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Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ❚❡❝❤♥✐❝❛❧✲❡❝♦♥♦♠✐❝ ❡✈❛❧✉❛t✐♦♥ ♦❢ s♣❡❝tr✉♠ r❡❣✐♠❡s✿ st❛t✐❝✱ ❞②♥❛♠✐❝ ❛♥❞ ✉t♦♣✐❛♥ ❛❧❧♦❝❛t✐♦♥s ❢♦r ❇✸● ❈❉▼❆ ✇✐t❤ ❝♦❡①✐st✐♥❣ ❯❲❇ ❞❡✈✐❝❡s ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ■♥st✐t✉t ❢ür ◆❛❝❤r✐❝❤t❡♥t❡❝❤♥✐❦ ❯♥✐✈❡rs✐tät ❑❛r❧sr✉❤❡ ❑❛r❧sr✉❤❡✱ ●❡r♠❛♥② ■❊❊❊ P■▼❘❈ ✭❆t❤❡♥s✱ ●r❡❡❝❡✮✱ ✸✲✼ ❙❡♣t✱ ✷✵✵✼ ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮

Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ❲❡ t❤❛♥❦ t❤❡ ❊✉r♦♣❡❛♥ ❈♦♠♠✐ss✐♦♥ ❢♦r ✜♥❛♥❝✐❛❧ s✉♣♣♦rt t❤r♦✉❣❤ t❤❡ ♣r♦❥❡❝ts P❯▲❙❊❘❙✲■■✳ ❍♦✇❡✈❡r✱ t❤✐s ♠❛t❡r✐❛❧ s❤♦✉❧❞ ♥♦t ❜❡ ❝♦♥str✉❡❞ ❛s ♦✣❝✐❛❧ ♣♦s✐t✐♦♥ ♦❢ ❛♥② ♣r♦❥❡❝t ♦r ❛❣❡♥❝②✳ ✐♥st✐t✉t✐♦♥✲❧♦❣♦ ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮

Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ❖✉t❧✐♥❡ Pr❡❛♠❜❧❡ ✶ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ✷ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✸ ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ✹ ✐♥st✐t✉t✐♦♥✲❧♦❣♦ ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮

Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ✏P❛② ❛s ②♦✉ ❣♦✑ s♣❡❝tr✉♠ ❆t st❛rt ♦❢ ❛ ❞②♥❛♠✐❝ s♣❡❝tr✉♠ ❛❧❧♦❝❛t✐♦♥ ✭❉❙❆✮ ♣❡r✐♦❞✱ ❛ ✏s♣❡❝tr✉♠ ♠❛♥❛❣❡r✑ ❛✉❝t✐♦♥s ✭s❡❧❧s❄✮ s♣❡❝tr✉♠ ❧✐❝❡♥s❡s ◆❡t✇♦r❦s ❝♦♥s✐❞❡r t❤❡ ✐♥t❡r❡sts ♦❢ t❤❡✐r ❛❝t✐✈❡ ✉s❡rs ❛♥❞ ♣✉r❝❤❛s❡ ✭❜✐❞ ❢♦r✮ s♣❡❝tr✉♠ ▼❛♥❛❣❡r ✐ss✉❡s s❤♦rt✲t❡r♠ ❧✐❝❡♥s❡s t♦ ❡❛❝❤ ♥❡t✇♦r❦ ❆t t❤❡ ❡♥❞ ♦❢ ❛ s❤♦rt ♣❡r✐♦❞✱ ❛❧❧ ❧✐❝❡♥s❡s ❡①♣✐r❡ ❛♥❞ t❤❡ ✇❤♦❧❡ ♣r♦❝❡ss ✐s r❡✲✐♥✐t✐❛t❡❞ ✏❢r♦♠ s❝r❛t❝❤✑ ❆❜♦✈❡ ❝❛♥ ❜❡ ❞♦♥❡ ✏❝❡❧❧ ❜② ❝❡❧❧✑ ❛♠♦♥❣ ❈❉▼❆ ♥❡t✇♦r❦s ❜② ❡♠♣❧♦②✐♥❣ ✷✲❧❛②❡r s♣r❡❛❞✐♥❣ ❛s ✐♥ ❯▼❚❙ ❉♦✐♥❣ s♦ ✇❤❡♥ ♥♦♥✲❈❉▼❆ ♥❡t✇♦r❦s ❛r❡ ♣r❡s❡♥t ✐s ♠✉❝❤ tr✐❝❦✐❡r ❞✉❡ t♦ ✐♥t❡r❢❡r❡♥❝❡ ❝♦♥tr♦❧ ✐♥st✐t✉t✐♦♥✲❧♦❣♦ ▼❛♥❛❣❡r ❝❛♥ ❛r✐s❡ ❢r♦♠ ❛ ✏♣♦♦❧✐♥❣✑ ❜✉s✐♥❡ss ♠♦❞❡❧ ❙❡✈❡r❛❧ ♣✉❜❧✐❝❛t✐♦♥s ♦♥ t❤✐s ♠♦❞❡❧ ❛r❡ ❛✈❛✐❧❛❜❧❡ ✭❡✳❣✳✱ ❬✶✱ ✷❪✮ ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮

Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ❍♦✇ t❤❡ ♥❡t✇♦r❦ ❜✐❞s❄ ❙✉♣♣♦s❡ ♦♥❧② ♦♥❡ ❜❛♥❞ ✐s ❛✉❝t✐♦♥❡❞ ❆ss✉♠❡ ✷♥❞✲♣r✐❝❡ ✭❱✐❝❦r❡②✮ ❛✉❝t✐♦♥ ❍✐❣❤❡st ❜✐❞❞❡r ✇✐♥s✱ ❜✉t ♣❛②♠❡♥t ❡q✉❛❧s ❤✐❣❤❡st ❧♦s✐♥❣ ❜✐❞ ❖♣t✐♠❛❧ ❜✐❞ ❡q✉❛❧s t❤❡ ✏✈❛❧✉❡✑ ♦❢ t❤❡ ❜❛♥❞ ◆❡t✇♦r❦ r❡♣♦♥s❡ ❈♦♠♣✉t❡ r❡✈❡♥✉❡ t❤❡ ❜❛♥❞ ✇♦✉❧❞ ♣r♦❞✉❝❡ ✭✐❢ ✇♦♥✮ ❇✐❞ ❂ r❡✈❡♥✉❡ ■❢ ♠❛♥② ❜❛♥❞s ❛r❡ ❛✉❝t✐♦♥❡❞ t❤❡ ❛♥❛❧②s✐s ✐s ❛❧♠♦st t❤❡ s❛♠❡ ✭s❡❡ ♣❛♣❡r✮ ✐♥st✐t✉t✐♦♥✲❧♦❣♦ ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮

Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ❍♦✇ t♦ ❝♦♠♣✉t❡ r❡✈❡♥✉❡❄ ◆❡t✇♦r❦ s❡r✈❡s ❞❛t❛✲❞♦✇♥❧♦❛❞✐♥❣ t❡r♠✐♥❛❧s ❊❛❝❤ t❡r♠✐♥❛❧ ❤❛s ✸ ♣❛r❛♠❡t❡rs✿ ❞❛t❛ r❛t❡ ❘ ✐ ✱ ❝❤❛♥♥❡❧ ❣❛✐♥ ❤ ✐ ✱ ✏✇✐❧❧✐♥❣♥❡ss t♦ ♣❛②✑✱ β ✐ ❆ t❡r♠✐♥❛❧✬s ❜❡♥❡✜t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ β ✐ ❘ ✐ ( ▲ / ▼ ) ❢ ( ① ) ▲ ✐♥❢♦r♠❛t✐♦♥ ❜✐ts ✐♥ ▼ ✲❜✐t ♣❛❝❦❡t ❢ ( ① ) ✐s t❤❡ ♣❛❝❦❡t✲s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t②✱ ✇✐t❤ ① t❤❡ s✐❣♥❛❧✲t♦✲♥♦✐s❡ r❛t✐♦ ✭❙◆❘✮ ✭♥❡❣❧❡❝t ❞♦✇♥❧✐♥❦ ✐♥t❡r❢❡r❡♥❝❡✦✮ ◆❡t✇♦r❦ ❝❤❛r❣❡s t❡r♠✐♥❛❧ ♣❡r ✉♥✐t ❙◆❘ ❚❡r♠✐♥❛❧ ♠❛①✐♠✐s❡s ❜❡♥❡✜t ♠✐♥✉s ❝♦st✿ β ✐ ❘ ✐ ( ▲ / ▼ ) ❢ ( ① ) − ❝① ■❢ ♥❡t✇♦r❦ q✉♦t❡s ❛ ♣r✐❝❡ ❝ t❡r♠✐♥❛❧ ❜✉②s ❙◆❘ ① ( ❝ ) ✐♥st✐t✉t✐♦♥✲❧♦❣♦ ◆❡t✇♦r❦ ❝❤♦♦s❡s t❤❡ ❝ t❤❛t ♠❛①✐♠✐s❡s r❡✈❡♥✉❡ ✭ ❝ × ① ( ❝ ) ✮ ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮

Pr❡❛♠❜❧❡ ❋r❛♠❡✇♦r❦ ❢♦r s♣❡❝tr✉♠ r❡❣✐♠❡ ❡✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❈❧♦s✐♥❣ ❈♦♠♠❡♥ts ❖♣♣♦s✐♥❣ ✐♥t❡r❡sts ♠❡❡t c R x c * x xS’(x) S(x) ∝ f(x) T2 c 1 x T1 c 2 x c 1 x 1 c 2 x 2 x R x * ✐♥st✐t✉t✐♦♥✲❧♦❣♦ ❋✐❣✉r❡✿ ❚❡r♠✐♥❛❧ ♠❛①✐♠✐s❡s ❜❡♥❡✜t ♠✐♥✉s ❝♦st✿ ❙ ( ① ) − ❝① ✳ ◆❡t✇♦r❦ ❝❤♦♦s❡s ❝ = ❝ ∗ ❛♥❞ t❡r♠✐♥❛❧ ① = ① ∗ ✳ ❘❡✈❡♥✉❡✿ ❝ ∗ ① ∗ ∝ β ❘❢ ( ① ∗ ) ❱✐r❣✐❧✐♦ ❘❖❉❘■●❯❊❩ ❛♥❞ ❋r✐❡❞r✐❝❤ ❏❖◆❉❘❆▲ ❚❡❝❤✲❊❝♦♥♦♠✐❝ ❙♣❡❝tr✉♠ ❘❡❣✐♠❡ ❊✈❛❧✉❛t✐♦♥ ✭P■▼❘❈✬✵✼✮