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❖♥ t❤❡ ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ▼✉❧t✐❧✐♥❡❛r ▼❛♣ ❛♥❞ s♦♠❡ ❱❛r✐❛♥ts ▲é♦ ❉✉❝❛s ✶ ✱ ❆❧✐❝❡ P❡❧❧❡t✲▼❛r② ✷ ✶ ❈r②♣t♦❧♦❣② ●r♦✉♣✱ ❈❲■✱ ❆♠st❡r❞❛♠ ✷ ▲■P✱ ❊◆❙ ❞❡ ▲②♦♥ ❆s✐❛❝r②♣t ✷✵✶✽ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶ ✴ ✷✷

❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄ ❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ●●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱ ❊✉r♦❝r②♣t✳ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄ ❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ●●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱ ❊✉r♦❝r②♣t✳ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄ ❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ◮ ❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ ●●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱ ❊✉r♦❝r②♣t✳ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄ ❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ◮ ❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ●●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱ ❊✉r♦❝r②♣t✳ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❋✉♥❝t✐♦♥❛❧✐t②✿ ❆❞❞✐t✐♦♥✿ ❆❞❞✭❊♥❝ ✱ ❊♥❝ ✮ ❂ ❊♥❝ ✶ ✷ ✶ ✷ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ▼✉❧t✭❊♥❝ ✶ ✱ ❊♥❝ ✷ ✮ ❂ ❊♥❝ ✶ ✷ ✶ ✷ ✶ ✷ ✐❢ ✶ ✷ ❩❡r♦✲t❡st✿ ❩❡r♦✲t❡st✭❊♥❝ ✮ ❂ ❚r✉❡ ✐✛ ✵ ❙❡❝✉r✐t②✿ ♠✉❧t✐♣❧❡ s❡❝✉r✐t② ❞❡✜♥✐t✐♦♥s ❈r②♣t♦❣r❛♣❤✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡✜♥✐t✐♦♥✿ κ ❛s②♠♠❡tr✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉✐✛❡r❡♥t ❧❡✈❡❧s ♦❢ ❡♥❝♦❞✐♥❣s✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❜s❡ts ♦❢ { ✶ , . . . , κ } ✳ ❉❡♥♦t❡ ❜② ❊♥❝ ( a , S ) ❛ ❧❡✈❡❧✲ S ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ♠❡ss❛❣❡ a ✱ ❢♦r S ⊆ { ✶ , . . . , κ } =: [ κ ] ✳ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✸ ✴ ✷✷

❈r②♣t♦❣r❛♣❤✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡✜♥✐t✐♦♥✿ κ ❛s②♠♠❡tr✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉✐✛❡r❡♥t ❧❡✈❡❧s ♦❢ ❡♥❝♦❞✐♥❣s✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❜s❡ts ♦❢ { ✶ , . . . , κ } ✳ ❉❡♥♦t❡ ❜② ❊♥❝ ( a , S ) ❛ ❧❡✈❡❧✲ S ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ♠❡ss❛❣❡ a ✱ ❢♦r S ⊆ { ✶ , . . . , κ } =: [ κ ] ✳ ❋✉♥❝t✐♦♥❛❧✐t②✿ ❆❞❞✐t✐♦♥✿ ❆❞❞✭❊♥❝ ( a ✶ , S ) ✱ ❊♥❝ ( a ✷ , S ) ✮ ❂ ❊♥❝ ( a ✶ + a ✷ , S ) ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ▼✉❧t✭❊♥❝ ( a ✶ , S ✶ ) ✱ ❊♥❝ ( a ✷ , S ✷ ) ✮ ❂ ❊♥❝ ( a ✶ · a ✷ , S ✶ ∪ S ✷ ) ✐❢ S ✶ ∩ S ✷ = ∅ ❩❡r♦✲t❡st✿ ❩❡r♦✲t❡st✭❊♥❝ ( a , [ κ ]) ✮ ❂ ❚r✉❡ ✐✛ a = ✵ ❙❡❝✉r✐t②✿ ♠✉❧t✐♣❧❡ s❡❝✉r✐t② ❞❡✜♥✐t✐♦♥s ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✸ ✴ ✷✷

❚❤r❡❡ ♠❛✐♥ ❝❛♥❞✐❞❛t❡s ●●❍✶✸ ✱ ❈▲❚✶✸✱ ●●❍✶✺ ●●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐ ✭❊✉r♦❝r②♣t ✷✵✶✸✮ ❈▲❚✶✸✿ ❈♦r♦♥✱ ▲❡♣♦✐♥t✱ ❚✐❜♦✉❝❤✐ ✭❈r②♣t♦ ✷✵✶✸✮ ●●❍✶✺✿ ●❡♥tr②✱ ●♦r❜✉♥♦✈✱ ❍❛❧❡✈✐ ✭❚❈❈ ✷✵✶✺✮ ▼♠❛♣✿ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❝❛♥❞✐❞❛t❡s ❆♣♣❧✐❝❛t✐♦♥s✿ ❖♥❡✲r♦✉♥❞ ❦❡②✲❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ κ + ✶ ✉s❡rs ✭❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s✮ ❆ttr✐❜✉t❡ ❜❛s❡❞ ❡♥❝r②♣t✐♦♥✱ ✇✐t♥❡ss ❡♥❝r②♣t✐♦♥✱ . . . ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✭✐❖✮ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✹ ✴ ✷✷

▼♠❛♣✿ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❝❛♥❞✐❞❛t❡s ❆♣♣❧✐❝❛t✐♦♥s✿ ❖♥❡✲r♦✉♥❞ ❦❡②✲❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ κ + ✶ ✉s❡rs ✭❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s✮ ❆ttr✐❜✉t❡ ❜❛s❡❞ ❡♥❝r②♣t✐♦♥✱ ✇✐t♥❡ss ❡♥❝r②♣t✐♦♥✱ . . . ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✭✐❖✮ ❚❤r❡❡ ♠❛✐♥ ❝❛♥❞✐❞❛t❡s ●●❍✶✸ ✱ ❈▲❚✶✸✱ ●●❍✶✺ ●●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐ ✭❊✉r♦❝r②♣t ✷✵✶✸✮ ❈▲❚✶✸✿ ❈♦r♦♥✱ ▲❡♣♦✐♥t✱ ❚✐❜♦✉❝❤✐ ✭❈r②♣t♦ ✷✵✶✸✮ ●●❍✶✺✿ ●❡♥tr②✱ ●♦r❜✉♥♦✈✱ ❍❛❧❡✈✐ ✭❚❈❈ ✷✵✶✺✮ ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✹ ✴ ✷✷