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❚❤❡ ❉❡s✐❣♥ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊①❝❤❛♥❣❡s✿ ❙♦♠❡ ❖♣❡♥ ◗✉❡st✐♦♥s ❛t t❤❡ ■♥t❡rs❡❝t✐♦♥ ♦❢ ❊❝♦♥ ❛♥❞ ❈❙ ❊r✐❝ ❇✉❞✐s❤✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦ ❙✐♠♦♥s ■♥st✐t✉t❡ ❈♦♥❢❡r❡♥❝❡✿ ❆❧❣♦r✐t❤♠✐❝ ●❛♠❡ ❚❤❡♦r② ❛♥❞ Pr❛❝t✐❝❡ ◆♦✈ ✷✵✶✺

❖✈❡r✈✐❡✇ ✶✳ ❚❤❡ ❡❝♦♥♦♠✐❝ ❝❛s❡ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ tr❛❞✐♥❣ ◮ ❋✐♥❛♥❝✐❛❧ ❡①❝❤❛♥❣❡ ❞❡s✐❣♥ t❤❛t ✐s ♣r❡❞♦♠✐♥❛♥t ❛r♦✉♥❞ t❤❡ ✇♦r❧❞ ✕ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t ♦r❞❡r ❜♦♦❦ ✕ ✐s ❡❝♦♥♦♠✐❝❛❧❧② ✢❛✇❡❞ ◮ ❋❧❛✇✿ tr❡❛ts t✐♠❡ ❛s ❛ ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡ ✭s❡r✐❛❧ ♣r♦❝❡ss✐♥❣✮ ◮ ❙♦❧✉t✐♦♥✿ tr❡❛t t✐♠❡ ❛s ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡✱ ❜❛t❝❤ ♣r♦❝❡ss ✉s✐♥❣ ❛♥ ❛✉❝t✐♦♥✳ ✏❋r❡q✉❡♥t ❜❛t❝❤ ❛✉❝t✐♦♥s✑✳ ◮ ❊r✐❝ ❇✉❞✐s❤✱ P❡t❡r ❈r❛♠t♦♥ ❛♥❞ ❏♦❤♥ ❙❤✐♠ ✭✏❇❈❙✑✮ ✷✵✶✺✱ ◗✉❛rt❡r❧② ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s ✷✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝❛s❡ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ tr❛❞✐♥❣ ◮ ❉✐s❝r❡t❡ t✐♠❡ r❡s♣❡❝ts t❤❡ ❧✐♠✐ts ♦❢ ❝♦♠♣✉t❡rs ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s t❡❝❤♥♦❧♦❣②✳ ◆♦t ✐♥✜♥✐t❡❧② ❢❛st✳ ◮ ❇❡♥❡✜ts ❢♦r ❡①❝❤❛♥❣❡s✱ ❛❧❣♦ tr❛❞❡rs✱ r❡❣✉❧❛t♦rs ◮ ◗✉❛❧✐t❛t✐✈❡✴✐♥❢♦r♠❛❧ ❛r❣✉♠❡♥t ✐♥ ❇❈❙ ✷✵✶✺✱ ✇♦✉❧❞ ❜❡♥❡✜t ❣r❡❛t❧② ❢r♦♠ ❊❝♦♥✴❈❙ r❡s❡❛r❝❤ ✸✳ ❖t❤❡r ❊❝♦♥✴❈❙ ◗✉❡st✐♦♥s ❛❜♦✉t t❤❡ ❉❡s✐❣♥ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊①❝❤❛♥❣❡s ◮ ❋❧❛s❤ ❝r❛s❤❡s ◮ ❙♣❡❡❞ ✈s✳ ❙♠❛rts ❚r❛❞❡♦✛ ◮ ❈✐r❝✉✐t ❇r❡❛❦❡rs

❖✈❡r✈✐❡✇ ✶✳ ❚❤❡ ❡❝♦♥♦♠✐❝ ❝❛s❡ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ tr❛❞✐♥❣ ◮ ❋✐♥❛♥❝✐❛❧ ❡①❝❤❛♥❣❡ ❞❡s✐❣♥ t❤❛t ✐s ♣r❡❞♦♠✐♥❛♥t ❛r♦✉♥❞ t❤❡ ✇♦r❧❞ ✕ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t ♦r❞❡r ❜♦♦❦ ✕ ✐s ❡❝♦♥♦♠✐❝❛❧❧② ✢❛✇❡❞ ◮ ❋❧❛✇✿ tr❡❛ts t✐♠❡ ❛s ❛ ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡ ✭s❡r✐❛❧ ♣r♦❝❡ss✐♥❣✮ ◮ ❙♦❧✉t✐♦♥✿ tr❡❛t t✐♠❡ ❛s ❛ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡✱ ❜❛t❝❤ ♣r♦❝❡ss ✉s✐♥❣ ❛♥ ❛✉❝t✐♦♥✳ ✏❋r❡q✉❡♥t ❜❛t❝❤ ❛✉❝t✐♦♥s✑✳ ◮ ❊r✐❝ ❇✉❞✐s❤✱ P❡t❡r ❈r❛♠t♦♥ ❛♥❞ ❏♦❤♥ ❙❤✐♠ ✭✏❇❈❙✑✮ ✷✵✶✺✱ ◗✉❛rt❡r❧② ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s ✷✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝❛s❡ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ tr❛❞✐♥❣ ◮ ❉✐s❝r❡t❡ t✐♠❡ r❡s♣❡❝ts t❤❡ ❧✐♠✐ts ♦❢ ❝♦♠♣✉t❡rs ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s t❡❝❤♥♦❧♦❣②✳ ◆♦t ✐♥✜♥✐t❡❧② ❢❛st✳ ◮ ❇❡♥❡✜ts ❢♦r ❡①❝❤❛♥❣❡s✱ ❛❧❣♦ tr❛❞❡rs✱ r❡❣✉❧❛t♦rs ◮ ◗✉❛❧✐t❛t✐✈❡✴✐♥❢♦r♠❛❧ ❛r❣✉♠❡♥t ✐♥ ❇❈❙ ✷✵✶✺✱ ✇♦✉❧❞ ❜❡♥❡✜t ❣r❡❛t❧② ❢r♦♠ ❊❝♦♥✴❈❙ r❡s❡❛r❝❤ ✸✳ ❖t❤❡r ❊❝♦♥✴❈❙ ◗✉❡st✐♦♥s ❛❜♦✉t t❤❡ ❉❡s✐❣♥ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊①❝❤❛♥❣❡s ◮ ❋❧❛s❤ ❝r❛s❤❡s ◮ ❙♣❡❡❞ ✈s✳ ❙♠❛rts ❚r❛❞❡♦✛ ◮ ❈✐r❝✉✐t ❇r❡❛❦❡rs

❇r✐❡❢ ❉❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❈♦♥t✐♥✉♦✉s ▲✐♠✐t ❖r❞❡r ❇♦♦❦ ◮ ❇❛s✐❝ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦✿ ❧✐♠✐t ♦r❞❡r ◮ ❙♣❡❝✐✜❡s ❛ ♣r✐❝❡✱ q✉❛♥t✐t②✱ ❛♥❞ ❜✉②✴s❡❧❧ ✭❜✐❞✴❛s❦✮ ◮ ✏❇✉② ✶✵✵ s❤❛r❡s ♦❢ ❳❨❩ ❛t ✩✶✵✵✳✵✵✑ ◮ ❚r❛❞❡rs ♠❛② s✉❜♠✐t ❧✐♠✐t ♦r❞❡rs t♦ t❤❡ ♠❛r❦❡t ❛t ❛♥② t✐♠❡ ❞✉r✐♥❣ t❤❡ tr❛❞✐♥❣ ❞❛② ◮ ❆❧s♦ ♠❛② ❝❛♥❝❡❧ ♦r ♠♦❞✐❢② ♦✉tst❛♥❞✐♥❣ ❧✐♠✐t ♦r❞❡rs ❛t ❛♥② t✐♠❡ ◮ ❖r❞❡rs ❛♥❞ ❝❛♥❝❡❧❛t✐♦♥s ❛r❡ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❡①❝❤❛♥❣❡ ♦♥❡✲❛t✲❛✲t✐♠❡ ✐♥ ♦r❞❡r ♦❢ r❡❝❡✐♣t ✭s❡r✐❛❧ ♣r♦❝❡ss✮ ◮ ❙❡t ♦❢ ♦✉tst❛♥❞✐♥❣ ♦r❞❡rs ✐s ❦♥♦✇♥ ❛s t❤❡ ❧✐♠✐t ♦r❞❡r ❜♦♦❦ ◮ ❚r❛❞❡ ♦❝❝✉rs ✇❤❡♥❡✈❡r ❛ ♥❡✇ ❧✐♠✐t ♦r❞❡r ✐s s✉❜♠✐tt❡❞ t❤❛t ✐s ❡✐t❤❡r ✭✐✮ ❜✐❞ ≥ ❧♦✇❡st ❛s❦❀ ✭✐✐✮ ❛s❦ ≤ ❤✐❣❤❡st ❜✐❞ ◮ ◆❡✇ ❧✐♠✐t ♦r❞❡r ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❛❝❝❡♣t✐♥❣ ✭❢✉❧❧② ♦r ♣❛rt✐❛❧❧②✮ ♦♥❡ ♦r ♠♦r❡ ♦✉tst❛♥❞✐♥❣ ♦r❞❡rs

❇❈❙ ▼♦❞❡❧✿ ❇❛s✐❝s ♦❢ ❙❡t✉♣ ◮ ❚❤❡r❡ ✐s ❛ s❡❝✉r✐t②✱ x ✱ t❤❛t tr❛❞❡s ♦♥ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t✲♦r❞❡r ❜♦♦❦ ♠❛r❦❡t ◮ ❚❤❡r❡ ✐s ❛ ♣✉❜❧✐❝❧② ♦❜s❡r✈❛❜❧❡ s✐❣♥❛❧✱ y ✱ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ s❡❝✉r✐t② x ✳ y ❡✈♦❧✈❡s ❛s ❛ P♦✐ss♦♥ ❥✉♠♣ ♣r♦❝❡ss✳ ◮ P✉r♣♦s❡❢✉❧❧② str♦♥❣ ❛ss✉♠♣t✐♦♥✿ ◮ ❋✉♥❞❛♠❡♥t❛❧ ✈❛❧✉❡ ♦❢ x ✐s ♣❡r❢❡❝t❧② ❝♦rr❡❧❛t❡❞ t♦ t❤❡ ♣✉❜❧✐❝ s✐❣♥❛❧ y ◮ x ❝❛♥ ❛❧✇❛②s ❜❡ ❝♦st❧❡ss❧② ❧✐q✉✐❞❛t❡❞ ❛t t❤✐s ❢✉♥❞❛♠❡♥t❛❧ ✈❛❧✉❡ ◮ P❧❛②❡rs ◮ ■♥✈❡st♦rs✿ ❛rr✐✈❡ st♦❝❤❛st✐❝❛❧❧②✱ ♠❡❝❤❛♥✐❝❛❧❧② ❡✐t❤❡r ❜✉② ♦r s❡❧❧ x ❛t ♠❛r❦❡t ◮ ❚r❛❞✐♥❣ ❋✐r♠s✿ N ✱ ❛❧❧ ❡q✉❛❧❧② ❢❛st ✕ ③❡r♦ ❧❛t❡♥❝② ◮ ❖✈❡r❛❧❧✿ ✏❇❡st ❝❛s❡✑ s❝❡♥❛r✐♦ ❢♦r ♣r✐❝❡ ❞✐s❝♦✈❡r② ❛♥❞ ❧✐q✉✐❞✐t② ♣r♦✈✐s✐♦♥ ◮ ◆♦ ❛s②♠♠❡tr✐❝ ✐♥❢♦✱ ♥♦ ✐♥✈❡♥t♦r② ❝♦sts✱ ✐♥✈❡st♦rs ♠❡❝❤❛♥✐❝❛❧✱ tr❛❞✐♥❣ ✜r♠s ❤❛✈❡ ③❡r♦ ❧❛t❡♥❝②

✏❙♥✐♣✐♥❣✑ ◮ ●✐✈❡♥ t❤❡ ♠♦❞❡❧ s❡t✉♣ ✕ ♥♦ ❛s②♠♠❡tr✐❝ ✐♥❢♦r♠❛t✐♦♥✱ ♥♦ ✐♥✈❡♥t♦r② ❝♦sts✱ ❡t❝✳ ✕ ♦♥❡ ♠✐❣❤t ❝♦♥❥❡❝t✉r❡ t❤❛t ✭❇❡rtr❛♥❞✮ ❝♦♠♣❡t✐t✐♦♥ ❛♠♦♥❣ t❤❡ N tr❛❞✐♥❣ ✜r♠s ❧❡❛❞s t♦ ❡✛❡❝t✐✈❡❧② ✐♥✜♥✐t❡ ❧✐q✉✐❞✐t② ❢♦r ✐♥✈❡st♦rs ◮ ❚❤❛t ✐s✱ tr❛❞✐♥❣ ✜r♠s s❤♦✉❧❞ ♦✛❡r t♦ ❜✉② ♦r s❡❧❧ x ❛t ♣r✐❝❡ y ✐♥ ✉♥❧✐♠✐t❡❞ q✉❛♥t✐t② ❛t ③❡r♦ ❜✐❞✲❛s❦ s♣r❡❛❞ ◮ ❇✉t t❤❛t ✐s ♥♦t ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t ♦r❞❡r ❜♦♦❦ ♠❛r❦❡t✱ ❞✉❡ t♦ ❛ ♣❤❡♥♦♠❡♥♦♥ ✇❡ ❝❛❧❧ ✏s♥✐♣✐♥❣✑

✏❙♥✐♣✐♥❣✑ ◮ ❙✉♣♣♦s❡ y ❥✉♠♣s✱ ❡✳❣✳✱ ❢r♦♠ ② t♦ ¯ y ◮ ❚r❛❞✐♥❣ ✜r♠s ♣r♦✈✐❞✐♥❣ ❧✐q✉✐❞✐t② ✐♥ t❤❡ ♠❛r❦❡t ❢♦r x s❡♥❞ ❛ ♠❡ss❛❣❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ❧✐♠✐t ♦r❞❡r ❜♦♦❦ ◮ ❲✐t❤❞r❛✇ ♦❧❞ q✉♦t❡s ❜❛s❡❞ ♦♥ ② ◮ ❘❡♣❧❛❝❡ ✇✐t❤ ♥❡✇ q✉♦t❡s ❜❛s❡❞ ♦♥ ¯ y