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❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ❍❛r❛❧❞ ❍❛♠♠❛rstr♦♠ ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✶ ✴ ✷✼

❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ❆♥ ✐s♦❣❧♦ss ✐s t❤❡ ❣❡♦❣r❛♣❤✐❝❛❧ ❜♦✉♥❞❛r② ♦❢ ❛ ❝❡rt❛✐♥ ❧✐♥❣✉✐st✐❝ ❢❡❛t✉r❡✱ ✳ ✳ ✳ s✉❝❤ ❛s t❤❡ ♣r♦♥✉♥❝✐❛t✐♦♥ ♦❢ ❛ ✈♦✇❡❧✱ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❛ ✇♦r❞✱ ♦r ✉s❡ ♦❢ s♦♠❡ s②♥t❛❝t✐❝ ❢❡❛t✉r❡ ✭❲✐❦✐♣❡❞✐❛ ✽ ❏✉♥❡ ✷✵✶✵✮ ❲✐❞❡❧② ✉s❡❞ ✐♥ ❞✐❛❧❡❝t♦❧♦❣② ❊①❛♠♣❧❡✱ ♣✐♥✴♣❡♥ ♠❡r❣❡r ❛s ♦❢ ▲❛❜♦✈ ✭✶✾✾✼✮✿ ❤tt♣✿✴✴✇✇✇✳❧✐♥❣✳✉♣❡♥♥✳❡❞✉✴♣❤♦♥♦❴❛t❧❛s✴♠❛♣s✴▼❛♣✸✳❤t♠❧ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✷ ✴ ✷✼

❆♣♣r♦❛❝❤❡s t♦ ■s♦❣❧♦ss ▲✐♥❡s ❚❤❡r❡ ❛♣♣❡❛rs t♦ ❜❡ ♥♦ ♦❜❥❡❝t✐✈❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ✐s♦❣❧♦ss ❧✐♥❡✱ ❧❡t ❛❧♦♥❡ ❛♥ ❛✉t♦♠❛t❡❞ ♣r♦❝❡❞✉r❡ ❢♦r ❞r❛✇✐♥❣ ♦♥❡ ❉✐❛❧❡❝t♦❧♦❣✐sts t♦❞❛② ❞r❛✇ ✐s♦❣❧♦ss ❧✐♥❡s ❜② ❤❛♥❞✱ ❜❛s❡❞ ♦♥ ✐♥t✉✐t✐♦♥ ✭♣✳❝✳ ❇❡rt ❱❛✉① ✷✵✶✵✮ ■❢ ②♦✉ ❦♥♦✇ ♦❢ ❢♦r♠❛❧ ❛♣♣r♦❛❝❤❡s t♦ ❞r❛✇✐♥❣ ✐s♦❣❧♦ss ❧✐♥❡s✱ ❞♦ ❧❡t ✉s ❦♥♦✇✦ ❚❤✐s ✐s ✐♥✈♦❧✈❡s ❛ ❝❡rt❛✐♥ ❛♠♦✉♥t ♦❢ s✉❜❥❡❝t✐✈✐t② ❚♦❞❛② ✇❡ ✇✐❧❧ s✉❣❣❡st ❛ ♣❧❛✉s✐❜❧❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ❛ ♣r♦❝❡❞✉r❡ ❢♦r ❛❝t✉❛❧❧② ❞r❛✇✐♥❣ t❤❡ ❧✐♥❡ t❤❛t ✜ts t❤❡ ❞❡✜♥✐t✐♦♥ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✸ ✴ ✷✼

Pr♦❜❧❡♠ ❙❡tt✐♥❣ ★✶✿ ■♥♣✉t ●✐✈❡♥✿ ✷❉ ❣r✐❞ ♠❛♣ ✇✐t❤ r✐♥❣s ✭✏r❡❞✑✮ ❛♥❞ ❝r♦ss❡s ✭✏❣r❡❡♥✑✮ ❛♥❞ ❡♠♣t② ♣♦s✐t✐♦♥s ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✹ ✴ ✷✼

Pr♦❜❧❡♠ ❙❡tt✐♥❣ ★✷✿ ✏▲✐♥❡✑ ❆ss✉♠♣t✐♦♥s ▲❡❣❛❧ ❆ss✉♠♣t✐♦♥s ❛❜♦✉t ❛ ✏❧✐♥❡✑✿ ❆ ❧✐♥❡ ✐s ♥♦t ♥❡❝❡ss❛r✐❧② ❛ str❛✐❣❤t ❧✐♥❡ ❇✉t✱ ❡✐t❤❡r ▲❡❣❛❧ ❘✉♥s ❢r♦♠ t❤❡ ✇❡st ❡♥❞ t♦ t❤❡ ❡❛st ❡♥❞ ♦♥ t❤❡ ♠❛♣✱ ❝r♦ss✐♥❣ ❡❛❝❤ ❝♦❧✉♠♥ ❛t ❡①❛❝t❧② ♦♥❝❡ ❖❘ ❘✉♥s ❢r♦♠ t❤❡ ♥♦rt❤ ❡♥❞ t♦ t❤❡ s♦✉t❤ ❡♥❞ ◆❖❚ ▲❡❣❛❧ ♦♥ t❤❡ ♠❛♣✱ ❝r♦ss✐♥❣ ❡❛❝❤ r♦✇ ❛t ❡①❛❝t❧② ♦♥❝❡ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✺ ✴ ✷✼

❙♦♠❡ ▲❛❜❡❧❧✐♥❣ ❈♦♥✈❡♥t✐♦♥s ❋♦r ❛♥② ❧✐♥❡ ♦♥ t❤❡ ♠❛♣ s❡♣❛r❛t✐♥❣ r✐♥❣s ❛♥❞ ❝r♦ss❡s✱ ❧❛❜❡❧ ✭❜② ♠❛❥♦r✐t②✮ ♦♥❡ ♦❢ t❤❡ s✐❞❡s t❤❡ r✐♥❣s✲s✐❞❡ ❛♥❞ t❤❡ ♦t❤❡r t❤❡ ❝r♦ss❡s✲s✐❞❡ ▲❡t x ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✭❡✐t❤❡r r✐♥❣s ♦r ❝r♦ss❡s✮ ♦♥ t❤❡ ❝r♦ss❡s s✐❞❡ ▲❡t o ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✭❡✐t❤❡r r✐♥❣s ♦r ❝r♦ss❡s✮ ♦♥ t❤❡ r✐♥❣s s✐❞❡ ❆ ❝♦rr❡❝t❧② ❝❧❛ss✐✜❡❞ ♣♦✐♥t ✐s ❛ r✐♥❣ t❤❛t ♦❝❝✉rs ♦♥ t❤❡ r✐♥❣s✲s✐❞❡ ✭ c x ✮ ♦r ❛ ❝r♦ss t❤❛t ♦❝❝✉rs ♦♥ t❤❡ ❝r♦ss✲s✐❞❡ c o ❆ ♠✐s❝❧❛ss✐✜❡❞ ♣♦✐♥t ✐s ❛ r✐♥❣ t❤❛t ♦❝❝✉rs ♦♥ t❤❡ ❝r♦ss❡s✲s✐❞❡ ✭ m o ✮ ♦r ❛ ❝r♦ss t❤❛t ♦❝❝✉rs ♦♥ t❤❡ r✐♥❣s✲s✐❞❡ ✭ m x ✮ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✻ ✴ ✷✼

❉❡✜♥✐t✐♦♥ ♦❢ ❛♥ ■s♦❣❧♦ss ▲✐♥❡ ❆❜s♦❧✉t❡✲❖♣t✐♠❛❧ ❚❤❡ ❧✐♥❡ t❤❛t ♠❛①✐♠✐③❡s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦rr❡❝t❧② ❝❧❛ss✐✜❡❞ ♣♦✐♥ts✳ ■✳❡✳ t❤❡ ❧✐♥❡ t❤❛t ♠❛①✐♠✐③❡s c x + c o ✭❊q✉✐✈❛❧❡♥t t♦ ♠✐♥✐♠✐③✐♥❣ m x + m o ✮ Pr♦♣♦rt✐♦♥✲❖♣t✐♠❛❧ ❚❤❡ ❧✐♥❡ t❤❛t ♠❛①✐♠✐③❡s t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❝♦rr❡❝t❧② ❝❧❛ss✐✜❡❞ ♣♦✐♥ts t♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✱ ♦♥ ❜♦t❤ s✐❞❡s✳ ■✳❡✳ t❤❡ ❧✐♥❡ t❤❛t ♠❛①✐♠✐③❡s c x x + c o o ✭❊q✉✐✈❛❧❡♥t t♦ ♠✐♥✐♠✐③✐♥❣ m x x + m o o ✮ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✼ ✴ ✷✼

❊①❛♠♣❧❡ ✐ ❆❜s♦❧✉t❡✲❖♣t✐♠❛❧✿ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦rr❡❝t❧② ❝❧❛ss✐✜❡❞ ♣♦✐♥ts ✐ c x + c o ❂ ✷ ✰ ✷ ❂ ✹ ✐✐ c x + c o ❂ ✸ ✰ ✷ ❂ ✺ ❙♦ ❧✐♥❡ ✭✐✐✮ ✐s ❜❡tt❡r✳ Pr♦♣♦rt✐♦♥✲❖♣t✐♠❛❧✿ ❚❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❝♦rr❡❝t❧② ✐✐ ❝❧❛ss✐✜❡❞ ♣♦✐♥ts t♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✱ ♦♥ ❜♦t❤ s✐❞❡s c x x + c o ✐ o ❂ ✷✴✸ ✰ ✷✴✸ c x x + c o ✐✐ o ❂ ✸✴✸ ✰ ✷✴✸ ❙♦ ❧✐♥❡ ✭✐✐✮ ✐s ❜❡tt❡r✳ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✽ ✴ ✷✼

❋✐rst ◗✉❡st✐♦♥✿ ❊q✉✐✈❛❧❡♥❝❡ ❆r❡ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❡q✉✐✈❛❧❡♥t❄ ⇐ ⇒ ❋♦r ❛❧❧ ♣♦ss✐❜❧❡ ✷❉ ♠❛♣s ✇✐t❤ r✐♥❣s ❛♥❞ ❝r♦ss❡s✱ ❞♦ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❛❧✇❛②s ②✐❡❧❞ t❤❡ s❛♠❡ ✐s♦❣❧♦ss ❧✐♥❡✭s✮❄ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✾ ✴ ✷✼

❈♦✉♥t❡r❡①❛♠♣❧❡ ① ① ① ❚❤❡ ♦♣t✐♠❛❧ ❝✉t ❢♦r ❛✲✐s♦❣❧♦ss ❧✐♥❡s ❤❛s ① ♦♥❡ ♠✐s❝❧❛ss✐✜❡❞ x ① ① ❚❤❡ ♦♣t✐♠❛❧ ❝✉t ❢♦r ❜✲✐s♦❣❧♦ss ❧✐♥❡s ❤❛s ❛ ✸ / ✸ + ✶✶ / ✶✸ = ✶ . ✽✹✻ ✇❤✐❝❤ ✐s ❜❡tt❡r t❤❛♥ ♦ t❤❡ ❛✲❧✐♥❡ ✭✺ / ✻ + ✶✵ / ✶✵ = ✶ . ✽✸✮ ♦ ❙♦ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s ❛✴❜ ❛r❡ ♥♦t ① ❡q✉✐✈❛❧❡♥t ❜ ♦ ♦ ♦ ❍❛♠♠❛rstr♦♠ ❉r❛✇✐♥❣ ■s♦❣❧♦ss ▲✐♥❡s ✶✼ ❙❡♣ ✷✵✶✹✱ ❆♠st❡r❞❛♠ ✶✵ ✴ ✷✼