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❚♦♣♦❧♦❣② ♦❢ st❛t✐st✐❝❛❧ s②st❡♠s

❆ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ❏✉❛♥ P❛❜❧♦ ❱✐❣♥❡❛✉① ❆r✐③tí❛ ■▼❏✲P❘● ❯♥✐✈❡rs✐té P❛r✐s ❉✐❞❡r♦t P❛r✐s✱ ❏✉♥❡ ✶✹✱ ✷✵✶✾

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸ ✴ ✹✺

❙❤❛♥♥♦♥ ❡♥tr♦♣②

❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ ❢✉♥❝t✐♦♥s S✶ := S(n)

✶

: ∆n → R✱ ❣✐✈❡♥ ❜② S✶(p✵,...,pn) := −

n

• k=✵

pi lnpi, ✭✶✮ ❛s t❤❡ ♦♥❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✱ s✉❝❤ t❤❛t S✶(✶/n,...,✶/n) ✐s ♠♦♥♦t♦♥✐❝ ✐♥ n ❛♥❞ ✏■❢ ❛ ❝❤♦✐❝❡ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ✐♥t♦ t✇♦ s✉❝❝❡ss✐✈❡ ❝❤♦✐❝❡s✱ t❤❡ ♦r✐❣✐♥❛❧

✶

s❤♦✉❧❞ ❜❡ t❤❡ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✈❛❧✉❡s ♦❢

✶✳✑ ✶ ✷ ✸ ✶ ✷

✶ ✶ ✷ ✷ ✶ ✷

✸

✶

✶ ✶ ✷ ✸ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹ ✴ ✹✺

❙❤❛♥♥♦♥ ❡♥tr♦♣②

❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ ❢✉♥❝t✐♦♥s S✶ := S(n)

✶

: ∆n → R✱ ❣✐✈❡♥ ❜② S✶(p✵,...,pn) := −

n

• k=✵

pi lnpi, ✭✶✮ ❛s t❤❡ ♦♥❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✱ s✉❝❤ t❤❛t S✶(✶/n,...,✶/n) ✐s ♠♦♥♦t♦♥✐❝ ✐♥ n ❛♥❞ ✏■❢ ❛ ❝❤♦✐❝❡ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ✐♥t♦ t✇♦ s✉❝❝❡ss✐✈❡ ❝❤♦✐❝❡s✱ t❤❡ ♦r✐❣✐♥❛❧ S✶ s❤♦✉❧❞ ❜❡ t❤❡ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✈❛❧✉❡s ♦❢ S✶✳✑

✶ ✷ ✸ ✶ ✷

✶ ✶ ✷ ✷ ✶ ✷

✸

✶

✶ ✶ ✷ ✸ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹ ✴ ✹✺

❙❤❛♥♥♦♥ ❡♥tr♦♣②

❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ ❢✉♥❝t✐♦♥s S✶ := S(n)

✶

: ∆n → R✱ ❣✐✈❡♥ ❜② S✶(p✵,...,pn) := −

n

• k=✵

pi lnpi, ✭✶✮ ❛s t❤❡ ♦♥❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✱ s✉❝❤ t❤❛t S✶(✶/n,...,✶/n) ✐s ♠♦♥♦t♦♥✐❝ ✐♥ n ❛♥❞ ✏■❢ ❛ ❝❤♦✐❝❡ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ✐♥t♦ t✇♦ s✉❝❝❡ss✐✈❡ ❝❤♦✐❝❡s✱ t❤❡ ♦r✐❣✐♥❛❧ S✶ s❤♦✉❧❞ ❜❡ t❤❡ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✈❛❧✉❡s ♦❢ S✶✳✑ p✶ p✷ p✸ p✶ +p✷

p✶ p✶+p✷ p✷ p✶+p✷

p✸ ✶

S✶(p✶,p✷,p✸) = S✶(p✶+p✷,p✸)+(p✶+p✷)S✶

• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• ❏✉♥❡ ✶✹✱ ✷✵✶✾

✹ ✴ ✹✺

❙tr✉❝t✉r❛❧ α✲❡♥tr♦♣② ✭❚s❛❧❧✐s α✲❡♥tr♦♣②✮

❍❛✈r❞❛ ❛♥❞ ❈❤❛r✈át ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞ ❡♥tr♦♣②✱ ❢♦r ❡❛❝❤ α > ✵✱ α = ✶✿ Sα(p✵,...,pn) = Kα

• ✶−

n

• i=✵

pα

i

• .

✭✷✮ ■t s❛t✐s✜❡s ❛ ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s✳

✶ ✷ ✸ ✶ ✷

✶ ✶ ✷ ✷ ✶ ✷

✸

✶

✶ ✶ ✷ ✸ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✶ ✶ ✷ ✷ ✶ ✷

❲❤② t❤❡s❡ ❛①✐♦♠s❄ ❲❤❛t ✐s t❤❡✐r r♦❧❡ ✐♥ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②❄

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✺ ✴ ✹✺

❙tr✉❝t✉r❛❧ α✲❡♥tr♦♣② ✭❚s❛❧❧✐s α✲❡♥tr♦♣②✮

❍❛✈r❞❛ ❛♥❞ ❈❤❛r✈át ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞ ❡♥tr♦♣②✱ ❢♦r ❡❛❝❤ α > ✵✱ α = ✶✿ Sα(p✵,...,pn) = Kα

• ✶−

n

• i=✵

pα

i

• .

✭✷✮ ■t s❛t✐s✜❡s ❛ ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s✳ p✶ p✷ p✸ p✶ +p✷

p✶ p✶+p✷ p✷ p✶+p✷

p✸ ✶

S✶(p✶,p✷,p✸) = S✶(p✶+p✷,p✸)+(p✶ +p✷)αS✶

• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• ❲❤② t❤❡s❡ ❛①✐♦♠s❄ ❲❤❛t ✐s t❤❡✐r r♦❧❡ ✐♥ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②❄

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✺ ✴ ✹✺

❙tr✉❝t✉r❛❧ α✲❡♥tr♦♣② ✭❚s❛❧❧✐s α✲❡♥tr♦♣②✮

❍❛✈r❞❛ ❛♥❞ ❈❤❛r✈át ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞ ❡♥tr♦♣②✱ ❢♦r ❡❛❝❤ α > ✵✱ α = ✶✿ Sα(p✵,...,pn) = Kα

• ✶−

n

• i=✵

pα

i

• .

✭✷✮ ■t s❛t✐s✜❡s ❛ ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s✳ p✶ p✷ p✸ p✶ +p✷

p✶ p✶+p✷ p✷ p✶+p✷

p✸ ✶

S✶(p✶,p✷,p✸) = S✶(p✶+p✷,p✸)+(p✶ +p✷)αS✶

• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• ❲❤② t❤❡s❡ ❛①✐♦♠s❄ ❲❤❛t ✐s t❤❡✐r r♦❧❡ ✐♥ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②❄

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✺ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✻ ✴ ✹✺

▼✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts

❚❤❡ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t

• n

k✶,...,ks

• :=

n! k✶!···ks! =

Γ(n +✶) Γ(k✶ +✶)···Γ(ks +✶)

❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✇♦r❞s w ∈ Σn✱ ✇✐t❤ Σ = {σ✶,...,σs}✱ s✉❝❤ t❤❛t σi ❛♣♣❡❛rs ki t✐♠❡s✳ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❙❤❛♥♥♦♥ ❡♥tr♦♣②

✶ ✶ ✶

❛♣♣❡❛rs ♥❛t✉r❛❧❧② ✐♥ t❤❡ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛

✶ ✶ ✶

✭✸✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✼ ✴ ✹✺

▼✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts

❚❤❡ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t

• n

k✶,...,ks

• :=

n! k✶!···ks! =

Γ(n +✶) Γ(k✶ +✶)···Γ(ks +✶)

❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✇♦r❞s w ∈ Σn✱ ✇✐t❤ Σ = {σ✶,...,σs}✱ s✉❝❤ t❤❛t σi ❛♣♣❡❛rs ki t✐♠❡s✳ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❙❤❛♥♥♦♥ ❡♥tr♦♣② S✶(p✶,...,ps) = −s

i=✶pi lnpi ❛♣♣❡❛rs ♥❛t✉r❛❧❧② ✐♥ t❤❡ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛

• n

p✶n,...,psn

• = exp(nS✶(p✶,...,ps)+O(lnn))

✭✸✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✼ ✴ ✹✺

q✲♠✉❧t✐♥♦♠✐❛❧s

▲❡t q ❜❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡✳ ❉❡✜♥❡

✶ q✲✐♥t❡❣❡rs [n]q = qn−✶

q−✶ ✱

✷ q✲❢❛❝t♦r✐❛❧s✿ [n]q! := [n]q[n −✶]q ···[✶]q✳ ✸ q✲♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❜②

• n

k✶,...,ks

• q

:= [n]q! [k✶]q!···[ks]q!, ✭✹✮ ✇❤❡r❡ k✶,...,ks ❛r❡ s✉❝❤ t❤❛t s

i=✶ki = n✳

Pr♦♣♦s✐t✐♦♥

❲❤❡♥ ✐s ❛ ♣r✐♠❡ ♣♦✇❡r✱

✶

❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✢❛❣s ♦❢ ✈❡❝t♦r s♣❛❝❡s

✶ ✷

s✉❝❤ t❤❛t

✶

✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✽ ✴ ✹✺

q✲♠✉❧t✐♥♦♠✐❛❧s

▲❡t q ❜❡ ❛♥ ✐♥❞❡t❡r♠✐♥❛t❡✳ ❉❡✜♥❡

✶ q✲✐♥t❡❣❡rs [n]q = qn−✶

q−✶ ✱

✷ q✲❢❛❝t♦r✐❛❧s✿ [n]q! := [n]q[n −✶]q ···[✶]q✳ ✸ q✲♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❜②

• n

k✶,...,ks

• q

:= [n]q! [k✶]q!···[ks]q!, ✭✹✮ ✇❤❡r❡ k✶,...,ks ❛r❡ s✉❝❤ t❤❛t s

i=✶ki = n✳

Pr♦♣♦s✐t✐♦♥

❲❤❡♥ q ✐s ❛ ♣r✐♠❡ ♣♦✇❡r✱

• n

k✶,...,ks

• q ❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✢❛❣s ♦❢ ✈❡❝t♦r

s♣❛❝❡s V✶ ⊂ V✷ ⊂ ... ⊂ Vs = Fn

q s✉❝❤ t❤❛t dimVi = i j=✶kj✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✽ ✴ ✹✺

❚s❛❧❧✐s ✷✲❡♥tr♦♣② ✐♥ ❝♦♠❜✐♥❛t♦r✐❝s

Pr♦♣♦s✐t✐♦♥ ✭❱✳✱ ✷✵✶✽✮

▲❡t (p✶,...,ps) ❜❡ ❛ ♣r♦❜❛❜✐❧✐t②✳ ❚❤❡♥✱

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷).

✭✺✮ ❍❡r❡ S✷(p✶,...,ps) := ✶−

s

• i=✶

p✷

i

✭✻✮ ✐s ❚s❛❧❧✐s ✷✲❡♥tr♦♣② ✭✇✐t❤ t❤❡ ❛♣♣r♦♣r✐❛t❡ ❧❡❛❞✐♥❣ ❝♦♥st❛♥t✮✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✾ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭❙❤❛♥♥♦♥ ❡♥tr♦♣②✮

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ✐❞❡♥t✐t②

• n

p✶n,p✷n,p✸n

• =
• n

(p✶ +p✷)n,p✸n

• (p✶ +p✷)n

p✶n,p✷n

• ❜❡❝♦♠❡s ❛s②♠♣t♦t✐❝❛❧❧②

✶ ✶ ✷ ✸ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✵ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭❙❤❛♥♥♦♥ ❡♥tr♦♣②✮

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ✐❞❡♥t✐t②

• n

p✶n,p✷n,p✸n

• =
• n

(p✶ +p✷)n,p✸n

• (p✶ +p✷)n

p✶n,p✷n

• ❜❡❝♦♠❡s ❛s②♠♣t♦t✐❝❛❧❧②

exp(nS✶(p✶,p✷,p✸)+o(n)) = exp

• n
• S✶(p✶ +p✷,p✸)+(p✶ +p✷)S✶
• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• +o(n)
• .

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✵ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭α✲❡♥tr♦♣②✮

❙✐♥❝❡

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷),

t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r❡❧❛t✐♦♥

✶ ✷ ✸ ✶ ✷ ✸ ✶ ✷ ✶ ✷

✐♠♣❧✐❡s

✷ ✶ ✷ ✸ ✷ ✶ ✷ ✸ ✶ ✷ ✷ ✷ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✶ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭α✲❡♥tr♦♣②✮

❙✐♥❝❡

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷),

t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r❡❧❛t✐♦♥

• n

p✶n,p✷n,p✸n

• q

=

• n

(p✶ +p✷)n,p✸n

• q
• (p✶ +p✷)n

p✶n,p✷n

• q

,

✐♠♣❧✐❡s

✷ ✶ ✷ ✸ ✷ ✶ ✷ ✸ ✶ ✷ ✷ ✷ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✶ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭α✲❡♥tr♦♣②✮

❙✐♥❝❡

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷),

t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r❡❧❛t✐♦♥

• n

p✶n,p✷n,p✸n

• q

=

• n

(p✶ +p✷)n,p✸n

• q
• (p✶ +p✷)n

p✶n,p✷n

• q

,

✐♠♣❧✐❡s S✷(p✶,p✷,p✸) = S✷(p✶ +p✷,p✸)+(p✶ +p✷)✷S✷

• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• .

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✶ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✷ ✴ ✹✺

✏❬❚❪❤❡ ❛❝t✉❛❧ ♠❡ss❛❣❡ ✐s ♦♥❡ s❡❧❡❝t❡❞ ❢r♦♠ ❛ s❡t ♦❢ ♣♦ss✐❜❧❡ ♠❡ss❛❣❡s✳ ❚❤❡ s②st❡♠ ♠✉st ❜❡ ❞❡s✐❣♥❡❞ t♦ ♦♣❡r❛t❡ ❢♦r ❡❛❝❤ ♣♦ss✐❜❧❡ s❡❧❡❝t✐♦♥✱ ♥♦t ❥✉st t❤❡ ♦♥❡ ✇❤✐❝❤ ✇✐❧❧ ❛❝t✉❛❧❧② ❜❡ ❝❤♦s❡♥ s✐♥❝❡ t❤✐s ✐s ✉♥❦♥♦✇♥ ❛t t❤❡ t✐♠❡ ♦❢ ❞❡s✐❣♥✳✑ ✭❙❤❛♥♥♦♥✮ ❚❤❡ s♦✉r❝❡ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ ♣r♦❜❛❜✐❧✐st✐❝ ♠♦❞❡❧ t❤❛t q✉❛♥t✐✜❡s t❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ❛♥② ♣♦ss✐❜❧❡ ♠❡ss❛❣❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✸ ✴ ✹✺

❆ q✲❞❡❢♦r♠❛t✐♦♥ ♦❢ ❙❤❛♥♥♦♥✬s t❤❡♦r②

❈♦♥❝❡♣t ❙❤❛♥♥♦♥ ❝❛s❡ q✲❝❛s❡ ▼❡ss❛❣❡ ❛t t✐♠❡ n ✭n✲♠❡ss❛❣❡✮ ❲♦r❞ w ∈ {✵,✶}n ❱❡❝t♦r s✉❜s♣❛❝❡ v ⊂ Fn

q

❚②♣❡ ◆✉♠❜❡r ♦❢ ♦♥❡s ❉✐♠❡♥s✐♦♥ ◆✉♠❜❡r ♦❢ n✲♠❡ss❛❣❡s ♦❢ t②♣❡ k

• n

k

• n

k

• q

Pr♦❜❛❜✐❧✐t② ♦❢ ❛ n✲♠❡ss❛❣❡ ♦❢ t②♣❡ k

ξk(✶−ξ)n−k θkqk(k−✶)/✷

(−θ;q)n

❲❤❡r❡ q ✐s ❛ ♣r✐♠❡ ♣♦✇❡r✱ ❛♥❞ θ > ✵✱ ξ ∈ [✵,✶] ❛r❡ ❛r❜✐tr❛r② ♣❛r❛♠❡t❡rs✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✹ ✴ ✹✺

❚❤❡ q✲❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥

❚❤❡ ❜✐♥♦♠✐❛❧ t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t✱ ❢♦r ❛♥② ξ ∈ [✵,✶]✱ ✶ =

n

• k=✵
• n

k

• ξk(✶−ξ)n−k,

✭✼✮ ❛♥❞ ❛❧s♦ t❤❛t Y ∼ Bin(n,ξ) ❛♣♣❡❛rs ❛s t❤❡ s✉♠ Z✶ +···Zn✱ ✇✐t❤ Zi ∼ Ber(ξ)✳ ■♥ t✉r♥✱ ●❛✉ss✬ ❜✐♥♦♠✐❛❧ ❢♦r♠✉❧❛ s❛②s t❤❛t (✶+θ)(✶+θq)···(✶+θqn−✶)

• =:(−θ;q)n

=

n

• k=✵
• n

k

• q

θkqk(k−✶)/✷.

✭✽✮ ❚❤❡ q✲❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ❤❛s ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥ k →

n

k

• q

θkqk(k−✶)/✷ (−θ;q)n ✳ ❆ ✈❛r✐❛❜❧❡ Y ∼ Binq(n,θ) ❡q✉❛❧s ✭✐♥ ❧❛✇✮ X✶ +···+Xn✱

✇❤❡r❡ Xi ∼ Ber

• θqi−✶

✶+θqi−✶

• ✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✺ ✴ ✹✺

• r❛ss♠❛♥♥✐❛♥ ♣r♦❝❡ss ■

❋✐① ❛ s❡q✉❡♥❝❡ ♦❢ ❡♠❜❡❞❞✐♥❣s F✶

q

→ F✷

q

→ ...✳

❲❡ ✐♥tr♦❞✉❝❡ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❛t ❣❡♥❡r❛t❡s ❛t t✐♠❡ n ❛ ❣❡♥❡r❛❧✐③❡❞ ♠❡ss❛❣❡ Vn ⊂ Fn

q s✉❝❤ t❤❛t

✶ Vn−✶ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❢r♦♠ Vn ✭❛s Vn ∩Fn−✶

q

✮❀

✷ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ Vn = v✱ ✇❤❡♥ dimv = k✱ ❡q✉❛❧s

θkqk(k−✶)/✷

(−θ;q)n

.

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✻ ✴ ✹✺

• r❛ss♠❛♥♥✐❛♥ ♣r♦❝❡ss ■■✿ ❍♦✇❄

❋♦r w ⊂ Fn−✶

q

✱ Diln(w) := {v ⊂ Fn

q | dimv −dimw = ✶,

w ⊂ v ❛♥❞ v ⊂ Fn−✶

q

}. {Xi}i∈N ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ Xi ∼ Ber

• θqi−✶

✶+θqi−✶

• ✳ ❙❡t V✵ = ✵ ❛♥❞✱ ❛t t✐♠❡ n✱

✶ ✐❢ Xn = ✵✱ ❞♦ ♥♦t❤✐♥❣ Vn = Vn−✶❀ ✷ ✐❢ Xn = ✶✱ ✐♥❝r❡❛s❡ ❞✐♠❡♥s✐♦♥✿ ♣✐❝❦ Vn ❛t r❛♥❞♦♠✱ ✉♥✐❢♦r♠❧②✱ ❢r♦♠

Diln(Vn−✶)✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✼ ✴ ✹✺

❚❤❡♦r❡♠ ✭●❡♥❡r❛❧✐③❡❞ ❆❊P✮

❋♦r ❡✈❡r② δ > ✵ ❛♥❞ ❛❧♠♦st ❡✈❡r② ε > ✵ ✭❡①❝❡♣t ❛ ❝♦✉♥t❛❜❧❡ s❡t✮✱ t❤❡r❡ ❡①✐st n✵ ∈ N ❛♥❞ s❡ts An = ∆(ε)

k=✵ Gr(n −k,n)✱ ❢♦r ❛❧❧ n ≥ n✵✱ s✉❝❤ t❤❛t

✶

✐s ❛♥ ✐♥t❡❣❡r t❤❛t ❥✉st ❞❡♣❡♥❞s ♦♥ ❀

✷

❀

✸ ❢♦r ❛♥②

s✉❝❤ t❤❛t ✱

✶

✷

✷

✭✾✮ ▼♦r❡♦✈❡r✱ t❤❡ s✐③❡ ♦❢ ✐s ♦♣t✐♠❛❧✱ ✉♣ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧✿ ✐❢ ✐s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ s♠❛❧❧❡st s❡t ♦❢ s✉❜s♣❛❝❡s ♦❢ t❤❛t ❛❝❝✉♠✉❧❛t❡ ♣r♦❜❛❜✐❧✐t② ✶ ✱ t❤❡♥ ✶ ✶ ✷

✷

✭✶✵✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✽ ✴ ✹✺

❚❤❡♦r❡♠ ✭●❡♥❡r❛❧✐③❡❞ ❆❊P✮

❋♦r ❡✈❡r② δ > ✵ ❛♥❞ ❛❧♠♦st ❡✈❡r② ε > ✵ ✭❡①❝❡♣t ❛ ❝♦✉♥t❛❜❧❡ s❡t✮✱ t❤❡r❡ ❡①✐st n✵ ∈ N ❛♥❞ s❡ts An = ∆(ε)

k=✵ Gr(n −k,n)✱ ❢♦r ❛❧❧ n ≥ n✵✱ s✉❝❤ t❤❛t

✶

∆(ε) ✐s ❛♥ ✐♥t❡❣❡r t❤❛t ❥✉st ❞❡♣❡♥❞s ♦♥ ε❀

✷

❀

✸ ❢♦r ❛♥②

s✉❝❤ t❤❛t ✱

✶

✷

✷

✭✾✮ ▼♦r❡♦✈❡r✱ t❤❡ s✐③❡ ♦❢ ✐s ♦♣t✐♠❛❧✱ ✉♣ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧✿ ✐❢ ✐s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ s♠❛❧❧❡st s❡t ♦❢ s✉❜s♣❛❝❡s ♦❢ t❤❛t ❛❝❝✉♠✉❧❛t❡ ♣r♦❜❛❜✐❧✐t② ✶ ✱ t❤❡♥ ✶ ✶ ✷

✷

✭✶✵✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✽ ✴ ✹✺

❚❤❡♦r❡♠ ✭●❡♥❡r❛❧✐③❡❞ ❆❊P✮

❋♦r ❡✈❡r② δ > ✵ ❛♥❞ ❛❧♠♦st ❡✈❡r② ε > ✵ ✭❡①❝❡♣t ❛ ❝♦✉♥t❛❜❧❡ s❡t✮✱ t❤❡r❡ ❡①✐st n✵ ∈ N ❛♥❞ s❡ts An = ∆(ε)

k=✵ Gr(n −k,n)✱ ❢♦r ❛❧❧ n ≥ n✵✱ s✉❝❤ t❤❛t

✶

∆(ε) ✐s ❛♥ ✐♥t❡❣❡r t❤❛t ❥✉st ❞❡♣❡♥❞s ♦♥ ε❀

✷

P(Vn ∈ Ac

n) ≤ ε❀

✸ ❢♦r ❛♥②

s✉❝❤ t❤❛t ✱

✶

✷

✷

✭✾✮ ▼♦r❡♦✈❡r✱ t❤❡ s✐③❡ ♦❢ ✐s ♦♣t✐♠❛❧✱ ✉♣ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧✿ ✐❢ ✐s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ s♠❛❧❧❡st s❡t ♦❢ s✉❜s♣❛❝❡s ♦❢ t❤❛t ❛❝❝✉♠✉❧❛t❡ ♣r♦❜❛❜✐❧✐t② ✶ ✱ t❤❡♥ ✶ ✶ ✷

✷

✭✶✵✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✽ ✴ ✹✺

❚❤❡♦r❡♠ ✭●❡♥❡r❛❧✐③❡❞ ❆❊P✮

❋♦r ❡✈❡r② δ > ✵ ❛♥❞ ❛❧♠♦st ❡✈❡r② ε > ✵ ✭❡①❝❡♣t ❛ ❝♦✉♥t❛❜❧❡ s❡t✮✱ t❤❡r❡ ❡①✐st n✵ ∈ N ❛♥❞ s❡ts An = ∆(ε)

k=✵ Gr(n −k,n)✱ ❢♦r ❛❧❧ n ≥ n✵✱ s✉❝❤ t❤❛t

✶

∆(ε) ✐s ❛♥ ✐♥t❡❣❡r t❤❛t ❥✉st ❞❡♣❡♥❞s ♦♥ ε❀

✷

P(Vn ∈ Ac

n) ≤ ε❀

✸ ❢♦r ❛♥② v ∈ An s✉❝❤ t❤❛t dimv = k✱

• logq(P(Vn = v)−✶)

n

− n

✷S✷(k/n)

• ≤ δ.

✭✾✮ ▼♦r❡♦✈❡r✱ t❤❡ s✐③❡ ♦❢ ✐s ♦♣t✐♠❛❧✱ ✉♣ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧✿ ✐❢ ✐s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ s♠❛❧❧❡st s❡t ♦❢ s✉❜s♣❛❝❡s ♦❢ t❤❛t ❛❝❝✉♠✉❧❛t❡ ♣r♦❜❛❜✐❧✐t② ✶ ✱ t❤❡♥ ✶ ✶ ✷

✷

✭✶✵✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✽ ✴ ✹✺

❚❤❡♦r❡♠ ✭●❡♥❡r❛❧✐③❡❞ ❆❊P✮

❋♦r ❡✈❡r② δ > ✵ ❛♥❞ ❛❧♠♦st ❡✈❡r② ε > ✵ ✭❡①❝❡♣t ❛ ❝♦✉♥t❛❜❧❡ s❡t✮✱ t❤❡r❡ ❡①✐st n✵ ∈ N ❛♥❞ s❡ts An = ∆(ε)

k=✵ Gr(n −k,n)✱ ❢♦r ❛❧❧ n ≥ n✵✱ s✉❝❤ t❤❛t

✶

∆(ε) ✐s ❛♥ ✐♥t❡❣❡r t❤❛t ❥✉st ❞❡♣❡♥❞s ♦♥ ε❀

✷

P(Vn ∈ Ac

n) ≤ ε❀

✸ ❢♦r ❛♥② v ∈ An s✉❝❤ t❤❛t dimv = k✱

• logq(P(Vn = v)−✶)

n

− n

✷S✷(k/n)

• ≤ δ.

✭✾✮ ▼♦r❡♦✈❡r✱ t❤❡ s✐③❡ ♦❢ An ✐s ♦♣t✐♠❛❧✱ ✉♣ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧✿ ✐❢ s(n,ε) ✐s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ s♠❛❧❧❡st s❡t ♦❢ s✉❜s♣❛❝❡s ♦❢ Fn

q t❤❛t

❛❝❝✉♠✉❧❛t❡ ♣r♦❜❛❜✐❧✐t② ✶−ε✱ t❤❡♥ lim

n

✶ n logq |An| = lim

n

✶ n logq s(n,ε) = lim

n

n ✷S✷(∆(ε)/n) = ∆(ε). ✭✶✵✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✽ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶✾ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✵ ✴ ✹✺

❉❡✜♥✐t✐♦♥

❆♥ ✐♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡ ✐s ❛ ❝♦✉♣❧❡ (❙,M)✱ ✇❤❡r❡ ❙ ✭♦❜s❡r✈❛❜❧❡s✱ ❡①♣❡r✐♠❡♥ts✱ ✈❛r✐❛❜❧❡s✳✳✳✮ ✐s ❛ s♠❛❧❧ ❝❛t❡❣♦r② s✉❝❤ t❤❛t

✶ ❙ ❤❛s ❛ t❡r♠✐♥❛❧ ♦❜❥❡❝t✱ ❞❡♥♦t❡❞ ✶❀ ✷ ❙ ✐s ❛ s❦❡❧❡t❛❧ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ s❡t ✭♣♦s❡t✮❀ ✸ ❢♦r ♦❜❥❡❝ts X,Y ,Z ∈ ObS✱ ✐❢ Z → X ❛♥❞ Z → Y ✱ t❤❡♥ t❤❡ ❝❛t❡❣♦r✐❝❛❧

♣r♦❞✉❝t X ∧Y ❡①✐sts❀ ❛♥❞ M ✐s ❛ ❝♦♥s❡r✈❛t✐✈❡ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦r ✭t❤❡ ♣♦ss✐❜❧❡ ♦✉t♣✉ts✮ ❢r♦♠ ❙ ✐♥t♦ t❤❡ ❝❛t❡❣♦r② ▼❡❛s✉r❛❜❧❡❙♣❛❝❡surj✱ X → M(X) = (E(X),B(X))✱ t❤❛t s❛t✐s✜❡s

✹ E(✶) ∼

= {∗}❀

✺ ❢♦r ❡✈❡r② X ∈ Ob❙ ❛♥❞ ❛♥② x ∈ E(X)✱ t❤❡ σ✲❛❧❣❡❜r❛ B(X) ❝♦♥t❛✐♥s t❤❡

s✐♥❣❧❡t♦♥ {x}❀

✻ ❢♦r ❡✈❡r② ❞✐❛❣r❛♠ X

X ∧Y Y

← →

π

← →

σ

t❤❡ ♠❡❛s✉r❛❜❧❡ ♠❛♣ E(X ∧Y )

→ E(X)×E(Y ),z → (x(z),y(z)) := (π∗(z),σ∗(z)) ✐s ❛♥

✐♥❥❡❝t✐♦♥✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✶ ✴ ✹✺

X ∧P P P ∧V X ✶ X ∧V V

← → ← → ← → ← → ← → ← → ← → ← → ← → ← →

(EX ×EP,✷EX ⊗P) (EP,P) (M,M) (EX ,✷EX )

{∗}

(EX ×EV ,✷EX ⊗V) (EV ,V)

← → ← → ← → ← → ← → ← → ← → ← → ← → ← →

✇❤❡r❡ M

→ EP ×EV ❛♥❞ EX := { , , , , , }✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✷ ✴ ✹✺

X✶ X✷ X✸ X✶ ∧X✸ X✷ ∧X✸ X✶ ∧X✷ X✶ ∧X✷ ∧X✸ ✇✐t❤ E(Xi) ❛r❜✐tr❛r②✱ E(Xi ∧Xj) = E(Xi)×E(Xj)✱ ❛♥❞ E(X✶ ∧X✷ ∧X✸) = E(X✶)×E(X✷)×E(X✷)✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✸ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ t♦♣♦s

❚❤❡ ❝❛t❡❣♦r② ♦❢ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦rs [❙,❙❡ts] ❛s ✇❡❧❧ ❛s t❤❛t ♦❢ ❝♦♥tr❛✈❛r✐❛♥t ❢✉♥❝t♦rs [❙op,❙❡ts] ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛♣♣❧✐❝❛t✐♦♥s✳ ✭❋♦r ❡①❛♠♣❧❡✱ ♣r♦❜❛❜✐❧✐t✐❡s ❞❡✜♥❡ ❛ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦r❀ t❤❡ ♣r♦❜❛❜✐❧✐t✐❝ ❢✉♥❝t✐♦♥❛❧s✖❧✐❦❡ ❡♥tr♦♣②✖✱ ❛ ❝♦♥tr❛✈❛r✐❛♥t ♦♥❡✳✮ ❇♦t❤ ❝❛t❡❣♦r✐❡s ❛r❡ ●r♦t❤❡♥❞✐❡❝❦ t♦♣♦✐ ✭r❡❧❛t❡❞ t♦ ❣❡♦♠❡tr✐❝ ✐♥t✉✐t✐♦♥s✱ ❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥s✱ ❣❡♥❡r❛❧ ❛❧❣❡❜r❛✐❝ ❝♦♥str✉❝t✐♦♥s✳✳✳✮✳ ❚❤❡r❡ ✐s ❛♥ ❛♣♣r♦♣r✐❛t❡ ♥♦t✐♦♥ ♦❢ ❧♦❝❛❧✐③❛t✐♦♥✿ ♠❛r❣✐♥❛❧✐③❛t✐♦♥s ✐♥❞✉❝❡❞ ❜② t❤❡ s✉r❥❡❝t✐♦♥s ♦❢ ✳ ❲❡ ✇❛♥t t♦ st✉❞② t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♥✈❛r✐❛♥ts ❛tt❛❝❤❡❞ t♦ t❤❡s❡ ❢✉♥❝t♦rs ✭t❤❛t ❛r❡ ❛❧❣❡❜r❛✐❝ ❛♥❛❧♦❣s ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s✮✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✹ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ t♦♣♦s

❚❤❡ ❝❛t❡❣♦r② ♦❢ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦rs [❙,❙❡ts] ❛s ✇❡❧❧ ❛s t❤❛t ♦❢ ❝♦♥tr❛✈❛r✐❛♥t ❢✉♥❝t♦rs [❙op,❙❡ts] ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛♣♣❧✐❝❛t✐♦♥s✳ ✭❋♦r ❡①❛♠♣❧❡✱ ♣r♦❜❛❜✐❧✐t✐❡s ❞❡✜♥❡ ❛ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦r❀ t❤❡ ♣r♦❜❛❜✐❧✐t✐❝ ❢✉♥❝t✐♦♥❛❧s✖❧✐❦❡ ❡♥tr♦♣②✖✱ ❛ ❝♦♥tr❛✈❛r✐❛♥t ♦♥❡✳✮ ❇♦t❤ ❝❛t❡❣♦r✐❡s ❛r❡ ●r♦t❤❡♥❞✐❡❝❦ t♦♣♦✐ ✭r❡❧❛t❡❞ t♦ ❣❡♦♠❡tr✐❝ ✐♥t✉✐t✐♦♥s✱ ❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥s✱ ❣❡♥❡r❛❧ ❛❧❣❡❜r❛✐❝ ❝♦♥str✉❝t✐♦♥s✳✳✳✮✳ ❚❤❡r❡ ✐s ❛♥ ❛♣♣r♦♣r✐❛t❡ ♥♦t✐♦♥ ♦❢ ❧♦❝❛❧✐③❛t✐♦♥✿ ♠❛r❣✐♥❛❧✐③❛t✐♦♥s ✐♥❞✉❝❡❞ ❜② t❤❡ s✉r❥❡❝t✐♦♥s ♦❢ ✳ ❲❡ ✇❛♥t t♦ st✉❞② t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♥✈❛r✐❛♥ts ❛tt❛❝❤❡❞ t♦ t❤❡s❡ ❢✉♥❝t♦rs ✭t❤❛t ❛r❡ ❛❧❣❡❜r❛✐❝ ❛♥❛❧♦❣s ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s✮✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✹ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ t♦♣♦s

❚❤❡ ❝❛t❡❣♦r② ♦❢ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦rs [❙,❙❡ts] ❛s ✇❡❧❧ ❛s t❤❛t ♦❢ ❝♦♥tr❛✈❛r✐❛♥t ❢✉♥❝t♦rs [❙op,❙❡ts] ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛♣♣❧✐❝❛t✐♦♥s✳ ✭❋♦r ❡①❛♠♣❧❡✱ ♣r♦❜❛❜✐❧✐t✐❡s ❞❡✜♥❡ ❛ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦r❀ t❤❡ ♣r♦❜❛❜✐❧✐t✐❝ ❢✉♥❝t✐♦♥❛❧s✖❧✐❦❡ ❡♥tr♦♣②✖✱ ❛ ❝♦♥tr❛✈❛r✐❛♥t ♦♥❡✳✮ ❇♦t❤ ❝❛t❡❣♦r✐❡s ❛r❡ ●r♦t❤❡♥❞✐❡❝❦ t♦♣♦✐ ✭r❡❧❛t❡❞ t♦ ❣❡♦♠❡tr✐❝ ✐♥t✉✐t✐♦♥s✱ ❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥s✱ ❣❡♥❡r❛❧ ❛❧❣❡❜r❛✐❝ ❝♦♥str✉❝t✐♦♥s✳✳✳✮✳ ❚❤❡r❡ ✐s ❛♥ ❛♣♣r♦♣r✐❛t❡ ♥♦t✐♦♥ ♦❢ ❧♦❝❛❧✐③❛t✐♦♥✿ ♠❛r❣✐♥❛❧✐③❛t✐♦♥s ✐♥❞✉❝❡❞ ❜② t❤❡ s✉r❥❡❝t✐♦♥s ♦❢ M✳ ❲❡ ✇❛♥t t♦ st✉❞② t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♥✈❛r✐❛♥ts ❛tt❛❝❤❡❞ t♦ t❤❡s❡ ❢✉♥❝t♦rs ✭t❤❛t ❛r❡ ❛❧❣❡❜r❛✐❝ ❛♥❛❧♦❣s ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s✮✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✹ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ t♦♣♦s

❚❤❡ ❝❛t❡❣♦r② ♦❢ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦rs [❙,❙❡ts] ❛s ✇❡❧❧ ❛s t❤❛t ♦❢ ❝♦♥tr❛✈❛r✐❛♥t ❢✉♥❝t♦rs [❙op,❙❡ts] ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛♣♣❧✐❝❛t✐♦♥s✳ ✭❋♦r ❡①❛♠♣❧❡✱ ♣r♦❜❛❜✐❧✐t✐❡s ❞❡✜♥❡ ❛ ❝♦✈❛r✐❛♥t ❢✉♥❝t♦r❀ t❤❡ ♣r♦❜❛❜✐❧✐t✐❝ ❢✉♥❝t✐♦♥❛❧s✖❧✐❦❡ ❡♥tr♦♣②✖✱ ❛ ❝♦♥tr❛✈❛r✐❛♥t ♦♥❡✳✮ ❇♦t❤ ❝❛t❡❣♦r✐❡s ❛r❡ ●r♦t❤❡♥❞✐❡❝❦ t♦♣♦✐ ✭r❡❧❛t❡❞ t♦ ❣❡♦♠❡tr✐❝ ✐♥t✉✐t✐♦♥s✱ ❧♦❣✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥s✱ ❣❡♥❡r❛❧ ❛❧❣❡❜r❛✐❝ ❝♦♥str✉❝t✐♦♥s✳✳✳✮✳ ❚❤❡r❡ ✐s ❛♥ ❛♣♣r♦♣r✐❛t❡ ♥♦t✐♦♥ ♦❢ ❧♦❝❛❧✐③❛t✐♦♥✿ ♠❛r❣✐♥❛❧✐③❛t✐♦♥s ✐♥❞✉❝❡❞ ❜② t❤❡ s✉r❥❡❝t✐♦♥s ♦❢ M✳ ❲❡ ✇❛♥t t♦ st✉❞② t❤❡ ❣❡♦♠❡tr✐❝❛❧ ✐♥✈❛r✐❛♥ts ❛tt❛❝❤❡❞ t♦ t❤❡s❡ ❢✉♥❝t♦rs ✭t❤❛t ❛r❡ ❛❧❣❡❜r❛✐❝ ❛♥❛❧♦❣s ♦❢ ✈❡❝t♦r ❜✉♥❞❧❡s✮✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✹ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ ❉❡✜♥✐t✐♦♥

❋♦r ❡❛❝❤ X ∈ Ob❙, t❤❡ s❡t SX := {Y |X → Y } ✐s ❛ ♠♦♥♦✐❞ ✉♥❞❡r t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ (Y ,Z) → Y ∧Z ✭✬❥♦✐♥t ✈❛r✐❛❜❧❡✬✮✳ ❊❛❝❤ ❛rr♦✇ X → Y ✐♥❞✉❝❡s ❛♥ ✐♥❝❧✉s✐♦♥ SY → SX✱ ✇❤✐❝❤ ❞❡✜♥❡s ❛ ♣❛rt✐❝✉❧❛r ♣r❡s❤❡❛❢ ✭❝♦♥tr❛✈❛r✐❛♥t ❢✉♥❝t♦r✮✳ ▲❡t A ❞❡♥♦t❡ t❤❡ ♣r❡s❤❡❛❢ ♦❢ ❛❧❣❡❜r❛s X → R[SX] ✭✜♥✐t❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✮✳ ❚❤❡ ❝❛t❡❣♦r② ▼♦❞ ♦❢ ✲♠♦❞✉❧❡s ✐s ❛♥ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✭✐✳❡✳ ✐t ❜❡❤❛✈❡s ❧✐❦❡ t❤❡ ❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ r✐♥❣✮✳ ❲❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ❞❡r✐✈❡❞ ❢✉♥❝t♦rs ♦❢ ✱ ❞❡♥♦t❡❞ ✳

❉❡✜♥✐t✐♦♥

❚❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❛♥ ✲♠♦❞✉❧❡ ✐s ❙

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✺ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ ❉❡✜♥✐t✐♦♥

❋♦r ❡❛❝❤ X ∈ Ob❙, t❤❡ s❡t SX := {Y |X → Y } ✐s ❛ ♠♦♥♦✐❞ ✉♥❞❡r t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ (Y ,Z) → Y ∧Z ✭✬❥♦✐♥t ✈❛r✐❛❜❧❡✬✮✳ ❊❛❝❤ ❛rr♦✇ X → Y ✐♥❞✉❝❡s ❛♥ ✐♥❝❧✉s✐♦♥ SY → SX✱ ✇❤✐❝❤ ❞❡✜♥❡s ❛ ♣❛rt✐❝✉❧❛r ♣r❡s❤❡❛❢ ✭❝♦♥tr❛✈❛r✐❛♥t ❢✉♥❝t♦r✮✳ ▲❡t A ❞❡♥♦t❡ t❤❡ ♣r❡s❤❡❛❢ ♦❢ ❛❧❣❡❜r❛s X → R[SX] ✭✜♥✐t❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✮✳ ❚❤❡ ❝❛t❡❣♦r② ▼♦❞(A ) ♦❢ A ✲♠♦❞✉❧❡s ✐s ❛♥ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✭✐✳❡✳ ✐t ❜❡❤❛✈❡s ❧✐❦❡ t❤❡ ❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ r✐♥❣✮✳ ❲❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ❞❡r✐✈❡❞ ❢✉♥❝t♦rs ♦❢ HomA (R,−)✱ ❞❡♥♦t❡❞ Ext•(R,−)✳

❉❡✜♥✐t✐♦♥

❚❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❛♥ A ✲♠♦❞✉❧❡ M ✐s H•(❙,M) := Ext•(R,M).

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✺ ✴ ✹✺

❘❡s♦❧✉t✐♦♥

❚❤❡r❡ ✐s ❛ ❜❛r r❡s♦❧✉t✐♦♥ B• → R t❤❛t ❛❧❧♦✇s ✉s t♦ ❝♦♠♣✉t❡ H•(❙,M) ❛s t❤❡ ❤♦♠♦❧♦❣② ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❝♦♠♣❧❡① (Nat(B•,M),δ)✳ ❲❡ ❣❡t ❝♦❝❤❛✐♥s✱ ❝♦❝②❝❧❡s ❛♥❞ ❝♦❜♦✉♥❞❛r✐❡s✳ ■♠♣♦rt❛♥t t♦ r❡t❛✐♥✿ ✲❝♦❝❤❛✐♥s ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❝♦❤❡r❡♥t✴❢✉♥❝t♦r✐❛❧ ❝♦❧❧❡❝t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ✐♥ ✐♥❞❡①❡❞ ❜② ✲t✉♣❧❡s ♦❢ ❡❧❡♠❡♥ts ✐♥ ❙ ✭s✐♥❝❡ ❡❛❝❤ ✐s ❛ ❢r❡❡ ♠♦❞✉❧❡✮❀ t❤❡② ❛r❡ ✲❝♦❝②❧❡s ✇❤❡♥ t❤❡② s❛t✐s❢② ✵❀ ❛♥ ✲❝♦❜♦✉♥❞❛r② ✐s ❛♥ ✲❝♦❝❤❛✐♥ t❤❛t ❝♦♠❡s ❢r♦♠ ❛♥ ✶ ✲❝♦❝❤❛✐♥ ✱ s✉❝❤ t❤❛t ✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ✶✲❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥ ❡♥❝♦❞❡s t❤❡ r❡❝✉rr❡♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ ❡♥tr♦♣✐❡s ❛♥❞ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✻ ✴ ✹✺

❘❡s♦❧✉t✐♦♥

❚❤❡r❡ ✐s ❛ ❜❛r r❡s♦❧✉t✐♦♥ B• → R t❤❛t ❛❧❧♦✇s ✉s t♦ ❝♦♠♣✉t❡ H•(❙,M) ❛s t❤❡ ❤♦♠♦❧♦❣② ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❝♦♠♣❧❡① (Nat(B•,M),δ)✳ ❲❡ ❣❡t ❝♦❝❤❛✐♥s✱ ❝♦❝②❝❧❡s ❛♥❞ ❝♦❜♦✉♥❞❛r✐❡s✳ ■♠♣♦rt❛♥t t♦ r❡t❛✐♥✿ n✲❝♦❝❤❛✐♥s ϕ ∈ Nat(Bn,M) ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❝♦❤❡r❡♥t✴❢✉♥❝t♦r✐❛❧ ❝♦❧❧❡❝t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ✐♥ M ✐♥❞❡①❡❞ ❜② n✲t✉♣❧❡s ♦❢ ❡❧❡♠❡♥ts ✐♥ Ob❙ ✭s✐♥❝❡ ❡❛❝❤ Bn(X) ✐s ❛ ❢r❡❡ ♠♦❞✉❧❡✮❀ t❤❡② ❛r❡ n✲❝♦❝②❧❡s ✇❤❡♥ t❤❡② s❛t✐s❢② δφ = ✵❀ ❛♥ n✲❝♦❜♦✉♥❞❛r② ✐s ❛♥ n✲❝♦❝❤❛✐♥ φ t❤❛t ❝♦♠❡s ❢r♦♠ ❛♥ (n −✶)✲❝♦❝❤❛✐♥ ψ✱ s✉❝❤ t❤❛t ϕ = δψ✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ✶✲❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥ ❡♥❝♦❞❡s t❤❡ r❡❝✉rr❡♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ ❡♥tr♦♣✐❡s ❛♥❞ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✻ ✴ ✹✺

❘❡s♦❧✉t✐♦♥

❚❤❡r❡ ✐s ❛ ❜❛r r❡s♦❧✉t✐♦♥ B• → R t❤❛t ❛❧❧♦✇s ✉s t♦ ❝♦♠♣✉t❡ H•(❙,M) ❛s t❤❡ ❤♦♠♦❧♦❣② ♦❢ ❛ ❞✐✛❡r❡♥t✐❛❧ ❝♦♠♣❧❡① (Nat(B•,M),δ)✳ ❲❡ ❣❡t ❝♦❝❤❛✐♥s✱ ❝♦❝②❝❧❡s ❛♥❞ ❝♦❜♦✉♥❞❛r✐❡s✳ ■♠♣♦rt❛♥t t♦ r❡t❛✐♥✿ n✲❝♦❝❤❛✐♥s ϕ ∈ Nat(Bn,M) ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❝♦❤❡r❡♥t✴❢✉♥❝t♦r✐❛❧ ❝♦❧❧❡❝t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ✐♥ M ✐♥❞❡①❡❞ ❜② n✲t✉♣❧❡s ♦❢ ❡❧❡♠❡♥ts ✐♥ Ob❙ ✭s✐♥❝❡ ❡❛❝❤ Bn(X) ✐s ❛ ❢r❡❡ ♠♦❞✉❧❡✮❀ t❤❡② ❛r❡ n✲❝♦❝②❧❡s ✇❤❡♥ t❤❡② s❛t✐s❢② δφ = ✵❀ ❛♥ n✲❝♦❜♦✉♥❞❛r② ✐s ❛♥ n✲❝♦❝❤❛✐♥ φ t❤❛t ❝♦♠❡s ❢r♦♠ ❛♥ (n −✶)✲❝♦❝❤❛✐♥ ψ✱ s✉❝❤ t❤❛t ϕ = δψ✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ✶✲❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥ ❡♥❝♦❞❡s t❤❡ r❡❝✉rr❡♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ ❡♥tr♦♣✐❡s ❛♥❞ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✻ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✼ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ❝❛s❡

❉✐s❝r❡t❡ ✈❛r✐❛❜❧❡s✿ ✇❡ s✉♣♣♦s❡ t❤❛t ❡❛❝❤ EX ✐s ✜♥✐t❡✳

✶ ✷ ✸

❳

✶ ✷

✶ ✶ ✷ ✷ ✶ ✷

✸

✶ ❨ ❳ ▲❡t ❙ ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ ❛ s❡t ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ ✭✐✳❡✳ ❢✉♥❝t✐♦♥s ✵ ✶ s✉❝❤ t❤❛t ✶✮ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♥❞✐t✐♦♥✐♥❣ ❜② ✈❛r✐❛❜❧❡s ✐♥ ❙✳ ❊✈❡r② ❛rr♦✇ ✐♥ ❙ tr❛♥s❧❛t❡s ✐♥t♦ ❛ s✉r❥❡❝t✐♦♥ t❤❛t ✐♥❞✉❝❡s ❛ ♠❛r❣✐♥❛❧✐③❛t✐♦♥ ❣✐✈❡♥ ❜②

✶

◆❇✿ ❲❤❡♥❡✈❡r ❙ ❤❛s ♥♦ ✐♥✐t✐❛❧ ♦❜❥❡❝t✱ ❛ s❡❝t✐♦♥

❙ ❙❡ts

✐s ❥✉st ❛ ❝♦❤❡r❡♥t ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s t❤❛t ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♠❡ ❢r♦♠ ❛ ❣❧♦❜❛❧ ❧❛✇✱ ❝❛❧❧❡❞ ✏♣s❡✉❞♦✲♠❛r❣✐♥❛❧✑ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✽ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ❝❛s❡

❉✐s❝r❡t❡ ✈❛r✐❛❜❧❡s✿ ✇❡ s✉♣♣♦s❡ t❤❛t ❡❛❝❤ EX ✐s ✜♥✐t❡✳ p✶ p✷ p✸ ❳ p✶ +p✷

p✶ p✶+p✷ p✷ p✶+p✷

p✸ ✶ ❨ ❳ ▲❡t Q : ❙ → ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ X ❛ s❡t Q(X) ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ EX ✭✐✳❡✳ ❢✉♥❝t✐♦♥s p : EX → [✵,✶] s✉❝❤ t❤❛t

x∈EX p(x) = ✶✮

❝❧♦s❡❞ ✉♥❞❡r ❝♦♥❞✐t✐♦♥✐♥❣ ❜② ✈❛r✐❛❜❧❡s ✐♥ ❙✳ ❊✈❡r② ❛rr♦✇ X → Y ✐♥ ❙ tr❛♥s❧❛t❡s ✐♥t♦ ❛ s✉r❥❡❝t✐♦♥ π : EX → EY t❤❛t ✐♥❞✉❝❡s ❛ ♠❛r❣✐♥❛❧✐③❛t✐♦♥ π∗ := Q(π) : Q(X) → Q(Y ) ❣✐✈❡♥ ❜②

π∗p(y) =

• x∈π−✶(y)

p(x).

◆❇✿ ❲❤❡♥❡✈❡r ❙ ❤❛s ♥♦ ✐♥✐t✐❛❧ ♦❜❥❡❝t✱ ❛ s❡❝t✐♦♥

❙ ❙❡ts

✐s ❥✉st ❛ ❝♦❤❡r❡♥t ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s t❤❛t ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♠❡ ❢r♦♠ ❛ ❣❧♦❜❛❧ ❧❛✇✱ ❝❛❧❧❡❞ ✏♣s❡✉❞♦✲♠❛r❣✐♥❛❧✑ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✽ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ❝❛s❡

❉✐s❝r❡t❡ ✈❛r✐❛❜❧❡s✿ ✇❡ s✉♣♣♦s❡ t❤❛t ❡❛❝❤ EX ✐s ✜♥✐t❡✳ p✶ p✷ p✸ ❳ p✶ +p✷

p✶ p✶+p✷ p✷ p✶+p✷

p✸ ✶ ❨ ❳ ▲❡t Q : ❙ → ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ X ❛ s❡t Q(X) ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s ♦♥ EX ✭✐✳❡✳ ❢✉♥❝t✐♦♥s p : EX → [✵,✶] s✉❝❤ t❤❛t

x∈EX p(x) = ✶✮

❝❧♦s❡❞ ✉♥❞❡r ❝♦♥❞✐t✐♦♥✐♥❣ ❜② ✈❛r✐❛❜❧❡s ✐♥ ❙✳ ❊✈❡r② ❛rr♦✇ X → Y ✐♥ ❙ tr❛♥s❧❛t❡s ✐♥t♦ ❛ s✉r❥❡❝t✐♦♥ π : EX → EY t❤❛t ✐♥❞✉❝❡s ❛ ♠❛r❣✐♥❛❧✐③❛t✐♦♥ π∗ := Q(π) : Q(X) → Q(Y ) ❣✐✈❡♥ ❜②

π∗p(y) =

• x∈π−✶(y)

p(x).

◆❇✿ ❲❤❡♥❡✈❡r ❙ ❤❛s ♥♦ ✐♥✐t✐❛❧ ♦❜❥❡❝t✱ ❛ s❡❝t✐♦♥ q ∈ Γ(Q) := Hom[❙,❙❡ts](∗,Q) ✐s ❥✉st ❛ ❝♦❤❡r❡♥t ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s t❤❛t ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♠❡ ❢r♦♠ ❛ ❣❧♦❜❛❧ ❧❛✇✱ ❝❛❧❧❡❞ ✏♣s❡✉❞♦✲♠❛r❣✐♥❛❧✑ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✽ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ❢✉♥❝t✐♦♥❛❧s

▲❡t F(X) ❜❡ t❤❡ ❛❞❞✐t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♠❡❛s✉r❛❜❧❡ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ Q(X)✱ ❛♥❞ F(π) : F(Y ) → F(X) ✭❝♦♥tr❛✈❛r✐❛♥t✮ s✉❝❤ t❤❛t F(π)(φ) = φ◦π∗✳ ■♥ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✿ π∗f (p✶,p✷,p✸) = f (p✶ +p✷,p✸)✳ ❋♦r ❡❛❝❤ ❛♥❞ ✱ ❞❡✜♥❡

✵

✭✶✶✮ ❚❤✐s t✉r♥s ✐♥t♦ ❛♥ ✲♠♦❞✉❧❡ t❤❛t ✇❡ ❞❡♥♦t❡ ✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✾ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ❢✉♥❝t✐♦♥❛❧s

▲❡t F(X) ❜❡ t❤❡ ❛❞❞✐t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♠❡❛s✉r❛❜❧❡ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ Q(X)✱ ❛♥❞ F(π) : F(Y ) → F(X) ✭❝♦♥tr❛✈❛r✐❛♥t✮ s✉❝❤ t❤❛t F(π)(φ) = φ◦π∗✳ ■♥ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✿ π∗f (p✶,p✷,p✸) = f (p✶ +p✷,p✸)✳ ❋♦r ❡❛❝❤ Y ∈ SX ❛♥❞ φ ∈ F(X)✱ ❞❡✜♥❡ (Y .φ)(P) =

• y∈EY

PX (Y =y)=✵

P(Y = y)αφ(PX |Y =y). ✭✶✶✮ ❚❤✐s t✉r♥s F ✐♥t♦ ❛♥ A ✲♠♦❞✉❧❡ t❤❛t ✇❡ ❞❡♥♦t❡ Fα✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷✾ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ H•(❙,Fα)

❚❤❡ ✶✲❝♦❝②❝❧❡s ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❝♦❧❧❡❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥❛❧s

• φ[X] : Q(X) → R
• X∈ObS s✉❝❤ t❤❛t

✵ = X.φ[Y ]−φ[XY ]+φ[X] ✭✶✷✮

Pr♦♣♦s✐t✐♦♥ ✭❇❛✉❞♦t✲❇❡♥♥❡q✉✐♥✱ ✷✵✶✺❀ ❱✳ ✷✵✶✼✮

❚❤❡ ♦♥❧② ✶✲❝♦❝②❝❧❡s ❛r❡ ❣✐✈❡♥ ❜② ♠✉❧t✐♣❧❡s ♦❢ Sα[X] =

• −

x∈EX P(x)logP(x)

✇❤❡♥ α = ✶

• x∈EX P(x)α −✶

✇❤❡♥ α = ✶

.

• ❧♦❜❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢r❡❡ ❝♦♥st❛♥ts ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r

✵ ♦❢

❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ ❙ ✶ ✱

✶ ❙ ✶

✵

✶ ❙

✵ ✶ ✇❤❡♥

✶

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✵ ✴ ✹✺

Pr♦❜❛❜✐❧✐st✐❝ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ H•(❙,Fα)

❚❤❡ ✶✲❝♦❝②❝❧❡s ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❝♦❧❧❡❝t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥❛❧s

• φ[X] : Q(X) → R
• X∈ObS s✉❝❤ t❤❛t

✵ = X.φ[Y ]−φ[XY ]+φ[X] ✭✶✷✮

Pr♦♣♦s✐t✐♦♥ ✭❇❛✉❞♦t✲❇❡♥♥❡q✉✐♥✱ ✷✵✶✺❀ ❱✳ ✷✵✶✼✮

❚❤❡ ♦♥❧② ✶✲❝♦❝②❝❧❡s ❛r❡ ❣✐✈❡♥ ❜② ♠✉❧t✐♣❧❡s ♦❢ Sα[X] =

• −

x∈EX P(x)logP(x)

✇❤❡♥ α = ✶

• x∈EX P(x)α −✶

✇❤❡♥ α = ✶

.

• ❧♦❜❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ ❢r❡❡ ❝♦♥st❛♥ts ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r β✵ ♦❢

❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ ❙\{✶}✱ H✶(❙,F✶) ∼

= Rβ✵;

H✶(❙,Fα) ∼

= Rβ✵−✶ ✇❤❡♥ α = ✶.

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✵ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ ❈♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡

▲❡t C : ❙ → ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ X t❤❡ s❡t C(X) ♦❢ ❢✉♥❝t✐♦♥s ν : EX → N s✉❝❤ t❤❛t ν :=

x∈EX ν(x) > ✵ ✭❝♦✉♥t✐♥❣ ❢✉♥❝t✐♦♥s✱

❤✐st♦❣r❛♠s✳✳✳✮✳

• ✐✈❡♥

✱ t❤❡ ❛rr♦✇ ✐s ❣✐✈❡♥ ❜②

✶

✳ ▲❡t ❜❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♠❡❛s✉r❛❜❧❡ ✵ ✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ ✱ ❛♥❞ ✭❝♦♥tr❛✈❛r✐❛♥t✮ s✉❝❤ t❤❛t ✳ ❋♦r ❡❛❝❤ ❛♥❞ ✱ ❞❡✜♥❡

✵

✭✶✸✮ ✇❤❡r❡ ✐s ❛ r❡str✐❝t✐♦♥✳ ❚❤✐s t✉r♥s ✐♥t♦ ❛♥ ✲♠♦❞✉❧❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✶ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ ❈♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡

▲❡t C : ❙ → ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ X t❤❡ s❡t C(X) ♦❢ ❢✉♥❝t✐♦♥s ν : EX → N s✉❝❤ t❤❛t ν :=

x∈EX ν(x) > ✵ ✭❝♦✉♥t✐♥❣ ❢✉♥❝t✐♦♥s✱

❤✐st♦❣r❛♠s✳✳✳✮✳

• ✐✈❡♥ π : X → Y ✱ t❤❡ ❛rr♦✇ π∗ := C(π) : C(X) → C(Y ) ✐s ❣✐✈❡♥ ❜②

π∗ν(y) =

x∈π−✶(y) ν(x)✳

▲❡t ❜❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♠❡❛s✉r❛❜❧❡ ✵ ✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ ✱ ❛♥❞ ✭❝♦♥tr❛✈❛r✐❛♥t✮ s✉❝❤ t❤❛t ✳ ❋♦r ❡❛❝❤ ❛♥❞ ✱ ❞❡✜♥❡

✵

✭✶✸✮ ✇❤❡r❡ ✐s ❛ r❡str✐❝t✐♦♥✳ ❚❤✐s t✉r♥s ✐♥t♦ ❛♥ ✲♠♦❞✉❧❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✶ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ ❈♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡

▲❡t C : ❙ → ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ X t❤❡ s❡t C(X) ♦❢ ❢✉♥❝t✐♦♥s ν : EX → N s✉❝❤ t❤❛t ν :=

x∈EX ν(x) > ✵ ✭❝♦✉♥t✐♥❣ ❢✉♥❝t✐♦♥s✱

❤✐st♦❣r❛♠s✳✳✳✮✳

• ✐✈❡♥ π : X → Y ✱ t❤❡ ❛rr♦✇ π∗ := C(π) : C(X) → C(Y ) ✐s ❣✐✈❡♥ ❜②

π∗ν(y) =

x∈π−✶(y) ν(x)✳

▲❡t G(X) ❜❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♠❡❛s✉r❛❜❧❡ (✵,∞)✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ C(X)✱ ❛♥❞ G(π) : G(Y ) → G(X) ✭❝♦♥tr❛✈❛r✐❛♥t✮ s✉❝❤ t❤❛t G(π)(φ) = φ◦π∗✳ ❋♦r ❡❛❝❤ ❛♥❞ ✱ ❞❡✜♥❡

✵

✭✶✸✮ ✇❤❡r❡ ✐s ❛ r❡str✐❝t✐♦♥✳ ❚❤✐s t✉r♥s ✐♥t♦ ❛♥ ✲♠♦❞✉❧❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✶ ✴ ✹✺

■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✿ ❈♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡

▲❡t C : ❙ → ❙❡ts ❜❡ ❛ ❢✉♥❝t♦r t❤❛t ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ X t❤❡ s❡t C(X) ♦❢ ❢✉♥❝t✐♦♥s ν : EX → N s✉❝❤ t❤❛t ν :=

x∈EX ν(x) > ✵ ✭❝♦✉♥t✐♥❣ ❢✉♥❝t✐♦♥s✱

❤✐st♦❣r❛♠s✳✳✳✮✳

• ✐✈❡♥ π : X → Y ✱ t❤❡ ❛rr♦✇ π∗ := C(π) : C(X) → C(Y ) ✐s ❣✐✈❡♥ ❜②

π∗ν(y) =

x∈π−✶(y) ν(x)✳

▲❡t G(X) ❜❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♠❡❛s✉r❛❜❧❡ (✵,∞)✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ♦♥ C(X)✱ ❛♥❞ G(π) : G(Y ) → G(X) ✭❝♦♥tr❛✈❛r✐❛♥t✮ s✉❝❤ t❤❛t G(π)(φ) = φ◦π∗✳ ❋♦r ❡❛❝❤ Y ∈ SX ❛♥❞ φ ∈ G(X)✱ ❞❡✜♥❡ (Y .φ)(ν) =

• y∈EY

ν(Y =y)=✵

φ(ν|Y =yi).

✭✶✸✮ ✇❤❡r❡ ν|Y =yi ✐s ❛ r❡str✐❝t✐♦♥✳ ❚❤✐s t✉r♥s G ✐♥t♦ ❛♥ A ✲♠♦❞✉❧❡✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✶ ✴ ✹✺

❈♦♠♣✉t✐♥❣ H•(❙,G)

Pr♦♣♦s✐t✐♦♥ ✭❱✳ ✷✵✶✾✮

✶ H✵(❙,G) ❤❛s ❞✐♠❡♥s✐♦♥ ✶ ❛♥❞ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡①♣♦♥❡♥t✐❛❧

❢✉♥❝t✐♦♥✳

✷ ❚❤❡ ✶✲❝♦❝②❝❧❡s ❛r❡ ❣❡♥❡r❛❧✐③❡❞ ✭❋♦♥t❡♥é✲❲❛r❞✮ ♠✉❧t✐♥♦♠✐❛❧

❝♦❡✣❝✐❡♥ts✿

φ[Y ](ν) =

[ν]D!

• y∈EY [ν(y)]D!

✇❤❡r❡ [✵]D! = ✶ ❛♥❞ [n]D! = DnDn−✶ ···D✶✱ ❢♦r ❛♥② s❡q✉❡♥❝❡ {Di}i≥✶ s✉❝❤ t❤❛t D✶ = ✶✳

❚❤❡ ✵✲❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥ r❡❛❞s✿

✶ ✷

✳ ❚❤❡ ✶✲❝♦❝②❝❧❡ ❝♦♥✜t✐♦♥ r❡❛❞s✿ ❡✳❣✳

✶ ✷ ✸ ✶ ✷ ✸ ✶ ✷ ✶ ✷

✿ ✉s✉❛❧ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts❀

✶ ✶ ✿ t❤❡

✲♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✷ ✴ ✹✺

❈♦♠♣✉t✐♥❣ H•(❙,G)

Pr♦♣♦s✐t✐♦♥ ✭❱✳ ✷✵✶✾✮

✶ H✵(❙,G) ❤❛s ❞✐♠❡♥s✐♦♥ ✶ ❛♥❞ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡①♣♦♥❡♥t✐❛❧

❢✉♥❝t✐♦♥✳

✷ ❚❤❡ ✶✲❝♦❝②❝❧❡s ❛r❡ ❣❡♥❡r❛❧✐③❡❞ ✭❋♦♥t❡♥é✲❲❛r❞✮ ♠✉❧t✐♥♦♠✐❛❧

❝♦❡✣❝✐❡♥ts✿

φ[Y ](ν) =

[ν]D!

• y∈EY [ν(y)]D!

✇❤❡r❡ [✵]D! = ✶ ❛♥❞ [n]D! = DnDn−✶ ···D✶✱ ❢♦r ❛♥② s❡q✉❡♥❝❡ {Di}i≥✶ s✉❝❤ t❤❛t D✶ = ✶✳

❚❤❡ ✵✲❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥ r❡❛❞s✿ ϕ(ν) = ϕ(ν✶)ϕ(ν✷)···ϕ(νs)✳ ❚❤❡ ✶✲❝♦❝②❝❧❡ ❝♦♥✜t✐♦♥ r❡❛❞s✿ φ[XY ] = (X.φ[Y ])φ[X] ❡✳❣✳

• n

k✶,k✷,k✸

• =
• n

k✶ +k✷,k✸

• k✶ +k✷

k✶,k✷

• Dn = n✿ ✉s✉❛❧ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts❀ Dn = qn−✶

q−✶ ✿ t❤❡ q✲♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✷ ✴ ✹✺

❆s②♠♣t♦t✐❝ r❡❧❛t✐♦♥

❊①❛♠♣❧❡✿ ✵✲❝♦❝②❝❧❡s✿ t❤❡ ❡①♣♦♥❡♥t✐❛❧ exp(k ν) ✐s ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ✵✲❝♦❝②❝❧❡✱ t❤❡ ❝♦♥st❛♥t k ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✵✲❝♦❝②❝❧❡✳ ✶✲❝♦❝②❝❧❡s✿

• n

p✶n,...,psn

• = exp(nS✶(p✶,...,ps)+o(n))

❛♥❞

• n

p✶n,...,psn

• q

= exp(n✷ lnq

✷ S✷(p✶,...,ps)+o(n✷)).

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✸ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭❙❤❛♥♥♦♥ ❡♥tr♦♣②✮

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ✐❞❡♥t✐t②

• n

p✶n,p✷n,p✸n

• =
• n

(p✶ +p✷)n,p✸n

• (p✶ +p✷)n

p✶n,p✷n

• ❜❡❝♦♠❡s ❛s②♠♣t♦t✐❝❛❧❧②

✶ ✶ ✷ ✸ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✹ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭❙❤❛♥♥♦♥ ❡♥tr♦♣②✮

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ✐❞❡♥t✐t②

• n

p✶n,p✷n,p✸n

• =
• n

(p✶ +p✷)n,p✸n

• (p✶ +p✷)n

p✶n,p✷n

• ❜❡❝♦♠❡s ❛s②♠♣t♦t✐❝❛❧❧②

exp(nS✶(p✶,p✷,p✸)+o(n)) = exp

• n
• S✶(p✶ +p✷,p✸)+(p✶ +p✷)S✶
• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• +o(n)
• .

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✹ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭α✲❡♥tr♦♣②✮

❙✐♥❝❡

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷),

t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r❡❧❛t✐♦♥

✶ ✷ ✸ ✶ ✷ ✸ ✶ ✷ ✶ ✷

✐♠♣❧✐❡s

✷ ✶ ✷ ✸ ✷ ✶ ✷ ✸ ✶ ✷ ✷ ✷ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✺ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭α✲❡♥tr♦♣②✮

❙✐♥❝❡

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷),

t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r❡❧❛t✐♦♥

• n

p✶n,p✷n,p✸n

• q

=

• n

(p✶ +p✷)n,p✸n

• q
• (p✶ +p✷)n

p✶n,p✷n

• q

,

✐♠♣❧✐❡s

✷ ✶ ✷ ✸ ✷ ✶ ✷ ✸ ✶ ✷ ✷ ✷ ✶ ✶ ✷ ✷ ✶ ✷

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✺ ✴ ✹✺

❘❡❝✉rr❡♥❝❡ ✭α✲❡♥tr♦♣②✮

❙✐♥❝❡

• n

p✶n,...,psn

• q

= qn✷S✷(p✶,...,ps)/✷+o(n✷),

t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r❡❧❛t✐♦♥

• n

p✶n,p✷n,p✸n

• q

=

• n

(p✶ +p✷)n,p✸n

• q
• (p✶ +p✷)n

p✶n,p✷n

• q

,

✐♠♣❧✐❡s S✷(p✶,p✷,p✸) = S✷(p✶ +p✷,p✸)+(p✶ +p✷)✷S✷

• p✶

p✶ +p✷

,

p✷ p✶ +p✷

• .

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✺ ✴ ✹✺

❆s②♠♣t♦t✐❝ r❡❧❛t✐♦♥ ■

Pr♦♣♦s✐t✐♦♥ ✭❱✳ ✷✵✶✾✮

▲❡t φ ❜❡ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ✶✲❝♦❝②❝❧❡✳ ❙✉♣♣♦s❡ t❤❛t✱ ❢♦r ❡✈❡r② ✈❛r✐❛❜❧❡ X✱ t❤❡r❡ ❡①✐sts ❛ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ ψ[X] : ∆(X) → R ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❝♦✉♥t✐♥❣ ❢✉♥❝t✐♦♥s {νn}n≥✶ ⊂ CX s✉❝❤ t❤❛t

✶

νn → ∞✱ ❛♥❞

✷ ❢♦r ❡✈❡r② x ∈ EX✱ νn(x)/νn → p(x) ❛s n → ∞

t❤❡ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛

φ[X](νn) = exp(νnαψ[X](p)+o(νnα))

❤♦❧❞s✳ ❚❤❡♥ ψ ✐s ❛ ✶✲❝♦❝②❝❧❡ ♦❢ t②♣❡ α✱ ✐✳❡✳ f ∈ Z ✶(❙,Fα)✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✻ ✴ ✹✺

❖✉t❧✐♥❡

✶

■♥tr♦❞✉❝t✐♦♥ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s

✷

• ❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

✸

■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✼ ✴ ✹✺

❈♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s

▲❡t E ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡✱ ❛♥❞ ❙ ❛ ❝❛t❡❣♦r② ♦❢ s✉❜s♣❛❝❡s ♦❢ E✱ ✇✐t❤ ❛rr♦✇s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✐♥❝❧✉s✐♦♥s✳ ❲❡ s✉♣♣♦s❡ ✐t ✐s ❝♦♥❞✐t✐♦♥❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r ✐♥t❡rs❡❝t✐♦♥s✿ ✐❢ Z,V ,W ❛r❡ ♦❜❥❡❝ts ♦❢ ❙ s✉❝❤ t❤❛t Z ⊂ V ❛♥❞ Z ⊂ W ✱ t❤❡♥ V ∩W ∈ Ob❙✳ ▲❡t E ❜❡ t❤❡ ❢✉♥❝t♦r V → EV := E/V ✱ s❡♥❞✐♥❣ V ⊂ W t♦ t❤❡ ❝❛♥♦♥✐❝❛❧ ♣r♦❥❡❝t✐♦♥ πWV : EV → EW ✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❢✉♥❝t♦r N ♦❢ s✉♣♣♦rts✱ s✉❝❤ t❤❛t NV ❝♦♥t❛✐♥s ❛✣♥❡ s✉❜s♣❛❝❡s ♦❢ E ❛♥❞ N (πWV ) ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ s✉❜s♣❛❝❡s ✉♥❞❡r πWV ✳ ❲❡ s✉♣♣♦s❡ N t♦ ❜❡ ❝❧♦s❡❞ ✉♥❞❡r ❝❡rt❛✐♥ ♦♣❡r❛t✐♦♥s✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✽ ✴ ✹✺

❈♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s

❲❡ t❤❡♥ ✐♥tr♦❞✉❝❡ ❛ s❤❡❛❢ P ♦❢ ❣❛✉ss✐❛♥ ♣r♦❜❛❜✐❧✐t② ❧❛✇s✿ ρ ∈ PV ✐❢ ✐t ✐s s✉♣♣♦rt❡❞ ♦♥ A ∈ NV ✱ ❛♥❞ ✐t ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ❛♥❞ ✇✐t❤ ●❛✉ss✐❛♥ ❞❡♥s✐t② ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❣✐✈❡♥ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ A✳ ❚❤❡ ♠❡❛♥ ❛♥❞ ❝♦✈❛r✐❛♥❝❡ ♦❢ s✉❝❤ ρ ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ✇✐t❤♦✉t ✜①✐♥❣ ❝♦♦r❞✐♥❛t❡s ♦r ❛ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❆s ❜❡❢♦r❡✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ s❤❡❛❢ F ♦❢ ❢✉♥❝t✐♦♥❛❧s ♦❢ ♣r♦❜❛❜✐❧✐t② ❧❛✇s ✇✐t❤ ❙❤❛♥♥♦♥✬s ❛❝t✐♦♥✿ ❢♦r ϕ ∈ FV ✱ (W .ϕ)(ρ) :=

• πWV (A(ρ))

ϕ(ρ|XW =w)dπWV

∗

ρ(w),

✭✶✹✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸✾ ✴ ✹✺

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✵ ✴ ✹✺

❉✐♠❡♥s✐♦♥ ✐s ❛ ❝♦❝②❝❧❡

■❢ A ✐s t❤❡ s✉♣♣♦rt ♦❢ ρ ∈ PV ✱ t❤❡♥ πWV (A) ✐s t❤❡ s✉♣♣♦rt ♦❢ t❤❡ ♠❛r❣✐♥❛❧ ❧❛✇ πWV

∗

ρ ∈ PW ✱ ❛♥❞ (πWV )−✶(w) ✐s t❤❡ s✉♣♣♦rt ♦❢ ρ|XW =w✳ ❖♥❡ ❤❛s t❤❡

❡q✉❛❧✐t②✿ dim(A) = dim(πWV (A))+

• πWV (A)dim((πWV )−✶(w))dπWV

∗

ρ(w) = dim(imπWV |A)+dim(kerπWV |A).

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✶ ✴ ✹✺

❊♥tr♦♣② ✐s ❛ ❝♦❝②❝❧❡

❉✐✛❡r❡♥t✐❛❧ ❡♥tr♦♣② S(ρ) = −

• A(ρ)

dρ dλ ln dρ dλ dλ

✐s ♥♦t ✐♥✈❛r✐❛♥t ✉♥❞❡r ❝❤❛♥❣❡ ♦❢ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛ s❤❡❛❢ X a t❤❛t ❡♥❝♦❞❡s ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❆ s❡❝t✐♦♥ ❝♦rr❡s♣♦♥❞ t♦ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s {φV }V ∈Ob❙✱ s✉❝❤ t❤❛t φV ❞❡♣❡♥❞s ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ρ ♦♥ EV ❛♥❞ ❛ r❡❢❡r❡♥❝❡ ♠❡❛s✉r❡ λ ♦♥ ✐ts s✉♣♣♦rt ✭✇✐t❤ ρ ≪ λ✮✱ ❛♥❞

∀C > ✵, φV (ρ,Cλ) = φV (ρ,λ)+alnC.

✭✶✺✮ ❚❤❡ ❡♥tr♦♣② S ❞❡✜♥❡s ❛ s❡❝t✐♦♥ ♦❢ X −✶✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✷ ✴ ✹✺

❚❤❡♦r❡♠ ✭❱✳✱ ✷✵✶✾✮

❋♦r ❡✈❡r② a ∈ R✱ t❤❡ ❝♦❤♦♠♦❧♦❣② H✶(❙,X a) ♦✈❡r ❛ s✉✣❝✐❡♥t❧② r✐❝❤ ❣r❛ss♠❛♥♥✐❛♥ ✐♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡ ✐s t❤❡ ❛✣♥❡ s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡ ♠❛❞❡ ❜② t❤❡ ❢✉♥❝t✐♦♥s

ΦV (ρ) = −aS(ρ)+c.dim(A(ρ)),

✭✶✻✮ ✇❤❡r❡ c ❝❛♥ ❜❡ ❛♥② r❡❛❧ ❝♦♥st❛♥t✳ ❋♦r ❣❛✉ss✐❛♥ ♣r♦❜❛❜✐❧✐t✐❡s✱ t❤❡ ❢❛❝t t❤❛t ❞✐✛❡r❡♥t✐❛❧ ❡♥tr♦♣② ✐s ❛ ✶✲❝♦❝②❝❧❡ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❙❝❤✉r✬s ❞❡t❡r♠✐♥❛♥t❛❧ ❢♦r♠✉❧❛ det

A

B C D

• = det(A)det(D −BA−✶C);

✭✶✼✮

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✸ ✴ ✹✺

❊①t❡♥s✐♦♥s ❛♥❞ ♦♣❡♥ ♣r♦❜❧❡♠s

❚❤❡r❡ ✐s ❛ q✉❛♥t✉♠ ✈❡rs✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✱ ✇❤❡r❡ ❱♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣② ❛♣♣❡❛rs ❛s ❛ ✵✲❝♦❝❤❛✐♥✱ ✇❤♦s❡ ❝♦❜♦✉♥❞❛r② ✐s r❡❧❛t❡❞ t♦ ❙❤❛♥♥♦♥ ❡♥tr♦♣②✳ ❲❤❛t ✐s t❤❡ r♦❧❡ ♦❢ ♦t❤❡r q✉❛♥t✉♠ ❡♥tr♦♣✐❡s❄ ❘❡❧❛t✐♦♥s ✇✐t❤ ❡♥t❛♥❣❧❡♠❡♥t❄ ❋✉♥❝t♦r✐❛❧ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❝❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ✐♥❢♦r♠❛t✐♦♥ ❡✳❣✳ t❤r♦✉❣❤ ❣❡♦♠❡tr✐❝ q✉❛♥t✐③❛t✐♦♥✳ ❚❤❡ s❛♠❡ ❢♦r♠❛❧✐s♠ ❣✐✈❡s ♦t❤❡r ❞❡r✐✈❡❞ ❢✉♥❝t♦rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❢✉♥❝t♦rs ♦❢ t❤❡ ❣❧♦❜❛❧ s❡❝t✐♦♥s ❢✉♥❝t♦r Γ❙(−)✳ ❲❤❛t ✐s t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣② ❛♥❞ t❤✐s ❝♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t❡①t✉❛❧✐t②❄ ■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣② ✐♥ ❤✐❣❤❡r ❞❡❣r❡❡s❄ ❄ Pr♦❞✉❝ts ✐♥ ❝♦❤♦♠♦❧♦❣②❄ ✲❞❡❢♦r♠❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ❢♦r ✢❛❣s❄ ❆r❡ t❤❡r❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ♠♦❞❡❧s ❢♦r ♦t❤❡r ✲❡♥tr♦♣✐❡s❄

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✹ ✴ ✹✺

❊①t❡♥s✐♦♥s ❛♥❞ ♦♣❡♥ ♣r♦❜❧❡♠s

❚❤❡r❡ ✐s ❛ q✉❛♥t✉♠ ✈❡rs✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣②✱ ✇❤❡r❡ ❱♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣② ❛♣♣❡❛rs ❛s ❛ ✵✲❝♦❝❤❛✐♥✱ ✇❤♦s❡ ❝♦❜♦✉♥❞❛r② ✐s r❡❧❛t❡❞ t♦ ❙❤❛♥♥♦♥ ❡♥tr♦♣②✳ ❲❤❛t ✐s t❤❡ r♦❧❡ ♦❢ ♦t❤❡r q✉❛♥t✉♠ ❡♥tr♦♣✐❡s❄ ❘❡❧❛t✐♦♥s ✇✐t❤ ❡♥t❛♥❣❧❡♠❡♥t❄ ❋✉♥❝t♦r✐❛❧ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❝❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ✐♥❢♦r♠❛t✐♦♥ ❡✳❣✳ t❤r♦✉❣❤ ❣❡♦♠❡tr✐❝ q✉❛♥t✐③❛t✐♦♥✳ ❚❤❡ s❛♠❡ ❢♦r♠❛❧✐s♠ ❣✐✈❡s ♦t❤❡r ❞❡r✐✈❡❞ ❢✉♥❝t♦rs✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❢✉♥❝t♦rs ♦❢ t❤❡ ❣❧♦❜❛❧ s❡❝t✐♦♥s ❢✉♥❝t♦r Γ❙(−)✳ ❲❤❛t ✐s t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ✐♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣② ❛♥❞ t❤✐s ❝♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t❡①t✉❛❧✐t②❄ ■♥❢♦r♠❛t✐♦♥ ❝♦❤♦♠♦❧♦❣② ✐♥ ❤✐❣❤❡r ❞❡❣r❡❡s❄ Ext(M,N)❄ Pr♦❞✉❝ts ✐♥ ❝♦❤♦♠♦❧♦❣②❄ q✲❞❡❢♦r♠❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ❢♦r ✢❛❣s❄ ❆r❡ t❤❡r❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ♠♦❞❡❧s ❢♦r ♦t❤❡r α✲❡♥tr♦♣✐❡s❄

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✹ ✴ ✹✺

P✳ ❇❛✉❞♦t ❛♥❞ ❉✳ ❇❡♥♥❡q✉✐♥✱ ❚❤❡ ❤♦♠♦❧♦❣✐❝❛❧ ♥❛t✉r❡ ♦❢ ❡♥tr♦♣②✱ ❊♥tr♦♣②✱ ✶✼ ✭✷✵✶✺✮✱ ♣♣✳ ✸✷✺✸✕✸✸✶✽✳ ❏✳ P✳ ❱✐❣♥❡❛✉①✱ ❆ ❝♦♠❜✐♥❛t♦r✐❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r ts❛❧❧✐s ✷✲❡♥tr♦♣②✱ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✽✵✼✳✵✺✶✺✷✱ ✭✷✵✶✽✮✳ ❏✳ P✳ ❱✐❣♥❡❛✉①✱ ■♥❢♦r♠❛t✐♦♥ t❤❡♦r② ✇✐t❤ ✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡s✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ ✭✷✵✶✾✮✳ ❚♦ ❛♣♣❡❛r✳ ❏✳ P✳ ❱✐❣♥❡❛✉①✱ ❚♦♣♦❧♦❣② ♦❢ ❙t❛t✐st✐❝❛❧ ❙②st❡♠s✿ ❆ ❈♦❤♦♠♦❧♦❣✐❝❛❧ ❆♣♣r♦❛❝❤ t♦ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ P❤❉ t❤❡s✐s✱ ❯♥✐✈❡rs✐té P❛r✐s ❉✐❞❡r♦t✱ ✷✵✶✾✳

❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹✺ ✴ ✹✺