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■♥ ❧♦✇ ❞✐♠❡♥s✐♦♥s ✿ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ✲ ❊①♣❧✐❝✐t ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❝t✐♦♥s ♦❢ ❛ ♠♦♥♦✐❞ ♦♥ ❝❛t❡❣♦r✐❡s ✭r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✮✱ ✲ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s ❢♦r ♠♦♥♦✐❞s✳ ▼♦t✐✈❛t✐♦♥ ◮ ●✐✈❡♥ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮ ❈ ⊲ ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s ⊲ ✇❡ ✇♦✉❧❞ ❧✐❦❡ ❜✉✐❧❞ ❛ ✭s♠❛❧❧ ✦✮ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ( ∞ , ✶ ) ✲❝❛t❡❣♦r✐❡s✱ ✲ t❤❛t ✐s✱ ❛ ❢r❡❡ ( ∞ , ✶ ) ✲❝❛t❡❣♦r② ❤♦♠♦t♦♣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ◮ P♦❧②❣r❛♣❤✐❝ r❡s♦❧✉t✐♦♥s ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ r❡✇r✐t✐♥❣ s②st❡♠ t❤❛t ♣r❡s❡♥ts ❈ ✱ ✭●✉✐r❛✉❞✲▼✳✱ ✷✵✶✷✮✳ ⊲ ❆♣♣❧✐❝❛t✐♦♥s✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts ✲ ❇❛✉❡s✲❲✐rs❝❤✐♥❣ ✭❝♦✮❤♦♠♦❧♦❣② ♦❢ ❝❛t❡❣♦r② ❈ ✳

❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ✲ ❊①♣❧✐❝✐t ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❝t✐♦♥s ♦❢ ❛ ♠♦♥♦✐❞ ♦♥ ❝❛t❡❣♦r✐❡s ✭r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✮✱ ✲ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s ❢♦r ♠♦♥♦✐❞s✳ ▼♦t✐✈❛t✐♦♥ ◮ ●✐✈❡♥ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮ ❈ ⊲ ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s ⊲ ✇❡ ✇♦✉❧❞ ❧✐❦❡ ❜✉✐❧❞ ❛ ✭s♠❛❧❧ ✦✮ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ( ∞ , ✶ ) ✲❝❛t❡❣♦r✐❡s✱ ✲ t❤❛t ✐s✱ ❛ ❢r❡❡ ( ∞ , ✶ ) ✲❝❛t❡❣♦r② ❤♦♠♦t♦♣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ◮ P♦❧②❣r❛♣❤✐❝ r❡s♦❧✉t✐♦♥s ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ r❡✇r✐t✐♥❣ s②st❡♠ t❤❛t ♣r❡s❡♥ts ❈ ✱ ✭●✉✐r❛✉❞✲▼✳✱ ✷✵✶✷✮✳ ⊲ ❆♣♣❧✐❝❛t✐♦♥s✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts ✲ ❇❛✉❡s✲❲✐rs❝❤✐♥❣ ✭❝♦✮❤♦♠♦❧♦❣② ♦❢ ❝❛t❡❣♦r② ❈ ✳ ◮ ■♥ ❧♦✇ ❞✐♠❡♥s✐♦♥s ✿ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s✿ ✲ ❊①♣❧✐❝✐t ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❝t✐♦♥s ♦❢ ❛ ♠♦♥♦✐❞ ♦♥ ❝❛t❡❣♦r✐❡s ✭r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✮✱ ✲ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s ❢♦r ♠♦♥♦✐❞s✳ ▼♦t✐✈❛t✐♦♥ ◮ ●✐✈❡♥ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮ ❈ ⊲ ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s ⊲ ✇❡ ✇♦✉❧❞ ❧✐❦❡ ❜✉✐❧❞ ❛ ✭s♠❛❧❧ ✦✮ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ( ∞ , ✶ ) ✲❝❛t❡❣♦r✐❡s✱ ✲ t❤❛t ✐s✱ ❛ ❢r❡❡ ( ∞ , ✶ ) ✲❝❛t❡❣♦r② ❤♦♠♦t♦♣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ◮ P♦❧②❣r❛♣❤✐❝ r❡s♦❧✉t✐♦♥s ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ r❡✇r✐t✐♥❣ s②st❡♠ t❤❛t ♣r❡s❡♥ts ❈ ✱ ✭●✉✐r❛✉❞✲▼✳✱ ✷✵✶✷✮✳ ⊲ ❆♣♣❧✐❝❛t✐♦♥s✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts ✲ ❇❛✉❡s✲❲✐rs❝❤✐♥❣ ✭❝♦✮❤♦♠♦❧♦❣② ♦❢ ❝❛t❡❣♦r② ❈ ✳ ◮ ■♥ ❧♦✇ ❞✐♠❡♥s✐♦♥s ✿ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ⊲ ❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ✳

▼♦t✐✈❛t✐♦♥ ◮ ●✐✈❡♥ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮ ❈ ⊲ ♣r❡s❡♥t❡❞ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s ⊲ ✇❡ ✇♦✉❧❞ ❧✐❦❡ ❜✉✐❧❞ ❛ ✭s♠❛❧❧ ✦✮ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ( ∞ , ✶ ) ✲❝❛t❡❣♦r✐❡s✱ ✲ t❤❛t ✐s✱ ❛ ❢r❡❡ ( ∞ , ✶ ) ✲❝❛t❡❣♦r② ❤♦♠♦t♦♣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ◮ P♦❧②❣r❛♣❤✐❝ r❡s♦❧✉t✐♦♥s ❝♦♥str✉❝t❡❞ ❢r♦♠ ❛ r❡✇r✐t✐♥❣ s②st❡♠ t❤❛t ♣r❡s❡♥ts ❈ ✱ ✭●✉✐r❛✉❞✲▼✳✱ ✷✵✶✷✮✳ ⊲ ❆♣♣❧✐❝❛t✐♦♥s✿ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts ✲ ❇❛✉❡s✲❲✐rs❝❤✐♥❣ ✭❝♦✮❤♦♠♦❧♦❣② ♦❢ ❝❛t❡❣♦r② ❈ ✳ ◮ ■♥ ❧♦✇ ❞✐♠❡♥s✐♦♥s ✿ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ⊲ ❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ✳ ⊲ ❆♣♣❧✐❝❛t✐♦♥s✿ ✲ ❊①♣❧✐❝✐t ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❝t✐♦♥s ♦❢ ❛ ♠♦♥♦✐❞ ♦♥ ❝❛t❡❣♦r✐❡s ✭r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✮✱ ✲ ❈♦❤❡r❡♥❝❡ t❤❡♦r❡♠s ❢♦r ♠♦♥♦✐❞s✳

❋♦r❣❡tt✐♥❣ t❤❡ ✐♥✈♦❧✉t✐✈❡ ❝❤❛r❛❝t❡r ♦❢ ❣❡♥❡r❛t♦rs✱ ♦♥❡ ❣❡ts t❤❡ ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ ❆rt ❲ ❙ tstst ststs ♦❢ t❤❡ ❆rt✐♥ ♠♦♥♦✐❞ ❇ ❲ ✳ ❖❜❥❡❝t✐✈❡✳ P✉s❤ ❢✉rt❤❡r ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ st✉❞② t❤❡ r❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ ❜r❛✐❞ r❡❧❛t✐♦♥s ✳ ✭❇r✐❡s❦♦r♥✲❙❛✐t♦✱ ✶✾✼✷✱ ❉❡❧✐❣♥❡✱ ✶✾✼✷✱ ❉❡❧✐❣♥❡✱ ✶✾✾✼✱ ❚✐ts✱ ✶✾✽✶✱ ▼✐❝❤❡❧✱ ✶✾✾✾✮✳ ▼♦t✐✈❛t✐♦♥ ◮ ❆ ❈♦①❡t❡r s②st❡♠ ( ❲ , ❙ ) ✐s ❛ ❞❛t❛ ♠❛❞❡ ♦❢ ❛ ❣r♦✉♣ ❲ ✇✐t❤ ❛ ♣r❡s❡♥t❛t✐♦♥ ❜② ❛ ✭✜♥✐t❡✮ s❡t ❙ ♦❢ ✐♥✈♦❧✉t✐♦♥s✱ s ✷ = ✶✱ s❛t✐s❢②✐♥❣ ❜r❛✐❞ r❡❧❛t✐♦♥s tstst . . . = ststs . . .

❖❜❥❡❝t✐✈❡✳ P✉s❤ ❢✉rt❤❡r ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ st✉❞② t❤❡ r❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ ❜r❛✐❞ r❡❧❛t✐♦♥s ✳ ✭❇r✐❡s❦♦r♥✲❙❛✐t♦✱ ✶✾✼✷✱ ❉❡❧✐❣♥❡✱ ✶✾✼✷✱ ❉❡❧✐❣♥❡✱ ✶✾✾✼✱ ❚✐ts✱ ✶✾✽✶✱ ▼✐❝❤❡❧✱ ✶✾✾✾✮✳ ▼♦t✐✈❛t✐♦♥ ◮ ❆ ❈♦①❡t❡r s②st❡♠ ( ❲ , ❙ ) ✐s ❛ ❞❛t❛ ♠❛❞❡ ♦❢ ❛ ❣r♦✉♣ ❲ ✇✐t❤ ❛ ♣r❡s❡♥t❛t✐♦♥ ❜② ❛ ✭✜♥✐t❡✮ s❡t ❙ ♦❢ ✐♥✈♦❧✉t✐♦♥s✱ s ✷ = ✶✱ s❛t✐s❢②✐♥❣ ❜r❛✐❞ r❡❧❛t✐♦♥s tstst . . . = ststs . . . ◮ ❋♦r❣❡tt✐♥❣ t❤❡ ✐♥✈♦❧✉t✐✈❡ ❝❤❛r❛❝t❡r ♦❢ ❣❡♥❡r❛t♦rs✱ ♦♥❡ ❣❡ts t❤❡ ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ � � ❆rt ( ❲ ) = ❙ | tstst . . . = ststs . . . ♦❢ t❤❡ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❲ ) ✳

▼♦t✐✈❛t✐♦♥ ◮ ❆ ❈♦①❡t❡r s②st❡♠ ( ❲ , ❙ ) ✐s ❛ ❞❛t❛ ♠❛❞❡ ♦❢ ❛ ❣r♦✉♣ ❲ ✇✐t❤ ❛ ♣r❡s❡♥t❛t✐♦♥ ❜② ❛ ✭✜♥✐t❡✮ s❡t ❙ ♦❢ ✐♥✈♦❧✉t✐♦♥s✱ s ✷ = ✶✱ s❛t✐s❢②✐♥❣ ❜r❛✐❞ r❡❧❛t✐♦♥s tstst . . . = ststs . . . ◮ ❋♦r❣❡tt✐♥❣ t❤❡ ✐♥✈♦❧✉t✐✈❡ ❝❤❛r❛❝t❡r ♦❢ ❣❡♥❡r❛t♦rs✱ ♦♥❡ ❣❡ts t❤❡ ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ � � ❆rt ( ❲ ) = ❙ | tstst . . . = ststs . . . ♦❢ t❤❡ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❲ ) ✳ ❖❜❥❡❝t✐✈❡✳ ⊲ P✉s❤ ❢✉rt❤❡r ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ st✉❞② t❤❡ r❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ ❜r❛✐❞ r❡❧❛t✐♦♥s ✳ ✭❇r✐❡s❦♦r♥✲❙❛✐t♦✱ ✶✾✼✷✱ ❉❡❧✐❣♥❡✱ ✶✾✼✷✱ ❉❡❧✐❣♥❡✱ ✶✾✾✼✱ ❚✐ts✱ ✶✾✽✶✱ ▼✐❝❤❡❧✱ ✶✾✾✾✮✳

❚❤❡ ❛ss♦❝✐❛t❡❞ ❆rt✐♥ ♠♦♥♦✐❞ ❇ ❙ ✹ ✐s t❤❡ ♠♦♥♦✐❞ ♦❢ ❜r❛✐❞s ♦♥ ✹ str❛♥❞s✿ ❆rt ✷ ❙ ✹ r s t rsr srs rt tr tst sts ❚❤❡ r❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ ❜r❛✐❞ r❡❧❛t✐♦♥s ♦♥ ✹ str❛♥❞s ❛r❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❩❛♠♦❧♦❞❝❤✐❦♦✈ r❡❧❛t✐♦♥ ✭❉❡❧✐❣♥❡✱ ✶✾✾✼✮✳ stsrst strsrt srtstr srstsr rsrtsr ❩ r s t tstrst rstrsr tsrtst tsrsts trsrts rtstrs rstsrs ▼♦t✐✈❛t✐♦♥ ◮ ❙❡t ❲ = ❙ ✹ t❤❡ ❣r♦✉♣ ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ { ✶ , ✷ , ✸ , ✹ } ✱ ✇✐t❤ ❙ = { r , s , t } ✇❤❡r❡ r = s = t =

❚❤❡ r❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ ❜r❛✐❞ r❡❧❛t✐♦♥s ♦♥ ✹ str❛♥❞s ❛r❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❩❛♠♦❧♦❞❝❤✐❦♦✈ r❡❧❛t✐♦♥ ✭❉❡❧✐❣♥❡✱ ✶✾✾✼✮✳ stsrst strsrt srtstr srstsr rsrtsr ❩ r s t tstrst rstrsr tsrtst tsrsts trsrts rtstrs rstsrs ▼♦t✐✈❛t✐♦♥ ◮ ❙❡t ❲ = ❙ ✹ t❤❡ ❣r♦✉♣ ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ { ✶ , ✷ , ✸ , ✹ } ✱ ✇✐t❤ ❙ = { r , s , t } ✇❤❡r❡ r = s = t = ◮ ❚❤❡ ❛ss♦❝✐❛t❡❞ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❙ ✹ ) ✐s t❤❡ ♠♦♥♦✐❞ ♦❢ ❜r❛✐❞s ♦♥ ✹ str❛♥❞s✿ � � � � ❆rt ✷ ( ❙ ✹ ) = rsr = srs , rt = tr , tst = sts r , s , t = = =

▼♦t✐✈❛t✐♦♥ ◮ ❙❡t ❲ = ❙ ✹ t❤❡ ❣r♦✉♣ ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ { ✶ , ✷ , ✸ , ✹ } ✱ ✇✐t❤ ❙ = { r , s , t } ✇❤❡r❡ r = s = t = ◮ ❚❤❡ ❛ss♦❝✐❛t❡❞ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❙ ✹ ) ✐s t❤❡ ♠♦♥♦✐❞ ♦❢ ❜r❛✐❞s ♦♥ ✹ str❛♥❞s✿ � � � � ❆rt ✷ ( ❙ ✹ ) = rsr = srs , rt = tr , tst = sts r , s , t = = = ◮ ❚❤❡ r❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ ❜r❛✐❞ r❡❧❛t✐♦♥s ♦♥ ✹ str❛♥❞s ❛r❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❩❛♠♦❧♦❞❝❤✐❦♦✈ r❡❧❛t✐♦♥ ✭❉❡❧✐❣♥❡✱ ✶✾✾✼✮✳ stsrst strsrt srtstr srstsr rsrtsr ❩ r , s , t tstrst rstrsr tsrtst tsrsts trsrts rtstrs rstsrs

❛❧❣❡❜r❛✐❝ ❝♦♠❜✐♥❛t♦r✐❝✱ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ✷ P ✷ ✶ ✷ ✷✶✶ ✶✷✶ ✷✷✶ ✷✶✷ ❘❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ r❡❧❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ✷✷✶✶ ✷✶✷✶ ❋♦r ♥ ✸✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ✬❡①♣❧♦s✐♦♥✬ ✇✐t❤ t❤❡ ❑♥✉t❤✬s ♣r❡s❡♥t❛t✐♦♥✳ ❖❜❥❡❝t✐✈❡✳ ❈♦♠♣✉t❡ ✜♥✐t❡ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ♥ ✳ ▼♦t✐✈❛t✐♦♥ ◮ P❧❛❝t✐❝ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ♥ � � � ❢♦r ❛❧❧ ✶ � ① � ② < ③ � ♥ ③①② = ①③② � P ♥ = ✶ , . . . , ♥ ②③① = ②①③ ❢♦r ❛❧❧ ✶ � ① < ② � ③ � ♥

❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ✷ P ✷ ✶ ✷ ✷✶✶ ✶✷✶ ✷✷✶ ✷✶✷ ❘❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ r❡❧❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ✷✷✶✶ ✷✶✷✶ ❋♦r ♥ ✸✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ✬❡①♣❧♦s✐♦♥✬ ✇✐t❤ t❤❡ ❑♥✉t❤✬s ♣r❡s❡♥t❛t✐♦♥✳ ❖❜❥❡❝t✐✈❡✳ ❈♦♠♣✉t❡ ✜♥✐t❡ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ♥ ✳ ▼♦t✐✈❛t✐♦♥ ◮ P❧❛❝t✐❝ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ♥ � � � ❢♦r ❛❧❧ ✶ � ① � ② < ③ � ♥ ③①② = ①③② � P ♥ = ✶ , . . . , ♥ ②③① = ②①③ ❢♦r ❛❧❧ ✶ � ① < ② � ③ � ♥ ⊲ ❛❧❣❡❜r❛✐❝ ❝♦♠❜✐♥❛t♦r✐❝✱ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳

❘❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ r❡❧❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ✷✷✶✶ ✷✶✷✶ ❋♦r ♥ ✸✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ✬❡①♣❧♦s✐♦♥✬ ✇✐t❤ t❤❡ ❑♥✉t❤✬s ♣r❡s❡♥t❛t✐♦♥✳ ❖❜❥❡❝t✐✈❡✳ ❈♦♠♣✉t❡ ✜♥✐t❡ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ♥ ✳ ▼♦t✐✈❛t✐♦♥ ◮ P❧❛❝t✐❝ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ♥ � � � ❢♦r ❛❧❧ ✶ � ① � ② < ③ � ♥ ③①② = ①③② � P ♥ = ✶ , . . . , ♥ ②③① = ②①③ ❢♦r ❛❧❧ ✶ � ① < ② � ③ � ♥ ⊲ ❛❧❣❡❜r❛✐❝ ❝♦♠❜✐♥❛t♦r✐❝✱ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳ ◮ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ✷ � � � � ✷✶✶ = ✶✷✶ , ✷✷✶ = ✷✶✷ P ✷ = ✶ , ✷

❋♦r ♥ ✸✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ✬❡①♣❧♦s✐♦♥✬ ✇✐t❤ t❤❡ ❑♥✉t❤✬s ♣r❡s❡♥t❛t✐♦♥✳ ❖❜❥❡❝t✐✈❡✳ ❈♦♠♣✉t❡ ✜♥✐t❡ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ♥ ✳ ▼♦t✐✈❛t✐♦♥ ◮ P❧❛❝t✐❝ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ♥ � � � ❢♦r ❛❧❧ ✶ � ① � ② < ③ � ♥ ③①② = ①③② � P ♥ = ✶ , . . . , ♥ ②③① = ②①③ ❢♦r ❛❧❧ ✶ � ① < ② � ③ � ♥ ⊲ ❛❧❣❡❜r❛✐❝ ❝♦♠❜✐♥❛t♦r✐❝✱ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳ ◮ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ✷ � � � � ✷✶✶ = ✶✷✶ , ✷✷✶ = ✷✶✷ P ✷ = ✶ , ✷ ⊲ ❘❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ r❡❧❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ✷✷✶✶ ✷✶✷✶

❖❜❥❡❝t✐✈❡✳ ❈♦♠♣✉t❡ ✜♥✐t❡ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ♥ ✳ ▼♦t✐✈❛t✐♦♥ ◮ P❧❛❝t✐❝ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ♥ � � � ❢♦r ❛❧❧ ✶ � ① � ② < ③ � ♥ ③①② = ①③② � P ♥ = ✶ , . . . , ♥ ②③① = ②①③ ❢♦r ❛❧❧ ✶ � ① < ② � ③ � ♥ ⊲ ❛❧❣❡❜r❛✐❝ ❝♦♠❜✐♥❛t♦r✐❝✱ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳ ◮ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ✷ � � � � ✷✶✶ = ✶✷✶ , ✷✷✶ = ✷✶✷ P ✷ = ✶ , ✷ ⊲ ❘❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ r❡❧❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ✷✷✶✶ ✷✶✷✶ ◮ ❋♦r ♥ � ✸✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ✬❡①♣❧♦s✐♦♥✬ ✇✐t❤ t❤❡ ❑♥✉t❤✬s ♣r❡s❡♥t❛t✐♦♥✳

▼♦t✐✈❛t✐♦♥ ◮ P❧❛❝t✐❝ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ♥ � � � ❢♦r ❛❧❧ ✶ � ① � ② < ③ � ♥ ③①② = ①③② � P ♥ = ✶ , . . . , ♥ ②③① = ②①③ ❢♦r ❛❧❧ ✶ � ① < ② � ③ � ♥ ⊲ ❛❧❣❡❜r❛✐❝ ❝♦♠❜✐♥❛t♦r✐❝✱ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✳ ◮ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ✷ � � � � ✷✶✶ = ✶✷✶ , ✷✷✶ = ✷✶✷ P ✷ = ✶ , ✷ ⊲ ❘❡❧❛t✐♦♥s ❛♠♦♥❣st t❤❡ r❡❧❛t✐♦♥s ❣❡♥❡r❛t❡❞ ❜② ✷✷✶✶ ✷✶✷✶ ◮ ❋♦r ♥ � ✸✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ✬❡①♣❧♦s✐♦♥✬ ✇✐t❤ t❤❡ ❑♥✉t❤✬s ♣r❡s❡♥t❛t✐♦♥✳ ❖❜❥❡❝t✐✈❡✳ ⊲ ❈♦♠♣✉t❡ ✜♥✐t❡ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r P ♥ ✳

P ✹ ♦♥ t❤❡ ❣❡♥❡r❛t♦rs ✶✱ ✷✱ ✸✱ ✹✱ ✭❑✉❜❛t✲❖❦♥✐➠s❦✐✱ ✷✵✶✹✮✳ ❈♦♠♣✉t❛t✐♦♥s ♦❢ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r ♠♦♥♦✐❞s ❇ ♥ ♦❢ P ♥ ♥❡❡❞ ♥❡✇ ❣❡♥❡r❛t♦rs✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❛❞❞s ❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ❛♥❞ ❛ ✇❛② t♦ ❤♦♠♦t♦♣✐❝❛❧❧② r❡❞✉❝❡ t❤❡♠✳ ▼♦t✐✈❛t✐♦♥ ◮ ❚❤❡ ❑♥✉t❤✲❇❡♥❞✐① ♣r♦❝❡❞✉r❡ ❞♦❡s ♥♦t t❡r♠✐♥❛t❡ ❢♦r ⊲ ❇ + ✸ = � s , t | sts = tst � ♦♥ t❤❡ t✇♦ ❣❡♥❡r❛t♦rs s ❛♥❞ t ✱ ✭❑❛♣✉r✲◆❛r❡♥❞r❛♥✱ ✶✾✽✺✮

❈♦♠♣✉t❛t✐♦♥s ♦❢ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r ♠♦♥♦✐❞s ❇ ♥ ♦❢ P ♥ ♥❡❡❞ ♥❡✇ ❣❡♥❡r❛t♦rs✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❛❞❞s ❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ❛♥❞ ❛ ✇❛② t♦ ❤♦♠♦t♦♣✐❝❛❧❧② r❡❞✉❝❡ t❤❡♠✳ ▼♦t✐✈❛t✐♦♥ ◮ ❚❤❡ ❑♥✉t❤✲❇❡♥❞✐① ♣r♦❝❡❞✉r❡ ❞♦❡s ♥♦t t❡r♠✐♥❛t❡ ❢♦r ⊲ ❇ + ✸ = � s , t | sts = tst � ♦♥ t❤❡ t✇♦ ❣❡♥❡r❛t♦rs s ❛♥❞ t ✱ ✭❑❛♣✉r✲◆❛r❡♥❞r❛♥✱ ✶✾✽✺✮ ⊲ P ✹ ♦♥ t❤❡ ❣❡♥❡r❛t♦rs ✶✱ ✷✱ ✸✱ ✹✱ ✭❑✉❜❛t✲❖❦♥✐➠s❦✐✱ ✷✵✶✹✮✳

❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❛❞❞s ❣❡♥❡r❛t♦rs✱ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ♦r✐❡♥t❡❞ s②③②❣✐❡s ❛♥❞ ❛ ✇❛② t♦ ❤♦♠♦t♦♣✐❝❛❧❧② r❡❞✉❝❡ t❤❡♠✳ ▼♦t✐✈❛t✐♦♥ ◮ ❚❤❡ ❑♥✉t❤✲❇❡♥❞✐① ♣r♦❝❡❞✉r❡ ❞♦❡s ♥♦t t❡r♠✐♥❛t❡ ❢♦r ⊲ ❇ + ✸ = � s , t | sts = tst � ♦♥ t❤❡ t✇♦ ❣❡♥❡r❛t♦rs s ❛♥❞ t ✱ ✭❑❛♣✉r✲◆❛r❡♥❞r❛♥✱ ✶✾✽✺✮ ⊲ P ✹ ♦♥ t❤❡ ❣❡♥❡r❛t♦rs ✶✱ ✷✱ ✸✱ ✹✱ ✭❑✉❜❛t✲❖❦♥✐➠s❦✐✱ ✷✵✶✹✮✳ ◮ ❈♦♠♣✉t❛t✐♦♥s ♦❢ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r ♠♦♥♦✐❞s ❇ + ♥ ♦❢ P ♥ ♥❡❡❞ ♥❡✇ ❣❡♥❡r❛t♦rs✳

❛♥❞ ❛ ✇❛② t♦ ❤♦♠♦t♦♣✐❝❛❧❧② r❡❞✉❝❡ t❤❡♠✳ ▼♦t✐✈❛t✐♦♥ ◮ ❚❤❡ ❑♥✉t❤✲❇❡♥❞✐① ♣r♦❝❡❞✉r❡ ❞♦❡s ♥♦t t❡r♠✐♥❛t❡ ❢♦r ⊲ ❇ + ✸ = � s , t | sts = tst � ♦♥ t❤❡ t✇♦ ❣❡♥❡r❛t♦rs s ❛♥❞ t ✱ ✭❑❛♣✉r✲◆❛r❡♥❞r❛♥✱ ✶✾✽✺✮ ⊲ P ✹ ♦♥ t❤❡ ❣❡♥❡r❛t♦rs ✶✱ ✷✱ ✸✱ ✹✱ ✭❑✉❜❛t✲❖❦♥✐➠s❦✐✱ ✷✵✶✹✮✳ ◮ ❈♦♠♣✉t❛t✐♦♥s ♦❢ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r ♠♦♥♦✐❞s ❇ + ♥ ♦❢ P ♥ ♥❡❡❞ ♥❡✇ ❣❡♥❡r❛t♦rs✳ ◮ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❛❞❞s ⊲ ❣❡♥❡r❛t♦rs✱ ⊲ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ⊲ ♦r✐❡♥t❡❞ s②③②❣✐❡s

▼♦t✐✈❛t✐♦♥ ◮ ❚❤❡ ❑♥✉t❤✲❇❡♥❞✐① ♣r♦❝❡❞✉r❡ ❞♦❡s ♥♦t t❡r♠✐♥❛t❡ ❢♦r ⊲ ❇ + ✸ = � s , t | sts = tst � ♦♥ t❤❡ t✇♦ ❣❡♥❡r❛t♦rs s ❛♥❞ t ✱ ✭❑❛♣✉r✲◆❛r❡♥❞r❛♥✱ ✶✾✽✺✮ ⊲ P ✹ ♦♥ t❤❡ ❣❡♥❡r❛t♦rs ✶✱ ✷✱ ✸✱ ✹✱ ✭❑✉❜❛t✲❖❦♥✐➠s❦✐✱ ✷✵✶✹✮✳ ◮ ❈♦♠♣✉t❛t✐♦♥s ♦❢ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❢♦r ♠♦♥♦✐❞s ❇ + ♥ ♦❢ P ♥ ♥❡❡❞ ♥❡✇ ❣❡♥❡r❛t♦rs✳ ◮ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❛❞❞s ⊲ ❣❡♥❡r❛t♦rs✱ ⊲ ♦r✐❡♥t❡❞ r❡❧❛t✐♦♥s✱ ⊲ ♦r✐❡♥t❡❞ s②③②❣✐❡s ❛♥❞ ❛ ✇❛② t♦ ❤♦♠♦t♦♣✐❝❛❧❧② r❡❞✉❝❡ t❤❡♠✳

P❧❛♥ ■✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ✲ P♦❧②❣r❛♣❤s ❛s ❤✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ r❡✇r✐t✐♥❣ s②st❡♠s ✲ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ❛s ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥s ■■✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ✲ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ✲ ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ♣♦❧②❣r❛♣❤s ✲ ❈♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ ■■■✳ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❆rt✐♥ ❛♥❞ ♣❧❛❝t✐❝ ♠♦♥♦✐❞s ❘❡❢❡r❡♥❝❡s ✲ ❍❛❣❡✲▼✳✱ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ♣❧❛❝t✐❝ ♠♦♥♦✐❞s ✱ ✷✵✶✺✳ ✲ ●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❆rt✐♥ ♠♦♥♦✐❞s ✱ ✷✵✶✺✳ ✲ ●✉✐r❛✉❞✲▼✳✲▼✐♠r❛♠✱ ❆ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❝♦❤❡r❡♥❝❡ ♦❢ ♠♦♥♦✐❞s ✱ ✷✵✶✸✳

P❛rt ■✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s

❆ ✶✲ ♣♦❧②❣r❛♣❤ ✐s ❛♥ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✵ ✶ s ✵ ✵ ✶ t ✵ ❆ ✷✲ ♣♦❧②❣r❛♣❤ ✐s ❛ tr✐♣❧❡ ✇❤❡r❡ ✵ ✶ ✷ ✐s ❛ ✶✲♣♦❧②❣r❛♣❤✱ ✵ ✶ ✷ ✐s ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❡❡ ✶✲❝❛t❡❣♦r② ✶ ✳ s ✶ s ✵ s ✶ s ✵ s ✶ t ✵ s ✶ ✵ ✷ ✶ s ✵ t ✶ t ✵ t ✶ t ✵ t ✶ t ✶ ❆ r❡✇r✐t✐♥❣ st❡♣ ✐s ❛ ✷✲❝❡❧❧ ♦❢ t❤❡ ❢r❡❡ ✷✲❝❛t❡❣♦r② ✷ ♦✈❡r ✇✐t❤ s❤❛♣❡ ✇❤❡r❡ ✉ ✈ ✐s ❛ ✷✲❝❡❧❧ ♦❢ ✷ ❛♥❞ ✇ ✱ ✇ ❛r❡ ✶✲❝❡❧❧s ♦❢ ✶ ✳ P♦❧②❣r❛♣❤s

❆ ✷✲ ♣♦❧②❣r❛♣❤ ✐s ❛ tr✐♣❧❡ ✇❤❡r❡ ✵ ✶ ✷ ✐s ❛ ✶✲♣♦❧②❣r❛♣❤✱ ✵ ✶ ✷ ✐s ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❡❡ ✶✲❝❛t❡❣♦r② ✶ ✳ s ✶ s ✵ s ✶ s ✵ s ✶ t ✵ s ✶ ✵ ✷ ✶ s ✵ t ✶ t ✵ t ✶ t ✵ t ✶ t ✶ ❆ r❡✇r✐t✐♥❣ st❡♣ ✐s ❛ ✷✲❝❡❧❧ ♦❢ t❤❡ ❢r❡❡ ✷✲❝❛t❡❣♦r② ✷ ♦✈❡r ✇✐t❤ s❤❛♣❡ ✇❤❡r❡ ✉ ✈ ✐s ❛ ✷✲❝❡❧❧ ♦❢ ✷ ❛♥❞ ✇ ✱ ✇ ❛r❡ ✶✲❝❡❧❧s ♦❢ ✶ ✳ � � P♦❧②❣r❛♣❤s ◮ ❆ ✶✲ ♣♦❧②❣r❛♣❤ ✐s ❛♥ ❞✐r❡❝t❡❞ ❣r❛♣❤ ( Σ ✵ , Σ ✶ ) s ✵ Σ ✵ Σ ✶ t ✵

❆ r❡✇r✐t✐♥❣ st❡♣ ✐s ❛ ✷✲❝❡❧❧ ♦❢ t❤❡ ❢r❡❡ ✷✲❝❛t❡❣♦r② ✷ ♦✈❡r ✇✐t❤ s❤❛♣❡ ✇❤❡r❡ ✉ ✈ ✐s ❛ ✷✲❝❡❧❧ ♦❢ ✷ ❛♥❞ ✇ ✱ ✇ ❛r❡ ✶✲❝❡❧❧s ♦❢ ✶ ✳ � � � � � � � � � P♦❧②❣r❛♣❤s ◮ ❆ ✶✲ ♣♦❧②❣r❛♣❤ ✐s ❛♥ ❞✐r❡❝t❡❞ ❣r❛♣❤ ( Σ ✵ , Σ ✶ ) s ✵ Σ ✵ Σ ✶ t ✵ ◮ ❆ ✷✲ ♣♦❧②❣r❛♣❤ ✐s ❛ tr✐♣❧❡ Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ✇❤❡r❡ ⊲ ( Σ ✵ , Σ ✶ ) ✐s ❛ ✶✲♣♦❧②❣r❛♣❤✱ ⊲ Σ ✷ ✐s ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❡❡ ✶✲❝❛t❡❣♦r② Σ ∗ ✶ ✳ s ✶ ( α ) s ✵ s ✶ s ✵ s ✶ ( α ) t ✵ s ✶ ( α ) Σ ∗ Σ ✵ Σ ✷ = α = ✶ s ✵ t ✶ ( α ) t ✵ t ✶ ( α ) t ✵ t ✶ t ✶ ( α )

� � � � � � � � � � � � � � P♦❧②❣r❛♣❤s ◮ ❆ ✶✲ ♣♦❧②❣r❛♣❤ ✐s ❛♥ ❞✐r❡❝t❡❞ ❣r❛♣❤ ( Σ ✵ , Σ ✶ ) s ✵ Σ ✵ Σ ✶ t ✵ ◮ ❆ ✷✲ ♣♦❧②❣r❛♣❤ ✐s ❛ tr✐♣❧❡ Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ✇❤❡r❡ ⊲ ( Σ ✵ , Σ ✶ ) ✐s ❛ ✶✲♣♦❧②❣r❛♣❤✱ ⊲ Σ ✷ ✐s ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❡❡ ✶✲❝❛t❡❣♦r② Σ ∗ ✶ ✳ s ✶ ( α ) s ✵ s ✶ s ✵ s ✶ ( α ) t ✵ s ✶ ( α ) Σ ∗ Σ ✵ Σ ✷ = α = ✶ s ✵ t ✶ ( α ) t ✵ t ✶ ( α ) t ✵ t ✶ t ✶ ( α ) ◮ ❆ r❡✇r✐t✐♥❣ st❡♣ ✐s ❛ ✷✲❝❡❧❧ ♦❢ t❤❡ ❢r❡❡ ✷✲❝❛t❡❣♦r② Σ ∗ ✷ ♦✈❡r Σ ✇✐t❤ s❤❛♣❡ ✉ ✇ ′ ✇ α ✈ α � ✈ ✐s ❛ ✷✲❝❡❧❧ ♦❢ Σ ✷ ❛♥❞ ✇ ✱ ✇ ′ ❛r❡ ✶✲❝❡❧❧s ♦❢ Σ ∗ ✇❤❡r❡ ✉ ✶ ✳

� � � � � � � � � � � � � � � � � P♦❧②❣r❛♣❤s ◮ ❆ ✶✲ ♣♦❧②❣r❛♣❤ ✐s ❛♥ ❞✐r❡❝t❡❞ ❣r❛♣❤ ( Σ ✵ , Σ ✶ ) s ✵ Σ ✵ Σ ✶ t ✵ ◮ ❆ ✷✲ ♣♦❧②❣r❛♣❤ ✐s ❛ tr✐♣❧❡ Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ✇❤❡r❡ ⊲ ( Σ ✵ , Σ ✶ ) ✐s ❛ ✶✲♣♦❧②❣r❛♣❤✱ ⊲ Σ ✷ ✐s ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❢r❡❡ ✶✲❝❛t❡❣♦r② Σ ∗ ✶ ✳ s ✶ ( α ) s ✵ s ✶ s ✵ s ✶ ( α ) t ✵ s ✶ ( α ) Σ ∗ Σ ✵ Σ ✷ = α = ✶ s ✵ t ✶ ( α ) t ✵ t ✶ ( α ) t ✵ t ✶ t ✶ ( α ) ◮ ❆ r❡✇r✐t✐♥❣ st❡♣ ✐s ❛ ✷✲❝❡❧❧ ♦❢ t❤❡ ❢r❡❡ ✷✲❝❛t❡❣♦r② Σ ∗ ✷ ♦✈❡r Σ ✇✐t❤ s❤❛♣❡ ✇✉✇ ′ ✉ ✇ ′ ✇ t ✵ ( ✇ ′ ) α s ✵ ( ✇ ) ✇ α ✇ ′ ✈ ✇✈✇ ′ α � ✈ ✐s ❛ ✷✲❝❡❧❧ ♦❢ Σ ✷ ❛♥❞ ✇ ✱ ✇ ′ ❛r❡ ✶✲❝❡❧❧s ♦❢ Σ ∗ ✇❤❡r❡ ✉ ✶ ✳

▲❡t ❈ ❜❡ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮✳ ❆ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛ ✷✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t ❈ ✶ ✷ ❆♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛ ✸ ✶ ✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t ❈ ✷ ✶ � � � � � � � � � � � P♦❧②❣r❛♣❤s ◮ ❆ ( ✸ , ✶ ) ✲ ♣♦❧②❣r❛♣❤ ✐s ❛ ♣❛✐r Σ = ( Σ ✷ , Σ ✸ ) ♠❛❞❡ ♦❢ ⊲ ❛ ✷✲♣♦❧②❣r❛♣❤ Σ ✷ ✱ ⊲ ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ♦❢ t❤❡ ❢r❡❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ ✳ ✉ s ✵ s ✶ s ✷ ❆ Σ ∗ • α � • Σ ⊤ � � β Σ ✵ Σ ✸ ✶ ✷ t ✵ t ✶ t ✷ ✈

❆♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛ ✸ ✶ ✲♣♦❧②❣r❛♣❤ s✉❝❤ t❤❛t ❈ ✷ ✶ � � � � � � � � � � � P♦❧②❣r❛♣❤s ◮ ❆ ( ✸ , ✶ ) ✲ ♣♦❧②❣r❛♣❤ ✐s ❛ ♣❛✐r Σ = ( Σ ✷ , Σ ✸ ) ♠❛❞❡ ♦❢ ⊲ ❛ ✷✲♣♦❧②❣r❛♣❤ Σ ✷ ✱ ⊲ ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ♦❢ t❤❡ ❢r❡❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ ✳ ✉ s ✵ s ✶ s ✷ ❆ Σ ∗ • α � • Σ ⊤ � � β Σ ✵ Σ ✸ ✶ ✷ t ✵ t ✶ t ✷ ✈ ▲❡t ❈ ❜❡ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮✳ ◮ ❆ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛ ✷✲♣♦❧②❣r❛♣❤ Σ s✉❝❤ t❤❛t ❈ ≃ Σ ∗ ✶ /Σ ✷

� � � � � � � � � � � P♦❧②❣r❛♣❤s ◮ ❆ ( ✸ , ✶ ) ✲ ♣♦❧②❣r❛♣❤ ✐s ❛ ♣❛✐r Σ = ( Σ ✷ , Σ ✸ ) ♠❛❞❡ ♦❢ ⊲ ❛ ✷✲♣♦❧②❣r❛♣❤ Σ ✷ ✱ ⊲ ❛ ❣❧♦❜✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ♦❢ t❤❡ ❢r❡❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ ✳ ✉ s ✵ s ✶ s ✷ ❆ Σ ∗ • α � • Σ ⊤ � � β Σ ✵ Σ ✸ ✶ ✷ t ✵ t ✶ t ✷ ✈ ▲❡t ❈ ❜❡ ❛ ❝❛t❡❣♦r② ✭♦r ❛ ♠♦♥♦✐❞✮✳ ◮ ❆ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛ ✷✲♣♦❧②❣r❛♣❤ Σ s✉❝❤ t❤❛t ❈ ≃ Σ ∗ ✶ /Σ ✷ ◮ ❆♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ s✉❝❤ t❤❛t ❈ ≃ Σ ∗ ✶ /Σ ✷

■♥ ♦t❤❡r ✇♦r❞s✿ t❤❡ q✉♦t✐❡♥t ✷ ✶ ✲❝❛t❡❣♦r② ✸ ✐s ❛s♣❤❡r✐❝❛❧✱ ✷ t❤❡ ❝♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② ✸ ♦♥ t❤❡ ✷ ✶ ✲❝❛t❡❣♦r② ✷ ❝♦♥t❛✐♥s ❡✈❡r② ♣❛✐r ♦❢ ♣❛r❛❧❧❡❧ ✷✲❝❡❧❧s✳ ✸✲❝❡❧❧s ♦❢ ✸ ❣❡♥❡r❛t❡ ❛ t✐❧✐♥❣ ♦❢ ✷ ✳ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ▲❡t ❜❡ ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✳ ❋♦r t❤❡ ▲❛❝❦✬s ♠♦❞❡❧ str✉❝t✉r❡ ♦♥ ✷✲❝❛t❡❣♦r✐❡s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐✮ ❚❤❡ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ ✐✐✮ ❚❤❡ ✷ ✶ ✲❝❛t❡❣♦r② ✸ ✐s ❛ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✱ t❤❛t ✐s✱ ❛ ❝♦✜❜r❛♥t ✷ ✷✲❝❛t❡❣♦r② ✇❡❛❦❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ◮ ❆ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❈ s✉❝❤ t❤❛t t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ✐s ❛ ❤♦♠♦t♦♣② ❜❛s✐s ✳

t❤❡ ❝♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② ✸ ♦♥ t❤❡ ✷ ✶ ✲❝❛t❡❣♦r② ✷ ❝♦♥t❛✐♥s ❡✈❡r② ♣❛✐r ♦❢ ♣❛r❛❧❧❡❧ ✷✲❝❡❧❧s✳ ✸✲❝❡❧❧s ♦❢ ✸ ❣❡♥❡r❛t❡ ❛ t✐❧✐♥❣ ♦❢ ✷ ✳ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ▲❡t ❜❡ ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✳ ❋♦r t❤❡ ▲❛❝❦✬s ♠♦❞❡❧ str✉❝t✉r❡ ♦♥ ✷✲❝❛t❡❣♦r✐❡s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐✮ ❚❤❡ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ ✐✐✮ ❚❤❡ ✷ ✶ ✲❝❛t❡❣♦r② ✸ ✐s ❛ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✱ t❤❛t ✐s✱ ❛ ❝♦✜❜r❛♥t ✷ ✷✲❝❛t❡❣♦r② ✇❡❛❦❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ◮ ❆ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❈ s✉❝❤ t❤❛t t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ✐s ❛ ❤♦♠♦t♦♣② ❜❛s✐s ✳ ■♥ ♦t❤❡r ✇♦r❞s✿ ⊲ t❤❡ q✉♦t✐❡♥t ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ /Σ ✸ ✐s ❛s♣❤❡r✐❝❛❧✱

✸✲❝❡❧❧s ♦❢ ✸ ❣❡♥❡r❛t❡ ❛ t✐❧✐♥❣ ♦❢ ✷ ✳ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ▲❡t ❜❡ ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✳ ❋♦r t❤❡ ▲❛❝❦✬s ♠♦❞❡❧ str✉❝t✉r❡ ♦♥ ✷✲❝❛t❡❣♦r✐❡s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐✮ ❚❤❡ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ ✐✐✮ ❚❤❡ ✷ ✶ ✲❝❛t❡❣♦r② ✸ ✐s ❛ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✱ t❤❛t ✐s✱ ❛ ❝♦✜❜r❛♥t ✷ ✷✲❝❛t❡❣♦r② ✇❡❛❦❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ◮ ❆ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❈ s✉❝❤ t❤❛t t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ✐s ❛ ❤♦♠♦t♦♣② ❜❛s✐s ✳ ■♥ ♦t❤❡r ✇♦r❞s✿ ⊲ t❤❡ q✉♦t✐❡♥t ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ /Σ ✸ ✐s ❛s♣❤❡r✐❝❛❧✱ ⊲ t❤❡ ❝♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② Σ ✸ ♦♥ t❤❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ ❝♦♥t❛✐♥s ❡✈❡r② ♣❛✐r ♦❢ ♣❛r❛❧❧❡❧ ✷✲❝❡❧❧s✳

❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ▲❡t ❜❡ ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✳ ❋♦r t❤❡ ▲❛❝❦✬s ♠♦❞❡❧ str✉❝t✉r❡ ♦♥ ✷✲❝❛t❡❣♦r✐❡s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐✮ ❚❤❡ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ ✐✐✮ ❚❤❡ ✷ ✶ ✲❝❛t❡❣♦r② ✸ ✐s ❛ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✱ t❤❛t ✐s✱ ❛ ❝♦✜❜r❛♥t ✷ ✷✲❝❛t❡❣♦r② ✇❡❛❦❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳ ❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ◮ ❆ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❈ s✉❝❤ t❤❛t t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ✐s ❛ ❤♦♠♦t♦♣② ❜❛s✐s ✳ ■♥ ♦t❤❡r ✇♦r❞s✿ ⊲ t❤❡ q✉♦t✐❡♥t ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ /Σ ✸ ✐s ❛s♣❤❡r✐❝❛❧✱ ⊲ t❤❡ ❝♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② Σ ✸ ♦♥ t❤❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ ❝♦♥t❛✐♥s ❡✈❡r② ♣❛✐r ♦❢ ♣❛r❛❧❧❡❧ ✷✲❝❡❧❧s✳ ⊲ ✸✲❝❡❧❧s ♦❢ Σ ✸ ❣❡♥❡r❛t❡ ❛ t✐❧✐♥❣ ♦❢ Σ ⊤ ✷ ✳

❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ◮ ❆ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✐s ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❈ s✉❝❤ t❤❛t t❤❡ ❝❡❧❧✉❧❛r ❡①t❡♥s✐♦♥ Σ ✸ ✐s ❛ ❤♦♠♦t♦♣② ❜❛s✐s ✳ ■♥ ♦t❤❡r ✇♦r❞s✿ ⊲ t❤❡ q✉♦t✐❡♥t ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ /Σ ✸ ✐s ❛s♣❤❡r✐❝❛❧✱ ⊲ t❤❡ ❝♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② Σ ✸ ♦♥ t❤❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ ❝♦♥t❛✐♥s ❡✈❡r② ♣❛✐r ♦❢ ♣❛r❛❧❧❡❧ ✷✲❝❡❧❧s✳ ⊲ ✸✲❝❡❧❧s ♦❢ Σ ✸ ❣❡♥❡r❛t❡ ❛ t✐❧✐♥❣ ♦❢ Σ ⊤ ✷ ✳ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ▲❡t Σ ❜❡ ❛♥ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✳ ❋♦r t❤❡ ▲❛❝❦✬s ♠♦❞❡❧ str✉❝t✉r❡ ♦♥ ✷✲❝❛t❡❣♦r✐❡s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss❡rt✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐✮ ❚❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ ✐✐✮ ❚❤❡ ( ✷ , ✶ ) ✲❝❛t❡❣♦r② Σ ⊤ ✷ /Σ ✸ ✐s ❛ ❝♦✜❜r❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❈ ✱ t❤❛t ✐s✱ ❛ ❝♦✜❜r❛♥t ✷✲❝❛t❡❣♦r② ✇❡❛❦❧② ❡q✉✐✈❛❧❡♥t t♦ ❈ ✳

❋r❡❡ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ✸✿ t❤❡ ❢✉❧❧ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥✿ ❛❧❧ t❤❡ rs st rt r s t sr rs ts st tr rt ✷✲s♣❤❡r❡s ❛ ❤♦♠♦t♦♣② ❜❛s✐s ❝❛♥ ❜❡ ♠❛❞❡ ✇✐t❤ ♦♥❧② ♦♥❡ ✸✲❝❡❧❧ rs st rt r s t sr rs ts st tr rt ❩ r s t ✇❤❡r❡ t❤❡ ✸✲❝❡❧❧ ❩ r s t ✐s t❤❡ ♣❡r♠✉t♦❤❡❞r♦♥ s rt st r str srt rs t tsr ❩ r s t rst r t trs rts st rs rt s ❊①❛♠♣❧❡s ◮ ❋r❡❡ ♠♦♥♦✐❞ ✿ ♥♦ r❡❧❛t✐♦♥✱ ❛♥ ❡♠♣t② ❤♦♠♦t♦♣② ❜❛s✐s✿ � � � � � � ∅ ∅ ① ✶ , . . . , ① ♥

❛ ❤♦♠♦t♦♣② ❜❛s✐s ❝❛♥ ❜❡ ♠❛❞❡ ✇✐t❤ ♦♥❧② ♦♥❡ ✸✲❝❡❧❧ rs st rt r s t sr rs ts st tr rt ❩ r s t ✇❤❡r❡ t❤❡ ✸✲❝❡❧❧ ❩ r s t ✐s t❤❡ ♣❡r♠✉t♦❤❡❞r♦♥ s rt st r str srt rs t tsr ❩ r s t rst r t trs rts st rs rt s � � ❊①❛♠♣❧❡s ◮ ❋r❡❡ ♠♦♥♦✐❞ ✿ ♥♦ r❡❧❛t✐♦♥✱ ❛♥ ❡♠♣t② ❤♦♠♦t♦♣② ❜❛s✐s✿ � � � � � � ∅ ∅ ① ✶ , . . . , ① ♥ ◮ ❋r❡❡ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ✸✿ ⊲ t❤❡ ❢✉❧❧ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥✿ � � γ rs � rs , ts γ st � st , tr γ rt � rt · � � · � ❛❧❧ t❤❡ � | r , s , t sr ✷✲s♣❤❡r❡s

✇❤❡r❡ t❤❡ ✸✲❝❡❧❧ ❩ r s t ✐s t❤❡ ♣❡r♠✉t♦❤❡❞r♦♥ s rt st r str srt rs t tsr ❩ r s t rst r t trs rts st rs rt s � � ❊①❛♠♣❧❡s ◮ ❋r❡❡ ♠♦♥♦✐❞ ✿ ♥♦ r❡❧❛t✐♦♥✱ ❛♥ ❡♠♣t② ❤♦♠♦t♦♣② ❜❛s✐s✿ � � � � � � ∅ ∅ ① ✶ , . . . , ① ♥ ◮ ❋r❡❡ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ✸✿ ⊲ t❤❡ ❢✉❧❧ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥✿ � � γ rs � rs , ts γ st � st , tr γ rt � rt · � � · � ❛❧❧ t❤❡ � | r , s , t sr ✷✲s♣❤❡r❡s ⊲ ❛ ❤♦♠♦t♦♣② ❜❛s✐s ❝❛♥ ❜❡ ♠❛❞❡ ✇✐t❤ ♦♥❧② ♦♥❡ ✸✲❝❡❧❧ � � γ rs � rs , ts γ st � st , tr γ rt � rt � � � � r , s , t sr ❩ r , s , t

� � � � � � � � ❊①❛♠♣❧❡s ◮ ❋r❡❡ ♠♦♥♦✐❞ ✿ ♥♦ r❡❧❛t✐♦♥✱ ❛♥ ❡♠♣t② ❤♦♠♦t♦♣② ❜❛s✐s✿ � � � � � � ∅ ∅ ① ✶ , . . . , ① ♥ ◮ ❋r❡❡ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ♦❢ r❛♥❦ ✸✿ ⊲ t❤❡ ❢✉❧❧ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥✿ � � γ rs � rs , ts γ st � st , tr γ rt � rt · � � · � ❛❧❧ t❤❡ � | r , s , t sr ✷✲s♣❤❡r❡s ⊲ ❛ ❤♦♠♦t♦♣② ❜❛s✐s ❝❛♥ ❜❡ ♠❛❞❡ ✇✐t❤ ♦♥❧② ♦♥❡ ✸✲❝❡❧❧ � � γ rs � rs , ts γ st � st , tr γ rt � rt � � � � r , s , t sr ❩ r , s , t ✇❤❡r❡ t❤❡ ✸✲❝❡❧❧ ❩ r , s , t ✐s t❤❡ ♣❡r♠✉t♦❤❡❞r♦♥ s γ rt � srt γ st r str γ rs t tsr ❩ r , s , t rst γ rt s � rts r γ st t γ rs trs

st ❆rt ✸ ❙ ✸ s t tst sts ❆rt✐♥ ♠♦♥♦✐❞ ❇ ❙ ✹ sr tr st ❆rt ✸ ❙ ✹ r s t rsr srs rt tr tst sts ❩ r s t s rt s sr st r rt srtstr strsrt srstsr st rs t rs tsr stsrst rsrtsr rs rt sr st rst tstrst ❩ r s t rstrsr ts rt st rst rs tsrtst rstsrs r st rs tsr st tsrsts t rs ts trsrts rtstrs rt s rt s ❊①❛♠♣❧❡s ◮ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❙ ✸ ) s = t = ⇒

❆rt✐♥ ♠♦♥♦✐❞ ❇ ❙ ✹ sr tr st ❆rt ✸ ❙ ✹ r s t rsr srs rt tr tst sts ❩ r s t s rt s sr st r rt srtstr strsrt srstsr st rs t rs tsr stsrst rsrtsr rs rt sr st rst tstrst ❩ r s t rstrsr ts rt st rst rs tsrtst rstsrs r st rs tsr st tsrsts t rs ts trsrts rtstrs rt s rt s ❊①❛♠♣❧❡s ◮ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❙ ✸ ) s = t = ⇒ � � � γ st � sts � � � ❆rt ✸ ( ❙ ✸ ) = s , t tst ∅

� � � � � � � � � � ❊①❛♠♣❧❡s ◮ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❙ ✸ ) s = t = ⇒ � � � γ st � sts � � � ❆rt ✸ ( ❙ ✸ ) = s , t tst ∅ ◮ ❆rt✐♥ ♠♦♥♦✐❞ ❇ + ( ❙ ✹ ) � � γ sr � srs , rt γ tr � tr , tst γ st � sts � � � � ❆rt ✸ ( ❙ ✹ ) = r , s , t rsr ❩ r , s , t s γ rt s γ − sr γ st r � rt � srtstr strsrt srstsr st γ rs t γ rs tsr stsrst rsrtsr rs γ rt sr γ st rst tstrst ❩ r , s , t rstrsr ts γ rt st rst γ rs tsrtst rstsrs r γ st rs tsr γ st tsrsts t γ rs ts � trsrts � rtstrs γ rt s γ − rt s

❈♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥s Pr♦❜❧❡♠s✳ ✶✳ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❄ ✷✳ ❍♦✇ t♦ tr❛♥s❢♦r♠ ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ❄

P❛rt ■■✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥✲r❡❞✉❝t✐♦♥ ♣r♦❝❡❞✉r❡

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s

✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ ✶ ✳ ✶ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸

❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ✉ ✐♥ Σ ∗ ✶ ✱ ✉

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ✉ ✐♥ Σ ∗ ✶ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✶✲❝❡❧❧ ① ✉ ①

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ✉ ✐♥ Σ ∗ ✶ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✶✲❝❡❧❧ ① ❛♥❞ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ δ � ① ✉

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ✉ ✐♥ Σ ∗ ✶ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✶✲❝❡❧❧ ① ❛♥❞ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ δ � ① ✉ ◮ r❡♠♦✈❡ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ α ✐♥ Σ ✷ ✇✐t❤ ① ✐♥ Σ ✶ ✱ α � ① ✉

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ✉ ✐♥ Σ ∗ ✶ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✶✲❝❡❧❧ ① ❛♥❞ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ δ � ① ✉ ◮ r❡♠♦✈❡ ❛ ❣❡♥❡r❛t♦r ✿ ❢♦r ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ α ✐♥ Σ ✷ ✇✐t❤ ① ✐♥ Σ ✶ ✱ r❡♠♦✈❡ ① ❛♥❞ α ✚ ❩ � ✁ α ✉ ❆ ①

� ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✷✲❝❡❧❧ ❢ ✐♥ Σ ⊤ ✷ ✱ ❢ ✉ � ✈

� ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✷✲❝❡❧❧ ❢ ✐♥ Σ ⊤ ✷ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ α ❢ ❢ ✉ � � ✈ α ❢

� ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✷✲❝❡❧❧ ❢ ✐♥ Σ ⊤ ✷ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ α ❢ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ ❢ ❢ ✉ � ❆ ❢ � � � ✈ α ❢

� � ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✷✲❝❡❧❧ ❢ ✐♥ Σ ⊤ ✷ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ α ❢ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ ❢ ❢ ✉ � ❆ ❢ � � � ✈ α ❢ ◮ r❡♠♦✈❡ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✸✲❝❡❧❧ ❆ ✇✐t❤ α ✐♥ Σ ✷ ✱ ❢ ❆ � � ✉ � � ✈ α

� � ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✷✲❝❡❧❧ ❢ ✐♥ Σ ⊤ ✷ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✷✲❝❡❧❧ α ❢ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ ❢ ❢ ✉ � ❆ ❢ � � � ✈ α ❢ ◮ r❡♠♦✈❡ ❛ r❡❧❛t✐♦♥ ✿ ❢♦r ❛ ✸✲❝❡❧❧ ❆ ✇✐t❤ α ✐♥ Σ ✷ ✱ r❡♠♦✈❡ α ❛♥❞ ❆ ❢ ❆ � � ✉ � ✁ ❆ � ✈ ✚ ❩ α

� � � ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ✸ ✲❝❡❧❧ ✿ ❢♦r ✸✲❝❡❧❧s ❇ ✱ ❢ ✉ � � ✈ ❇ � � ❣

� � � � � ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ✸ ✲❝❡❧❧ ✿ ❢♦r ✸✲❝❡❧❧s ❇ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ : ❢ ⇛ ❣ ❢ ✉ � � ✈ ❇ � � ❆ ❣

� � � � � � � � � � ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ✸ ✲❝❡❧❧ ✿ ❢♦r ✸✲❝❡❧❧s ❇ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ : ❢ ⇛ ❣ ❢ ✉ � � ✈ ❇ � � ❆ ❣ ◮ r❡♠♦✈❡ ❛ ✸ ✲❝❡❧❧ ✿ ❢♦r ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ : ❢ ⇛ ❣ ❢ ✉ � � ✈ ❇ � � ❆ ❣

� � � � � � � � � � ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ◮ ❚✇♦ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡✲❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ✷✲❝❛t❡❣♦r✐❡s ≈ Σ ⊤ → Υ ⊤ ✷ /Σ ✸ − ✷ /Υ ✸ ✐♥❞✉❝✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦♥ ♣r❡s❡♥t❡❞ ❝❛t❡❣♦r✐❡s✿ Σ ∗ ✶ ≃ Υ ∗ ✶ ✳ ◮ ❆♥ ❡❧❡♠❡♥t❛r② ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ Σ ✐s ❛ ✸✲❢✉♥❝t♦r ✇✐t❤ s♦✉r❝❡ Σ ⊤ ✸ t❤❛t ❜❡❧♦♥❣s t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛✐rs ♦❢ ❞✉❛❧ ♦♣❡r❛t✐♦♥s✿ ◮ ❛❞❞ ❛ ✸ ✲❝❡❧❧ ✿ ❢♦r ✸✲❝❡❧❧s ❇ ✱ ❛❞❞ ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ : ❢ ⇛ ❣ ❢ ✉ � � ✈ ❇ � � ❆ ❣ ◮ r❡♠♦✈❡ ❛ ✸ ✲❝❡❧❧ ✿ ❢♦r ❛ ❣❡♥❡r❛t✐♥❣ ✸✲❝❡❧❧ ❆ : ❢ ⇛ ❣ r❡♠♦✈❡ ❆ ❢ ✉ � ✁ ❆ � ✈ ❇ � � ❆ ❣

❈♦♥s❡q✉❡♥❝❡✳ ■❢ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ❛♥❞ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ t❤❡♥ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ❚✇♦ ✭✜♥✐t❡✮ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡ ❡q✉✐✈❛❧❡♥t ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ t❤❡r❡ ❡①✐sts ❛ ✭✜♥✐t❡✮ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ T : Σ ⊤ − → Υ ⊤

❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ❚✇♦ ✭✜♥✐t❡✮ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ ❛♥❞ Υ ❛r❡ ❚✐❡t③❡ ❡q✉✐✈❛❧❡♥t ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ t❤❡r❡ ❡①✐sts ❛ ✭✜♥✐t❡✮ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ T : Σ ⊤ − → Υ ⊤ ❈♦♥s❡q✉❡♥❝❡✳ ■❢ Σ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ❛♥❞ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥ T : Σ ⊤ − → Υ ⊤ t❤❡♥ Υ ✐s ❛ ❝♦❤❡r❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳

t❡r♠✐♥❛t❡s ✐❢ ✐t ❞♦❡s ♥♦t ❣❡♥❡r❛t❡ ❛♥② ✐♥✜♥✐t❡ r❡❞✉❝t✐♦♥ s❡q✉❡♥❝❡ ✉ ✶ ✉ ✷ ✉ ♥ ❆ ❜r❛♥❝❤✐♥❣ ♦❢ ✐s ❛ ♣❛✐r ❢ ❣ ♦❢ ✷✲❝❡❧❧s ♦❢ ✷ ✇✐t❤ ❛ ❝♦♠♠♦♥ s♦✉r❝❡ ❢ ✈ ✉ ✇ ❣ ✐s ❝♦♥✢✉❡♥t ✐❢ ❛❧❧ ♦❢ ✐ts ❜r❛♥❝❤✐♥❣s ❛r❡ ❝♦♥✢✉❡♥t✿ ❢ ❢ ✈ ✉ ✉ ❣ ✇ ❣ ✐s ❝♦♥✈❡r❣❡♥t ✐❢ ✐t t❡r♠✐♥❛t❡s ❛♥❞ ✐t ✐s ❝♦♥✢✉❡♥t✳ ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ▲❡t Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ❜❡ ❛ ✷✲♣♦❧②❣r❛♣❤✳

❆ ❜r❛♥❝❤✐♥❣ ♦❢ ✐s ❛ ♣❛✐r ❢ ❣ ♦❢ ✷✲❝❡❧❧s ♦❢ ✷ ✇✐t❤ ❛ ❝♦♠♠♦♥ s♦✉r❝❡ ❢ ✈ ✉ ✇ ❣ ✐s ❝♦♥✢✉❡♥t ✐❢ ❛❧❧ ♦❢ ✐ts ❜r❛♥❝❤✐♥❣s ❛r❡ ❝♦♥✢✉❡♥t✿ ❢ ❢ ✈ ✉ ✉ ❣ ✇ ❣ ✐s ❝♦♥✈❡r❣❡♥t ✐❢ ✐t t❡r♠✐♥❛t❡s ❛♥❞ ✐t ✐s ❝♦♥✢✉❡♥t✳ ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ▲❡t Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ❜❡ ❛ ✷✲♣♦❧②❣r❛♣❤✳ ◮ Σ t❡r♠✐♥❛t❡s ✐❢ ✐t ❞♦❡s ♥♦t ❣❡♥❡r❛t❡ ❛♥② ✐♥✜♥✐t❡ r❡❞✉❝t✐♦♥ s❡q✉❡♥❝❡ � ✉ ✷ � · · · � ✉ ♥ � · · · ✉ ✶

✐s ❝♦♥✢✉❡♥t ✐❢ ❛❧❧ ♦❢ ✐ts ❜r❛♥❝❤✐♥❣s ❛r❡ ❝♦♥✢✉❡♥t✿ ❢ ❢ ✈ ✉ ✉ ❣ ✇ ❣ ✐s ❝♦♥✈❡r❣❡♥t ✐❢ ✐t t❡r♠✐♥❛t❡s ❛♥❞ ✐t ✐s ❝♦♥✢✉❡♥t✳ � ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ▲❡t Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ❜❡ ❛ ✷✲♣♦❧②❣r❛♣❤✳ ◮ Σ t❡r♠✐♥❛t❡s ✐❢ ✐t ❞♦❡s ♥♦t ❣❡♥❡r❛t❡ ❛♥② ✐♥✜♥✐t❡ r❡❞✉❝t✐♦♥ s❡q✉❡♥❝❡ � ✉ ✷ � · · · � ✉ ♥ � · · · ✉ ✶ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♦❢ Σ ✐s ❛ ♣❛✐r ( ❢ , ❣ ) ♦❢ ✷✲❝❡❧❧s ♦❢ Σ ∗ ✷ ✇✐t❤ ❛ ❝♦♠♠♦♥ s♦✉r❝❡ ❢ ✈ ✉ � ✇ ❣

✐s ❝♦♥✈❡r❣❡♥t ✐❢ ✐t t❡r♠✐♥❛t❡s ❛♥❞ ✐t ✐s ❝♦♥✢✉❡♥t✳ � � � � � ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ▲❡t Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ❜❡ ❛ ✷✲♣♦❧②❣r❛♣❤✳ ◮ Σ t❡r♠✐♥❛t❡s ✐❢ ✐t ❞♦❡s ♥♦t ❣❡♥❡r❛t❡ ❛♥② ✐♥✜♥✐t❡ r❡❞✉❝t✐♦♥ s❡q✉❡♥❝❡ � ✉ ✷ � · · · � ✉ ♥ � · · · ✉ ✶ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♦❢ Σ ✐s ❛ ♣❛✐r ( ❢ , ❣ ) ♦❢ ✷✲❝❡❧❧s ♦❢ Σ ∗ ✷ ✇✐t❤ ❛ ❝♦♠♠♦♥ s♦✉r❝❡ ❢ ✈ ✉ � ✇ ❣ ◮ Σ ✐s ❝♦♥✢✉❡♥t ✐❢ ❛❧❧ ♦❢ ✐ts ❜r❛♥❝❤✐♥❣s ❛r❡ ❝♦♥✢✉❡♥t✿ ❢ ′ ❢ ✈ ✉ ′ ✉ ❣ ✇ ❣ ′

� � � � � ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ▲❡t Σ = ( Σ ✵ , Σ ✶ , Σ ✷ ) ❜❡ ❛ ✷✲♣♦❧②❣r❛♣❤✳ ◮ Σ t❡r♠✐♥❛t❡s ✐❢ ✐t ❞♦❡s ♥♦t ❣❡♥❡r❛t❡ ❛♥② ✐♥✜♥✐t❡ r❡❞✉❝t✐♦♥ s❡q✉❡♥❝❡ � ✉ ✷ � · · · � ✉ ♥ � · · · ✉ ✶ ◮ ❆ ❜r❛♥❝❤✐♥❣ ♦❢ Σ ✐s ❛ ♣❛✐r ( ❢ , ❣ ) ♦❢ ✷✲❝❡❧❧s ♦❢ Σ ∗ ✷ ✇✐t❤ ❛ ❝♦♠♠♦♥ s♦✉r❝❡ ❢ ✈ ✉ � ✇ ❣ ◮ Σ ✐s ❝♦♥✢✉❡♥t ✐❢ ❛❧❧ ♦❢ ✐ts ❜r❛♥❝❤✐♥❣s ❛r❡ ❝♦♥✢✉❡♥t✿ ❢ ′ ❢ ✈ ✉ ′ ✉ ❣ ✇ ❣ ′ ◮ Σ ✐s ❝♦♥✈❡r❣❡♥t ✐❢ ✐t t❡r♠✐♥❛t❡s ❛♥❞ ✐t ✐s ❝♦♥✢✉❡♥t✳

❆ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ✐s ❛ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ t❤❡ ❢♦r♠ ❢ ❣ � ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❜r❛♥❝❤✐♥❣ ❢ ✈ ✉ � ✇ ❣ ✐s ❧♦❝❛❧ ✐❢ ❢ ❛♥❞ ❣ ❛r❡ r❡✇r✐t✐♥❣ st❡♣s✳

� ❘❡✇r✐t✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✷ ✲♣♦❧②❣r❛♣❤s ◮ ❆ ❜r❛♥❝❤✐♥❣ ❢ ✈ ✉ � ✇ ❣ ✐s ❧♦❝❛❧ ✐❢ ❢ ❛♥❞ ❣ ❛r❡ r❡✇r✐t✐♥❣ st❡♣s✳ ◮ ❆ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ✐s ❛ ❧♦❝❛❧ ❜r❛♥❝❤✐♥❣ ♦❢ t❤❡ ❢♦r♠ ❢ � ❣ �

❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣✿ ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ st � sts � ❆rt ✷ ( ❙ ✸ ) = s , t tst

❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ st � sts � ❆rt ✷ ( ❙ ✸ ) = s , t tst ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣✿

❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ st � sts � ❆rt ✷ ( ❙ ✸ ) = s , t tst ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣✿ tstst

� ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ st � sts � ❆rt ✷ ( ❙ ✸ ) = s , t tst ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣✿ γ st st sts st tst st

� ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ st � sts � ❆rt ✷ ( ❙ ✸ ) = s , t tst ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣✿ γ st st stsst ts tst � ts sts ts γ st

■t ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ rs � rs , ts γ st � st , tr γ rt � rt � ❆rt ✷ ( ❙ ✹ ) = r , s , t sr

❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ rs � rs , ts γ st � st , tr γ rt � rt � ❆rt ✷ ( ❙ ✹ ) = r , s , t sr ⊲ ■t ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ tsr

� ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ rs � rs , ts γ st � st , tr γ rt � rt � ❆rt ✷ ( ❙ ✹ ) = r , s , t sr ⊲ ■t ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ st r γ st r ts r

� ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ rs � rs , ts γ st � st , tr γ rt � rt � ❆rt ✷ ( ❙ ✹ ) = r , s , t sr ⊲ ■t ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ str γ st r t sr t γ rs � t rs

� ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ rs � rs , ts γ st � st , tr γ rt � rt � ❆rt ✷ ( ❙ ✹ ) = r , s , t sr ⊲ ■t ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ s γ rt � s rt s tr γ st r tsr t γ rs � tr s γ rt s � rt s

� � � � ❊①❛♠♣❧❡ ◮ ❚❤❡ ✷✲♣♦❧②❣r❛♣❤ � � � γ rs � rs , ts γ st � st , tr γ rt � rt � ❆rt ✷ ( ❙ ✹ ) = r , s , t sr ⊲ ■t ❤❛s ♦♥❧② ♦♥❡ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ s γ rt � sr t str γ rs t γ st r tsr rst t γ rs r γ st γ rt s � r ts trs

❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ ✐s t❤❡ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ♦❜t❛✐♥❡❞ ❢r♦♠ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❢♦❧❧♦✇✐♥❣ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r ❡✈❡r② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❢ ✈ ✉ ✇ ❣ ❝♦♠♣✉t❡ ❢ ❛♥❞ ❣ r❡❞✉❝✐♥❣ t♦ s♦♠❡ ♥♦r♠❛❧ ❢♦r♠s✳ ❢ ❢ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ ❛ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ✈ ✇ ✇ ❣ ❣ ❢ ❢ ✈ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ t❤❡ ✷✲❝❡❧❧ ❢ ❣ ❛♥❞ t❤❡ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ❢ ❣ ✇ ❣ ✇ ❣ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ▲❡t Σ ❜❡ ❛ t❡r♠✐♥❛t✐♥❣ ✷✲♣♦❧②❣r❛♣❤ ✭✇✐t❤ ❛ t♦t❛❧ t❡r♠✐♥❛t✐♦♥ ♦r❞❡r✮✳

❢♦r ❡✈❡r② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❢ ✈ ✉ ✇ ❣ ❝♦♠♣✉t❡ ❢ ❛♥❞ ❣ r❡❞✉❝✐♥❣ t♦ s♦♠❡ ♥♦r♠❛❧ ❢♦r♠s✳ ❢ ❢ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ ❛ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ✈ ✇ ✇ ❣ ❣ ❢ ❢ ✈ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ t❤❡ ✷✲❝❡❧❧ ❢ ❣ ❛♥❞ t❤❡ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ❢ ❣ ✇ ❣ ✇ ❣ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ▲❡t Σ ❜❡ ❛ t❡r♠✐♥❛t✐♥❣ ✷✲♣♦❧②❣r❛♣❤ ✭✇✐t❤ ❛ t♦t❛❧ t❡r♠✐♥❛t✐♦♥ ♦r❞❡r✮✳ ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ S ( Σ ) ♦❜t❛✐♥❡❞ ❢r♦♠ Σ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❢♦❧❧♦✇✐♥❣ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s

❝♦♠♣✉t❡ ❢ ❛♥❞ ❣ r❡❞✉❝✐♥❣ t♦ s♦♠❡ ♥♦r♠❛❧ ❢♦r♠s✳ ❢ ❢ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ ❛ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ✈ ✇ ✇ ❣ ❣ ❢ ❢ ✈ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ t❤❡ ✷✲❝❡❧❧ ❢ ❣ ❛♥❞ t❤❡ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ❢ ❣ ✇ ❣ ✇ ❣ � ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ▲❡t Σ ❜❡ ❛ t❡r♠✐♥❛t✐♥❣ ✷✲♣♦❧②❣r❛♣❤ ✭✇✐t❤ ❛ t♦t❛❧ t❡r♠✐♥❛t✐♦♥ ♦r❞❡r✮✳ ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ S ( Σ ) ♦❜t❛✐♥❡❞ ❢r♦♠ Σ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❢♦❧❧♦✇✐♥❣ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ⊲ ❢♦r ❡✈❡r② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❢ ✈ ✉ � ✇ ❣

❢ ❢ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ ❛ ✸✲❝❡❧❧ ❆ ❢ ❣ ✉ ❆ ❢ ❣ ✈ ✇ ✇ ❣ ❣ ❢ ❢ ✈ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ t❤❡ ✷✲❝❡❧❧ ❢ ❣ ❛♥❞ t❤❡ ✸✲❝❡❧❧ ❆ ❢ ❣ ❆ ❢ ❣ ✉ ❢ ❣ ✇ ✇ ❣ ❣ � ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ▲❡t Σ ❜❡ ❛ t❡r♠✐♥❛t✐♥❣ ✷✲♣♦❧②❣r❛♣❤ ✭✇✐t❤ ❛ t♦t❛❧ t❡r♠✐♥❛t✐♦♥ ♦r❞❡r✮✳ ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ S ( Σ ) ♦❜t❛✐♥❡❞ ❢r♦♠ Σ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❢♦❧❧♦✇✐♥❣ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ⊲ ❢♦r ❡✈❡r② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❢ ′ ❢ � � ✈ ✈ ✉ � � � ✇ ✇ ❣ ❣ ′ ❝♦♠♣✉t❡ ❢ ′ ❛♥❞ ❣ ′ r❡❞✉❝✐♥❣ t♦ s♦♠❡ ♥♦r♠❛❧ ❢♦r♠s✳

❢ ❢ ✈ ✈ ✐❢ ✈ ✇ ✱ ❛❞❞ t❤❡ ✷✲❝❡❧❧ ❢ ❣ ❛♥❞ t❤❡ ✸✲❝❡❧❧ ❆ ❢ ❣ ❆ ❢ ❣ ✉ ❢ ❣ ✇ ✇ ❣ ❣ � � � � � � � ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ▲❡t Σ ❜❡ ❛ t❡r♠✐♥❛t✐♥❣ ✷✲♣♦❧②❣r❛♣❤ ✭✇✐t❤ ❛ t♦t❛❧ t❡r♠✐♥❛t✐♦♥ ♦r❞❡r✮✳ ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ S ( Σ ) ♦❜t❛✐♥❡❞ ❢r♦♠ Σ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❢♦❧❧♦✇✐♥❣ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ⊲ ❢♦r ❡✈❡r② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❢ ′ ❢ � � ✈ ✈ ✉ � � � ✇ ✇ ❣ ❣ ′ ❝♦♠♣✉t❡ ❢ ′ ❛♥❞ ❣ ′ r❡❞✉❝✐♥❣ t♦ s♦♠❡ ♥♦r♠❛❧ ❢♦r♠s✳ ❢ ′ ❢ ✈ ⊲ ✐❢ � ✈ = � ✇ ✱ ❛❞❞ ❛ ✸✲❝❡❧❧ ❆ ❢ , ❣ ✉ ❆ ❢ , ❣ ✈ = � � ✇ ✇ ❣ ❣ ′

� � � � � � � � � � � � ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ▲❡t Σ ❜❡ ❛ t❡r♠✐♥❛t✐♥❣ ✷✲♣♦❧②❣r❛♣❤ ✭✇✐t❤ ❛ t♦t❛❧ t❡r♠✐♥❛t✐♦♥ ♦r❞❡r✮✳ ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ S ( Σ ) ♦❜t❛✐♥❡❞ ❢r♦♠ Σ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❢♦❧❧♦✇✐♥❣ ❚✐❡t③❡ tr❛♥s❢♦r♠❛t✐♦♥s ⊲ ❢♦r ❡✈❡r② ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣ ❢ ′ ❢ � � ✈ ✈ ✉ � � � ✇ ✇ ❣ ❣ ′ ❝♦♠♣✉t❡ ❢ ′ ❛♥❞ ❣ ′ r❡❞✉❝✐♥❣ t♦ s♦♠❡ ♥♦r♠❛❧ ❢♦r♠s✳ ❢ ′ ❢ ✈ ⊲ ✐❢ � ✈ = � ✇ ✱ ❛❞❞ ❛ ✸✲❝❡❧❧ ❆ ❢ , ❣ ✉ ❆ ❢ , ❣ � ✈ = � ✇ ✇ ❣ ❣ ′ ❢ ′ ❢ ✈ � ✈ α ❢ , ❣ ⊲ ✐❢ � ✈ < � ✇ ✱ ❛❞❞ t❤❡ ✷✲❝❡❧❧ α ❢ , ❣ ❛♥❞ t❤❡ ✸✲❝❡❧❧ ❆ ❢ , ❣ ✉ ❆ ❢ , ❣ � � � ✇ ✇ ❣ ❣ ′

❚❤✐s ❞❡✜♥❡s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✸ ✶ ✲♣♦❧②❣r❛♣❤s ♥ ♥ ✵ ✶ ✶ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ ✐s t❤❡ ✸ ✶ ✲♣♦❧②❣r❛♣❤ ♥ ♥ ✵ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ❋♦r ❛ t❡r♠✐♥❛t✐♥❣ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✱ t❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ ✐s ❛ ❝♦❤❡r❡♥t ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ Pr♦♦❢✳ ♦❜t❛✐♥❡❞ ❢r♦♠ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❑♥✉t❤✲❇❡♥❞✐①✬s ♣r♦❝❡❞✉r❡ ❙q✉✐❡r✬s ❝♦❤❡r❡♥❝❡ t❤❡♦r❡♠✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ◮ P♦t❡♥t✐❛❧ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s α ❢ , ❣ ❝❛♥ ❝r❡❛t❡ ♥❡✇ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s✱ ⊲ ✇❤♦s❡ ❝♦♥✢✉❡♥❝❡ ♠✉st ❛❧s♦ ❜❡ ❡①❛♠✐♥❡❞✱ ⊲ ♣♦ss✐❜❧② ❣❡♥❡r❛t✐♥❣ t❤❡ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s ❛♥❞ ✸✲❝❡❧❧s ⊲ ✳✳✳

❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ❋♦r ❛ t❡r♠✐♥❛t✐♥❣ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✱ t❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ ✐s ❛ ❝♦❤❡r❡♥t ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ Pr♦♦❢✳ ♦❜t❛✐♥❡❞ ❢r♦♠ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❑♥✉t❤✲❇❡♥❞✐①✬s ♣r♦❝❡❞✉r❡ ❙q✉✐❡r✬s ❝♦❤❡r❡♥❝❡ t❤❡♦r❡♠✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ◮ P♦t❡♥t✐❛❧ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s α ❢ , ❣ ❝❛♥ ❝r❡❛t❡ ♥❡✇ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s✱ ⊲ ✇❤♦s❡ ❝♦♥✢✉❡♥❝❡ ♠✉st ❛❧s♦ ❜❡ ❡①❛♠✐♥❡❞✱ ⊲ ♣♦ss✐❜❧② ❣❡♥❡r❛t✐♥❣ t❤❡ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s ❛♥❞ ✸✲❝❡❧❧s ⊲ ✳✳✳ ◮ ❚❤✐s ❞❡✜♥❡s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ = Σ ✵ ⊆ Σ ✶ ⊆ · · · ⊆ Σ ♥ ⊆ Σ ♥ + ✶ ⊆ · · · ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ � Σ ♥ . S ( Σ ) = ♥ � ✵

Pr♦♦❢✳ ♦❜t❛✐♥❡❞ ❢r♦♠ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❑♥✉t❤✲❇❡♥❞✐①✬s ♣r♦❝❡❞✉r❡ ❙q✉✐❡r✬s ❝♦❤❡r❡♥❝❡ t❤❡♦r❡♠✳ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ◮ P♦t❡♥t✐❛❧ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s α ❢ , ❣ ❝❛♥ ❝r❡❛t❡ ♥❡✇ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s✱ ⊲ ✇❤♦s❡ ❝♦♥✢✉❡♥❝❡ ♠✉st ❛❧s♦ ❜❡ ❡①❛♠✐♥❡❞✱ ⊲ ♣♦ss✐❜❧② ❣❡♥❡r❛t✐♥❣ t❤❡ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s ❛♥❞ ✸✲❝❡❧❧s ⊲ ✳✳✳ ◮ ❚❤✐s ❞❡✜♥❡s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ = Σ ✵ ⊆ Σ ✶ ⊆ · · · ⊆ Σ ♥ ⊆ Σ ♥ + ✶ ⊆ · · · ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ � Σ ♥ . S ( Σ ) = ♥ � ✵ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ❋♦r ❛ t❡r♠✐♥❛t✐♥❣ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✱ t❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ S ( Σ ) ♦❢ Σ ✐s ❛ ❝♦❤❡r❡♥t ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳

❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ◮ P♦t❡♥t✐❛❧ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s α ❢ , ❣ ❝❛♥ ❝r❡❛t❡ ♥❡✇ ❝r✐t✐❝❛❧ ❜r❛♥❝❤✐♥❣s✱ ⊲ ✇❤♦s❡ ❝♦♥✢✉❡♥❝❡ ♠✉st ❛❧s♦ ❜❡ ❡①❛♠✐♥❡❞✱ ⊲ ♣♦ss✐❜❧② ❣❡♥❡r❛t✐♥❣ t❤❡ ❛❞❥✉♥❝t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ✷✲❝❡❧❧s ❛♥❞ ✸✲❝❡❧❧s ⊲ ✳✳✳ ◮ ❚❤✐s ❞❡✜♥❡s ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤s Σ = Σ ✵ ⊆ Σ ✶ ⊆ · · · ⊆ Σ ♥ ⊆ Σ ♥ + ✶ ⊆ · · · ◮ ❚❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♦❢ Σ ✐s t❤❡ ( ✸ , ✶ ) ✲♣♦❧②❣r❛♣❤ � Σ ♥ . S ( Σ ) = ♥ � ✵ ❚❤❡♦r❡♠✳ ❬●❛✉ss❡♥t✲●✉✐r❛✉❞✲▼✳✱ ✷✵✶✺❪ ❋♦r ❛ t❡r♠✐♥❛t✐♥❣ ♣r❡s❡♥t❛t✐♦♥ Σ ♦❢ ❛ ❝❛t❡❣♦r② ❈ ✱ t❤❡ ❤♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ S ( Σ ) ♦❢ Σ ✐s ❛ ❝♦❤❡r❡♥t ❝♦♥✈❡r❣❡♥t ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❈ ✳ Pr♦♦❢✳ ⊲ S ( Σ ) ♦❜t❛✐♥❡❞ ❢r♦♠ Σ ❜② s✉❝❝❡ss✐✈❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❑♥✉t❤✲❇❡♥❞✐①✬s ♣r♦❝❡❞✉r❡ ⊲ ❙q✉✐❡r✬s ❝♦❤❡r❡♥❝❡ t❤❡♦r❡♠✳

❑◆ ❚❤❡ ❞❡❣❧❡① ♦r❞❡r ❣❡♥❡r❛t❡❞ ❜② t s ❛ ♣r♦✈❡s t❤❡ t❡r♠✐♥❛t✐♦♥ ♦❢ ✳ ✷ ❑◆ s t ❛ t❛ ❛s st ❛ s❛s ❛❛ s❛❛ ❛❛t ❆ ❇ ❈ ❉ ✷ ❛ t ❛s ❛❛ ❛❛❛s ❛❛t s❛s❛s st❛ s❛st s❛❛❛ s❛s s❛ s s❛❛ s❛ ❛❛ ❛❛❛❛ s❛s❛❛ s❛❛❛t s❛ ❑◆ ❍♦✇❡✈❡r✳ ❚❤❡ ❡①t❡♥❞❡❞ ♣r❡s❡♥t❛t✐♦♥ ♦❜t❛✐♥❡❞ ✐s ❜✐❣❣❡r t❤❛♥ ♥❡❝❡ss❛r②✳ ✷ ❍♦♠♦t♦♣✐❝❛❧ ❝♦♠♣❧❡t✐♦♥ ♣r♦❝❡❞✉r❡ ❊①❛♠♣❧❡✳ ❚❤❡ ❑❛♣✉r✲◆❛r❡♥❞r❛♥✬s ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❇ + ( ❙ ✸ ) ✱ ♦❜t❛✐♥❡❞ ❢r♦♠ ❆rt✐♥✬s ♣r❡s❡♥t❛t✐♦♥ ❜② ❝♦❤❡r❡♥t ❛❞❥✉♥❝t✐♦♥ ♦❢ t❤❡ ❈♦①❡t❡r ❡❧❡♠❡♥t st � � β � ❛ � α � ❛s , st � Σ ❑◆ = s , t , ❛ t❛ ✷