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❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s

❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❙❡r❣✐ ❊❧✐③❛❧❞❡

❉❛rt♠♦✉t❤ ❈♦❧❧❡❣❡

❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼❛rt✐♥ ❘✉❜❡② ❈❛♥❛❉❆▼ ✷✵✶✸

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❉②❝❦ ♣❛t❤s

P ❚ ❇

t(P) = ✸ ❜(P) = ✷

❋♦r P ∈ D♥ ✭❉②❝❦ ♣❛t❤s ✇✐t❤ ✷♥ st❡♣s✮✱ ❧❡t t(P) = # ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ = ✏❤❡✐❣❤t✑ ♦❢ t❤❡ ❧❛st ✏♣❡❛❦✑ ❜(P) = # ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ = ♥✉♠❜❡r ♦❢ r❡t✉r♥s

❚❤❡♦r❡♠ ✭❉❡✉ts❝❤ ✬✾✽✮

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛✐r t ❜ ♦✈❡r

♥ ✐s s②♠♠❡tr✐❝✱ ✐✳❡✳✱ P

♥

①t P ②❜ P

P

♥

①❜ P ②t P Pr♦♦❢ ✶ ✭❉❡✉ts❝❤✮✿ ❘❡❝✉rs✐✈❡ ❜✐❥❡❝t✐♦♥✳ Pr♦♦❢ ✷✿ ●❡♥❡r❛t✐♥❣ ❢❝ts✳ ❇♦t❤ ♣r♦♦❢s r❡❧② ♦♥ t❤❡ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ♦❢ ❉②❝❦ ♣❛t❤s✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❉②❝❦ ♣❛t❤s

P ❚ ❇

t(P) = ✸ ❜(P) = ✷

❋♦r P ∈ D♥ ✭❉②❝❦ ♣❛t❤s ✇✐t❤ ✷♥ st❡♣s✮✱ ❧❡t t(P) = # ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ = ✏❤❡✐❣❤t✑ ♦❢ t❤❡ ❧❛st ✏♣❡❛❦✑ ❜(P) = # ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ = ♥✉♠❜❡r ♦❢ r❡t✉r♥s

❚❤❡♦r❡♠ ✭❉❡✉ts❝❤ ✬✾✽✮

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛✐r (t, ❜) ♦✈❡r D♥ ✐s s②♠♠❡tr✐❝✱ ✐✳❡✳✱

• P∈D♥

①t(P)②❜(P) =

• P∈D♥

①❜(P)②t(P). Pr♦♦❢ ✶ ✭❉❡✉ts❝❤✮✿ ❘❡❝✉rs✐✈❡ ❜✐❥❡❝t✐♦♥✳ Pr♦♦❢ ✷✿ ●❡♥❡r❛t✐♥❣ ❢❝ts✳ ❇♦t❤ ♣r♦♦❢s r❡❧② ♦♥ t❤❡ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ♦❢ ❉②❝❦ ♣❛t❤s✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❉②❝❦ ♣❛t❤s

P ❚ ❇

t(P) = ✸ ❜(P) = ✷

❋♦r P ∈ D♥ ✭❉②❝❦ ♣❛t❤s ✇✐t❤ ✷♥ st❡♣s✮✱ ❧❡t t(P) = # ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ = ✏❤❡✐❣❤t✑ ♦❢ t❤❡ ❧❛st ✏♣❡❛❦✑ ❜(P) = # ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ = ♥✉♠❜❡r ♦❢ r❡t✉r♥s

❚❤❡♦r❡♠ ✭❉❡✉ts❝❤ ✬✾✽✮

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛✐r (t, ❜) ♦✈❡r D♥ ✐s s②♠♠❡tr✐❝✱ ✐✳❡✳✱

• P∈D♥

①t(P)②❜(P) =

• P∈D♥

①❜(P)②t(P). Pr♦♦❢ ✶ ✭❉❡✉ts❝❤✮✿ ❘❡❝✉rs✐✈❡ ❜✐❥❡❝t✐♦♥✳ Pr♦♦❢ ✷✿ ●❡♥❡r❛t✐♥❣ ❢❝ts✳ ❇♦t❤ ♣r♦♦❢s r❡❧② ♦♥ t❤❡ r❡❝✉rs✐✈❡ str✉❝t✉r❡ ♦❢ ❉②❝❦ ♣❛t❤s✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❛r❜✐tr❛r② ❜♦✉♥❞❛r✐❡s

P ❚ ❇ ❖ ❋

t(P) = ✹ ❜(P) = ✸

❚ ❛♥❞ ❇ ♣❛t❤s ❢r♦♠ ❖ t♦ ❋ ✇✐t❤ st❡♣s ◆ ❛♥❞ ❊✱ ✇✐t❤ ❚ ✇❡❛❦❧② ❛❜♦✈❡ ❇ P ∈ P(❚, ❇) = s❡t ♦❢ ♣❛t❤s ❢r♦♠ ❖ t♦ ❋ ✇❡❛❦❧② ❜❡t✇❡❡♥ ❚ ❛♥❞ ❇ t(P) =# ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ ✭t♦♣ ❝♦♥t❛❝ts ♦❢ P✮ ❜(P) =# ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ ✭❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P✮

❚❤❡♦r❡♠

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t ❜ ♦✈❡r ❚ ❇ ✐s s②♠♠❡tr✐❝✱ ✐✳❡✳✱

P ❚ ❇

①t P ②❜ P

P ❚ ❇

①❜ P ②t P

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❛r❜✐tr❛r② ❜♦✉♥❞❛r✐❡s

P ❚ ❇ ❖ ❋

t(P) = ✹ ❜(P) = ✸

❚ ❛♥❞ ❇ ♣❛t❤s ❢r♦♠ ❖ t♦ ❋ ✇✐t❤ st❡♣s ◆ ❛♥❞ ❊✱ ✇✐t❤ ❚ ✇❡❛❦❧② ❛❜♦✈❡ ❇ P ∈ P(❚, ❇) = s❡t ♦❢ ♣❛t❤s ❢r♦♠ ❖ t♦ ❋ ✇❡❛❦❧② ❜❡t✇❡❡♥ ❚ ❛♥❞ ❇ t(P) =# ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ ✭t♦♣ ❝♦♥t❛❝ts ♦❢ P✮ ❜(P) =# ♦❢ ❊ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ ✭❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P✮

❚❤❡♦r❡♠

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ (t, ❜) ♦✈❡r P(❚, ❇) ✐s s②♠♠❡tr✐❝✱ ✐✳❡✳✱

• P∈P(❚,❇)

①t(P)②❜(P) =

• P∈P(❚,❇)

①❜(P)②t(P).

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❊①❛♠♣❧❡

①✸ ①✷ ① ①✷ ① ✶ ①② ② ②✷ ①✷② ①② ② ①②✷ ②✷ ②✸

• P∈P(❚,❇)

①t(P)② ❜(P) = ①✸ + ①✷② + ①② ✷ + ② ✸ + ✷①✷ + ✷①② + ✷② ✷ + ✷① + ✷② + ✶

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

Pr♦♦❢

❚❤❡ ❦♥♦✇♥ ♣r♦♦❢s ❢♦r ❉②❝❦ ♣❛t❤s ❞♦ ♥♦t s❡❡♠ t♦ ❣❡♥❡r❛❧✐③❡ t♦ ❛r❜✐tr❛r② ❜♦✉♥❞❛r✐❡s✳ ❲❡ ❣✐✈❡ ❛♥ ✐♥✈♦❧✉t✐♦♥ ❚ ❇ ❚ ❇ ✇✐t❤ t❤❡ ♣r♦♣❡rt② t P ❜ P ❛♥❞ ❜ P t P ✳ ■❞❡❛✿ ●✐✈❡♥ P ❚ ❇ ✇✐t❤ t P ❜ P ✱ t✉r♥ s♦♠❡ ♦❢ ✐ts t♦♣ ❝♦♥t❛❝ts ✐♥t♦ ❜♦tt♦♠ ❝♦♥t❛❝ts✱ ♦♥❡ ❛t ❛ t✐♠❡✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

Pr♦♦❢

❚❤❡ ❦♥♦✇♥ ♣r♦♦❢s ❢♦r ❉②❝❦ ♣❛t❤s ❞♦ ♥♦t s❡❡♠ t♦ ❣❡♥❡r❛❧✐③❡ t♦ ❛r❜✐tr❛r② ❜♦✉♥❞❛r✐❡s✳ ❲❡ ❣✐✈❡ ❛♥ ✐♥✈♦❧✉t✐♦♥ Φ : P(❚, ❇) → P(❚, ❇) ✇✐t❤ t❤❡ ♣r♦♣❡rt② t(Φ(P)) = ❜(P) ❛♥❞ ❜(Φ(P)) = t(P)✳ ■❞❡❛✿ ●✐✈❡♥ P ∈ P(❚, ❇) ✇✐t❤ t(P) > ❜(P)✱ t✉r♥ s♦♠❡ ♦❢ ✐ts t♦♣ ❝♦♥t❛❝ts ✐♥t♦ ❜♦tt♦♠ ❝♦♥t❛❝ts✱ ♦♥❡ ❛t ❛ t✐♠❡✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❊①❛♠♣❧❡

❲❡ ❞❡✜♥❡ t❤❡ ✐♥✈♦❧✉t✐♦♥ Φ ❜② ✐t❡r❛t✐♥❣ ❛ ♠❛♣ φ✱ ✇❤✐❝❤ t✉r♥s ♦♥❡ t♦♣ ❝♦♥t❛❝t ✐♥t♦ ♦♥❡ ❜♦tt♦♠ ❝♦♥t❛❝t✳

P → φ φ(P) → φ φ✷(P) = Φ(P)

(t, ❜) = (✹, ✷) (t, ❜) = (✸, ✸) (t, ❜) = (✷, ✹)

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❋r♦♠ ♣❛t❤s t♦ ✇♦r❞s

❚♦ ❞❡✜♥❡ φ(P)✱ ✇❡ ✜rst ✜♥❞ t❤❡ t♦♣ ❝♦♥t❛❝t t❤❛t ✇✐❧❧ ❜❡ ❝❤❛♥❣❡❞ ✐♥t♦ ❛ ❜♦tt♦♠ ❝♦♥t❛❝t✳ ✶✳ ❘❡❝♦r❞ t♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P ❛s ❛ ✇♦r❞ ✇ ♦✈❡r t ❜ ✿ P ✇ ❜tt❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❋r♦♠ ♣❛t❤s t♦ ✇♦r❞s

❚♦ ❞❡✜♥❡ φ(P)✱ ✇❡ ✜rst ✜♥❞ t❤❡ t♦♣ ❝♦♥t❛❝t t❤❛t ✇✐❧❧ ❜❡ ❝❤❛♥❣❡❞ ✐♥t♦ ❛ ❜♦tt♦♠ ❝♦♥t❛❝t✳ ✶✳ ❘❡❝♦r❞ t♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P ❛s ❛ ✇♦r❞ ✇ ♦✈❡r {t, ❜}✿ P ✇ = ❜tt❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❋r♦♠ ♣❛t❤s t♦ ✇♦r❞s

✷✳ ❍❛✈✐♥❣ ❜✉✐❧t ✇✱ s❡❧❡❝t ❛ t♦♣ ❝♦♥t❛❝t ❛s ❢♦❧❧♦✇s✿ ❉r❛✇ ❛ ♣❛t❤ ✇✐t❤ ❛ st❡♣ ✶ ✶ ❢♦r ❡❛❝❤ t✱ ❛♥❞ ❛ st❡♣ ✶ ✶ ❢♦r ❡❛❝❤ ❜✳ ▼❛t❝❤ t✬s ❛♥❞ ❜✬s t❤❛t ✏❢❛❝❡✑ ❡❛❝❤ ♦t❤❡r ✐♥ t❤❡ ♣❛t❤✳ ❙❡❧❡❢t t❤❡ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t ❛s t❤❡ t♦♣ ❝♦♥t❛❝t t❤❛t ✇✐❧❧ ❜❡ ❝❤❛♥❣❡❞✳ ✇ = ❜tt❜t❜❜❜tt❜tt❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❋r♦♠ ♣❛t❤s t♦ ✇♦r❞s

✷✳ ❍❛✈✐♥❣ ❜✉✐❧t ✇✱ s❡❧❡❝t ❛ t♦♣ ❝♦♥t❛❝t ❛s ❢♦❧❧♦✇s✿

◮ ❉r❛✇ ❛ ♣❛t❤ ✇✐t❤ ❛ st❡♣ (✶, ✶) ❢♦r ❡❛❝❤ t✱ ❛♥❞ ❛ st❡♣ (✶, −✶)

❢♦r ❡❛❝❤ ❜✳ ▼❛t❝❤ t✬s ❛♥❞ ❜✬s t❤❛t ✏❢❛❝❡✑ ❡❛❝❤ ♦t❤❡r ✐♥ t❤❡ ♣❛t❤✳ ❙❡❧❡❢t t❤❡ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t ❛s t❤❡ t♦♣ ❝♦♥t❛❝t t❤❛t ✇✐❧❧ ❜❡ ❝❤❛♥❣❡❞✳ ✇ = ❜tt❜t❜❜❜tt❜tt❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❋r♦♠ ♣❛t❤s t♦ ✇♦r❞s

✷✳ ❍❛✈✐♥❣ ❜✉✐❧t ✇✱ s❡❧❡❝t ❛ t♦♣ ❝♦♥t❛❝t ❛s ❢♦❧❧♦✇s✿

◮ ❉r❛✇ ❛ ♣❛t❤ ✇✐t❤ ❛ st❡♣ (✶, ✶) ❢♦r ❡❛❝❤ t✱ ❛♥❞ ❛ st❡♣ (✶, −✶)

❢♦r ❡❛❝❤ ❜✳

◮ ▼❛t❝❤ t✬s ❛♥❞ ❜✬s t❤❛t ✏❢❛❝❡✑ ❡❛❝❤ ♦t❤❡r ✐♥ t❤❡ ♣❛t❤✳

❙❡❧❡❢t t❤❡ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t ❛s t❤❡ t♦♣ ❝♦♥t❛❝t t❤❛t ✇✐❧❧ ❜❡ ❝❤❛♥❣❡❞✳ ✇ = ❜tt❜t❜❜❜tt❜tt❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❋r♦♠ ♣❛t❤s t♦ ✇♦r❞s

✷✳ ❍❛✈✐♥❣ ❜✉✐❧t ✇✱ s❡❧❡❝t ❛ t♦♣ ❝♦♥t❛❝t ❛s ❢♦❧❧♦✇s✿

◮ ❉r❛✇ ❛ ♣❛t❤ ✇✐t❤ ❛ st❡♣ (✶, ✶) ❢♦r ❡❛❝❤ t✱ ❛♥❞ ❛ st❡♣ (✶, −✶)

❢♦r ❡❛❝❤ ❜✳

◮ ▼❛t❝❤ t✬s ❛♥❞ ❜✬s t❤❛t ✏❢❛❝❡✑ ❡❛❝❤ ♦t❤❡r ✐♥ t❤❡ ♣❛t❤✳ ◮ ❙❡❧❡❢t t❤❡ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t ❛s t❤❡ t♦♣ ❝♦♥t❛❝t t❤❛t ✇✐❧❧

❜❡ ❝❤❛♥❣❡❞✳ ✇ = ❜tt❜t❜❜❜tt❜tt❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ♠❛♣ φ

• ✐✈❡♥ P ∈ P(❚, ❇)✱ ❞❡✜♥❡ φ(P) ❛s ❢♦❧❧♦✇s✿

◮ ❘❡❝♦r❞ t♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P ❛s ❛ ✇♦r❞ ✇ ♦✈❡r {t, ❜}✳

❋✐♥❞ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t❀ ❧❡t ❊ ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❡♣✳ ❲r✐t❡ P ❳❨ ❊❩✱ ✇❤❡r❡ ❨ t♦✉❝❤❡s ❇ ♦♥❧② ❛t ✐ts ❧❡❢t ❡♥❞♣♦✐♥t✳ ▲❡t P ❳❊❨ ❩✳ P ✇ = ❜tt❜tt ❜❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ♠❛♣ φ

• ✐✈❡♥ P ∈ P(❚, ❇)✱ ❞❡✜♥❡ φ(P) ❛s ❢♦❧❧♦✇s✿

◮ ❘❡❝♦r❞ t♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P ❛s ❛ ✇♦r❞ ✇ ♦✈❡r {t, ❜}✳ ◮ ❋✐♥❞ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t❀ ❧❡t ❊ ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❡♣✳

❲r✐t❡ P ❳❨ ❊❩✱ ✇❤❡r❡ ❨ t♦✉❝❤❡s ❇ ♦♥❧② ❛t ✐ts ❧❡❢t ❡♥❞♣♦✐♥t✳ ▲❡t P ❳❊❨ ❩✳ P ❊ ✇ = ❜tt❜tt ❜❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ♠❛♣ φ

• ✐✈❡♥ P ∈ P(❚, ❇)✱ ❞❡✜♥❡ φ(P) ❛s ❢♦❧❧♦✇s✿

◮ ❘❡❝♦r❞ t♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P ❛s ❛ ✇♦r❞ ✇ ♦✈❡r {t, ❜}✳ ◮ ❋✐♥❞ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t❀ ❧❡t ❊ ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❡♣✳ ◮ ❲r✐t❡ P = ❳❨ ❊❩✱ ✇❤❡r❡ ❨ t♦✉❝❤❡s ❇ ♦♥❧② ❛t ✐ts ❧❡❢t

❡♥❞♣♦✐♥t✳ ▲❡t P ❳❊❨ ❩✳ P ❊ ❨ ❳ ❩ ✇ = ❜tt❜tt ❜❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ♠❛♣ φ

• ✐✈❡♥ P ∈ P(❚, ❇)✱ ❞❡✜♥❡ φ(P) ❛s ❢♦❧❧♦✇s✿

◮ ❘❡❝♦r❞ t♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ♦❢ P ❛s ❛ ✇♦r❞ ✇ ♦✈❡r {t, ❜}✳ ◮ ❋✐♥❞ ❧❡❢t♠♦st ✉♥♠❛t❝❤❡❞ t❀ ❧❡t ❊ ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❡♣✳ ◮ ❲r✐t❡ P = ❳❨ ❊❩✱ ✇❤❡r❡ ❨ t♦✉❝❤❡s ❇ ♦♥❧② ❛t ✐ts ❧❡❢t

❡♥❞♣♦✐♥t✳

◮ ▲❡t φ(P) = ❳❊❨ ❩✳

❊ ❨ ❳ ❩ φ(P) ✇ = ❜tt❜tt ❜❜t❜tt

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ✐♥✈♦❧✉t✐♦♥ Φ

❋♦r P ∈ P(❚, ❇) ✇✐t❤ t(P) = ❡ ❛♥❞ ❜(P) = ❢ ✱ ❞❡✜♥❡ Φ(P) = φ❡−❢ (P).

❚❤❡♦r❡♠

✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ♦♥ ❚ ❇ t❤❛t s❛t✐s✜❡s t P ❜ P ❛♥❞ ❜ P t P ✳

P P

✷ P

P

t ❜ ✹ ✷ t ❜ ✸ ✸ t ❜ ✷ ✹

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ✐♥✈♦❧✉t✐♦♥ Φ

❋♦r P ∈ P(❚, ❇) ✇✐t❤ t(P) = ❡ ❛♥❞ ❜(P) = ❢ ✱ ❞❡✜♥❡ Φ(P) = φ❡−❢ (P).

❚❤❡♦r❡♠

Φ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ♦♥ P(❚, ❇) t❤❛t s❛t✐s✜❡s t(Φ(P)) = ❜(P) ❛♥❞ ❜(Φ(P)) = t(P)✳

P P

✷ P

P

t ❜ ✹ ✷ t ❜ ✸ ✸ t ❜ ✷ ✹

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊ ❚❤❡ ❜✐❥❡❝t✐♦♥

❚❤❡ ✐♥✈♦❧✉t✐♦♥ Φ

❋♦r P ∈ P(❚, ❇) ✇✐t❤ t(P) = ❡ ❛♥❞ ❜(P) = ❢ ✱ ❞❡✜♥❡ Φ(P) = φ❡−❢ (P).

❚❤❡♦r❡♠

Φ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ♦♥ P(❚, ❇) t❤❛t s❛t✐s✜❡s t(Φ(P)) = ❜(P) ❛♥❞ ❜(Φ(P)) = t(P)✳

P → φ φ(P) → φ φ✷(P) = Φ(P)

(t, ❜) = (✹, ✷) (t, ❜) = (✸, ✸) (t, ❜) = (✷, ✹)

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣❛t❤s ✇✐t❤ ❙ st❡♣s

P ❚ ❇ ❖ ❋

• P(❚, ❇) = s❡t ♦❢ ♣❛t❤s ❢r♦♠ ❖ t♦ ❋

✇✐t❤ st❡♣s ◆✱ ❊ ❛♥❞ ❙ ✇❡❛❦❧② ❜❡t✇❡❡♥ ❚ ❛♥❞ ❇✳ ❋♦r P ∈ P(❚, ❇)✱ ❞❡✜♥❡ t(P) ❛♥❞ ❜(P) ❛s ❜❡❢♦r❡✳ ❚❤❡ ❞❡s❝❡♥t s❡t ♦❢ P ✐s t❤❡ s❡t ♦❢ ①✲❝♦♦r❞✐♥❛t❡s ✇❤❡r❡ ❙ st❡♣s ♦❝❝✉r✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ❚ ❇ ❚ ❇ t❤❛t s✇✐t❝❤❡s t❤❡ st❛t✐st✐❝s t ❜ ❛♥❞ ♣r❡s❡r✈❡s t❤❡ ❞❡s❝❡♥t s❡t✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♣❛t❤s ✇✐t❤ ❙ st❡♣s

P ❚ ❇ ❖ ❋

• P(❚, ❇) = s❡t ♦❢ ♣❛t❤s ❢r♦♠ ❖ t♦ ❋

✇✐t❤ st❡♣s ◆✱ ❊ ❛♥❞ ❙ ✇❡❛❦❧② ❜❡t✇❡❡♥ ❚ ❛♥❞ ❇✳ ❋♦r P ∈ P(❚, ❇)✱ ❞❡✜♥❡ t(P) ❛♥❞ ❜(P) ❛s ❜❡❢♦r❡✳ ❚❤❡ ❞❡s❝❡♥t s❡t ♦❢ P ✐s t❤❡ s❡t ♦❢ ①✲❝♦♦r❞✐♥❛t❡s ✇❤❡r❡ ❙ st❡♣s ♦❝❝✉r✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ P(❚, ❇) → P(❚, ❇) t❤❛t s✇✐t❝❤❡s t❤❡ st❛t✐st✐❝s (t, ❜) ❛♥❞ ♣r❡s❡r✈❡s t❤❡ ❞❡s❝❡♥t s❡t✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ ❡①❛♠♣❧❡s

❚❤❡ ♠❛♣ φ ❢♦r ♣❛t❤s ✇✐t❤ ❙ st❡♣s✿

φ

→

φ

→

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ ❡①❛♠♣❧❡s

❚❤❡ ✐♥✈♦❧✉t✐♦♥ Φ ❢♦r ♣❛t❤s ✇✐t❤ ❙ st❡♣s✿

φ

→

φ

→

φ

→

φ

→

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ r❡❧❛t❡❞ t❤❡♦r❡♠

P ❚ ❇ ❋♦r P ∈ P(❚, ❇)✱ ❧❡t ℓ(P) = # ♦❢ ◆ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ r(P) = # ♦❢ ◆ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ ❊①❛♠♣❧❡✿ t(P) = ✹✱ ❜(P) = ✸✱ ℓ(P) = ✷✱ r(P) = ✶✳

❚❤❡♦r❡♠

❚❤❡ ♣❛✐rs ❜ ❛♥❞ t r ❤❛✈❡ t❤❡ s❛♠❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦✈❡r ❚ ❇ ✱ ✐✳❡✳✱

P ❚ ❇

①❜ P ②

P P ❚ ❇

①t P ②r P ❲❡ ❞♦ ♥♦t ❦♥♦✇ ♦❢ ❛ ❜✐❥❡❝t✐✈❡ ♣r♦♦❢ s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ r❡❧❛t❡❞ t❤❡♦r❡♠

P ❚ ❇ ❋♦r P ∈ P(❚, ❇)✱ ❧❡t ℓ(P) = # ♦❢ ◆ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ r(P) = # ♦❢ ◆ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ ❊①❛♠♣❧❡✿ t(P) = ✹✱ ❜(P) = ✸✱ ℓ(P) = ✷✱ r(P) = ✶✳

❚❤❡♦r❡♠

❚❤❡ ♣❛✐rs (❜, ℓ) ❛♥❞ (t, r) ❤❛✈❡ t❤❡ s❛♠❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦✈❡r P(❚, ❇)✱ ✐✳❡✳✱

• P∈P(❚,❇)

①❜(P)②ℓ(P) =

• P∈P(❚,❇)

①t(P)②r(P). ❲❡ ❞♦ ♥♦t ❦♥♦✇ ♦❢ ❛ ❜✐❥❡❝t✐✈❡ ♣r♦♦❢ s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❆ r❡❧❛t❡❞ t❤❡♦r❡♠

P ❚ ❇ ❋♦r P ∈ P(❚, ❇)✱ ❧❡t ℓ(P) = # ♦❢ ◆ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❚ r(P) = # ♦❢ ◆ st❡♣s ✐♥ ❝♦♠♠♦♥ ✇✐t❤ ❇ ❊①❛♠♣❧❡✿ t(P) = ✹✱ ❜(P) = ✸✱ ℓ(P) = ✷✱ r(P) = ✶✳

❚❤❡♦r❡♠

❚❤❡ ♣❛✐rs (❜, ℓ) ❛♥❞ (t, r) ❤❛✈❡ t❤❡ s❛♠❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦✈❡r P(❚, ❇)✱ ✐✳❡✳✱

• P∈P(❚,❇)

①❜(P)②ℓ(P) =

• P∈P(❚,❇)

①t(P)②r(P). ❲❡ ❞♦ ♥♦t ❦♥♦✇ ♦❢ ❛ ❜✐❥❡❝t✐✈❡ ♣r♦♦❢ s✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

Pr♦♦❢ ✐❞❡❛

❇♦t❤

• P∈P(❚,❇)

①❜(P)②ℓ(P) ❛♥❞

• P∈P(❚,❇)

①t(P)②r(P) ❡q✉❛❧ t❤❡ ❚✉tt❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛ ❧❛tt✐❝❡ ♣❛t❤ ♠❛tr♦✐❞✱ ❛s ❞❡✜♥❡❞ ❜② ❇♦♥✐♥✕❉❡ ▼✐❡r✕◆♦② ✬✵✸✳ ❚❤❡ st❛t✐st✐❝s ❜ ❛♥❞ ℓ ✭t ❛♥❞ r✮ ❛r❡ ✐♥t❡r♥❛❧ ❛♥❞ ❡①t❡r♥❛❧ ❛❝t✐✈✐t✐❡s ✇✐t❤ r❡s♣❡❝t t♦ ❞✐✛❡r❡♥t ❧✐♥❡❛r ♦r❞❡r✐♥❣s ♦❢ t❤❡ ❣r♦✉♥❞ s❡t✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❦✲❢❛♥s ♦❢ ♣❛t❤s

P✶P✷ P✵ = ❚ P✸ = ❇ ❤✵ = ✹ ❤✶ = ✹ ❤✷ = ✻ P✶, P✷, . . . , P❦ ∈ P(❚, ❇)✱ P✐ ✇❡❛❦❧② ❛❜♦✈❡ P✐+✶ ❢♦r ❛❧❧ ✐✳ ▲❡t P✵ = ❚✱ P❦+✶ = ❇✳ ❋♦r ✵ ≤ ✐ ≤ ❦✱ ❧❡t ❤✐ =# ♦❢ ❊ st❡♣s ✇❤❡r❡ P✐ ❛♥❞ P✐+✶ ❝♦♥✐♥❝✐❞❡

❚❤❡♦r❡♠

❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❤✵ ❤✶ ❤❦ ♦✈❡r ❦✲❢❛♥s ♦❢ ♣❛t❤s ❛s ❛❜♦✈❡ ✐s s②♠♠❡tr✐❝✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s P❛t❤s ✇✐t❤ st❡♣s ◆, ❊, ❙ ▲❡❢t ❛♥❞ r✐❣❤t ❝♦♥t❛❝ts ❆♥♦t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

❦✲❢❛♥s ♦❢ ♣❛t❤s

P✶P✷ P✵ = ❚ P✸ = ❇ ❤✵ = ✹ ❤✶ = ✹ ❤✷ = ✻ P✶, P✷, . . . , P❦ ∈ P(❚, ❇)✱ P✐ ✇❡❛❦❧② ❛❜♦✈❡ P✐+✶ ❢♦r ❛❧❧ ✐✳ ▲❡t P✵ = ❚✱ P❦+✶ = ❇✳ ❋♦r ✵ ≤ ✐ ≤ ❦✱ ❧❡t ❤✐ =# ♦❢ ❊ st❡♣s ✇❤❡r❡ P✐ ❛♥❞ P✐+✶ ❝♦♥✐♥❝✐❞❡

❚❤❡♦r❡♠

❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ (❤✵, ❤✶, . . . , ❤❦) ♦✈❡r ❦✲❢❛♥s ♦❢ ♣❛t❤s ❛s ❛❜♦✈❡ ✐s s②♠♠❡tr✐❝✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

▲❡t ❚ = ◆◆ . . . ◆❊❊ . . . ❊✳ ✉✶ ✷ ✉✷ ✷ ✉✸ ✶ ✉✹ ✶ ❤✐ = # ❊ st❡♣s ✐♥ P✐ ∩ P✐+✶ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸ ✉❥ ♦❢ ✉♥✉s❡❞ ❊ st❡♣s ❛t ❧❡✈❡❧ ❥ ✻ ✹ ✸ ✸ ✶ ❚ ❛♥❞ ❇ ❢♦r♠ t❤❡ s❤❛♣❡ ♦❢ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♦❢ ❛ ♣❛rt✐t✐♦♥ ✳ ❉❡❢✿ ❆ ❙❙❨❚ ♦❢ s❤❛♣❡ ✐s ❝❛❧❧❡❞ ❦✲✢❛❣❣❡❞ ✐❢ t❤❡ ❡♥tr✐❡s ✐♥ r♦✇ r ❛r❡ ❦ r ❢♦r ❡❛❝❤ r✳ ✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ✹ ✺ ✻ ✼ ✽ ✇❡✐❣❤t ✶s ✷s ✷ ✸ ✸ ✸ ✷ ✷ ✶ ✶

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

▲❡t ❚ = ◆◆ . . . ◆❊❊ . . . ❊✳ ✉✶ = ✷ ✉✷ = ✷ ✉✸ = ✶ ✉✹ = ✶ ❤✐ = # ❊ st❡♣s ✐♥ P✐ ∩ P✐+✶ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸ ✉❥ = # ♦❢ ✉♥✉s❡❞ ❊ st❡♣s ❛t ❧❡✈❡❧ ❥ ✻ ✹ ✸ ✸ ✶ ❚ ❛♥❞ ❇ ❢♦r♠ t❤❡ s❤❛♣❡ ♦❢ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♦❢ ❛ ♣❛rt✐t✐♦♥ ✳ ❉❡❢✿ ❆ ❙❙❨❚ ♦❢ s❤❛♣❡ ✐s ❝❛❧❧❡❞ ❦✲✢❛❣❣❡❞ ✐❢ t❤❡ ❡♥tr✐❡s ✐♥ r♦✇ r ❛r❡ ❦ r ❢♦r ❡❛❝❤ r✳ ✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ✹ ✺ ✻ ✼ ✽ ✇❡✐❣❤t ✶s ✷s ✷ ✸ ✸ ✸ ✷ ✷ ✶ ✶

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

▲❡t ❚ = ◆◆ . . . ◆❊❊ . . . ❊✳ ✉✶ = ✷ ✉✷ = ✷ ✉✸ = ✶ ✉✹ = ✶ ❤✐ = # ❊ st❡♣s ✐♥ P✐ ∩ P✐+✶ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸ ✉❥ = # ♦❢ ✉♥✉s❡❞ ❊ st❡♣s ❛t ❧❡✈❡❧ ❥ λ = (✻, ✹, ✸, ✸, ✶) ❚ ❛♥❞ ❇ ❢♦r♠ t❤❡ s❤❛♣❡ ♦❢ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♦❢ ❛ ♣❛rt✐t✐♦♥ λ✳ ❉❡❢✿ ❆ ❙❙❨❚ ♦❢ s❤❛♣❡ ✐s ❝❛❧❧❡❞ ❦✲✢❛❣❣❡❞ ✐❢ t❤❡ ❡♥tr✐❡s ✐♥ r♦✇ r ❛r❡ ❦ r ❢♦r ❡❛❝❤ r✳ ✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ✹ ✺ ✻ ✼ ✽ ✇❡✐❣❤t ✶s ✷s ✷ ✸ ✸ ✸ ✷ ✷ ✶ ✶

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

▲❡t ❚ = ◆◆ . . . ◆❊❊ . . . ❊✳ ✉✶ = ✷ ✉✷ = ✷ ✉✸ = ✶ ✉✹ = ✶ ❤✐ = # ❊ st❡♣s ✐♥ P✐ ∩ P✐+✶ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸ ✉❥ = # ♦❢ ✉♥✉s❡❞ ❊ st❡♣s ❛t ❧❡✈❡❧ ❥ λ = (✻, ✹, ✸, ✸, ✶) ❚ ❛♥❞ ❇ ❢♦r♠ t❤❡ s❤❛♣❡ ♦❢ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♦❢ ❛ ♣❛rt✐t✐♦♥ λ✳ ❉❡❢✿ ❆ ❙❙❨❚ ♦❢ s❤❛♣❡ λ ✐s ❝❛❧❧❡❞ ❦✲✢❛❣❣❡❞ ✐❢ t❤❡ ❡♥tr✐❡s ✐♥ r♦✇ r ❛r❡ ≤ ❦ + r ❢♦r ❡❛❝❤ r✳ ✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ≤ ✹ ≤ ✺ ≤ ✻ ≤ ✼ ≤ ✽ ✇❡✐❣❤t ✶s ✷s ✷ ✸ ✸ ✸ ✷ ✷ ✶ ✶

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

▲❡t ❚ = ◆◆ . . . ◆❊❊ . . . ❊✳ ✉✶ = ✷ ✉✷ = ✷ ✉✸ = ✶ ✉✹ = ✶ ❤✐ = # ❊ st❡♣s ✐♥ P✐ ∩ P✐+✶ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸ ✉❥ = # ♦❢ ✉♥✉s❡❞ ❊ st❡♣s ❛t ❧❡✈❡❧ ❥ λ = (✻, ✹, ✸, ✸, ✶) ❚ ❛♥❞ ❇ ❢♦r♠ t❤❡ s❤❛♣❡ ♦❢ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♦❢ ❛ ♣❛rt✐t✐♦♥ λ✳ ❉❡❢✿ ❆ ❙❙❨❚ ♦❢ s❤❛♣❡ λ ✐s ❝❛❧❧❡❞ ❦✲✢❛❣❣❡❞ ✐❢ t❤❡ ❡♥tr✐❡s ✐♥ r♦✇ r ❛r❡ ≤ ❦ + r ❢♦r ❡❛❝❤ r✳ ✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ≤ ✹ ≤ ✺ ≤ ✻ ≤ ✼ ≤ ✽ ✇❡✐❣❤t = (#✶s, #✷s, . . . ) = (✷, ✸, ✸, ✸, ✷, ✷, ✶, ✶)

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛♥ ❡①♣❧✐❝✐t ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥

◮ ❦✲❢❛♥s ♦❢ ♣❛t❤s ✐♥ P(❚, ❇) ✇✐t❤ st❛t✐st✐❝s ❤✐ ❛♥❞ ✉❥✱ ❛♥❞ ◮ ❦✲✢❛❣❣❡❞ ❙❙❨❚ ♦❢ s❤❛♣❡ λ ❛♥❞ ✇❡✐❣❤t

(λ✶ − ❤✵, λ✶ − ❤✶, . . . , λ✶ − ❤❦, ✉✶, ✉✷, . . . , ✉r).

✉✶ = ✷ ✉✷ = ✷ ✉✸ = ✶ ✉✹ = ✶

→

Ψ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸

✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ≤ ✹ ≤ ✺ ≤ ✻ ≤ ✼ ≤ ✽

λ✶ = ✻ ✇❡✐❣❤t = (✷, ✸, ✸, ✸, ✷, ✷, ✶, ✶)

❚❤❡ ❜✐❥❡❝t✐♦♥ ✉s❡s ❛ ✈❛r✐❛t✐♦♥ ♦❢ ❥❡✉ ❞❡ t❛q✉✐♥✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ✢❛❣❣❡❞ ❙❙❨❚

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛♥ ❡①♣❧✐❝✐t ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥

◮ ❦✲❢❛♥s ♦❢ ♣❛t❤s ✐♥ P(❚, ❇) ✇✐t❤ st❛t✐st✐❝s ❤✐ ❛♥❞ ✉❥✱ ❛♥❞ ◮ ❦✲✢❛❣❣❡❞ ❙❙❨❚ ♦❢ s❤❛♣❡ λ ❛♥❞ ✇❡✐❣❤t

(λ✶ − ❤✵, λ✶ − ❤✶, . . . , λ✶ − ❤❦, ✉✶, ✉✷, . . . , ✉r).

✉✶ = ✷ ✉✷ = ✷ ✉✸ = ✶ ✉✹ = ✶

→

Ψ ❤✵ = ✹ ❤✶ = ✸ ❤✷ = ✸ ❤✸ = ✸

✶ ✶ ✷ ✷ ✸ ✹ ✷ ✸ ✸ ✹ ✹ ✺ ✻ ✺ ✻ ✼ ✽ ≤ ✹ ≤ ✺ ≤ ✻ ≤ ✼ ≤ ✽

λ✶ = ✻ ✇❡✐❣❤t = (✷, ✸, ✸, ✸, ✷, ✷, ✶, ✶)

❚❤❡ ❜✐❥❡❝t✐♦♥ ✉s❡s ❛ ✈❛r✐❛t✐♦♥ ♦❢ ❥❡✉ ❞❡ t❛q✉✐♥✳

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡♦r❡♠ ✭❝♦♥❥❡❝t✉r❡❞ ❜② ❈✳ ◆✐❝♦❧ás ✬✵✾✮

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❞❡❣r❡❡s ♦❢ ❦ + ✶ ❝♦♥s❡❝✉t✐✈❡ ✈❡rt✐❝❡s ✐♥ ❛ ❦✲tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛ ❝♦♥✈❡① ♥✲❣♦♥ ❡q✉❛❧s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ (❤✵, ❤✶, . . . , ❤❦) ♦✈❡r ❦✲❢❛♥s ♦❢ ❉②❝❦ ♣❛t❤s ♦❢ s❡♠✐❧❡♥❣t❤ ♥ − ✷❦✳ ❚❤❡ ♣r♦♦❢ ✉s❡s t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❉②❝❦ ♣❛t❤s✱ t♦❣❡t❤❡r ✇✐t❤ ❛ ❜✐❥❡❝t✐♦♥ ♦❢ ❙❡rr❛♥♦✕❙t✉♠♣ ❜❡t✇❡❡♥ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s ❛♥❞ ❦✲✢❛❣❣❡❞ ❙❙❨❚✳

❤✵ ✶ ❤✶ ✷ ❤✷ ✷ ❙✲❙

✶ ✶ ✷ ✸ ✹ ✹

✶ ❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s

❚♦♣ ❛♥❞ ❜♦tt♦♠ ❝♦♥t❛❝ts ❱❛r✐❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❆♣♣❧✐❝❛t✐♦♥s ❋❧❛❣❣❡❞ ❙❙❚❨ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❈♦♥♥❡❝t✐♦♥ t♦ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s

❚❤❡♦r❡♠ ✭❝♦♥❥❡❝t✉r❡❞ ❜② ❈✳ ◆✐❝♦❧ás ✬✵✾✮

❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❞❡❣r❡❡s ♦❢ ❦ + ✶ ❝♦♥s❡❝✉t✐✈❡ ✈❡rt✐❝❡s ✐♥ ❛ ❦✲tr✐❛♥❣✉❧❛t✐♦♥ ♦❢ ❛ ❝♦♥✈❡① ♥✲❣♦♥ ❡q✉❛❧s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ (❤✵, ❤✶, . . . , ❤❦) ♦✈❡r ❦✲❢❛♥s ♦❢ ❉②❝❦ ♣❛t❤s ♦❢ s❡♠✐❧❡♥❣t❤ ♥ − ✷❦✳ ❚❤❡ ♣r♦♦❢ ✉s❡s t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❉②❝❦ ♣❛t❤s✱ t♦❣❡t❤❡r ✇✐t❤ ❛ ❜✐❥❡❝t✐♦♥ ♦❢ ❙❡rr❛♥♦✕❙t✉♠♣ ❜❡t✇❡❡♥ ❦✲tr✐❛♥❣✉❧❛t✐♦♥s ❛♥❞ ❦✲✢❛❣❣❡❞ ❙❙❨❚✳

❤✵ = ✶ ❤✶ = ✷ ❤✷ = ✷

→

❙✲❙

✶ ✶ ✷ ✸ ✹ ✹

→

Ψ−✶

❙❡r❣✐ ❊❧✐③❛❧❞❡ ❇✐❥❡❝t✐♦♥s ❢♦r ❧❛tt✐❝❡ ♣❛t❤s ❜❡t✇❡❡♥ t✇♦ ❜♦✉♥❞❛r✐❡s