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♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✷ ✸ ✷ ✸ ✸ ❲❡ ✇❛♥t t♦ ♣❡r❢♦r♠ ✐t ♦♥ ❛♥❞ ✳ ❖❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛❜❧❡✳ ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ●✐✈❡♥ ❛ ✇♦r❞ w ✿ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✭ Σ ℓ , Σ r ❛r❡ ❞✐s❥♦✐♥t✮ r❡♣❧❛❝❡ ❡❛❝❤ ab ∈ Σ ℓ Σ r ✐♥ w ✇✐t❤ ❛ ❢r❡s❤ c ab Σ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ r❡♣❧❛❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❜❧♦❝❦ a ℓ ∈ Σ ∗ ✐♥ w ❜② ❛ ❢r❡s❤ a ℓ ✳ ✭♠❛①✐♠❛❧ ❜❧♦❝❦✿ a ℓ t❤❛t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✮✳ { b , c } ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ aaabbcccbbcccbbb ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✹ ✴ ✸✺

♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❲❡ ✇❛♥t t♦ ♣❡r❢♦r♠ ✐t ♦♥ ❛♥❞ ✳ ❖❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛❜❧❡✳ ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ●✐✈❡♥ ❛ ✇♦r❞ w ✿ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✭ Σ ℓ , Σ r ❛r❡ ❞✐s❥♦✐♥t✮ r❡♣❧❛❝❡ ❡❛❝❤ ab ∈ Σ ℓ Σ r ✐♥ w ✇✐t❤ ❛ ❢r❡s❤ c ab Σ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ r❡♣❧❛❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❜❧♦❝❦ a ℓ ∈ Σ ∗ ✐♥ w ❜② ❛ ❢r❡s❤ a ℓ ✳ ✭♠❛①✐♠❛❧ ❜❧♦❝❦✿ a ℓ t❤❛t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✮✳ { b , c } ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ aaabbcccbbcccbbb aaab ✷ c ✸ b ✷ c ✸ b ✸ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✹ ✴ ✸✺

❲❡ ✇❛♥t t♦ ♣❡r❢♦r♠ ✐t ♦♥ ❛♥❞ ✳ ❖❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛❜❧❡✳ ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ●✐✈❡♥ ❛ ✇♦r❞ w ✿ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✭ Σ ℓ , Σ r ❛r❡ ❞✐s❥♦✐♥t✮ r❡♣❧❛❝❡ ❡❛❝❤ ab ∈ Σ ℓ Σ r ✐♥ w ✇✐t❤ ❛ ❢r❡s❤ c ab Σ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ r❡♣❧❛❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❜❧♦❝❦ a ℓ ∈ Σ ∗ ✐♥ w ❜② ❛ ❢r❡s❤ a ℓ ✳ ✭♠❛①✐♠❛❧ ❜❧♦❝❦✿ a ℓ t❤❛t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✮✳ { b , c } ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ { a , c } , { b } ♣❛✐r ❝♦♠♣r❡ss✐♦♥ aaabbcccbbcccbbb aaab ✷ c ✸ b ✷ c ✸ b ✸ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✹ ✴ ✸✺

❲❡ ✇❛♥t t♦ ♣❡r❢♦r♠ ✐t ♦♥ ❛♥❞ ✳ ❖❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛❜❧❡✳ ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ●✐✈❡♥ ❛ ✇♦r❞ w ✿ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✭ Σ ℓ , Σ r ❛r❡ ❞✐s❥♦✐♥t✮ r❡♣❧❛❝❡ ❡❛❝❤ ab ∈ Σ ℓ Σ r ✐♥ w ✇✐t❤ ❛ ❢r❡s❤ c ab Σ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ r❡♣❧❛❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❜❧♦❝❦ a ℓ ∈ Σ ∗ ✐♥ w ❜② ❛ ❢r❡s❤ a ℓ ✳ ✭♠❛①✐♠❛❧ ❜❧♦❝❦✿ a ℓ t❤❛t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✮✳ { b , c } ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ { a , c } , { b } ♣❛✐r ❝♦♠♣r❡ss✐♦♥ aaabbcccbbcccbbb aaabbcccbbcccbbb aaab ✷ c ✸ b ✷ c ✸ b ✸ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✹ ✴ ✸✺

❲❡ ✇❛♥t t♦ ♣❡r❢♦r♠ ✐t ♦♥ ❛♥❞ ✳ ❖❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛❜❧❡✳ ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ●✐✈❡♥ ❛ ✇♦r❞ w ✿ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✭ Σ ℓ , Σ r ❛r❡ ❞✐s❥♦✐♥t✮ r❡♣❧❛❝❡ ❡❛❝❤ ab ∈ Σ ℓ Σ r ✐♥ w ✇✐t❤ ❛ ❢r❡s❤ c ab Σ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ r❡♣❧❛❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❜❧♦❝❦ a ℓ ∈ Σ ∗ ✐♥ w ❜② ❛ ❢r❡s❤ a ℓ ✳ ✭♠❛①✐♠❛❧ ❜❧♦❝❦✿ a ℓ t❤❛t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✮✳ { b , c } ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ { a , c } , { b } ♣❛✐r ❝♦♠♣r❡ss✐♦♥ aaabbcccbbcccbbb aaabbcccbbcccbbb aaab ✷ c ✸ b ✷ c ✸ b ✸ aa d bcc e bcc e bb ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✹ ✴ ✸✺

❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ❈♦♠♣r❡ss✐♦♥ ♦♣❡r❛t✐♦♥s ●✐✈❡♥ ❛ ✇♦r❞ w ✿ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✭ Σ ℓ , Σ r ❛r❡ ❞✐s❥♦✐♥t✮ r❡♣❧❛❝❡ ❡❛❝❤ ab ∈ Σ ℓ Σ r ✐♥ w ✇✐t❤ ❛ ❢r❡s❤ c ab Σ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ r❡♣❧❛❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❜❧♦❝❦ a ℓ ∈ Σ ∗ ✐♥ w ❜② ❛ ❢r❡s❤ a ℓ ✳ ✭♠❛①✐♠❛❧ ❜❧♦❝❦✿ a ℓ t❤❛t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✮✳ { b , c } ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ { a , c } , { b } ♣❛✐r ❝♦♠♣r❡ss✐♦♥ aaabbcccbbcccbbb aaabbcccbbcccbbb aaab ✷ c ✸ b ✷ c ✸ b ✸ aa d bcc e bcc e bb ❲❡ ✇❛♥t t♦ ♣❡r❢♦r♠ ✐t ♦♥ S ( U ) ❛♥❞ S ( V ) ✳ ❖❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ♣❛rt✐❛❧❧② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛❜❧❡✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✹ ✴ ✸✺

P❤❛s❡✿ ♦♥❡ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❧♦♦♣✳ ❙❤♦rt❡♥✐♥❣ ❈♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ✐♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡ ❝♦♠♣r❡ss❡❞ ❛s ❛ ❜❧♦❝❦ ❝♦♥s✐❞❡r❡❞ ❛♥❞ ❝♦♠♣r❡ss❡❞✱ ♦r ♦♥❡ ♦❢ t❤❡♠ ✇❛s ❝♦♠♣r❡ss❡❞ ❡❛r❧✐❡r Pr❡❧✐♠✐♥❛r✐❡s✿ ❡①♣❧✐❝✐t ✇♦r❞ ❈❤❡❝❦✐♥❣ ❡q✉❛❧✐t② ♦❢ t✇♦ ❡①♣❧✐❝✐t ✇♦r❞s ❘❡q✉✐r❡✿ t✇♦ ✇♦r❞s u , v t♦ ❜❡ t❡st❡❞ ❢♦r ❡q✉❛❧✐t② ✶✿ ✇❤✐❧❡ | u | ❃ ✶ ♦r | v | ❃ ✶ ❞♦ Σ ← ❧❡tt❡rs ✐♥ u , v ✷✿ ♣❡r❢♦r♠ Σ ✲❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ✸✿ ✇❤✐❧❡ s♦♠❡ ♣❛✐r ✐♥ Σ ✷ ✇❛s ♥♦t ❝♦♥s✐❞❡r❡❞ ❞♦ ✹✿ ❣✉❡ss ♣❛rt✐t✐♦♥ ♦❢ Σ t♦ (Σ ℓ , Σ r ) ✺✿ ♣❡r❢♦r♠ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✻✿ ✼✿ t❡st ❡q✉❛❧✐t② ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✺ ✴ ✸✺

❙❤♦rt❡♥✐♥❣ ❈♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ✐♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡ ❝♦♠♣r❡ss❡❞ ❛s ❛ ❜❧♦❝❦ ❝♦♥s✐❞❡r❡❞ ❛♥❞ ❝♦♠♣r❡ss❡❞✱ ♦r ♦♥❡ ♦❢ t❤❡♠ ✇❛s ❝♦♠♣r❡ss❡❞ ❡❛r❧✐❡r Pr❡❧✐♠✐♥❛r✐❡s✿ ❡①♣❧✐❝✐t ✇♦r❞ ❈❤❡❝❦✐♥❣ ❡q✉❛❧✐t② ♦❢ t✇♦ ❡①♣❧✐❝✐t ✇♦r❞s ❘❡q✉✐r❡✿ t✇♦ ✇♦r❞s u , v t♦ ❜❡ t❡st❡❞ ❢♦r ❡q✉❛❧✐t② ✶✿ ✇❤✐❧❡ | u | ❃ ✶ ♦r | v | ❃ ✶ ❞♦ Σ ← ❧❡tt❡rs ✐♥ u , v ✷✿ ♣❡r❢♦r♠ Σ ✲❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ✸✿ ✇❤✐❧❡ s♦♠❡ ♣❛✐r ✐♥ Σ ✷ ✇❛s ♥♦t ❝♦♥s✐❞❡r❡❞ ❞♦ ✹✿ ❣✉❡ss ♣❛rt✐t✐♦♥ ♦❢ Σ t♦ (Σ ℓ , Σ r ) ✺✿ ♣❡r❢♦r♠ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✻✿ ✼✿ t❡st ❡q✉❛❧✐t② P❤❛s❡✿ ♦♥❡ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❧♦♦♣✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✺ ✴ ✸✺

Pr❡❧✐♠✐♥❛r✐❡s✿ ❡①♣❧✐❝✐t ✇♦r❞ ❈❤❡❝❦✐♥❣ ❡q✉❛❧✐t② ♦❢ t✇♦ ❡①♣❧✐❝✐t ✇♦r❞s ❘❡q✉✐r❡✿ t✇♦ ✇♦r❞s u , v t♦ ❜❡ t❡st❡❞ ❢♦r ❡q✉❛❧✐t② ✶✿ ✇❤✐❧❡ | u | ❃ ✶ ♦r | v | ❃ ✶ ❞♦ Σ ← ❧❡tt❡rs ✐♥ u , v ✷✿ ♣❡r❢♦r♠ Σ ✲❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ✸✿ ✇❤✐❧❡ s♦♠❡ ♣❛✐r ✐♥ Σ ✷ ✇❛s ♥♦t ❝♦♥s✐❞❡r❡❞ ❞♦ ✹✿ ❣✉❡ss ♣❛rt✐t✐♦♥ ♦❢ Σ t♦ (Σ ℓ , Σ r ) ✺✿ ♣❡r❢♦r♠ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✻✿ ✼✿ t❡st ❡q✉❛❧✐t② P❤❛s❡✿ ♦♥❡ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❧♦♦♣✳ ❙❤♦rt❡♥✐♥❣ ❈♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ab ✐♥ u , v ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡ a = b ❝♦♠♣r❡ss❡❞ ❛s ❛ ❜❧♦❝❦ a � = b ❝♦♥s✐❞❡r❡❞ ❛♥❞ ❝♦♠♣r❡ss❡❞✱ ♦r ♦♥❡ ♦❢ t❤❡♠ ✇❛s ❝♦♠♣r❡ss❡❞ ❡❛r❧✐❡r ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✺ ✴ ✸✺

✐t ✐s ❢r♦♠ t❤❡ s✉❜st✐t✉t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡✿ ❖❑✱ s♦❧✉t✐♦♥ ❝❤❛♥❣❡s ♣❛rt✐❛❧❧② ❤❡r❡ ❛♥❞ t❤❡r❡✿ ❥✉st ♣♦♣ t❤❡ ♣r♦❜❧❡♠❛t✐❝ ❧❡tt❡r ♦✉t P❛✐r❈♦♠♣r❡ss✐♦♥ ✶✿ ❢♦r ❞♦ ❧❡t ✿ ✜rst ❧❡tt❡r ♦❢ ●✉❡ss ✷✿ ✐❢ t❤❡♥ ✸✿ r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ ❜② P♦♣ ✹✿ ✐❢ t❤❡♥ ●✉❡ss ✺✿ r❡♠♦✈❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✻✿ ❧❡t ✿ ❧❛st ✳ ✳ ✳ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ ✼✿ ✽✿ ♣❡r❢♦r♠ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ P❛✐r ❈♦♠♣r❡ss✐♦♥ ♦♥ ✇♦r❞ ❡q✉❛t✐♦♥ ■♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ S ( U ) ♦r S ( V ) ✿ ♣❛✐r ✐s ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥✿ ❖❑✱ ✇❡ r❡♣❧❛❝❡ ✐t ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✻ ✴ ✸✺

♣❛rt✐❛❧❧② ❤❡r❡ ❛♥❞ t❤❡r❡✿ ❥✉st ♣♦♣ t❤❡ ♣r♦❜❧❡♠❛t✐❝ ❧❡tt❡r ♦✉t P❛✐r❈♦♠♣r❡ss✐♦♥ ✶✿ ❢♦r ❞♦ ❧❡t ✿ ✜rst ❧❡tt❡r ♦❢ ●✉❡ss ✷✿ ✐❢ t❤❡♥ ✸✿ r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ ❜② P♦♣ ✹✿ ✐❢ t❤❡♥ ●✉❡ss ✺✿ r❡♠♦✈❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✻✿ ❧❡t ✿ ❧❛st ✳ ✳ ✳ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ ✼✿ ✽✿ ♣❡r❢♦r♠ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ P❛✐r ❈♦♠♣r❡ss✐♦♥ ♦♥ ✇♦r❞ ❡q✉❛t✐♦♥ ■♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ S ( U ) ♦r S ( V ) ✿ ♣❛✐r ✐s ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥✿ ❖❑✱ ✇❡ r❡♣❧❛❝❡ ✐t ✐t ✐s ❢r♦♠ t❤❡ s✉❜st✐t✉t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡✿ ❖❑✱ s♦❧✉t✐♦♥ ❝❤❛♥❣❡s ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✻ ✴ ✸✺

P❛✐r❈♦♠♣r❡ss✐♦♥ ✶✿ ❢♦r ❞♦ ❧❡t ✿ ✜rst ❧❡tt❡r ♦❢ ●✉❡ss ✷✿ ✐❢ t❤❡♥ ✸✿ r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ ❜② P♦♣ ✹✿ ✐❢ t❤❡♥ ●✉❡ss ✺✿ r❡♠♦✈❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✻✿ ❧❡t ✿ ❧❛st ✳ ✳ ✳ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ ✼✿ ✽✿ ♣❡r❢♦r♠ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ P❛✐r ❈♦♠♣r❡ss✐♦♥ ♦♥ ✇♦r❞ ❡q✉❛t✐♦♥ ■♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ S ( U ) ♦r S ( V ) ✿ ♣❛✐r ✐s ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥✿ ❖❑✱ ✇❡ r❡♣❧❛❝❡ ✐t ✐t ✐s ❢r♦♠ t❤❡ s✉❜st✐t✉t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡✿ ❖❑✱ s♦❧✉t✐♦♥ ❝❤❛♥❣❡s ♣❛rt✐❛❧❧② ❤❡r❡ ❛♥❞ t❤❡r❡✿ ❥✉st ♣♦♣ t❤❡ ♣r♦❜❧❡♠❛t✐❝ ❧❡tt❡r ♦✉t ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✻ ✴ ✸✺

P❛✐r ❈♦♠♣r❡ss✐♦♥ ♦♥ ✇♦r❞ ❡q✉❛t✐♦♥ ■♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ S ( U ) ♦r S ( V ) ✿ ♣❛✐r ✐s ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥✿ ❖❑✱ ✇❡ r❡♣❧❛❝❡ ✐t ✐t ✐s ❢r♦♠ t❤❡ s✉❜st✐t✉t✐♦♥ ❢♦r ❛ ✈❛r✐❛❜❧❡✿ ❖❑✱ s♦❧✉t✐♦♥ ❝❤❛♥❣❡s ♣❛rt✐❛❧❧② ❤❡r❡ ❛♥❞ t❤❡r❡✿ ❥✉st ♣♦♣ t❤❡ ♣r♦❜❧❡♠❛t✐❝ ❧❡tt❡r ♦✉t P❛✐r❈♦♠♣r❡ss✐♦♥ ✶✿ ❢♦r X ∈ X ❞♦ ❧❡t b ✿ ✜rst ❧❡tt❡r ♦❢ S ( X ) ⊲ ●✉❡ss ✷✿ ✐❢ b ∈ Σ r t❤❡♥ ✸✿ r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ❜② bX ⊲ P♦♣ ✹✿ ✐❢ S ( X ) = ǫ t❤❡♥ ⊲ ●✉❡ss ✺✿ r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✻✿ ❧❡t a ✿ ❧❛st ✳ ✳ ✳ ⊲ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ Σ ℓ ✼✿ ✽✿ ♣❡r❢♦r♠ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✻ ✴ ✸✺

❇❧♦❝❦ ❈♦♠♣r❡ss✐♦♥ ❇❧♦❝❦❈♦♠♣r❡ss✐♦♥ ✶✿ ❢♦r X ∈ X ❞♦ ❧❡t S ( X ) = a ℓ wb r ⊲ ●✉❡ss ✷✿ r❡♣❧❛❝❡ X ✇✐t❤ a ℓ Xb r ✸✿ ✐❢ S ( X ) = ǫ t❤❡♥ ⊲ ●✉❡ss ✹✿ r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✺✿ ✻✿ ♣❡r❢♦r♠ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ♦♥ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✼ ✴ ✸✺

❆ ♣❤❛s❡ ✐s ♦♥❡ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❧♦♦♣ ❚❤❡ ❛❧❣♦r✐t❤♠ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ✶✿ ✇❤✐❧❡ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ ♥♦♥tr✐✈✐❛❧ ❞♦ Σ ← ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✷✿ ♣❡r❢♦r♠ Σ ✲❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ✸✿ ✇❤✐❧❡ s♦♠❡ ♣❛✐r ✐♥ Σ ✷ ✇❛s ♥♦t ❝♦♥s✐❞❡r❡❞ ❞♦ ✹✿ ❣✉❡ss ♣❛rt✐t✐♦♥ ♦❢ Σ t♦ (Σ ℓ , Σ r ) ⊲ ■♠♣♦rt❛♥t ✺✿ ♣❡r❢♦r♠ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✻✿ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✽ ✴ ✸✺

❚❤❡ ❛❧❣♦r✐t❤♠ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ✶✿ ✇❤✐❧❡ s✐❞❡s ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛r❡ ♥♦♥tr✐✈✐❛❧ ❞♦ Σ ← ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✷✿ ♣❡r❢♦r♠ Σ ✲❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ✸✿ ✇❤✐❧❡ s♦♠❡ ♣❛✐r ✐♥ Σ ✷ ✇❛s ♥♦t ❝♦♥s✐❞❡r❡❞ ❞♦ ✹✿ ❣✉❡ss ♣❛rt✐t✐♦♥ ♦❢ Σ t♦ (Σ ℓ , Σ r ) ⊲ ■♠♣♦rt❛♥t ✺✿ ♣❡r❢♦r♠ (Σ ℓ , Σ r ) ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ✻✿ ❆ ♣❤❛s❡ ✐s ♦♥❡ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❧♦♦♣ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✽ ✴ ✸✺

■♥ ◆▲✐♥❙P❆❈❊ ✇❡ ❝❛♥ ❛♥❛❧②s❡ ♦♥❧② ✏❣♦♦❞ ❝❤♦✐❝❡s✑✿ ✐❢ ✇❡ ❡①❝❡❡❞ t❤❡ s♣❛❝❡ t❤❡♥ ✇❡ r❡❥❡❝t✳ ◆♦t❡s ♦♥ ❛♥❛❧②s✐s ❆ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣r♦❝❡❞✉r❡ ✐s✿ s♦✉♥❞ tr❛♥s❢♦r♠s s❛t✐s✜❛❜❧❡ t♦ s❛t✐s✜❛❜❧❡✱ r❡❣❛r❞❧❡ss ♦❢ ❝❤♦✐❝❡s ❝♦♠♣❧❡t❡ ❣✐✈❡♥ ❛ s❛t✐s✜❛❜❧❡ ✐♥st❛♥❝❡ ✐t tr❛♥s❢♦r♠s ✐t t♦ ❛ s❛t✐s✜❛❜❧❡ ♦♥❡ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✾ ✴ ✸✺

◆♦t❡s ♦♥ ❛♥❛❧②s✐s ❆ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣r♦❝❡❞✉r❡ ✐s✿ s♦✉♥❞ tr❛♥s❢♦r♠s s❛t✐s✜❛❜❧❡ t♦ s❛t✐s✜❛❜❧❡✱ r❡❣❛r❞❧❡ss ♦❢ ❝❤♦✐❝❡s ❝♦♠♣❧❡t❡ ❣✐✈❡♥ ❛ s❛t✐s✜❛❜❧❡ ✐♥st❛♥❝❡ ✐t tr❛♥s❢♦r♠s ✐t t♦ ❛ s❛t✐s✜❛❜❧❡ ♦♥❡ ■♥ ◆▲✐♥❙P❆❈❊ ✇❡ ❝❛♥ ❛♥❛❧②s❡ ♦♥❧② ✏❣♦♦❞ ❝❤♦✐❝❡s✑✿ ✐❢ ✇❡ ❡①❝❡❡❞ t❤❡ s♣❛❝❡ t❤❡♥ ✇❡ r❡❥❡❝t✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✾ ✴ ✸✺

❚❤❡♥ st✐❝❦ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥✳ ❈♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ●✐✈❡♥ t❤❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ❝♦rr❡s♣♦♥❞ t♦ ✱ ✐❢ t❤❡② ❛r❡ ❞♦♥❡ ❛s ✐❢ t❤❡ ❛❧❣♦r✐t❤♠ ❦♥❡✇ ✳ t❤❡ ✜rst✴❧❛st ❧❡tt❡r ♦❢ ❧❡♥❣t❤ ♦❢ ✲♣r❡✜①✴s✉✣① ✇❤❡t❤❡r ✳ ◆♦t✿ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❙♦❧✉t✐♦♥s ❙♦❧✉t✐♦♥ ■❢ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✱ t❤❡r❡ ✐s ♦♥❡ ♦✈❡r Σ ❂ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥✿ ▼❛♣ ❛❧❧ ❧❡tt❡rs ♦✉ts✐❞❡ Σ t♦ ❛ ✜①❡❞ ♦♥❡ ✐♥ Σ ✳ ❉♦♥❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✵ ✴ ✸✺

❈♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ●✐✈❡♥ t❤❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ❝♦rr❡s♣♦♥❞ t♦ ✱ ✐❢ t❤❡② ❛r❡ ❞♦♥❡ ❛s ✐❢ t❤❡ ❛❧❣♦r✐t❤♠ ❦♥❡✇ ✳ t❤❡ ✜rst✴❧❛st ❧❡tt❡r ♦❢ ❧❡♥❣t❤ ♦❢ ✲♣r❡✜①✴s✉✣① ✇❤❡t❤❡r ✳ ◆♦t✿ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❙♦❧✉t✐♦♥s ❙♦❧✉t✐♦♥ ■❢ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✱ t❤❡r❡ ✐s ♦♥❡ ♦✈❡r Σ ❂ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥✿ ▼❛♣ ❛❧❧ ❧❡tt❡rs ♦✉ts✐❞❡ Σ t♦ ❛ ✜①❡❞ ♦♥❡ ✐♥ Σ ✳ ❉♦♥❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ❚❤❡♥ st✐❝❦ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✵ ✴ ✸✺

t❤❡ ✜rst✴❧❛st ❧❡tt❡r ♦❢ ❧❡♥❣t❤ ♦❢ ✲♣r❡✜①✴s✉✣① ✇❤❡t❤❡r ✳ ◆♦t✿ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❙♦❧✉t✐♦♥s ❙♦❧✉t✐♦♥ ■❢ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✱ t❤❡r❡ ✐s ♦♥❡ ♦✈❡r Σ ❂ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥✿ ▼❛♣ ❛❧❧ ❧❡tt❡rs ♦✉ts✐❞❡ Σ t♦ ❛ ✜①❡❞ ♦♥❡ ✐♥ Σ ✳ ❉♦♥❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ❚❤❡♥ st✐❝❦ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥✳ ❈♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ●✐✈❡♥ S t❤❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ❝♦rr❡s♣♦♥❞ t♦ S ✱ ✐❢ t❤❡② ❛r❡ ❞♦♥❡ ❛s ✐❢ t❤❡ ❛❧❣♦r✐t❤♠ ❦♥❡✇ S ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✵ ✴ ✸✺

❙♦❧✉t✐♦♥s ❙♦❧✉t✐♦♥ ■❢ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✱ t❤❡r❡ ✐s ♦♥❡ ♦✈❡r Σ ❂ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥✿ ▼❛♣ ❛❧❧ ❧❡tt❡rs ♦✉ts✐❞❡ Σ t♦ ❛ ✜①❡❞ ♦♥❡ ✐♥ Σ ✳ ❉♦♥❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ❚❤❡♥ st✐❝❦ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ s♦❧✉t✐♦♥✳ ❈♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ●✐✈❡♥ S t❤❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ❝♦rr❡s♣♦♥❞ t♦ S ✱ ✐❢ t❤❡② ❛r❡ ❞♦♥❡ ❛s ✐❢ t❤❡ ❛❧❣♦r✐t❤♠ ❦♥❡✇ S ✳ t❤❡ ✜rst✴❧❛st ❧❡tt❡r ♦❢ S ( X ) ❧❡♥❣t❤ ♦❢ a ✲♣r❡✜①✴s✉✣① ✇❤❡t❤❡r S ( X ) = ǫ ✳ ◆♦t✿ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ♣❛rt✐t✐♦♥✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✵ ✴ ✸✺

❚♦ ❜❡ ♠♦r❡ ♣r❡❝✐s❡✿ ■❢ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ ❛❢t❡r P❛✐r❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ♣♦♣♣❡❞ ❧❡tt❡rs ❢r♦♠ ❛♥❞ ♣❡r❢♦r♠✐♥❣ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ ✳ Pr♦♦❢✳ ❙♦✉♥❞♥❡ss✿ ❧❡t ❤❛✈❡ ❛ s♦❧✉t✐♦♥ ✳ ❈r❡❛t❡ ✿ t❛❦❡ ✱ r❡♣❧❛❝❡ ✇✐t❤ ❛♥❞ r❡❛tt❛❝❤ t❤❡ ♣♦♣♣❡❞ ❧❡tt❡rs❀ t❤✐s ✐s ✳ ❚❤❡♥ ✐s ✇✐t❤ r❡♣❧❛❝❡❞ ✇✐t❤ ✳ ❈♦♠♣❧❡t❡♥❡ss✿ ❢♦r t❤♦s❡ ❝❤♦✐❝❡s ❛❢t❡r ♣♦♣♣✐♥❣ ❡❛❝❤ ✐s ❡✐t❤❡r ✇✐t❤✐♥ ✈❛r✐❛❜❧❡ ♦r ♦✉ts✐❞❡ ✐t✳ ❙♦ t❤❡ ❝♦♠♣r❡ss✐♦♥ ✇♦r❦s✳ P❛✐r ❈♦♠♣r❡ss✐♦♥✿ ❝♦rr❡❝t♥❡ss ▲❡♠♠❛ P❛✐r❈♦♠♠♣r❡ss✐♦♥ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✶ ✴ ✸✺

Pr♦♦❢✳ ❙♦✉♥❞♥❡ss✿ ❧❡t ❤❛✈❡ ❛ s♦❧✉t✐♦♥ ✳ ❈r❡❛t❡ ✿ t❛❦❡ ✱ r❡♣❧❛❝❡ ✇✐t❤ ❛♥❞ r❡❛tt❛❝❤ t❤❡ ♣♦♣♣❡❞ ❧❡tt❡rs❀ t❤✐s ✐s ✳ ❚❤❡♥ ✐s ✇✐t❤ r❡♣❧❛❝❡❞ ✇✐t❤ ✳ ❈♦♠♣❧❡t❡♥❡ss✿ ❢♦r t❤♦s❡ ❝❤♦✐❝❡s ❛❢t❡r ♣♦♣♣✐♥❣ ❡❛❝❤ ✐s ❡✐t❤❡r ✇✐t❤✐♥ ✈❛r✐❛❜❧❡ ♦r ♦✉ts✐❞❡ ✐t✳ ❙♦ t❤❡ ❝♦♠♣r❡ss✐♦♥ ✇♦r❦s✳ P❛✐r ❈♦♠♣r❡ss✐♦♥✿ ❝♦rr❡❝t♥❡ss ▲❡♠♠❛ P❛✐r❈♦♠♠♣r❡ss✐♦♥ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡✳ ❚♦ ❜❡ ♠♦r❡ ♣r❡❝✐s❡✿ ■❢ U = V ❤❛s ❛ s♦❧✉t✐♦♥ S t❤❡♥ ❛❢t❡r P❛✐r❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ♣♦♣♣❡❞ ❧❡tt❡rs ❢r♦♠ S ( X ) ❛♥❞ ♣❡r❢♦r♠✐♥❣ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ S ( X ) ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✶ ✴ ✸✺

❈♦♠♣❧❡t❡♥❡ss✿ ❢♦r t❤♦s❡ ❝❤♦✐❝❡s ❛❢t❡r ♣♦♣♣✐♥❣ ❡❛❝❤ ✐s ❡✐t❤❡r ✇✐t❤✐♥ ✈❛r✐❛❜❧❡ ♦r ♦✉ts✐❞❡ ✐t✳ ❙♦ t❤❡ ❝♦♠♣r❡ss✐♦♥ ✇♦r❦s✳ P❛✐r ❈♦♠♣r❡ss✐♦♥✿ ❝♦rr❡❝t♥❡ss ▲❡♠♠❛ P❛✐r❈♦♠♠♣r❡ss✐♦♥ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡✳ ❚♦ ❜❡ ♠♦r❡ ♣r❡❝✐s❡✿ ■❢ U = V ❤❛s ❛ s♦❧✉t✐♦♥ S t❤❡♥ ❛❢t❡r P❛✐r❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ♣♦♣♣❡❞ ❧❡tt❡rs ❢r♦♠ S ( X ) ❛♥❞ ♣❡r❢♦r♠✐♥❣ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ S ( X ) ✳ Pr♦♦❢✳ ❙♦✉♥❞♥❡ss✿ ❧❡t U ′ = V ′ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ S ′ ✳ ❈r❡❛t❡ S ✿ t❛❦❡ S ′ ( X ) ✱ r❡♣❧❛❝❡ c ab ✇✐t❤ ab ❛♥❞ r❡❛tt❛❝❤ t❤❡ ♣♦♣♣❡❞ ❧❡tt❡rs❀ t❤✐s ✐s S ( X ) ✳ ❚❤❡♥ S ( U ) ✐s S ′ ( U ′ ) ✇✐t❤ c ab r❡♣❧❛❝❡❞ ✇✐t❤ ab ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✶ ✴ ✸✺

P❛✐r ❈♦♠♣r❡ss✐♦♥✿ ❝♦rr❡❝t♥❡ss ▲❡♠♠❛ P❛✐r❈♦♠♠♣r❡ss✐♦♥ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡✳ ❚♦ ❜❡ ♠♦r❡ ♣r❡❝✐s❡✿ ■❢ U = V ❤❛s ❛ s♦❧✉t✐♦♥ S t❤❡♥ ❛❢t❡r P❛✐r❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ♣♦♣♣❡❞ ❧❡tt❡rs ❢r♦♠ S ( X ) ❛♥❞ ♣❡r❢♦r♠✐♥❣ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ♦♥ S ( X ) ✳ Pr♦♦❢✳ ❙♦✉♥❞♥❡ss✿ ❧❡t U ′ = V ′ ❤❛✈❡ ❛ s♦❧✉t✐♦♥ S ′ ✳ ❈r❡❛t❡ S ✿ t❛❦❡ S ′ ( X ) ✱ r❡♣❧❛❝❡ c ab ✇✐t❤ ab ❛♥❞ r❡❛tt❛❝❤ t❤❡ ♣♦♣♣❡❞ ❧❡tt❡rs❀ t❤✐s ✐s S ( X ) ✳ ❚❤❡♥ S ( U ) ✐s S ′ ( U ′ ) ✇✐t❤ c ab r❡♣❧❛❝❡❞ ✇✐t❤ ab ✳ ❈♦♠♣❧❡t❡♥❡ss✿ ❢♦r t❤♦s❡ ❝❤♦✐❝❡s ❛❢t❡r ♣♦♣♣✐♥❣ ❡❛❝❤ ab ✐s ❡✐t❤❡r ✇✐t❤✐♥ ✈❛r✐❛❜❧❡ ♦r ♦✉ts✐❞❡ ✐t✳ ❙♦ t❤❡ ❝♦♠♣r❡ss✐♦♥ ✇♦r❦s✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✶ ✴ ✸✺

❇❧♦❝❦ ❈♦♠♣r❡ss✐♦♥✿ ❝♦rr❡❝t♥❡ss ▲❡♠♠❛ ❇❧♦❝❦❈♦♠♣r❡ss✐♦♥ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡✳ ❚♦ ❜❡ ♠♦r❡ ♣r❡❝✐s❡✿ ■❢ U = V ❤❛s ❛ s♦❧✉t✐♦♥ S t❤❡♥ ❛❢t❡r ❇❧♦❝❦❈♦♠♣r❡ss✐♦♥ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ♣♦♣♣❡❞ ❧❡tt❡rs ❢r♦♠ S ( X ) ❛♥❞ ♣❡r❢♦r♠✐♥❣ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ♦♥ S ( X ) ✳ Pr♦♦❢✳ Pr♦♦❢ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ P❛✐r ❈♦♠♣r❡ss✐♦♥✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✷ ✴ ✸✺

Pr♦♦❢✿ ❛s ✐♥ t❤❡ ✇♦r❞ ❝❛s❡ ❈♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ✐♥ t❤❡ s♦❧✉t✐♦♥ ✇♦r❞✿ ❝♦♠♣r❡ss❡❞ ❛s ❛ ❜❧♦❝❦ ❝♦♥s✐❞❡r❡❞ ❛♥❞ ❝♦♠♣r❡ss❡❞✱ ♦r ♦♥❡ ♦❢ t❤❡♠ ✇❛s ❝♦♠♣r❡ss❡❞ ❡❛r❧✐❡r ❙❤♦rt❡♥✐♥❣ ♣r♦♣❡rt② ▲❡♠♠❛ ❋♦r S ♦✈❡r Σ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❤♦✐❝❡s ❛❢t❡r ♦♥❡ ♣❤❛s❡ ❛♠♦♥❣ ❡❛❝❤ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧❡tt❡rs ✐♥ S ( U ) ❛t ❧❡❛st ♦♥❡ ✐s ❝♦♠♣r❡ss❡❞✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✸ ✴ ✸✺

❙❤♦rt❡♥✐♥❣ ♣r♦♣❡rt② ▲❡♠♠❛ ❋♦r S ♦✈❡r Σ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❤♦✐❝❡s ❛❢t❡r ♦♥❡ ♣❤❛s❡ ❛♠♦♥❣ ❡❛❝❤ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❧❡tt❡rs ✐♥ S ( U ) ❛t ❧❡❛st ♦♥❡ ✐s ❝♦♠♣r❡ss❡❞✳ Pr♦♦❢✿ ❛s ✐♥ t❤❡ ✇♦r❞ ❝❛s❡ ❈♦♥s✐❞❡r ❝♦♥s❡❝✉t✐✈❡ ab ✐♥ t❤❡ s♦❧✉t✐♦♥ ✇♦r❞✿ a = b ❝♦♠♣r❡ss❡❞ ❛s ❛ ❜❧♦❝❦ a � = b ❝♦♥s✐❞❡r❡❞ ❛♥❞ ❝♦♠♣r❡ss❡❞✱ ♦r ♦♥❡ ♦❢ t❤❡♠ ✇❛s ❝♦♠♣r❡ss❡❞ ❡❛r❧✐❡r ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✸ ✴ ✸✺

▲❡♠♠❛ ❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ s♣❛❝❡ ❧✐♥❡❛r ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ st♦r❡❞ ❡q✉❛t✐♦♥✳ ❙♣❛❝❡ ❝♦♥s✉♠♣t✐♦♥✿ ✐♥✐t✐❛❧ ♥♦t❡s ❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ▲♦♥❣ ❜❧♦❝❦s ❛r❡ ❛ ♣r♦❜❧❡♠❀ ❛ ✜① ✐s ❛❧r❡❛❞② ❦♥♦✇♥✿ ✇❡ ❞♦ ♥♦t ❣✉❡ss ❡①♣❧✐❝✐t ❧❡♥❣t❤s✱ r❛t❤❡r ❞❡♥♦t❡ t❤❡♠ ❛s ✐♥t❡❣❡r ✈❛r✐❛❜❧❡s ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ❜❧♦❝❦s❀ ❧❡♥❣t❤s ❞❡♣❡♥❞s ♦♥ t❤♦s❡ ✈❛r✐❛❜❧❡s ✇❡ ✐❞❡♥t✐❢② t❤❡ s❛♠❡ ❧❡♥❣t❤s✿ ❡q✉❛❧✐t✐❡s ♦❢ ❧✐♥❡❛r ❡①♣r❡ss✐♦♥s ✐♥ t❡r♠s ♦❢ ✈❛r✐❛❜❧❡s ✈❡r✐❢② t❤❡ s②st❡♠ ♦❢ s✉❝❤ ✐♥t❡❣❡r✲❡q✉❛t✐♦♥s ❝♦♠♣r❡ss ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✹ ✴ ✸✺

❙♣❛❝❡ ❝♦♥s✉♠♣t✐♦♥✿ ✐♥✐t✐❛❧ ♥♦t❡s ❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ▲♦♥❣ ❜❧♦❝❦s ❛r❡ ❛ ♣r♦❜❧❡♠❀ ❛ ✜① ✐s ❛❧r❡❛❞② ❦♥♦✇♥✿ ✇❡ ❞♦ ♥♦t ❣✉❡ss ❡①♣❧✐❝✐t ❧❡♥❣t❤s✱ r❛t❤❡r ❞❡♥♦t❡ t❤❡♠ ❛s ✐♥t❡❣❡r ✈❛r✐❛❜❧❡s ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ❜❧♦❝❦s❀ ❧❡♥❣t❤s ❞❡♣❡♥❞s ♦♥ t❤♦s❡ ✈❛r✐❛❜❧❡s ✇❡ ✐❞❡♥t✐❢② t❤❡ s❛♠❡ ❧❡♥❣t❤s✿ ❡q✉❛❧✐t✐❡s ♦❢ ❧✐♥❡❛r ❡①♣r❡ss✐♦♥s ✐♥ t❡r♠s ♦❢ ✈❛r✐❛❜❧❡s ✈❡r✐❢② t❤❡ s②st❡♠ ♦❢ s✉❝❤ ✐♥t❡❣❡r✲❡q✉❛t✐♦♥s ❝♦♠♣r❡ss ▲❡♠♠❛ ❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ s♣❛❝❡ ❧✐♥❡❛r ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ st♦r❡❞ ❡q✉❛t✐♦♥✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✹ ✴ ✸✺

❙♣❛❝❡ ❝♦♥s✉♠♣t✐♦♥✿ ✐♥✐t✐❛❧ ♥♦t❡s ❍✉✛♠❛♥ ❝♦❞✐♥❣ ❲❡ ♥❡❡❞ t♦ r❡❝❛❧❝✉❧❛t❡ ❍✉✛♠❛♥ ❝♦❞✐♥❣✳ ✇❡ ❜✉✐❧❞ ❛ ❧❛❜❡❧❧❡❞ tr❡❡✱ ❧❛❜❡❧s t♦ ❛ ❧❡❛❢ ❣✐✈❡ t❤❡ ❡♥❝♦❞✐♥❣ ❝❛❧❝✉❧❛t❡ ❢r❡q✉❡♥❝✐❡s ♠❡r❣❡ t✇♦ ❧❡❛st ❝♦♠♠♦♥ s②♠❜♦❧s ❝r❡❛t❡ ❛ ♥❡✇ ♥♦❞❡ ✇✐t❤ t✇♦ ❡❞❣❡s t♦ t❤♦s❡ s②♠❜♦❧s✱ ❧❛❜❡❧❧❡❞ ✇✐t❤ ✵ ❛♥❞ ✶ ❚❤✐s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ s♣❛❝❡ ❧✐♥❡❛r ✐♥ t❤❡ ✐♥♣✉t✳ ❙♣❛❝❡ ❜♦✉♥❞ ✐s ❖❑✱ ❥✉st ✏❞❡❧❛②❡❞✑ ❜② ♦♥❡ st❡♣✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✺ ✴ ✸✺

❉❡✜♥✐t✐♦♥ ✭❉❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧✮ ❆♥ ✐♥t❡r✈❛❧ ♦❢ ♣♦s✐t✐♦♥s ✐♥ t❤❡ ✐♥♣✉t ❡q✉❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❞❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧ ✭❞❡♣✐♥t✮❀ ❜❛s✐❝ ❞❡♣✐♥t ❤❛s ✶ ♣♦s✐t✐♦♥✳ ❲❡ ❛ss♦❝✐❛t❡ ❛ ❞❡♣✐♥t t♦ ❡❛❝❤ s②♠❜♦❧ ✐♥ t❤❡ ❡q✉❛t✐♦♥❀ ❞❡♣ ✳ ✿ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛❝t♦rs ♦❢ ✐♥✐t✐❛❧ ❡q✉❛t✐♦♥ ❛r❡ ❡q✉❛❧✿ ✭❛s s❡q✉❡♥❝❡ ♦❢ ❧❡tt❡rs ❛♥❞ ✈❛r✐❛❜❧❡s✮ ✇❡ t❛❦❡ t❤❡✐r ✉♥✐♦♥s ✭♦♥❧② ✇❤❡♥ r❡s✉❧t ✐s ❛♥ ✐♥t❡r✈❛❧✮ ❞❡♣ ❞❡♣ ✉s❡ ❤❛✈❡ st❛♥❞❛r❞ ♠❡❛♥✐♥❣ ✭❞❡♣ ❞❡♣ ✮ ❉❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧ ❋♦r ❛ ❧❡tt❡r ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❞❡✜♥❡ ❛ ❢❛❝t♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✱ ♦♥ ✇❤✐❝❤ ✐t ❞❡♣❡♥❞s✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✻ ✴ ✸✺

✿ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛❝t♦rs ♦❢ ✐♥✐t✐❛❧ ❡q✉❛t✐♦♥ ❛r❡ ❡q✉❛❧✿ ✭❛s s❡q✉❡♥❝❡ ♦❢ ❧❡tt❡rs ❛♥❞ ✈❛r✐❛❜❧❡s✮ ✇❡ t❛❦❡ t❤❡✐r ✉♥✐♦♥s ✭♦♥❧② ✇❤❡♥ r❡s✉❧t ✐s ❛♥ ✐♥t❡r✈❛❧✮ ❞❡♣ ❞❡♣ ✉s❡ ❤❛✈❡ st❛♥❞❛r❞ ♠❡❛♥✐♥❣ ✭❞❡♣ ❞❡♣ ✮ ❉❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧ ❋♦r ❛ ❧❡tt❡r ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❞❡✜♥❡ ❛ ❢❛❝t♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✱ ♦♥ ✇❤✐❝❤ ✐t ❞❡♣❡♥❞s✳ ❉❡✜♥✐t✐♦♥ ✭❉❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧✮ ❆♥ ✐♥t❡r✈❛❧ ♦❢ ♣♦s✐t✐♦♥s ✐♥ t❤❡ ✐♥♣✉t ❡q✉❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❞❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧ ✭❞❡♣✐♥t✮❀ ❜❛s✐❝ ❞❡♣✐♥t ❤❛s ✶ ♣♦s✐t✐♦♥✳ ❲❡ ❛ss♦❝✐❛t❡ ❛ ❞❡♣✐♥t t♦ ❡❛❝❤ s②♠❜♦❧ ✐♥ t❤❡ ❡q✉❛t✐♦♥❀ D = ❞❡♣ ( i ) ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✻ ✴ ✸✺

❉❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧ ❋♦r ❛ ❧❡tt❡r ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❞❡✜♥❡ ❛ ❢❛❝t♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✱ ♦♥ ✇❤✐❝❤ ✐t ❞❡♣❡♥❞s✳ ❉❡✜♥✐t✐♦♥ ✭❉❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧✮ ❆♥ ✐♥t❡r✈❛❧ ♦❢ ♣♦s✐t✐♦♥s ✐♥ t❤❡ ✐♥♣✉t ❡q✉❛t✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❞❡♣❡♥❞❡♥❝② ✐♥t❡r✈❛❧ ✭❞❡♣✐♥t✮❀ ❜❛s✐❝ ❞❡♣✐♥t ❤❛s ✶ ♣♦s✐t✐♦♥✳ ❲❡ ❛ss♦❝✐❛t❡ ❛ ❞❡♣✐♥t t♦ ❡❛❝❤ s②♠❜♦❧ ✐♥ t❤❡ ❡q✉❛t✐♦♥❀ D = ❞❡♣ ( i ) ✳ D ∼ D ′ ✿ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢❛❝t♦rs ♦❢ ✐♥✐t✐❛❧ ❡q✉❛t✐♦♥ ❛r❡ ❡q✉❛❧✿ UV [ D ] = UV [ D ′ ] ✭❛s s❡q✉❡♥❝❡ ♦❢ ❧❡tt❡rs ❛♥❞ ✈❛r✐❛❜❧❡s✮ ✇❡ t❛❦❡ t❤❡✐r ✉♥✐♦♥s ✭♦♥❧② ✇❤❡♥ r❡s✉❧t ✐s ❛♥ ✐♥t❡r✈❛❧✮ ❞❡♣ ( i ) ∪ ❞❡♣ ( j ) ✉s❡ ⊇ , ⊆ ❤❛✈❡ st❛♥❞❛r❞ ♠❡❛♥✐♥❣ ✭❞❡♣ ( i ) ⊇ ❞❡♣ ( j ) ✮ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✻ ✴ ✸✺

❢♦r♠❛❧❧② ♥♦t ❡♥❝♦❞✐♥❣✿ ❛ss✐❣♥s ❞✐✛❡r❡♥t ❝♦❞❡s t♦ t❤❡ s❛♠❡ ❧❡tt❡r ♥❡✈❡r ❛ss✐❣♥s t❤❡ s❛♠❡ ❝♦❞❡ t♦ ❞✐✛❡r❡♥t ❧❡tt❡rs ✇♦rs❡ t❤❛♥ ❍✉✛♠❛♥ ❝♦❞✐♥❣❀ ❡♥♦✉❣❤ t♦ ❡st✐♠❛t❡ ✐ts ❜✐t✲s✐③❡ ❉❡♣✐♥ts✿ ✐❞❡❛ ❉❡♣✐♥ts ❛ss✐❣♥ t♦ ❡❛❝❤ ❧❡tt❡r ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛ ❢❛❝t♦r ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡q✉❛t✐♦♥ UV [ D ] ❧❡tt❡rs ✇✐t❤ t❤✐s ❢r❛❣♠❡♥t ❛ss✐❣♥❡❞ ❛r❡ ♥✉♠❜❡r❡❞ ✶ , ✷ , . . . , k ✇❡ ❛ss✐❣♥ t♦ t❤❡♠ ❝♦❞❡s UV [ D ]# ✶ , UV [ D ]# ✷ , . . . , UV [ D ]# k ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✼ ✴ ✸✺

❢♦r♠❛❧❧② ♥♦t ❡♥❝♦❞✐♥❣✿ ❛ss✐❣♥s ❞✐✛❡r❡♥t ❝♦❞❡s t♦ t❤❡ s❛♠❡ ❧❡tt❡r ♥❡✈❡r ❛ss✐❣♥s t❤❡ s❛♠❡ ❝♦❞❡ t♦ ❞✐✛❡r❡♥t ❧❡tt❡rs ✇♦rs❡ t❤❛♥ ❍✉✛♠❛♥ ❝♦❞✐♥❣❀ ❡♥♦✉❣❤ t♦ ❡st✐♠❛t❡ ✐ts ❜✐t✲s✐③❡ ❉❡♣✐♥ts✿ ✐❞❡❛ ❉❡♣✐♥ts ❛ss✐❣♥ t♦ ❡❛❝❤ ❧❡tt❡r ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛ ❢❛❝t♦r ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡q✉❛t✐♦♥ UV [ D ] ❧❡tt❡rs ✇✐t❤ t❤✐s ❢r❛❣♠❡♥t ❛ss✐❣♥❡❞ ❛r❡ ♥✉♠❜❡r❡❞ ✶ , ✷ , . . . , k ✇❡ ❛ss✐❣♥ t♦ t❤❡♠ ❝♦❞❡s UV [ D ]# ✶ , UV [ D ]# ✷ , . . . , UV [ D ]# k ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✼ ✴ ✸✺

❉❡♣✐♥ts✿ ✐❞❡❛ ❉❡♣✐♥ts ❛ss✐❣♥ t♦ ❡❛❝❤ ❧❡tt❡r ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛ ❢❛❝t♦r ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡q✉❛t✐♦♥ UV [ D ] ❧❡tt❡rs ✇✐t❤ t❤✐s ❢r❛❣♠❡♥t ❛ss✐❣♥❡❞ ❛r❡ ♥✉♠❜❡r❡❞ ✶ , ✷ , . . . , k ✇❡ ❛ss✐❣♥ t♦ t❤❡♠ ❝♦❞❡s UV [ D ]# ✶ , UV [ D ]# ✷ , . . . , UV [ D ]# k ❢♦r♠❛❧❧② ♥♦t ❡♥❝♦❞✐♥❣✿ ❛ss✐❣♥s ❞✐✛❡r❡♥t ❝♦❞❡s t♦ t❤❡ s❛♠❡ ❧❡tt❡r ♥❡✈❡r ❛ss✐❣♥s t❤❡ s❛♠❡ ❝♦❞❡ t♦ ❞✐✛❡r❡♥t ❧❡tt❡rs ✇♦rs❡ t❤❛♥ ❍✉✛♠❛♥ ❝♦❞✐♥❣❀ ❡♥♦✉❣❤ t♦ ❡st✐♠❛t❡ ✐ts ❜✐t✲s✐③❡ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✼ ✴ ✸✺

s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡ ❢♦r ❝♦♠♣r❡ss❡❞ str✐♥❣s✿ ✇❤❡♥ ✇❡ ❝♦♠♣r❡ss ✐♥s✐❞❡ ✇✐t❤ ❞❡♣✐♥ts ✶ ✷ t❤❡♥ ❡❛❝❤ ❣❡ts ❛ ❞❡♣✐♥t ✳ ✶ ✷ ❝♦♠♣r❡ss✐♦♥✿ ✇✐t❤ ❞❡♣ ✶ ❛♥❞ ❞❡♣ ✶ ✷ ❣❡ts ❛ ❞❡♣✐♥t ✶ ✷ s②♠♠❡tr✐❝❛❧❧② ❢♦r ✳ ❍♦✇ ❛r❡ ❞❡♣❡♥❞❡♥❝② ❢❛❝t♦rs ❞❡✜♥❡❞ ❞❡♣ ( j ) ❢♦r j ✿ ♣♦s✐t✐♦♥ ✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥ ✐♥✐t✐❛❧❧②✿ ❞❡♣ ( UV [ i ]) = { i } ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✽ ✴ ✸✺

❝♦♠♣r❡ss✐♦♥✿ ✇✐t❤ ❞❡♣ ✶ ❛♥❞ ❞❡♣ ✶ ✷ ❣❡ts ❛ ❞❡♣✐♥t ✶ ✷ s②♠♠❡tr✐❝❛❧❧② ❢♦r ✳ ❍♦✇ ❛r❡ ❞❡♣❡♥❞❡♥❝② ❢❛❝t♦rs ❞❡✜♥❡❞ ❞❡♣ ( j ) ❢♦r j ✿ ♣♦s✐t✐♦♥ ✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥ ✐♥✐t✐❛❧❧②✿ ❞❡♣ ( UV [ i ]) = { i } s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡ ❢♦r ❝♦♠♣r❡ss❡❞ str✐♥❣s✿ ✇❤❡♥ ✇❡ ❝♦♠♣r❡ss a ℓ ✐♥s✐❞❡ ba ℓ c ✇✐t❤ ❞❡♣✐♥ts D b , D ✶ , D ✷ , . . . , D ℓ , D c t❤❡♥ ❡❛❝❤ a ❣❡ts ❛ ❞❡♣✐♥t D b ∪ D ✶ ∪ D ✷ ∪ · · · ∪ D ℓ ∪ D c ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✽ ✴ ✸✺

❍♦✇ ❛r❡ ❞❡♣❡♥❞❡♥❝② ❢❛❝t♦rs ❞❡✜♥❡❞ ❞❡♣ ( j ) ❢♦r j ✿ ♣♦s✐t✐♦♥ ✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥ ✐♥✐t✐❛❧❧②✿ ❞❡♣ ( UV [ i ]) = { i } s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡ ❢♦r ❝♦♠♣r❡ss❡❞ str✐♥❣s✿ ✇❤❡♥ ✇❡ ❝♦♠♣r❡ss a ℓ ✐♥s✐❞❡ ba ℓ c ✇✐t❤ ❞❡♣✐♥ts D b , D ✶ , D ✷ , . . . , D ℓ , D c t❤❡♥ ❡❛❝❤ a ❣❡ts ❛ ❞❡♣✐♥t D b ∪ D ✶ ∪ D ✷ ∪ · · · ∪ D ℓ ∪ D c ✳ (Σ ℓ , Σ r ) ❝♦♠♣r❡ss✐♦♥✿ a = UV [ i ] ∈ Σ ℓ ✇✐t❤ ❞❡♣ ( i ) = D ✶ ❛♥❞ ❞❡♣ ( i + ✶ ) = D ✷ ❣❡ts ❛ ❞❡♣✐♥t D ✶ ∪ D ✷ s②♠♠❡tr✐❝❛❧❧② ❢♦r Σ r ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✽ ✴ ✸✺

✭❉✶✮ P♦s ✐s ❛♥ ✐♥t❡r✈❛❧ ✭✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥✮✳ ✭❉✷✮ ❋♦r t❤❛t ❤❛✈❡ s②♠❜♦❧s ✐♥ t❤❡ ❡q✉❛t✐♦♥✱ ❡✐t❤❡r✿ ♦r ✳ ✭❉✸✮ ■❢ t❤❡♥ P♦s P♦s ✳ ❈r✉❝✐❛❧ ♣r♦♣❡rt✐❡s ❋♦r ❛ ❞❡♣✐♥t D ❝❛❧❧ P♦s ( D ) = { i | ❞❡♣ ( i ) = D } ✭✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥✮ ⇒ i ≤ i ′ ❛♥❞ j ≤ j ′ [ i , j ] ≤ [ i ′ , j ′ ] ⇐ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✾ ✴ ✸✺

❈r✉❝✐❛❧ ♣r♦♣❡rt✐❡s ❋♦r ❛ ❞❡♣✐♥t D ❝❛❧❧ P♦s ( D ) = { i | ❞❡♣ ( i ) = D } ✭✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥✮ ⇒ i ≤ i ′ ❛♥❞ j ≤ j ′ [ i , j ] ≤ [ i ′ , j ′ ] ⇐ ✭❉✶✮ P♦s ( D ) ✐s ❛♥ ✐♥t❡r✈❛❧ ✭✐♥ t❤❡ ❝✉rr❡♥t ❡q✉❛t✐♦♥✮✳ ✭❉✷✮ ❋♦r D , D ′ t❤❛t ❤❛✈❡ s②♠❜♦❧s ✐♥ t❤❡ ❡q✉❛t✐♦♥✱ ❡✐t❤❡r✿ D ≤ D ′ ♦r D ≥ D ′ ✳ ✭❉✸✮ ■❢ D ∼ D ′ t❤❡♥ UV [ P♦s ( D )] = UV [ P♦s ( D ′ )] ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✶✾ ✴ ✸✺

▲❡♠♠❛ P♦s ✐s ❛♥ ✐♥t❡r✈❛❧✳ Pr♦♦❢✳ ❲❡ ♣r♦✈❡ ✐t t♦❣❡t❤❡r ✇✐t❤ ❉✶✕❉✸✳ ❊✈❡r②t❤✐♥❣ ✐s ❡❛s② ✐♥❞✉❝t✐♦♥ ❡①❝❡♣t ❉✸✿ P♦s P♦s ✳ ❚❤❡ ♣r♦♦❢ ✐s s✐♠♣❧❡ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❛♣♣r♦❛❝❤✿ t❤r♦✉❣❤ P♦s ❉✉❛❧ ✈✐❡✇ P♦s ⊇ ( D ) = { j | ❞❡♣ ( j ) ⊇ D } P♦s ⊆ ( D ) = { j | ❞❡♣ ( j ) ⊆ D } ❲❡ ❢♦❝✉s ♦♥ P♦s ⊇ ( D ) ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✵ ✴ ✸✺

Pr♦♦❢✳ ❲❡ ♣r♦✈❡ ✐t t♦❣❡t❤❡r ✇✐t❤ ❉✶✕❉✸✳ ❊✈❡r②t❤✐♥❣ ✐s ❡❛s② ✐♥❞✉❝t✐♦♥ ❡①❝❡♣t ❉✸✿ P♦s P♦s ✳ ❚❤❡ ♣r♦♦❢ ✐s s✐♠♣❧❡ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❛♣♣r♦❛❝❤✿ t❤r♦✉❣❤ P♦s ❉✉❛❧ ✈✐❡✇ P♦s ⊇ ( D ) = { j | ❞❡♣ ( j ) ⊇ D } P♦s ⊆ ( D ) = { j | ❞❡♣ ( j ) ⊆ D } ❲❡ ❢♦❝✉s ♦♥ P♦s ⊇ ( D ) ✳ ▲❡♠♠❛ P♦s ⊇ ( D ) ✐s ❛♥ ✐♥t❡r✈❛❧✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✵ ✴ ✸✺

❉✉❛❧ ✈✐❡✇ P♦s ⊇ ( D ) = { j | ❞❡♣ ( j ) ⊇ D } P♦s ⊆ ( D ) = { j | ❞❡♣ ( j ) ⊆ D } ❲❡ ❢♦❝✉s ♦♥ P♦s ⊇ ( D ) ✳ ▲❡♠♠❛ P♦s ⊇ ( D ) ✐s ❛♥ ✐♥t❡r✈❛❧✳ Pr♦♦❢✳ ❲❡ ♣r♦✈❡ ✐t t♦❣❡t❤❡r ✇✐t❤ ❉✶✕❉✸✳ ❊✈❡r②t❤✐♥❣ ✐s ❡❛s② ✐♥❞✉❝t✐♦♥ ❡①❝❡♣t ❉✸✿ D ∼ D ′ ⇒ UV [ P♦s ( D )] = UV [ P♦s ( D ′ )] ✳ ❚❤❡ ♣r♦♦❢ ✐s s✐♠♣❧❡ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❛♣♣r♦❛❝❤✿ t❤r♦✉❣❤ P♦s ⊆ ( D ) ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✵ ✴ ✸✺

✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r ✷ ✳ ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ■♥❞✉❝t✐♦♥✿ s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r ✷ ✳ ■♥❞✉❝t✐♦♥✿ s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r ✷ ✳ s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ■♥❞✉❝t✐♦♥✿ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r ✷ ✳ s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ■♥❞✉❝t✐♦♥✿ ◮ ✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r ✷ ✳ s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ■♥❞✉❝t✐♦♥✿ ◮ ✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ◮ ✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ■♥❞✉❝t✐♦♥✿ ◮ ✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ◮ ✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ◮ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ( a ) ∈ D ′ ⊆ D ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ Σ r ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ⇒ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r D ✷ ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

Pr♦♦❢✖❝t❞✳ Pr♦♦❢✳ ❋✐① ❞❡♣✐♥t D ∼ D ′ ❀ ❝♦♥s✐❞❡r P♦s ⊆ ( D ) ❛♥❞ P♦s ⊆ ( D ′ ) ❈❧❛✐♠✿ ❚❤♦s❡ ❛r❡ ✐♥t❡r✈❛❧s✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡tt❡rs ❛r❡ t❤❡ s❛♠❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡♣✐♥ts ❛r❡ s✐♠✐❧❛r✳ ■♥❞✉❝t✐♦♥✿ ◮ ✐♥t❡r✈❛❧s✿ ✇❡ ♦♥❧② ❧♦♦s❡ ❧❡tt❡rs ❢r♦♠ ❜♦t❤ ❡♥❞s ❛♥❞ ♣❡r❤❛♣s ❣❛✐♥ ❢r♦♠ ✈❛r✐❛❜❧❡s ◮ ✐♥s✐❞❡✿ ❡✈❡r②t❤✐♥❣ ✐s t❤❡ s❛♠❡ ◮ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ♦✉ts✐❞❡✿ ✭♣r♦♦❢ ❜② ❡①❛♠♣❧❡✮ ❞❡♣ ( a ) ∈ D ′ ⊆ D ✶ ✐s ❧❡❢t✲♠♦st ✇✐t❤ t❤✐s ♣r♦♣❡rt②✱ ❛♥❞ ✐♥ Σ r ✐ts ❞❡♣✐♥t ✐s ❡①t❡♥❞❡❞ ⇒ ✇❡ ♥♦ ❧♦♥❣❡r ❝❛r❡ ❛❜♦✉t ✐t ✐t ✇♦r❦s t❤❡ s❛♠❡ ❢♦r D ✷ ✳ s♦♠❡ ✐♥❝❧✉s✐♦♥✲❡①❝❧✉s✐♦♥ ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✶ ✴ ✸✺

❢♦r♠❛❧❧② ♥♦t ❡♥❝♦❞✐♥❣✿ ❛ss✐❣♥s ❞✐✛❡r❡♥t ❝♦❞❡s t♦ t❤❡ s❛♠❡ ❧❡tt❡r ♥❡✈❡r ❛ss✐❣♥s t❤❡ s❛♠❡ ❝♦❞❡ t♦ ❞✐✛❡r❡♥t ❧❡tt❡rs ✐s ✇♦rs❡ t❤❛♥ ❍✉✛♠❛♥ ❝♦❞✐♥❣ ❡♥♦✉❣❤ t♦ ❡st✐♠❛t❡ ✐ts ❜✐t✲s✐③❡ ❊♥❝♦❞✐♥❣ ❉❡✜♥✐t✐♦♥ ✭❊♥❝♦❞✐♥❣✮ ❋✐① ❞❡♣✐♥t D ✱ ❡♥❝♦❞❡ ❧❡tt❡rs ✇✐t❤ t❤✐s ❞❡♣✐♥t ❛s U ✵ V ✵ [ D ]# ✶ # , U ✵ V ✵ [ D ]# ✷ # , . . . i ✐♥ ❜✐♥❛r② U ✵ V ✵ [ D ] ❛s ✐♥ t❤❡ ✐♥♣✉t ❡q✉❛t✐♦♥ ❢♦r D ∼ D ′ t❤❡ ❡♥❝♦❞✐♥❣ ✐s t❤❡ s❛♠❡ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✷ ✴ ✸✺

❊♥❝♦❞✐♥❣ ❉❡✜♥✐t✐♦♥ ✭❊♥❝♦❞✐♥❣✮ ❋✐① ❞❡♣✐♥t D ✱ ❡♥❝♦❞❡ ❧❡tt❡rs ✇✐t❤ t❤✐s ❞❡♣✐♥t ❛s U ✵ V ✵ [ D ]# ✶ # , U ✵ V ✵ [ D ]# ✷ # , . . . i ✐♥ ❜✐♥❛r② U ✵ V ✵ [ D ] ❛s ✐♥ t❤❡ ✐♥♣✉t ❡q✉❛t✐♦♥ ❢♦r D ∼ D ′ t❤❡ ❡♥❝♦❞✐♥❣ ✐s t❤❡ s❛♠❡ ❢♦r♠❛❧❧② ♥♦t ❡♥❝♦❞✐♥❣✿ ❛ss✐❣♥s ❞✐✛❡r❡♥t ❝♦❞❡s t♦ t❤❡ s❛♠❡ ❧❡tt❡r ♥❡✈❡r ❛ss✐❣♥s t❤❡ s❛♠❡ ❝♦❞❡ t♦ ❞✐✛❡r❡♥t ❧❡tt❡rs ✐s ✇♦rs❡ t❤❛♥ ❍✉✛♠❛♥ ❝♦❞✐♥❣ ❡♥♦✉❣❤ t♦ ❡st✐♠❛t❡ ✐ts ❜✐t✲s✐③❡ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✷ ✴ ✸✺

❚❤❡♥ ❢♦r ❛ ❜❛s✐❝ ❞❡♣✐♥t P♦s ✶ ✿ ♦❧❞ P♦s ❧♦♦s❡s ✶ ✸ ♦❢ ✐ts ♣♦s✐t✐♦♥s ✭❡✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞✮ ✷ ❜♦✉♥❞ ✸ ✸ ❚❤❡ s♣❛❝❡ ✐s ❧✐♥❡❛r ♦❝❝✉rs ✸ t✐♠❡s ✐♥ ❛❧❧ ❞❡♣✐♥ts ❧❡tt❡r ❢r♦♠ t❤❡ ✐♥♣✉t ♦❝❝✉rs ✸ t✐♠❡s ✐♥ t❤❡ ❡♥❝♦❞✐♥❣s ❡❛❝❤ ❞❡♣✐♥t ❤❛s ❧❡♥❣t❤ ✸❝ ♥✉♠❜❡rs ✐♥ t❤❡ ❡♥❝♦❞✐♥❣ ❛r❡ ❝♦♥st❛♥t✲❧❡♥❣t❤ ▼❛❦❡ t❤✐s ✐♥ ❡①♣❡❝t❛t✐♦♥✳ ■❞❡❛ ❊♥s✉r❡ t❤❛t ❡❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s O ( ✶ ) ❧❡tt❡rs ♣❡r ♣❤❛s❡ ❡❛❝❤ P♦s ⊇ ( D ) ❡①♣❛♥❞s ❜② O ( ✶ ) ♣♦s✐t✐♦♥s ♣❡r ♣❤❛s❡ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✸ ✴ ✸✺

❚❤❡ s♣❛❝❡ ✐s ❧✐♥❡❛r ♦❝❝✉rs ✸ t✐♠❡s ✐♥ ❛❧❧ ❞❡♣✐♥ts ❧❡tt❡r ❢r♦♠ t❤❡ ✐♥♣✉t ♦❝❝✉rs ✸ t✐♠❡s ✐♥ t❤❡ ❡♥❝♦❞✐♥❣s ❡❛❝❤ ❞❡♣✐♥t ❤❛s ❧❡♥❣t❤ ✸❝ ♥✉♠❜❡rs ✐♥ t❤❡ ❡♥❝♦❞✐♥❣ ❛r❡ ❝♦♥st❛♥t✲❧❡♥❣t❤ ▼❛❦❡ t❤✐s ✐♥ ❡①♣❡❝t❛t✐♦♥✳ ■❞❡❛ ❊♥s✉r❡ t❤❛t ❡❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s O ( ✶ ) ❧❡tt❡rs ♣❡r ♣❤❛s❡ ❡❛❝❤ P♦s ⊇ ( D ) ❡①♣❛♥❞s ❜② O ( ✶ ) ♣♦s✐t✐♦♥s ♣❡r ♣❤❛s❡ ❚❤❡♥ ❢♦r ❛ ❜❛s✐❝ ❞❡♣✐♥t | P♦s ⊇ ( D ) | = O ( ✶ ) ✿ ♦❧❞ P♦s ⊇ ( D ) ❧♦♦s❡s ✶ / ✸ ♦❢ ✐ts ♣♦s✐t✐♦♥s ✭❡✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞✮ k ′ ≤ ✷ ✸ k + c ⇒ ❜♦✉♥❞ ✸ c ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✸ ✴ ✸✺

▼❛❦❡ t❤✐s ✐♥ ❡①♣❡❝t❛t✐♦♥✳ ■❞❡❛ ❊♥s✉r❡ t❤❛t ❡❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s O ( ✶ ) ❧❡tt❡rs ♣❡r ♣❤❛s❡ ❡❛❝❤ P♦s ⊇ ( D ) ❡①♣❛♥❞s ❜② O ( ✶ ) ♣♦s✐t✐♦♥s ♣❡r ♣❤❛s❡ ❚❤❡♥ ❢♦r ❛ ❜❛s✐❝ ❞❡♣✐♥t | P♦s ⊇ ( D ) | = O ( ✶ ) ✿ ♦❧❞ P♦s ⊇ ( D ) ❧♦♦s❡s ✶ / ✸ ♦❢ ✐ts ♣♦s✐t✐♦♥s ✭❡✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞✮ k ′ ≤ ✷ ✸ k + c ⇒ ❜♦✉♥❞ ✸ c ❚❤❡ s♣❛❝❡ ✐s ❧✐♥❡❛r i ♦❝❝✉rs ✸ c t✐♠❡s ✐♥ ❛❧❧ ❞❡♣✐♥ts ⇒ ❧❡tt❡r ❢r♦♠ t❤❡ ✐♥♣✉t ♦❝❝✉rs ✸ c t✐♠❡s ✐♥ t❤❡ ❡♥❝♦❞✐♥❣s ❡❛❝❤ ❞❡♣✐♥t ❤❛s ❧❡♥❣t❤ ≤ ✸❝ ⇒ ♥✉♠❜❡rs ✐♥ t❤❡ ❡♥❝♦❞✐♥❣ ❛r❡ ❝♦♥st❛♥t✲❧❡♥❣t❤ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✸ ✴ ✸✺

■❞❡❛ ❊♥s✉r❡ t❤❛t ❡❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s O ( ✶ ) ❧❡tt❡rs ♣❡r ♣❤❛s❡ ❡❛❝❤ P♦s ⊇ ( D ) ❡①♣❛♥❞s ❜② O ( ✶ ) ♣♦s✐t✐♦♥s ♣❡r ♣❤❛s❡ ❚❤❡♥ ❢♦r ❛ ❜❛s✐❝ ❞❡♣✐♥t | P♦s ⊇ ( D ) | = O ( ✶ ) ✿ ♦❧❞ P♦s ⊇ ( D ) ❧♦♦s❡s ✶ / ✸ ♦❢ ✐ts ♣♦s✐t✐♦♥s ✭❡✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞✮ k ′ ≤ ✷ ✸ k + c ⇒ ❜♦✉♥❞ ✸ c ❚❤❡ s♣❛❝❡ ✐s ❧✐♥❡❛r i ♦❝❝✉rs ✸ c t✐♠❡s ✐♥ ❛❧❧ ❞❡♣✐♥ts ⇒ ❧❡tt❡r ❢r♦♠ t❤❡ ✐♥♣✉t ♦❝❝✉rs ✸ c t✐♠❡s ✐♥ t❤❡ ❡♥❝♦❞✐♥❣s ❡❛❝❤ ❞❡♣✐♥t ❤❛s ❧❡♥❣t❤ ≤ ✸❝ ⇒ ♥✉♠❜❡rs ✐♥ t❤❡ ❡♥❝♦❞✐♥❣ ❛r❡ ❝♦♥st❛♥t✲❧❡♥❣t❤ ▼❛❦❡ t❤✐s ✐♥ ❡①♣❡❝t❛t✐♦♥✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✸ ✴ ✸✺

▲❡tt❡rs ♦✉ts✐❞❡ ✭♦♥❡s r❡♣❧❛❝✐♥❣ ❝♦♠♣r❡ss❡❞ str✐♥❣s✮ ❛r❡ ♥♦t ♣♦♣♣❡❞✳ ❆ ✈❛r✐❛❜❧❡ ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡❢t✴r✐❣❤t✲♠♦st ♦r s❡❝♦♥❞ ❧❡❢t✴r✐❣❤t ♠♦st ❧❡tt❡r ✐♥ ✐s ♦✉ts✐❞❡ ✭♦r ✶✮✳ P♦s✐t✐♦♥s ✇✐t❤ ❧❡tt❡rs ♦✉ts✐❞❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡✐r ❞❡♣✐♥t ❝❤❛♥❣❡❞✳ ❆ ❞❡♣✐♥t ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡tt❡r ♦♥❡ ♦r t✇♦ t♦ t❤❡ ❧❡❢t✴r✐❣❤t ♦❢ P♦s ✐s ♦✉ts✐❞❡ ✭♦r t❤❡r❡ ✐s ♦♥❡ ♦♥❧② ♦♥❡ ❧❡tt❡r t♦ t❤❡ ❧❡❢t✴r✐❣❤t✮✳ ▲❡♠♠❛ ❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ✭❞❡♣✐♥t✮ ❜❡❝♦♠❡s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ t❤❡♥ ✐t st❛②s s♦ ✐♥ t❤✐s ♣❤❛s❡ ❛♥❞ ♣♦♣s ✭❡①t❡♥❞s t♦✮ ❛t ♠♦st ♦♥❡ ❧❡tt❡r✳ ❇❧♦❝❦✐♥❣ ❘❡❝❛❧❧ Σ ✿ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✹ ✴ ✸✺

❆ ✈❛r✐❛❜❧❡ ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡❢t✴r✐❣❤t✲♠♦st ♦r s❡❝♦♥❞ ❧❡❢t✴r✐❣❤t ♠♦st ❧❡tt❡r ✐♥ ✐s ♦✉ts✐❞❡ ✭♦r ✶✮✳ P♦s✐t✐♦♥s ✇✐t❤ ❧❡tt❡rs ♦✉ts✐❞❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡✐r ❞❡♣✐♥t ❝❤❛♥❣❡❞✳ ❆ ❞❡♣✐♥t ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡tt❡r ♦♥❡ ♦r t✇♦ t♦ t❤❡ ❧❡❢t✴r✐❣❤t ♦❢ P♦s ✐s ♦✉ts✐❞❡ ✭♦r t❤❡r❡ ✐s ♦♥❡ ♦♥❧② ♦♥❡ ❧❡tt❡r t♦ t❤❡ ❧❡❢t✴r✐❣❤t✮✳ ▲❡♠♠❛ ❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ✭❞❡♣✐♥t✮ ❜❡❝♦♠❡s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ t❤❡♥ ✐t st❛②s s♦ ✐♥ t❤✐s ♣❤❛s❡ ❛♥❞ ♣♦♣s ✭❡①t❡♥❞s t♦✮ ❛t ♠♦st ♦♥❡ ❧❡tt❡r✳ ❇❧♦❝❦✐♥❣ ❘❡❝❛❧❧ Σ ✿ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ▲❡tt❡rs ♦✉ts✐❞❡ Σ ✭♦♥❡s r❡♣❧❛❝✐♥❣ ❝♦♠♣r❡ss❡❞ str✐♥❣s✮ ❛r❡ ♥♦t ♣♦♣♣❡❞✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✹ ✴ ✸✺

P♦s✐t✐♦♥s ✇✐t❤ ❧❡tt❡rs ♦✉ts✐❞❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡✐r ❞❡♣✐♥t ❝❤❛♥❣❡❞✳ ❆ ❞❡♣✐♥t ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡tt❡r ♦♥❡ ♦r t✇♦ t♦ t❤❡ ❧❡❢t✴r✐❣❤t ♦❢ P♦s ✐s ♦✉ts✐❞❡ ✭♦r t❤❡r❡ ✐s ♦♥❡ ♦♥❧② ♦♥❡ ❧❡tt❡r t♦ t❤❡ ❧❡❢t✴r✐❣❤t✮✳ ▲❡♠♠❛ ❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ✭❞❡♣✐♥t✮ ❜❡❝♦♠❡s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ t❤❡♥ ✐t st❛②s s♦ ✐♥ t❤✐s ♣❤❛s❡ ❛♥❞ ♣♦♣s ✭❡①t❡♥❞s t♦✮ ❛t ♠♦st ♦♥❡ ❧❡tt❡r✳ ❇❧♦❝❦✐♥❣ ❘❡❝❛❧❧ Σ ✿ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ▲❡tt❡rs ♦✉ts✐❞❡ Σ ✭♦♥❡s r❡♣❧❛❝✐♥❣ ❝♦♠♣r❡ss❡❞ str✐♥❣s✮ ❛r❡ ♥♦t ♣♦♣♣❡❞✳ ❆ ✈❛r✐❛❜❧❡ ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡❢t✴r✐❣❤t✲♠♦st ♦r s❡❝♦♥❞ ❧❡❢t✴r✐❣❤t ♠♦st ❧❡tt❡r ✐♥ S ( X ) ✐s ♦✉ts✐❞❡ Σ ✭♦r | S ( X ) | = ✶✮✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✹ ✴ ✸✺

❆ ❞❡♣✐♥t ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡tt❡r ♦♥❡ ♦r t✇♦ t♦ t❤❡ ❧❡❢t✴r✐❣❤t ♦❢ P♦s ✐s ♦✉ts✐❞❡ ✭♦r t❤❡r❡ ✐s ♦♥❡ ♦♥❧② ♦♥❡ ❧❡tt❡r t♦ t❤❡ ❧❡❢t✴r✐❣❤t✮✳ ▲❡♠♠❛ ❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ✭❞❡♣✐♥t✮ ❜❡❝♦♠❡s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ t❤❡♥ ✐t st❛②s s♦ ✐♥ t❤✐s ♣❤❛s❡ ❛♥❞ ♣♦♣s ✭❡①t❡♥❞s t♦✮ ❛t ♠♦st ♦♥❡ ❧❡tt❡r✳ ❇❧♦❝❦✐♥❣ ❘❡❝❛❧❧ Σ ✿ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ▲❡tt❡rs ♦✉ts✐❞❡ Σ ✭♦♥❡s r❡♣❧❛❝✐♥❣ ❝♦♠♣r❡ss❡❞ str✐♥❣s✮ ❛r❡ ♥♦t ♣♦♣♣❡❞✳ ❆ ✈❛r✐❛❜❧❡ ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡❢t✴r✐❣❤t✲♠♦st ♦r s❡❝♦♥❞ ❧❡❢t✴r✐❣❤t ♠♦st ❧❡tt❡r ✐♥ S ( X ) ✐s ♦✉ts✐❞❡ Σ ✭♦r | S ( X ) | = ✶✮✳ P♦s✐t✐♦♥s ✇✐t❤ ❧❡tt❡rs ♦✉ts✐❞❡ Σ ❞♦ ♥♦t ❤❛✈❡ t❤❡✐r ❞❡♣✐♥t ❝❤❛♥❣❡❞✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✹ ✴ ✸✺

▲❡♠♠❛ ❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ✭❞❡♣✐♥t✮ ❜❡❝♦♠❡s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ t❤❡♥ ✐t st❛②s s♦ ✐♥ t❤✐s ♣❤❛s❡ ❛♥❞ ♣♦♣s ✭❡①t❡♥❞s t♦✮ ❛t ♠♦st ♦♥❡ ❧❡tt❡r✳ ❇❧♦❝❦✐♥❣ ❘❡❝❛❧❧ Σ ✿ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ▲❡tt❡rs ♦✉ts✐❞❡ Σ ✭♦♥❡s r❡♣❧❛❝✐♥❣ ❝♦♠♣r❡ss❡❞ str✐♥❣s✮ ❛r❡ ♥♦t ♣♦♣♣❡❞✳ ❆ ✈❛r✐❛❜❧❡ ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡❢t✴r✐❣❤t✲♠♦st ♦r s❡❝♦♥❞ ❧❡❢t✴r✐❣❤t ♠♦st ❧❡tt❡r ✐♥ S ( X ) ✐s ♦✉ts✐❞❡ Σ ✭♦r | S ( X ) | = ✶✮✳ P♦s✐t✐♦♥s ✇✐t❤ ❧❡tt❡rs ♦✉ts✐❞❡ Σ ❞♦ ♥♦t ❤❛✈❡ t❤❡✐r ❞❡♣✐♥t ❝❤❛♥❣❡❞✳ ❆ ❞❡♣✐♥t D ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡tt❡r ♦♥❡ ♦r t✇♦ t♦ t❤❡ ❧❡❢t✴r✐❣❤t ♦❢ P♦s ⊇ ( D ) ✐s ♦✉ts✐❞❡ Σ ✭♦r t❤❡r❡ ✐s ♦♥❡ ♦♥❧② ♦♥❡ ❧❡tt❡r t♦ t❤❡ ❧❡❢t✴r✐❣❤t✮✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✹ ✴ ✸✺

❇❧♦❝❦✐♥❣ ❘❡❝❛❧❧ Σ ✿ ❧❡tt❡rs ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✳ ▲❡tt❡rs ♦✉ts✐❞❡ Σ ✭♦♥❡s r❡♣❧❛❝✐♥❣ ❝♦♠♣r❡ss❡❞ str✐♥❣s✮ ❛r❡ ♥♦t ♣♦♣♣❡❞✳ ❆ ✈❛r✐❛❜❧❡ ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡❢t✴r✐❣❤t✲♠♦st ♦r s❡❝♦♥❞ ❧❡❢t✴r✐❣❤t ♠♦st ❧❡tt❡r ✐♥ S ( X ) ✐s ♦✉ts✐❞❡ Σ ✭♦r | S ( X ) | = ✶✮✳ P♦s✐t✐♦♥s ✇✐t❤ ❧❡tt❡rs ♦✉ts✐❞❡ Σ ❞♦ ♥♦t ❤❛✈❡ t❤❡✐r ❞❡♣✐♥t ❝❤❛♥❣❡❞✳ ❆ ❞❡♣✐♥t D ✐s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ ✇❤❡♥ t❤❡ ❧❡tt❡r ♦♥❡ ♦r t✇♦ t♦ t❤❡ ❧❡❢t✴r✐❣❤t ♦❢ P♦s ⊇ ( D ) ✐s ♦✉ts✐❞❡ Σ ✭♦r t❤❡r❡ ✐s ♦♥❡ ♦♥❧② ♦♥❡ ❧❡tt❡r t♦ t❤❡ ❧❡❢t✴r✐❣❤t✮✳ ▲❡♠♠❛ ❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ✭❞❡♣✐♥t✮ ❜❡❝♦♠❡s ❧❡❢t✴r✐❣❤t ❜❧♦❝❦❡❞ t❤❡♥ ✐t st❛②s s♦ ✐♥ t❤✐s ♣❤❛s❡ ❛♥❞ ♣♦♣s ✭❡①t❡♥❞s t♦✮ ❛t ♠♦st ♦♥❡ ❧❡tt❡r✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✹ ✴ ✸✺

❙tr❛t❡❣② ❈❤♦♦s❡ t❤❡ ♣❛rt✐t✐♦♥ t♦ ❛❧t❡r♥❛t✐✈❡❧② ❤❛❧✈❡ t❤❡ s✉♠s ❜❡❧♦✇✿ ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ✶ ✶ ❜❛s✐❝ ❞❡♣✐♥t ❜❛s✐❝ ❞❡♣✐♥t ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ t❤❡ ✜rst ❧✐♠✐ts t❤❡ ♥✉♠❜❡r ♦❢ ❧❡tt❡rs ♣♦♣♣❡❞ ❢r♦♠ ❡q✉❛t✐♦♥s t❤❡ s❡❝♦♥❞✿ ❡①t❡♥s✐♦♥s ♦❢ P♦s ❙tr❛t❡❣② ❖✉r ♦♥❧② ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡ ✐s t❤❡ ♣❛rt✐t✐♦♥✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✺ ✴ ✸✺

t❤❡ ✜rst ❧✐♠✐ts t❤❡ ♥✉♠❜❡r ♦❢ ❧❡tt❡rs ♣♦♣♣❡❞ ❢r♦♠ ❡q✉❛t✐♦♥s t❤❡ s❡❝♦♥❞✿ ❡①t❡♥s✐♦♥s ♦❢ P♦s ❙tr❛t❡❣② ❖✉r ♦♥❧② ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡ ✐s t❤❡ ♣❛rt✐t✐♦♥✳ ❙tr❛t❡❣② ❈❤♦♦s❡ t❤❡ ♣❛rt✐t✐♦♥ t♦ ❛❧t❡r♥❛t✐✈❡❧② ❤❛❧✈❡ t❤❡ s✉♠s ❜❡❧♦✇✿ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ � � ✶ + ✶ D : ❜❛s✐❝ ❞❡♣✐♥t D : ❜❛s✐❝ ❞❡♣✐♥t ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✺ ✴ ✸✺

❙tr❛t❡❣② ❖✉r ♦♥❧② ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡ ✐s t❤❡ ♣❛rt✐t✐♦♥✳ ❙tr❛t❡❣② ❈❤♦♦s❡ t❤❡ ♣❛rt✐t✐♦♥ t♦ ❛❧t❡r♥❛t✐✈❡❧② ❤❛❧✈❡ t❤❡ s✉♠s ❜❡❧♦✇✿ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ � � ✶ + ✶ D : ❜❛s✐❝ ❞❡♣✐♥t D : ❜❛s✐❝ ❞❡♣✐♥t ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ t❤❡ ✜rst ❧✐♠✐ts t❤❡ ♥✉♠❜❡r ♦❢ ❧❡tt❡rs ♣♦♣♣❡❞ ❢r♦♠ ❡q✉❛t✐♦♥s t❤❡ s❡❝♦♥❞✿ ❡①t❡♥s✐♦♥s ♦❢ P♦s ⊇ ( D ) ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✺ ✴ ✸✺

❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ ♦r ♦r ♥♦ ✐s ❧❡❢t✲❜❧♦❝❦❡❞ ♦r ♥♦ ✶ ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t ✐s ❧❡❢t✲♣♦♣♣❡❞ ✶ ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t ✐s ❝♦♠♣r❡ss❡❞✳ ❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

♦r ♦r ♥♦ ✐s ❧❡❢t✲❜❧♦❝❦❡❞ ♦r ♥♦ ✶ ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t ✐s ❧❡❢t✲♣♦♣♣❡❞ ✶ ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t ✐s ❝♦♠♣r❡ss❡❞✳ ❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ S ( X ) = abc . . . ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

♦r ♥♦ ✶ ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t ✐s ❧❡❢t✲♣♦♣♣❡❞ ✶ ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t ✐s ❝♦♠♣r❡ss❡❞✳ ❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ S ( X ) = abc . . . a / ∈ Σ ♦r b / ∈ Σ ♦r ♥♦ b X ✐s ❧❡❢t✲❜❧♦❝❦❡❞ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ S ( X ) = abc . . . a / ∈ Σ ♦r b / ∈ Σ ♦r ♥♦ b X ✐s ❧❡❢t✲❜❧♦❝❦❡❞ c / ∈ Σ ♦r ♥♦ c ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t a ✐s ❧❡❢t✲♣♦♣♣❡❞ a , b , c ∈ Σ ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t b ✐s ❝♦♠♣r❡ss❡❞✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ S ( X ) = abc . . . a / ∈ Σ ♦r b / ∈ Σ ♦r ♥♦ b X ✐s ❧❡❢t✲❜❧♦❝❦❡❞ c / ∈ Σ ♦r ♥♦ c ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t a ✐s ❧❡❢t✲♣♦♣♣❡❞ a , b , c ∈ Σ ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t b ✐s ❝♦♠♣r❡ss❡❞✳ ❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ S ( X ) = abc . . . a / ∈ Σ ♦r b / ∈ Σ ♦r ♥♦ b X ✐s ❧❡❢t✲❜❧♦❝❦❡❞ c / ∈ Σ ♦r ♥♦ c ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t a ✐s ❧❡❢t✲♣♦♣♣❡❞ a , b , c ∈ Σ ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t b ✐s ❝♦♠♣r❡ss❡❞✳ ❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

❙tr❛t❡❣② ❡①✐sts Pr♦♦❢✳ � � n X + n X X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆ r❛♥❞♦♠ ♣❛rt✐t✐♦♥ r❡❞✉❝❡s t❤✐s s✉♠ ❜② ✶✴✷✿ S ( X ) = abc . . . a / ∈ Σ ♦r b / ∈ Σ ♦r ♥♦ b X ✐s ❧❡❢t✲❜❧♦❝❦❡❞ c / ∈ Σ ♦r ♥♦ c ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t a ✐s ❧❡❢t✲♣♦♣♣❡❞ a , b , c ∈ Σ ✶ / ✷ ♣r♦❜❛❜✐❧✐t② t❤❛t b ✐s ❝♦♠♣r❡ss❡❞✳ ❙②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ●♦ ❢♦r ❡①♣❡❝t❛t✐♦♥ ♦✈❡r ❛❧❧ ✈❛r✐❛❜❧❡s✳ ❙✐♠✐❧❛r❧② ❢♦r ❜❛s✐❝ ❞❡♣✐♥ts✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✻ ✴ ✸✺

❚❤❡ ♣r♦♦❢ ✇♦r❦s ❢♦r ❛r❜✐tr❛r② ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ✐♥♣✉t✳ ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ✉s❡ ♦♥❧② ✜①✲s✐③❡ ❢♦r ❡❛❝❤ s②♠❜♦❧ ✭str❡❛♠❧✐♥✐♥❣✮✳ ▲❡t ✐s t❤❡ ❜✐t✲s✐③❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡♥❝♦❞✐♥❣✳ ❡ P♦s ❜❛s✐❝ ❞❡♣✐♥t ✷ P♦s ❧♦❣ P♦s ✶ ❜❛s✐❝ ❞❡♣✐♥t t❤❡ ✜rst ✉♣♣❡r ❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② t❤❡ s❡❝♦♥❞ ✉♣♣❡r✲❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② ♥✉♠❜❡rs ✐♥ ❯♣♣❡r✲❜♦✉♥❞✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ▲❡♠♠❛ ❚❤❡ ❞❡♣✐♥t✲❡♥❝♦❞✐♥❣ ♦❢ ❛♥ ❡q✉❛t✐♦♥ U , V ✐s ♥♦t ❧❛r❣❡r t❤❛♥ H ( U , V ) ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✼ ✴ ✸✺

▲❡t ✐s t❤❡ ❜✐t✲s✐③❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡♥❝♦❞✐♥❣✳ ❡ P♦s ❜❛s✐❝ ❞❡♣✐♥t ✷ P♦s ❧♦❣ P♦s ✶ ❜❛s✐❝ ❞❡♣✐♥t t❤❡ ✜rst ✉♣♣❡r ❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② t❤❡ s❡❝♦♥❞ ✉♣♣❡r✲❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② ♥✉♠❜❡rs ✐♥ ❯♣♣❡r✲❜♦✉♥❞✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ▲❡♠♠❛ ❚❤❡ ❞❡♣✐♥t✲❡♥❝♦❞✐♥❣ ♦❢ ❛♥ ❡q✉❛t✐♦♥ U , V ✐s ♥♦t ❧❛r❣❡r t❤❛♥ H ( U , V ) ✳ ❚❤❡ ♣r♦♦❢ ✇♦r❦s ❢♦r ❛r❜✐tr❛r② ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ✐♥♣✉t✳ ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ✉s❡ ♦♥❧② ✜①✲s✐③❡ ❢♦r ❡❛❝❤ s②♠❜♦❧ ✭str❡❛♠❧✐♥✐♥❣✮✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✼ ✴ ✸✺

t❤❡ ✜rst ✉♣♣❡r ❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② t❤❡ s❡❝♦♥❞ ✉♣♣❡r✲❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② ♥✉♠❜❡rs ✐♥ ❯♣♣❡r✲❜♦✉♥❞✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ▲❡♠♠❛ ❚❤❡ ❞❡♣✐♥t✲❡♥❝♦❞✐♥❣ ♦❢ ❛♥ ❡q✉❛t✐♦♥ U , V ✐s ♥♦t ❧❛r❣❡r t❤❛♥ H ( U , V ) ✳ ❚❤❡ ♣r♦♦❢ ✇♦r❦s ❢♦r ❛r❜✐tr❛r② ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ✐♥♣✉t✳ ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ✉s❡ ♦♥❧② ✜①✲s✐③❡ ❢♦r ❡❛❝❤ s②♠❜♦❧ ✭str❡❛♠❧✐♥✐♥❣✮✳ ▲❡t e ✐s t❤❡ ❜✐t✲s✐③❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡♥❝♦❞✐♥❣✳ � H d ( U , V ) = ❡ · | P♦s ⊇ ( D ) | D : ❜❛s✐❝ ❞❡♣✐♥t � H n ( U , V ) = ✷ | P♦s ⊇ ( D ) | · ❧♦❣ ( | P♦s ⊇ ( D ) | + ✶ ) , D : ❜❛s✐❝ ❞❡♣✐♥t H ( U , V ) = H d ( U , V ) + H n ( U , V ) ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✼ ✴ ✸✺

❯♣♣❡r✲❜♦✉♥❞✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ▲❡♠♠❛ ❚❤❡ ❞❡♣✐♥t✲❡♥❝♦❞✐♥❣ ♦❢ ❛♥ ❡q✉❛t✐♦♥ U , V ✐s ♥♦t ❧❛r❣❡r t❤❛♥ H ( U , V ) ✳ ❚❤❡ ♣r♦♦❢ ✇♦r❦s ❢♦r ❛r❜✐tr❛r② ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ✐♥♣✉t✳ ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ✉s❡ ♦♥❧② ✜①✲s✐③❡ ❢♦r ❡❛❝❤ s②♠❜♦❧ ✭str❡❛♠❧✐♥✐♥❣✮✳ ▲❡t e ✐s t❤❡ ❜✐t✲s✐③❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❡♥❝♦❞✐♥❣✳ � H d ( U , V ) = ❡ · | P♦s ⊇ ( D ) | D : ❜❛s✐❝ ❞❡♣✐♥t � H n ( U , V ) = ✷ | P♦s ⊇ ( D ) | · ❧♦❣ ( | P♦s ⊇ ( D ) | + ✶ ) , D : ❜❛s✐❝ ❞❡♣✐♥t H ( U , V ) = H d ( U , V ) + H n ( U , V ) t❤❡ ✜rst ✉♣♣❡r ❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② UV [ D ] t❤❡ s❡❝♦♥❞ ✉♣♣❡r✲❜♦✉♥❞s t❤❡ ❜✐ts ✉s❡❞ ❜② ♥✉♠❜❡rs ✐♥ UV [ D ]# i ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✼ ✴ ✸✺

P♦s P♦s ❞❡♣✐♥t ❜❛s✐❝ ❞❡♣✐♥t ❊❛s✐❡r t♦ ❝❛❧❝✉❧❛t❡✿ ❧♦❣ P♦s ❧♦❣ P♦s ✶ ✷ ❜❛s✐❝ ❞❡♣✐♥ts ❙✐③❡ ♦❢ ♥✉♠❜❡rs✿ ❲❤❡♥ ❞❡♣✐♥t ❤❛s ♣♦s✐t✐♦♥s✱ t❤❡② ✉s❡ ✷ ❧♦❣ ❜✐ts✳ Pr♦♦❢ ❙✐③❡ ♦❢ ❛❧❧ ❞❡♣✐♥ts✿ ▲❡t D = i ❜❡ ❛ ❜❛s✐❝ ❞❡♣✐♥t✳ U ✵ V ✵ [ i ] ❤❛s s✐③❡ e ❛♥❞ ♦❝❝✉rs ✐♥ ❝♦❞❡ ❢♦r | P♦s ⊇ ( D ) | ♠❛♥② s②♠❜♦❧s✳ ❚❤✐s ❣✐✈❡s � e · | P♦s ⊇ ( D ) | = H d ( U , V ) D : ❜❛s✐❝ ❞❡♣✐♥t ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✽ ✴ ✸✺

P♦s ❜❛s✐❝ ❞❡♣✐♥t ❊❛s✐❡r t♦ ❝❛❧❝✉❧❛t❡✿ ❧♦❣ P♦s ❧♦❣ P♦s ✶ ✷ ❜❛s✐❝ ❞❡♣✐♥ts Pr♦♦❢ ❙✐③❡ ♦❢ ❛❧❧ ❞❡♣✐♥ts✿ ▲❡t D = i ❜❡ ❛ ❜❛s✐❝ ❞❡♣✐♥t✳ U ✵ V ✵ [ i ] ❤❛s s✐③❡ e ❛♥❞ ♦❝❝✉rs ✐♥ ❝♦❞❡ ❢♦r | P♦s ⊇ ( D ) | ♠❛♥② s②♠❜♦❧s✳ ❚❤✐s ❣✐✈❡s � e · | P♦s ⊇ ( D ) | = H d ( U , V ) D : ❜❛s✐❝ ❞❡♣✐♥t ❙✐③❡ ♦❢ ♥✉♠❜❡rs✿ ❲❤❡♥ ❞❡♣✐♥t D ′ ❤❛s k ♣♦s✐t✐♦♥s✱ t❤❡② ✉s❡ ≤ ✷ k ❧♦❣ k = h ( k ) ❜✐ts✳ � h ( | # P♦s ( D ′ ) | ) D ′ : ❞❡♣✐♥t ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✽ ✴ ✸✺

P♦s ❜❛s✐❝ ❞❡♣✐♥t Pr♦♦❢ ❙✐③❡ ♦❢ ❛❧❧ ❞❡♣✐♥ts✿ ▲❡t D = i ❜❡ ❛ ❜❛s✐❝ ❞❡♣✐♥t✳ U ✵ V ✵ [ i ] ❤❛s s✐③❡ e ❛♥❞ ♦❝❝✉rs ✐♥ ❝♦❞❡ ❢♦r | P♦s ⊇ ( D ) | ♠❛♥② s②♠❜♦❧s✳ ❚❤✐s ❣✐✈❡s � e · | P♦s ⊇ ( D ) | = H d ( U , V ) D : ❜❛s✐❝ ❞❡♣✐♥t ❙✐③❡ ♦❢ ♥✉♠❜❡rs✿ ❲❤❡♥ ❞❡♣✐♥t D ′ ❤❛s k ♣♦s✐t✐♦♥s✱ t❤❡② ✉s❡ ≤ ✷ k ❧♦❣ k = h ( k ) ❜✐ts✳ � h ( | # P♦s ( D ′ ) | ) D ′ : ❞❡♣✐♥t ❊❛s✐❡r t♦ ❝❛❧❝✉❧❛t❡✿ D ′ = D ✶ ∪ D ✷ ∪ · · · ∪ D ℓ ⇒ ❧♦❣ | # P♦s ( D ) | ≤ ❧♦❣ | P♦s ⊇ ( D i ) | � �� � ❜❛s✐❝ ❞❡♣✐♥ts ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✽ ✴ ✸✺

Pr♦♦❢ ❙✐③❡ ♦❢ ❛❧❧ ❞❡♣✐♥ts✿ ▲❡t D = i ❜❡ ❛ ❜❛s✐❝ ❞❡♣✐♥t✳ U ✵ V ✵ [ i ] ❤❛s s✐③❡ e ❛♥❞ ♦❝❝✉rs ✐♥ ❝♦❞❡ ❢♦r | P♦s ⊇ ( D ) | ♠❛♥② s②♠❜♦❧s✳ ❚❤✐s ❣✐✈❡s � e · | P♦s ⊇ ( D ) | = H d ( U , V ) D : ❜❛s✐❝ ❞❡♣✐♥t ❙✐③❡ ♦❢ ♥✉♠❜❡rs✿ ❲❤❡♥ ❞❡♣✐♥t D ′ ❤❛s k ♣♦s✐t✐♦♥s✱ t❤❡② ✉s❡ ≤ ✷ k ❧♦❣ k = h ( k ) ❜✐ts✳ � � h ( | # P♦s ( D ′ ) | ) ≤ h ( | P♦s ⊇ ( D ) | ) = H n ( U , V ) D ′ : ❞❡♣✐♥t D : ❜❛s✐❝ ❞❡♣✐♥t ❊❛s✐❡r t♦ ❝❛❧❝✉❧❛t❡✿ D ′ = D ✶ ∪ D ✷ ∪ · · · ∪ D ℓ ⇒ ❧♦❣ | # P♦s ( D ) | ≤ ❧♦❣ | P♦s ⊇ ( D i ) | � �� � ❜❛s✐❝ ❞❡♣✐♥ts ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✽ ✴ ✸✺

❲❡ s❤♦✇ t❤❛t t❤❡ str❛t❡❣② ❣✉❛r❛♥t❡❡s t❤❛t ✺ ✵ ✵ ✻ ❇② ✐♥❞✉❝t✐♦♥✱ t❤✐s ❣✐✈❡s ❛♥ ❜♦✉♥❞ ♦♥ ✳ ✵ ✵ ❆♥❞ ✵ ✳ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ❙✐♠✐❧❛r ❜♦✉♥❞ ✐s s❤♦✇♥ ❢♦r ✳ ❙♦ t❤❡ ❡♥❝♦❞✐♥❣ ✐s ❧✐♥❡❛r✲s✐③❡✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢ ▲❡t ( U , V ) ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡ ( U ′ , V ′ ) ❡q✉❛t✐♦♥ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ♣❤❛s❡ ( U ✵ , V ✵ ) ✿ ✐♥♣✉t ❡q❛✉t✐♦♥ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✾ ✴ ✸✺

❇② ✐♥❞✉❝t✐♦♥✱ t❤✐s ❣✐✈❡s ❛♥ ❜♦✉♥❞ ♦♥ ✳ ✵ ✵ ❆♥❞ ✵ ✳ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ❙✐♠✐❧❛r ❜♦✉♥❞ ✐s s❤♦✇♥ ❢♦r ✳ ❙♦ t❤❡ ❡♥❝♦❞✐♥❣ ✐s ❧✐♥❡❛r✲s✐③❡✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢ ▲❡t ( U , V ) ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡ ( U ′ , V ′ ) ❡q✉❛t✐♦♥ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ♣❤❛s❡ ( U ✵ , V ✵ ) ✿ ✐♥♣✉t ❡q❛✉t✐♦♥ ❲❡ s❤♦✇ t❤❛t t❤❡ str❛t❡❣② ❣✉❛r❛♥t❡❡s t❤❛t H d ( U ′ , V ′ ) = ✺ ✻ H d ( U , V ) + α H d ( U ✵ , V ✵ ) ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✾ ✴ ✸✺

❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢ ▲❡t ( U , V ) ❡q✉❛t✐♦♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡ ( U ′ , V ′ ) ❡q✉❛t✐♦♥ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ♣❤❛s❡ ( U ✵ , V ✵ ) ✿ ✐♥♣✉t ❡q❛✉t✐♦♥ ❲❡ s❤♦✇ t❤❛t t❤❡ str❛t❡❣② ❣✉❛r❛♥t❡❡s t❤❛t H d ( U ′ , V ′ ) = ✺ ✻ H d ( U , V ) + α H d ( U ✵ , V ✵ ) ❇② ✐♥❞✉❝t✐♦♥✱ t❤✐s ❣✐✈❡s ❛♥ O ( H d ( U ✵ , V ✵ )) ❜♦✉♥❞ ♦♥ H d ✳ ❆♥❞ H ( U ✵ , V ✵ ) = H d ( U ✵ , V ✵ ) + H n ( U ✵ , V ✵ ) = O ( e | U ✵ V ✵ | ) ✳ ❙✐♠✐❧❛r ❜♦✉♥❞ ✐s s❤♦✇♥ ❢♦r H n ( U , V ) ✳ ❙♦ t❤❡ ❡♥❝♦❞✐♥❣ ✐s ❧✐♥❡❛r✲s✐③❡✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✷✾ ✴ ✸✺

❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✿ ❛t ♠♦st ✷ ✷ ✳ ✵ ✵ ❇② str❛t❡❣②✿ ❛t ❧❡❛st ❤❛❧✈❡❞ ❡✈❡r② ♦t❤❡r st❡♣✱ s♦ ❛t ♠♦st ✷ ✷ ✽ ▲✐♠✐t✐♥❣ H d ✿ ♥❡✇ ❧❡tt❡rs ❇✐t✲s✐③❡ ♦❢ ❧❡tt❡rs ♣♦♣♣❡❞ ✐♥ ♦♥❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❆t ♠♦st ✶ ♣❡r ♥♦t ❜❧♦❝❦❡❞ s✐❞❡ ♦❢ ♦❝❝✉rr❡♥❝❡ ♦❢ ✈❛r✐❛❜❧❡✱ ❜✐ts✐③❡ e ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✸✵ ✴ ✸✺

❆t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✿ ❛t ♠♦st ✷ ✷ ✳ ✵ ✵ ❇② str❛t❡❣②✿ ❛t ❧❡❛st ❤❛❧✈❡❞ ❡✈❡r② ♦t❤❡r st❡♣✱ s♦ ❛t ♠♦st ✷ ✷ ✽ ▲✐♠✐t✐♥❣ H d ✿ ♥❡✇ ❧❡tt❡rs ❇✐t✲s✐③❡ ♦❢ ❧❡tt❡rs ♣♦♣♣❡❞ ✐♥ ♦♥❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❆t ♠♦st ✶ ♣❡r ♥♦t ❜❧♦❝❦❡❞ s✐❞❡ ♦❢ ♦❝❝✉rr❡♥❝❡ ♦❢ ✈❛r✐❛❜❧❡✱ ❜✐ts✐③❡ e ✳   � �   e n X + n X     X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✸✵ ✴ ✸✺

❇② str❛t❡❣②✿ ❛t ❧❡❛st ❤❛❧✈❡❞ ❡✈❡r② ♦t❤❡r st❡♣✱ s♦ ❛t ♠♦st ✷ ✷ ✽ ▲✐♠✐t✐♥❣ H d ✿ ♥❡✇ ❧❡tt❡rs ❇✐t✲s✐③❡ ♦❢ ❧❡tt❡rs ♣♦♣♣❡❞ ✐♥ ♦♥❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❆t ♠♦st ✶ ♣❡r ♥♦t ❜❧♦❝❦❡❞ s✐❞❡ ♦❢ ♦❝❝✉rr❡♥❝❡ ♦❢ ✈❛r✐❛❜❧❡✱ ❜✐ts✐③❡ e ✳   � �   e n X + n X     X ∈X X ∈X ❧❡❢t✲✉♥❜❧♦❝❦❡❞ r✐❣❤t✲✉♥❜❧♦❝❦❡❞ ❆t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣❤❛s❡✿ ❛t ♠♦st ✷ e | U ✵ V ✵ | = ✷ n ✳ ❆✳ ❏❡➺ ❲♦r❞ ❊q✉❛t✐♦♥s ✐♥ ◆▲✐♥❙P❆❈❊ ✺✳✵✹✳✶✼ ✸✵ ✴ ✸✺