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s r rs rrs rs s s s Pttr t ,

  1. ❯s❡ ♦❢ r❛♥❞♦♠ ♥✉♠❜❡rs ✐♥ ♣r♦❣r❛♠s ❈❤✳✽✿ ❘❛♥❞♦♠ ♥✉♠❜❡rs ❛♥❞ s✐♠♣❧❡ ❣❛♠❡s ❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥ ✶ , ✷ ❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r② ✶ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s ✷ ❆✉❣ ✶✺✱ ✷✵✶✺ ❘❛♥❞♦♠ ♥✉♠❜❡rs ❛r❡ ✉s❡❞ t♦ s✐♠✉❧❛t❡ ✉♥❝❡rt❛✐♥ ❡✈❡♥ts ❉r❛✇✐♥❣ r❛♥❞♦♠ ♥✉♠❜❡rs ❉❡t❡r♠✐♥✐st✐❝ ♣r♦❜❧❡♠s P②t❤♦♥ ❤❛s ❛ r❛♥❞♦♠ ♠♦❞✉❧❡ ❢♦r ❞r❛✇✐♥❣ r❛♥❞♦♠ ♥✉♠❜❡rs✳ ❙♦♠❡ ♣r♦❜❧❡♠s ✐♥ s❝✐❡♥❝❡ ❛♥❞ t❡❝❤♥♦❧♦❣② ❛r❡ ❞❡sr❝r✐❜❡❞ ❜② r❛♥❞♦♠✳r❛♥❞♦♠✭✮ ❞r❛✇s r❛♥❞♦♠ ♥✉♠❜❡rs ✐♥ [ ✵ , ✶ ) ✿ ✏❡①❛❝t✑ ♠❛t❤❡♠❛t✐❝s✱ ❧❡❛❞✐♥❣ t♦ ✏♣r❡❝✐s❡✑ r❡s✉❧ts ❃❃❃ ✐♠♣♦rt r❛♥❞♦♠ ❊①❛♠♣❧❡✿ t❤r♦✇✐♥❣ ❛ ❜❛❧❧ ✉♣ ✐♥ t❤❡ ❛✐r ✭ ② ( t ) = ✈ ✵ t − ✶ ✷ ❣t ✷ ✮ ❃❃❃ r❛♥❞♦♠✳r❛♥❞♦♠✭✮ ✵✳✽✶✺✺✵✺✹✻✽✽✺✸✸✽✶✵✹ ❃❃❃ r❛♥❞♦♠✳r❛♥❞♦♠✭✮ ✵✳✹✹✾✶✸✸✷✻✽✵✾✵✷✾✽✺✷ ❙t♦❝❤❛st✐❝ ♣r♦❜❧❡♠s ❃❃❃ r❛♥❞♦♠✳r❛♥❞♦♠✭✮ ✵✳✽✽✸✷✵✻✺✸✶✶✻✸✻✼✹✺✹ ❙♦♠❡ ♣r♦❜❧❡♠s ❛♣♣❡❛r ♣❤②s✐❝❛❧❧② ✉♥❝❡rt❛✐♥ ❊①❛♠♣❧❡s✿ r♦❧❧✐♥❣ ❛ ❞✐❡✱ ♠♦❧❡❝✉❧❛r ♠♦t✐♦♥✱ ❣❛♠❡s ◆♦t✐❝❡ ❯s❡ r❛♥❞♦♠ ♥✉♠❜❡rs t♦ ♠✐♠✐❝ t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ♥✉♠❜❡rs ✐s ♣r♦❞✉❝❡❞ ❜② ❛ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ ✲ t❤❡ ♥✉♠❜❡rs ❥✉st ❛♣♣❡❛r r❛♥❞♦♠✳ ❡①♣❡r✐♠❡♥t✳ ❉✐str✐❜✉t✐♦♥ ♦❢ r❛♥❞♦♠ ♥✉♠❜❡rs ❉✐str✐❜✉t✐♦♥ ♦❢ r❛♥❞♦♠ ♥✉♠❜❡rs ✈✐s✉❛❧✐③❡❞ ◆ ❂ ✺✵✵ ★ ♥♦ ♦❢ s❛♠♣❧❡s ① ❂ r❛♥❣❡✭◆✮ ② ❂ ❬r❛♥❞♦♠✳✉♥✐❢♦r♠✭✲✶✱✶✮ ❢♦r ✐ ✐♥ ①❪ ❢r♦♠ s❝✐t♦♦❧s✳st❞ ✐♠♣♦rt ♣❧♦t ♣❧♦t✭①✱ ②✱ ✬✰✬✱ ❛①✐s❂❬✵✱◆✲✶✱✲✶✳✷✱✶✳✷❪✮ r❛♥❞♦♠✳r❛♥❞♦♠✭✮ ❣❡♥❡r❛t❡s r❛♥❞♦♠ ♥✉♠❜❡rs t❤❛t ❛r❡ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ✐♥ t❤❡ ✐♥t❡r✈❛❧ [ ✵ , ✶ ) 1 r❛♥❞♦♠✳✉♥✐❢♦r♠✭❛✱ ❜✮ ❣❡♥❡r❛t❡s r❛♥❞♦♠ ♥✉♠❜❡rs ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ✐♥ [ ❛ , ❜ ) 0.5 ✏❯♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞✑ ♠❡❛♥s t❤❛t ✐❢ ✇❡ ❣❡♥❡r❛t❡ ❛ ❧❛r❣❡ s❡t ♦❢ ♥✉♠❜❡rs✱ ♥♦ ♣❛rt ♦❢ [ ❛ , ❜ ) ❣❡ts ♠♦r❡ ♥✉♠❜❡rs t❤❛♥ ♦t❤❡rs 0 -0.5 -1 0 50 100 150 200 250 300 350 400 450

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