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  1. ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❖♥ t❤❡ ✉♥✐❢♦r♠ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ P✐❡rr❡ ▲✐ss② ▲❛❜♦r❛t♦✐r❡ ❏❛❝q✉❡s✲▲♦✉✐s ▲✐♦♥s✱ ❯♥✐✈❡rs✐té P✐❡rr❡ ❡t ▼❛r✐❡ ❈✉r✐❡ ◆❡✇ tr❡♥❞s ✐♥ ▼♦❞❡❧✐♥❣✱ ❈♦♥tr♦❧ ❛♥❞ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ❈■▼■✱ ❚♦✉❧♦✉s❡✱ ❏✉♥❡ ✶✽✱ ✷✵✶✹

  2. ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❝♦♥tr♦❧❧❡❞ ♦♥ ♦♥❡ s✐❞❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ tr❛♥s♣♦rt ❡q✉❛t✐♦♥ ❯♥✐❢♦r♠ ❝♦♥tr♦❧❧❛❜✐❧✐t② ✐♥ t❤❡ ✈❛♥✐s❤✐♥❣ ✈✐s❝♦s✐t② ❧✐♠✐t ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❝♦♥tr♦❧❧❡❞ ♦♥ ♦♥❡ s✐❞❡ ✷ ■♥tr♦❞✉❝t✐♦♥ ▲✐♥❦ ❜❡t✇❡❡♥ ❤❡❛t ❛♥❞ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤✿ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛❜✐❧✐t② ✐♥❡q✉❛❧✐t✐❡s ❈♦♥❝❧✉s✐♦♥ ✸ ❈♦♥❝❧✉s✐♦♥

  3. ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❝♦♥tr♦❧❧❡❞ ♦♥ ♦♥❡ s✐❞❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ tr❛♥s♣♦rt ❡q✉❛t✐♦♥ ❯♥✐❢♦r♠ ❝♦♥tr♦❧❧❛❜✐❧✐t② ✐♥ t❤❡ ✈❛♥✐s❤✐♥❣ ✈✐s❝♦s✐t② ❧✐♠✐t ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❝♦♥tr♦❧❧❡❞ ♦♥ ♦♥❡ s✐❞❡ ✷ ■♥tr♦❞✉❝t✐♦♥ ▲✐♥❦ ❜❡t✇❡❡♥ ❤❡❛t ❛♥❞ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤✿ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛❜✐❧✐t② ✐♥❡q✉❛❧✐t✐❡s ❈♦♥❝❧✉s✐♦♥ ✸ ❈♦♥❝❧✉s✐♦♥

  4. ▼ ✵ s♣❡❡❞✱ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱ ❧❡♥❣t❤ ▲ ✵✱ ▲ ✷ ✵ ▲ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ ✶ ✵ ▲ ✮✱ ② ✵ ▲ ✷ ✵ ❚ ✱ ❝♦♥tr♦❧ ✈ ◗ ✵ ❚ ✵ ▲ ✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱

  5. ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱ ❧❡♥❣t❤ ▲ ✵✱ ▲ ✷ ✵ ▲ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ ✶ ✵ ▲ ✮✱ ② ✵ ▲ ✷ ✵ ❚ ✱ ❝♦♥tr♦❧ ✈ ◗ ✵ ❚ ✵ ▲ ✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱ ▼ � = ✵ s♣❡❡❞✱

  6. ❧❡♥❣t❤ ▲ ✵✱ ▲ ✷ ✵ ▲ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ ✶ ✵ ▲ ✮✱ ② ✵ ▲ ✷ ✵ ❚ ✱ ❝♦♥tr♦❧ ✈ ◗ ✵ ❚ ✵ ▲ ✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱ ▼ � = ✵ s♣❡❡❞✱ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ε > ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱

  7. ▲ ✷ ✵ ▲ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ ✶ ✵ ▲ ✮✱ ② ✵ ▲ ✷ ✵ ❚ ✱ ❝♦♥tr♦❧ ✈ ◗ ✵ ❚ ✵ ▲ ✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱ ▼ � = ✵ s♣❡❡❞✱ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ε > ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱ ❧❡♥❣t❤ ▲ > ✵✱

  8. ▲ ✷ ✵ ❚ ✱ ❝♦♥tr♦❧ ✈ ◗ ✵ ❚ ✵ ▲ ✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱ ▼ � = ✵ s♣❡❡❞✱ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ε > ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱ ❧❡♥❣t❤ ▲ > ✵✱ ② ✵ ∈ ▲ ✷ ( ✵ , ▲ ) ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ − ✶ ( ✵ , ▲ ) ✮✱

  9. ◗ ✵ ❚ ✵ ▲ ✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱ ▼ � = ✵ s♣❡❡❞✱ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ε > ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱ ❧❡♥❣t❤ ▲ > ✵✱ ② ✵ ∈ ▲ ✷ ( ✵ , ▲ ) ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ − ✶ ( ✵ , ▲ ) ✮✱ ❝♦♥tr♦❧ ✈ ∈ ▲ ✷ ( ✵ , ❚ ) ✱

  10. ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ◆♦t❛t✐♦♥s ❚ > ✵ t✐♠❡✱ ▼ � = ✵ s♣❡❡❞✱ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ε > ✵ ✭❞❡st✐♥❡❞ t♦ t❡♥❞ t♦ ✵✮✱ ❧❡♥❣t❤ ▲ > ✵✱ ② ✵ ∈ ▲ ✷ ( ✵ , ▲ ) ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✭♦r ❍ − ✶ ( ✵ , ▲ ) ✮✱ ❝♦♥tr♦❧ ✈ ∈ ▲ ✷ ( ✵ , ❚ ) ✱ ◗ := ( ✵ , ❚ ) × ( ✵ , ▲ ) ✳

  11. ❉❡✜♥✐t✐♦♥ ▲❡t ❚ ▲ ▼ ❜❡ ✜①❡❞✳ ❊q✉❛t✐♦♥ ✭❚✲❉✮ ✐s ♥✉❧❧✲❝♦♥tr♦❧❧❛❜❧❡ ❛t t✐♠❡ ▲ ✷ ✵ ▲ ✱ t❤❡r❡ ❡①✐sts ✈ ▲ ✷ ✵ ❚ ❚ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡✈❡r② ② ✵ s✉❝❤ t❤❛t ② ❚ ✵ ✳ ◆✉❧❧✲❝♦♥tr♦❧❧❛❜✐❧✐t② ✐♥ ❛r❜✐tr❛r② s♠❛❧❧ t✐♠❡✿ ✇❡❧❧✲❦♥♦✇♥ ❢♦r ❛ ❧♦♥❣ t✐♠❡ ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✭❋❛tt♦r✐♥✐✲❘✉ss❡❧❧❵✼✶✮✳ ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❡q✉❛t✐♦♥ ❚❤❡ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥  ② t − ε ② ①① + ▼② ① = ✵ ✐♥ ◗ ,    ② ( ., ✵ ) = ✈ ( t ) ✐♥ ( ✵ , ❚ ) ,   ✭❚✲❉✮ ② ( ., ▲ ) = ✵ ✐♥ ( ✵ , ❚ ) ,   ② ( ✵ , . ) = ② ✵ ✐♥ ( ✵ , ▲ ) .   

  12. ❚❤❡ ❚✲❉ ❡q✉❛t✐♦♥ ❚❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❡q✉❛t✐♦♥ ❚❤❡ tr❛♥s♣♦rt✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥  ② t − ε ② ①① + ▼② ① = ✵ ✐♥ ◗ ,    ② ( ., ✵ ) = ✈ ( t ) ✐♥ ( ✵ , ❚ ) ,   ✭❚✲❉✮ ② ( ., ▲ ) = ✵ ✐♥ ( ✵ , ❚ ) ,   ② ( ✵ , . ) = ② ✵ ✐♥ ( ✵ , ▲ ) .    ❉❡✜♥✐t✐♦♥ ▲❡t ❚ , ▲ , ▼ , ε ❜❡ ✜①❡❞✳ ❊q✉❛t✐♦♥ ✭❚✲❉✮ ✐s ♥✉❧❧✲❝♦♥tr♦❧❧❛❜❧❡ ❛t t✐♠❡ ❚ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡✈❡r② ② ✵ ∈ ▲ ✷ ( ✵ , ▲ ) ✱ t❤❡r❡ ❡①✐sts ✈ ∈ ▲ ✷ ( ✵ , ❚ ) s✉❝❤ t❤❛t ② ( ❚ ) = ✵ ✳ ◆✉❧❧✲❝♦♥tr♦❧❧❛❜✐❧✐t② ✐♥ ❛r❜✐tr❛r② s♠❛❧❧ t✐♠❡✿ ✇❡❧❧✲❦♥♦✇♥ ❢♦r ❛ ❧♦♥❣ t✐♠❡ ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✭❋❛tt♦r✐♥✐✲❘✉ss❡❧❧❵✼✶✮✳

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