rr ss trtr sts - - PowerPoint PPT Presentation

• rö❜♥❡r ❇❛s❡s ♦❢ ❙tr✉❝t✉r❡❞ ❙②st❡♠s ❛♥❞

❆♣♣❧✐❝❛t✐♦♥s t♦ ❈r②♣t♦❧♦❣②

P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r

■◆❘■❆✴❈◆❘❙✴❯♥✐✈✳ ▲♦rr❛✐♥❡✱ ❈❛r❛♠❡❧ Pr♦❥❡❝t ❏♦✉r♥é❡s ❈✷✱ ▼❛r❝❤ ✷✻✱ ✷✵✶✹

• -;)

• -;N

• -[d]);printf

• A);}

CARAMEL

✶

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s

• ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦

▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞✳

❢✶ ❢♠ ①✶ ①♥ ✇❤❡r❡ ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ❢✶ ①✶ ①♥ ✵ ✳ ✳ ✳ ❢♠ ①✶ ①♥ ✵ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥

♥ ♥

◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥

✷

❇é③♦✉t t❤❡♦r❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥

♥✳

❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s✳

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s

• ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦

▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞✳

❢✶, . . . , ❢♠ P Kr①✶, . . . , ①♥s, ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ $ ’ & ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ù ñ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ K♥ K

♥

◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥

✷

❇é③♦✉t t❤❡♦r❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥

♥✳

❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s✳

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s

• ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦

▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞✳

❢✶, . . . , ❢♠ P Kr①✶, . . . , ①♥s, ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ $ ’ & ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ù ñ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ K♥ K

♥

◆P✲❤❛r❞ ♣r♦❜❧❡♠ " ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ K “ F✷ ❇é③♦✉t t❤❡♦r❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥ K

♥✳

❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s✳

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s

• ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦

▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞✳

❢✶, . . . , ❢♠ P Kr①✶, . . . , ①♥s, ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ $ ’ & ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ù ñ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ K♥ K

♥

◆P✲❤❛r❞ ♣r♦❜❧❡♠ " ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ K “ F✷ ❇é③♦✉t t❤❡♦r❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥ K

♥✳

❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ù ñ ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s✳

✷

❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s

• ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦

▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞✳

❢✶, . . . , ❢♠ P Kr①✶, . . . , ①♥s, ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ $ ’ & ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ù ñ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ K♥ K

♥

◆P✲❤❛r❞ ♣r♦❜❧❡♠ " ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ K “ F✷ ❇é③♦✉t t❤❡♦r❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥ K

♥✳

❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ù ñ ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s✳

✷

✵✲❞✐♠❡♥s✐♦♥❛❧ s♦❧✈✐♥❣ str❛t❡❣② ✇✐t❤ ●rö❜♥❡r ❜❛s❡s

❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵

❈♦♠♣❧❡①✐t② ❆❧❣♦r✐t❤♠s ó ✏❣r❡✈❧❡①✑ ●❜ ❘♦✇ ❊❝❤❡❧♦♥ ❢♦r♠s ♦❢ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✉♣ t♦ ❞❡❣r❡❡ ❞r❡❣ ❖ ´ ♠ `♥`❞r❡❣

♥

˘ω¯ ❇✉❝❤❜❡r❣❡r ✭✶✾✻✺✮ ❋✹ ✭❋❛✉❣èr❡ ✶✾✾✾✮ ❋✺ ✭❋❛✉❣èr❡ ✷✵✵✷✮ ó ✏❧❡①✑ ●❜ ▲✐♥❡❛r ❛❧❣❡❜r❛ ✐♥ Kr❳s

■

❛s ❛ K✲ ✈❡❝t✳ s♣❛❝❡ ♦❢ ❞✐♠✳ ❉❊●p■q ❣p✉q “ ✵, ①✐ “ ❤✐p✉q ❖ ` ♥❉❊●p■q✸˘ ❋●▲▼ ❋❛✉❣èr❡✱ ●✐❛♥♥✐✱ ▲❛③❛r❞✱ ▼♦r❛ ✭✶✾✾✸✮

▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞ ❢✶ ❢♠ ✵ ❞❡❣ ❢✐ ❞✐ ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t ▼♦♥♦♠✐❛❧s ❞ ❞✐ ✳ ❈♦❧✉♠♥s✿ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❞✳ ♠✶ ♠ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✿ ♠❛①✐♠❛❧ ❞❡❣r❡❡ r❡❛❝❤❡❞ ❍✐❧❜❡rt s❡r✐❡s✿ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ❍❙ t

❞

❞✐♠ ❳ ❞ ■❞ t❞ ❞r❡❣ ❞❡❣ ❍❙ ✶

✸

✵✲❞✐♠❡♥s✐♦♥❛❧ s♦❧✈✐♥❣ str❛t❡❣② ✇✐t❤ ●rö❜♥❡r ❜❛s❡s

❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵

❈♦♠♣❧❡①✐t② ❆❧❣♦r✐t❤♠s ó ✏❣r❡✈❧❡①✑ ●❜ ❘♦✇ ❊❝❤❡❧♦♥ ❢♦r♠s ♦❢ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✉♣ t♦ ❞❡❣r❡❡ ❞r❡❣ ❖ ´ ♠ `♥`❞r❡❣

♥

˘ω¯ ❇✉❝❤❜❡r❣❡r ✭✶✾✻✺✮ ❋✹ ✭❋❛✉❣èr❡ ✶✾✾✾✮ ❋✺ ✭❋❛✉❣èr❡ ✷✵✵✷✮ ó ✏❧❡①✑ ●❜ ▲✐♥❡❛r ❛❧❣❡❜r❛ ✐♥ Kr❳s

■

❛s ❛ K✲ ✈❡❝t✳ s♣❛❝❡ ♦❢ ❞✐♠✳ ❉❊●p■q ❣p✉q “ ✵, ①✐ “ ❤✐p✉q ❖ ` ♥❉❊●p■q✸˘ ❋●▲▼ ❋❛✉❣èr❡✱ ●✐❛♥♥✐✱ ▲❛③❛r❞✱ ▼♦r❛ ✭✶✾✾✸✮

▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞ ❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵, ❞❡❣p❢✐q “ ❞✐ ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t P ▼♦♥♦♠✐❛❧sp❞ ´ ❞✐q✳ ❈♦❧✉♠♥s✿ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❞✳ ♠✶ ą ¨ ¨ ¨ ą ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ¨ ˚ ˚ ˝ ˛ ‹ ‹ ‚ ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✿ ♠❛①✐♠❛❧ ❞❡❣r❡❡ r❡❛❝❤❡❞ ❍✐❧❜❡rt s❡r✐❡s✿ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ❍❙ t

❞

❞✐♠ ❳ ❞ ■❞ t❞ ❞r❡❣ ❞❡❣ ❍❙ ✶

✸

✵✲❞✐♠❡♥s✐♦♥❛❧ s♦❧✈✐♥❣ str❛t❡❣② ✇✐t❤ ●rö❜♥❡r ❜❛s❡s

❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵

❈♦♠♣❧❡①✐t② ❆❧❣♦r✐t❤♠s ó ✏❣r❡✈❧❡①✑ ●❜ ❘♦✇ ❊❝❤❡❧♦♥ ❢♦r♠s ♦❢ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✉♣ t♦ ❞❡❣r❡❡ ❞r❡❣ ❖ ´ ♠ `♥`❞r❡❣

♥

˘ω¯ ❇✉❝❤❜❡r❣❡r ✭✶✾✻✺✮ ❋✹ ✭❋❛✉❣èr❡ ✶✾✾✾✮ ❋✺ ✭❋❛✉❣èr❡ ✷✵✵✷✮ ó ✏❧❡①✑ ●❜ ▲✐♥❡❛r ❛❧❣❡❜r❛ ✐♥ Kr❳s

■

❛s ❛ K✲ ✈❡❝t✳ s♣❛❝❡ ♦❢ ❞✐♠✳ ❉❊●p■q ❣p✉q “ ✵, ①✐ “ ❤✐p✉q ❖ ` ♥❉❊●p■q✸˘ ❋●▲▼ ❋❛✉❣èr❡✱ ●✐❛♥♥✐✱ ▲❛③❛r❞✱ ▼♦r❛ ✭✶✾✾✸✮

▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞ ❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵, ❞❡❣p❢✐q “ ❞✐ ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t P ▼♦♥♦♠✐❛❧sp❞ ´ ❞✐q✳ ❈♦❧✉♠♥s✿ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❞✳ ♠✶ ą ¨ ¨ ¨ ą ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ¨ ˚ ˚ ˝ ˛ ‹ ‹ ‚ ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✿ ♠❛①✐♠❛❧ ❞❡❣r❡❡ r❡❛❝❤❡❞ ❍✐❧❜❡rt s❡r✐❡s✿ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ❍❙ptq “ ÿ

❞PN

❞✐♠pKr❳s❞{■❞qt❞ ❞r❡❣ “ ❞❡❣p❍❙q ` ✶

✸

❙tr✉❝t✉r❡❞ ❙②st❡♠s ✐♥ ❈r②♣t♦❧♦❣②

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ r❛♥❦ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ď r ▼✉❧t✐❤♦♠ s②st❡♠s ❳ ❨ ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥♦rsr

✶

❢✶ ✶ ❢✶ q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣ ✶ ❢♣ q ✵ ❙♣❛rs❡ s②st❡♠s ❢✶ ❢♣ ✵ ❢✐

▼ ❛ ✐

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ❢ ✷ ❢ ✵ ♠♦❞ ①✷

✶

①✶ ①✷

♥

①♥

• ❇ ♦❢

str✉❝t✉r❡❞ s②st❡♠s

❈♦♠♣❧❡①✐t②❄ ❉❡❞✐❝❛t❡❞ ❆❧❣♦r✐t❤♠s❄ ❊①♣❡r✐♠❡♥t❛❧ ✈❛❧✐❞❛t✐♦♥ ❛♥❞ ✐♠♣❛❝t ♦♥ ❛♣♣❧✐❝❛t✐♦♥s❄

✹

❙tr✉❝t✉r❡❞ ❙②st❡♠s ✐♥ ❈r②♣t♦❧♦❣②

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ r❛♥❦ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ď r ▼✉❧t✐❤♦♠ s②st❡♠s Kr❳s b Kr❨ s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚“ ✵ ❙♣❛rs❡ s②st❡♠s ❢✶ “ ¨ ¨ ¨ “ ❢♣ “ ✵ ❢✐ “ ř

µP▼ ❛pµ,✐qµ

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ❢ ✷ ❢ ✵ ♠♦❞ ①✷

✶

①✶ ①✷

♥

①♥

• ❇ ♦❢

str✉❝t✉r❡❞ s②st❡♠s

❈♦♠♣❧❡①✐t②❄ ❉❡❞✐❝❛t❡❞ ❆❧❣♦r✐t❤♠s❄ ❊①♣❡r✐♠❡♥t❛❧ ✈❛❧✐❞❛t✐♦♥ ❛♥❞ ✐♠♣❛❝t ♦♥ ❛♣♣❧✐❝❛t✐♦♥s❄

✹

❙tr✉❝t✉r❡❞ ❙②st❡♠s ✐♥ ❈r②♣t♦❧♦❣②

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ r❛♥❦ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ď r ▼✉❧t✐❤♦♠ s②st❡♠s Kr❳s b Kr❨ s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚“ ✵ ❙♣❛rs❡ s②st❡♠s ❢✶ “ ¨ ¨ ¨ “ ❢♣ “ ✵ ❢✐ “ ř

µP▼ ❛pµ,✐qµ

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ❢ ✷ ´ ❢ “ ✵ ♠♦❞ x①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥y

• ❇ ♦❢

str✉❝t✉r❡❞ s②st❡♠s

❈♦♠♣❧❡①✐t②❄ ❉❡❞✐❝❛t❡❞ ❆❧❣♦r✐t❤♠s❄ ❊①♣❡r✐♠❡♥t❛❧ ✈❛❧✐❞❛t✐♦♥ ❛♥❞ ✐♠♣❛❝t ♦♥ ❛♣♣❧✐❝❛t✐♦♥s❄

✹

❙tr✉❝t✉r❡❞ ❙②st❡♠s ✐♥ ❈r②♣t♦❧♦❣②

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ r❛♥❦ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ď r ▼✉❧t✐❤♦♠ s②st❡♠s Kr❳s b Kr❨ s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚“ ✵ ❙♣❛rs❡ s②st❡♠s ❢✶ “ ¨ ¨ ¨ “ ❢♣ “ ✵ ❢✐ “ ř

µP▼ ❛pµ,✐qµ

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ❢ ✷ ´ ❢ “ ✵ ♠♦❞ x①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥y

• ❇ ♦❢

str✉❝t✉r❡❞ s②st❡♠s

❈♦♠♣❧❡①✐t②❄ ❉❡❞✐❝❛t❡❞ ❆❧❣♦r✐t❤♠s❄ ❊①♣❡r✐♠❡♥t❛❧ ✈❛❧✐❞❛t✐♦♥ ❛♥❞ ✐♠♣❛❝t ♦♥ ❛♣♣❧✐❝❛t✐♦♥s❄

✹

❙tr✉❝t✉r❡❞ ❙②st❡♠s ✐♥ ❈r②♣t♦❧♦❣②

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ r❛♥❦ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ď r ▼✉❧t✐❤♦♠ s②st❡♠s Kr❳s b Kr❨ s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚“ ✵ ❙♣❛rs❡ s②st❡♠s ❢✶ “ ¨ ¨ ¨ “ ❢♣ “ ✵ ❢✐ “ ř

µP▼ ❛pµ,✐qµ

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ❢ ✷ ´ ❢ “ ✵ ♠♦❞ x①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥y

• ❇ ♦❢

str✉❝t✉r❡❞ s②st❡♠s

❈♦♠♣❧❡①✐t②❄ ❉❡❞✐❝❛t❡❞ ❆❧❣♦r✐t❤♠s❄ ❊①♣❡r✐♠❡♥t❛❧ ✈❛❧✐❞❛t✐♦♥ ❛♥❞ ✐♠♣❛❝t ♦♥ ❛♣♣❧✐❝❛t✐♦♥s❄

✹

▼✐♥❘❛♥❦

r P N✳ ▼✵, . . . , ▼♥✿ ♥ ` ✶ ♠❛tr✐❝❡s ♦❢ s✐③❡ ♣ ˆ q✳ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ❋✐♥❞ λ✶, . . . , λ♥ s✉❝❤ t❤❛t ❘❛♥❦ ˜ ▼✵ ´

♥

ÿ

✐“✶

λ✐▼✐ ¸ ď r ▼✉❧t✐✈❛r✐❛t❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❊✐❣❡♥❱❛❧✉❡ ♣r♦❜❧❡♠✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❝r②♣t♦❧♦❣②✱ ❝♦❞✐♥❣ t❤❡♦r②✱ ❣❡♦♠❡tr②✱ ✳✳✳ ❑✐♣♥✐s✴❙❤❛♠✐r ❈r②♣t♦✬✾✾ ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t✱ ❈r②♣t♦ ✬✵✽ ❈♦✉rt♦✐s ❈r②♣t♦✬✵✶

• ❛❜♦r✐t✴❘✉❛tt❛✴❙❝❤r❡❦✬✶✸

❋✉♥❞❛♠❡♥t❛❧ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❇✉ss✴❋r❛♥❞s❡♥✴❙❤❛❧❧✐t✱ ✶✾✾✾✳

✺

▼✐♥❘❛♥❦

r P N✳ ▼✵, . . . , ▼♥✿ ♥ ` ✶ ♠❛tr✐❝❡s ♦❢ s✐③❡ ♣ ˆ q✳ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ ❋✐♥❞ λ✶, . . . , λ♥ s✉❝❤ t❤❛t ❘❛♥❦ ˜ ▼✵ ´

♥

ÿ

✐“✶

λ✐▼✐ ¸ ď r ▼✉❧t✐✈❛r✐❛t❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❊✐❣❡♥❱❛❧✉❡ ♣r♦❜❧❡♠✳ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❝r②♣t♦❧♦❣②✱ ❝♦❞✐♥❣ t❤❡♦r②✱ ❣❡♦♠❡tr②✱ ✳✳✳ ❑✐♣♥✐s✴❙❤❛♠✐r ❈r②♣t♦✬✾✾ ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t✱ ❈r②♣t♦ ✬✵✽ ❈♦✉rt♦✐s ❈r②♣t♦✬✵✶

• ❛❜♦r✐t✴❘✉❛tt❛✴❙❝❤r❡❦✬✶✸

❋✉♥❞❛♠❡♥t❛❧ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❇✉ss✴❋r❛♥❞s❡♥✴❙❤❛❧❧✐t✱ ✶✾✾✾✳

✺

❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s

▲❡t r ă q ă ♣ ❜❡ ✐♥t❡❣❡rs ❛♥❞ ▼ ❜❡ t❤❡ ♣ ˆ q ♠❛tr✐① ▼p❳q “ » — – ❢✶,✶p❳q ¨ ¨ ¨ ¨ ¨ ¨ ❢✶,qp❳q ✳ ✳ ✳ ¨ ¨ ¨ ¨ ¨ ¨ ✳ ✳ ✳ ❢♣,✶p❳q ¨ ¨ ¨ ¨ ¨ ¨ ❢♣,qp❳q fi ffi fl ✇✐t❤ ❢✐,❥ P Kr①✶, . . . , ①♥s ♦❢ ❞❡❣r❡❡ ❉✳

• ❡♥❡r❛❧✐③❡❞ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠

❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ①

♥ s✉❝❤ t❤❛t r❛♥❦ ▼ ①

r✳ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣ ♣r♦❜❧❡♠✿ ▼✐♥♦rsr

✶ ▼ ❳

✵

✻

❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s

▲❡t r ă q ă ♣ ❜❡ ✐♥t❡❣❡rs ❛♥❞ ▼ ❜❡ t❤❡ ♣ ˆ q ♠❛tr✐① ▼p❳q “ » — – ❢✶,✶p❳q ¨ ¨ ¨ ¨ ¨ ¨ ❢✶,qp❳q ✳ ✳ ✳ ¨ ¨ ¨ ¨ ¨ ¨ ✳ ✳ ✳ ❢♣,✶p❳q ¨ ¨ ¨ ¨ ¨ ¨ ❢♣,qp❳q fi ffi fl ✇✐t❤ ❢✐,❥ P Kr①✶, . . . , ①♥s ♦❢ ❞❡❣r❡❡ ❉✳

• ❡♥❡r❛❧✐③❡❞ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠

❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① P K

♥ s✉❝❤ t❤❛t r❛♥❦p▼p①qq ď r✳

♣♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣ ♣r♦❜❧❡♠✿ ▼✐♥♦rsr`✶p▼p❳qq “ ✵

✻

▼❛✐♥ r❡s✉❧ts

✇✐t❤ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ▼✳ ❙❛❢❡② ❊❧ ❉✐♥ ♣ ˆ q ♠❛tr✐①✳ ♥ ✈❛r✐❛❜❧❡s✳ ❊♥tr✐❡s ♦❢ ❞❡❣r❡❡ ❉✳ ❩❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✭♥ “ p♣ ´ rqpq ´ rq✮✳

❙②st❡♠ Ý Ñ ❣r❡✈❧❡① ●❇ Ý Ñ

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ˆ´ ♣ r ` ✶ ¯´ q r ` ✶ ¯´♥ ` ❞r❡❣ ❞r❡❣ ¯ω˙ ❖ ` ♥ ¨ ❉❊●✸˘

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t② ✭✉♥❞❡r ❣❡♥❡r✐❝✐t② ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✮

❞r❡❣ ❉r q r ❉ ✶ ♣ r q r ✶ ❉❊● ❉ ♣

r q r q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐

❢❛♠✐❧✐❡s ♦❢ ●❡♥❡r❛❧✐③❡❞ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠s t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳

✼

▼❛✐♥ r❡s✉❧ts

✇✐t❤ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ▼✳ ❙❛❢❡② ❊❧ ❉✐♥ ♣ ˆ q ♠❛tr✐①✳ ♥ ✈❛r✐❛❜❧❡s✳ ❊♥tr✐❡s ♦❢ ❞❡❣r❡❡ ❉✳ ❩❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✭♥ “ p♣ ´ rqpq ´ rq✮✳

❙②st❡♠ Ý Ñ ❣r❡✈❧❡① ●❇ Ý Ñ

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ˆ´ ♣ r ` ✶ ¯´ q r ` ✶ ¯´♥ ` ❞r❡❣ ❞r❡❣ ¯ω˙ ❖ ` ♥ ¨ ❉❊●✸˘

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t② ✭✉♥❞❡r ❣❡♥❡r✐❝✐t② ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✮

❞r❡❣ “ ❉rpq ´ rq ` p❉ ´ ✶qp♣ ´ rqpq ´ rq ` ✶ ❉❊● ❉ ♣

r q r q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐

❢❛♠✐❧✐❡s ♦❢ ●❡♥❡r❛❧✐③❡❞ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠s t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳

✼

▼❛✐♥ r❡s✉❧ts

✇✐t❤ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ▼✳ ❙❛❢❡② ❊❧ ❉✐♥ ♣ ˆ q ♠❛tr✐①✳ ♥ ✈❛r✐❛❜❧❡s✳ ❊♥tr✐❡s ♦❢ ❞❡❣r❡❡ ❉✳ ❩❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✭♥ “ p♣ ´ rqpq ´ rq✮✳

❙②st❡♠ Ý Ñ ❣r❡✈❧❡① ●❇ Ý Ñ

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ˆ´ ♣ r ` ✶ ¯´ q r ` ✶ ¯´♥ ` ❞r❡❣ ❞r❡❣ ¯ω˙ ❖ ` ♥ ¨ ❉❊●✸˘

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t② ✭✉♥❞❡r ❣❡♥❡r✐❝✐t② ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✮

❞r❡❣ “ ❉rpq ´ rq ` p❉ ´ ✶qp♣ ´ rqpq ´ rq ` ✶ ❉❊● “ ❉p♣´rqpq´rq

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q!

❢❛♠✐❧✐❡s ♦❢ ●❡♥❡r❛❧✐③❡❞ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠s t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳

✼

▼❛✐♥ r❡s✉❧ts

✇✐t❤ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ▼✳ ❙❛❢❡② ❊❧ ❉✐♥ ♣ ˆ q ♠❛tr✐①✳ ♥ ✈❛r✐❛❜❧❡s✳ ❊♥tr✐❡s ♦❢ ❞❡❣r❡❡ ❉✳ ❩❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✭♥ “ p♣ ´ rqpq ´ rq✮✳

❙②st❡♠ Ý Ñ ❣r❡✈❧❡① ●❇ Ý Ñ

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ˆ´ ♣ r ` ✶ ¯´ q r ` ✶ ¯´♥ ` ❞r❡❣ ❞r❡❣ ¯ω˙ ❖ ` ♥ ¨ ❉❊●✸˘

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t② ✭✉♥❞❡r ❣❡♥❡r✐❝✐t② ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✮

❞r❡❣ “ ❉rpq ´ rq ` p❉ ´ ✶qp♣ ´ rqpq ´ rq ` ✶ ❉❊● “ ❉p♣´rqpq´rq

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q!

❢❛♠✐❧✐❡s ♦❢ ●❡♥❡r❛❧✐③❡❞ ▼✐♥❘❛♥❦ Pr♦❜❧❡♠s t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s✳

✼

❆♣♣❧✐❝❛t✐♦♥ ✐♥ ❈r②♣t♦❧♦❣②

❈♦✉rt♦✐s✱ ❈r②♣t♦✬✵✶ ❆✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡ ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ▼✐♥❘❛♥❦✳ Pr♦♣♦s❡❞ ♣❛r❛♠❡t❡rs✿ ♣ “ q✱ K “ F✻✺✺✷✶✱ r “ q ´ ✸✳ ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦✿ ❖ q✾ q s❡❝✉r✐t② ❋●❜ ❋✺✰❋●▲▼ ✻ ✷✶✵✻ ✷✳✽s ✼ ✷✶✷✷ ✶✸✵s ✶✶ ✷✶✸✽ ✷✸✽ ❞❛②s ✭❡st✳✮ ♦♥ ✻✹ q✉❛❞❝♦r❡ ♣r♦❝✳ ❇♦tt❧❡♥❡❝❦ ❢♦r q ✶✶✿ ❝♦♠♣✉t✐♥❣ t❤❡ ✐♥♣✉t s②st❡♠✳

✽

❆♣♣❧✐❝❛t✐♦♥ ✐♥ ❈r②♣t♦❧♦❣②

❈♦✉rt♦✐s✱ ❈r②♣t♦✬✵✶ ❆✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡ ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ▼✐♥❘❛♥❦✳ Pr♦♣♦s❡❞ ♣❛r❛♠❡t❡rs✿ ♣ “ q✱ K “ F✻✺✺✷✶✱ r “ q ´ ✸✳ ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦✿ ❖pq✾q q s❡❝✉r✐t② ❋●❜ ❋✺✰❋●▲▼ ✻ ✷✶✵✻ ✷✳✽s ✼ ✷✶✷✷ ✶✸✵s ✶✶ ✷✶✸✽ ✷✸✽ ❞❛②s ✭❡st✳✮ ♦♥ ✻✹ q✉❛❞❝♦r❡ ♣r♦❝✳ ❇♦tt❧❡♥❡❝❦ ❢♦r q ✶✶✿ ❝♦♠♣✉t✐♥❣ t❤❡ ✐♥♣✉t s②st❡♠✳

✽

❆♣♣❧✐❝❛t✐♦♥ ✐♥ ❈r②♣t♦❧♦❣②

❈♦✉rt♦✐s✱ ❈r②♣t♦✬✵✶ ❆✉t❤❡♥t✐❝❛t✐♦♥ s❝❤❡♠❡ ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ▼✐♥❘❛♥❦✳ Pr♦♣♦s❡❞ ♣❛r❛♠❡t❡rs✿ ♣ “ q✱ K “ F✻✺✺✷✶✱ r “ q ´ ✸✳ ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦✿ ❖pq✾q q s❡❝✉r✐t② ❋●❜ ❋✺✰❋●▲▼ ✻ ✷✶✵✻ ✷✳✽s ✼ ✷✶✷✷ ✶✸✵s ✶✶ ✷✶✸✽ ✷✸✽ ❞❛②s ✭❡st✳✮ ♦♥ ✻✹ q✉❛❞❝♦r❡ ♣r♦❝✳ ❇♦tt❧❡♥❡❝❦ ❢♦r q “ ✶✶✿ ❝♦♠♣✉t✐♥❣ t❤❡ ✐♥♣✉t s②st❡♠✳

✽

❘♦❛❞♠❛♣ ♦❢ t❤❡ ♣r♦♦❢

D “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ✈✶,✶ . . . ✈✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♣,✶ . . . ✈♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s ❚❤♦♠✴P♦rt❡♦✉s ✼✶✱ ●✐❛♠❜❡❧❧✐ ✵✹✱ ❍❛rr✐s✴❚✉ ✽✹ ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s

q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐ ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ❆❜❤②❛♥❦❛r ✬✽✽ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t t

r ✷

✶ t ♣q

♣ r q r

▼✐♥♦rsr

✶

❢✶ ✶ ❢✶ q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣ ✶ ❢♣ q ❊♥tr✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s ❉ ♣

r q r q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t❉ ✶ t❉

♣ r q r

t

r ✷

✶ t ♥

r ˆ r ♠❛tr✐①✿ ❆✐,❥ptq “ ř

ℓ

`♣´✐

ℓ

˘`q´❥

ℓ

˘ tℓ tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ ❜② ❛❞❞✐♥❣ ✈✐ ❥ ❢✐ ❥

✾

❘♦❛❞♠❛♣ ♦❢ t❤❡ ♣r♦♦❢

D “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ✈✶,✶ . . . ✈✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♣,✶ . . . ✈♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s ❚❤♦♠✴P♦rt❡♦✉s ✼✶✱ ●✐❛♠❜❡❧❧✐ ✵✹✱ ❍❛rr✐s✴❚✉ ✽✹ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ❆❜❤②❛♥❦❛r ✬✽✽ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙Dptq “ ❞❡tp❆ptqq t

´r

✷

¯

p✶ ´ tq♣q´p♣´rqpq´rq ▼✐♥♦rsr

✶

❢✶ ✶ ❢✶ q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣ ✶ ❢♣ q ❊♥tr✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s ❉ ♣

r q r q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t❉ ✶ t❉

♣ r q r

t

r ✷

✶ t ♥

r ˆ r ♠❛tr✐①✿ ❆✐,❥ptq “ ř

ℓ

`♣´✐

ℓ

˘`q´❥

ℓ

˘ tℓ tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ ❜② ❛❞❞✐♥❣ ✈✐ ❥ ❢✐ ❥

✾

❘♦❛❞♠❛♣ ♦❢ t❤❡ ♣r♦♦❢

D “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ✈✶,✶ . . . ✈✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♣,✶ . . . ✈♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s ❚❤♦♠✴P♦rt❡♦✉s ✼✶✱ ●✐❛♠❜❡❧❧✐ ✵✹✱ ❍❛rr✐s✴❚✉ ✽✹ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ❆❜❤②❛♥❦❛r ✬✽✽ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙Dptq “ ❞❡tp❆ptqq t

´r

✷

¯

p✶ ´ tq♣q´p♣´rqpq´rq I “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s ❉ ♣

r q r q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t❉ ✶ t❉

♣ r q r

t

r ✷

✶ t ♥

r ˆ r ♠❛tr✐①✿ ❆✐,❥ptq “ ř

ℓ

`♣´✐

ℓ

˘`q´❥

ℓ

˘ tℓ tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ ❜② ❛❞❞✐♥❣ ✈✐ ❥ ❢✐ ❥

✾

❘♦❛❞♠❛♣ ♦❢ t❤❡ ♣r♦♦❢

D “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ✈✶,✶ . . . ✈✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♣,✶ . . . ✈♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s ❚❤♦♠✴P♦rt❡♦✉s ✼✶✱ ●✐❛♠❜❡❧❧✐ ✵✹✱ ❍❛rr✐s✴❚✉ ✽✹ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ❆❜❤②❛♥❦❛r ✬✽✽ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙Dptq “ ❞❡tp❆ptqq t

´r

✷

¯

p✶ ´ tq♣q´p♣´rqpq´rq I “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❞❡❣r❡❡ ♦❢ ✐s ❉ ♣

r q r q r ✶ ✐ ✵

✐ ♣ ✐ q ✶ ✐ ♣ r ✐ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ✐s ❍❙ t ❞❡t ❆ t❉ ✶ t❉

♣ r q r

t

r ✷

✶ t ♥

r ˆ r ♠❛tr✐①✿ ❆✐,❥ptq “ ř

ℓ

`♣´✐

ℓ

˘`q´❥

ℓ

˘ tℓ tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ D ❜② ❛❞❞✐♥❣ x✈✐,❥ ´ ❢✐,❥y

✾

❘♦❛❞♠❛♣ ♦❢ t❤❡ ♣r♦♦❢

D “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ✈✶,✶ . . . ✈✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♣,✶ . . . ✈♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s ❚❤♦♠✴P♦rt❡♦✉s ✼✶✱ ●✐❛♠❜❡❧❧✐ ✵✹✱ ❍❛rr✐s✴❚✉ ✽✹ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ❆❜❤②❛♥❦❛r ✬✽✽ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙Dptq “ ❞❡tp❆ptqq t

´r

✷

¯

p✶ ´ tq♣q´p♣´rqpq´rq I “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❞❡❣r❡❡ ♦❢ I ✐s ❉p♣´rqpq´rq

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ I ✐s ❍❙Iptq “ ❞❡tp❆pt❉qqp✶ ´ t❉qp♣´rqpq´rq t

´r

✷

¯

p✶ ´ tq♥

r ˆ r ♠❛tr✐①✿ ❆✐,❥ptq “ ř

ℓ

`♣´✐

ℓ

˘`q´❥

ℓ

˘ tℓ tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ D ❜② ❛❞❞✐♥❣ x✈✐,❥ ´ ❢✐,❥y

✾

❘♦❛❞♠❛♣ ♦❢ t❤❡ ♣r♦♦❢

D “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ✈✶,✶ . . . ✈✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✈♣,✶ . . . ✈♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ✈❛r✐❛❜❧❡s ❚❤♦♠✴P♦rt❡♦✉s ✼✶✱ ●✐❛♠❜❡❧❧✐ ✵✹✱ ❍❛rr✐s✴❚✉ ✽✹ ❚❤❡ ❞❡❣r❡❡ ♦❢ D ✐s

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❈♦♥❝❛✴❍❡r③♦❣ ❆▼❙✬✾✹✱ ❆❜❤②❛♥❦❛r ✬✽✽ ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ D ✐s ❍❙Dptq “ ❞❡tp❆ptqq t

´r

✷

¯

p✶ ´ tq♣q´p♣´rqpq´rq I “ ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ ❊♥tr✐❡s ❛r❡ ♣♦❧②♥♦♠✐❛❧s ❚❤❡ ❞❡❣r❡❡ ♦❢ I ✐s ❉p♣´rqpq´rq

q´r´✶

ź

✐“✵

✐!p♣ ` ✐q! pq ´ ✶ ´ ✐q!p♣ ´ r ` ✐q! ❚❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ I ✐s ❍❙Iptq “ ❞❡tp❆pt❉qqp✶ ´ t❉qp♣´rqpq´rq t

´r

✷

¯

p✶ ´ tq♥

r ˆ r ♠❛tr✐①✿ ❆✐,❥ptq “ ř

ℓ

`♣´✐

ℓ

˘`q´❥

ℓ

˘ tℓ tr❛♥s❢❡r ♦❢ ♣r♦♣❡rt✐❡s ♦❢ D ❜② ❛❞❞✐♥❣ x✈✐,❥ ´ ❢✐,❥y ■♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢✿ ❈♦❤❡♥✕▼❛❝❛✉❧❛② r✐♥❣s❀ q✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s✳

✾

❙tr✉❝t✉r❡❞ ❙②st❡♠s ✐♥ ❈r②♣t♦❧♦❣②

❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ▼✐♥❘❛♥❦ Pr♦❜❧❡♠ r❛♥❦ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚ď r ▼✉❧t✐❤♦♠ s②st❡♠s Kr❳s b Kr❨ s ❉❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s ▼✐♥♦rsr`✶ ¨ ˚ ˝ ❢✶,✶ . . . ❢✶,q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,q ˛ ‹ ‚“ ✵ ❙♣❛rs❡ s②st❡♠s ❢✶ “ ¨ ¨ ¨ “ ❢♣ “ ✵ ❢✐ “ ř

µP▼ ❛pµ,✐qµ

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s ❢ ✷ ´ ❢ “ ✵ ♠♦❞ x①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥y ✶✵

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s

✇✐t❤ ▼✳ ❇❛r❞❡t✱ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ❇✳ ❙❛❧✈② ❇♦♦❧❡❛♥ ▼◗ Pr♦❜❧❡♠ ❢✶, . . . , ❢♠ P F✷r①✶, . . . , ①♥s q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❋✐♥❞ ♦♥❡✴❛❧❧ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ $ ’ ’ ’ & ’ ’ ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ❢✷p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❙❆❚✳ ❙❡❝✉r✐t② ♦❢ s❡✈❡r❛❧ ♠♦❞❡r♥ ❝r②♣t♦s②st❡♠s r❡❧✐❡s ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ❇♦♦❧❡❛♥ ▼◗ ✭◗❯❆❉✱✳✳✳✮✳ ❆s②♠♣t♦t✐❝❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ♦❢ ♣❛r❛♠❡t❡r ✷♥

♠

❢❡✇ s♦❧✉t✐♦♥s ❢♦r r❛♥❞♦♠ s②st❡♠s ✭❋✉s❝♦✴❇❛❝❤✱ ❚❆▼❈✬✵✼✮✳ ❇❡st ♣r♦✈❡♥ ✇♦rst ❝❛s❡ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ ❡①❤❛✉st✐✈❡ s❡❛r❝❤✱ ✹ ✷♥ ❧♦❣✷ ♥ ✭❇♦✉✐❧❧❛❣✉❡t✴❈❤❡♥✴❈❤❡♥❣✴❈❤♦✉✴◆✐❡❞❡r❤❛❣❡♥✴❙❤❛♠✐r✴❨❛♥❣ ❈❍❊❙✬✶✵✮✳

Pr♦❜❧❡♠✿ ❝♦♥str✉❝t ❛♥ ❖ ✷❝♥ ❛❧❣♦r✐t❤♠✱ ✇✐t❤ ❝ ✶✳

✶✶

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s

✇✐t❤ ▼✳ ❇❛r❞❡t✱ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ❇✳ ❙❛❧✈② ❇♦♦❧❡❛♥ ▼◗ Pr♦❜❧❡♠ ❢✶, . . . , ❢♠ P F✷r①✶, . . . , ①♥s q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❋✐♥❞ ♦♥❡✴❛❧❧ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ $ ’ ’ ’ & ’ ’ ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ❢✷p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❙❆❚✳ ❙❡❝✉r✐t② ♦❢ s❡✈❡r❛❧ ♠♦❞❡r♥ ❝r②♣t♦s②st❡♠s r❡❧✐❡s ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ❇♦♦❧❡❛♥ ▼◗ ✭◗❯❆❉✱✳✳✳✮✳ ❆s②♠♣t♦t✐❝❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ♦❢ ♣❛r❛♠❡t❡r ✷♥´♠ ❢❡✇ s♦❧✉t✐♦♥s ❢♦r r❛♥❞♦♠ s②st❡♠s ✭❋✉s❝♦✴❇❛❝❤✱ ❚❆▼❈✬✵✼✮✳ ❇❡st ♣r♦✈❡♥ ✇♦rst ❝❛s❡ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ ❡①❤❛✉st✐✈❡ s❡❛r❝❤✱ ✹ ¨ ✷♥ ❧♦❣✷ ♥ ✭❇♦✉✐❧❧❛❣✉❡t✴❈❤❡♥✴❈❤❡♥❣✴❈❤♦✉✴◆✐❡❞❡r❤❛❣❡♥✴❙❤❛♠✐r✴❨❛♥❣ ❈❍❊❙✬✶✵✮✳

Pr♦❜❧❡♠✿ ❝♦♥str✉❝t ❛♥ ❖ ✷❝♥ ❛❧❣♦r✐t❤♠✱ ✇✐t❤ ❝ ✶✳

✶✶

◗✉❛❞r❛t✐❝ ❜♦♦❧❡❛♥ s②st❡♠s

✇✐t❤ ▼✳ ❇❛r❞❡t✱ ❏✳✲❈✳ ❋❛✉❣èr❡✱ ❇✳ ❙❛❧✈② ❇♦♦❧❡❛♥ ▼◗ Pr♦❜❧❡♠ ❢✶, . . . , ❢♠ P F✷r①✶, . . . , ①♥s q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s✳ ❋✐♥❞ ♦♥❡✴❛❧❧ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ $ ’ ’ ’ & ’ ’ ’ % ❢✶p①✶, . . . , ①♥q “ ✵ ❢✷p①✶, . . . , ①♥q “ ✵ ✳ ✳ ✳ ❢♠p①✶, . . . , ①♥q “ ✵ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❙❆❚✳ ❙❡❝✉r✐t② ♦❢ s❡✈❡r❛❧ ♠♦❞❡r♥ ❝r②♣t♦s②st❡♠s r❡❧✐❡s ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ❇♦♦❧❡❛♥ ▼◗ ✭◗❯❆❉✱✳✳✳✮✳ ❆s②♠♣t♦t✐❝❛❧❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ♦❢ ♣❛r❛♠❡t❡r ✷♥´♠ ❢❡✇ s♦❧✉t✐♦♥s ❢♦r r❛♥❞♦♠ s②st❡♠s ✭❋✉s❝♦✴❇❛❝❤✱ ❚❆▼❈✬✵✼✮✳ ❇❡st ♣r♦✈❡♥ ✇♦rst ❝❛s❡ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ ❡①❤❛✉st✐✈❡ s❡❛r❝❤✱ ✹ ¨ ✷♥ ❧♦❣✷ ♥ ✭❇♦✉✐❧❧❛❣✉❡t✴❈❤❡♥✴❈❤❡♥❣✴❈❤♦✉✴◆✐❡❞❡r❤❛❣❡♥✴❙❤❛♠✐r✴❨❛♥❣ ❈❍❊❙✬✶✵✮✳

Pr♦❜❧❡♠✿ ❝♦♥str✉❝t ❛♥ ❖p✷❝♥q ❛❧❣♦r✐t❤♠✱ ✇✐t❤ ❝ ă ✶✳

✶✶

❘❡❧❛t❡❞ ✇♦r❦s

❆❧❣♦r✐t❤♠✐❝ ❚♦♦❧s ❊①❤❛✉st✐✈❡ s❡❛r❝❤✿ ❇♦✉✐❧❧❛❣✉❡t✴❈❤❡♥✴❈❤❡♥❣✴❈❤♦✉✴◆✐❡❞❡r❤❛❣❡♥✴❙❤❛♠✐r✴❨❛♥❣ ❈❍❊❙✬✶✵✱✳ ✳ ✳ ❀ ❙❆❚✲❙♦❧✈❡rs✿ ❉❛✈✐s✴P✉t♥❛♠✴▲♦❣❡♠❛♥♥✴▲♦✈❡❧❛♥❞ ❏✳ ♦❢ ❆❈▼✬✻✵✱ ❈♦♠♠✳ ♦❢ ❆❈▼✬✻✷❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s❀

❍②❜r✐❞ ❛♣♣r♦❛❝❤✿ ❇❡tt❛❧❡✴❋❛✉❣èr❡✴P❡rr❡t ❏✳ ♦❢ ▼❛t❤✳ ❈r②♣t♦✬✵✾✳

♦✈❡r

✷✱ r❛♥❞♦♠ s②st❡♠s

❣❡♥❡r✐❝ s②st❡♠s

✶✷

❘❡❧❛t❡❞ ✇♦r❦s

❆❧❣♦r✐t❤♠✐❝ ❚♦♦❧s ❊①❤❛✉st✐✈❡ s❡❛r❝❤✿ ❇♦✉✐❧❧❛❣✉❡t✴❈❤❡♥✴❈❤❡♥❣✴❈❤♦✉✴◆✐❡❞❡r❤❛❣❡♥✴❙❤❛♠✐r✴❨❛♥❣ ❈❍❊❙✬✶✵✱✳ ✳ ✳ ❀ ❙❆❚✲❙♦❧✈❡rs✿ ❉❛✈✐s✴P✉t♥❛♠✴▲♦❣❡♠❛♥♥✴▲♦✈❡❧❛♥❞ ❏✳ ♦❢ ❆❈▼✬✻✵✱ ❈♦♠♠✳ ♦❢ ❆❈▼✬✻✷❀

• rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s❀

❍②❜r✐❞ ❛♣♣r♦❛❝❤✿ ❇❡tt❛❧❡✴❋❛✉❣èr❡✴P❡rr❡t ❏✳ ♦❢ ▼❛t❤✳ ❈r②♣t♦✬✵✾✳

♦✈❡r F✷✱ r❛♥❞♦♠ s②st❡♠s ‰ ❣❡♥❡r✐❝ s②st❡♠s

✶✷

▼❛✐♥ r❡s✉❧ts

❢✶, . . . , ❢♠ P F✷r①✶, . . . , ①♥s. ❆❧❣♦r✐t❤♠✿ ✉s❡ ✭s♣❛rs❡✮ ❧✐♥❡❛r ❛❧❣❡❜r❛ t♦ ♣r✉♥❡ ✉s❡❧❡ss s✉❜tr❡❡s ✐♥ t❤❡ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ tr❡❡✳ ❈♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ❯♥❞❡r ♣r❡❝✐s❡ ❛❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s✱ ✐❢ ♠ ♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ❖ ✷✵ ✽✹✶♥ ✇✐t❤ ❛ ❞❡t❡r♠✐♥✐st✐❝ ✈❛r✐❛♥t❀ ♦❢ ❡①♣❡❝t❛t✐♦♥ ❖ ✷✵ ✼✾✶♥ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✈❛r✐❛♥t✳ ✰ ❣❡♥❡r❛❧✐③❛t✐♦♥s ✇❤❡♥ ♠ ♥ ✭ ✶✮✳ ❆❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s✿ ✈❛r✐❛♥t ♦❢ ❋rö❜❡r❣ ❈♦♥❥❡❝t✉r❡ ♦♥ t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♦❢ ❣❡♥❡r✐❝ ♦✈❡r❞❡t❡r♠✐♥❡❞ s②st❡♠s✳

✶✸

▼❛✐♥ r❡s✉❧ts

❢✶, . . . , ❢♠ P F✷r①✶, . . . , ①♥s. ❆❧❣♦r✐t❤♠✿ ✉s❡ ✭s♣❛rs❡✮ ❧✐♥❡❛r ❛❧❣❡❜r❛ t♦ ♣r✉♥❡ ✉s❡❧❡ss s✉❜tr❡❡s ✐♥ t❤❡ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ tr❡❡✳ ❈♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ❯♥❞❡r ♣r❡❝✐s❡ ❛❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s✱ ✐❢ ♠ “ ♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ❖ ` ✷✵.✽✹✶♥˘ ✇✐t❤ ❛ ❞❡t❡r♠✐♥✐st✐❝ ✈❛r✐❛♥t❀ ♦❢ ❡①♣❡❝t❛t✐♦♥ ❖ ` ✷✵.✼✾✶♥˘ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✈❛r✐❛♥t✳ ✰ ❣❡♥❡r❛❧✐③❛t✐♦♥s ✇❤❡♥ ♠ “ α♥ ✭α ě ✶✮✳ ❆❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s✿ ✈❛r✐❛♥t ♦❢ ❋rö❜❡r❣ ❈♦♥❥❡❝t✉r❡ ♦♥ t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♦❢ ❣❡♥❡r✐❝ ♦✈❡r❞❡t❡r♠✐♥❡❞ s②st❡♠s✳

✶✸

❆❧❣♦r✐t❤♠

❋ ❋p①✶, . . . , ①♥´✶, ✵q ❋p①✶, . . . , ①♥´✷, ✵, ✵q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✵ ❋p①✶, . . . , ①♥´✷, ✶, ✵q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✶ ①♥ “ ✵ ❋p①✶, . . . , ①♥´✶, ✶q ❋p①✶, . . . , ①♥´✷, ✵, ✶q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✵ ❋p①✶, . . . , ①♥´✷, ✶, ✶q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✶ ①♥ “ ✶ looooooooooooooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooon , / / / / / / / / / / / / / / / / / / / / / / / / / / . / / / / / / / / / / / / / / / / / / / / / / / / / /

• ❞❡♣t❤ ❦

✷❦ s②st❡♠s ✐♥ ♥ ´ ❦ ✈❛r✐❛❜❧❡s✱ ✇✐t❤ ♥♦ s♦❧✉t✐♦♥❄ ❍✐❧❜❡rt ◆✉❧❧st❡❧❧❡♥s❛t③ ❋ ①✶ ①♥

❦ ❛♥ ❦ ✶

❛♥ ❤❛s ♥♦ s♦❧✉t✐♦♥ ✐♥

♥ ❦ ✷

✶ ❋ ①✷

✶

①✶ ①✷

♥

①♥

❈❛♥ ❜❡ t❡st❡❞ ❜② s♦❧✈✐♥❣ ❛ ❧✐♥❡❛r s②st❡♠ ✐♥✈♦❧✈✐♥❣ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①

✶✹

❆❧❣♦r✐t❤♠

❋ ❋p①✶, . . . , ①♥´✶, ✵q ❋p①✶, . . . , ①♥´✷, ✵, ✵q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✵ ❋p①✶, . . . , ①♥´✷, ✶, ✵q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✶ ①♥ “ ✵ ❋p①✶, . . . , ①♥´✶, ✶q ❋p①✶, . . . , ①♥´✷, ✵, ✶q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✵ ❋p①✶, . . . , ①♥´✷, ✶, ✶q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✶ ①♥ “ ✶ looooooooooooooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooon , / / / / / / / / / / / / / / / / / / / / / / / / / / . / / / / / / / / / / / / / / / / / / / / / / / / / /

• ❞❡♣t❤ ❦

✷❦ s②st❡♠s ✐♥ ♥ ´ ❦ ✈❛r✐❛❜❧❡s✱ ✇✐t❤ ♥♦ s♦❧✉t✐♦♥❄ ❍✐❧❜❡rt ◆✉❧❧st❡❧❧❡♥s❛t③ ❋p①✶, . . . , ①♥´❦, ❛♥´❦`✶, . . . , ❛♥q ❤❛s ♥♦ s♦❧✉t✐♦♥ ✐♥ F♥´❦

✷

õ ✶ P x❋, ①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥y

❈❛♥ ❜❡ t❡st❡❞ ❜② s♦❧✈✐♥❣ ❛ ❧✐♥❡❛r s②st❡♠ ✐♥✈♦❧✈✐♥❣ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①

✶✹

❆❧❣♦r✐t❤♠

❋ ❋p①✶, . . . , ①♥´✶, ✵q ❋p①✶, . . . , ①♥´✷, ✵, ✵q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✵ ❋p①✶, . . . , ①♥´✷, ✶, ✵q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✶ ①♥ “ ✵ ❋p①✶, . . . , ①♥´✶, ✶q ❋p①✶, . . . , ①♥´✷, ✵, ✶q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✵ ❋p①✶, . . . , ①♥´✷, ✶, ✶q ✳ ✳ ✳ H❄ ✳ ✳ ✳ H❄ ①♥´✶ “ ✶ ①♥ “ ✶ looooooooooooooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooon , / / / / / / / / / / / / / / / / / / / / / / / / / / . / / / / / / / / / / / / / / / / / / / / / / / / / /

• ❞❡♣t❤ ❦

✷❦ s②st❡♠s ✐♥ ♥ ´ ❦ ✈❛r✐❛❜❧❡s✱ ✇✐t❤ ♥♦ s♦❧✉t✐♦♥❄ ❍✐❧❜❡rt ◆✉❧❧st❡❧❧❡♥s❛t③ ❋p①✶, . . . , ①♥´❦, ❛♥´❦`✶, . . . , ❛♥q ❤❛s ♥♦ s♦❧✉t✐♦♥ ✐♥ F♥´❦

✷

õ ✶ P x❋, ①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥y

❈❛♥ ❜❡ t❡st❡❞ ❜② s♦❧✈✐♥❣ ❛ ❧✐♥❡❛r s②st❡♠ ✐♥✈♦❧✈✐♥❣ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐①

✶✹

❇♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞

■ “ x❢✶, . . . , ❢♠y Ă F✷r①✶, . . . , ①♥s. ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t P ❙q✉❛r❡❋r❡❡▼♦♥♦♠✐❛❧sp❞ ´ ✷q✳ ❈♦❧✉♠♥s✿ ❙q✉❛r❡✲❢r❡❡ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞✳ ♠✶ ą ¨ ¨ ¨ ą ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ¨ ˚ ˚ ˝ ˛ ‹ ‹ ‚ “ ▼❛❝ ■❢ t❤❡ s②st❡♠ ❤❛s ♥♦ s♦❧✉t✐♦♥✱ t❤❡♥ ❣✶ ❣♠✱ s✳t✳

♠ ✐ ✶

❢✐❣✐ ✶ ♠♦❞ ①✷

✶

①✶ ①✷

♥

①♥ ✈ s✳t✳ ✈ ▼❛❝ ✵ ✵ ✶ ❉❡❝✐❞✐♥❣ ✐❢ t❤❡ s②st❡♠ ❤❛s s♦❧✉t✐♦♥s ♦r ♥♦t ✐s r❡❞✉❝❡❞ t♦ t❡st✐♥❣ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ❛ s♣❛rs❡ ❧✐♥❡❛r s②st❡♠✳ Pr♦❜❧❡♠✿ ✇❤✐❝❤ ❞❄

✶✺

❇♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞

■ “ x❢✶, . . . , ❢♠y Ă F✷r①✶, . . . , ①♥s. ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t P ❙q✉❛r❡❋r❡❡▼♦♥♦♠✐❛❧sp❞ ´ ✷q✳ ❈♦❧✉♠♥s✿ ❙q✉❛r❡✲❢r❡❡ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞✳ ♠✶ ą ¨ ¨ ¨ ą ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ¨ ˚ ˚ ˝ ˛ ‹ ‹ ‚ “ ▼❛❝ ■❢ t❤❡ s②st❡♠ ❤❛s ♥♦ s♦❧✉t✐♦♥✱ t❤❡♥ D❣✶, . . . , ❣♠✱ s✳t✳

♠

ÿ

✐“✶

❢✐❣✐ “ ✶ ♠♦❞ @ ①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥

D ñ D✈ s✳t✳ ✈ ¨ ▼❛❝ “ ` ✵ . . . ✵ ✶ ˘ . ❉❡❝✐❞✐♥❣ ✐❢ t❤❡ s②st❡♠ ❤❛s s♦❧✉t✐♦♥s ♦r ♥♦t ✐s r❡❞✉❝❡❞ t♦ t❡st✐♥❣ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ❛ s♣❛rs❡ ❧✐♥❡❛r s②st❡♠✳ Pr♦❜❧❡♠✿ ✇❤✐❝❤ ❞❄

✶✺

❇♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞

■ “ x❢✶, . . . , ❢♠y Ă F✷r①✶, . . . , ①♥s. ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t P ❙q✉❛r❡❋r❡❡▼♦♥♦♠✐❛❧sp❞ ´ ✷q✳ ❈♦❧✉♠♥s✿ ❙q✉❛r❡✲❢r❡❡ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞✳ ♠✶ ą ¨ ¨ ¨ ą ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ¨ ˚ ˚ ˝ ˛ ‹ ‹ ‚ “ ▼❛❝ ■❢ t❤❡ s②st❡♠ ❤❛s ♥♦ s♦❧✉t✐♦♥✱ t❤❡♥ D❣✶, . . . , ❣♠✱ s✳t✳

♠

ÿ

✐“✶

❢✐❣✐ “ ✶ ♠♦❞ @ ①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥

D ñ D✈ s✳t✳ ✈ ¨ ▼❛❝ “ ` ✵ . . . ✵ ✶ ˘ . ❉❡❝✐❞✐♥❣ ✐❢ t❤❡ s②st❡♠ ❤❛s s♦❧✉t✐♦♥s ♦r ♥♦t ✐s r❡❞✉❝❡❞ t♦ t❡st✐♥❣ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ❛ s♣❛rs❡ ❧✐♥❡❛r s②st❡♠✳ Pr♦❜❧❡♠✿ ✇❤✐❝❤ ❞❄

✶✺

❇♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞

■ “ x❢✶, . . . , ❢♠y Ă F✷r①✶, . . . , ①♥s. ❘♦✇s✿ ❛❧❧ ♣r♦❞✉❝ts t❢✐ ✇❤❡r❡ t P ❙q✉❛r❡❋r❡❡▼♦♥♦♠✐❛❧sp❞ ´ ✷q✳ ❈♦❧✉♠♥s✿ ❙q✉❛r❡✲❢r❡❡ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ❞✳ ♠✶ ą ¨ ¨ ¨ ą ♠ℓ t✶❢✶ ✳ ✳ ✳ t❦❢♠ ¨ ˚ ˚ ˝ ˛ ‹ ‹ ‚ “ ▼❛❝ ■❢ t❤❡ s②st❡♠ ❤❛s ♥♦ s♦❧✉t✐♦♥✱ t❤❡♥ D❣✶, . . . , ❣♠✱ s✳t✳

♠

ÿ

✐“✶

❢✐❣✐ “ ✶ ♠♦❞ @ ①✷

✶ ´ ①✶, . . . , ①✷ ♥ ´ ①♥

D ñ D✈ s✳t✳ ✈ ¨ ▼❛❝ “ ` ✵ . . . ✵ ✶ ˘ . ❉❡❝✐❞✐♥❣ ✐❢ t❤❡ s②st❡♠ ❤❛s s♦❧✉t✐♦♥s ♦r ♥♦t ✐s r❡❞✉❝❡❞ t♦ t❡st✐♥❣ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ❛ s♣❛rs❡ ❧✐♥❡❛r s②st❡♠✳ Pr♦❜❧❡♠✿ ✇❤✐❝❤ ❞❄

✶✺

❆❧❣♦r✐t❤♠ ❇♦♦❧❡❛♥❙♦❧✈❡

■♥♣✉t✿ ♠, ♥, ❦ P N s✉❝❤ t❤❛t ♠ ě ♥ ą ❦ ❢✶, . . . , ❢♠ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✐♥ F✷r①✶, . . . , ①♥s✳ ❖✉t♣✉t✿❚❤❡ s❡t ♦❢ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵✳ ❙ ✳ ❞✵ s♦♠❡ ✐♥t❡❣❡r✳ ✭❝❤♦✐❝❡ ♦❢ ❛ ❜♦✉♥❞✮ ❋♦r ❛❧❧ ❛♥

❦ ✶

❛♥

❦ ✷

❋♦r ✐ ❢r♦♠ ✶ t♦ ♠ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮ ❢✐ ①✶ ①♥

❦

❢✐ ①✶ ①♥

❦ ❛♥ ❦ ✶

❛♥

✷ ①✶

①♥

❦ ✳

❊♥❞❋♦r ▼ ❜♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ♦❢ ❢✶ ❢♠ ✐♥ ❞❡❣r❡❡ ❞✵✳ ■❢ t❤❡ s②st❡♠ ✉ ▼ ✵ ✵ ✶ ✐s ✐♥❝♦♥s✐st❡♥t ✭♣r✉♥✐♥❣✮ ❚ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❢✶ ❢♠ ✵✮ ✭❡①❤❛✉st✐✈❡ s❡❛r❝❤✮✳ ❋♦r ❛❧❧ t✶ t♥

❦

❚ ❙ ❙ t✶ t♥

❦ ❛♥ ❦ ✶

❛♥ ✳ ❊♥❞❋♦r ❊♥❞■❢ ❊♥❞❋♦r ❘❡t✉r♥ ❙✳

✶✻

❆❧❣♦r✐t❤♠ ❇♦♦❧❡❛♥❙♦❧✈❡

■♥♣✉t✿ ♠, ♥, ❦ P N s✉❝❤ t❤❛t ♠ ě ♥ ą ❦ ❢✶, . . . , ❢♠ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✐♥ F✷r①✶, . . . , ①♥s✳ ❖✉t♣✉t✿❚❤❡ s❡t ♦❢ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵✳ ❙ :“ H✳ ❞✵:“ s♦♠❡ ✐♥t❡❣❡r✳ ✭❝❤♦✐❝❡ ♦❢ ❛ ❜♦✉♥❞✮ ❋♦r ❛❧❧ ❛♥

❦ ✶

❛♥

❦ ✷

❋♦r ✐ ❢r♦♠ ✶ t♦ ♠ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮ ❢✐ ①✶ ①♥

❦

❢✐ ①✶ ①♥

❦ ❛♥ ❦ ✶

❛♥

✷ ①✶

①♥

❦ ✳

❊♥❞❋♦r ▼ ❜♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ♦❢ ❢✶ ❢♠ ✐♥ ❞❡❣r❡❡ ❞✵✳ ■❢ t❤❡ s②st❡♠ ✉ ▼ ✵ ✵ ✶ ✐s ✐♥❝♦♥s✐st❡♥t ✭♣r✉♥✐♥❣✮ ❚ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❢✶ ❢♠ ✵✮ ✭❡①❤❛✉st✐✈❡ s❡❛r❝❤✮✳ ❋♦r ❛❧❧ t✶ t♥

❦

❚ ❙ ❙ t✶ t♥

❦ ❛♥ ❦ ✶

❛♥ ✳ ❊♥❞❋♦r ❊♥❞■❢ ❊♥❞❋♦r ❘❡t✉r♥ ❙✳

✶✻

❆❧❣♦r✐t❤♠ ❇♦♦❧❡❛♥❙♦❧✈❡

■♥♣✉t✿ ♠, ♥, ❦ P N s✉❝❤ t❤❛t ♠ ě ♥ ą ❦ ❢✶, . . . , ❢♠ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✐♥ F✷r①✶, . . . , ①♥s✳ ❖✉t♣✉t✿❚❤❡ s❡t ♦❢ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵✳ ❙ :“ H✳ ❞✵:“ s♦♠❡ ✐♥t❡❣❡r✳ ✭❝❤♦✐❝❡ ♦❢ ❛ ❜♦✉♥❞✮ ❋♦r ❛❧❧ p❛♥´❦`✶, . . . , ❛♥q P F❦

✷

❋♦r ✐ ❢r♦♠ ✶ t♦ ♠ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮ ˜ ❢✐p①✶, . . . , ①♥´❦q :“ ❢✐p①✶, . . . , ①♥´❦, ❛♥´❦`✶, . . . , ❛♥q P F✷r①✶, . . . , ①♥´❦s✳ ❊♥❞❋♦r ▼ ❜♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ♦❢ ❢✶ ❢♠ ✐♥ ❞❡❣r❡❡ ❞✵✳ ■❢ t❤❡ s②st❡♠ ✉ ▼ ✵ ✵ ✶ ✐s ✐♥❝♦♥s✐st❡♥t ✭♣r✉♥✐♥❣✮ ❚ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭❢✶ ❢♠ ✵✮ ✭❡①❤❛✉st✐✈❡ s❡❛r❝❤✮✳ ❋♦r ❛❧❧ t✶ t♥

❦

❚ ❙ ❙ t✶ t♥

❦ ❛♥ ❦ ✶

❛♥ ✳ ❊♥❞❋♦r ❊♥❞■❢ ❊♥❞❋♦r ❘❡t✉r♥ ❙✳

✶✻

❆❧❣♦r✐t❤♠ ❇♦♦❧❡❛♥❙♦❧✈❡

■♥♣✉t✿ ♠, ♥, ❦ P N s✉❝❤ t❤❛t ♠ ě ♥ ą ❦ ❢✶, . . . , ❢♠ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧s ✐♥ F✷r①✶, . . . , ①♥s✳ ❖✉t♣✉t✿❚❤❡ s❡t ♦❢ ❜♦♦❧❡❛♥ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ❢✶ “ ¨ ¨ ¨ “ ❢♠ “ ✵✳ ❙ :“ H✳ ❞✵:“ s♦♠❡ ✐♥t❡❣❡r✳ ✭❝❤♦✐❝❡ ♦❢ ❛ ❜♦✉♥❞✮ ❋♦r ❛❧❧ p❛♥´❦`✶, . . . , ❛♥q P F❦

✷

❋♦r ✐ ❢r♦♠ ✶ t♦ ♠ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮ ˜ ❢✐p①✶, . . . , ①♥´❦q :“ ❢✐p①✶, . . . , ①♥´❦, ❛♥´❦`✶, . . . , ❛♥q P F✷r①✶, . . . , ①♥´❦s✳ ❊♥❞❋♦r ▼ :“ ❜♦♦❧❡❛♥ ▼❛❝❛✉❧❛② ♠❛tr✐① ♦❢ p˜ ❢✶, . . . , ˜ ❢♠q ✐♥ ❞❡❣r❡❡ ❞✵✳ ■❢ t❤❡ s②st❡♠ ✉ ¨ ▼ “ ` ✵ . . . ✵ ✶ ˘ ✐s ✐♥❝♦♥s✐st❡♥t ✭♣r✉♥✐♥❣✮ ❚ :“ s♦❧✉t✐♦♥s ♦❢ t❤❡ s②st❡♠ ✭˜ ❢✶ “ ¨ ¨ ¨ “ ˜ ❢♠ “ ✵✮ ✭❡①❤❛✉st✐✈❡ s❡❛r❝❤✮✳ ❋♦r ❛❧❧ pt✶, . . . , t♥´❦q P ❚ ❙ :“ ❙ Y tpt✶, . . . , t♥´❦, ❛♥´❦`✶, . . . , ❛♥qu✳ ❊♥❞❋♦r ❊♥❞■❢ ❊♥❞❋♦r ❘❡t✉r♥ ❙✳

✶✻

❲♦r❦♣❧❛♥

✶

❈❤♦✐❝❡ ♦❢ ❞✵ ✭✐♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ❦✮❄ ✐♥❞❡① ♦❢ t❤❡ ✜rst ♥♦♥✲♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥

p✶`tq♥´❦ p✶´tqp✶`t✷q♠

❞✵ „ ▼pγq♥ ✇❤❡♥ ❦ “ p✶ ´ γq♥ ▼ ✶ ✶ ✷ ✶ ✷ ✷

✷

✶✵ ✶ ✷ ✶ ✷ ✶ ✷

✷

❙✐③❡s ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄

✸

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❝♦♥s✐st❡♥❝② t❡sts ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄ ❖p✷p✶´γ`ω❋pγq`εq♥q

✹

❋✐♥❞ ♦♣t✐♠❛❧ ❦ ❢♦r ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t②❄

• ❛✉ss✿ ❦ “ ✵.✼✸♥❀

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞✿ ❦ “ ✵.✻✵♥ ❲✐❡❞❡♠❛♥♥✿ ❦ “ ✵.✹✺♥✳

✺

❉❡❣❡♥❡r❛❝② ♣❤❡♥♦♠❡♥♦♠s❄ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r✐t②✳

✶✼

❲♦r❦♣❧❛♥

✶

❈❤♦✐❝❡ ♦❢ ❞✵ ✭✐♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ❦✮❄ ✐♥❞❡① ♦❢ t❤❡ ✜rst ♥♦♥✲♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥

p✶`tq♥´❦ p✶´tqp✶`t✷q♠

❞✵ „ ▼pγq♥ ✇❤❡♥ ❦ “ p✶ ´ γq♥ ▼pγq “ ˆ ´✶{γ ` ✶{✷ ` ✶{✷ b ✷{γ✷ ´ ✶✵{γ ´ ✶ ` ✷p✶{γ ` ✷q a p✶{γ ` ✷q{γ ˙ γ.

✷

❙✐③❡s ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄

✸

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❝♦♥s✐st❡♥❝② t❡sts ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄ ❖p✷p✶´γ`ω❋pγq`εq♥q

✹

❋✐♥❞ ♦♣t✐♠❛❧ ❦ ❢♦r ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t②❄

• ❛✉ss✿ ❦ “ ✵.✼✸♥❀

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞✿ ❦ “ ✵.✻✵♥ ❲✐❡❞❡♠❛♥♥✿ ❦ “ ✵.✹✺♥✳

✺

❉❡❣❡♥❡r❛❝② ♣❤❡♥♦♠❡♥♦♠s❄ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r✐t②✳

✶✼

❲♦r❦♣❧❛♥

✶

❈❤♦✐❝❡ ♦❢ ❞✵ ✭✐♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ❦✮❄ ✐♥❞❡① ♦❢ t❤❡ ✜rst ♥♦♥✲♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥

p✶`tq♥´❦ p✶´tqp✶`t✷q♠

❞✵ „ ▼pγq♥ ✇❤❡♥ ❦ “ p✶ ´ γq♥ ▼pγq “ ˆ ´✶{γ ` ✶{✷ ` ✶{✷ b ✷{γ✷ ´ ✶✵{γ ´ ✶ ` ✷p✶{γ ` ✷q a p✶{γ ` ✷q{γ ˙ γ.

✷

❙✐③❡s ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄

✸

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❝♦♥s✐st❡♥❝② t❡sts ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄ ❖p✷p✶´γ`ω❋pγq`εq♥q

✹

❋✐♥❞ ♦♣t✐♠❛❧ ❦ ❢♦r ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t②❄

• ❛✉ss✿ ❦ “ ✵.✼✸♥❀

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞✿ ❦ “ ✵.✻✵♥ ❲✐❡❞❡♠❛♥♥✿ ❦ “ ✵.✹✺♥✳

✺

❉❡❣❡♥❡r❛❝② ♣❤❡♥♦♠❡♥♦♠s❄ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r✐t②✳

✶✼

❲♦r❦♣❧❛♥

✶

❈❤♦✐❝❡ ♦❢ ❞✵ ✭✐♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ❦✮❄ ✐♥❞❡① ♦❢ t❤❡ ✜rst ♥♦♥✲♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥

p✶`tq♥´❦ p✶´tqp✶`t✷q♠

❞✵ „ ▼pγq♥ ✇❤❡♥ ❦ “ p✶ ´ γq♥ ▼pγq “ ˆ ´✶{γ ` ✶{✷ ` ✶{✷ b ✷{γ✷ ´ ✶✵{γ ´ ✶ ` ✷p✶{γ ` ✷q a p✶{γ ` ✷q{γ ˙ γ.

✷

❙✐③❡s ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄

✸

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❝♦♥s✐st❡♥❝② t❡sts ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄ ❖p✷p✶´γ`ω❋pγq`εq♥q

✹

❋✐♥❞ ♦♣t✐♠❛❧ ❦ ❢♦r ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t②❄

• ❛✉ss✿ ❦ “ ✵.✼✸♥❀

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞✿ ❦ “ ✵.✻✵♥ ❲✐❡❞❡♠❛♥♥✿ ❦ “ ✵.✹✺♥✳

✺

❉❡❣❡♥❡r❛❝② ♣❤❡♥♦♠❡♥♦♠s❄ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r✐t②✳

✶✼

❲♦r❦♣❧❛♥

✶

❈❤♦✐❝❡ ♦❢ ❞✵ ✭✐♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ❦✮❄ ✐♥❞❡① ♦❢ t❤❡ ✜rst ♥♦♥✲♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥

p✶`tq♥´❦ p✶´tqp✶`t✷q♠

❞✵ „ ▼pγq♥ ✇❤❡♥ ❦ “ p✶ ´ γq♥ ▼pγq “ ˆ ´✶{γ ` ✶{✷ ` ✶{✷ b ✷{γ✷ ´ ✶✵{γ ´ ✶ ` ✷p✶{γ ` ✷q a p✶{γ ` ✷q{γ ˙ γ.

✷

❙✐③❡s ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄

✸

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❝♦♥s✐st❡♥❝② t❡sts ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄ ❖p✷p✶´γ`ω❋pγq`εq♥q

✹

❋✐♥❞ ♦♣t✐♠❛❧ ❦ ❢♦r ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t②❄

• ❛✉ss✿ ❦ “ ✵.✼✸♥❀

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞✿ ❦ “ ✵.✻✵♥ ❲✐❡❞❡♠❛♥♥✿ ❦ “ ✵.✹✺♥✳

✺

❉❡❣❡♥❡r❛❝② ♣❤❡♥♦♠❡♥♦♠s❄ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r✐t②✳

✶✼

❲♦r❦♣❧❛♥

✶

❈❤♦✐❝❡ ♦❢ ❞✵ ✭✐♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s♣❡❝✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ❦✮❄ ✐♥❞❡① ♦❢ t❤❡ ✜rst ♥♦♥✲♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥

p✶`tq♥´❦ p✶´tqp✶`t✷q♠

❞✵ „ ▼pγq♥ ✇❤❡♥ ❦ “ p✶ ´ γq♥ ▼pγq “ ˆ ´✶{γ ` ✶{✷ ` ✶{✷ b ✷{γ✷ ´ ✶✵{γ ´ ✶ ` ✷p✶{γ ` ✷q a p✶{γ ` ✷q{γ ˙ γ.

✷

❙✐③❡s ♦❢ t❤❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄

✸

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❝♦♥s✐st❡♥❝② t❡sts ✭❢✉♥❝t✐♦♥ ♦❢ ❦✮❄ ❖p✷p✶´γ`ω❋pγq`εq♥q

✹

❋✐♥❞ ♦♣t✐♠❛❧ ❦ ❢♦r ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t②❄

• ❛✉ss✿ ❦ “ ✵.✼✸♥❀

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞✿ ❦ “ ✵.✻✵♥ ❲✐❡❞❡♠❛♥♥✿ ❦ “ ✵.✹✺♥✳

✺

❉❡❣❡♥❡r❛❝② ♣❤❡♥♦♠❡♥♦♠s❄ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r✐t②✳

✶✼

❈♦♠♣❧❡①✐t②

❈♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ❯♥❞❡r ♣r❡❝✐s❡ ❛❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s✱ ✐❢ ♠ “ ♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ❖ ` ✷✵.✽✹✶♥˘ ✇✐t❤ ❛ ❞❡t❡r♠✐♥✐st✐❝ ✈❛r✐❛♥t❀ ❖ ` ✷✵.✼✾✶♥˘ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ▲❛s ❱❡❣❛s ✈❛r✐❛♥t✳ ✰ ❣❡♥❡r❛❧✐③❛t✐♦♥s ✇❤❡♥ ♠ “ α♥ ✭α ě ✶✮✳ ❊①♣❡r✐♠❡♥ts ❆❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s ❛r❡ ✈❡r✐✜❡❞ ✇✐t❤ ♣r♦❜✳ ❝❧♦s❡ t♦ ✶✳ Pr♦❜❛❜✐❧✐st✐❝ ✈❛r✐❛♥t✿ ✇❤❡♥ ♥ ♠✱ ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ✇❤❡♥ ♥ ✷✵✵ ❈r②♣t♦ ❛♣♣❧✐❝❛t✐♦♥s ✭◗❯❆❉✮✳ ❱❛r✐❛♥t ♦❢ ❋rö❜❡r❣ ❝♦♥❥❡❝t✉r❡ ❚❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r s②st❡♠s t❡♥❞s t♦ ✶ ✇❤❡♥ ♥ ✳

✶✽

❈♦♠♣❧❡①✐t②

❈♦♠♣❧❡①✐t② ❛♥❛❧②s✐s ❯♥❞❡r ♣r❡❝✐s❡ ❛❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s✱ ✐❢ ♠ “ ♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ❖ ` ✷✵.✽✹✶♥˘ ✇✐t❤ ❛ ❞❡t❡r♠✐♥✐st✐❝ ✈❛r✐❛♥t❀ ❖ ` ✷✵.✼✾✶♥˘ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ▲❛s ❱❡❣❛s ✈❛r✐❛♥t✳ ✰ ❣❡♥❡r❛❧✐③❛t✐♦♥s ✇❤❡♥ ♠ “ α♥ ✭α ě ✶✮✳ ❊①♣❡r✐♠❡♥ts ❆❧❣❡❜r❛✐❝ ❛ss✉♠♣t✐♦♥s ❛r❡ ✈❡r✐✜❡❞ ✇✐t❤ ♣r♦❜✳ ❝❧♦s❡ t♦ ✶✳ Pr♦❜❛❜✐❧✐st✐❝ ✈❛r✐❛♥t✿ ✇❤❡♥ ♥ “ ♠✱ ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ✇❤❡♥ ♥ ě ✷✵✵ ù ❈r②♣t♦ ❛♣♣❧✐❝❛t✐♦♥s ✭◗❯❆❉✮✳ ❱❛r✐❛♥t ♦❢ ❋rö❜❡r❣ ❝♦♥❥❡❝t✉r❡ ❚❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ γ✲str♦♥❣ s❡♠✐✲r❡❣✉❧❛r s②st❡♠s t❡♥❞s t♦ ✶ ✇❤❡♥ ♥ Ñ 8✳

✶✽

❈♦♠♣❧❡①✐t② ✭■■✮

Solving αn equations in n variables: 2c n

1 1.82 3 4 5

0.31 0.79 1

α β = 0 W i e d e m a n n c: exponent of the complexity Exhaustive search New Algorithm Gröbner Bases

0.84

2.48 Coppersmith = Deterministic

❋✐❣✉r❡✿ ❊①♣♦♥❡♥t ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② ✐♥ t❡r♠s ♦❢ α

✶✾

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❙tr✉❝t✉r❡s ❤❛✈❡ ❛♥ ✐♠♣❛❝t ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s✳ ❆❧❣♦r✐t❤♠✐❝ ✐♠♣r♦✈❡♠❡♥ts ▼✐♥r❛♥❦ ❈❤❛❧❧❡♥❣❡ ✽ ✾ ✺

❬❈r②♣t♦ ✷✵✵✽❪ ✸✷✽✷✸✸s ❬■ss❛❝ ✷✵✶✵❪ ✾✸✺s ❬P❛s❝♦ ✷✵✶✵❪ ✼✸s ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t ❋❛✉❣èr❡✴❙✳✴❙❛❢❡② ❋❛✉❣èr❡✴▲❛❝❤❛rtr❡

❉❡❞✐❝❛t❡❞ ❋✺ ❛❧❣♦r✐t❤♠ ❢♦r ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s✴s♣❛rs❡ s②st❡♠s ✭s♣❡❡❞✲✉♣ ♠♦r❡ t❤❛♥ ✶✵✵ ❢♦r s♦♠❡ ♦✈❡r❞❡t❡r♠✐♥❡❞ ❜✐❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✮ ❋❛✉❣èr❡✴❙✳✴❙✈❛rt③✱ ❛r❳✐✈✿✶✹✵✷✳✼✷✵✺✱ ✷✵✶✹ P❡rs♣❡❝t✐✈❡s ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s❄

✷✵

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❙tr✉❝t✉r❡s ❤❛✈❡ ❛♥ ✐♠♣❛❝t ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s✳ ❆❧❣♦r✐t❤♠✐❝ ✐♠♣r♦✈❡♠❡♥ts ▼✐♥r❛♥❦ ❈❤❛❧❧❡♥❣❡ p✽, ✾, ✺q

❬❈r②♣t♦ ✷✵✵✽❪ ✸✷✽✷✸✸s Ý Ñ❬■ss❛❝ ✷✵✶✵❪ ✾✸✺sÝ Ñ ❬P❛s❝♦ ✷✵✶✵❪ ✼✸s ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t ❋❛✉❣èr❡✴❙✳✴❙❛❢❡② ❋❛✉❣èr❡✴▲❛❝❤❛rtr❡

❉❡❞✐❝❛t❡❞ ❋✺ ❛❧❣♦r✐t❤♠ ❢♦r ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s✴s♣❛rs❡ s②st❡♠s ✭s♣❡❡❞✲✉♣ ♠♦r❡ t❤❛♥ ✶✵✵ ❢♦r s♦♠❡ ♦✈❡r❞❡t❡r♠✐♥❡❞ ❜✐❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✮ ❋❛✉❣èr❡✴❙✳✴❙✈❛rt③✱ ❛r❳✐✈✿✶✹✵✷✳✼✷✵✺✱ ✷✵✶✹ P❡rs♣❡❝t✐✈❡s ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s❄

✷✵

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❙tr✉❝t✉r❡s ❤❛✈❡ ❛♥ ✐♠♣❛❝t ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s✳ ❆❧❣♦r✐t❤♠✐❝ ✐♠♣r♦✈❡♠❡♥ts ▼✐♥r❛♥❦ ❈❤❛❧❧❡♥❣❡ p✽, ✾, ✺q

❬❈r②♣t♦ ✷✵✵✽❪ ✸✷✽✷✸✸s Ý Ñ❬■ss❛❝ ✷✵✶✵❪ ✾✸✺sÝ Ñ ❬P❛s❝♦ ✷✵✶✵❪ ✼✸s ❋❛✉❣èr❡✴▲❡✈②✲❞✐t✲❱❡❤❡❧✴P❡rr❡t ❋❛✉❣èr❡✴❙✳✴❙❛❢❡② ❋❛✉❣èr❡✴▲❛❝❤❛rtr❡

❉❡❞✐❝❛t❡❞ ❋✺ ❛❧❣♦r✐t❤♠ ❢♦r ♠✉❧t✐✲❤♦♠♦❣❡♥❡♦✉s✴s♣❛rs❡ s②st❡♠s ✭s♣❡❡❞✲✉♣ ♠♦r❡ t❤❛♥ ✶✵✵ ❢♦r s♦♠❡ ♦✈❡r❞❡t❡r♠✐♥❡❞ ❜✐❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✮ ❋❛✉❣èr❡✴❙✳✴❙✈❛rt③✱ ❛r❳✐✈✿✶✹✵✷✳✼✷✵✺✱ ✷✵✶✹ P❡rs♣❡❝t✐✈❡s ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ s②st❡♠s❄

✷✵

▲♦♥❣ t❡r♠ ♣❡rs♣❡❝t✐✈❡s

❈❤❛❧❧❡♥❣❡s ❙②st❡♠❛t✐❝ ♠❡t❤♦❞♦❧♦❣✐❡s ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ str✉❝t✉r❡s ✭❍✐❧❜❡rt s❡r✐❡s✱ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✱ t♦♦❧s ❢♦r ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✱ ✐♥✈❛r✐❛♥ts✱✳ ✳ ✳ ✮✳ ■♠♣❛❝t ♦❢ r✐♥❣ ✐s♦♠♦r♣❤✐s♠s ♦♥ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s ✭ ♠✉❧t✐✈❛r✐❛t❡ ❈r②♣t♦❣r❛♣❤②✮✳ ❆❧❣♦r✐t❤♠✐❝ ❢r❛♠❡✇♦r❦ ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ♣❛r❛❧❧❡❧✐s♠✱✳ ✳ ✳ ✮✳

❚❤❛♥❦ ②♦✉✦

✷✶

▲♦♥❣ t❡r♠ ♣❡rs♣❡❝t✐✈❡s

❈❤❛❧❧❡♥❣❡s ❙②st❡♠❛t✐❝ ♠❡t❤♦❞♦❧♦❣✐❡s ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ str✉❝t✉r❡s ✭❍✐❧❜❡rt s❡r✐❡s✱ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✱ t♦♦❧s ❢♦r ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✱ ✐♥✈❛r✐❛♥ts✱✳ ✳ ✳ ✮✳ ■♠♣❛❝t ♦❢ r✐♥❣ ✐s♦♠♦r♣❤✐s♠s ♦♥ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s ✭ ♠✉❧t✐✈❛r✐❛t❡ ❈r②♣t♦❣r❛♣❤②✮✳ ❆❧❣♦r✐t❤♠✐❝ ❢r❛♠❡✇♦r❦ ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ♣❛r❛❧❧❡❧✐s♠✱✳ ✳ ✳ ✮✳

❚❤❛♥❦ ②♦✉✦

✷✶

▲♦♥❣ t❡r♠ ♣❡rs♣❡❝t✐✈❡s

❈❤❛❧❧❡♥❣❡s ❙②st❡♠❛t✐❝ ♠❡t❤♦❞♦❧♦❣✐❡s ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ str✉❝t✉r❡s ✭❍✐❧❜❡rt s❡r✐❡s✱ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✱ t♦♦❧s ❢♦r ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✱ ✐♥✈❛r✐❛♥ts✱✳ ✳ ✳ ✮✳ ■♠♣❛❝t ♦❢ r✐♥❣ ✐s♦♠♦r♣❤✐s♠s ♦♥ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s ✭ ♠✉❧t✐✈❛r✐❛t❡ ❈r②♣t♦❣r❛♣❤②✮✳ ❆❧❣♦r✐t❤♠✐❝ ❢r❛♠❡✇♦r❦ ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ♣❛r❛❧❧❡❧✐s♠✱✳ ✳ ✳ ✮✳

❚❤❛♥❦ ②♦✉✦

✷✶

▲♦♥❣ t❡r♠ ♣❡rs♣❡❝t✐✈❡s

❈❤❛❧❧❡♥❣❡s ❙②st❡♠❛t✐❝ ♠❡t❤♦❞♦❧♦❣✐❡s ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ str✉❝t✉r❡s ✭❍✐❧❜❡rt s❡r✐❡s✱ ❞❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t②✱ t♦♦❧s ❢♦r ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✱ ✐♥✈❛r✐❛♥ts✱✳ ✳ ✳ ✮✳ ■♠♣❛❝t ♦❢ r✐♥❣ ✐s♦♠♦r♣❤✐s♠s ♦♥ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s ✭ ♠✉❧t✐✈❛r✐❛t❡ ❈r②♣t♦❣r❛♣❤②✮✳ ❆❧❣♦r✐t❤♠✐❝ ❢r❛♠❡✇♦r❦ ❢♦r str✉❝t✉r❡❞ s②st❡♠s ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s✱ ♣❛r❛❧❧❡❧✐s♠✱✳ ✳ ✳ ✮✳

❚❤❛♥❦ ②♦✉✦

✷✶