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r ss C P r r r r

  1. ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳ ❋r❛♥❦✐ ❉✐❧❧❡♥ ❆♣r✐❧ ✷✹✱ ✷✵✵✾ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  2. ■♥tr♦❞✉❝t✐♦♥ ▲❡t ❢ : ▼ ♥ → � ▼ ♠ ❜❡ ❛♥ ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥ ♦❢ ❛ ❘✐❡♠❛♥♥✐❛♥ ♥ ✲♠❛♥✐❢♦❧❞ ▼ ♥ ✐♥t♦ ❛ ❑❛❡❤❧❡r ♠ ✲♠❛♥✐❢♦❧❞ � ▼ ♠ ♦❢ ❝♦♠♣❧❡① ❞✐♠❡♥s✐♦♥ ♠ ✳ ❚❤❡♥ ❢ ✐s s❛✐❞ t♦ ❜❡ t♦t❛❧❧② r❡❛❧ ✐❢ t❤❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❏ ♦❢ � ▼ ♠ ❝❛rr✐❡s ❡❛❝❤ t❛♥❣❡♥t s♣❛❝❡ ♦❢ ▼ ♥ ✐♥t♦ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦r♠❛❧ s♣❛❝❡✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) ⊂ ❚ ⊥ ♣ ▼ ✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ ♥ = ♠ ✱ t❤❡♥ ▼ ♥ ✐s s❛✐❞ t♦ ❜❡ ▲❛❣r❛♥❣✐❛♥ ✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) = ❚ ⊥ ♣ ▼ ✳ ■❢ ♥♦ ❝♦♥❢✉s✐♦♥ ✐s ♣♦ss✐❜❧❡✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ▼ ♥ ❡♠❜❡❞❞❡❞ ❛♥❞ ✇❡ ✐❞❡♥t✐❢② ▼ ♥ ❛♥❞ ❢ ( ▼ ♥ ) ✳ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  3. ■♥tr♦❞✉❝t✐♦♥ ▲❡t ❢ : ▼ ♥ → � ▼ ♠ ❜❡ ❛♥ ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥ ♦❢ ❛ ❘✐❡♠❛♥♥✐❛♥ ♥ ✲♠❛♥✐❢♦❧❞ ▼ ♥ ✐♥t♦ ❛ ❑❛❡❤❧❡r ♠ ✲♠❛♥✐❢♦❧❞ � ▼ ♠ ♦❢ ❝♦♠♣❧❡① ❞✐♠❡♥s✐♦♥ ♠ ✳ ❚❤❡♥ ❢ ✐s s❛✐❞ t♦ ❜❡ t♦t❛❧❧② r❡❛❧ ✐❢ t❤❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❏ ♦❢ � ▼ ♠ ❝❛rr✐❡s ❡❛❝❤ t❛♥❣❡♥t s♣❛❝❡ ♦❢ ▼ ♥ ✐♥t♦ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦r♠❛❧ s♣❛❝❡✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) ⊂ ❚ ⊥ ♣ ▼ ✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ ♥ = ♠ ✱ t❤❡♥ ▼ ♥ ✐s s❛✐❞ t♦ ❜❡ ▲❛❣r❛♥❣✐❛♥ ✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) = ❚ ⊥ ♣ ▼ ✳ ■❢ ♥♦ ❝♦♥❢✉s✐♦♥ ✐s ♣♦ss✐❜❧❡✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ▼ ♥ ❡♠❜❡❞❞❡❞ ❛♥❞ ✇❡ ✐❞❡♥t✐❢② ▼ ♥ ❛♥❞ ❢ ( ▼ ♥ ) ✳ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  4. ■♥tr♦❞✉❝t✐♦♥ ▲❡t ❢ : ▼ ♥ → � ▼ ♠ ❜❡ ❛♥ ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥ ♦❢ ❛ ❘✐❡♠❛♥♥✐❛♥ ♥ ✲♠❛♥✐❢♦❧❞ ▼ ♥ ✐♥t♦ ❛ ❑❛❡❤❧❡r ♠ ✲♠❛♥✐❢♦❧❞ � ▼ ♠ ♦❢ ❝♦♠♣❧❡① ❞✐♠❡♥s✐♦♥ ♠ ✳ ❚❤❡♥ ❢ ✐s s❛✐❞ t♦ ❜❡ t♦t❛❧❧② r❡❛❧ ✐❢ t❤❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❏ ♦❢ � ▼ ♠ ❝❛rr✐❡s ❡❛❝❤ t❛♥❣❡♥t s♣❛❝❡ ♦❢ ▼ ♥ ✐♥t♦ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦r♠❛❧ s♣❛❝❡✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) ⊂ ❚ ⊥ ♣ ▼ ✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ ♥ = ♠ ✱ t❤❡♥ ▼ ♥ ✐s s❛✐❞ t♦ ❜❡ ▲❛❣r❛♥❣✐❛♥ ✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) = ❚ ⊥ ♣ ▼ ✳ ■❢ ♥♦ ❝♦♥❢✉s✐♦♥ ✐s ♣♦ss✐❜❧❡✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ▼ ♥ ❡♠❜❡❞❞❡❞ ❛♥❞ ✇❡ ✐❞❡♥t✐❢② ▼ ♥ ❛♥❞ ❢ ( ▼ ♥ ) ✳ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  5. ■♥tr♦❞✉❝t✐♦♥ ▲❡t ❢ : ▼ ♥ → � ▼ ♠ ❜❡ ❛♥ ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥ ♦❢ ❛ ❘✐❡♠❛♥♥✐❛♥ ♥ ✲♠❛♥✐❢♦❧❞ ▼ ♥ ✐♥t♦ ❛ ❑❛❡❤❧❡r ♠ ✲♠❛♥✐❢♦❧❞ � ▼ ♠ ♦❢ ❝♦♠♣❧❡① ❞✐♠❡♥s✐♦♥ ♠ ✳ ❚❤❡♥ ❢ ✐s s❛✐❞ t♦ ❜❡ t♦t❛❧❧② r❡❛❧ ✐❢ t❤❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❏ ♦❢ � ▼ ♠ ❝❛rr✐❡s ❡❛❝❤ t❛♥❣❡♥t s♣❛❝❡ ♦❢ ▼ ♥ ✐♥t♦ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦r♠❛❧ s♣❛❝❡✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) ⊂ ❚ ⊥ ♣ ▼ ✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ ♥ = ♠ ✱ t❤❡♥ ▼ ♥ ✐s s❛✐❞ t♦ ❜❡ ▲❛❣r❛♥❣✐❛♥ ✱ ✐✳❡✳ ❏ ( ❢ ∗ ❚ ♣ ▼ ) = ❚ ⊥ ♣ ▼ ✳ ■❢ ♥♦ ❝♦♥❢✉s✐♦♥ ✐s ♣♦ss✐❜❧❡✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ▼ ♥ ❡♠❜❡❞❞❡❞ ❛♥❞ ✇❡ ✐❞❡♥t✐❢② ▼ ♥ ❛♥❞ ❢ ( ▼ ♥ ) ✳ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  6. ■♥tr♦❞✉❝t✐♦♥ ▼ ♥ = C P ♥ ( ✹ ) ❋r♦♠ ♥♦✇ ♦♥✿ ♥ = ♠ ✱ � ▼ ♥ ⊂ C P ♥ ( ✹ ) ▲❛❣r❛♥❣✐❛♥ ❙✉❜♠❛♥✐❢♦❧❞ ❈♦♥t❡♥ts✿ ▲✐❢t t♦ ❙ ✷ ♥ + ✶ ( ✶ ) ✱ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡s ♦❢ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ❈✉r✈❛t✉r❡ ✐♥❡q✉❛❧✐t✐❡s Ps❡✉❞♦✲♣❛r❛❧❧❡❧ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏♦❤♥ ❇♦❧t♦♥✱ ❇❛♥❣✲❨❡♥ ❈❤❡♥✱ ❏♦❤❛♥ ❋❛st❡♥❛❦❡❧s✱ ❏♦❡r✐ ❱❛♥ ❞❡r ❱❡❦❡♥✱ P♦❧ ❱❡rstr❛❡❧❡♥✱ ▲✉❝ ❱r❛♥❝❦❡♥ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  7. ■♥tr♦❞✉❝t✐♦♥ ▼ ♥ = C P ♥ ( ✹ ) ❋r♦♠ ♥♦✇ ♦♥✿ ♥ = ♠ ✱ � ▼ ♥ ⊂ C P ♥ ( ✹ ) ▲❛❣r❛♥❣✐❛♥ ❙✉❜♠❛♥✐❢♦❧❞ ❈♦♥t❡♥ts✿ ▲✐❢t t♦ ❙ ✷ ♥ + ✶ ( ✶ ) ✱ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡s ♦❢ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ❈✉r✈❛t✉r❡ ✐♥❡q✉❛❧✐t✐❡s Ps❡✉❞♦✲♣❛r❛❧❧❡❧ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏♦❤♥ ❇♦❧t♦♥✱ ❇❛♥❣✲❨❡♥ ❈❤❡♥✱ ❏♦❤❛♥ ❋❛st❡♥❛❦❡❧s✱ ❏♦❡r✐ ❱❛♥ ❞❡r ❱❡❦❡♥✱ P♦❧ ❱❡rstr❛❡❧❡♥✱ ▲✉❝ ❱r❛♥❝❦❡♥ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  8. ■♥tr♦❞✉❝t✐♦♥ ▼ ♥ = C P ♥ ( ✹ ) ❋r♦♠ ♥♦✇ ♦♥✿ ♥ = ♠ ✱ � ▼ ♥ ⊂ C P ♥ ( ✹ ) ▲❛❣r❛♥❣✐❛♥ ❙✉❜♠❛♥✐❢♦❧❞ ❈♦♥t❡♥ts✿ ▲✐❢t t♦ ❙ ✷ ♥ + ✶ ( ✶ ) ✱ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠✳ ❊①❛♠♣❧❡s ♦❢ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ❈✉r✈❛t✉r❡ ✐♥❡q✉❛❧✐t✐❡s Ps❡✉❞♦✲♣❛r❛❧❧❡❧ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏♦❤♥ ❇♦❧t♦♥✱ ❇❛♥❣✲❨❡♥ ❈❤❡♥✱ ❏♦❤❛♥ ❋❛st❡♥❛❦❡❧s✱ ❏♦❡r✐ ❱❛♥ ❞❡r ❱❡❦❡♥✱ P♦❧ ❱❡rstr❛❡❧❡♥✱ ▲✉❝ ❱r❛♥❝❦❡♥ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

  9. ❇❛s✐❝ ❡q✉❛t✐♦♥s ❢♦r ▲❛❣r❛♥❣✐❛♥ ❙✉❜♠❛♥✐❢♦❧❞s ❋♦r♠✉❧❛s ♦❢ ●❛✉ss ❛♥❞ ❲❡✐♥❣❛rt❡♥✿ � ∇ ❳ ❨ = ∇ ❳ ❨ + ❤ ( ❳ , ❨ ) , � ∇ ❳ ◆ = − ❆ ◆ ❳ + ∇ ⊥ ❳ ◆ , ❘❡❧❛t✐♦♥s✿ ❏ ( ❆ ❏❳ ❨ ) = ❤ ( ❳ , ❨ ) ❛♥❞ ∇ ⊥ ❳ ❏❨ = ❏ ∇ ❳ ❨ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ●❛✉ss✿ ❘ ( ❳ , ❨ ) = ❳ ∧ ❨ + [ ❆ ❏❳ , ❆ ❏❨ ] . ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ ❈♦❞❛③③✐✿ ( ∇ ❤ )( ❳ , ❨ , ❩ ) := ∇ ⊥ ❳ ❤ ( ❨ , ❩ ) − ❤ ( ∇ ❳ ❨ , ❩ ) − ❤ ( ❨ , ∇ ❳ ❩ ) ✐s t♦t❛❧❧② s②♠♠❡tr✐❝ ❋r❛♥❦✐ ❉✐❧❧❡♥ ▲❛❣r❛♥❣✐❛♥ s✉❜♠❛♥✐❢♦❧❞s ♦❢ C P ♥ ✳

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