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❇❛♥❛❝❤ ❆❧❣❡❜r❛s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ●♦t❤❡♥❜✉r❣ ✷✾✳✵✼✳✷✵✶✸ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❇❛rt♦s③ ❑♦s♠❛ ❑✇❛➧♥✐❡✇s❦✐

❯♥✐✈❡rs✐t② ♦❢ ❇✐❛➟②st♦❦ ✭s♦♦♥ ■▼P❆◆ ❲❛rs❛✇✮ ❈❤❡❝❦ ♦✉t ❛❧s♦ ♠② ♣♦st❡r ♦♥ ❲❡❞♥❡s❞❛② ✦✦✦

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐✱ ❆✳ ❱✳ ▲❡❜❡❞❡✈ ✏❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❡♥❞♦♠♦r♣❤✐s♠s ❛♥❞ r❡❞✉❝t✐♦♥ ♦❢ r❡❧❛t✐♦♥s ✐♥ r❡❧❛t✐✈❡ ❈✉♥t③✲P✐♠s♥❡r ❛❧❣❡❜r❛s✑ ❏✳ ♦❢ ❋✉♥❝t✳ ❆♥❛❧②s✐s✱ ✷✻✹ ✭✷✵✶✸✮✱ ♥♦✳ ✽✱ ✶✽✵✻✲✶✽✹✼

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

◗✉❡st✐♦♥ ♦❢ t❤❡ ❉❛②✿ ❲❤❛t ✐s ❝r♦ss❡❞ ♣r♦❞✉❝t❄

❚❤r♦✉❣❤♦✉t ❆ ✐s ✉♥✐t❛❧ ❈ ∗✲❛❧❣❡❜r❛✳ ❈r♦ss❡❞ ♣r♦❞✉❝t ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ α : ❆ → ❆ ✐s ❛ ✉♥✐✈❡rs❛❧ ❈ ∗✲❛❧❣❡❜r❛ ❈ ∗(❆, ✉) ❣❡♥❡r❛t❡❞ ❜② ❆ ❛♥❞ ✉ s✉❜❥❡❝t t♦ r❡❧❛t✐♦♥s✿ α(❛) = ✉❛✉∗, α−✶(❛) = ✉∗❛✉, ❛ ∈ ❆ Pr♦❜❧❡♠ ■❢ α : ❆ → ❆ ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠✱ t❤❡♥ α−✶(❛) = ✉∗❛✉✳ ❲❤❛t r❡❧❛t✐♦♥ s❤♦✉❧❞ ✇❡ ✉s❡ ✐♥st❡❛❞❄ ▲❡t ❆ ⊂ ❇ ❜❡ ❈ ∗✲❛❧❣❡❜r❛s ✇✐t❤ ❛ ❝♦♠♠♦♥ ✉♥✐t ✶✱ ❯ ∈ ❇✳ Pr♦♣♦s✐t✐♦♥ ✭t❤❡ ❍✐♥t✮✳ ▲❡t α : ❆ → ❆ ❜❡ ❛ ♠❛♣ ♦❢ t❤❡ ❢♦r♠ α(❛) = ❯❛❯∗✳ ❚❤❡♥ α ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ⇐ ⇒ ❯ ♣❛rt✐❛❧ ✐s♦♠❡tr②✱ ❯∗❯ ∈ ❆′, ✇❤❡r❡ ❆′ ✐s t❤❡ ❝♦♠♠✉t❛♥t ♦❢ ❆✳

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❚♦✇❛r❞s t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥

❆ss✉♠❡ α(❛) = ❯❛❯∗ ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ♦❢ ❆✳ Pr♦♣♦s✐t✐♦♥✳ ✶✮ ❈ ∗(❆, ❯) = s♣❛♥{❯∗♥❛❯♠ : ❛ ∈ ❆, ♥, ♠ ∈ N} ✷✮ ❏ = {❛ ∈ ❆ : ❯∗❯❛ = ❛} = ❯∗❯❆ ∩ ❆ ✐s ❛♥ ✐❞❡❛❧ ✐♥ ❆ s✉❝❤ t❤❛t ❏ ∩ ❦❡r α = {✵} ✭❏ ⊂ (❦❡r α)⊥✮ ❘❡♠✳ ■♥ t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ t❤❡ ❡❧❡♠❡♥ts ❯∗♥❛❯♠ ❛r❡ t❤❡ ✬❜r✐❝❦s✬ t❤❡ ✐❞❡❛❧ ❏ = {❛ ∈ ❆ : ❯∗❯❛ = ❛} ✐s t❤❡ ✬❝❡♠❡♥t✬

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

✶✮ ❘❡❛❞✐♥❣ ♦✛ t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❢r♦♠ ❈ ∗(❆, ❯)

❈♦♥s✐❞❡r ✐♥✜♥✐t❡ ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ❧❛❜❡❧❡❞ ❜② N = {✵, ✶, ✷, ...} M(❆) := {[❛♥,♠] : ❛♥,♠ ∈ ❆ ♦♥❧② ✜♥✐t❡ ❡♥tr✐❡s ♥♦♥ ③❡r♦} ❛♥❞ ❛ ♠❛♣♣✐♥❣ Ψ : M(❆) → ❈ ∗(❆, ❯) ❣✐✈❡♥ ❜② Ψ([❛♥,♠]) =

• ♥,♠∈N

❯∗♥❛♥,♠❯♠ Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ♠❛♣ Ψ ❜❡❝♦♠❡s ❛ ∗✲❤♦♠♦♠♦r♣❤✐s♠ ✐❢ ✇❡ ❞❡✜♥❡ ♦♥ M(❆) t❤❡ ∗✲❛❧❣❡❜r❛ str✉❝t✉r❡ (M(❆), +, ·, ∗, ⋆) ❛s ❢♦❧❧♦✇s (❛ + ❜)♠,♥ := ❛♠,♥ + ❜♠,♥, (✶) (λ❛)♠,♥ := λ❛♠,♥ (✷) (❛∗)♠,♥ := ❛∗

♥,♠

(✸) ❛♥❞ ❛ ⋆ ❜ := ❛ ·

∞

• ❥=✵

Λ❥(❜) +

∞

• ❥=✶

Λ❥(❛) · ❜ (✹)

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

M(❆) := {[❛♥,♠] : ❛♥,♠ ∈ ❆ ♦♥❧② ✜♥✐t❡ ❡♥tr✐❡s ♥♦♥ ③❡r♦} Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ♠❛♣ Ψ ❜❡❝♦♠❡s ❛ ∗✲❤♦♠♦♠♦r♣❤✐s♠ ✐❢ ✇❡ ❞❡✜♥❡ ♦♥ M(❆) t❤❡ ∗✲❛❧❣❡❜r❛ str✉❝t✉r❡ (M(❆), +, ·, ∗, ⋆) ❛s ❢♦❧❧♦✇s (❛ + ❜)♠,♥ := ❛♠,♥ + ❜♠,♥ ✭✶✮ (λ❛)♠,♥ := λ❛♠,♥ ✭✷✮ (❛∗)♠,♥ := ❛∗

♥,♠

✭✸✮ ❛♥❞ ❛ ⋆ ❜ = ❛ ·

∞

• ❥=✵

Λ❥(❜) +

∞

• ❥=✶

Λ❥(❛) · ❜ ✭✹✮ ✇❤❡r❡ · ✐s ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ Λ : M(❆) → M(❆) ✐s ❣✐✈❡♥ ❜② Λ([❛♥,♠]) :=        ✵ ✵ ✵ ✵ · · · ✵ α(❛✵,✵) α(❛✵,✶) α(❛✵,✷) · · · ✵ α(❛✶,✵) α(❛✶,✶) α(❛✶,✷) · · · ✵ α(❛✷,✵) α(❛✷,✶) α(❛✷,✷) · · · ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳        .

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

✷✮ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ♥♦r♠ ♦❢ ❡❧❡♠❡♥ts ✐♥ ❈ ∗(❆, ❯)

❆♥ ❡❧❡♠❡♥t [❛♥,♠] ∈ M(❆) ✐s ❦✲❞✐❛❣♦♥❛❧✱ ✇❤❡r❡ ❦ ∈ Z✱ ✐❢ ✐t ✐s ♦❢ t❤❡ ❢♦r♠ ✐❢ ❦ ≥ ✵✱ ♦r          ✵ ✵         

❛❦,✵ ❛r+❦,r

• ❦

         r + ✶

       ✵ ✵       

❛✵,−❦ ❛r+❦,r

−❦ r+❦+✶

• ✱ ✐❢ ❦ < ✵✳

Pr♦♣♦s✐t✐♦♥✳ ■❢ ❛ = Ψ([❛♥,♠]) ✇❤❡r❡ [❛♥,♠] ✐s ❦✲❞✐❛❣♦♥❛❧✱ t❤❡♥

❛ = ❧✐♠

♥→∞ ♠❛①

     ♠❛①

✐=✶,...,♥

• ❞
• ✐
• ❥=✵,

❥+❦≥✵

α✐−❥(❛❥+❦,❥), ❏

• , ❞(❛♥+❦,♥, ❦❡r α)

     (✺)

✇❤❡r❡ ❏ = {❛ ∈ ❆ : ❯∗❯❛ = ❛} ❛♥❞ ❞(❛, ■) = ✐♥❢❜∈■ ❛ − ❜✳

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❚❤❡♦r❡♠ ✭❝r♦ss❡❞ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

▲❡t α : ❆ → ❆ ❜❡ ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ❛♥❞ ❏ ⊂ (❦❡r α)⊥✳ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❈ ∗✲s❡♠✐♥♦r♠ · ♦♥ t❤❡ ∗✲❛❧❣❡❜r❛ (M(❆), +, ·, ∗, ⋆) s✉❝❤ t❤❛t ✭✺✮ ❤♦❧❞s ❛♥❞ ❡✐t❤❡r ✶)

• ❦

❛❦ =

• ❦

λ❦❛❦, ❢♦r ❛❧❧ λ ∈ T, ♦r ✷) ❛✵ ≤

• ❦

❛❦ ❢♦r ❛❧❧ ❦✲❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ❛❦ ∈ M(❆)✳ ❚❤✐s ❈ ∗✲s❡♠✐♥♦r♠ ②✐❡❧❞s ❛ ❈ ∗✲❛❧❣❡❜r❛ ❈ ∗(❆, α, ❏) := M(❆)/ · , ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡❧❡♠❡♥ts ✉ :=   

✵ α(✶) ✵ · · · ✵ ✵ ✵ · · · ✵ ✵ ✵ · · · ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳

  

❛♥❞

• ❛ :=

  

❛ ✵ ✵ · · · ✵ ✵ ✵ · · · ✵ ✵ ✵ · · · ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳

   , ❛ ∈ ❆, ❛♥❞ ✉♥✐✈❡rs❛❧ s✉❜❥❡❝t t♦ r❡❧❛t✐♦♥s✿ α(❛) = ✉ ❛✉∗✱ ❏ = {❛ ∈ ❆ : ✉∗✉ ❛ = ❛}.

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❉❡✜♥✐t✐♦♥ ▲❡t α : ❆ → ❆ ❜❡ ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ❛♥❞ ❏ ⊂ (❦❡r α)⊥✳ ❲❡ ❝❛❧❧ ❈ ∗(❆, α, ❏) t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t ♦❢ ❆ ❜② α r❡❧❛t✐✈❡ t♦ ❏✳ ■❢ ❏ = (❦❡r α)⊥ ✇❡ ✇r✐t❡ ❈ ∗(❆, α) ❛♥❞ ❝❛❧❧ ✐t ❝r♦ss❡❞ ♣r♦❞✉❝t ♦❢ ❆ ❜② α✳ ❈ ∗(❆, α, ❏) ✐s ❛ ✉♥✐✈❡rs❛❧ ❈ ∗✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❆ ❛♥❞ ✉ s✉❜❥❡❝t t♦ α(❛) = ✉❛✉∗, ❛ ∈ ❆, ❏ = {❛ ∈ ❆ : ✉∗✉❛ = ❛} = ✉∗✉❆ ∩ ❆. Pr♦♣♦s✐t✐♦♥ ❦❡r α ✉♥✐t❛❧ = ⇒ ❈ ∗(❆, α) ✐s ✉♥✐✈❡rs❛❧ ❣❡♥❡r❛t❡❞ ❜② ❆ ❛♥❞ ✉ s✉❜❥❡❝t t♦ α(❛) = ✉❛✉∗, ❛ ∈ ❆, ✉∗✉ ∈ ❆. α ♠♦♥♦♠♦r♣❤✐s♠ = ⇒ ❈ ∗(❆, α) = ❆ ⋊α N ❙t❛❝❡②✬s ❝r♦ss❡❞ ♣r♦❞✉❝t✿ α(❛) = ✉❛✉∗, ❛ ∈ ❆, ✉∗✉ = ✶. α ❛✉t♦♠♦r♣❤✐s♠ = ⇒ ❈ ∗(❆, α) = ❆ ⋊α Z ❝❧❛ss✐❝❛❧ ❝r♦ss❡❞ ♣r♦❞✉❝t✿ α(❛) = ✉❛✉∗, α−✶(❛) = ✉∗❛✉, ❛ ∈ ❆.

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❘❡❧❡✈❛♥t ❝♦♥str✉❝t✐♦♥s

❏✲❘❡❞✉❝t✐♦♥ ♦❢ ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ ❙✉♣♣♦s❡ ❏ ✐s ❛♥ ❛r❜✐tr❛r② ✐❞❡❛❧ ✐♥ ❆✳ ▲❡t ❏∞ ❜❡ t❤❡ s♠❛❧❧❡st α✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ s✳t✳ ♣✉tt✐♥❣ q : ❆ → ❆/❏∞ ❛♥❞ ❆r := q(❆), αr ◦q := q◦α, ❏r := q(❏). ✇❡ ❤❛✈❡ ❏r ⊂ (❦❡r αr)⊥✳ ❈ ∗(❆r, αr, ❏r) ✐s ❣❡♥❡r❛t❡❞ ❜② ❛♥ ✐♠❛❣❡ ♦❢ ❆ ❛♥❞ ✉ s✉❜❥❡❝t t♦ r❡❧❛t✐♦♥s α(❛) = ✉❛✉∗, ❛ ∈ ❆, ❏ ⊂ ✉∗✉❆ ∩ ❆ ❏✲❯♥✐t✐③❛t✐♦♥ ♦❢ ❦❡r♥❡❧ ■❢ ❏ ⊂ (❦❡r α)⊥ ♦♥❡ ❝❛♥ ❝♦♥str✉❝t ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ α❏ : ❆❏ → ❆❏ s✉❝❤ t❤❛t ✶✮ ❆ ⊂ ❆❏✱ α❏|❆ = α✱ ❦❡r α❏ ✐s ✉♥✐t❛❧ ✷✮ ❆❏ = ❆ ⇐ ⇒ ❦❡r α ✐s ✉♥✐t❛❧ ❏ = (❦❡r α)⊥

• ✸✮ ❈ ∗(❆, α, ❏) ∼

= ❈ ∗(❆❏, α❏) α❏(❛) = ✉❛✉∗, ❛ ∈ ❆❏, ✉∗✉ ∈ ❆❏ ❍❡r❡❞✐t❛t✐♦♥ ♦❢ r❛♥❣❡✳ ❙✉♣♣♦s❡ ❦❡r α ✐s ✉♥✐t❛❧✳ ❚❤❡r❡ ✐s ❛♥ ❡♥❞♦♠♦r♣❤✐s♠ β : ❇ → ❇ ❡①t❡♥❞✐♥❣ α : ❆ → ❆ s✉❝❤ t❤❛t ❦❡r β ✐s ✉♥✐t❛❧✱ β(❇) ✐s ❤❡r❡❞✐t❛r② ❛♥❞ ❈ ∗(❆, α) ∼ = ❈ ∗(❇, β).

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❘❡♠❛r❦s ♦♥ ❊①❡❧✬s ❝r♦ss❡❞ ♣r♦❞✉❝t ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ❢r❡❡♥❡ss

❙t❛♥❞✐♥❣ ❛ss✉♠♣t✐♦♥s✿ ❦❡r α ✐s ✉♥✐t❛❧ ❛♥❞ α(❆) ✐s ❤❡r❡❞✐t❛r② ✐♥ ❆✳ Pr♦♣♦s✐t✐♦♥ ❬❑✇❛✶✱ ❑▲✱ ❆❇▲❪ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ tr❛♥s❢❡r ♦♣❡r❛t♦r L : ❆ → ❆ ❢♦r α : ❆ → ❆ ❛♥❞ ❈ ∗(❆, α) ✐s ❛ ✉♥✐✈❡rs❛❧ ❈ ∗✲❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② ❆ ❛♥❞ ✉ s✉❜❥❡❝t t♦✿ α(❛) = ✉❛✉∗, L(❛) = ✉∗❛✉, ❛ ∈ ❆. ▼♦r❡♦✈❡r✱ ❈ ∗(❆, α) ∼ = ❆ ⋊α,L N ✕ ❊①❡❧✬s ❝r♦ss❡❞ ♣r♦❞✉❝t✳ ❘❡♠❛r❦✳ α : L(❆) → α(❆) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ✐ts ❞✉❛❧

• α :

α(❆) → L(❆) ♠❛② ❜❡ tr❡❛t❡❞ ❛s ❛ ♣❛rt✐❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠ ♦❢ ❆✿

• ❆
• α(❆)
• L(❆)
• α

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

• ❆
• α(❆)
• L(❆)
• α

❯♥✐q✉❡♥❡ss ❚❤❡♦r❡♠ ❬❑✇❛✷❪✳ ■❢ α ✐s t♦♣♦❧♦❣✐❝❛❧❧② ❢r❡❡

✭t❤❡ s❡t ♦❢ ♣❡r✐♦❞✐❝ ♣♦✐♥ts ♦❢ ♣❡r✐♦❞ ♥ ∈ N ❤❛s ❡♠♣t② ✐♥t❡r✐♦r✮✱ t❤❡♥ ❢♦r

❛♥② ❢❛✐t❤❢✉❧ r❡♣r❡s❡♥t❛t✐♦♥ π : ❆ → ❇ ❛♥❞ ❯ ∈ ❇ s✉❝❤ t❤❛t π(α(❛)) = ❯π(❛)❯∗, π(L(❛)) = ❯∗π(❛)❯, ❛ ∈ ❆, t❤❡ ♠❛♣♣✐♥❣s ❛ → π(❛), ❛ ∈ ❆, ✉ → ❯ ②✐❡❧❞ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❈ ∗(❆, α) ∼ = ❈ ∗(π(❆), ❯)✳ ❖♣❡♥ ♣r♦❜❧❡♠✿ ❍♦✇ t♦ ❞❡✜♥❡ ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ❢r❡❡♥❡ss ❢♦r ❛♥ ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠❄

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s

❘❡❢❡r❡♥❝❡s

❬❑▲❪ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐✱ ❆✳ ❱✳ ▲❡❜❡❞❡✈ ✏❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❡♥❞♦♠♦r♣❤✐s♠s ❛♥❞ r❡❞✉❝t✐♦♥ ♦❢ r❡❧❛t✐♦♥s ✐♥ r❡❧❛t✐✈❡ ❈✉♥t③✲P✐♠s♥❡r ❛❧❣❡❜r❛s✑ ❏✳ ♦❢ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ✭✷✵✶✸✮ ❬❊①❡❧❪ ❘✳ ❊①❡❧✿ ✧❆ ♥❡✇ ❧♦♦❦ ❛t t❤❡ ❝r♦ss❡❞✲♣r♦❞✉❝t ♦❢ ❛ ❈ ∗✲❛❧❣❡❜r❛ ❜② ❛♥ ❡♥❞♦♠♦r♣❤✐s♠✧✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ❉②♥❛♠✳ ❙②st❡♠s✱ ✭✷✵✵✸✮ ❬❆❇▲❪ ❆✳❇✳ ❆♥t♦♥❡✈✐❝❤✱ ❱✳■✳ ❇❛❦❤t✐♥✱ ❆✳❱✳ ▲❡❜❡❞❡✈✱ ✧❈r♦ss❡❞ ♣r♦❞✉❝t ♦❢ ❈ ∗✲❛❧❣❡❜r❛ ❜② ❛♥ ❡♥❞♦♠♦r♣❤✐s♠✱ ❝♦❡✣❝✐❡♥t ❛❧❣❡❜r❛s ❛♥❞ tr❛♥s❢❡r ♦♣❡r❛t♦rs✧✱ ❙❜✳ ▼❛t❤✳ ✭✷✵✶✶✮ ❬❑✇❛✶❪ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐✱ ✏❖♥ tr❛♥s❢❡r ♦♣❡r❛t♦rs ❢♦r ❈✯✲❞②♥❛♠✐❝❛❧ s②st❡♠s✑ ❘♦❝❦② ❏✳ ▼❛t❤✳ ✭✷✵✶✷✮✱ ❬❑✇❛✷❪ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ✏❉②♥❛♠✐❝❛❧ s②st❡♠ ❞✉❛❧ t♦ ✐♥t❡r❛❝t✐♦♥s ❛♥❞ ❣r❛♣❤ ❛❧❣❡❜r❛s✑ ❛r❳✐✈✿✶✸✵✶✳✺✶✷✺ ❬❑✇❛✸❪ ❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ✏ ❊①t❡♥s✐♦♥s ♦❢ ❈ ∗✲❞②♥❛♠✐❝❛❧ s②st❡♠s t♦ s②st❡♠s ✇✐t❤ ❝♦♠♣❧❡t❡ tr❛♥s❢❡r ♦♣❡r❛t♦rs✧ ❛r❳✐✈✿❖❆✴✵✼✵✸✽✵✵

❇✳ ❑✳ ❑✇❛➧♥✐❡✇s❦✐ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❜② ❛r❜✐tr❛r② ❡♥❞♦♠♦r♣❤✐s♠s