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r str rs tt strts stt rr

  1. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ■◆❘■❆✲❙❛❝❧❛② ✫ ❈▼❆P✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ❋r❛♥❝❡ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆✉❞r❡② ❍❡r♠❛♥t✱ ■◆❘■❆✲❙❛❝❧❛② ✫ ❈▼❆P ❏♦✉r♥é❡s ❋r❛♥❝♦✲❈❤✐❧✐❡♥♥❡s ❞✬❖♣t✐♠✐s❛t✐♦♥ ✲ ❚♦✉❧♦♥✱ ✶✾✲✷✶ ♠❛✐✱ ✷✵✵✽ ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  2. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ✶ ❋r❛♠❡✇♦r❦ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ✷ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❲❡❧❧✲♣♦s❡❞♥❡ss ❙t❛t❡ ❝♦♥str❛✐♥ts ✸ ❋r❛♠❡✇♦r❦ ✭❆❧t❡r♥❛t✐✈❡✮ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ❙❡♥s✐t✐✈✐t② ✹ ❙❡♥s✐t✐✈✐t②✿ ❢r❛♠❡✇♦r❦ ▼❛✐♥ r❡s✉❧t ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  3. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ❉❛t❛ ♦❢ t❤❡ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ � ✶ � ✶ ✷ ✉ ✷ ( t ) + ❣ ( t ) ② ( t ) � ( P ) ▼✐♥ ❞ t ✵ s✳t✳ ② ( t ) = ✉ ( t ) , ˙ ② ( ✵ ) = ② ( ✶ ) = ✵ , ② ( t ) ≥ ❤ ✇✐t❤ ❣ ( t ) := ( ❝ − s✐♥ ( α t )) ❣ ✵ , ❝ > ✵ , α > ✵ . ❚✐♠❡ ✈✐❡✇❡❞ ❛s s❡❝♦♥❞ st❛t❡ ✈❛r✐❛❜❧❡ ✭ ˙ τ = ✶✮ µ = ( ❤ − ❤ ✵ ) / ( ❤ ✶ − ❤ ✵ ) ❤♦♠♦t♦♣② ♣❛r❛♠❡t❡r❀ ❤ ✵ = ♠✐♥ ¯ ② ( t ) ✱ ✇❤❡r❡ ¯ ② ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠ ❤ ✶ = ❤ t❛r❣❡t ✈❛❧✉❡❀ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s ❛r❡ ❣ ✵ := ✶✵ , α = ✶✵ π, ❝ = ✵ . ✶ , ❤ ✶ = − ✵ . ✵✵✶ . ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  4. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ❯♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠✿ ♦♣t✐♠❛❧ st❛t❡ k = 0 0.01 -0.01 -0.03 -0.05 -0.07 -0.09 -0.11 -0.13 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  5. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆❡✐❣❜♦r❤♦♦❞ ♦❢ ❧✐♠✐t✐♥❣ ♣r♦❜❧❡♠✿ ✇❤❡♥ µ > ✵ ✐s s♠❛❧❧ ❋♦r µ > ✵ t❤❡ st❛t❡ ❝♦♥str❛✐♥t ✐s ❛❝t✐✈❡ ✭❝♦♥✈❡① ♣r♦❜❧❡♠✮ ❚❤❡ ❝♦♥t❛❝t s❡t ❝♦✉❧❞ ❜❡ t❤❡♥ ❢♦r s♠❛❧❧ µ > ✵✿ ✶ ❖♥❡ ♣♦✐♥t ✷ ❆ s♠❛❧❧ ✐♥t❡r✈❛❧ ✸ ❆ ♥♦♥ ❝♦♥♥❡❝t❡❞ s❡t ❨♦✉r ❣✉❡ss ❄ ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  6. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆❡✐❣❜♦r❤♦♦❞ ♦❢ ❧✐♠✐t✐♥❣ ♣r♦❜❧❡♠✿ ✇❤❡♥ µ > ✵ ✐s s♠❛❧❧ ❋♦r µ > ✵ t❤❡ st❛t❡ ❝♦♥str❛✐♥t ✐s ❛❝t✐✈❡ ✭❝♦♥✈❡① ♣r♦❜❧❡♠✮ ❙tr✉❝t✉r❛❧ r❡s✉❧t✿ t❤❡ ❝♦♥t❛❝t s❡t ✐s ❛♥ ✐♥t❡r✈❛❧ ◗✉❛♥t✐t❛t✐✈❡ r❡s✉❧t✿ ✜rst✲♦r❞❡r ❡①♣❛♥s✐♦♥ ♦❢ ✈❛❧✉❡ ♦❢ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ t❤❛t ✐♥t❡r✈❛❧ ✦ ◆❡①t✿ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ✉s✐♥❣ ❛ s❤♦♦t✐♥❣ ❛❧❣♦r✐t❤♠ t❤❛t ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ❧❛t❡r✳ ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  7. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ■■ k = 1 0.01 -0.01 -0.03 -0.05 -0.07 -0.09 -0.11 -0.13 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  8. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ■■■ k = 2 0.01 -0.01 -0.03 -0.05 -0.07 -0.09 -0.11 -0.13 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  9. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ■❱ k = 3 0.01 -0.01 -0.03 -0.05 -0.07 -0.09 -0.11 -0.13 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  10. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❱ k = 4 0.01 -0.01 -0.03 -0.05 -0.07 -0.09 -0.11 -0.13 -0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

  11. ❆♥ ❛❝❛❞❡♠✐❝ ❡①❛♠♣❧❡ ●❡♥❡r❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♣r♦❜❧❡♠s ❋r❛♠❡✇♦r❦ ❙t❛t❡ ❝♦♥str❛✐♥ts ❙❡♥s✐t✐✈✐t② ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❱■ k = 5 0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 -0.12 -0.14 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ❏✳ ❋ré❞ér✐❝ ❇♦♥♥❛♥s ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r st❛t❡✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

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