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❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s

t❤❡ ●❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛ ▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ❲❡②❧✬s ❚✉❜❡ ❋♦r♠✉❧❛ ❛♥❞ ❍✐st♦r✐❝❛❧ ▼♦t✐✈❛t✐♦♥s ❚❤❡ ●❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛ ❊①❛♠♣❧❡s ❖♣❡♥ Pr♦❜❧❡♠s

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷ ✴ ✸✶

■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✸ ✴ ✸✶

■♥tr♦❞✉❝t✐♦♥

❉❡✜♥✐t✐♦♥

▲❡t ▼ ❜❡ ❛ t♦♣♦❧♦❣✐❝❛❧ ♠❛♥✐❢♦❧❞ ❛♥❞ ∀♣ ∈ ▼ ❞❡♥♦t❡ ❜② ❚♣▼ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ❛t ♣✳ ❚❤❡♥ ▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♠❛♣ ❣ s✉❝❤ t❤❛t ∀♣ ∈ ▼ ❣♣ : ❚♣▼ × ❚♣▼ − → R s✉❝❤ t❤❛t ❣♣ ✐s s②♠♠❡tr✐❝✱ ❣♣ ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ✐✳❡✳ ∀❳ ∈ ❚♣▼ ❣♣(❳, ❳) ≥ ✵ ❛♥❞ t❤❡ ❡q✉❛❧✐t② ❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❳ = ✵✱ ❣ ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ♦❢ ♣✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✹ ✴ ✸✶

■♥tr♦❞✉❝t♦♥

❉❡✜♥✐t✐♦♥

• ✐✈❡♥ (Ω, F, P) ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ❛♥❞ ▼ ❛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛

♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ ❢ : Ω × ▼ − → R ✐s s❛✐❞ t♦ ❜❡ ❛ r❡❛❧ r❛♥❞♦♠ ✜❡❧❞✳

❉❡✜♥✐t✐♦♥

❆ r❛♥❞♦♠ ✜❡❧❞ ❢ ✐s ❝❛❧❧❡❞ ●❛✉ss✐❛♥ ✐❢ ✐ts ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛r❡

• ❛✉ss✐❛♥✱ ✐✳ ❡✳ ✐❢ (❢t✶, . . . , ❢t♥) ✐s ●❛✉ss✐❛♥ ❞✐str✐❜✉t❡❞ ❢♦r ❡✈❡r② ♥ ∈ N ❛♥❞

(t✶, . . . , t♥) ∈ ▼♥✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✺ ✴ ✸✶

❆♣♣❧✐❝❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s

❇r❛✐♥ ♠❛♣♣✐♥❣ ❈♦s♠♦❧♦❣② ■♥ ❜♦t❤ ❝❛s❡s ✇❡ t❛❦❡ ▼ = ❙✷ ❛s ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❤✉♠❛♥✬s ❜r❛✐♥ ❛♥❞ ✉♥✐✈❡rs❡ r❡s♣❡❝t✐✈❡❧②✳ ❆♥❛❧②③❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❡①❝✉rs✐♦♥✬s s❡ts ❆✉ ❢ ▼ t ▼ ❢t ✉ ❢

✶ ✉

▼

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✻ ✴ ✸✶

❆♣♣❧✐❝❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s

❇r❛✐♥ ♠❛♣♣✐♥❣ ❈♦s♠♦❧♦❣② ■♥ ❜♦t❤ ❝❛s❡s ✇❡ t❛❦❡ ▼ = ❙✷ ❛s ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❤✉♠❛♥✬s ❜r❛✐♥ ❛♥❞ ✉♥✐✈❡rs❡ r❡s♣❡❝t✐✈❡❧②✳ ❆♥❛❧②③❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❡①❝✉rs✐♦♥✬s s❡ts ❆✉ (❢ , ▼) = {t ∈ ▼ : ❢t ≥ ✉} = ❢ −✶ [✉, +∞) ⊂ ▼

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✻ ✴ ✸✶

❲❡②❧✬s ❚✉❜❡ ❋♦r♠✉❧❛ ❛♥❞ ❍✐st♦r✐❝❛❧ ▼♦t✐✈❛t✐♦♥s

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✼ ✴ ✸✶

▼♦t✐✈❛t✐♦♥

❊✉❧❡r✲P♦✐♥❝❛ré ❈❤❛r❛❝t❡r✐st✐❝

■s t❤❡ ✉♥✐q✉❡ ❢✉♥❝t✐♦♥❛❧ t❤❛t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

✶

ϕ(▼) =

• ✵

✐❢ ▼ = ∅ ✶ ✐❢ ▼ = ∅ ❛♥❞ ❝♦♥✈❡①

✷

ϕ (▼✶ ∪ ▼✷) = ϕ (▼✶) + ϕ (▼✷) − ϕ (▼✶ ∩ ▼✷)

■♥ ❛ ♣✉r❡❧② ❛❧❣❡❜r❛✐❝ ✇❛② ✐t ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ▼

▼ ❥ ✵

✶ ❥

❥

✇❤❡r❡ ❍✵ ❍

▼ ❛r❡ t❤❡ ❤♦♠♦❧♦❣② ❣r♦✉♣s ♦❢ ▼ ❛♥❞ t❤❡

❞✐♠❡♥s✐♦♥ ♦❢ ❍❦ ✐s ❇❡tt✐✬s ♥✉♠❜❡r

❦✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✽ ✴ ✸✶

▼♦t✐✈❛t✐♦♥

❊✉❧❡r✲P♦✐♥❝❛ré ❈❤❛r❛❝t❡r✐st✐❝

■s t❤❡ ✉♥✐q✉❡ ❢✉♥❝t✐♦♥❛❧ t❤❛t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

✶

ϕ(▼) =

• ✵

✐❢ ▼ = ∅ ✶ ✐❢ ▼ = ∅ ❛♥❞ ❝♦♥✈❡①

✷

ϕ (▼✶ ∪ ▼✷) = ϕ (▼✶) + ϕ (▼✷) − ϕ (▼✶ ∩ ▼✷)

■♥ ❛ ♣✉r❡❧② ❛❧❣❡❜r❛✐❝ ✇❛② ✐t ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ϕ(▼) =

dim ▼

• ❥=✵

(−✶)❥ β❥ ✇❤❡r❡ ❍✵, . . . , ❍dim ▼ ❛r❡ t❤❡ ❤♦♠♦❧♦❣② ❣r♦✉♣s ♦❢ ▼ ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ ❍❦ ✐s ❇❡tt✐✬s ♥✉♠❜❡r β❦✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✽ ✴ ✸✶

❋r♦♠ ❛ ❣❡♦♠❡tr✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐❢ ▼ ✐s ❛♥ ♦r✐❡♥t❛❜❧❡ s✉r❢❛❝❡ ✇✐t❤♦✉t ❜♦✉♥❞❛r② ❡♠❜❡❞❞❡❞ ✐♥ R✸✱ t❤❡♥ ϕ(▼) = ✶ ✷π

• ▼

❑(①)❞① ✇❤❡r❡ ❑ (①) ❞❡♥♦t❡s t❤❡ ●❛✉ss✐❛♥ ❝✉r✈❛t✉r❡ ❛t t❤❡ ♣♦✐♥t ① ∈ ▼✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✾ ✴ ✸✶

❲❡②❧✬s ❚✉❜❡ ❋♦r♠✉❧❛

▲❡t (▼, ❞, µ) ❜❡ ❛ ♠❡❛s✉r❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❆ ⊂ ▼ ❛ s✉❜s❡t✳ ❚❤❡♥ t❤❡ t✉❜❡ ♦❢ r❛❞✐✉s ǫ ❛r♦✉♥❞ ❆ ✐s t❤❡ s✉❜s❡t ♦❢ ▼ ❞❡✜♥❡❞ ❜② ❚✉❜❡(❆, ǫ) ˙ = {① ∈ ▼ : ❞(①, ❆) ≤ ǫ}

❲❡②❧✬s ❚✉❜❡ ❋♦r♠✉❧❛

▲❡t ❆

♥ ❜❡ ❛ ❝♦♥✈❡① s❡t✳ ❚❤❡♥✱ ❢♦r

❡♥♦✉❣❤ s♠❛❧❧✱ ❚✉❜❡ ❆

♥ ✐ ✵ ♥ ✐ ♥ ✐ ✐ ❆

✇❤❡r❡ ✐s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✱

✵ ❆ ✐s t❤❡ ❊P❈ ♦❢ ❆ ❛♥❞ ❦ t❤❡

▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ✉♥✐t❛r② ❜❛❧❧ ✐♥

❦✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ✐ ❆ ❛r❡

❝❛❧❧❡❞ ▲✐♣s❤✐t③ ❑✐❧❧✐♥❣ ❝✉r✈❛t✉r❡s ♦❢ ❆✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✵ ✴ ✸✶

❲❡②❧✬s ❚✉❜❡ ❋♦r♠✉❧❛

▲❡t (▼, ❞, µ) ❜❡ ❛ ♠❡❛s✉r❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❆ ⊂ ▼ ❛ s✉❜s❡t✳ ❚❤❡♥ t❤❡ t✉❜❡ ♦❢ r❛❞✐✉s ǫ ❛r♦✉♥❞ ❆ ✐s t❤❡ s✉❜s❡t ♦❢ ▼ ❞❡✜♥❡❞ ❜② ❚✉❜❡(❆, ǫ) ˙ = {① ∈ ▼ : ❞(①, ❆) ≤ ǫ}

❲❡②❧✬s ❚✉❜❡ ❋♦r♠✉❧❛

▲❡t ❆ ⊂ R♥ ❜❡ ❛ ❝♦♥✈❡① s❡t✳ ❚❤❡♥✱ ❢♦r ǫ ❡♥♦✉❣❤ s♠❛❧❧✱ λ(❚✉❜❡(❆, ǫ)) =

♥

• ✐=✵

ǫ♥−✐µ♥−✐L✐(❆) ✇❤❡r❡ λ ✐s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✱ L✵(❆) ✐s t❤❡ ❊P❈ ♦❢ ❆ ❛♥❞ µ❦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ✉♥✐t❛r② ❜❛❧❧ ✐♥ R❦✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts L✐(❆) ❛r❡ ❝❛❧❧❡❞ ▲✐♣s❤✐t③ ❑✐❧❧✐♥❣ ❝✉r✈❛t✉r❡s ♦❢ ❆✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✵ ✴ ✸✶

❊①❛♠♣❧❡

▲❡t ▼ ❜❡ t❤❡ ♣❧❛♥❡ ❛♥❞ ❆ ❛ tr✐❛♥❣❧❡✳ ■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ✷−❞✐♠❡♥s✐♦♥❛❧ ✈♦❧✉♠❡ ♦❢ t❤❡ t✉❜❡✱ ✐✳ ❡✳ ✐ts ❛r❡❛✱ ✇❡ ♥❡❡❞ t♦ s✉♠ ✉♣✿ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ tr✐❛♥❣❧❡✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ t❤r❡❡ r❡❝t❛♥❣❧❡s✳ ❚❤❡ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝✉❧❛r s❡❝t♦rs✱ ✇❤♦s❡ ✉♥✐♦♥ ✐s ❛ ❞✐s❦ ♦❢ r❛❞✐✉s ǫ ❛♥❞ ❊P❈✭❝✐r❝❧❡✮ ❂ ✶ ✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✶ ✴ ✸✶

❍✐st♦r✐❝❛❧ ▼♦t✐✈❛t✐♦♥s

❘✐❝❡✬s ❋♦r♠✉❧❛

▲❡t ❢ ❜❡ ❛ r❛♥❞♦♠ ✜❡❧❞ ❞❡✜♥❡❞ ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ [✵, ❚] ❛♥❞ ✉ ∈ R ❛ ✜①❡❞ ♥✉♠❜❡r✳ ❚❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ✉♣❝r♦ss✐♥❣ ✐s ❞❡✜♥❡❞ ❛s ◆+

✉ (✵, ❚) ˙

= #

• t ∈ [✵, ❚] : ❢ (t) = ✉, ❢ ′(t) > ✵
• ■❢ ♣t ❞❡♥♦t❡s t❤❡ ❧❛✇ ♦❢ t❤❡ ✈❡❝t♦r (❢t, ❢ ′

t ) ✇❡ ❤❛✈❡

E

• ◆+

✉ (✵, ❚)

• =

❚

✵

❞t +∞

✵

② ♣t(✉, ②)❞② ◆♦t✐❝✐♥❣ t❤❛t ϕ

• ❢ −✶[✉, +∞)
• ≡ ✶❢✵≥✉ + ◆+

✉ (✵, ❚),

✇❡ ❣❡t E

• ϕ
• ❢ −✶[✉, +∞)
• = P (❢✵ ≥ ✉) + E
• ◆+

✉ (✵, ❚)

• ▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐

❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✷ ✴ ✸✶

• ❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✸ ✴ ✸✶

• ❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛
• ❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛

E [L❥ (❆✉(❢ , ▼))] =

dim ▼−❥

• ❧=✵

❥ + ❧ ❧

• ρ❧(✉) L❥+❧(▼)

❆s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡✱ ✇❤❡♥ ❥ ✵✱ ✇❡ ❤❛✈❡✿ ❆✉ ❢ ▼

▼ ❥ ✵ ❥ ✉ ❥ ▼

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✹ ✴ ✸✶

• ❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛
• ❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛

E [L❥ (❆✉(❢ , ▼))] =

dim ▼−❥

• ❧=✵

❥ + ❧ ❧

• ρ❧(✉) L❥+❧(▼)

❆s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡✱ ✇❤❡♥ ❥ = ✵✱ ✇❡ ❤❛✈❡✿ E[ϕ (❆✉(❢ , ▼))] =

dim ▼

• ❥=✵

ρ❥(✉)L❥(▼)

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✹ ✴ ✸✶

❈♦❡✣❝✐❡♥ts ρ❥(✉)

❋♦r ❡✈❡r② ❥ ≥ ✵ ✇❡ ❤❛✈❡ ρ❥(✉) = (✷π)− ❥+✶

✷ ❍❥−✶(✉)❡− ✉✷ ✷

✇❤❡r❡ ❍❦ ✐s t❤❡ ❦✲t❤ ❍❡r♠✐t❡✬s ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❍−✶(①) ˙ = √ ✷π Ψ(①)❡

①✷ ✷

✇❤❡r❡ Ψ(①) = +∞

①

✶ √ ✷π ❡− t✷

✷ ❞t ❞❡♥♦t❡s t❤❡ t❛✐❧ ❢✉♥❝t✐♦♥ ♦❢ ❛ st❛♥❞❛r❞

• ❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✺ ✴ ✸✶

▲✐♣s❝❤✐t③ ✲ ❑✐❧❧✐♥❣ ❈✉r✈❛t✉r❡s

❉❡✜♥✐t✐♦♥

• ✐✈❡♥ ❛♥ ♥ ❞✐♠❡♥s✐♦♥❛❧ ♦r✐❡♥t❛❜❧❡ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ (▼, ❣)✱ t❤❡ ✈❛❧✉❡s

L❣

❥ (▼) =

  

(−✷π)− ♥−❥

✷

( ♥−❥

✷ )!

• ▼ ❚r▼

❘

♥−❥ ✷

• ❞❱♦❧❣

✐❢ ♥ − ❥ ✐s ❡✈❡♥ ✵ ✐❢ ♥ − ❥ ✐s ♦❞❞ ❛r❡ ❝❛❧❧❡❞ ▲✐♣s❝❤✐t③ ✲ ❑✐❧❧❧✐♥❣ ❝✉r✈❛t✉r❡s ♦❢ ▼✳ ❋♦r ❡✈❡r② t ∈ ▼✱ ❘ ❞❡♥♦t❡s t❤❡ ❝✉r✈❛t✉r❡ t❡♥s♦r ❛♥❞ ❱♦❧❣ t❤❡ ✈♦❧✉♠❡ ❢♦r♠✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✻ ✴ ✸✶

▼❡tr✐❝ ✐♥❞✉❝❡❞ ❜② t❤❡ ✜❡❧❞

❉❡✜♥✐t✐♦♥

• ✐✈❡♥ ❛ r❛♥❞♦♠ ✜❡❧❞ ❢ ♦♥ ❛ ♠❛♥✐❢♦❧❞ ▼✱ ✇❡ ❞❡✜♥❡✿

❣t(❳t, ❨t) ˙ = E[(❳t❢ ) · (❨t❢ )] ✇❤❡r❡ ❳t, ❨t ∈ ❚t▼✱ t❤❡ t❛♥❣❡♥t s♣❛❝❡ t♦ ▼ ❛t t✳ ❲❡ ❝❛♥ ❛❧s♦ ✇r✐t❡ ❣ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✜❡❧❞✱ ♠♦r❡ ♣r❡❝✐s❡❧② ❣t(❳t, ❨t) = ❨s❳t❈(s, t)|s=t ✇❡ r❡❢❡r t♦ ❣ ❛s t❤❡ ♠❡tr✐❝ ✐♥❞✉❝❡❞ ♦♥ ▼ ❜② ❢ ✳

❘❡♠❛r❦

❣ ✐s ❛ ♠❡tr✐❝ ♦♥ ▼✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✼ ✴ ✸✶

• ❛✉ss✐❛♥ ❑✐♥❡♠❛t✐❝ ❋♦r♠✉❧❛

E [L❦ (❆✉(❢ , ▼))] =

dim ▼−❦

• ❥=✵

❦ + ❥ ❥

• ρ❥(✉) L❢

❦+❥(▼)

■❢ ❦ = ✵✿ E[ϕ (❆✉(❢ , ▼))] =

dim ▼

• ❥=✵

ρ❥(✉)L❢

❥ (▼)

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✽ ✴ ✸✶

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

Pr♦✈❡ ❝❛s❡ ❦ = ✵✱ ✐✳❡✳ t❤❡ ❊P❈ ❝❛s❡✳ ❯s❡ ▼♦rs❡✬s t❤❡♦r❡♠ ❛♥❞ ✭❧♦t ♦❢✮ ❝♦♠♣✉t❛t✐♦♥s✳ ❚♦ ♣r♦✈❡ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✉s❡ ❛ st♦❝❤❛st✐❝ ✈❡rs✐♦♥ ♦❢ t❤❡ ❈r♦❢t♦♥✬s ❢♦r♠✉❧❛✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✶✾ ✴ ✸✶

❈❧❛ss✐❝ ❈r♦❢t♦♥✬s ❋♦r♠✉❧❛

❚❤❡♦r❡♠

❙✉♣♣♦s❡ ▼ ⊂ R♥ ✐s ❛ ❝♦♠♣❛❝t s♣❛❝❡✳ ❚❤❡♥ L❦(▼) =

• r❛✛ (♥,♥−❦)

L✵ (▼ ∩ ❱ ) ❞λ♥

♥−❦

✇❤❡r❡✿

• r❛✛ (♥, ❥) ∼

= ●r♥(❱ , ❥) × R♥ ✐s t❤❡ ❛✣♥❡ ❣r❛ss♠❛♥✐❛♥ λ♥

❥ = ν♥ ❥ × ▲❡❜♥ ✐s t❤❡ ♣r♦❞✉❝t ♠❡❛s✉r❡ ♦♥ ●r❛✛ (♥, ❥) ❛♥❞ ν♥ ❥ t❤❡

♠❡❛s✉r❡ ♦♥ ●r♥(❱ , ❥) ✇✐t❤ t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ ν♥

❥ (●r❛✛ (♥, ❥)) =

♥

❥

• ▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐

❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✵ ✴ ✸✶

❊①❛♠♣❧❡s

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✶ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❉❡✜♥✐t✐♦♥✴❚❤❡♦r❡♠

❊✈❡r② r❛♥❞♦♠ ✜❡❧❞ ♦♥ t❤❡ s♣❤❡r❡ ❚ : ❙✷ × Ω → R ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❚(①, ω) =

+∞

• ℓ=✵

ℓ

• ♠=−ℓ

❛ℓ♠(ω)❨ℓ♠(①) ✇❤❡r❡ t❤❡ r❛♥❞♦♠ ❝♦❡✣❝✐❡♥ts s❛t✐s❢② ❛ ♠❛ ♠

♠ ♠

✶ ♠ ❛♥❞ ❨ ♠ ❙✷ ✱ ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✱ ❛r❡ ❞❡✜♥❡❞ ❜② ❨ ♠ ✷ ✶ ✹ ♠ ♠ P ♠ ❡✐♠ ✇❤❡r❡ P ♠ ❛r❡ t❤❡ ❛ss♦❝✐❛t❡❞ ▲❡❣❡♥❞r❡ ❢✉♥❝t✐♦♥s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✷ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❉❡✜♥✐t✐♦♥✴❚❤❡♦r❡♠

❊✈❡r② r❛♥❞♦♠ ✜❡❧❞ ♦♥ t❤❡ s♣❤❡r❡ ❚ : ❙✷ × Ω → R ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❚(①, ω) =

+∞

• ℓ=✵

ℓ

• ♠=−ℓ

❛ℓ♠(ω)❨ℓ♠(①) ✇❤❡r❡ t❤❡ r❛♥❞♦♠ ❝♦❡✣❝✐❡♥ts s❛t✐s❢② E [❛ℓ♠¯ ❛ℓ′♠′] = Cℓδℓ′

ℓ δ♠′ ♠

ℓ ≥ ✶, ♠ = −ℓ, . . . , ℓ ❛♥❞ ❨ ♠ ❙✷ ✱ ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✱ ❛r❡ ❞❡✜♥❡❞ ❜② ❨ ♠ ✷ ✶ ✹ ♠ ♠ P ♠ ❡✐♠ ✇❤❡r❡ P ♠ ❛r❡ t❤❡ ❛ss♦❝✐❛t❡❞ ▲❡❣❡♥❞r❡ ❢✉♥❝t✐♦♥s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✷ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❉❡✜♥✐t✐♦♥✴❚❤❡♦r❡♠

❊✈❡r② r❛♥❞♦♠ ✜❡❧❞ ♦♥ t❤❡ s♣❤❡r❡ ❚ : ❙✷ × Ω → R ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❚(①, ω) =

+∞

• ℓ=✵

ℓ

• ♠=−ℓ

❛ℓ♠(ω)❨ℓ♠(①) ✇❤❡r❡ t❤❡ r❛♥❞♦♠ ❝♦❡✣❝✐❡♥ts s❛t✐s❢② E [❛ℓ♠¯ ❛ℓ′♠′] = Cℓδℓ′

ℓ δ♠′ ♠

ℓ ≥ ✶, ♠ = −ℓ, . . . , ℓ ❛♥❞ ❨ℓ♠ : ❙✷ → C✱ ❝❛❧❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✱ ❛r❡ ❞❡✜♥❡❞ ❜② ❨ℓ♠(θ, φ) ˙ =

• ✷ℓ + ✶

✹π (ℓ − ♠)! (ℓ + ♠)!Pℓ♠(cos θ)❡✐♠φ ✇❤❡r❡ Pℓ♠ ❛r❡ t❤❡ ❛ss♦❝✐❛t❡❞ ▲❡❣❡♥❞r❡ ❢✉♥❝t✐♦♥s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✷ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❚❤❡♦r❡♠

❚❤❡ ✐♥❞✉❝❡❞ ♠❡tr✐❝ ♦♥ ❙✷ ❜② t❤❡ ✜❡❧❞ ❚ = +∞

ℓ=✵

ℓ

♠=−ℓ ❛ℓ♠❨ℓ♠ ✐s

❞s✷ = r✷ sin✷ θ ❞φ✷ + ❞θ✷ ✇❤❡r❡ r✷ =

+∞

• ℓ=✶

✷ℓ + ✶ ✹π · ℓ(ℓ + ✶) ✷ ❱❛r [ ❛ℓ♠ ] ❲❤② ✐s t❤✐s ❛♠❛③✐♥❣❄ ❇❡❝❛✉s❡ ✇❡ ❤❛✈❡ ❢♦✉♥❞ t❤❡ ❊✉❝❧✐❞✐❛♥ ♠❡tr✐❝ ♦❢ ❙✷

r ✦

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✸ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❚❤❡♦r❡♠

❚❤❡ ✐♥❞✉❝❡❞ ♠❡tr✐❝ ♦♥ ❙✷ ❜② t❤❡ ✜❡❧❞ ❚ = +∞

ℓ=✵

ℓ

♠=−ℓ ❛ℓ♠❨ℓ♠ ✐s

❞s✷ = r✷ sin✷ θ ❞φ✷ + ❞θ✷ ✇❤❡r❡ r✷ =

+∞

• ℓ=✶

✷ℓ + ✶ ✹π · ℓ(ℓ + ✶) ✷ ❱❛r [ ❛ℓ♠ ] ❲❤② ✐s t❤✐s ❛♠❛③✐♥❣❄ ❇❡❝❛✉s❡ ✇❡ ❤❛✈❡ ❢♦✉♥❞ t❤❡ ❊✉❝❧✐❞✐❛♥ ♠❡tr✐❝ ♦❢ ❙✷

r ✦

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✸ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

❲❡ ❛r❡ ❣✐✈❡♥ t❤❡ ✐❞❡♥t✐t② ❋(θ) ˙ =

♥

• ♠=−♥

|❨♥♠|✷ =

♥

• ♠=−♥

❝✷

♥♠P✷ ♥♠(cos θ) = ✷♥ + ✶

✹π ❛♥❞ t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚♥ = ♥

ℓ=✵

❧

♠=−ℓ ❛ℓ♠❨ℓ♠✳

❲❡ ✇♦r❦ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ♥✿ ♥ ✶ tr✐✈✐❛❧❀ ♥ ✷ ❚❤❡ st❛t❡♠❡♥t ✐s ❡q✉✐✈❛❧❡♥t t♦ ♣r♦✈❡ t❤✐s r❡❧❛t✐♦♥ ❛❜♦✉t t❤❡ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✿

♥ ♠ ♥

❨♥♠

✷ ♠✷

✷♥ ✶ ✹ ♥ ♥ ✶ ✷

✷

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✹ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

❲❡ ❛r❡ ❣✐✈❡♥ t❤❡ ✐❞❡♥t✐t② ❋(θ) ˙ =

♥

• ♠=−♥

|❨♥♠|✷ =

♥

• ♠=−♥

❝✷

♥♠P✷ ♥♠(cos θ) = ✷♥ + ✶

✹π ❛♥❞ t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚♥ = ♥

ℓ=✵

❧

♠=−ℓ ❛ℓ♠❨ℓ♠✳

❲❡ ✇♦r❦ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ♥✿ ♥ = ✶ tr✐✈✐❛❧❀ ♥ ≥ ✷ ❚❤❡ st❛t❡♠❡♥t ✐s ❡q✉✐✈❛❧❡♥t t♦ ♣r♦✈❡ t❤✐s r❡❧❛t✐♦♥ ❛❜♦✉t t❤❡ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✿

♥ ♠ ♥

❨♥♠

✷ ♠✷

✷♥ ✶ ✹ ♥ ♥ ✶ ✷

✷

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✹ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢

❲❡ ❛r❡ ❣✐✈❡♥ t❤❡ ✐❞❡♥t✐t② ❋(θ) ˙ =

♥

• ♠=−♥

|❨♥♠|✷ =

♥

• ♠=−♥

❝✷

♥♠P✷ ♥♠(cos θ) = ✷♥ + ✶

✹π ❛♥❞ t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚♥ = ♥

ℓ=✵

❧

♠=−ℓ ❛ℓ♠❨ℓ♠✳

❲❡ ✇♦r❦ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ♥✿ ♥ = ✶ tr✐✈✐❛❧❀ ♥ ≥ ✷ ❚❤❡ st❛t❡♠❡♥t ✐s ❡q✉✐✈❛❧❡♥t t♦ ♣r♦✈❡ t❤✐s r❡❧❛t✐♦♥ ❛❜♦✉t t❤❡ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✿

♥

• ♠=−♥

|❨♥♠ (θ, φ)|✷ ♠✷ = ✷♥ + ✶ ✹π · ♥(♥ + ✶) ✷ sin✷ θ

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✹ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❚❤❡♦r❡♠

• ✐✈❡♥ t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚ = +∞

ℓ=✵

ℓ

♠=−ℓ ❛ℓ♠❨ℓ♠ = +∞ ℓ=✵ ❚ℓ ♦♥ ❙✷✱ ✇❡

❤❛✈❡ t❤❛t ❆r❡❛ ❆✉ ❚ ❙✷ ✹ ✉ ❆✉ ❚ ❙✷ ✷ ✉ ✶ ✷

✸

✉ ❡

✉✷ ✷ ✷ ✹

r✷

✷

✇❤❡r❡ r✷

✶

✷ ✶ ✹ ✶ ✷

✷ ✵

✷ ✶ ✹

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✺ ✴ ✸✶

❚✇♦ ❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡ ❙✷

❚❤❡♦r❡♠

• ✐✈❡♥ t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚ = +∞

ℓ=✵

ℓ

♠=−ℓ ❛ℓ♠❨ℓ♠ = +∞ ℓ=✵ ❚ℓ ♦♥ ❙✷✱ ✇❡

❤❛✈❡ t❤❛t E

• ❆r❡❛
• ❆✉(❚, ❙✷)
• = ✹π Ψ

✉ σ

• E
• ϕ
• ❆✉
• ❚, ❙✷

= ✷Ψ ✉ σ

• +

✶

• (✷π)✸

✉ σ❡− ✉✷

✷σ✷ ✹π r✷

σ✷ ✇❤❡r❡ r✷ =

+∞

• ℓ=✶

✷ℓ + ✶ ✹π · ℓ(ℓ + ✶) ✷ Cℓ σ✷ =

+∞

• ℓ=✵

✷ℓ + ✶ ✹π Cℓ

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✺ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚ℓ ♦♥ ❙♥✱ ❣✐✈❡♥ ❜② t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❊ [❚ℓ(①)❚ℓ(②)] = ●ℓ,♥ (cos ❞(①, ②)) , ①, ② ∈ ❙♥ ✇❤❡r❡ ❞ ① ② ✐s t❤❡ s♣❤❡r✐❝❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ① ② ❙♥✳

• ♥

✶ ✶ ✐s t❤❡ ❧✬ ✲t❤ ●❡❣❡♥❜❛✉❡r✬s ♣♦❧②♥♦♠✐❛❧✱ ♥♦r♠❛❧✐③❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ● ♥ ✶ ✶✳

• ♥

P

♥ ✷

✶ ♥

✷

✶

❏❛❝♦❜✐✬s ♣♦❧②♥♦♠✐❛❧✳

• ✷

P ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✻ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚ℓ ♦♥ ❙♥✱ ❣✐✈❡♥ ❜② t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❊ [❚ℓ(①)❚ℓ(②)] = ●ℓ,♥ (cos ❞(①, ②)) , ①, ② ∈ ❙♥ ✇❤❡r❡ ❞(①, ②) ✐s t❤❡ s♣❤❡r✐❝❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ①, ② ∈ ❙♥✳

• ℓ,♥ : [−✶, ✶] −

→ R ✐s t❤❡ ❧✬ℓ✲t❤ ●❡❣❡♥❜❛✉❡r✬s ♣♦❧②♥♦♠✐❛❧✱ ♥♦r♠❛❧✐③❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ●ℓ,♥(✶) = ✶✳

• ♥

P

♥ ✷

✶ ♥

✷

✶

❏❛❝♦❜✐✬s ♣♦❧②♥♦♠✐❛❧✳

• ✷

P ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✻ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚ℓ ♦♥ ❙♥✱ ❣✐✈❡♥ ❜② t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❊ [❚ℓ(①)❚ℓ(②)] = ●ℓ,♥ (cos ❞(①, ②)) , ①, ② ∈ ❙♥ ✇❤❡r❡ ❞(①, ②) ✐s t❤❡ s♣❤❡r✐❝❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ①, ② ∈ ❙♥✳

• ℓ,♥ : [−✶, ✶] −

→ R ✐s t❤❡ ❧✬ℓ✲t❤ ●❡❣❡♥❜❛✉❡r✬s ♣♦❧②♥♦♠✐❛❧✱ ♥♦r♠❛❧✐③❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ●ℓ,♥(✶) = ✶✳

• ℓ,♥ = P

( ♥

✷ −✶, ♥ ✷ −✶)

ℓ

← → ❏❛❝♦❜✐✬s ♣♦❧②♥♦♠✐❛❧✳

• ✷

P ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✻ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✜❡❧❞ ❚ℓ ♦♥ ❙♥✱ ❣✐✈❡♥ ❜② t❤❡ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❊ [❚ℓ(①)❚ℓ(②)] = ●ℓ,♥ (cos ❞(①, ②)) , ①, ② ∈ ❙♥ ✇❤❡r❡ ❞(①, ②) ✐s t❤❡ s♣❤❡r✐❝❛❧ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ①, ② ∈ ❙♥✳

• ℓ,♥ : [−✶, ✶] −

→ R ✐s t❤❡ ❧✬ℓ✲t❤ ●❡❣❡♥❜❛✉❡r✬s ♣♦❧②♥♦♠✐❛❧✱ ♥♦r♠❛❧✐③❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ●ℓ,♥(✶) = ✶✳

• ℓ,♥ = P

( ♥

✷ −✶, ♥ ✷ −✶)

ℓ

← → ❏❛❝♦❜✐✬s ♣♦❧②♥♦♠✐❛❧✳

• ℓ,✷ = Pℓ ←

→ ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✻ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

■♥❞✉❝❡❞ ♠❡tr✐❝ ❜② ❚ℓ ♦♥ ❙♥

❞s✷ = r✷

♥−✶

• ❥=✶

sin✷ θ❥ ❞φ✷ + r✷

♥−✷

• ❦=✶

 

♥−✶

• ❥=❦+✶

sin✷ θ❥   ❞θ✷

❥ + r✷ ❞θ✷ ♥−✶

✇❤❡r❡ r✷ = ●

′

ℓ,♥(✶) = ℓ

♥ (ℓ + ♥ − ✶)

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✼ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

❚❤❡♦r❡♠

▲❡t ❙♥ t❤❡ ♥✲❞✐♠❡♥s✐♦♥❛❧ s♣❤❡r❡✱ t❤❡♥ E [λ (❆✉ (❚ℓ, ❙♥))] = Ψ(✉) ω♥ E

• ϕ
• ❆✉
• ❚ℓ, ❙✷♠

=

♠

• ❦=✵

ρ✷❦(✉) ✷ ♠! (♠ − ❦)! (✷❦)! (✹π)❦ r✷❦ E

• ϕ
• ❆✉
• ❚ℓ, ❙✷♠−✶

=

♠

• ❦=✶

ρ✷❦−✶(✉) ✷π❦ (✷♠ − ✶)! (✷❦ − ✶)! (♠ − ✶)! r✷❦−✶ ✹♠−❦ (♠ − ❦)! ✇❤❡r❡ ϕ ✐s t❤❡ ❊P❈✱ λ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✱ r✷ =

ℓ ♥ (ℓ + ♥ − ✶)✱ Ψ t❤❡

t❛✐❧ ❢✉♥❝t✐♦♥ ♦❢ ❛ st❛♥❞❛r❞ ❣❛✉ss✐❛♥ ❛♥❞ ω♥ t❤❡ s✉♣❡r✜❝✐❛❧ ✈♦❧✉♠❡ ♦❢ ❙♥✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✽ ✴ ✸✶

• ❡♥❡r❛❧ s♣❤❡r❡ ❙♥

❈❛s❡ ✉❂✵

E [λ (❆✵ (❚ℓ, ❙♥))] = ✶ ✷ ω♥ E

• ϕ
• ❆✵
• ❚ℓ, ❙✷♠

= ✶ E

• ϕ
• ❆✵
• ❚ℓ, ❙✷♠−✶

= (✷♠ − ✶)! ((♠ − ✶)!)✷ ✶ ✹♠−✶ ■♠−✶

• ′

ℓ,✷♠−✶(✶)

• ✇❤❡r❡ ■t(r) =

r

✵ (✶ − ✇✷)t ❞✇✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✷✾ ✴ ✸✶

❖♣❡♥ Pr♦❜❧❡♠s

✶ ❲❤❛t ❛❜♦✉t ❱❛r [L❥ (❆✉ (❢ , ▼))]❄ ✷ ❈▲❚ ❢♦r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s

❥ ❆✉ ❢ ▼

❀

✸ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ✭♦❢

♥♦♥ ❝♦♠♣❛❝t✲t②♣❡✮✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✸✵ ✴ ✸✶

❖♣❡♥ Pr♦❜❧❡♠s

✶ ❲❤❛t ❛❜♦✉t ❱❛r [L❥ (❆✉ (❢ , ▼))]❄ ✷ ❈▲❚ ❢♦r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s L❥ (❆✉ (❢ , ▼))❀ ✸ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ✭♦❢

♥♦♥ ❝♦♠♣❛❝t✲t②♣❡✮✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✸✵ ✴ ✸✶

❖♣❡♥ Pr♦❜❧❡♠s

✶ ❲❤❛t ❛❜♦✉t ❱❛r [L❥ (❆✉ (❢ , ▼))]❄ ✷ ❈▲❚ ❢♦r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s L❥ (❆✉ (❢ , ▼))❀ ✸ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ♦♥ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s ✭♦❢

♥♦♥ ❝♦♠♣❛❝t✲t②♣❡✮✳

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✸✵ ✴ ✸✶

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

▼❛r❝♦ ❈❛r❢❛❣♥✐♥✐ ❘❛♥❞♦♠ ❋✐❡❧❞s ♦♥ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s ❯♥✐✈❡rs✐t② ♦❢ ❈♦♥♥❡❝t✐❝✉t✱ ❖❝t♦❜❡r ✶✷t❤ ✷✵✶✽ ✸✶ ✴ ✸✶