rs ts - PowerPoint PPT Presentation

❇❛♥❛❝❤ ❆❧❣❡❜r❛s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ●♦t❤❡♥❜✉r❣ ✷✾✳✵✼✳✷✵✶✸ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s ❇❛rt♦s③ ❑♦s♠❛ ❑✇❛➧♥✐❡✇s❦✐ ❯♥✐✈❡rs✐t② ♦❢ ❇✐❛➟②st♦❦ ✭s♦♦♥ ■▼P❆◆ ❲❛rs❛✇✮ ❈❤❡❝❦ ♦✉t ❛❧s♦ ♠② ♣♦st❡r ♦♥ ❲❡❞♥❡s❞❛② ✦✦✦ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐✱ ❆✳ ❱✳ ▲❡❜❡❞❡✈ ✏❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❡♥❞♦♠♦r♣❤✐s♠s ❛♥❞ r❡❞✉❝t✐♦♥ ♦❢ r❡❧❛t✐♦♥s ✐♥ r❡❧❛t✐✈❡ ❈✉♥t③✲P✐♠s♥❡r ❛❧❣❡❜r❛s✑ ❏✳ ♦❢ ❋✉♥❝t✳ ❆♥❛❧②s✐s✱ ✷✻✹ ✭✷✵✶✸✮✱ ♥♦✳ ✽✱ ✶✽✵✻✲✶✽✹✼ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

◗✉❡st✐♦♥ ♦❢ t❤❡ ❉❛②✿ ❲❤❛t ✐s ❝r♦ss❡❞ ♣r♦❞✉❝t❄ ❚❤r♦✉❣❤♦✉t ❆ ✐s ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✳ ❈r♦ss❡❞ ♣r♦❞✉❝t ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ α : ❆ → ❆ ✐s ❛ ✉♥✐✈❡rs❛❧ ❈ ∗ ✲❛❧❣❡❜r❛ ❈ ∗ ( ❆ , ✉ ) ❣❡♥❡r❛t❡❞ ❜② ❆ ❛♥❞ ✉ s✉❜❥❡❝t t♦ r❡❧❛t✐♦♥s✿ α ( ❛ ) = ✉❛✉ ∗ , α − ✶ ( ❛ ) = ✉ ∗ ❛✉ , ❛ ∈ ❆ Pr♦❜❧❡♠ ■❢ α : ❆ → ❆ ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠✱ t❤❡♥ α − ✶ ( ❛ ) = ✉ ∗ ❛✉ ✳ ❲❤❛t r❡❧❛t✐♦♥ s❤♦✉❧❞ ✇❡ ✉s❡ ✐♥st❡❛❞❄ ▲❡t ❆ ⊂ ❇ ❜❡ ❈ ∗ ✲❛❧❣❡❜r❛s ✇✐t❤ ❛ ❝♦♠♠♦♥ ✉♥✐t ✶✱ ❯ ∈ ❇ ✳ Pr♦♣♦s✐t✐♦♥ ✭t❤❡ ❍✐♥t✮✳ ▲❡t α : ❆ → ❆ ❜❡ ❛ ♠❛♣ ♦❢ t❤❡ ❢♦r♠ α ( ❛ ) = ❯❛❯ ∗ ✳ ❚❤❡♥ ⇒ ❯ ♣❛rt✐❛❧ ✐s♦♠❡tr②✱ ❯ ∗ ❯ ∈ ❆ ′ , α ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ⇐ ✇❤❡r❡ ❆ ′ ✐s t❤❡ ❝♦♠♠✉t❛♥t ♦❢ ❆ ✳ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❚♦✇❛r❞s t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ ❆ss✉♠❡ α ( ❛ ) = ❯❛❯ ∗ ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ♦❢ ❆ ✳ Pr♦♣♦s✐t✐♦♥✳ ✶✮ ❈ ∗ ( ❆ , ❯ ) = s♣❛♥ { ❯ ∗ ♥ ❛❯ ♠ : ❛ ∈ ❆ , ♥ , ♠ ∈ N } ✷✮ ❏ = { ❛ ∈ ❆ : ❯ ∗ ❯❛ = ❛ } = ❯ ∗ ❯❆ ∩ ❆ ✐s ❛♥ ✐❞❡❛❧ ✐♥ ❆ s✉❝❤ t❤❛t ❏ ∩ ❦❡r α = { ✵ } ✭ ❏ ⊂ ( ❦❡r α ) ⊥ ✮ ❘❡♠✳ ■♥ t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ t❤❡ ❡❧❡♠❡♥ts ❯ ∗ ♥ ❛❯ ♠ ❛r❡ t❤❡ ✬❜r✐❝❦s✬ t❤❡ ✐❞❡❛❧ ❏ = { ❛ ∈ ❆ : ❯ ∗ ❯❛ = ❛ } ✐s t❤❡ ✬❝❡♠❡♥t✬ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

✶✮ ❘❡❛❞✐♥❣ ♦✛ t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❢r♦♠ ❈ ∗ ( ❆ , ❯ ) ❈♦♥s✐❞❡r ✐♥✜♥✐t❡ ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ❧❛❜❡❧❡❞ ❜② N = { ✵ , ✶ , ✷ , ... } M ( ❆ ) := { [ ❛ ♥ , ♠ ] : ❛ ♥ , ♠ ∈ ❆ ♦♥❧② ✜♥✐t❡ ❡♥tr✐❡s ♥♦♥ ③❡r♦ } ❛♥❞ ❛ ♠❛♣♣✐♥❣ Ψ : M ( ❆ ) → ❈ ∗ ( ❆ , ❯ ) ❣✐✈❡♥ ❜② � ❯ ∗ ♥ ❛ ♥ , ♠ ❯ ♠ Ψ([ ❛ ♥ , ♠ ]) = ♥ , ♠ ∈ N Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ♠❛♣ Ψ ❜❡❝♦♠❡s ❛ ∗ ✲❤♦♠♦♠♦r♣❤✐s♠ ✐❢ ✇❡ ❞❡✜♥❡ ♦♥ M ( ❆ ) t❤❡ ∗ ✲❛❧❣❡❜r❛ str✉❝t✉r❡ ( M ( ❆ ) , + , · , ∗ , ⋆ ) ❛s ❢♦❧❧♦✇s ( ❛ + ❜ ) ♠ , ♥ := ❛ ♠ , ♥ + ❜ ♠ , ♥ , ( ✶ ) ( λ ❛ ) ♠ , ♥ := λ ❛ ♠ , ♥ ( ✷ ) ( ❛ ∗ ) ♠ , ♥ := ❛ ∗ ( ✸ ) ♥ , ♠ ❛♥❞ ∞ ∞ � � Λ ❥ ( ❜ ) + Λ ❥ ( ❛ ) · ❜ ❛ ⋆ ❜ := ❛ · ( ✹ ) ❥ = ✵ ❥ = ✶ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

M ( ❆ ) := { [ ❛ ♥ , ♠ ] : ❛ ♥ , ♠ ∈ ❆ ♦♥❧② ✜♥✐t❡ ❡♥tr✐❡s ♥♦♥ ③❡r♦ } Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ♠❛♣ Ψ ❜❡❝♦♠❡s ❛ ∗ ✲❤♦♠♦♠♦r♣❤✐s♠ ✐❢ ✇❡ ❞❡✜♥❡ ♦♥ M ( ❆ ) t❤❡ ∗ ✲❛❧❣❡❜r❛ str✉❝t✉r❡ ( M ( ❆ ) , + , · , ∗ , ⋆ ) ❛s ❢♦❧❧♦✇s ( ❛ + ❜ ) ♠ , ♥ := ❛ ♠ , ♥ + ❜ ♠ , ♥ ✭✶✮ ( λ ❛ ) ♠ , ♥ := λ ❛ ♠ , ♥ ✭✷✮ ( ❛ ∗ ) ♠ , ♥ := ❛ ∗ ✭✸✮ ♥ , ♠ ❛♥❞ � ∞ � ∞ Λ ❥ ( ❜ ) + Λ ❥ ( ❛ ) · ❜ ❛ ⋆ ❜ = ❛ · ✭✹✮ ❥ = ✵ ❥ = ✶ ✇❤❡r❡ · ✐s ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ Λ : M ( ❆ ) → M ( ❆ ) ✐s ❣✐✈❡♥ ❜②   ✵ ✵ ✵ ✵ · · ·   ✵ α ( ❛ ✵ , ✵ ) α ( ❛ ✵ , ✶ ) α ( ❛ ✵ , ✷ ) · · ·     ✵ α ( ❛ ✶ , ✵ ) α ( ❛ ✶ , ✶ ) α ( ❛ ✶ , ✷ ) · · · Λ([ ❛ ♥ , ♠ ]) :=   .   ✵ α ( ❛ ✷ , ✵ ) α ( ❛ ✷ , ✶ ) α ( ❛ ✷ , ✷ ) · · ·   ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

✷✮ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ♥♦r♠ ♦❢ ❡❧❡♠❡♥ts ✐♥ ❈ ∗ ( ❆ , ❯ ) ❆♥ ❡❧❡♠❡♥t [ ❛ ♥ , ♠ ] ∈ M ( ❆ ) ✐s ❦ ✲ ❞✐❛❣♦♥❛❧ ✱ ✇❤❡r❡ ❦ ∈ Z ✱ ✐❢ ✐t ✐s ♦❢ t❤❡ ❢♦r♠   � − ❦ r + ❦ + ✶ ✵  � �� � � �� �  ❦   ❛ ✵ , − ❦ ✵     ❛ ❦ , ✵              ✐❢ ❦ ≥ ✵✱ ♦r   ✱ ✐❢ ❦ < ✵✳ r + ✶     ❛ r + ❦ , r         ❛ r + ❦ , r     ✵ ✵ Pr♦♣♦s✐t✐♦♥✳ ■❢ ❛ = Ψ([ ❛ ♥ , ♠ ]) ✇❤❡r❡ [ ❛ ♥ , ♠ ] ✐s ❦ ✲❞✐❛❣♦♥❛❧✱ t❤❡♥       ✐ � � � �� α ✐ − ❥ ( ❛ ❥ + ❦ , ❥ ) , ❏ � ❛ � = ❧✐♠ ♥ →∞ ♠❛① ♠❛① , ❞ ( ❛ ♥ + ❦ , ♥ , ❦❡r α ) ( ✺ ) ❞   ✐ = ✶ ,..., ♥   ❥ = ✵ , ❥ + ❦ ≥ ✵ ✇❤❡r❡ ❏ = { ❛ ∈ ❆ : ❯ ∗ ❯❛ = ❛ } ❛♥❞ ❞ ( ❛ , ■ ) = ✐♥❢ ❜ ∈ ■ � ❛ − ❜ � ✳ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❚❤❡♦r❡♠ ✭❝r♦ss❡❞ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮ ▲❡t α : ❆ → ❆ ❜❡ ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ❛♥❞ ❏ ⊂ ( ❦❡r α ) ⊥ ✳ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❈ ∗ ✲s❡♠✐♥♦r♠ � · � ♦♥ t❤❡ ∗ ✲❛❧❣❡❜r❛ ( M ( ❆ ) , + , · , ∗ , ⋆ ) s✉❝❤ t❤❛t ✭✺✮ ❤♦❧❞s ❛♥❞ ❡✐t❤❡r � � λ ❦ ❛ ❦ � , ❢♦r ❛❧❧ λ ∈ T , ✶ ) � ❛ ❦ � = � ❦ ❦ ♦r � ✷ ) � ❛ ✵ � ≤ � ❛ ❦ � ❦ ❢♦r ❛❧❧ ❦ ✲❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ❛ ❦ ∈ M ( ❆ ) ✳ ❚❤✐s ❈ ∗ ✲s❡♠✐♥♦r♠ ②✐❡❧❞s ❛ ❈ ∗ ✲❛❧❣❡❜r❛ ❈ ∗ ( ❆ , α, ❏ ) := M ( ❆ ) / � · � , ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡❧❡♠❡♥ts     α ( ✶ ) ✵ ✵ · · · ❛ ✵ ✵ · · · ✵ ✵ ✵ · · · ✵ ✵ ✵ · · ·     � ✉ :=   ❛ :=   , ❛ ∈ ❆ , ❛♥❞ ✵ ✵ ✵ · · · ✵ ✵ ✵ · · · ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❛♥❞ ✉♥✐✈❡rs❛❧ s✉❜❥❡❝t t♦ r❡❧❛t✐♦♥s✿ � ❛✉ ∗ ✱ ❏ = { ❛ ∈ ❆ : ✉ ∗ ✉ � α ( ❛ ) = ✉ � ❛ = � ❛ } . ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s