tr ts - - PowerPoint PPT Presentation

❈✉❜✐❝❛❧✱ ❈❛t❡❣♦r✐❝❛❧ ❛♥❞ ❈♦♥t✐♥✉♦✉s ✖ ❈♦♠♣❛r✐♥❣ ❈♦❤♦♠♦❧♦❣✐❡s ❈♦♠✐♥❣ ❋r♦♠ k✲●r❛♣❤s

❏✐❛♥❝❤❛♦ ❲✉

❚❡①❛s ❆✫▼ ❯♥✐✈❡rs✐t② ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊❧✐③❛❜❡t❤ ●✐❧❧❛s♣② ✭❯♥✐✈❡rs✐t② ♦❢ ▼♦♥t❛♥❛✮

◆♦✈❡♠❜❡r ✶✵✱ ✷✵✶✾

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶ ✴ ✷✶

▼♦t✐✈❛t✐♦♥s ❛♥❞ ♦✈❡r✈✐❡✇

❞✐r❡❝t❡❞ ❣r❛♣❤ Λ

→♣r♦✈✐❞❡ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡s→ ←❤❡❧♣ ✉♥❞❡rst❛♥❞ ❞②♥❛♠✐❝s←

• ✏✐♥✜♥✐t❡ ♣❛t❤ s♣❛❝❡

❝♦♥str✉❝t✐♦♥✑

• C ∗(Λ)
• ✭❣r❛♣❤ C∗✲❛❧❣❡❜r❛✮

s②♠❜♦❧✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠

• ❣r♦✉♣♦✐❞ GΛ

st❛♥❞❛r❞ C ∗(GΛ)

∼ =

• ✭❣r♦✉♣♦✐❞ C∗✲❛❧❣❡❜r❛✮ .

❆ ❣r❛♣❤ ✐s ❛ ✏✶✲❞✐♠❡♥s✐♦♥❛❧ ♦❜❥❡❝t✑✳ ❍✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ k✲❣r❛♣❤ Λ, ✷✲❝♦❝②❝❧❡ ω

→❛ ❧♦t ♠♦r❡ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡s✱ ❡✳❣✳✱ ◆❈ t♦r✐→

✭❆❧❧ ❯❈❚ ❑✐r❝❤❜❡r❣ ❛❧❣❡❜r❛s ❂ ❞✐r❡❝t ❧✐♠✐ts ♦❢ t❤❡s❡✮

• ✏✐♥✜♥✐t❡ ♣❛t❤ s♣❛❝❡

❝♦♥str✉❝t✐♦♥✑

• t✇✐st❡❞ k✲❣r❛♣❤ C∗✲❛❧❣

C ∗(Λ, ω)

• ✭❞②♥❛♠✐❝❛❧ s②st❡♠✮
• ❣r♦✉♣♦✐❞ GΛ, ✷✲❝♦❝②❝❧❡ cω

C ∗(GΛ, cω)

t✇✐st❡❞ ❣r♦✉♣♦✐❞ C∗✲❛❧❣

∼ =

• ⇒ ❉❡✜♥❡ ❛♥❞ st✉❞② t❤❡ ❝♦❤♦♠♦❧♦❣✐❡s ♦❢ Λ ❛♥❞ GΛ✳ ❱❛r✐♦✉s ❝♦❤♦♠♦❧♦❣✐❡s✿

❈✉❜✐❝❛❧ ❝♦❤♦♠♦❧♦❣② ❢♦r Λ✿ ❡❛s② t♦ ❝♦♠♣✉t❡✳ ❈❛t❡❣♦r✐❝❛❧ ❝♦❤♦♠♦❧♦❣② ❢♦r Λ✿ ❜❡tt❡r ❢✉♥❝t♦r✐❛❧✐t②✳ ❈♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣② ❢♦r GΛ✿ ♠♦st ✢❡①✐❜❧❡✳ ❲❡ ✐❞❡♥t✐❢② t❤❡ ✜rst t✇♦ ❛♥❞ ✐♥❞✐❝❛t❡ t❤❡ r❡❧❛t✐♦♥ t♦ t❤❡ t❤✐r❞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✷ ✴ ✷✶

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶ ❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ✭k✲❣r❛♣❤s✮ ✷ ❈✉❜✐❝❛❧ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ✸ ●r♦✉♣♦✐❞✱ C ∗✲❛❧❣❡❜r❛s ❛♥❞ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣②

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✸ ✴ ✷✶

❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s

❈♦♥✈❡♥t✐♦♥✿ ✵ ∈ N✳ ▲❡t k ∈ Z+✳

❉❡✜♥✐t✐♦♥ ✭❑✉♠❥✐❛♥✲P❛s❦✮

❆ k✲❣r❛♣❤ ✐s ❛ ❝♦✉♥t❛❜❧❡ ❝❛t❡❣♦r② Λ ✇✐t❤ ❛ ❞❡❣r❡❡ ♠❛♣ d : Λ → Nk s❛t✐s❢✳ d(µν) = d(µ) + d(ν) ✭✇r✐t❡ Λn := d−✶(n)✮ ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt②✿ ✐❢ λ ∈ Λm+n t❤❡♥ t❤❡r❡ ❛r❡ ✉♥✐q✉❡ µ ∈ Λm ❛♥❞ ν ∈ Λn s✳t✳ λ = µν✳ ❍♦✇ t♦ t❤✐♥❦ ♦❢ t❤❡♠ ❛s ❣r❛♣❤s❄ ❈♦♥str✉❝t ❛ k✲❝♦❧♦r❡❞ ❞✐r❡❝t❡❞ ❣r❛♣❤✿ ④✈❡rt✐❝❡s⑥ ❂ Λ✵✱ ④❡❞❣❡s ✇✐t❤ ❝♦❧♦r i⑥ ❂ Λei✳ ✭e✶, . . . , ek ❛r❡ t❤❡ st❛♥❞❛r❞ ❣❡♥❡r❛t♦rs ♦❢ Nk✮✳ ❊❛❝❤ λ ∈ Λei+ej = Λej+ei ❞❡t❡r♠✐♥❡s ❛ ❝♦♠♠✉t✐♥❣ sq✉❛r❡ ✈✐❛ t❤❡ t✇♦ ❢❛❝t♦r✐③❛t✐♦♥s✿ λ = µ ν = ν′ µ′ ⋔ ⋔ ⋔ ⋔ Λei Λej Λej Λei , ❞✐❛❣r❛♠✿ ν µ ν′ µ′ λ ✳ ❊❛❝❤ λ ∈ Λei+ej+ek = Λej+ei+ek = . . . ❞❡t❡r♠✐♥❡s ❛ ❝♦♠♠✉t✐♥❣ ❝✉❜❡ ✈✐❛ t❤❡ s✐① ❢❛❝t♦r✐③❛t✐♦♥s✱ ❛♥❞ s♦ ♦♥✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✹ ✴ ✷✶

❉❡✜♥✐t✐♦♥ ✭❑✉♠❥✐❛♥✲P❛s❦✱ r❡♣❡❛t❡❞✮

❆ k✲❣r❛♣❤ ✐s ❛ ❝♦✉♥t❛❜❧❡ ❝❛t❡❣♦r② Λ ✇✐t❤ ❛ ❞❡❣r❡❡ ♠❛♣ d : Λ → Nk s❛t✐s❢✳ d(µν) = d(µ) + d(ν) ✭✇r✐t❡ Λn := d−✶(n)✮ ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt②✿ ✐❢ λ ∈ Λm+n t❤❡♥ t❤❡r❡ ❛r❡ ✉♥✐q✉❡ µ ∈ Λm ❛♥❞ ν ∈ Λn s✳t✳ λ = µν✳ ❊①❛♠♣❧❡s✿ ✶✲❣r❛♣❤ ❂ ✭t❤❡ ♣❛t❤ ❝❛t❡❣♦r② ♦❢✮ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤✱ d = ❧❡♥❣t❤✳ ❚❤❡ s❡♠✐❣r♦✉♣ Nk ✇✐t❤ ❛ s✐♥❣❧❡ ♦❜❥❡❝t ❛♥❞ d = ✐❞✳ ❚❤❡ ♣♦s❡t ❝❛t❡❣♦r② (Nk, ≤) =: Ωk ✇✐t❤ t❤❡ ❞❡❣ ♦❢ t❤❡ ✉♥✐q✉❡ ♠♦r♣❤✐s♠ (− → n , − → m) ❢r♦♠ − → m t♦ − → n ❞❡✜♥❡❞ t♦ ❜❡ − → m − − → n ✳ ✭❛ k✶✲❣r❛♣❤✮ × ✭❛ k✷✲❣r❛♣❤✮ ❂ ✭❛ (k✶ + k✷)✲❣r❛♣❤✮✳ ❍♦✇ ❞♦ ✇❡ ✧❞r❛✇✧ ❛ k✲❣r❛♣❤❄

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✺ ✴ ✷✶

❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ❛s ✏❝✉❜❡ ❝♦♠♣❧❡①❡s✑✳

✶✳ ❈♦♥str✉❝t ❛ k✲❝♦❧♦r❡❞ ❞✐r❡❝t❡❞ ❣r❛♣❤✿ ④✈❡rt✐❝❡s⑥ ❂ Λ✵✱ ④❡❞❣❡s ✇✐t❤ ❝♦❧♦r i⑥ ❂ Λei✱ ❛s ❜❡❢♦r❡✳ ✷✳ ❋♦r n = ✵, ✶, . . . , k✱ ❞❡✜♥❡ t❤❡ s❡t ♦❢ n✲❝✉❜❡s Qn(Λ) :=

• ✶≤ei✶<...<ein≤k

Λei✶+...+ein . ∀ i ∈ {✶, . . . , n}✱ ❡❛❝❤ λ ∈ Λei✶+...+ein ⊂ Qn ❤❛s ❛ ❢r♦♥t ❢❛❝❡ F ✵

i (λ) ∈ Qn−✶ ❛♥❞ ❛ ❜❛❝❦ ❢❛❝❡ F ✶ i (λ) ∈ Qn−✶ s❛t✐s❢②✐♥❣

λ = F ✵

i (λ)µ = νF ✶ i (λ) ✇✐t❤ d(µ) = d(ν) = ei✳

✸✳ ❇✉✐❧❞ ❛ ❝♦♠♣❧❡① ♦♥ t❤❡ k✲❝♦❧♦r❡❞ ❣r❛♣❤✿ ✐♥❞✉❝t✐✈❡❧② ❢♦r n = ✷, ✸, . . . , k✱ ❢♦r ❡❛❝❤ λ ∈ Qn✱ ❣❧✉❡ ❛♥ n✲❝✉❜❡ t♦ ✐ts ❢r♦♥t ❛♥❞ ❜❛❝❦ ❢❛❝❡s✱ ✐✳❡✳ F ε

i (λ) ❢♦r i ∈ {✶, . . . , n} ❛♥❞ ε ∈ {✵, ✶}✳

✭◆♦t❡✿ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt② ❢♦r❝❡s t❤❡ ♣❧❛❝❡♠❡♥t ♦❢ ♠❛♥② ❝✉❜❡s✦✮ ❚❤❡r❡❢♦r❡✱ Λ ✐ts t♦♣♦❧♦❣✐❝❛❧ r❡❛❧✐③❛t✐♦♥ |Λ|✳ ❊①❛♠♣❧❡s✿ |Nk| ∼ = Tk✳ ❍♦✇ t♦ ❣❡t t❤❡ ❑❧❡✐♥ ❜♦tt❧❡ ❛s |Λ|❄

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✻ ✴ ✷✶

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶ ❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ✭k✲❣r❛♣❤s✮ ✷ ❈✉❜✐❝❛❧ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ✸ ●r♦✉♣♦✐❞✱ C ∗✲❛❧❣❡❜r❛s ❛♥❞ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣②

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✼ ✴ ✷✶

❈✉❜✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ❬❑✉♠❥✐❛♥✲P❛s❦✲❙✐♠s❪

∀n ∈ Z✱ Qn(Λ) = ④n✲❝✉❜❡s⑥ ❛s ❜❡❢♦r❡ ✭= ∅ ✐❢ n < ✵ ♦r n > k✮✳ ❈✉❜✐❝❛❧ n✲❝❤❛✐♥s✿ Cn(Λ) := ZQn(Λ)✳ ❇♦✉♥❞❛r② ♠❛♣s✿ ∂n : Cn(Λ) → Cn−✶(Λ) s✳t✳ λ ∈ Qn(Λ) → n

j=✶

✶

ε=✵(−✶)j+εF ε j (λ)✳

✭= ⇒ ∂n−✶ ◦ ∂n = ✵✮✳

❉❡✜♥✐t✐♦♥✿ ❚❤❡ n✲t❤ ❝✉❜✐❝❛❧ ❤♦♠♦❧♦❣② ❣r♦✉♣ ♦❢ Λ

H❝✉❜

n

(Λ) = H❝✉❜

n

(Λ, Z) := ❦❡r(∂n)/✐♠(∂n+✶) =: Z ❝✉❜

n

(Λ)/B❝✉❜

n

(Λ) . ❈✉❜✐❝❛❧ ✭T✲✈❛❧✉❡❞✮ n✲❝♦❝❤❛✐♥s✿ ❍♦♠(Cn(Λ), T)✳ ❈♦❜♦✉♥❞❛r② ♠❛♣s✿ ∂n+✶ δn : ❍♦♠(Cn(Λ), T) → ❍♦♠(Cn+✶(Λ), T)✳

❉❡✜♥✐t✐♦♥✿ ❚❤❡ n✲t❤ ❝✉❜✐❝❛❧ ✭T✲✈❛❧✉❡❞✮ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣ ♦❢ Λ

Hn

❝✉❜(Λ, T) := ❦❡r(δn)/✐♠(δn−✶) =: Z n ❝✉❜(Λ, T)/Bn ❝✉❜(Λ, T) .

Pr♦♣❡rt✐❡s✿ ✶✳ ✈❡r② ❝♦♠♣✉t❛❜❧❡✦ ✷✳ ✈❛♥✐s❤✐♥❣ ✇❤❡♥ n < ✵ ♦r n > k✳ ✸✳ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ s✐♠♣❧✐❝✐❛❧✴s✐♥❣✉❧❛r ✭❝♦✲✮❤♦♠♦❧♦❣② ♦❢ |Λ|✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✽ ✴ ✷✶

❈❛t❡❣♦r✐❝❛❧ ❤♦♠♦❧♦❣②

n✲❝♦♠♣♦s❛❜❧❡ t✉♣❧❡s✿ ❢♦r n ∈ Z+✱ Λ(n) := {(λ✶, . . . , λn) ∈ Λn : s(λi) = r(λi+✶), ∀i}✳ ❆❧s♦ Λ(✵) := ❖❜❥ Λ = {λ ∈ Λ: d(λ) = ✵} ❛♥❞ Λ(n) := ∅ ✐❢ n < ✵✳ ❈❛t❡❣♦r✐❝❛❧ n✲❝❤❛✐♥s✿ Fn(Λ) := ZΛ(n) ❢♦r n ∈ Z✳ ❇♦✉♥❞❛r② ♠❛♣s✿ ∂n : Fn(Λ) → Fn−✶(Λ) s✳t✳ (λ✶, . . . , λn) ∈ Λ(n) → (λ✷, . . . , λn)+

n−✶

• i=✶

(−✶)i(λ✶, . . . , λiλi+✶, . . . , λn)+(−✶)n(λ✶, . . . , λn−✶) ❈❤❡❝❦✿ ∂n−✶ ◦ ∂n = ✵✳

❉❡✜♥✐t✐♦♥✿ ❚❤❡ n✲t❤ ❝❛t❡❣♦r✐❝❛❧ ❤♦♠♦❧♦❣② ❣r♦✉♣ ♦❢ Λ

H❝❛t

n (Λ) = H❝❛t n (Λ, Z) := ❦❡r(∂n)/✐♠(∂n+✶) =: Z ❝❛t n (Λ)/B❝❛t n (Λ) .

❚❤✐♥❦✐♥❣ t♦♣♦❧♦❣✐❝❛❧❧②✿ ❢♦r♠ ❛ ✭t②♣✐❝❛❧❧② ∞✲❞✐♠❡♥s✐♦♥❛❧✮ s✐♠♣❧✐❝✐❛❧ ❝♦♠♣❧❡① ∆(Λ) s✳t✳ ④✈❡rt✐❝❡s⑥ ❂ ❖❜❥ Λ ❛♥❞ ❡❛❝❤ (λ✶, . . . , λn) ∈ Λ(n) ❞❡t❡r♠✐♥❡s ❛♥ n✲s✐♠♣❧❡① ✇✐t❤ ✈❡rt✐❝❡s r(λ✶), . . . , r(λn) ❛♥❞ s(λn)✳ ❚❤❡♥ H❝❛t

n (Λ) ❂ t❤❡ ✭s✐♠♣❧✐❝✐❛❧✮ ❤♦♠♦❧♦❣② ♦❢ ∆(Λ)✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✾ ✴ ✷✶

❈❛t❡❣♦r✐❝❛❧ ❝♦❤♦♠♦❧♦❣②

❈❛t❡❣♦r✐❝❛❧ ✭T✲✈❛❧✉❡❞✮ n✲❝♦❝❤❛✐♥s✿ ❍♦♠(Fn(Λ), T)✳ ❈♦❜♦✉♥❞❛r② ♠❛♣s✿ ∂n+✶ δn : ❍♦♠(Fn(Λ), T) → ❍♦♠(Fn+✶(Λ), T)✳

❉❡✜♥✐t✐♦♥✿ ❚❤❡ n✲t❤ ❝❛t❡❣♦r✐❝❛❧ ✭T✲✈❛❧✉❡❞✮ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣ ♦❢ Λ

Hn

❝❛t(Λ, T) := ❦❡r(δn)/✐♠(δn−✶) =: Z n ❝❛t(Λ, T)/Bn ❝❛t(Λ, T) .

■♥ ❧♦✇ ❞✐♠❡♥s✐♦♥s✿ Z ✶

❝❛t = {f : Λ → T | f (λ✶)f (λ✷) = f (λ✶λ✷), ∀(λ✶, λ✷) ∈ Λ(✷)} ,

Z ✷

❝❛t = {f : Λ(✷) → T | f (λ✶λ✷, λ✸)f (λ✶, λ✷) = f (λ✶, λ✷λ✸)f (λ✷, λ✸),

∀(λ✶, λ✷, λ✸) ∈ Λ(✸)} . Pr♦♣❡rt✐❡s✿ ✶✳ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ k✲❣r❛♣❤ str✉❝t✉r❡❀ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❢♦r ❛♥② ❝❛t❡❣♦r②❀ ✷✳ ❜❡tt❡r ❢✉♥❝t♦r❛❧✐t② ❛♥❞ ✢❡①✐❜✐❧✐t②❀ ✸✳ ✷✲❝♦❝②❝❧❡s ❝❛♥ ❜❡ ✉s❡❞ t♦ t✇✐st t❤❡ ♣❛t❤ ❛❧❣❡❜r❛ ✴ C ∗✲❛❧❣❡❜r❛❀ ✹✳ ❤❛r❞ t♦ ❝♦♠♣✉t❡✦

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✵ ✴ ✷✶

❈✉❜✐❝❛❧ ∼ = ❝❛t❡❣♦r✐❝❛❧

❚❤❡♦r❡♠ ✭❑✉♠❥✐❛♥✲P❛s❦✲❙✐♠s✮

❚❤❡r❡ ❛r❡ ✐s♦♠♦r♣❤✐s♠s Hn

❝✉❜(Λ, T) ∼

= Hn

❝❛t(Λ, T) ❢♦r n = ✵, ✶, ✷✳

■❞❡❛✿ ❊❛s② ✈❡r✐✜❝❛t✐♦♥ ❢♦r n = ✵, ✶✳ ❋♦r n = ✷✱ ✐❞❡♥t✐❢② ❜♦t❤ ✇✐t❤ ❊①t(Λ, T) ✿❂ ④✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ❝❡♥tr❛❧ ❡①t❡♥s✐♦♥s ♦❢ Λ ❜② T⑥✳ ◗✿ ❖t❤❡r n❄ ❆ ♠♦r❡ ♥❛t✉r❛❧ ♣r♦♦❢❄

❚❤❡♦r❡♠ ✭●✐❧❧❛s♣②✲❲✮

❚❤❡r❡ ❛r❡ ✐s♦♠♦r♣❤✐s♠s H❝✉❜

n

(Λ) ∼ = H❝❛t

n (Λ) ❛♥❞ Hn ❝✉❜(Λ, T) ∼

= Hn

❝❛t(Λ, T)✱

❢♦r n ∈ Z✱ ❛♥❞ t❤❡② ❛r❡ ✐♥❞✉❝❡❞ ❜② ❝❤❛✐♥ ♠❛♣s ▽n : Cn(Λ) → Fn(Λ) ❛♥❞ n : Fn(Λ) → Cn(Λ) ✭✐✳❡✳✱ t❤❡② ✐♥t❡rt✇✐♥❡ t❤❡ ❜♦✉♥❞❛r② ♠❛♣s✱ ❡✳❣✳✱ ▽n ◦ ∂n+✶ = ∂n+✶ ◦ ▽n+✶✮✳

❈♦r♦❧❧❛r②

❚❤❡ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ♦❢ ❛ k✲❣r❛♣❤ ♦♥❧② ❞❡♣❡♥❞s ♦♥ ✐ts t♦♣♦❧♦❣✐❝❛❧ r❡❛❧✐③❛t✐♦♥✱ ❛♥❞ ✈❛♥✐s❤❡s ✇❤❡♥ n > k ♦r n < ✵✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✶ ✴ ✷✶

❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❝✉❜❡s ▽n : Cn(Λ) → Fn(Λ)

∀ λ ∈ Qn ✇✐t❤ d(λ) = ei✶ + . . . + ein ✭i✶ < . . . < in✮✱ ✇❡ ❞❡✜♥❡ ▽n(λ) :=

• σ∈Sn

s❣♥(σ)(λσ

✶, . . . , λσ n)

✇❤❡r❡ λ = λσ

✶ · · · λσ n ❛♥❞ d(λσ j ) = eiσ(j)✱ ∀j✳

λ µ′ µ ν ν′ → µ ν − µ′ ν′ ■t ❣✐✈❡s ❛ ❝♦♥t✐♥✉♦✉s ♠❛♣ ❢r♦♠ t❤❡ t♦♣♦❧✳ r❡❛❧✳ |Λ| t♦ |∆(Λ)|✳ ▼♦r❡♦✈❡r✱ ✐t ✐♥❞✉❝❡s ▽n : ❍♦♠(Fn(Λ), T) → ❍♦♠(Cn(Λ), T)✳

❊①❛♠♣❧❡

❲❤❡♥ n = ✷✱ ω ∈ ❍♦♠(F✷(Λ), T) ❛♥❞ λ ∈ C✷(Λ)✱ ▽n(ω)(λ) = ω(µ, ν)ω(ν′, µ′) ✇❤❡r❡ λ = µν = ν′µ′ ❛s ❛❜♦✈❡✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✷ ✴ ✷✶

❈✉❜✉❧❛t✐♦♥ ♦❢ tr✐❛♥❣❧❡s ✭s✐♠♣❧✐❝❡s✮ n : Fn(Λ) → Cn(Λ)

❋♦r (λ✶, . . . , λn) ∈ Λ(n)✱ ✇❡ ❞❡✜♥❡ n(λ✶, . . . , λn) :=

• k✵,...,kn−✶∈N

✶≤k✵<...<kn−✶≤k

λ

• n
• i=✶

k

• j=ki+✶

dj(λi)ej,

n

• i=✶

k

• j=ki

dj(λi)ej

• ✇❤❡r❡ dj(−) ✐s t❤❡ j✲t❤ ❝♦♦r❞✳ ♦❢ d(−)✱ ❛♥❞ ❡❛❝❤ s✉♠♠❛♥❞ r❡♣r❡s❡♥ts ❛♥

n✲❞✐♠ r❡❝t❛♥❣❧❡✱ ✇❤✐❝❤ ✇❡ ✈✐❡✇ ❛s ❛ s✉♠ ♦❢ t❤❡ n✲❝✉❜❡s ✐t ❝♦♥t❛✐♥s✳ λ → λ µ λµ → → →

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✸ ✴ ✷✶

❖❜s❡r✈❡ t❤❛t

• ▽ = ✐❞C∗(Λ)✳

▽ ◦ ✐s ❤♦♠♦t♦♣✐❝ t♦ ✐❞F∗(Λ) ✭❜② ❜❛s✐❝ ❤♦♠♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛✮✳ ⇒ ❚❤❡② ✐♥❞✉❝❡ ∼ = ❜❡t✇❡❡♥ t❤❡ ✭❝♦✲✮❤♦♠♦❧♦❣✐❡s✳ ■♥ ❢❛❝t✱ t❤❡r❡ ✐s ❛ ♠♦r❡ ❞✐r❡❝t ✇❛② t♦ s❤♦✇ ∼ = ❜❡t✇❡❡♥ t❤❡ ✭❝♦✲✮❤♦♠♦❧♦❣✐❡s✿ ✇❡ ❝❛♥ ❛❝t✉❛❧❧② s❤♦✇ t❤❛t C∗(Λ) ❝❛♥ ❜❡ ❧✐❢t❡❞ t♦ ❛ ❢r❡❡ r❡s♦❧✉t✐♦♥ ♦❢ ZΛ ✉s✐♥❣ ❢✉t✉r❡✲♣❛t❤ k✲❣r❛♣❤s✱ ❛♥❞ t❤✉s ✐t ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣✉t❡s t❤❡ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣②✳ ❘❡♠❛r❦ ✶✿ ❲❡ ❝❛♥ ✉s❡ ❛r❜✐tr❛r② Λ✲♠♦❞✉❧❡s ❛s ❝♦❡✣❝✐❡♥ts✳ ❘❡♠❛r❦ ✷✿ ❖♥❡ ❝❛♥ s❤♦✇ |Λ| ❛♥❞ |∆(Λ)| ❛r❡ ❤♦♠♦t♦♣② ❡q✉✐✈❛❧❡♥t✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✹ ✴ ✷✶

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶ ❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ✭k✲❣r❛♣❤s✮ ✷ ❈✉❜✐❝❛❧ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ✸ ●r♦✉♣♦✐❞✱ C ∗✲❛❧❣❡❜r❛s ❛♥❞ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣②

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✺ ✴ ✷✶

k✲●r❛♣❤ C ∗✲❛❧❣❡❜r❛s

❆ss✉♠❡ Λ ✐s r♦✇✲✜♥✐t❡ ❛♥❞ s♦✉r❝❡✲❢r❡❡✿ ∀v ∈ ❖❜❥ Λ✱ ∀n ∈ Nk✱ ✇❡ ❤❛✈❡ ✵ < |{λ ∈ Λ: d(λ) = n, r(λ) = v}| < ∞✳

❉❡✜♥✐t✐♦♥ ✭❑✉♠❥✐❛♥✲P❛s❦✮

C ∗(Λ) ✐s t❤❡ ✉♥✐✈❡rs❛❧ C ∗✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ♣r♦❥❡❝t✐♦♥s pv✿ v ∈ ❖❜❥ Λ ❛♥❞ ♣❛rt✐❛❧ ✐s♦♠❡tr✐❡s sλ✿ λ ✐s ❛♥ ❡❞❣❡ ✐♥ Λ ✭✐✳❡✳✱ |d(λ)|✶ = ✶✮✱ s✉❜❥❡❝t t♦ t❤❡ r❡❧❛t✐♦♥s ✇❤❡♥❡✈❡r λ✷µ✶ = µ✷λ✶✱ ✇❡ ❤❛✈❡ sλ✷sµ✶ = sµ✷sλ✶ ✭ µ✶ λ✷ µ✷ λ✶ ✮✱ pv = s∗

λsλ ✇❤❡♥❡✈❡r s(λ) = v✱

pv =

• λ∈Λei ,r(λ)=v

sλs∗

λ ❢♦r ❛♥② ♦❜❥❡❝t v ❛♥❞ ❝♦❧♦r i✳

❊①❛♠♣❧❡✿ C ∗(Λ × Γ) ∼ = C ∗(Λ) ⊗ C ∗(Γ)✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✻ ✴ ✷✶

❚✇✐st❡❞ k✲❣r❛♣❤ C ∗✲❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥ ✭❑✉♠❥✐❛♥✲P❛s❦✲❙✐♠s✮

▲❡t ω ∈ Z ✷

❝❛t(Λ, T)✳ ❚❤❡♥ C ∗(Λ, ω) ✐s t❤❡ ✉♥✐✈❡rs❛❧ C ∗✲❛❧❣ ❣❡♥❡r❛t❡❞ ❜②

♣r♦❥❡❝t✐♦♥s pv✿ v ∈ ❖❜❥ Λ ❛♥❞ ♣❛rt✐❛❧ ✐s♦♠❡tr✐❡s sλ✿ λ ∈ Λ ✇✐t❤ |d(λ)|✶ = ✶✱ s✉❜❥❡❝t t♦ t❤❡ r❡❧❛t✐♦♥s ✇❤❡♥❡✈❡r λ✷µ✶ = µ✷λ✶✱ ✇❡ ❤❛✈❡ ω(λ✷, µ✶)sλ✷sµ✶ = ω(µ✷, λ✶)sµ✷sλ✶✱ pv = s∗

λsλ ✇❤❡♥❡✈❡r s(λ) = v✱

pv =

• λ∈Λei ,r(λ)=v

sλs∗

λ ❢♦r ❛♥② ♦❜❥❡❝t v ❛♥❞ ❝♦❧♦r i✳

❘❡♠❛r❦✿ ■❢ ω ∈ Z ✷

❝✉❜(Λ, T) ✐♥st❡❛❞ ♦❢ Z ✷ ❝❛t(Λ, T)✱ ✉s❡ t❤❡ r❡❧❛t✐♦♥

ω(λ✷µ✶)sλ✷sµ✶ = sµ✷sλ✶ ✐♥st❡❛❞✱ ✇❤❡r❡ λ✷µ✶ = µ✷λ✶ ✐s ❛ ✷✲❝✉❜❡✳ ❚❤✐s ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❝❤❛✐♥ ♠❛♣ ▽✷✳ ❊①❛♠♣❧❡✿ C ∗(Nk, ω) ✐s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✼ ✴ ✷✶

❚❤❡ ❣r♦✉♣♦✐❞ GΛ

❉❡✜♥✐t✐♦♥

❚❤❡ ✭❣❧♦❜❛❧✮ ♦✉ts♣❧✐tt✐♥❣ ♦❢ ❛ k✲❣r❛♣❤ Λ✱ ✇r✐tt❡♥ Λ(✶)✱ ✐s t❤❡ k✲❣r❛♣❤ ✇❤❡r❡ ④♦❜❥❡❝ts⑥ ✿❂ ④ k✲❝✉❜❡s ✐♥ Λ ⑥ ❂ ④♠♦r♣❤✐s♠s ✐♥ Λ ✇✐t❤ ❞❡❣r❡❡ − → ✶ ⑥✱ ❛ ♠♦r♣❤✐s♠ λ ♦❢ ❞❡❣r❡❡ − → n ✐s ❣✐✈❡♥ ❜② λ ∈ Λ ✇✐t❤ d(λ) = − → ✶ + − → n ✱ ✐ts r❛♥❣❡ r( λ) ✐s λ(− → ✵ , − → ✶ ) ❛♥❞ ✐ts s♦✉r❝❡ s( λ) ✐s λ(d(λ) − − → ✶ , d(λ)) ✳ ■♥❞✉❝t✐✈❡❧②✱ ✇❡ ❞❡✜♥❡ t❤❡ m✲st❡♣ ♦✉ts♣❧✐tt✐♥❣ Λ(m) := (Λ(m−✶))(✶)✳ ❚❤❡r❡ ✐s ❛ ❢♦r❣❡t❢✉❧ ❢✉♥❝t♦r F : Λ(✶) → Λ✱ λ → λ(✵, d(λ) − − → ✶ )✳ ✐♥✈❡rs❡ s②st❡♠ Λ = Λ(✵) ← Λ(✶) ← Λ(✷) ← . . .✳ C ∗(Λ, ω) ∼ = C ∗(Λ(✶), F ∗ω) ∼ = C ∗(Λ(✷), (F ∗)✷ω) ∼ = . . .✳ ❚❤❡ ✐♥✈❡rs❡ ❧✐♠✐t ❧✐♠ ← − Λ(m) =: HΛ ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ s❡♠✐❣r♦✉♣♦✐❞ ✭✐♥ ❢❛❝t ❛ t♦♣♦❧♦❣✐❝❛❧ k✲❣r❛♣❤✮✱ ❝❛❧❧❡❞ t❤❡ ❉❡❛❝♦♥✉✲❘❡♥❛✉❧t s❡♠✐❣r♦✉♣♦✐❞✳

✏●r♦t❤❡♥❞✐❡❝❦✬s ❝♦♥str✉❝t✐♦♥✑

• ❣r♦✉♣♦✐❞ GΛ✱ ✇✐t❤ ❛ ❞❡❣r❡❡ ♠❛♣ d : GΛ → Zk✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✽ ✴ ✷✶

❚❤❡ ❣r♦✉♣♦✐❞ GΛ

k✲❣r❛♣❤ Λ, ✷✲❝♦❝②❝❧❡ ω

t✇✐st❡❞ k✲❣r❛♣❤ C ∗✲❛❧❣❡❜r❛

• m✲st❡♣ ♦✉ts♣❧✐tt✐♥❣
• C ∗(Λ, ω)
• Λ(m), (F ∗)mω

t✇✐st❡❞ k✲❣r❛♣❤ C ∗✲❛❧❣❡❜r❛

• ✐♥✈❡rs❡ ❧✐♠✐t ❛s m → ∞
• C ∗(Λ(m), (F ∗)mω)
• ∼

=

• s❡♠✐❣r♦✉♣♦✐❞ HΛ, F ∗

∞ω

✏●r♦t❤❡♥❞✐❡❝❦✑

❣r♦✉♣♦✐❞ GΛ,

?

cω

❘❡♥❛✉❧t

C ∗(GΛ,

?

cω)

∼ =

• ❚♦ HΛ ❛♥❞ GΛ ✇❡ ❝❛♥ ❛ss♦❝✐❛t❡ t❤❡✐r ❝♦♥t✐♥✉♦✉s ✭❝❛t❡❣♦r✐❝❛❧✮ ❝♦❤♦♠♦❧♦❣②✳

❚❤❡ ♥❛t✉r❛❧ ❢♦r❣❡t❢✉❧ ♠❛♣ F∞ : HΛ → Λ ✐♥❞✉❝❡s F ∗

∞ : Hn ❝❛t(Λ, T) → Hn ❝♦♥t(HΛ, T),

ω → F ∗

∞ω✳

❚❤❡ ♥❛t✉r❛❧ ✐♥❝❧✉s✐♦♥ HΛ → GΛ ✐♥❞✉❝❡s Hn

❝♦♥t(GΛ, T) → Hn ❝♦♥t(HΛ, T)✱

❜✉t ✇❡ ✇❛♥t F ∗

∞ω → cω✳

❚❤❡♦r❡♠ ✭●✐❧❧❛s♣②✲❲✮

❚❤❡ ♥❛t✉r❛❧ ❡♠❜❡❞❞✐♥❣ HΛ ֒ → GΛ ✐♥❞✉❝❡s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣② ♦❢ HΛ ❛♥❞ t❤❛t ♦❢ GΛ✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶✾ ✴ ✷✶

❚❤❡♦r❡♠ ✭●✐❧❧❛s♣②✲❑✉♠❥✐❛♥✲❲ ✭r❡♣❡❛t❡❞✮✮

❚❤❡ ♥❛t✉r❛❧ ❡♠❜❡❞❞✐♥❣ HΛ ֒ → GΛ ✐♥❞✉❝❡s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣② ♦❢ HΛ ❛♥❞ t❤❛t ♦❢ GΛ✳ ∃ ❝♦❝❤❛✐♥ ♠❛♣ βn : C n

❝♦♥t(HΛ, T) → C n ❝♦♥t(GΛ, T) ✉s✐♥❣ t❤❡ ✐❞❡❛ ♦❢

❜❛r②❝❡♥tr✐❝ s✉❜❞✐✈✐s✐♦♥s✳ ❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣✐♣❡❧✐♥❡ ✭k✲❣r❛♣❤ ❝♦❝②❝❧❡s ❣r♦✉♣♦✐❞ ❝♦❝②❝❧❡s✮✿ C n(Λ, T)F ∗ → C n(Λ(✶), T) F ∗ → C n(Λ(✷), T) . . . → C n

❝♦♥t(HΛ, T) βn

→ C n

❝♦♥t(GΛ, T) .

❲❤❡♥ n = ✷✱ t❤✐s r❡❝♦✈❡rs t❤❡ ❑P❙ ❝♦♥str✉❝t✐♦♥ ✭✉♣ t♦ ❛ s✐❣♥✮ ❛♥❞ ❡①♣❧❛✐♥s ❤♦✇ t✇✐st❡❞ k✲❣r❛♣❤ C ∗✲❛❧❣❡❜r❛s ❛r❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t✇✐st❡❞ ❣r♦✉♣♦✐❞ C ∗✲❛❧❣❡❜r❛s✳

❆♣♣❧✐❝❛t✐♦♥✿ ✭✐♥✮❞❡♣❡♥❞❡♥❝❡ ♦❢ K∗(C ∗(Λ, ω)) ✇✳r✳t✳ ω

❘❡❝❛❧❧ ❢r♦♠ ❈❤r✐st✐❛♥✬s t❛❧❦✿ ✐♥❞❡♣❡♥❞❡♥❝❡ ❢♦❧❧♦✇s ❢r♦♠ ❤♦♠♦t♦♣② ❜❡t✇❡❡♥ ❝♦❝②❝❧❡s✳ ❙♦ ✇❤❡♥ ❝❛♥ ✇❡ ❣❡t ❤♦♠♦t♦♣②❄ ◆♦t ❛❧✇❛②s✦ ❚♦rs✐♦♥s ✐♥ Hn(Λ) ♦❜str✉❝ts ❤♦♠♦t♦♣②✳ ❍♦✇❡✈❡r✱ ♥♦♥✲❤♦♠♦t♦♣✐❝ ❝♦❝②❝❧❡s ✐♥ Hn(Λ, T) ♦❢t❡♥ ❜❡❝♦♠❡ ❤♦♠♦t♦♣✐❝ ❛❢t❡r ♦✉t✲s♣❧✐tt✐♥❣✳ ❲❡ ❞♦♥✬t ❦♥♦✇ ❤♦✇ ♦❢t❡♥✳

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✷✵ ✴ ✷✶

❍❛♣♣② ❇✐rt❤❞❛②✱ ❏❡❛♥✦

❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✷✶ ✴ ✷✶