rt ts rs r - - PowerPoint PPT Presentation

❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ♦❢ ❣r❛♣❤s

❆♣♦♦r✈❛ ❑❤❛r❡

❙t❛♥❢♦r❞ ❯♥✐✈❡rs✐t② ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❉♦♠✐♥✐q✉❡ ●✉✐❧❧♦t ✭❯✳ ❉❡❧❛✇❛r❡✮ ❛♥❞ ❇❛❧❛ ❘❛❥❛r❛t♥❛♠ ✭❙t❛♥❢♦r❞✮

❆▼❙ ❙❡❝t✐♦♥❛❧ ▼❡❡t✐♥❣✱ ▲♦②♦❧❛ ❖❝t♦❜❡r ✸✱ ✷✵✶✺

❲♦r❦✐♥❣ ❡①❛♠♣❧❡

◗✉❡st✐♦♥✳ ❙✉♣♣♦s❡ A =       1 0.6 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.6 1       . ❘❛✐s❡ ❡❛❝❤ ❡♥tr② t♦ t❤❡ t❤ ♣♦✇❡r ❢♦r s♦♠❡ ✳ ❲❤❡♥ ✐s t❤❡ r❡s✉❧t✐♥❣ ♠❛tr✐① ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡❄

✷

❲♦r❦✐♥❣ ❡①❛♠♣❧❡

◗✉❡st✐♦♥✳ ❙✉♣♣♦s❡ A =       1 0.6 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.6 1       . ❘❛✐s❡ ❡❛❝❤ ❡♥tr② t♦ t❤❡ αt❤ ♣♦✇❡r ❢♦r s♦♠❡ α > 0✳ ❲❤❡♥ ✐s t❤❡ r❡s✉❧t✐♥❣ ♠❛tr✐① ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡❄

✷

▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s

• r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞

❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X1, . . . , Xp ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t ✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✈❡❝t♦r ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ ■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ ❣✐✈❡♥ ❞❛t❛ ♦❢ ✳

✸

▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s

• r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞

❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X1, . . . , Xp ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ Xi ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t Xi✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✈❡❝t♦r ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ ■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ ❣✐✈❡♥ ❞❛t❛ ♦❢ ✳

✸

▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s

• r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞

❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X1, . . . , Xp ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ Xi ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t Xi✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ ♦❢ t❤❡ ✈❡❝t♦r (X1, . . . , Xp) ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ Σ = (σjk)p

j,k=1 = (Cov(Xj, Xk))p j,k=1,

■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ ❣✐✈❡♥ ❞❛t❛ ♦❢ ✳

✸

▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s

• r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞

❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X1, . . . , Xp ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ Xi ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t Xi✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ ♦❢ t❤❡ ✈❡❝t♦r (X1, . . . , Xp) ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ Σ = (σjk)p

j,k=1 = (Cov(Xj, Xk))p j,k=1,

■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ Σ ❣✐✈❡♥ ❞❛t❛ x1, . . . , xn ∈ Rp ♦❢ (X1, . . . , Xp)✳

✸

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥

❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ S := 1 n − 1

n

• j=1

(xj − x)(xj − x)T . ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ ❤❛s r❛♥❦ ❛t ♠♦st ✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ ✱ s♠❛❧❧ ✑ ♣r♦❜❧❡♠s✱ ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳

✹

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥

❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ S := 1 n − 1

n

• j=1

(xj − x)(xj − x)T . S ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ S ❤❛s r❛♥❦ ❛t ♠♦st n✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ ✱ s♠❛❧❧ ✑ ♣r♦❜❧❡♠s✱ ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳

✹

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥

❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ S := 1 n − 1

n

• j=1

(xj − x)(xj − x)T . S ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ S ❤❛s r❛♥❦ ❛t ♠♦st n✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ p✱ s♠❛❧❧ n✑ ♣r♦❜❧❡♠s✱ S ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ Σ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳

✹

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥

❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ S := 1 n − 1

n

• j=1

(xj − x)(xj − x)T . S ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ S ❤❛s r❛♥❦ ❛t ♠♦st n✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ p✱ s♠❛❧❧ n✑ ♣r♦❜❧❡♠s✱ S ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ Σ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ Σ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳

✹

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥

❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ S := 1 n − 1

n

• j=1

(xj − x)(xj − x)T . S ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ S ❤❛s r❛♥❦ ❛t ♠♦st n✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ p✱ s♠❛❧❧ n✑ ♣r♦❜❧❡♠s✱ S ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ Σ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ Σ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ 100, 000+ ✈❛r✐❛❜❧❡s✳

✹

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ✭❝♦♥t✳✮

❆❧t❡r♥❛t❡ ❛♣♣r♦❛❝❤✿ ❚❤r❡s❤♦❧❞✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❚r✉❡ Σ =   1 0.2 0.2 1 0.5 0.5 1   S =   0.95 0.18 0.02 0.18 0.96 0.47 0.02 0.47 0.98   ◆❛t✉r❛❧ t♦ t❤r❡s❤♦❧❞ s♠❛❧❧ ❡♥tr✐❡s ✭t❤✐♥❦✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✿ ❈❛♥ ❜❡ s✐❣♥✐✜❝❛♥t ✐❢ ❛♥❞ ♦♥❧②✱ s❛②✱ ♦❢ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ tr✉❡ ❛r❡ ♥♦♥③❡r♦✳ ❍✐❣❤❧② s❝❛❧❛❜❧❡✳ ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦♥❡ ✲ ♥♦ ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s t❤✐s ♣r♦❝❡❞✉r❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈❡ ✭s❡♠✐✮❞❡✜♥✐t❡♥❡ss❄ ❈r✐t✐❝❛❧ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✱ s✐♥❝❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳

✺

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ✭❝♦♥t✳✮

❆❧t❡r♥❛t❡ ❛♣♣r♦❛❝❤✿ ❚❤r❡s❤♦❧❞✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❚r✉❡ Σ =   1 0.2 0.2 1 0.5 0.5 1   S =   0.95 0.18 0.02 0.18 0.96 0.47 0.02 0.47 0.98   ◆❛t✉r❛❧ t♦ t❤r❡s❤♦❧❞ s♠❛❧❧ ❡♥tr✐❡s ✭t❤✐♥❦✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✿ ˜ S =   0.95 0.18 0.18 0.96 0.47 0.47 0.98   ❈❛♥ ❜❡ s✐❣♥✐✜❝❛♥t ✐❢ ❛♥❞ ♦♥❧②✱ s❛②✱ ♦❢ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ tr✉❡ ❛r❡ ♥♦♥③❡r♦✳ ❍✐❣❤❧② s❝❛❧❛❜❧❡✳ ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦♥❡ ✲ ♥♦ ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s t❤✐s ♣r♦❝❡❞✉r❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈❡ ✭s❡♠✐✮❞❡✜♥✐t❡♥❡ss❄ ❈r✐t✐❝❛❧ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✱ s✐♥❝❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳

✺

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ✭❝♦♥t✳✮

❆❧t❡r♥❛t❡ ❛♣♣r♦❛❝❤✿ ❚❤r❡s❤♦❧❞✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❚r✉❡ Σ =   1 0.2 0.2 1 0.5 0.5 1   S =   0.95 0.18 0.02 0.18 0.96 0.47 0.02 0.47 0.98   ◆❛t✉r❛❧ t♦ t❤r❡s❤♦❧❞ s♠❛❧❧ ❡♥tr✐❡s ✭t❤✐♥❦✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✿ ˜ S =   0.95 0.18 0.18 0.96 0.47 0.47 0.98   ❈❛♥ ❜❡ s✐❣♥✐✜❝❛♥t ✐❢ p = 1, 000, 000 ❛♥❞ ♦♥❧②✱ s❛②✱ ∼ 1% ♦❢ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ tr✉❡ Σ ❛r❡ ♥♦♥③❡r♦✳ ❍✐❣❤❧② s❝❛❧❛❜❧❡✳ ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦♥❡ ✲ ♥♦ ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s t❤✐s ♣r♦❝❡❞✉r❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈❡ ✭s❡♠✐✮❞❡✜♥✐t❡♥❡ss❄ ❈r✐t✐❝❛❧ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✱ s✐♥❝❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳

✺

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ✭❝♦♥t✳✮

❆❧t❡r♥❛t❡ ❛♣♣r♦❛❝❤✿ ❚❤r❡s❤♦❧❞✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❚r✉❡ Σ =   1 0.2 0.2 1 0.5 0.5 1   S =   0.95 0.18 0.02 0.18 0.96 0.47 0.02 0.47 0.98   ◆❛t✉r❛❧ t♦ t❤r❡s❤♦❧❞ s♠❛❧❧ ❡♥tr✐❡s ✭t❤✐♥❦✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✿ ˜ S =   0.95 0.18 0.18 0.96 0.47 0.47 0.98   ❈❛♥ ❜❡ s✐❣♥✐✜❝❛♥t ✐❢ p = 1, 000, 000 ❛♥❞ ♦♥❧②✱ s❛②✱ ∼ 1% ♦❢ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ tr✉❡ Σ ❛r❡ ♥♦♥③❡r♦✳ ❍✐❣❤❧② s❝❛❧❛❜❧❡✳ ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦♥❡ ✲ ♥♦ ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s t❤✐s ♣r♦❝❡❞✉r❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈❡ ✭s❡♠✐✮❞❡✜♥✐t❡♥❡ss❄ ❈r✐t✐❝❛❧ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✱ s✐♥❝❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳

✺

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ✭❝♦♥t✳✮

❆❧t❡r♥❛t❡ ❛♣♣r♦❛❝❤✿ ❚❤r❡s❤♦❧❞✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❚r✉❡ Σ =   1 0.2 0.2 1 0.5 0.5 1   S =   0.95 0.18 0.02 0.18 0.96 0.47 0.02 0.47 0.98   ◆❛t✉r❛❧ t♦ t❤r❡s❤♦❧❞ s♠❛❧❧ ❡♥tr✐❡s ✭t❤✐♥❦✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✿ ˜ S =   0.95 0.18 0.18 0.96 0.47 0.47 0.98   ❈❛♥ ❜❡ s✐❣♥✐✜❝❛♥t ✐❢ p = 1, 000, 000 ❛♥❞ ♦♥❧②✱ s❛②✱ ∼ 1% ♦❢ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ tr✉❡ Σ ❛r❡ ♥♦♥③❡r♦✳ ❍✐❣❤❧② s❝❛❧❛❜❧❡✳ ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦♥❡ ✲ ♥♦ ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s t❤✐s ♣r♦❝❡❞✉r❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈❡ ✭s❡♠✐✮❞❡✜♥✐t❡♥❡ss❄ ❈r✐t✐❝❛❧ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✱ s✐♥❝❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳

✺

❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ✭❝♦♥t✳✮

❆❧t❡r♥❛t❡ ❛♣♣r♦❛❝❤✿ ❚❤r❡s❤♦❧❞✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❚r✉❡ Σ =   1 0.2 0.2 1 0.5 0.5 1   S =   0.95 0.18 0.02 0.18 0.96 0.47 0.02 0.47 0.98   ◆❛t✉r❛❧ t♦ t❤r❡s❤♦❧❞ s♠❛❧❧ ❡♥tr✐❡s ✭t❤✐♥❦✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✿ ˜ S =   0.95 0.18 0.18 0.96 0.47 0.47 0.98   ❈❛♥ ❜❡ s✐❣♥✐✜❝❛♥t ✐❢ p = 1, 000, 000 ❛♥❞ ♦♥❧②✱ s❛②✱ ∼ 1% ♦❢ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ tr✉❡ Σ ❛r❡ ♥♦♥③❡r♦✳ ❍✐❣❤❧② s❝❛❧❛❜❧❡✳ ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦♥❡ ✲ ♥♦ ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s t❤✐s ♣r♦❝❡❞✉r❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈❡ ✭s❡♠✐✮❞❡✜♥✐t❡♥❡ss❄ ❈r✐t✐❝❛❧ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s✱ s✐♥❝❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛r❡ ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳

✺

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ ❢♦r ❛❧❧ ✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ ✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r ✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s ✇❤❡r❡ ✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ ✭❢♦r ✜①❡❞ ✮❄

✻

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ Pn ❢♦r ❛❧❧ n✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ ✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r ✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s ✇❤❡r❡ ✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ ✭❢♦r ✜①❡❞ ✮❄

✻

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ Pn ❢♦r ❛❧❧ n✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ n✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r n ≥ 3✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s ✇❤❡r❡ ✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ ✭❢♦r ✜①❡❞ ✮❄

✻

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ Pn ❢♦r ❛❧❧ n✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ n✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r n ≥ 3✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s ✇❤❡r❡ ✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ ✭❢♦r ✜①❡❞ ✮❄

✻

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ Pn ❢♦r ❛❧❧ n✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ n✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r n ≥ 3✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s xα ✇❤❡r❡ α > 0✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ ✭❢♦r ✜①❡❞ ✮❄

✻

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ Pn ❢♦r ❛❧❧ n✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ n✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r n ≥ 3✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s xα ✇❤❡r❡ α > 0✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ ✭❢♦r ✜①❡❞ ✮❄

✻

❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ●✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ f : I → R✱ ✇❤❡♥ ✐s ✐t tr✉❡ t❤❛t f[A] := (f(ajk)) ∈ Pn ❢♦r ❛❧❧ A ∈ Pn(I)? Pr♦❜❧❡♠✿ ❲❤❛t ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤✐s ♣r♦♣❡rt②❄ ❊♥tr②✇✐s❡ ❢✉♥❝t✐♦♥s ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ♦♥ Pn ❢♦r ❛❧❧ n✿ ❙❝❤♦❡♥❜❡r❣ ✭❉✉❦❡✱ ✶✾✹✷✮✱ ❘✉❞✐♥ ✭❉✉❦❡✱ ✶✾✺✾✮✱ ❛♥❞ ♦t❤❡rs✳ ❆♥❛❧②t✐❝✱ ✇✐t❤ ♥♦♥♥❡❣❛t✐✈❡ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts✳ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✜①❡❞ n✿ ❍❛r❞ ♣r♦❜❧❡♠✱ ♦♣❡♥ ❢♦r n ≥ 3✳ ❘❡❝❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ♣♦❧②♥♦♠✐❛❧s✿ ❬❇❡❧t♦♥✲●✉✐❧❧♦t✲❑❤❛r❡✲P✉t✐♥❛r✱ ✷✵✶✺❪✳ ❋♦❝✉s ♦♥ ❞✐st✐♥❣✉✐s❤❡❞ ❢❛♠✐❧✐❡s t♦ ❣❡t ✐♥s✐❣❤ts ✐♥t♦ ❣❡♥❡r❛❧ ❝❛s❡✳ ❲❡❧❧✲st✉❞✐❡❞ ❢❛♠✐❧② ✐♥ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦✇❡r ❢✉♥❝t✐♦♥s xα ✇❤❡r❡ α > 0✳ ✭❆♣♣❧✐❝❛t✐♦♥s ✉s❡ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❤❛r❞✲ ❛♥❞ s♦❢t✲ t❤r❡s❤♦❧❞✐♥❣✱ ❛♥❞ ♣♦✇❡rs✱ t♦ r❡❣✉❧❛r✐③❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳✮ ◗✉❡st✐♦♥✿ ❲❤✐❝❤ ♣♦✇❡r ❢✉♥❝t✐♦♥s ❛♣♣❧✐❡❞ ❡♥tr②✇✐s❡ ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ♦♥ Pn ✭❢♦r ✜①❡❞ n✮❄

✻

P♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

❚❤❡♦r❡♠ ✭❋✐t③●❡r❛❧❞ ❛♥❞ ❍♦r♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✶✾✼✼✮ ▲❡t n ≥ 2✳ ❚❤❡♥✿

✶ f(x) = xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pn((0, ∞)) ✐❢ α ≥ n − 2✳ ✷ ■❢

✐s ♥♦t ❛♥ ✐♥t❡❣❡r✱ t❤❡r❡ ✐s ❛ ♠❛tr✐① s✉❝❤ t❤❛t ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥t✿ s♠❛❧❧❡st s✉❝❤ t❤❛t ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

❙♦ ❢♦r ✱ ❛❧❧ ♣♦✇❡rs ✇♦r❦✳ ❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

✼

P♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

❚❤❡♦r❡♠ ✭❋✐t③●❡r❛❧❞ ❛♥❞ ❍♦r♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✶✾✼✼✮ ▲❡t n ≥ 2✳ ❚❤❡♥✿

✶ f(x) = xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pn((0, ∞)) ✐❢ α ≥ n − 2✳ ✷ ■❢ α < n − 2 ✐s ♥♦t ❛♥ ✐♥t❡❣❡r✱ t❤❡r❡ ✐s ❛ ♠❛tr✐①

A = (ajk) ∈ Pn s✉❝❤ t❤❛t A◦α := (aα

jk) ∈ Pn✳

■♥ ♦t❤❡r ✇♦r❞s✱ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥t✿ s♠❛❧❧❡st s✉❝❤ t❤❛t ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

❙♦ ❢♦r ✱ ❛❧❧ ♣♦✇❡rs ✇♦r❦✳ ❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

✼

P♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

❚❤❡♦r❡♠ ✭❋✐t③●❡r❛❧❞ ❛♥❞ ❍♦r♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✶✾✼✼✮ ▲❡t n ≥ 2✳ ❚❤❡♥✿

✶ f(x) = xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pn((0, ∞)) ✐❢ α ≥ n − 2✳ ✷ ■❢ α < n − 2 ✐s ♥♦t ❛♥ ✐♥t❡❣❡r✱ t❤❡r❡ ✐s ❛ ♠❛tr✐①

A = (ajk) ∈ Pn s✉❝❤ t❤❛t A◦α := (aα

jk) ∈ Pn✳

■♥ ♦t❤❡r ✇♦r❞s✱ f(x) = xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pn((0, ∞)) ✐❢ ❛♥❞ ♦♥❧② ✐❢ α ∈ N ∪ [n − 2, ∞)✳ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥t✿ n − 2 = s♠❛❧❧❡st α0 s✉❝❤ t❤❛t α ≥ α0 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

❙♦ ❢♦r ✱ ❛❧❧ ♣♦✇❡rs ✇♦r❦✳ ❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

✼

P♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②

❚❤❡♦r❡♠ ✭❋✐t③●❡r❛❧❞ ❛♥❞ ❍♦r♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✶✾✼✼✮ ▲❡t n ≥ 2✳ ❚❤❡♥✿

✶ f(x) = xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pn((0, ∞)) ✐❢ α ≥ n − 2✳ ✷ ■❢ α < n − 2 ✐s ♥♦t ❛♥ ✐♥t❡❣❡r✱ t❤❡r❡ ✐s ❛ ♠❛tr✐①

A = (ajk) ∈ Pn s✉❝❤ t❤❛t A◦α := (aα

jk) ∈ Pn✳

■♥ ♦t❤❡r ✇♦r❞s✱ f(x) = xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pn((0, ∞)) ✐❢ ❛♥❞ ♦♥❧② ✐❢ α ∈ N ∪ [n − 2, ∞)✳ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥t✿ n − 2 = s♠❛❧❧❡st α0 s✉❝❤ t❤❛t α ≥ α0 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

❙♦ ❢♦r A =       1 0.6 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.6 1       ✱ ❛❧❧ ♣♦✇❡rs α ∈ N ∪ [3, ∞) ✇♦r❦✳ ❈❛♥ ✇❡ ❞♦ ❜❡tt❡r❄

✼

▼❛tr✐❝❡s ✇✐t❤ str✉❝t✉r❡s ♦❢ ③❡r♦s

❘❡✜♥❡ t❤❡ ❋✐t③●❡r❛❧❞✕❍♦r♥ ♣r♦❜❧❡♠ ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ③❡r♦s✳ ❆ ❣r❛♣❤ ✐s ❛ s❡t ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✿ ❚❤❡ ♣❛tt❡r♥ ♦❢ ③❡r♦s ♦❢ ❛ s②♠♠❡tr✐❝ ♠❛tr✐① ✐s ♥❛t✉r❛❧❧② ❡♥❝♦❞❡❞ ❜② ❛ ❣r❛♣❤ ♦♥ ✿ ❊❞❣❡ ❜❡t✇❡❡♥ ❛♥❞

✽

▼❛tr✐❝❡s ✇✐t❤ str✉❝t✉r❡s ♦❢ ③❡r♦s

❘❡✜♥❡ t❤❡ ❋✐t③●❡r❛❧❞✕❍♦r♥ ♣r♦❜❧❡♠ ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ③❡r♦s✳ ❆ ❣r❛♣❤ G = (V, E) ✐s ❛ s❡t ♦❢ ✈❡rt✐❝❡s V ❛♥❞ ❡❞❣❡s E ⊂ V × V ✿ ❚❤❡ ♣❛tt❡r♥ ♦❢ ③❡r♦s ♦❢ ❛ s②♠♠❡tr✐❝ ♠❛tr✐① ✐s ♥❛t✉r❛❧❧② ❡♥❝♦❞❡❞ ❜② ❛ ❣r❛♣❤ ♦♥ ✿ ❊❞❣❡ ❜❡t✇❡❡♥ ❛♥❞

✽

▼❛tr✐❝❡s ✇✐t❤ str✉❝t✉r❡s ♦❢ ③❡r♦s

❘❡✜♥❡ t❤❡ ❋✐t③●❡r❛❧❞✕❍♦r♥ ♣r♦❜❧❡♠ ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ③❡r♦s✳ ❆ ❣r❛♣❤ G = (V, E) ✐s ❛ s❡t ♦❢ ✈❡rt✐❝❡s V ❛♥❞ ❡❞❣❡s E ⊂ V × V ✿ ❚❤❡ ♣❛tt❡r♥ ♦❢ ③❡r♦s ♦❢ ❛ s②♠♠❡tr✐❝ n × n ♠❛tr✐① ✐s ♥❛t✉r❛❧❧② ❡♥❝♦❞❡❞ ❜② ❛ ❣r❛♣❤ ♦♥ V = {1, 2, . . . , n}✿ ❊❞❣❡ ❜❡t✇❡❡♥ j ❛♥❞ k ⇐ ⇒ ajk = 0. A =     1 3 2 3 1 5 5 1 4 2 4 1     ← →

✽

❚❤❡ ❝♦♥❡ PG

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V, E) ✇✐t❤ V = {1, . . . , n} ✇❡ ❞❡✜♥❡ ❛ s✉❜s❡t

♦❢ Pn ❜② PG := {A ∈ Pn : ajk = 0 ✐❢ (j, k) ∈ E ❛♥❞ j = k}. ◆♦t❡✿ ❝❛♥ ❜❡ ③❡r♦ ✐❢ ✳ ❊①❛♠♣❧❡✿ ❉❡✜♥❡ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✿ ❢♦r ❛❧❧ s♠❛❧❧❡st s✳t✳ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pr♦❜❧❡♠✿ ❍♦✇ ❞♦❡s t❤❡ str✉❝t✉r❡ ♦❢ r❡❧❛t❡ t♦ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②❄ ✭❋✐t③●❡r❛❧❞✲❍♦r♥ st✉❞✐❡❞ t❤❡ ❝❛s❡ ✳✮

✾

❚❤❡ ❝♦♥❡ PG

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V, E) ✇✐t❤ V = {1, . . . , n} ✇❡ ❞❡✜♥❡ ❛ s✉❜s❡t

♦❢ Pn ❜② PG := {A ∈ Pn : ajk = 0 ✐❢ (j, k) ∈ E ❛♥❞ j = k}. ◆♦t❡✿ ajk ❝❛♥ ❜❡ ③❡r♦ ✐❢ (j, k) ∈ E✳ ❊①❛♠♣❧❡✿    

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

    ❉❡✜♥❡ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ✿ ❢♦r ❛❧❧ s♠❛❧❧❡st s✳t✳ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ Pr♦❜❧❡♠✿ ❍♦✇ ❞♦❡s t❤❡ str✉❝t✉r❡ ♦❢ r❡❧❛t❡ t♦ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②❄ ✭❋✐t③●❡r❛❧❞✲❍♦r♥ st✉❞✐❡❞ t❤❡ ❝❛s❡ ✳✮

✾

❚❤❡ ❝♦♥❡ PG

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V, E) ✇✐t❤ V = {1, . . . , n} ✇❡ ❞❡✜♥❡ ❛ s✉❜s❡t

♦❢ Pn ❜② PG := {A ∈ Pn : ajk = 0 ✐❢ (j, k) ∈ E ❛♥❞ j = k}. ◆♦t❡✿ ajk ❝❛♥ ❜❡ ③❡r♦ ✐❢ (j, k) ∈ E✳ ❊①❛♠♣❧❡✿    

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

    ❉❡✜♥❡ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r G✿ HG := {α ≥ 0 : A◦α ∈ PG ❢♦r ❛❧❧ A ∈ PG([0, ∞))} CE(G) := s♠❛❧❧❡st α0 s✳t✳ xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ PG, ∀α ≥ α0. Pr♦❜❧❡♠✿ ❍♦✇ ❞♦❡s t❤❡ str✉❝t✉r❡ ♦❢ r❡❧❛t❡ t♦ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②❄ ✭❋✐t③●❡r❛❧❞✲❍♦r♥ st✉❞✐❡❞ t❤❡ ❝❛s❡ ✳✮

✾

❚❤❡ ❝♦♥❡ PG

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V, E) ✇✐t❤ V = {1, . . . , n} ✇❡ ❞❡✜♥❡ ❛ s✉❜s❡t

♦❢ Pn ❜② PG := {A ∈ Pn : ajk = 0 ✐❢ (j, k) ∈ E ❛♥❞ j = k}. ◆♦t❡✿ ajk ❝❛♥ ❜❡ ③❡r♦ ✐❢ (j, k) ∈ E✳ ❊①❛♠♣❧❡✿    

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

    ❉❡✜♥❡ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r G✿ HG := {α ≥ 0 : A◦α ∈ PG ❢♦r ❛❧❧ A ∈ PG([0, ∞))} CE(G) := s♠❛❧❧❡st α0 s✳t✳ xα ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t② ♦♥ PG, ∀α ≥ α0. Pr♦❜❧❡♠✿ ❍♦✇ ❞♦❡s t❤❡ str✉❝t✉r❡ ♦❢ G r❡❧❛t❡ t♦ t❤❡ s❡t ♦❢ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t②❄ ✭❋✐t③●❡r❛❧❞✲❍♦r♥ st✉❞✐❡❞ t❤❡ ❝❛s❡ G = Kn✳✮

✾

❆ ✜rst ❡①❛♠♣❧❡✿ tr❡❡s

❉❡✜♥✐t✐♦♥✿ ❆ tr❡❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ ♥♦ ❝②❝❧❡s✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❚❆▼❙✲✶✱ ✷✵✶✺✮ ▲❡t ❜❡ ❛ tr❡❡ ✇✐t❤ ❛t ❧❡❛st ✈❡rt✐❝❡s✳ ❚❤❡♥ ✳ ❚❤❡ ♣r♦♦❢ ✉s❡s ✐♥❞✉❝t✐♦♥ ♦♥ ✱ ❛♥❞ ❙❝❤✉r ❝♦♠♣❧❡♠❡♥ts✿

✶✵

❆ ✜rst ❡①❛♠♣❧❡✿ tr❡❡s

❉❡✜♥✐t✐♦♥✿ ❆ tr❡❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ ♥♦ ❝②❝❧❡s✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❚❆▼❙✲✶✱ ✷✵✶✺✮ ▲❡t T ❜❡ ❛ tr❡❡ ✇✐t❤ ❛t ❧❡❛st 3 ✈❡rt✐❝❡s✳ ❚❤❡♥ HT = [1, ∞)✳ ❚❤❡ ♣r♦♦❢ ✉s❡s ✐♥❞✉❝t✐♦♥ ♦♥ ✱ ❛♥❞ ❙❝❤✉r ❝♦♠♣❧❡♠❡♥ts✿

✶✵

❆ ✜rst ❡①❛♠♣❧❡✿ tr❡❡s

❉❡✜♥✐t✐♦♥✿ ❆ tr❡❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❝♦♥t❛✐♥✐♥❣ ♥♦ ❝②❝❧❡s✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❚❆▼❙✲✶✱ ✷✵✶✺✮ ▲❡t T ❜❡ ❛ tr❡❡ ✇✐t❤ ❛t ❧❡❛st 3 ✈❡rt✐❝❡s✳ ❚❤❡♥ HT = [1, ∞)✳ ❚❤❡ ♣r♦♦❢ ✉s❡s ✐♥❞✉❝t✐♦♥ ♦♥ n✱ ❛♥❞ ❙❝❤✉r ❝♦♠♣❧❡♠❡♥ts✿ SM◦α − (SM)◦α ∈ Pn−1.

✶✵

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢

❤❛s ✈❡rt✐❝❡s t❤❡♥ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

✷ ■❢

❝♦♥t❛✐♥s ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭ ✮✳ ❈♦♥s❡q✉❡♥❝❡✿ ✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✭ ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢

❤❛s ✈❡rt✐❝❡s t❤❡♥ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

✷ ■❢

❝♦♥t❛✐♥s ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭ ✮✳ ❈♦♥s❡q✉❡♥❝❡✿ ✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✭ ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢

❤❛s ✈❡rt✐❝❡s t❤❡♥ ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳

✷ ■❢

❝♦♥t❛✐♥s ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭ ✮✳ ❈♦♥s❡q✉❡♥❝❡✿ ✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✭ ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢ G ❤❛s n ✈❡rt✐❝❡s t❤❡♥ α ≥ n − 2 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳ ✷ ■❢

❝♦♥t❛✐♥s ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭ ✮✳ ❈♦♥s❡q✉❡♥❝❡✿ ✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✭ ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢ G ❤❛s n ✈❡rt✐❝❡s t❤❡♥ α ≥ n − 2 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳ ✷ ■❢ G ❝♦♥t❛✐♥s Km ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ α < m − 2

❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭α ∈ N✮✳ ❈♦♥s❡q✉❡♥❝❡✿ ✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✭ ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢ G ❤❛s n ✈❡rt✐❝❡s t❤❡♥ α ≥ n − 2 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳ ✷ ■❢ G ❝♦♥t❛✐♥s Km ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ α < m − 2

❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭α ∈ N✮✳ ❈♦♥s❡q✉❡♥❝❡✿ m − 2 ≤ CE(G) ≤ n − 2✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ G ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✭ ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢ G ❤❛s n ✈❡rt✐❝❡s t❤❡♥ α ≥ n − 2 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳ ✷ ■❢ G ❝♦♥t❛✐♥s Km ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ α < m − 2

❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭α ∈ N✮✳ ❈♦♥s❡q✉❡♥❝❡✿ m − 2 ≤ CE(G) ≤ n − 2✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ G ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ G = K(1)

4

✭K4 ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s ✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t ✳

✶✶

• ❡♥❡r❛❧ ❣r❛♣❤s

CE(T) = 1 ❢♦r ❛❧❧ tr❡❡s T✱ ❛♥❞ CE(Kn) = n − 2✳ ❲❤❛t ✐s CE(G) ✐♥ ❣❡♥❡r❛❧❄ ❙♦♠❡ ♣r❡❧✐♠✐♥❛r② ♦❜s❡r✈❛t✐♦♥s✿

✶ ■❢ G ❤❛s n ✈❡rt✐❝❡s t❤❡♥ α ≥ n − 2 ♣r❡s❡r✈❡s ♣♦s✐t✐✈✐t②✳ ✷ ■❢ G ❝♦♥t❛✐♥s Km ❛s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤✱ t❤❡♥ α < m − 2

❞♦❡s ♥♦t ♣r❡s❡r✈❡ ♣♦s✐t✐✈✐t② ✭α ∈ N✮✳ ❈♦♥s❡q✉❡♥❝❡✿ m − 2 ≤ CE(G) ≤ n − 2✳ ◗✉❡st✐♦♥✿ ■s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ G ❡q✉❛❧ t♦ t❤❡ ❝❧✐q✉❡ ♥✉♠❜❡r ♠✐♥✉s ✷❄ ❆♥s✇❡r✿ ◆♦✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ G = K(1)

4

✭K4 ♠✐♥✉s ❛ ❝❤♦r❞✮✳ ❈❧❡❛r❧②✱ t❤❡ ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐s K3✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t HK(1)

4

= {1} ∪ [2, ∞)✳

✶✶

❈❤♦r❞❛❧ ❣r❛♣❤s

❚r❡❡s ❛r❡ ❣r❛♣❤s ✇✐t❤ ♥♦ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ n ≥ 3✳ ❉❡✜♥✐t✐♦♥✿ ❆ ❣r❛♣❤ ✐s ❝❤♦r❞❛❧ ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥ ✐♥❞✉❝❡❞ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ ✳ ❈❤♦r❞❛❧ ◆♦t ❈❤♦r❞❛❧ ◆❛♠❡s✿ ❚r✐❛♥❣✉❧❛t❡❞✱ ❞❡❝♦♠♣♦s❛❜❧❡✱ r✐❣✐❞ ❝✐r❝✉✐t ❣r❛♣❤s✳ ✳ ✳ ❊①❛♠♣❧❡s✿ ❚r❡❡s✱ ❈♦♠♣❧❡t❡ ❣r❛♣❤s✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛♥② ❣r❛♣❤✱ ❆♣♦❧❧♦♥✐❛♥ ❣r❛♣❤s✱ ❇❛♥❞ ❣r❛♣❤s✱ ❙♣❧✐t ❣r❛♣❤s✱ ❡t❝✳ ❖❝❝✉r ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠s✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❣r❛♣❤✐❝❛❧ ♠♦❞❡❧s✱

• ❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✱ ❡t❝✳

✶✷

❈❤♦r❞❛❧ ❣r❛♣❤s

❚r❡❡s ❛r❡ ❣r❛♣❤s ✇✐t❤ ♥♦ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ n ≥ 3✳ ❉❡✜♥✐t✐♦♥✿ ❆ ❣r❛♣❤ ✐s ❝❤♦r❞❛❧ ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥ ✐♥❞✉❝❡❞ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ n ≥ 4✳ ❈❤♦r❞❛❧ ◆♦t ❈❤♦r❞❛❧ ◆❛♠❡s✿ ❚r✐❛♥❣✉❧❛t❡❞✱ ❞❡❝♦♠♣♦s❛❜❧❡✱ r✐❣✐❞ ❝✐r❝✉✐t ❣r❛♣❤s✳ ✳ ✳ ❊①❛♠♣❧❡s✿ ❚r❡❡s✱ ❈♦♠♣❧❡t❡ ❣r❛♣❤s✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛♥② ❣r❛♣❤✱ ❆♣♦❧❧♦♥✐❛♥ ❣r❛♣❤s✱ ❇❛♥❞ ❣r❛♣❤s✱ ❙♣❧✐t ❣r❛♣❤s✱ ❡t❝✳ ❖❝❝✉r ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠s✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❣r❛♣❤✐❝❛❧ ♠♦❞❡❧s✱

• ❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✱ ❡t❝✳

✶✷

❈❤♦r❞❛❧ ❣r❛♣❤s

❚r❡❡s ❛r❡ ❣r❛♣❤s ✇✐t❤ ♥♦ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ n ≥ 3✳ ❉❡✜♥✐t✐♦♥✿ ❆ ❣r❛♣❤ ✐s ❝❤♦r❞❛❧ ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥ ✐♥❞✉❝❡❞ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ n ≥ 4✳ ❈❤♦r❞❛❧ ◆♦t ❈❤♦r❞❛❧ ◆❛♠❡s✿ ❚r✐❛♥❣✉❧❛t❡❞✱ ❞❡❝♦♠♣♦s❛❜❧❡✱ r✐❣✐❞ ❝✐r❝✉✐t ❣r❛♣❤s✳ ✳ ✳ ❊①❛♠♣❧❡s✿ ❚r❡❡s✱ ❈♦♠♣❧❡t❡ ❣r❛♣❤s✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛♥② ❣r❛♣❤✱ ❆♣♦❧❧♦♥✐❛♥ ❣r❛♣❤s✱ ❇❛♥❞ ❣r❛♣❤s✱ ❙♣❧✐t ❣r❛♣❤s✱ ❡t❝✳ ❖❝❝✉r ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠s✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❣r❛♣❤✐❝❛❧ ♠♦❞❡❧s✱

• ❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✱ ❡t❝✳

✶✷

❈❤♦r❞❛❧ ❣r❛♣❤s

❚r❡❡s ❛r❡ ❣r❛♣❤s ✇✐t❤ ♥♦ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ n ≥ 3✳ ❉❡✜♥✐t✐♦♥✿ ❆ ❣r❛♣❤ ✐s ❝❤♦r❞❛❧ ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥ ✐♥❞✉❝❡❞ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ n ≥ 4✳ ❈❤♦r❞❛❧ ◆♦t ❈❤♦r❞❛❧ ◆❛♠❡s✿ ❚r✐❛♥❣✉❧❛t❡❞✱ ❞❡❝♦♠♣♦s❛❜❧❡✱ r✐❣✐❞ ❝✐r❝✉✐t ❣r❛♣❤s✳ ✳ ✳ ❊①❛♠♣❧❡s✿ ❚r❡❡s✱ ❈♦♠♣❧❡t❡ ❣r❛♣❤s✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛♥② ❣r❛♣❤✱ ❆♣♦❧❧♦♥✐❛♥ ❣r❛♣❤s✱ ❇❛♥❞ ❣r❛♣❤s✱ ❙♣❧✐t ❣r❛♣❤s✱ ❡t❝✳ ❖❝❝✉r ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠s✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❣r❛♣❤✐❝❛❧ ♠♦❞❡❧s✱

• ❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✱ ❡t❝✳

✶✷

❈❤♦r❞❛❧ ❣r❛♣❤s

❚r❡❡s ❛r❡ ❣r❛♣❤s ✇✐t❤ ♥♦ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ n ≥ 3✳ ❉❡✜♥✐t✐♦♥✿ ❆ ❣r❛♣❤ ✐s ❝❤♦r❞❛❧ ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥ ✐♥❞✉❝❡❞ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ n ≥ 4✳ ❈❤♦r❞❛❧ ◆♦t ❈❤♦r❞❛❧ ◆❛♠❡s✿ ❚r✐❛♥❣✉❧❛t❡❞✱ ❞❡❝♦♠♣♦s❛❜❧❡✱ r✐❣✐❞ ❝✐r❝✉✐t ❣r❛♣❤s✳ ✳ ✳ ❊①❛♠♣❧❡s✿ ❚r❡❡s✱ ❈♦♠♣❧❡t❡ ❣r❛♣❤s✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛♥② ❣r❛♣❤✱ ❆♣♦❧❧♦♥✐❛♥ ❣r❛♣❤s✱ ❇❛♥❞ ❣r❛♣❤s✱ ❙♣❧✐t ❣r❛♣❤s✱ ❡t❝✳ ❖❝❝✉r ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠s✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❣r❛♣❤✐❝❛❧ ♠♦❞❡❧s✱

• ❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✱ ❡t❝✳

✶✷

❈❤♦r❞❛❧ ❣r❛♣❤s

❚r❡❡s ❛r❡ ❣r❛♣❤s ✇✐t❤ ♥♦ ❝②❝❧❡s ♦❢ ❧❡♥❣t❤ n ≥ 3✳ ❉❡✜♥✐t✐♦♥✿ ❆ ❣r❛♣❤ ✐s ❝❤♦r❞❛❧ ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥ ✐♥❞✉❝❡❞ ❝②❝❧❡ ♦❢ ❧❡♥❣t❤ n ≥ 4✳ ❈❤♦r❞❛❧ ◆♦t ❈❤♦r❞❛❧ ◆❛♠❡s✿ ❚r✐❛♥❣✉❧❛t❡❞✱ ❞❡❝♦♠♣♦s❛❜❧❡✱ r✐❣✐❞ ❝✐r❝✉✐t ❣r❛♣❤s✳ ✳ ✳ ❊①❛♠♣❧❡s✿ ❚r❡❡s✱ ❈♦♠♣❧❡t❡ ❣r❛♣❤s✱ ❚r✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛♥② ❣r❛♣❤✱ ❆♣♦❧❧♦♥✐❛♥ ❣r❛♣❤s✱ ❇❛♥❞ ❣r❛♣❤s✱ ❙♣❧✐t ❣r❛♣❤s✱ ❡t❝✳ ❖❝❝✉r ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✿ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ♣r♦❜❧❡♠s✱ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❣r❛♣❤✐❝❛❧ ♠♦❞❡❧s✱

• ❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✱ ❡t❝✳

✶✷

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ▲❡t G ❜❡ ❛♥② ❝❤♦r❞❛❧ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st 2 ✈❡rt✐❝❡s ❛♥❞ ❧❡t r ❜❡ t❤❡ ❧❛r❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t ❡✐t❤❡r Kr ♦r K(1)

r

✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ G✳ ❚❤❡♥ HG = N ∪ [r − 2, ∞). ■♥ ♣❛rt✐❝✉❧❛r✱ CE(G) = r − 2✳ ❊✳❣✳✱ ❢♦r ❜❛♥❞ ❣r❛♣❤s ✇✐t❤ ❜❛♥❞✇✐❞t❤ ✱ ✳ ❙♦ ❢♦r ✱ ❛❧❧ ♣♦✇❡rs ✇♦r❦✳

✶✸

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ▲❡t G ❜❡ ❛♥② ❝❤♦r❞❛❧ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st 2 ✈❡rt✐❝❡s ❛♥❞ ❧❡t r ❜❡ t❤❡ ❧❛r❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t ❡✐t❤❡r Kr ♦r K(1)

r

✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ G✳ ❚❤❡♥ HG = N ∪ [r − 2, ∞). ■♥ ♣❛rt✐❝✉❧❛r✱ CE(G) = r − 2✳ ❊✳❣✳✱ ❢♦r ❜❛♥❞ ❣r❛♣❤s ✇✐t❤ ❜❛♥❞✇✐❞t❤ d✱ CE(G) = min(d, n − 2)✳ ❙♦ ❢♦r A =       1 0.6 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.5 0.6 1 0.6 0.5 0.6 1       ✱ ❛❧❧ ♣♦✇❡rs ≥ 2 = d ✇♦r❦✳

✶✸

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❝♦♥t✳✮

❙♦♠❡ ❦❡② ✐❞❡❛s ❢♦r t❤❡ ♣r♦♦❢✿

✶ ▼❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥s✿ ■❢

✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ✱ ❡✈❡r② ❞❡❝♦♠♣♦s❡s ❛s ✇✐t❤ ❛♥❞ ✿

✷ ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ❢✉♥❝t✐♦♥s✿

▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ♣♦✇❡rs ❝❧❛ss✐✜❡❞ ✐♥ ❬●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠❪✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✵✶✺✳

✸ ■♥❞✉❝t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❞❡❝♦♠♣♦s✐t✐♦♥✱

♦r❞❡r✐♥❣ ♦❢ ❝❧✐q✉❡s✱ ❡t❝✳✮✳

✶✹

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❝♦♥t✳✮

❙♦♠❡ ❦❡② ✐❞❡❛s ❢♦r t❤❡ ♣r♦♦❢✿

✶ ▼❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥s✿ ■❢ (A, S, B) ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ G✱

❡✈❡r② M ∈ PG ❞❡❝♦♠♣♦s❡s ❛s M = M1 + M2 ✇✐t❤ M1 ∈ PA∪S ❛♥❞ M2 ∈ PB∪S✿

✷ ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ❢✉♥❝t✐♦♥s✿

▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ♣♦✇❡rs ❝❧❛ss✐✜❡❞ ✐♥ ❬●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠❪✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✵✶✺✳

✸ ■♥❞✉❝t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❞❡❝♦♠♣♦s✐t✐♦♥✱

♦r❞❡r✐♥❣ ♦❢ ❝❧✐q✉❡s✱ ❡t❝✳✮✳

✶✹

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❝♦♥t✳✮

❙♦♠❡ ❦❡② ✐❞❡❛s ❢♦r t❤❡ ♣r♦♦❢✿

✶ ▼❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥s✿ ■❢ (A, S, B) ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ G✱

❡✈❡r② M ∈ PG ❞❡❝♦♠♣♦s❡s ❛s M = M1 + M2 ✇✐t❤ M1 ∈ PA∪S ❛♥❞ M2 ∈ PB∪S✿

  MAA MAS MT

AS

MSS MSB MT

SB

MBB   =   MAA MAS MT

AS

MT

ASM−1 AAMAS

  +   MSS − MT

ASM−1 AAMAS

MSB MT

SB

MBB   .

✷ ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ❢✉♥❝t✐♦♥s✿

▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ♣♦✇❡rs ❝❧❛ss✐✜❡❞ ✐♥ ❬●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠❪✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✵✶✺✳

✸ ■♥❞✉❝t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❞❡❝♦♠♣♦s✐t✐♦♥✱

♦r❞❡r✐♥❣ ♦❢ ❝❧✐q✉❡s✱ ❡t❝✳✮✳

✶✹

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❝♦♥t✳✮

❙♦♠❡ ❦❡② ✐❞❡❛s ❢♦r t❤❡ ♣r♦♦❢✿

✶ ▼❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥s✿ ■❢ (A, S, B) ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ G✱

❡✈❡r② M ∈ PG ❞❡❝♦♠♣♦s❡s ❛s M = M1 + M2 ✇✐t❤ M1 ∈ PA∪S ❛♥❞ M2 ∈ PB∪S✿

  MAA MAS MT

AS

MSS MSB MT

SB

MBB   =   MAA MAS MT

AS

MT

ASM−1 AAMAS

  +   MSS − MT

ASM−1 AAMAS

MSB MT

SB

MBB   .

✷ ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ❢✉♥❝t✐♦♥s✿

f[A + B] − (f[A] + f[B]) ∈ Pn ∀A, B ∈ Pn. ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ♣♦✇❡rs ❝❧❛ss✐✜❡❞ ✐♥ ❬●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠❪✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✵✶✺✳

✸ ■♥❞✉❝t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❞❡❝♦♠♣♦s✐t✐♦♥✱

♦r❞❡r✐♥❣ ♦❢ ❝❧✐q✉❡s✱ ❡t❝✳✮✳

✶✹

Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❝♦♥t✳✮

❙♦♠❡ ❦❡② ✐❞❡❛s ❢♦r t❤❡ ♣r♦♦❢✿

✶ ▼❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥s✿ ■❢ (A, S, B) ✐s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ G✱

❡✈❡r② M ∈ PG ❞❡❝♦♠♣♦s❡s ❛s M = M1 + M2 ✇✐t❤ M1 ∈ PA∪S ❛♥❞ M2 ∈ PB∪S✿

  MAA MAS MT

AS

MSS MSB MT

SB

MBB   =   MAA MAS MT

AS

MT

ASM−1 AAMAS

  +   MSS − MT

ASM−1 AAMAS

MSB MT

SB

MBB   .

✷ ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ❢✉♥❝t✐♦♥s✿

f[A + B] − (f[A] + f[B]) ∈ Pn ∀A, B ∈ Pn. ▲♦❡✇♥❡r s✉♣❡r✲❛❞❞✐t✐✈❡ ♣♦✇❡rs ❝❧❛ss✐✜❡❞ ✐♥ ❬●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠❪✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✵✶✺✳

✸ ■♥❞✉❝t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝❤♦r❞❛❧ ❣r❛♣❤s ✭❞❡❝♦♠♣♦s✐t✐♦♥✱

♦r❞❡r✐♥❣ ♦❢ ❝❧✐q✉❡s✱ ❡t❝✳✮✳

✶✹

◆♦♥✲❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❋♦r ❛❧❧ n ≥ 3✱ HCn = [1, ∞)✳ ❘❡♠❛r❦✿ ✐s t❤❡ ❜✐❣❣❡st s✉❝❤ t❤❛t ♦r ✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❙✉♣♣♦s❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st ✈❡rt✐❝❡s✳ ❚❤❡♥ ✳ Pr♦♦❢ ✉s❡s ❛ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ ❢❛❝t t❤❛t✱ ❢♦r ✇❤❡r❡ s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ ✳ ❘❡♠❛r❦✿ ✐s t❤❡ ❜✐❣❣❡st s✉❝❤ t❤❛t ♦r ✳

✶✺

◆♦♥✲❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❋♦r ❛❧❧ n ≥ 3✱ HCn = [1, ∞)✳ ❘❡♠❛r❦✿ 1 ✐s t❤❡ ❜✐❣❣❡st r − 2 s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ Cn✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❙✉♣♣♦s❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st ✈❡rt✐❝❡s✳ ❚❤❡♥ ✳ Pr♦♦❢ ✉s❡s ❛ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ ❢❛❝t t❤❛t✱ ❢♦r ✇❤❡r❡ s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ ✳ ❘❡♠❛r❦✿ ✐s t❤❡ ❜✐❣❣❡st s✉❝❤ t❤❛t ♦r ✳

✶✺

◆♦♥✲❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❋♦r ❛❧❧ n ≥ 3✱ HCn = [1, ∞)✳ ❘❡♠❛r❦✿ 1 ✐s t❤❡ ❜✐❣❣❡st r − 2 s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ Cn✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❙✉♣♣♦s❡ G ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st 3 ✈❡rt✐❝❡s✳ ❚❤❡♥ HG = [1, ∞)✳ Pr♦♦❢ ✉s❡s ❛ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ ❢❛❝t t❤❛t✱ ❢♦r ✇❤❡r❡ s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ ✳ ❘❡♠❛r❦✿ ✐s t❤❡ ❜✐❣❣❡st s✉❝❤ t❤❛t ♦r ✳

✶✺

◆♦♥✲❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❋♦r ❛❧❧ n ≥ 3✱ HCn = [1, ∞)✳ ❘❡♠❛r❦✿ 1 ✐s t❤❡ ❜✐❣❣❡st r − 2 s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ Cn✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❙✉♣♣♦s❡ G ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st 3 ✈❡rt✐❝❡s✳ ❚❤❡♥ HG = [1, ∞)✳ Pr♦♦❢ ✉s❡s ❛ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ ❢❛❝t t❤❛t✱ ρ(A◦α) ≤ ρ(A)α ❢♦r A ∈ Pn, α ≥ 1, ✇❤❡r❡ ρ(M) = s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ M✳ ❘❡♠❛r❦✿ ✐s t❤❡ ❜✐❣❣❡st s✉❝❤ t❤❛t ♦r ✳

✶✺

◆♦♥✲❝❤♦r❞❛❧ ❣r❛♣❤s

❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❋♦r ❛❧❧ n ≥ 3✱ HCn = [1, ∞)✳ ❘❡♠❛r❦✿ 1 ✐s t❤❡ ❜✐❣❣❡st r − 2 s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ Cn✳ ❚❤❡♦r❡♠ ✭●✉✐❧❧♦t✱ ❑❤❛r❡✱ ❘❛❥❛r❛t♥❛♠✱ ❏❈❚✲❆ ✷✵✶✺✮ ❙✉♣♣♦s❡ G ✐s ❛ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st 3 ✈❡rt✐❝❡s✳ ❚❤❡♥ HG = [1, ∞)✳ Pr♦♦❢ ✉s❡s ❛ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ ❢❛❝t t❤❛t✱ ρ(A◦α) ≤ ρ(A)α ❢♦r A ∈ Pn, α ≥ 1, ✇❤❡r❡ ρ(M) = s♣❡❝tr❛❧ r❛❞✐✉s ♦❢ M✳ ❘❡♠❛r❦✿ 1 ✐s t❤❡ ❜✐❣❣❡st r − 2 s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ G✳

✶✺

❖♣❡♥ ♣r♦❜❧❡♠s

✶ ❲❤❛t ✐s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣✐✈❡♥ ❣r❛♣❤❄ ✷ ❋♦r ❛♥② ❣r❛♣❤

✱ ✐s ✱ ✇❤❡r❡ ✐s t❤❡ ❜✐❣❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t ♦r ❄

✸ ❚❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣r❛♣❤ ❛❧✇❛②s ❛♣♣❡❛rs t♦ ❜❡ ❛♥

✐♥t❡❣❡r✳ ❈❛♥ t❤✐s ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ✭✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ❡①♣❧✐❝✐t❧②✮❄

✹ ❱❛r✐❛♥ts ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡♥tr✐❡s✳

◆❡❡❞ t♦ ✇♦r❦ ✇✐t❤ t❤❡s❡✱ ❡✈❡♥ ❢♦r ❛❜♦✈❡ q✉❡st✐♦♥s✳

✺ ❈♦♥♥❡❝t✐♦♥s t♦ ♦t❤❡r ✭♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧✮ ❣r❛♣❤ ✐♥✈❛r✐❛♥ts❄ ✶✻

❖♣❡♥ ♣r♦❜❧❡♠s

✶ ❲❤❛t ✐s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣✐✈❡♥ ❣r❛♣❤❄ ✷ ❋♦r ❛♥② ❣r❛♣❤ G✱ ✐s HG = N ∪ [r − 2, ∞)✱ ✇❤❡r❡ r ✐s t❤❡

❜✐❣❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ G❄

✸ ❚❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣r❛♣❤ ❛❧✇❛②s ❛♣♣❡❛rs t♦ ❜❡ ❛♥

✐♥t❡❣❡r✳ ❈❛♥ t❤✐s ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ✭✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ❡①♣❧✐❝✐t❧②✮❄

✹ ❱❛r✐❛♥ts ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡♥tr✐❡s✳

◆❡❡❞ t♦ ✇♦r❦ ✇✐t❤ t❤❡s❡✱ ❡✈❡♥ ❢♦r ❛❜♦✈❡ q✉❡st✐♦♥s✳

✺ ❈♦♥♥❡❝t✐♦♥s t♦ ♦t❤❡r ✭♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧✮ ❣r❛♣❤ ✐♥✈❛r✐❛♥ts❄ ✶✻

❖♣❡♥ ♣r♦❜❧❡♠s

✶ ❲❤❛t ✐s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣✐✈❡♥ ❣r❛♣❤❄ ✷ ❋♦r ❛♥② ❣r❛♣❤ G✱ ✐s HG = N ∪ [r − 2, ∞)✱ ✇❤❡r❡ r ✐s t❤❡

❜✐❣❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ G❄

✸ ❚❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣r❛♣❤ ❛❧✇❛②s ❛♣♣❡❛rs t♦ ❜❡ ❛♥

✐♥t❡❣❡r✳ ❈❛♥ t❤✐s ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ✭✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ❡①♣❧✐❝✐t❧②✮❄

✹ ❱❛r✐❛♥ts ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡♥tr✐❡s✳

◆❡❡❞ t♦ ✇♦r❦ ✇✐t❤ t❤❡s❡✱ ❡✈❡♥ ❢♦r ❛❜♦✈❡ q✉❡st✐♦♥s✳

✺ ❈♦♥♥❡❝t✐♦♥s t♦ ♦t❤❡r ✭♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧✮ ❣r❛♣❤ ✐♥✈❛r✐❛♥ts❄ ✶✻

❖♣❡♥ ♣r♦❜❧❡♠s

✶ ❲❤❛t ✐s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣✐✈❡♥ ❣r❛♣❤❄ ✷ ❋♦r ❛♥② ❣r❛♣❤ G✱ ✐s HG = N ∪ [r − 2, ∞)✱ ✇❤❡r❡ r ✐s t❤❡

❜✐❣❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ G❄

✸ ❚❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣r❛♣❤ ❛❧✇❛②s ❛♣♣❡❛rs t♦ ❜❡ ❛♥

✐♥t❡❣❡r✳ ❈❛♥ t❤✐s ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ✭✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ❡①♣❧✐❝✐t❧②✮❄

✹ ❱❛r✐❛♥ts ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡♥tr✐❡s✳

◆❡❡❞ t♦ ✇♦r❦ ✇✐t❤ t❤❡s❡✱ ❡✈❡♥ ❢♦r ❛❜♦✈❡ q✉❡st✐♦♥s✳

✺ ❈♦♥♥❡❝t✐♦♥s t♦ ♦t❤❡r ✭♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧✮ ❣r❛♣❤ ✐♥✈❛r✐❛♥ts❄ ✶✻

❖♣❡♥ ♣r♦❜❧❡♠s

✶ ❲❤❛t ✐s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣✐✈❡♥ ❣r❛♣❤❄ ✷ ❋♦r ❛♥② ❣r❛♣❤ G✱ ✐s HG = N ∪ [r − 2, ∞)✱ ✇❤❡r❡ r ✐s t❤❡

❜✐❣❣❡st ✐♥t❡❣❡r s✉❝❤ t❤❛t Kr ♦r K(1)

r

⊂ G❄

✸ ❚❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ♦❢ ❛ ❣r❛♣❤ ❛❧✇❛②s ❛♣♣❡❛rs t♦ ❜❡ ❛♥

✐♥t❡❣❡r✳ ❈❛♥ t❤✐s ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ✭✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t ❡①♣❧✐❝✐t❧②✮❄

✹ ❱❛r✐❛♥ts ❢♦r ♠❛tr✐❝❡s ✇✐t❤ ♥❡❣❛t✐✈❡ ❡♥tr✐❡s✳

◆❡❡❞ t♦ ✇♦r❦ ✇✐t❤ t❤❡s❡✱ ❡✈❡♥ ❢♦r ❛❜♦✈❡ q✉❡st✐♦♥s✳

✺ ❈♦♥♥❡❝t✐♦♥s t♦ ♦t❤❡r ✭♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧✮ ❣r❛♣❤ ✐♥✈❛r✐❛♥ts❄ ✶✻

❇✐❜❧✐♦❣r❛♣❤②

❬✶❪ ❉✳ ●✉✐❧❧♦t✱ ❆✳ ❑❤❛r❡✱ ❇✳ ❘❛❥❛r❛t♥❛♠✱ ❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ♦❢ ❣r❛♣❤s✱ ❛❝✲ ❝❡♣t❡❞ ✐♥ ❏✳ ❈♦♠❜✐♥✳ ❚❤❡♦r②✱ ❙❡r✳ ❆✱ ✷✵✶✺✳ ❬✷❪ ❉✳ ●✉✐❧❧♦t✱ ❆✳ ❑❤❛r❡✱ ❛♥❞ ❇✳ ❘❛❥❛r❛t♥❛♠✱ ❈♦♠♣❧❡t❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❍❛❞❛♠❛r❞ ♣♦✇❡rs ♣r❡s❡r✈✐♥❣ ▲♦❡✇♥❡r ♣♦s✐t✐✈✐t②✱ ♠♦♥♦t♦♥✐❝✐t②✱ ❛♥❞ ❝♦♥✈❡①✲ ✐t②✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✹✷✺✭✶✮✿✹✽✾✲✺✵✼✱ ✷✵✶✺✳ ❬✸❪ ❉✳ ●✉✐❧❧♦t✱ ❆✳ ❑❤❛r❡✱ ❇✳ ❘❛❥❛r❛t♥❛♠✱ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r ♠❛tr✐❝❡s ✇✐t❤ s♣❛rs✐t② ❝♦♥str❛✐♥ts✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✐♥ ♣r❡ss✱ ✷✵✶✺✳ ❬✹❪ ❉✳ ●✉✐❧❧♦t✱ ❆✳ ❑❤❛r❡✱ ❇✳ ❘❛❥❛r❛t♥❛♠✱ Pr❡s❡r✈✐♥❣ ♣♦s✐t✐✈✐t② ❢♦r r❛♥❦✲ ❝♦♥str❛✐♥❡❞ ♠❛tr✐❝❡s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✐♥ ♣r❡ss✱ ✷✵✶✺✳ ❬✺❪ ❆✳ ❇❡❧t♦♥✱ ❉✳ ●✉✐❧❧♦t✱ ❆✳ ❑❤❛r❡✱ ❛♥❞ ▼✳ P✉t✐♥❛r✱ ▼❛tr✐① ♣♦s✐t✐✈✐t② ♣r❡✲ s❡r✈❡rs ✐♥ ✜①❡❞ ❞✐♠❡♥s✐♦♥✳ ■✱ ❆❞✈✳ ▼❛t❤✳✱ ✉♥❞❡r r❡✈✐❡✇✱ ✷✵✶✺✳ P❛♣❡rs ❛✈❛✐❧❛❜❧❡ ❛t✿ ❤tt♣✿✴✴✇❡❜✳st❛♥❢♦r❞✳❡❞✉✴∼❦❤❛r❡✴

✶✼