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❊❊✸✻✺✿ ❘✐s❦ ❆✈❡rs❡ ❈♦♥tr♦❧ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ❊①♣♦♥❡♥t✐❛❧ r✐s❦ ❛✈❡rs✐♦♥ ❘✐s❦ ❛✈❡rs❡ ❝♦♥tr♦❧ ✶

❖✉t❧✐♥❡ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ❊①♣♦♥❡♥t✐❛❧ r✐s❦ ❛✈❡rs✐♦♥ ❘✐s❦ ❛✈❡rs❡ ❝♦♥tr♦❧ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✷

❘✐s❦ ♠❡❛s✉r❡s ■ s✉♣♣♦s❡ ❢ ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇❡✬❞ ❧✐❦❡ t♦ ❜❡ s♠❛❧❧ ✭ ✐✳❡✳ ✱ ❛♥ ♦❜❥❡❝t✐✈❡ ♦r ❝♦st✮ ■ ❊ ❢ ❣✐✈❡s ❛✈❡r❛❣❡ ♦r ♠❡❛♥ ✈❛❧✉❡ ■ ♠❛♥② ✇❛②s t♦ q✉❛♥t✐❢② r✐s❦ ✭♦❢ ❛ ❧❛r❣❡ ✈❛❧✉❡ ♦❢ ❢ ✮ ■ Pr♦❜ ✭ ❢ ✕ ❢ ❜❛❞ ✮ ✭✈❛❧✉❡✲❛t✲r✐s❦✱ ❱❆❘✮ ■ ❊ ✭ ❢ � ❢ ❜❛❞ ✮ ✰ ✭❝♦♥❞✐t✐♦♥❛❧ ✈❛❧✉❡✲❛t✲r✐s❦✱ ❈❱❆❘✮ ■ ✈❛r ❢ ❂ ❊ ✭ ❢ � ❊ ❢ ✮ ✷ ✭✈❛r✐❛♥❝❡✮ ■ ❊ ✭ ❢ � ❊ ❢ ✮ ✷ ✭❞♦✇♥s✐❞❡ ✈❛r✐❛♥❝❡✮ ✰ ■ ❊ ✣ ✭ ❢ ✮ ✱ ✇❤❡r❡ ✣ ✐s ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❝♦♥✈❡① ✭✇❤❡♥ ❧❛r❣❡ ❢ ✐s ❣♦♦❞✿ ❡①♣❡❝t❡❞ ✉t✐❧✐t② ❊ ❯ ✭ ❢ ✮ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❝♦♥❝❛✈❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ❯ ✮ ■ r✐s❦ ❛✈❡rs✐♦♥✿ ✇❡ ✇❛♥t ❊ ❢ s♠❛❧❧ ❛♥❞ ❧♦✇ r✐s❦ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✸

❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ■ ♥♦✇ s✉♣♣♦s❡ r❛♥❞♦♠ ❝♦st ❢ ✭ ① ❀ ✦ ✮ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡ ① ❛♥❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✦ ■ ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ ① ❧❡❛❞ t♦ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ♠❡❛♥ ❝♦st ❊ ❢ ✭ ① ❀ ✦ ✮ ❛♥❞ r✐s❦ ❘ ✭ ❢ ✭ ① ❀ ✦ ✮✮ ■ t❤❡r❡ ✐s t②♣✐❝❛❧❧② ❛ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ ♠✐♥✐♠✐③✐♥❣ ♠❡❛♥ ❝♦st ❛♥❞ r✐s❦ ■ st❛♥❞❛r❞ ❛♣♣r♦❛❝❤✿ ♠✐♥✐♠✐③❡ ❊ ❢ ✭ ① ❀ ✦ ✮ ✰ ✕ ❘ ✭ ❢ ✭ ① ❀ ✦ ✮✮ ■ ❊ ❢ ✭ ① ❀ ✦ ✮ ✰ ✕ ❘ ✭ ❢ ✭ ① ❀ ✦ ✮✮ ✐s t❤❡ r✐s❦✲❛❞❥✉st❡❞ ♠❡❛♥ ❝♦st ■ ✕ ❃ ✵ ✐s ❝❛❧❧❡❞ t❤❡ r✐s❦ ❛✈❡rs✐♦♥ ♣❛r❛♠❡t❡r ■ ✈❛r②✐♥❣ ✕ ♦✈❡r ✭ ✵ ❀ ✶ ✮ ❣✐✈❡s tr❛❞❡✲♦✛ ♦❢ ♠❡❛♥ ❝♦st ❛♥❞ r✐s❦ ■ ♠❡❛♥✲✈❛r✐❛♥❝❡ ♦♣t✐♠✐③❛t✐♦♥ ✿ ❝❤♦♦s❡ ① t♦ ♠✐♥✐♠✐③❡ ❊ ❢ ✭ ① ❀ ✦ ✮ ✰ ✕ ✈❛r ❢ ✭ ① ❀ ✦ ✮ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✹

❊①❛♠♣❧❡✿ ❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ ■ ✜♥❞ ♣❛t❤ ✐♥ ❞✐r❡❝t❡❞ ❣r❛♣❤ ❢r♦♠ ✈❡rt❡① ❆ t♦ ✈❡rt❡① ❇ ■ ❡❞❣❡ ✇❡✐❣❤ts ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ❦♥♦✇♥ ❞✐str✐❜✉t✐♦♥s ■ ❝♦♠♠✐t t♦ ♣❛t❤ ❜❡❢♦r❡❤❛♥❞✱ ✇✐t❤ ♥♦ ❦♥♦✇❧❡❞❣❡ ♦❢ ✇❡✐❣❤t ✈❛❧✉❡s ■ ♣❛t❤ ❧❡♥❣t❤ ▲ ✐s r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ■ ♠✐♥✐♠✐③❡ ❊ ▲ ✰ ✕ ✈❛r ▲ ✱ ✇✐t❤ ✕ ✕ ✵ ■ ❢♦r ✜①❡❞ ✕ ✱ r❡❞✉❝❡s t♦ ❞❡t❡r♠✐♥✐st✐❝ s❤♦rt❡st ♣❛t❤ ♣r♦❜❧❡♠ ✇✐t❤ ❡❞❣❡ ✇❡✐❣❤ts ❊ ✇ ❡ ✰ ✕ ✈❛r ✇ ❡ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✺

❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ (9 , 35) (8 , 125) 2 4 6 (7 , 40) (6 , 200) 1 8 (5 , 22) (7 , 23) (1 , 5) (6 , 12) (4 , 9) (8 , 8) (14 , 3) (7 , 12) 3 5 7 (8 , 2) (11 , 8) ■ ✜♥❞ ♣❛t❤ ❢r♦♠ ✈❡rt❡① ❆ ❂ ✶ t♦ ✈❡rt❡① ❇ ❂ ✽ ■ ❡❞❣❡ ✇❡✐❣❤ts ❛r❡ ❧♦❣♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ■ ❡❞❣❡s ❧❛❜❡❧❡❞ ✇✐t❤ ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡✿ ✭ ❊ ✇ ❡ ❀ ✈❛r ✇ ❡ ✮ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✻

❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ ✕ ❂ ✵ ✿ ❊ ▲ ❂ ✸✵ ✱ ✈❛r ▲ ❂ ✹✵✵ (9 , 35) (8 , 125) 2 4 6 (7 , 40) (6 , 200) 1 8 (5 , 22) (7 , 23) (1 , 5) (6 , 12) (4 , 9) (8 , 8) (14 , 3) (7 , 12) 3 5 7 (8 , 2) (11 , 8) ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✼

❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ ✕ ❂ ✵ ✿ ✵✺ ✿ ❊ ▲ ❂ ✸✺ ✱ ✈❛r ▲ ❂ ✶✵✵ (9 , 35) (8 , 125) 2 4 6 (7 , 40) (6 , 200) 1 8 (5 , 22) (7 , 23) (1 , 5) (6 , 12) (4 , 9) (8 , 8) (14 , 3) (7 , 12) 3 5 7 (8 , 2) (11 , 8) ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✼

❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ ✕ ❂ ✶✵ ✿ ❊ ▲ ❂ ✹✵ ✱ ✈❛r ▲ ❂ ✷✺ (9 , 35) (8 , 125) 2 4 6 (7 , 40) (6 , 200) 1 8 (5 , 22) (7 , 23) (1 , 5) (6 , 12) (4 , 9) (8 , 8) (14 , 3) (7 , 12) 3 5 7 (8 , 2) (11 , 8) ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✼

❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ tr❛❞❡✲♦✛ ❝✉r✈❡✿ ✕ ❂ ✵ ✱ ✕ ❂ ✵ ✿ ✵✺ ✱ ✕ ❂ ✶✵ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✽

❙t♦❝❤❛st✐❝ s❤♦rt❡st ♣❛t❤ ❞✐str✐❜✉t✐♦♥ ♦❢ ▲ ✿ ✕ ❂ ✵ ✱ ✕ ❂ ✵ ✿ ✵✺ ✱ ✕ ❂ ✶✵ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✾

❊①❛♠♣❧❡✿ ▼❡❛♥✲✈❛r✐❛♥❝❡ ✭▼❛r❦♦✇✐t③✮ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ■ ❝❤♦♦s❡ ♣♦rt❢♦❧✐♦ ① ✷ ❘ ♥ ■ ① ✐ ✐s ❛♠♦✉♥t ♦❢ ❛ss❡t ✐ ❤❡❧❞ ✭s❤♦rt ♣♦s✐t✐♦♥ ✇❤❡♥ ① ✐ ❁ ✵ ✮ ■ ✭r❛♥❞♦♠✮ ❛ss❡t r❡t✉r♥ r ✷ ❘ ♥ ❤❛s ❦♥♦✇♥ ♠❡❛♥ ❊ r ❂ ✖ ✱ ❝♦✈❛r✐❛♥❝❡ ❊ ✭ r � ✖ ✮✭ r � ✖ ✮ ❚ ❂ ✝ ■ ♣♦rt❢♦❧✐♦ r❡t✉r♥ ✐s ✭r❛♥❞♦♠ ✈❛r✐❛❜❧❡✮ ❘ ❂ r ❚ ① ■ ♠❡❛♥ r❡t✉r♥ ✐s ❊ ❘ ❂ ✖ ❚ ① ■ r❡t✉r♥ ✈❛r✐❛♥❝❡ ✐s ✈❛r ❘ ❂ ① ❚ ✝ ① ■ ♠❛①✐♠✐③❡ ❊ ❘ � ✌ ✈❛r ❘ ❂ ✖ ❚ ① � ✌ ① ❚ ✝ ① ✱ ❵r✐s❦ ❛❞❥✉st❡❞ ✭♠❡❛♥✮ r❡t✉r♥✬ ■ ✌ ❃ ✵ ✐s r✐s❦ ❛✈❡rs✐♦♥ ♣❛r❛♠❡t❡r ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✶✵

❊①❛♠♣❧❡✿ ▼❡❛♥✲✈❛r✐❛♥❝❡ ✭▼❛r❦♦✇✐t③✮ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ■ ❝❛♥ ❛❞❞ ❝♦♥str❛✐♥ts s✉❝❤ ❛s ■ ✶ ❚ ① ❂ ✶ ✭❜✉❞❣❡t ❝♦♥str❛✐♥t✮ ■ ① ✕ ✵ ✭❧♦♥❣ ♣♦s✐t✐♦♥s ♦♥❧②✮ ■ ❝❛♥ ❜❡ s♦❧✈❡❞ ❛s ❛ ✭❝♦♥✈❡①✮ q✉❛❞r❛t✐❝ ♣r♦❣r❛♠ ✭◗P✮ ✖ ❚ ① � ✌ ① ❚ ✝ ① ♠❛①✐♠✐③❡ ✶ ❚ ① ❂ ✶ ❀ s✉❜❥❡❝t t♦ ① ✕ ✵ ✭♦r ❛♥❛❧②t✐❝❛❧❧② ✇✐t❤♦✉t ❧♦♥❣✲♦♥❧② ❝♦♥str❛✐♥t✮ ■ ✈❛r②✐♥❣ ✌ ❣✐✈❡s tr❛❞❡✲♦✛ ♦❢ ♠❡❛♥ r❡t✉r♥ ❛♥❞ r✐s❦ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✶✶

❊①❛♠♣❧❡✿ ▼❡❛♥✲✈❛r✐❛♥❝❡ ✭▼❛r❦♦✇✐t③✮ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡✿ ♥ ❂ ✸✵ ✱ r ✘ ◆ ✭ ✖❀ ✝✮ tr❛❞❡✲♦✛ ❝✉r✈❡✿ ✌ ❂ ✶✵ � ✷ ✱ ✌ ❂ ✶✵ � ✶ ✱ ✌ ❂ ✶ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✶✷

❊①❛♠♣❧❡✿ ▼❡❛♥✲✈❛r✐❛♥❝❡ ✭▼❛r❦♦✇✐t③✮ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡✿ ♥ ❂ ✸✵ ✱ r ✘ ◆ ✭ ✖❀ ✝✮ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣♦rt❢♦❧✐♦ r❡t✉r♥✿ ✌ ❂ ✶✵ � ✷ ✱ ✌ ❂ ✶✵ � ✶ ✱ ✌ ❂ ✶ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✶✸

❖✉t❧✐♥❡ ❘✐s❦ ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ❊①♣♦♥❡♥t✐❛❧ r✐s❦ ❛✈❡rs✐♦♥ ❘✐s❦ ❛✈❡rs❡ ❝♦♥tr♦❧ ❊①♣♦♥❡♥t✐❛❧ r✐s❦ ❛✈❡rs✐♦♥ ✶✹