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  1. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❊❢❢✐❝✐❡♥❝② ❜❡t✇❡❡♥ ❧❛r❣❡ ❛♥❞ s♠❛❧❧ ❜❛♥❦s✿ t❤❡ r♦❧❡ ♦❢ ♥♦♥❧✐♥❡❛r✐t② ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❝❛♣✐t❛❧ r❡❣✉❧❛t✐♦♥ ❙❡r❡♥❛ ❇r✐❛♥③♦♥✐ ✶ ●✐♦✈❛♥♥✐ ❈❛♠♣✐s✐ ✷ ❆♥♥❛r✐t❛ ❈♦❧❛s❛♥t❡ ✸ ✶ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛♥❛❣❡♠❡♥t P♦❧②t❡❝❤♥✐❝ ❯♥✐✈❡rs✐t② ♦❢ ▼❛r❝❤❡ ✷ ❉❡♣❛rt♠❡♥t ♦❢ ❊❝♦♥♦♠✐❝s ▼❛r❝♦ ❇✐❛❣✐ ❯♥✐✈❡rs✐t② ♦❢ ▼♦❞❡♥❛ ❛♥❞ ❘❡❣❣✐♦ ❊♠✐❧✐❛ ✸ ▲❛❜♦r❛t♦r② ♦❢ ❊①♣❡r✐♠❡♥t❛❧ ❊❝♦♥♦♠✐❝s ❯♥✐✈❡rs✐t❛t ❏❛✉♠❡ ■ ❚❤❡ ✶✶t❤ ◆♦♥❧✐♥❡❛r ❊❝♦♥♦♠✐❝ ❉②♥❛♠✐❝s ❝♦♥❢❡r❡♥❝❡ ✲ ❑②✐✈ ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s ✭❑❙❊✮✱ ❙❡♣t❡♠❜❡r✱ ✹✲✻✱ ✷✵✶✾

  2. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❛❜❧❡ ♦❢ ❝♦♥t❡♥ts ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s

  3. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❇❛♥❦✐♥❣ ✐♥❞✉str② ✐♥ ❡❝♦♥♦♠✐❝s ■♥ t❤✐s ✇♦r❦ ✇❡ ❢♦❝✉s ♦♥ t❤❡ ❝♦st✲❡✣❝✐❡♥❝② ♦❢ t❤❡ ✐t❛❧✐❛♥ ❜❛♥❦✐♥❣ s②st❡♠ ❝♦♥s✐❞❡r✐♥❣ ❜❛♥❦s ♦❢ ❞✐✛❡r❡♥t s✐③❡ ✭❧❛r❣❡ ❛♥❞ s♠❛❧❧✮ ❛s ❇r✐❛♥③♦♥✐ ❡t ❛❧✳✭✷✵✶✽✮✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ str❡ss t❤❡ ❤✐❣❤❡r ❡✣❝✐❡♥❝② ♦❢ s♠❛❧❧ ❜❛♥❦s ✇✐t❤ r❡s♣❡❝t t♦ ❧❛r❣❡ ❜❛♥❦s ❢♦❧❧♦✇✐♥❣ ❡♠♣✐r✐❝❛❧ ❡✈✐❞❡♥❝❡✳

  4. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❊♠♣✐r✐❝❛❧ ❧✐t❡r❛t✉r❡ ♦♥ ✐t❛❧✐❛♥ ❜❛♥❦✐♥❣ s❡❝t♦r • t❤❡ r♦❧❡ ♦❢ s♠❛❧❧ ❜❛♥❦s ✐♥ s✉♣♣♦rt✐♥❣ ❧♦❝❛❧ ✜r♠s ❛♥❞ ❢❛♠✐❧✐❡s t❤❛♥❦s t♦ t❤❡✐r ❧♦♥❣ ❛♥❞ st❛❜❧❡ r❡❧❛t✐♦♥s❤✐♣ ✭❆❧❡ss❛♥❞r✐♥✐ ❡t ❛❧✳✭✷✵✶✽✮✮❀ • ■♥ ♣❡r✐♦❞ ♦❢ ❝r✐s✐s ❧♦❝❛❧ ❜❛♥❦s ✐♥❝r❡❛s❡❞ t❤❡✐r ❧♦❛♥s s✉♣♣❧② ✐♥ ❢❛✈♦✉r ♦❢ ❢❛♠✐❧✐❡s ❛♥❞ s♠❛❧❧ ✜r♠s ✭❙t❡❢❛♥✐ ❡t ❛❧✳✭✷✵✶✻✮✮❀ • t❤❛♥❦s t♦ s♦❢t ✐♥❢♦r♠❛t✐♦♥ ❧♦❝❛❧ ❜❛♥❦s ♠✐t✐❣❛t❡ ✐♥❢♦r♠❛t✐♦♥ ❛s②♠♠❡tr✐❡s ✐♥ ❝r❡❞✐t ♠❛r❦❡ts ❛❧❧♦✇✐♥❣ t❤❡♠ ❛ ❧♦✇❡r ❝r❡❞✐t r❛t✐♦♥✐♥❣ ✭❇❛r❜♦♥✐ ❛♥❞ ❘♦ss✐ ✭✷✵✶✾✮✮✳

  5. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❢✉rt❤❡r ❡♠♣✐r✐❝❛❧ r❡s✉❧ts • ❡✣❝✐❡♥❝② t❡♥❞s t♦ ❞❡❝r❡❛s❡ ✇✐t❤ s✐③❡❀ • ❝♦♦♣❡r❛t✐✈❡s ♣❡r❢♦r♠ ❜❡tt❡r t❤❛♥ ♦t❤❡rs • ❣❡♦❣r❛♣❤✐❝ ❧♦❝❛❧✐s❛t✐♦♥ ♠❛tt❡r ✭❆✐❡❧❧♦ ❡t ❛❧✳ ✭✷✵✶✸✮✮✳ • r❡❣✉❧❛t✐♦♥ ♣❧❛②s ❛ ♣r❡❞♦♠✐♥❛♥t r♦❧❡ ✐♥ ♣❡r✐♦❞ ♦❢ ✜♥❛♥❝✐❛❧ ❞✐str❡ss ✭❆❧❡ss❛♥❞r✐♥✐ ❡t ❛❧✳ ✭✷✵✶✻✱✷✵✶✽✮✮✳

  6. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❆ss✉♠♣t✐♦♥s ❛♥❞ ♠♦t✐✈❛t✐♦♥s ❚❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ✐t❛❧✐❛♥ ❜❛♥❦✐♥❣ s❡❝t♦r ✇❡ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s✿ • ❲❡ ❝♦♥s✐❞❡r ❧❛r❣❡ ❜❛♥❦s ❛♥❞ s♠❛❧❧ ❜❛♥❦s✳ • ❲❡ ❛ss✉♠❡ ❛ q✉❛❞r❛t✐❝ ❝♦st ❢✉♥❝t✐♦♥ ❢♦r ❧❛r❣❡ ❜❛♥❦s ❛♥❞ ❛ ❧✐♥❡❛r ❝♦st ❢✉♥❝t✐♦♥ ❢♦r s♠❛❧❧ ❜❛♥❦s✳ • ❲❡ ✐♥tr♦❞✉❝❡ ❛ ♥♦♥❧✐♥❡❛r ❞❡♠❛♥❞ ❢✉♥❝t✐♦♥ ❛s ✐♥ ❚r❛♠♦♥t❛♥❛✭✷✵✶✵✮✳ • ❊♠♣✐r✐❝❛❧ ❡✈✐❞❡♥❝❡ s❤♦✇s t❤❛t ❡✣❝✐❡♥❝② t❡♥❞s t♦ ❞❡❝r❡❛s❡ ✇✐t❤ s✐③❡✳

  7. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❆ ❢✐rst ❜❛♥❦✐♥❣ ♠♦❞❡❧ ✇✐t❤ ❝❛♣✐t❛❧ r❡❣✉❧❛t✐♦♥ ❋❛♥t✐✭✷✵✶✹✮ • ❜❛♥❦s ♦❢ t❤❡ s❛♠❡ s✐③❡❀ • ✐❞❡♥t✐❝❛❧ ♠❛r❣✐♥❛❧ ❝♦sts❀ • ❢♦❝✉s ♦♥ t❤❡ r♦❧❡ ♦❢ r❡❣✉❧❛t✐♦♥❀ • ❞✐✛❡r❡♥t ❜✐❢✉r❝❛t✐♦♥ str✉❝t✉r❡✳

  8. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❤❡ ❜❡♥❝❤♠❛r❦ ♠♦❞❡❧ ❇r✐❛♥③♦♥✐ ❡t ❛❧✳ ✭✷✵✶✽✮ • ❧❛r❣❡ ❛♥❞ s♠❛❧❧ ❜❛♥❦s❀ • q✉❛❞r❛t✐❝ ❝♦sts ❢♦r ❧❛r❣❡ ❜❛♥❦s ❛♥❞ ❧✐♥❡❛r ❝♦sts ❢♦r s♠❛❧❧ ❜❛♥❦s❀ • ❏✉st✐✜❝❛t✐♦♥ ♦❢ t❤❡ s♣❡❝✐✜❝ ✉s❡ ♦❢ ❡①♣❡❝t❛t✐♦♥s❀ • ❢♦❝✉s ♦♥ t❤❡ r♦❧❡ ♦❢ ❡✣❝✐❡♥❝②❀ • ❆♥ ✐♥t❡r❡st✐♥❣ ❡❝♦♥♦♠✐❝ s❝❡♥❛r✐♦ ❡♠❡r❣❡s ❢r♦♠ t❤❡ ❧♦❝❛❧ st❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ✜①❡❞ ♣♦✐♥ts✳

  9. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❋✉rt❤❡r ❡①t❡♥s✐♦♥s ❖✉r ♠♦❞❡❧ ❲❡ st❛rt ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ❇r✐❛♥③♦♥✐ ❡t ❛❧✳ ✭✷✵✶✽✮ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡ ♥♦♥❧✐♥❡❛r✐t② ✐♥ t❤❡ ❞❡♠❛♥❞ ❢✉♥❝t✐♦♥ t♦♦ ❛s ✐♥ ❚r❛♠♦♥t❛♥❛✭✷✵✶✵✮✳ ●✐✈❡♥ t❤❛t t❤❡ ♠♦❞❡❧ ✐s ❤❛r❞ t♦ s♦❧✈❡ ❛♥❛❧②t✐❝❛❧❧② ✇❡ st✉❞② ✐t ❢r♦♠ ❛ ♥✉♠❡r✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❲❡ ❢♦❝✉s ♦♥ ❞✐✛❡r❡♥❝❡s ❛♥❞ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ r❡s♣❡❝t t❤❡ ✇♦r❦ ♦❢ ❇r✐❛♥③♦♥✐ ❡t ❛❧✳ ✭✷✵✶✽✮✳

  10. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼♦❞❡❧ ❆♥❛❧②t✐❝❛❧ ❆♥❛❧②s✐s ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s ▼♦♥t❡❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥s ❚❤❡ ♣r✐♥❝✐♣❛❧ ✐♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♠♦❞❡❧ ■♥✈❡rs❡ ❞❡♠❛♥❞ ❢✉♥❝t✐♦♥ ❢♦r ❧♦❛♥s � � ✶ r L ( L ✶ , t + L ✷ , t ) = a − b L ✶ , t + L ✷ , t ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡✿ L i , t = K i , t + D i , t ❛♥❞ K i , t = γ L i , t ❢♦r ❡✈❡r② i = ✶ , ✷

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