ β ✐ ( ▼ , Z ✷ ) ∑ ✐ = ✵ ✇❤❡r❡ β ✐ ( ▼ , Z ✷ ) ✐s t❤❡ ✐✲t❤ ❇❡tt✐ ♥✉♠❜❡r ♦❢ ▼ ✇✐t❤ Z ✷ ✲❝♦❡✣❝✐❡♥ts✳ ▸ ■❢ ❢ ❛tt❛✐♥s ❡q✉❛❧✐t②✱ ✐t ✐s ❝❛❧❧❡❞ ♣❡r❢❡❝t ▸ ✏❉✐r❡❝t✐♦♥ ❛❧♦♥❣ ✇❤✐❝❤ ❛♥ ♦❜❥❡❝t ❛♣♣❡❛rs t✐❣❤t✑ ▸ ❊♠❜❡❞❞✐♥❣ ✐s t✐❣❤t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ♣r♦❥❡❝t✐♦♥s ❛r❡ ♣❡r❢❡❝t ▼♦rs❡ ❢✉♥❝t✐♦♥s✳ ▸ ❉❡❝✐❞✐♥❣ ✇❤❡t❤❡r s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❡①✐st ✐s ◆P✲❝♦♠♣❧❡t❡ ✳ ✷ ▸ ■s ❞❡❝✐❞✐♥❣ ✇❤❡t❤❡r ✏❛❧❧ ▼♦rs❡ ❢✉♥❝t✐♦♥s ❛r❡ ♣❡r❢❡❝t✑ ❝♦◆P✲❝♦♠♣❧❡t❡ ❄ ✷ ❏♦s✇✐❣ ❛♥❞ P❢❡ts❝❤ ✷✵✵✻✳

❊✉❧❡r ❈❤❛r❛❝t❡r✐st✐❝ ✱ ✜rst ✷ ✲❇❡tt✐ ♥✉♠❜❡r✭❄✮ ❋✐♥✐t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♠❛♥✐❢♦❧❞s ▸ ❘❡♣r❡s❡♥t ♠❛♥✐❢♦❧❞s ✭s✉r❢❛❝❡s✮ ❛s s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s 2 6 5 2 �� 1 , 2 , 4 � , � 1 , 2 , 6 � , � 1 , 3 , 4 � , � 1 , 3 , 7 � , 7 7 3 � 1 , 5 , 6 � , � 1 , 5 , 7 � , � 2 , 3 , 5 � , � 2 , 3 , 7 � , � 2 , 4 , 5 � , � 2 , 6 , 7 � , � 3 , 4 , 6 � , � 3 , 5 , 6 � , 1 � 4 , 5 , 7 � , � 4 , 6 , 7 �� 4 4 2 2 6 5 ▸ ❆ P▲ tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛ ♠❛♥✐❢♦❧❞ ▼ ✐s ❛ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❈ s✉❝❤ t❤❛t ❡✈❡r② ✈❡rt❡① ❧✐♥❦ ✐s P▲ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ st❛♥❞❛r❞ s♣❤❡r❡✳ a 2 1 st C (6) b c 4 6 3 3 lk C (6) 5 C a a d 1 2 ▸ ◗✉❡st✐♦♥✿ ❝❛♥ ✇❡ ❧✐♥❦ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❈ t♦ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ▼ ❄

✱ ✜rst ✷ ✲❇❡tt✐ ♥✉♠❜❡r✭❄✮ ❋✐♥✐t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♠❛♥✐❢♦❧❞s ▸ ❘❡♣r❡s❡♥t ♠❛♥✐❢♦❧❞s ✭s✉r❢❛❝❡s✮ ❛s s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s 2 6 5 2 �� 1 , 2 , 4 � , � 1 , 2 , 6 � , � 1 , 3 , 4 � , � 1 , 3 , 7 � , 7 7 3 � 1 , 5 , 6 � , � 1 , 5 , 7 � , � 2 , 3 , 5 � , � 2 , 3 , 7 � , � 2 , 4 , 5 � , � 2 , 6 , 7 � , � 3 , 4 , 6 � , � 3 , 5 , 6 � , 1 � 4 , 5 , 7 � , � 4 , 6 , 7 �� 4 4 2 2 6 5 ▸ ❆ P▲ tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛ ♠❛♥✐❢♦❧❞ ▼ ✐s ❛ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❈ s✉❝❤ t❤❛t ❡✈❡r② ✈❡rt❡① ❧✐♥❦ ✐s P▲ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ st❛♥❞❛r❞ s♣❤❡r❡✳ a 2 1 st C (6) b c 4 6 3 3 lk C (6) 5 C a a d 1 2 ▸ ◗✉❡st✐♦♥✿ ❝❛♥ ✇❡ ❧✐♥❦ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❈ t♦ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ▼ ❄ ❊✉❧❡r ❈❤❛r❛❝t❡r✐st✐❝ ▸

❋✐♥✐t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♠❛♥✐❢♦❧❞s ▸ ❘❡♣r❡s❡♥t ♠❛♥✐❢♦❧❞s ✭s✉r❢❛❝❡s✮ ❛s s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡①❡s 2 6 5 2 �� 1 , 2 , 4 � , � 1 , 2 , 6 � , � 1 , 3 , 4 � , � 1 , 3 , 7 � , 7 7 3 � 1 , 5 , 6 � , � 1 , 5 , 7 � , � 2 , 3 , 5 � , � 2 , 3 , 7 � , � 2 , 4 , 5 � , � 2 , 6 , 7 � , � 3 , 4 , 6 � , � 3 , 5 , 6 � , 1 � 4 , 5 , 7 � , � 4 , 6 , 7 �� 4 4 2 2 6 5 ▸ ❆ P▲ tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛ ♠❛♥✐❢♦❧❞ ▼ ✐s ❛ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ❈ s✉❝❤ t❤❛t ❡✈❡r② ✈❡rt❡① ❧✐♥❦ ✐s P▲ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ st❛♥❞❛r❞ s♣❤❡r❡✳ a 2 1 st C (6) b c 4 6 3 3 lk C (6) 5 C a a d 1 2 ▸ ◗✉❡st✐♦♥✿ ❝❛♥ ✇❡ ❧✐♥❦ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❈ t♦ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ ▼ ❄ ❊✉❧❡r ❈❤❛r❛❝t❡r✐st✐❝ ✱ ✜rst Z ✷ ✲❇❡tt✐ ♥✉♠❜❡r✭❄✮ ▸

❆❜str❛❝t ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t✐❣❤t♥❡ss ●♦❛❧✿ ✐♥tr✐♥s✐❝ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ ❝♦♠❜✐♥❛t♦r✐❝s ❛♥❞ t♦♣♦❧♦❣②✳ ▸ ❈ ❝♦♥♥❡❝t❡❞ ❛❜str❛❝t s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ✇✐t❤ ✈❡rt❡① s❡t ❱ ( ❈ ) ✳ ▸ ❲ ⊂ ❱ ( ❈ ) ✱ ❈ [ ❲ ] s✉❜❝♦♠♣❧❡① ♦❢ ❈ ✏✐♥❞✉❝❡❞ ❜② ❲ ✑✳ ▸ ❈ ✐s t✐❣❤t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀ ❲ ⊂ ❱ ( ❈ ) ✱ ❈ [ ❲ ] ❞♦❡s ♥♦t ✐♥tr♦❞✉❝❡ ❛♥② ♥❡✇ t♦♣♦❧♦❣✐❝❛❧ ❢❡❛t✉r❡s✳ ✸ ▸ ❚✐❣❤t tr✐❛♥❣✉❧❛t✐♦♥s ❛r❡ ❝♦♥❥❡❝t✉r❡❞ t♦ ❜❡ str♦♥❣❧② ♠✐♥✐♠❛❧ ✹ ✸ ❇❛♥❝❤♦✛ ✶✾✼✵✱ ❑ü❤♥❡❧ ✶✾✾✺ ✹ ❑ü❤♥❡❧ ▲✉t③