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❈♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✉s✐♥❣ ●rö❜♥❡r ❇❛s❡s✿ ❈♦♠♣❧❡①✐t② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s

❏❡❛♥✲❈❤❛r❧❡s ❋❛✉❣èr❡ ▼♦❤❛❜ ❙❛❢❡② ❊❧ ❉✐♥ P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r

❯P▼❈ ✕ ❈◆❘❙ ✕ ■◆❘■❆ P❛r✐s ✲ ❘♦❝q✉❡♥❝♦✉rt ▲■P✻ ✕ ❙❆▲❙❆ t❡❛♠

❙é♠✐♥❛✐r❡ ❆❧❣♦r✐t❤♠s ❏❛♥✉❛r② ✸✶✱ ✷✵✶✶

✶✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

Pr♦❜❧❡♠ st❛t❡♠❡♥t

(① − ✷)✷ + (② − ✷)✷ + ③✷ − ✶ = ✵.

▼ =

• ✷(① − ✷)

✷(② − ✷) ✷③ ✶ ✵ ✵

• ❈r✐t✐❝❛❧ ♣♦✐♥ts

▲❡t ❢✶ ❢♣ ①✶ ①♥ ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭♣ ♥ ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ ♥ ✶ ♠❛tr✐①

❢✶ ①✷ ❢✶ ①♥

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

❢♣ ①✷ ❢♣ ①♥

❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ①

♥ s✉❝❤ t❤❛t ❢✶ ①

❢♣ ① ✵ ❛♥❞ ❘❛♥❦ ▼ ♣✳ ❑✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣✳

✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

Pr♦❜❧❡♠ st❛t❡♠❡♥t

(① − ✷)✷ + (② − ✷)✷ + ③✷ − ✶ = ✵.

▼ =

• ✷(② − ✷)

✷③

• =
• ✵

✵

• ❈r✐t✐❝❛❧ ♣♦✐♥ts

▲❡t ❢✶ ❢♣ ①✶ ①♥ ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭♣ ♥ ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ ♥ ✶ ♠❛tr✐①

❢✶ ①✷ ❢✶ ①♥

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

❢♣ ①✷ ❢♣ ①♥

❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ①

♥ s✉❝❤ t❤❛t ❢✶ ①

❢♣ ① ✵ ❛♥❞ ❘❛♥❦ ▼ ♣✳ ❑✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣✳

✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

Pr♦❜❧❡♠ st❛t❡♠❡♥t

(① − ✷)✷ + (② − ✷)✷ + ③✷ − ✶ = ✵.

▼ =

• ✷(② − ✷)

✷③

• =
• ✵

✵

• ❈r✐t✐❝❛❧ ♣♦✐♥ts

▲❡t ❢✶, . . . , ❢♣ ∈ Q[①✶, . . . , ①♥] ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭♣ ≤ ♥ − ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ × (♥ − ✶) ♠❛tr✐①    

∂❢✶ ∂①✷

. . .

∂❢✶ ∂①♥

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

∂❢♣ ∂①✷

. . .

∂❢♣ ∂①♥

    . ❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① ∈ R♥ s✉❝❤ t❤❛t ❢✶(①) = · · · = ❢♣(①) = ✵ ❛♥❞ ❘❛♥❦(▼) < ♣✳ ❑✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣✳

✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

Pr♦❜❧❡♠ st❛t❡♠❡♥t

(① − ✷)✷ + (② − ✷)✷ + ③✷ − ✶ = ✵.

▼ =

• ✷(② − ✷)

✷③

• =
• ✵

✵

• ❈r✐t✐❝❛❧ ♣♦✐♥ts

▲❡t ❢✶, . . . , ❢♣ ∈ Q[①✶, . . . , ①♥] ❜❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❉ ✭♣ ≤ ♥ − ✶✮ ❛♥❞ ▼ ❜❡ t❤❡ ♣ × (♥ − ✶) ♠❛tr✐①    

∂❢✶ ∂①✷

. . .

∂❢✶ ∂①♥

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

∂❢♣ ∂①✷

. . .

∂❢♣ ∂①♥

    . ❈♦♠♣✉t❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ① ∈ R♥ s✉❝❤ t❤❛t ❢✶(①) = · · · = ❢♣(①) = ✵ ❛♥❞ ❘❛♥❦(▼) < ♣✳ ❑✵ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳ P♦❧②♥♦♠✐❛❧ s②st❡♠ s♦❧✈✐♥❣✳

✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s

P♦❧②♥♦♠✐❛❧ ▼✐♥✐♠✐③❛t✐♦♥    ♠✐♥

(①✶,...,①♥)∈R♥①✶

s✳t✳ ❢✶(①✶, . . . , ①♥) = · · · = ❢♣(①✶, . . . , ①♥) = ✵. ❆ss✉♠♣t✐♦♥✿ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ♦♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❡①❛❝t s♦❧✉t✐♦♥✴❝❡rt✐✜❡❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ❖t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✿ ❙❛♠♣❧✐♥❣ ♣♦✐♥ts ❈♦♥♥❡❝t✐✈✐t② q✉❡r✐❡s ◗✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥ ✳ ✳ ✳ ❈❛♥♥②✱ ●r✐❣♦r✐❡✈✱ ❱♦r♦❜❥♦✈✱ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ P❛r❞♦✱ ❘♦✉✐❧❧✐❡r✱ ❙❛❢❡②✱ ❙❝❤♦st✱ ❍♦♥❣✱ ✳ ✳ ✳

✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s

P♦❧②♥♦♠✐❛❧ ▼✐♥✐♠✐③❛t✐♦♥    ♠✐♥

(①✶,...,①♥)∈R♥ℓ(①✶, . . . , ①♥)

s✳t✳ ❢✶(①✶, . . . , ①♥) = · · · = ❢♣(①✶, . . . , ①♥) = ✵. ❆ss✉♠♣t✐♦♥✿ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ♦♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❡①❛❝t s♦❧✉t✐♦♥✴❝❡rt✐✜❡❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ❖t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✿ ❙❛♠♣❧✐♥❣ ♣♦✐♥ts ❈♦♥♥❡❝t✐✈✐t② q✉❡r✐❡s ◗✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥ ✳ ✳ ✳ ❈❛♥♥②✱ ●r✐❣♦r✐❡✈✱ ❱♦r♦❜❥♦✈✱ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ P❛r❞♦✱ ❘♦✉✐❧❧✐❡r✱ ❙❛❢❡②✱ ❙❝❤♦st✱ ❍♦♥❣✱ ✳ ✳ ✳

✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s

P♦❧②♥♦♠✐❛❧ ▼✐♥✐♠✐③❛t✐♦♥    ♠✐♥

(①✶,...,①♥)∈R♥ℓ(①✶, . . . , ①♥)

s✳t✳ ❢✶(①✶, . . . , ①♥) = · · · = ❢♣(①✶, . . . , ①♥) = ✵. ❆ss✉♠♣t✐♦♥✿ t❤❡ ♠✐♥✐♠✉♠ ✐s r❡❛❝❤❡❞ ♦♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❡①❛❝t s♦❧✉t✐♦♥✴❝❡rt✐✜❡❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ❖t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✿ ❙❛♠♣❧✐♥❣ ♣♦✐♥ts ❈♦♥♥❡❝t✐✈✐t② q✉❡r✐❡s ◗✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥ ✳ ✳ ✳ ❈❛♥♥②✱ ●r✐❣♦r✐❡✈✱ ❱♦r♦❜❥♦✈✱ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ P❛r❞♦✱ ❘♦✉✐❧❧✐❡r✱ ❙❛❢❡②✱ ❙❝❤♦st✱ ❍♦♥❣✱ ✳ ✳ ✳

✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s ✭■■✮

• ❡♥❡r✐❝❛❧❧②✱ ♥❜✳ ❝r✐t✳ ♣♦✐♥ts✿

♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣✳
• ❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠s✿ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ ▼✬❇❛❦♦♣✱ P❛r❞♦✱

❙❝❤♦st✱ ❙❛❢❡②✱✳✳✳

• rö❜♥❡r ❜❛s✐s

✇❡❧❧✲s✉✐t❡❞ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳ ■❙❙❆❈ ✷✵✶✵✮✳ ❊①♣❡r✐♠❡♥t❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ ●❇ ❛❧❣♦r✐t❤♠s ✭❘❆●❧✐❜✱ ❋●❜ ✐♥ ▼❛♣❧❡✮ ❝❛♥ s♦❧✈❡ ❧❛r❣❡ ♣r♦❜❧❡♠s✳

• ♦❛❧ ✶

❆r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

• ♦❛❧ ✷

❯s❡ t❤❡ str✉❝t✉r❡ ♦❢ ❋ ▼❛①▼✐♥♦rs ▼ t♦ s♣❡❡❞✲✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s ✭■■✮

• ❡♥❡r✐❝❛❧❧②✱ ♥❜✳ ❝r✐t✳ ♣♦✐♥ts✿

♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣✳
• ❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠s✿ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ ▼✬❇❛❦♦♣✱ P❛r❞♦✱

❙❝❤♦st✱ ❙❛❢❡②✱✳✳✳

• rö❜♥❡r ❜❛s✐s

✇❡❧❧✲s✉✐t❡❞ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳ ■❙❙❆❈ ✷✵✶✵✮✳ ❊①♣❡r✐♠❡♥t❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ ●❇ ❛❧❣♦r✐t❤♠s ✭❘❆●❧✐❜✱ ❋●❜ ✐♥ ▼❛♣❧❡✮ ❝❛♥ s♦❧✈❡ ❧❛r❣❡ ♣r♦❜❧❡♠s✳

• ♦❛❧ ✶

❆r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

• ♦❛❧ ✷

❯s❡ t❤❡ str✉❝t✉r❡ ♦❢ ❋ ▼❛①▼✐♥♦rs ▼ t♦ s♣❡❡❞✲✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s ✭■■✮

• ❡♥❡r✐❝❛❧❧②✱ ♥❜✳ ❝r✐t✳ ♣♦✐♥ts✿

♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣✳
• ❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠s✿ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ ▼✬❇❛❦♦♣✱ P❛r❞♦✱

❙❝❤♦st✱ ❙❛❢❡②✱✳✳✳

• rö❜♥❡r ❜❛s✐s → ✇❡❧❧✲s✉✐t❡❞ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳ ■❙❙❆❈ ✷✵✶✵✮✳

❊①♣❡r✐♠❡♥t❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ ●❇ ❛❧❣♦r✐t❤♠s ✭❘❆●❧✐❜✱ ❋●❜ ✐♥ ▼❛♣❧❡✮ ❝❛♥ s♦❧✈❡ ❧❛r❣❡ ♣r♦❜❧❡♠s✳

• ♦❛❧ ✶

❆r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

• ♦❛❧ ✷

❯s❡ t❤❡ str✉❝t✉r❡ ♦❢ ❋ ▼❛①▼✐♥♦rs ▼ t♦ s♣❡❡❞✲✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s ✭■■✮

• ❡♥❡r✐❝❛❧❧②✱ ♥❜✳ ❝r✐t✳ ♣♦✐♥ts✿

♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣✳
• ❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠s✿ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ ▼✬❇❛❦♦♣✱ P❛r❞♦✱

❙❝❤♦st✱ ❙❛❢❡②✱✳✳✳

• rö❜♥❡r ❜❛s✐s → ✇❡❧❧✲s✉✐t❡❞ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳ ■❙❙❆❈ ✷✵✶✵✮✳

❊①♣❡r✐♠❡♥t❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ ●❇ ❛❧❣♦r✐t❤♠s ✭❘❆●❧✐❜✱ ❋●❜ ✐♥ ▼❛♣❧❡✮ ❝❛♥ s♦❧✈❡ ❧❛r❣❡ ♣r♦❜❧❡♠s✳

• ♦❛❧ ✶

❆r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

• ♦❛❧ ✷

❯s❡ t❤❡ str✉❝t✉r❡ ♦❢ ❋ ▼❛①▼✐♥♦rs ▼ t♦ s♣❡❡❞✲✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼♦t✐✈❛t✐♦♥s ✭■■✮

• ❡♥❡r✐❝❛❧❧②✱ ♥❜✳ ❝r✐t✳ ♣♦✐♥ts✿

♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣✳
• ❡♦♠❡tr✐❝ ❛❧❣♦r✐t❤♠s✿ ❇❛s✉✱ P♦❧❧❛❝❦✱ ❘♦②✱ ❇❛♥❦✱ ●✐✉st✐✱ ❍❡✐♥t③✱ ▼✬❇❛❦♦♣✱ P❛r❞♦✱

❙❝❤♦st✱ ❙❛❢❡②✱✳✳✳

• rö❜♥❡r ❜❛s✐s → ✇❡❧❧✲s✉✐t❡❞ ❢♦r ❞❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s ✭❋❛✉❣èr❡✱ ❙❛❢❡②✱ ❙✳ ■❙❙❆❈ ✷✵✶✵✮✳

❊①♣❡r✐♠❡♥t❛❧❧②✱ ❝♦♠♣✉t✐♥❣ ❝r✐t✐❝❛❧ ♣♦✐♥ts ✇✐t❤ ●❇ ❛❧❣♦r✐t❤♠s ✭❘❆●❧✐❜✱ ❋●❜ ✐♥ ▼❛♣❧❡✮ ❝❛♥ s♦❧✈❡ ❧❛r❣❡ ♣r♦❜❧❡♠s✳

• ♦❛❧ ✶

❆r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

• ♦❛❧ ✷

❯s❡ t❤❡ str✉❝t✉r❡ ♦❢ ❋, ▼❛①▼✐♥♦rs(▼) t♦ s♣❡❡❞✲✉♣ t❤❡ ❝♦♠♣✉t❛t✐♦♥s✳

✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❙tr❛t❡❣② ♦❢ r❡s♦❧✉t✐♦♥ ✇✐t❤ ●rö❜♥❡r ❜❛s❡s

❲❡ ❢♦❝✉s ♦♥ t❤❡ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ I = ❋, ▼❛①▼✐♥♦rs(▼) ❈♦♠♣❧❡①✐t②

❋, ▼❛①▼✐♥♦rs(▼) = ✵

⇓ ✏❣r❡✈❧❡①✑ ●❜ ❘♦✇ ❊❝❤❡❧♦♥ ❢♦r♠s ♦❢ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s ✉♣ t♦ ❞❡❣r❡❡ Dr❡❣ ❖ ♥+Dr❡❣

♥

ω ⇓ ✏❧❡①✑ ●❜ ▲✐♥❡❛r ❛❧❣❡❜r❛ ✐♥

Q[❳] I

❛s Q✲✈❡❝t✳ s♣❛❝❡ ♦❢ ❞✐♠✳ ❉❡❣(I) ❖ (♥❉❡❣(I)ω)

✺✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼❛✐♥ r❡s✉❧ts

I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼) ❈♦♠♣❧❡①✐t② ♦❢ ❣r❡✈❧❡① ●❇✿ ❖ ♥+Dr❡❣

♥

ω ✳ ❇♦✉♥❞s ♦♥ Dr❡❣ ❢✶, . . . , ❢♣ ❣❡♥❡r✐❝ ❛♥❞ ❤♦♠✳✿ Dr❡❣ = ❉(♣ − ✶) + (❉ − ✷)♥ + ✷✳ ❢✶, . . . , ❢♣ ❣❡♥❡r✐❝✿ Dr❡❣ ≤ ❉(♣ − ✶) + (❉ − ✷)♥ + ✷✳ ❈♦♠♣❧❡①✐t② ❲❤❡♥ ❉ ❛♥❞ ♣ ❛r❡ ❝♦♥st❛♥t✱ t❤❡ ❛r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♦❢ ❝♦♠♣✉t✐♥❣ ❛ ❧❡① ●❇ ♦❢ ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜②✿ ❖ ♥✷♣ ✐❢ ❉ ✷❀ ❖

✶ ♥

❉ ✶ ❡ ♥ ✐❢ ❉ ✷✳ ♣♦❧②♥♦♠✐❛❧ ✐♥ ❉❊● ✳

✻✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

▼❛✐♥ r❡s✉❧ts

I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼) ❈♦♠♣❧❡①✐t② ♦❢ ❣r❡✈❧❡① ●❇✿ ❖ ♥+Dr❡❣

♥

ω ✳ ❇♦✉♥❞s ♦♥ Dr❡❣ ❢✶, . . . , ❢♣ ❣❡♥❡r✐❝ ❛♥❞ ❤♦♠✳✿ Dr❡❣ = ❉(♣ − ✶) + (❉ − ✷)♥ + ✷✳ ❢✶, . . . , ❢♣ ❣❡♥❡r✐❝✿ Dr❡❣ ≤ ❉(♣ − ✶) + (❉ − ✷)♥ + ✷✳ ❈♦♠♣❧❡①✐t② ❲❤❡♥ ❉ ❛♥❞ ♣ ❛r❡ ❝♦♥st❛♥t✱ t❤❡ ❛r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ♦❢ ❝♦♠♣✉t✐♥❣ ❛ ❧❡① ●❇ ♦❢ I ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜②✿ ❖(♥✷♣ω) ✐❢ ❉ = ✷❀ ❖( ✶

√♥((❉ − ✶)❡)♥ω) ✐❢ ❉ > ✷✳

♣♦❧②♥♦♠✐❛❧ ✐♥ ❉❊●(I)✳

✻✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❲♦r❦♣❧❛♥

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − →

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ♥ + Dr❡❣ Dr❡❣ ω ❖ (♥ · ❉❊●ω)

❙t❡♣ ✶✿ ❙t❛rt ✇✐t❤ ❛♥ ❛✣♥❡ ❝r✐t✐❝❛❧ ♣♦✐♥t s②st❡♠ ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼)✳ ❈♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ❢ (❤)

✶

, . . . , ❢ (❤)

♣

, ▼❛①▼✐♥♦rs(▼(❤)) ❍♦♠♦❣❡♥❡♦✉s s②st❡♠ ✐s ✵✲❞✐♠ → Dr❡❣ ❛✣♥❡ ≤ Dr❡❣ ❤♦♠✳ ❛♥❞ ❉❊● ❛✣♥❡ ≤ ❉❊● ❤♦♠✳ ❢♦r t❤❡ ❛♥❛❧②s✐s✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❢✶ ❢♣ ❛r❡ ❤♦♠♦❣❡♥❡♦✉s✳ ❙t❡♣ ✷✿ ❋✐♥❞

r❡❣ ✐s t❤❡ ❤♦♠♦❣❡♥❡♦✉s s❡tt✐♥❣✳ ✼✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❲♦r❦♣❧❛♥

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − →

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ♥ + Dr❡❣ Dr❡❣ ω ❖ (♥ · ❉❊●ω)

❙t❡♣ ✶✿ ❙t❛rt ✇✐t❤ ❛♥ ❛✣♥❡ ❝r✐t✐❝❛❧ ♣♦✐♥t s②st❡♠ ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼)✳ ❈♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ❢ (❤)

✶

, . . . , ❢ (❤)

♣

, ▼❛①▼✐♥♦rs(▼(❤)) ❍♦♠♦❣❡♥❡♦✉s s②st❡♠ ✐s ✵✲❞✐♠ → Dr❡❣ ❛✣♥❡ ≤ Dr❡❣ ❤♦♠✳ ❛♥❞ ❉❊● ❛✣♥❡ ≤ ❉❊● ❤♦♠✳ ⇒ ❢♦r t❤❡ ❛♥❛❧②s✐s✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❢✶, . . . , ❢♣ ❛r❡ ❤♦♠♦❣❡♥❡♦✉s✳ ❙t❡♣ ✷✿ ❋✐♥❞

r❡❣ ✐s t❤❡ ❤♦♠♦❣❡♥❡♦✉s s❡tt✐♥❣✳ ✼✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❲♦r❦♣❧❛♥

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − →

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ♥ + Dr❡❣ Dr❡❣ ω ❖ (♥ · ❉❊●ω)

❙t❡♣ ✶✿ ❙t❛rt ✇✐t❤ ❛♥ ❛✣♥❡ ❝r✐t✐❝❛❧ ♣♦✐♥t s②st❡♠ ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼)✳ ❈♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ❢ (❤)

✶

, . . . , ❢ (❤)

♣

, ▼❛①▼✐♥♦rs(▼(❤)) ❍♦♠♦❣❡♥❡♦✉s s②st❡♠ ✐s ✵✲❞✐♠ → Dr❡❣ ❛✣♥❡ ≤ Dr❡❣ ❤♦♠✳ ❛♥❞ ❉❊● ❛✣♥❡ ≤ ❉❊● ❤♦♠✳ ⇒ ❢♦r t❤❡ ❛♥❛❧②s✐s✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❢✶, . . . , ❢♣ ❛r❡ ❤♦♠♦❣❡♥❡♦✉s✳ ❙t❡♣ ✷✿ ❋✐♥❞ Dr❡❣ ✐s t❤❡ ❤♦♠♦❣❡♥❡♦✉s s❡tt✐♥❣✳

✼✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❙♦♠❡t❤✐♥❣ s♣❡❝✐❛❧ ❤❛♣♣❡♥s ✇✐t❤ ♠✐♥♦rs✳✳✳

❲❡ ❝♦♥s✐❞❡r ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ♣♦❧②♥♦♠✐❛❧ ♠❛tr✐❝❡s✳ ❇❡r♥st❡✐♥✴❙t✉r♠❢❡❧s✴❩❡❧❡✈✐♥s❦✐ ✶✾✾✸✱ ✳✳✳ ▲❡t ❯ = (✉✐,❥) ❜❡ ❛♥ ♣ × (♥ − ✶) ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✐♥❞❡t❡r♠✐♥❛t❡s✳ ❚❤❡ s❡t ♦❢ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❯ ✐s ❛ ✉♥✐✈❡rs❛❧ ●rö❜♥❡r ❜❛s✐s✳ ❚❤✐s ❣✐✈❡s t❤❡ ✐♥t✉✐t✐♦♥ t❤❛t ❣♦♦❞ t❤✐♥❣s ♠❛② ❤❛♣♣❡♥ ✐♥ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s✳ ▼❛①▼✐♥♦rs ▼ ✭❋❛✉❣èr❡✴❙❛❢❡②✴❙✳✮ ▲❡t ▼ ✇✐t❤ ❜❡ ❛♥ ♣ ♥ ✶ ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ❧✐♥❡❛r ❢♦r♠s ✐♥ ①✵ ①♥

✶ ♣ ✳

❆ ❣r❡✈❧❡① ●rö❜♥❡r ❜❛s✐s ▼✐♥♦rs ▼ ♣ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ♣ ♣ ♠✐♥♦rs✳ ❍❡r❡

r❡❣

♣ ✭t♦ ❜❡ ❝♦♠♣❛r❡❞ ✇✐t❤ ▼❛❝❛✉❧❛②✬s ❜♦✉♥❞ ♥ ♣ ♣ ❉ ✶ ✶✮✳

✽✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❙♦♠❡t❤✐♥❣ s♣❡❝✐❛❧ ❤❛♣♣❡♥s ✇✐t❤ ♠✐♥♦rs✳✳✳

❲❡ ❝♦♥s✐❞❡r ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ♣♦❧②♥♦♠✐❛❧ ♠❛tr✐❝❡s✳ ❇❡r♥st❡✐♥✴❙t✉r♠❢❡❧s✴❩❡❧❡✈✐♥s❦✐ ✶✾✾✸✱ ✳✳✳ ▲❡t ❯ = (✉✐,❥) ❜❡ ❛♥ ♣ × (♥ − ✶) ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✐♥❞❡t❡r♠✐♥❛t❡s✳ ❚❤❡ s❡t ♦❢ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❯ ✐s ❛ ✉♥✐✈❡rs❛❧ ●rö❜♥❡r ❜❛s✐s✳ ❚❤✐s ❣✐✈❡s t❤❡ ✐♥t✉✐t✐♦♥ t❤❛t ❣♦♦❞ t❤✐♥❣s ♠❛② ❤❛♣♣❡♥ ✐♥ ●rö❜♥❡r ❜❛s❡s ❝♦♠♣✉t❛t✐♦♥s✳ ▼❛①▼✐♥♦rs(▼) ✭❋❛✉❣èr❡✴❙❛❢❡②✴❙✳✮ ▲❡t ▼ ✇✐t❤ ❜❡ ❛♥ ♣ × (♥ − ✶) ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ❧✐♥❡❛r ❢♦r♠s ✐♥ Q[①✵, . . . , ①♥−✶−♣]✳ ❆ ❣r❡✈❧❡① ●rö❜♥❡r ❜❛s✐s ▼✐♥♦rs(▼, ♣) ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ♣ × ♣ ♠✐♥♦rs✳ ❍❡r❡ Dr❡❣ = ♣ ✭t♦ ❜❡ ❝♦♠♣❛r❡❞ ✇✐t❤ ▼❛❝❛✉❧❛②✬s ❜♦✉♥❞ (♥ − ♣)♣(❉ − ✶) + ✶✮✳

✽✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❇❛s✐❝ t♦♦❧✿ ❍✐❧❜❡rt s❡r✐❡s

• ❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ s✉❝❝❡ss✐✈❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳

❳ ❞ ❢ ❳ ❞❡❣ ❢ ❞ ❢ ❤♦♠♦❣❡♥❡♦✉s ❳

❞ ✵

❳ ❞ ▲❡t ■ ❜❡ ❛ ❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧ ♦❢ ❳ ①✶ ①♥ ✳ ■❞ ❢ ■ ❞❡❣ ❢ ❞ ❢ ❤♦♠♦❣❡♥❡♦✉s ✱ ■

❞ ✵■❞ ❳ ❞ ■❞

✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✲✈❡❝t♦r s♣❛❝❡✳ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ■✿ ❍❙■ t

❞ ✵ ❞✐♠ ❳ ❞ ■❞

t❞ ❞✐♠ ■ ❛♥❞ ❉❊● ■ ✭ P ✶ ✮ ❝❛♥ ❜❡ r❡❛❞ ♦♥ ❍❙■ t

P t ✶ t ❞✐♠ ■ ✳

■❢ ■ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✱

r❡❣ ■

❞❡❣ ❍❙■ t ✶✳ ■❢ ❢ ✐s ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ❉ ❛♥❞ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ ❛♥② ♣r✐♠❡ ❛ss♦❝✐❛t❡❞ t♦ ■✱ ❍❙■

❢

t ✶ t❉ ❍❙■ t ✳

✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❇❛s✐❝ t♦♦❧✿ ❍✐❧❜❡rt s❡r✐❡s

• ❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ s✉❝❝❡ss✐✈❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳

Q[❳]❞ = {❢ ∈ Q[❳] | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s } Q[❳] = ⊕❞≥✵Q[❳]❞ ▲❡t ■ ❜❡ ❛ ❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧ ♦❢ ❳ ①✶ ①♥ ✳ ■❞ ❢ ■ ❞❡❣ ❢ ❞ ❢ ❤♦♠♦❣❡♥❡♦✉s ✱ ■

❞ ✵■❞ ❳ ❞ ■❞

✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✲✈❡❝t♦r s♣❛❝❡✳ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ■✿ ❍❙■ t

❞ ✵ ❞✐♠ ❳ ❞ ■❞

t❞ ❞✐♠ ■ ❛♥❞ ❉❊● ■ ✭ P ✶ ✮ ❝❛♥ ❜❡ r❡❛❞ ♦♥ ❍❙■ t

P t ✶ t ❞✐♠ ■ ✳

■❢ ■ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✱

r❡❣ ■

❞❡❣ ❍❙■ t ✶✳ ■❢ ❢ ✐s ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ❉ ❛♥❞ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ ❛♥② ♣r✐♠❡ ❛ss♦❝✐❛t❡❞ t♦ ■✱ ❍❙■

❢

t ✶ t❉ ❍❙■ t ✳

✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❇❛s✐❝ t♦♦❧✿ ❍✐❧❜❡rt s❡r✐❡s

• ❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ s✉❝❝❡ss✐✈❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳

Q[❳]❞ = {❢ ∈ Q[❳] | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s } Q[❳] = ⊕❞≥✵Q[❳]❞ ▲❡t ■ ❜❡ ❛ ❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧ ♦❢ Q[❳] = Q[①✶, . . . , ①♥]✳ ■❞ = {❢ ∈ ■ | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s }✱ ■ = ⊕❞≥✵■❞

Q[❳]❞ ■❞

✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ Q✲✈❡❝t♦r s♣❛❝❡✳ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ■✿ ❍❙■ t

❞ ✵ ❞✐♠ ❳ ❞ ■❞

t❞ ❞✐♠ ■ ❛♥❞ ❉❊● ■ ✭ P ✶ ✮ ❝❛♥ ❜❡ r❡❛❞ ♦♥ ❍❙■ t

P t ✶ t ❞✐♠ ■ ✳

■❢ ■ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✱

r❡❣ ■

❞❡❣ ❍❙■ t ✶✳ ■❢ ❢ ✐s ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ❉ ❛♥❞ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ ❛♥② ♣r✐♠❡ ❛ss♦❝✐❛t❡❞ t♦ ■✱ ❍❙■

❢

t ✶ t❉ ❍❙■ t ✳

✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❇❛s✐❝ t♦♦❧✿ ❍✐❧❜❡rt s❡r✐❡s

• ❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ s✉❝❝❡ss✐✈❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳

Q[❳]❞ = {❢ ∈ Q[❳] | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s } Q[❳] = ⊕❞≥✵Q[❳]❞ ▲❡t ■ ❜❡ ❛ ❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧ ♦❢ Q[❳] = Q[①✶, . . . , ①♥]✳ ■❞ = {❢ ∈ ■ | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s }✱ ■ = ⊕❞≥✵■❞

Q[❳]❞ ■❞

✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ Q✲✈❡❝t♦r s♣❛❝❡✳ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ■✿ ❍❙■(t) =

❞=✵ ❞✐♠

• Q[❳]❞

■❞

• t❞

❞✐♠(■) ❛♥❞ ❉❊●(■) ✭= P(✶)✮ ❝❛♥ ❜❡ r❡❛❞ ♦♥ ❍❙■(t) =

P(t) (✶−t)❞✐♠(■) ✳

■❢ ■ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✱ Dr❡❣(■) = ❞❡❣(❍❙■(t)) + ✶✳ ■❢ ❢ ✐s ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ❉ ❛♥❞ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ ❛♥② ♣r✐♠❡ ❛ss♦❝✐❛t❡❞ t♦ ■✱ ❍❙■

❢

t ✶ t❉ ❍❙■ t ✳

✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❇❛s✐❝ t♦♦❧✿ ❍✐❧❜❡rt s❡r✐❡s

• ❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♦❢ s✉❝❝❡ss✐✈❡ ▼❛❝❛✉❧❛② ♠❛tr✐❝❡s✳

Q[❳]❞ = {❢ ∈ Q[❳] | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s } Q[❳] = ⊕❞≥✵Q[❳]❞ ▲❡t ■ ❜❡ ❛ ❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧ ♦❢ Q[❳] = Q[①✶, . . . , ①♥]✳ ■❞ = {❢ ∈ ■ | ❞❡❣(❢ ) = ❞, ❢ ❤♦♠♦❣❡♥❡♦✉s }✱ ■ = ⊕❞≥✵■❞

Q[❳]❞ ■❞

✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ Q✲✈❡❝t♦r s♣❛❝❡✳ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ■✿ ❍❙■(t) =

❞=✵ ❞✐♠

• Q[❳]❞

■❞

• t❞

❞✐♠(■) ❛♥❞ ❉❊●(■) ✭= P(✶)✮ ❝❛♥ ❜❡ r❡❛❞ ♦♥ ❍❙■(t) =

P(t) (✶−t)❞✐♠(■) ✳

■❢ ■ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✱ Dr❡❣(■) = ❞❡❣(❍❙■(t)) + ✶✳ ■❢ ❢ ✐s ❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ❉ ❛♥❞ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ ❛♥② ♣r✐♠❡ ❛ss♦❝✐❛t❡❞ t♦ ■✱ ❍❙■+❢ (t) = (✶ − t❉)❍❙■(t)✳

✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❙tr❛t❡❣②

❢✶, . . . , ❢♣ ❤♦♠♦❣❡♥❡♦✉s✳ I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼)✳

• ♦❛❧✿

♦❜t❛✐♥ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❍❙I(t)✳ ❙tr❛t❡❣②✿ ❙t❛rt ✇✐t❤ t❤❡ ♣ ♥ ✶ ♠❛tr✐① ❯ ✉✐ ❥ ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✳ ▲❡t ❜❡ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❯✳ ❆❜❤②❛♥❦❛r✬s ❢♦r♠✉❧❛ ❍❙ t ✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ❳ ✳ ❍❛♥❞❧❡ t❤❡ ❢❛❝t t❤❛t ✉✐ ❥

❢✐ ①❥ ✐s ♥♦t ❤♦♠♦❣❡♥❡♦✉s✳

❖❜t❛✐♥ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ ❛♥❞

✶✵✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❙tr❛t❡❣②

❢✶, . . . , ❢♣ ❤♦♠♦❣❡♥❡♦✉s✳ I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼)✳

• ♦❛❧✿

♦❜t❛✐♥ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❍❙I(t)✳ ❙tr❛t❡❣②✿ ❙t❛rt ✇✐t❤ t❤❡ ♣ × (♥ − ✶) ♠❛tr✐① ❯ = (✉✐,❥) ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✳ ▲❡t U ❜❡ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♠❛①✐♠❛❧ ♠✐♥♦rs ♦❢ ❯✳ ❆❜❤②❛♥❦❛r✬s ❢♦r♠✉❧❛ ❍❙U (t)✳ I = (U + {✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . , ❢♣) ∩ Q[❳]✳

❍❛♥❞❧❡ t❤❡ ❢❛❝t t❤❛t ✉✐,❥ − ∂❢✐

∂①❥ ✐s ♥♦t ❤♦♠♦❣❡♥❡♦✉s✳

❖❜t❛✐♥ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❍✐❧❜❡rt s❡r✐❡s ♦❢ U ❛♥❞ I

✶✵✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❉❡t❡r♠✐♥❛♥t❛❧ ✐❞❡❛❧s

❯ = (✉✐,❥)✿ ♣ × (♥ − ✶) ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✳ U = ▼❛①▼✐♥♦rs(❯) . ❆❜❤②❛♥❦❛r✬s ❢♦r♠✉❧❛ ❆❜❤②❛♥❦❛r ✶✾✽✽✱ ❈♦♥❝❛✴❍❡r③♦❣ ✶✾✾✹✳ ❍❙U (t) = ❞❡t(❆(t)) t(♣−✶

✷ )(✶ − t)♥(♣−✶) ,

✇❤❡r❡ ❆(t) ✐s t❤❡ (♣ − ✶) × (♣ − ✶) ♠❛tr✐① ✇❤❡r❡ ❆✐,❥(t) = ♣−✐

ℓ

♥−✶−❥

ℓ

• tℓ✳

Pr♦♣❡rt✐❡s ♦❢ U ❝♦❞✐♠(U ) = ♥ − ♣✳ ❉❊●(U ) = ♥−✶

♣−✶

• .

✶✶✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt②

❚❤❡♦r❡♠ ✭▼❛❝❛✉❧❛②✱ ❍♦❝❤st❡r✱ ❊❛❣♦♥✮ ❉❡t❡r♠✐♥❛♥t❛❧ r✐♥❣s ❛r❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✉✐ ❥ ❢✐ ①❥ ❢✶ ❢♣ ❳ ▲❡♠♠❛ ❆❧❣❡❜r❛✐❝ ❙❛r❞✬s ❚❤❡♦r❡♠ ❞✐♠ ✵✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ✉✐ ❥

❢✐ ①❥

❳ ❯ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ❤❛s ❝♦❞✐♠❡♥s✐♦♥ ♥♣ ❛s ❛♥ ✐❞❡❛❧ ♦❢ ❳ ❯ ✳ r❡❣✉❧❛r s❡q✉❡♥❝❡ ✐♥ ❳ ❯ ✭❯♥♠✐①❡❞♥❡ss ❚❤❡♦r❡♠✮✳ ■❢ ✉✐ ❥

❢✐ ①❥ ✇❡r❡ ❤♦♠♦❣❡♥❡♦✉s✱ ✇❡ ❝♦✉❧❞ ❝♦♥❝❧✉❞❡✳

✶✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt②

❚❤❡♦r❡♠ ✭▼❛❝❛✉❧❛②✱ ❍♦❝❤st❡r✱ ❊❛❣♦♥✮ ❉❡t❡r♠✐♥❛♥t❛❧ r✐♥❣s ❛r❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ I =

• U +
• ✉✐,❥ − ∂❢✐

∂①❥

• + ❢✶, . . . , ❢♣
• ∩ Q[❳]

▲❡♠♠❛ ❆❧❣❡❜r❛✐❝ ❙❛r❞✬s ❚❤❡♦r❡♠ ❞✐♠ ✵✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ✉✐ ❥

❢✐ ①❥

❳ ❯ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ❤❛s ❝♦❞✐♠❡♥s✐♦♥ ♥♣ ❛s ❛♥ ✐❞❡❛❧ ♦❢ ❳ ❯ ✳ r❡❣✉❧❛r s❡q✉❡♥❝❡ ✐♥ ❳ ❯ ✭❯♥♠✐①❡❞♥❡ss ❚❤❡♦r❡♠✮✳ ■❢ ✉✐ ❥

❢✐ ①❥ ✇❡r❡ ❤♦♠♦❣❡♥❡♦✉s✱ ✇❡ ❝♦✉❧❞ ❝♦♥❝❧✉❞❡✳

✶✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt②

❚❤❡♦r❡♠ ✭▼❛❝❛✉❧❛②✱ ❍♦❝❤st❡r✱ ❊❛❣♦♥✮ ❉❡t❡r♠✐♥❛♥t❛❧ r✐♥❣s ❛r❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ I =

• U +
• ✉✐,❥ − ∂❢✐

∂①❥

• + ❢✶, . . . , ❢♣
• ∩ Q[❳]

▲❡♠♠❛ ❆❧❣❡❜r❛✐❝ ❙❛r❞✬s ❚❤❡♦r❡♠ ⇒ ❞✐♠(I) = ✵✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ✉✐ ❥

❢✐ ①❥

❳ ❯ ❤❛s ❞✐♠❡♥s✐♦♥ ✵✳ ✉✐ ❥

❢✐ ①❥

❢✶ ❢♣ ❤❛s ❝♦❞✐♠❡♥s✐♦♥ ♥♣ ❛s ❛♥ ✐❞❡❛❧ ♦❢ ❳ ❯ ✳ r❡❣✉❧❛r s❡q✉❡♥❝❡ ✐♥ ❳ ❯ ✭❯♥♠✐①❡❞♥❡ss ❚❤❡♦r❡♠✮✳ ■❢ ✉✐ ❥

❢✐ ①❥ ✇❡r❡ ❤♦♠♦❣❡♥❡♦✉s✱ ✇❡ ❝♦✉❧❞ ❝♦♥❝❧✉❞❡✳

✶✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt②

❚❤❡♦r❡♠ ✭▼❛❝❛✉❧❛②✱ ❍♦❝❤st❡r✱ ❊❛❣♦♥✮ ❉❡t❡r♠✐♥❛♥t❛❧ r✐♥❣s ❛r❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ I =

• U +
• ✉✐,❥ − ∂❢✐

∂①❥

• + ❢✶, . . . , ❢♣
• ∩ Q[❳]

▲❡♠♠❛ ❆❧❣❡❜r❛✐❝ ❙❛r❞✬s ❚❤❡♦r❡♠ ⇒ ❞✐♠(I) = ✵✳ (U + {✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . , ❢♣) = I + ✉✐,❥ − ∂❢✐ ∂①❥ ⊂ Q[❳, ❯] ❤❛s

❞✐♠❡♥s✐♦♥ ✵✳ {✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . , ❢♣ ❤❛s ❝♦❞✐♠❡♥s✐♦♥ ♥♣ ❛s ❛♥ ✐❞❡❛❧ ♦❢ Q[❳, ❯]/U ✳

r❡❣✉❧❛r s❡q✉❡♥❝❡ ✐♥ ❳ ❯ ✭❯♥♠✐①❡❞♥❡ss ❚❤❡♦r❡♠✮✳ ■❢ ✉✐ ❥

❢✐ ①❥ ✇❡r❡ ❤♦♠♦❣❡♥❡♦✉s✱ ✇❡ ❝♦✉❧❞ ❝♦♥❝❧✉❞❡✳

✶✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt②

❚❤❡♦r❡♠ ✭▼❛❝❛✉❧❛②✱ ❍♦❝❤st❡r✱ ❊❛❣♦♥✮ ❉❡t❡r♠✐♥❛♥t❛❧ r✐♥❣s ❛r❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ I =

• U +
• ✉✐,❥ − ∂❢✐

∂①❥

• + ❢✶, . . . , ❢♣
• ∩ Q[❳]

▲❡♠♠❛ ❆❧❣❡❜r❛✐❝ ❙❛r❞✬s ❚❤❡♦r❡♠ ⇒ ❞✐♠(I) = ✵✳ (U + {✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . , ❢♣) = I + ✉✐,❥ − ∂❢✐ ∂①❥ ⊂ Q[❳, ❯] ❤❛s

❞✐♠❡♥s✐♦♥ ✵✳ {✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . , ❢♣ ❤❛s ❝♦❞✐♠❡♥s✐♦♥ ♥♣ ❛s ❛♥ ✐❞❡❛❧ ♦❢ Q[❳, ❯]/U ✳

r❡❣✉❧❛r s❡q✉❡♥❝❡ ✐♥ Q[❳, ❯]/U ✭❯♥♠✐①❡❞♥❡ss ❚❤❡♦r❡♠✮✳ ■❢ ✉✐,❥ − ∂❢✐

∂①❥ ✇❡r❡ ❤♦♠♦❣❡♥❡♦✉s✱ ✇❡ ❝♦✉❧❞ ❝♦♥❝❧✉❞❡✳

✶✷✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❇❛❝❦ t♦ t❤❡ ✏❤♦♠♦❣❡♥❡♦✉s✑ ❝❛s❡

❲❡ ❝♦♥s✐❞❡r U ❛s ❛♥ ✐❞❡❛❧ ♦❢ Q[❯, ❳]✳ ❍❙U (t) ✐s ❦♥♦✇♥ ❛♥❞ ✉✐,❥ − ∂❢✐

∂①❥ ✐s ♥♦t ❤♦♠♦❣❡♥❡♦✉s✳

✇❡ ❝♦♥s✐❞❡r ❛❧❧ ✈❛r✐❛❜❧❡s ✉✐,❥ ❛s ✈❛r✐❛❜❧❡s ♦❢ ✇❡✐❣❤t ❉ − ✶ ❲✐t❤ s✉❝❤ ❛ ✇❡✐❣❤t✱ ✉✐,❥ − ∂❢✐

∂①❥ ♠❛② ❜❡ s❡❡♥ ❛s ❛ ✏❤♦♠♦❣❡♥❡♦✉s✑ ♣♦❧②♥♦♠✐❛❧✳

q✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧ ❢ ♦❢ ✇❡✐❣❤t ❞❡❣r❡❡ ❞ ❛♥❞ t②♣❡ (❉ − ✶, ✶)✿ ❢ (λ❉−✶❯, λ❳) = λ❞❢ (❯, ❳) ❚❤✐s ❛❧❧♦✇s t♦ ❞❡✜♥❡ ❛ ❣r❛❞✐♥❣ ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ Q[❯, ❳]✳ ❲❡ ✉s❡ t❤✐s ❣r❛❞✐♥❣

✶✸✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❜❤②❛♥❦❛r✬s ❢♦r♠✉❧❛ ✇✐t❤ ✇❡✐❣❤ts ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ U

Q[❯, ❳] = ⊕❞∈NQ[❯, ❳]❞ ✭✇✐t❤ ✇❡✐❣❤t ❞❡❣r❡❡s✮ ◗✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧s ❛r❡ t❤♦s❡ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛t❡❞ ❜② ❛ s❡t ♦❢ q✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s✳ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ✇❍❙■(t) =

❞∈N ❞✐♠

• Q[❯,❳]❞

■❞

• t❞✳

❋✐rst Pr♦♣❡rt✐❡s ✭✇❡❧❧✲❦♥♦✇♥✮ ■♥ ❯ ✱ t❤❡ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ♦❢ t❤❡ ✐❞❡❛❧ ✭✇✐t❤ ✉✐ ❥ ♦❢ ✇❡✐❣❤t ❉ ✶✮ ✐s ✇❍❙ t❉

✶

■♥ ❳ ❯ ✱ ❚❤❡ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ♦❢ t❤❡ ✐❞❡❛❧ ✭✇✐t❤ ✉✐ ❥ ♦❢ ✇❡✐❣❤t ❉ ✶✮ ✐s ✇❍❙ t❉

✶

✶ t ♥

✭▼✐❧♥♦r✱ ❖❧✐♥✐❦✱ ❲❛❧r❡✐❝❤✱ s❡❡ ✏P♦✐♥❝❛ré s❡r✐❡s ♦❢ ❛ ❣r❛❞❡❞ ❛❧❣❡❜r❛✑✱ ❇♦✉r❜❛❦✐✮

■♥ ❳ ❯ ✇❍❙ t ❞❡t ❆ t❉

✶

t ❉

✶

♣ ✶ ✷

✶ t❉

✶ ♥ ♣ ✶ ✶

t ♥

✶✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❜❤②❛♥❦❛r✬s ❢♦r♠✉❧❛ ✇✐t❤ ✇❡✐❣❤ts ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ U

Q[❯, ❳] = ⊕❞∈NQ[❯, ❳]❞ ✭✇✐t❤ ✇❡✐❣❤t ❞❡❣r❡❡s✮ ◗✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧s ❛r❡ t❤♦s❡ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛t❡❞ ❜② ❛ s❡t ♦❢ q✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s✳ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ✇❍❙■(t) =

❞∈N ❞✐♠

• Q[❯,❳]❞

■❞

• t❞✳

❋✐rst Pr♦♣❡rt✐❡s ✭✇❡❧❧✲❦♥♦✇♥✮ ■♥ Q[❯]✱ t❤❡ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ♦❢ t❤❡ ✐❞❡❛❧ U ✭✇✐t❤ ✉✐,❥ ♦❢ ✇❡✐❣❤t ❉ − ✶✮ ✐s ✇❍❙U (t❉−✶) ■♥ Q[❳, ❯]✱ ❚❤❡ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ♦❢ t❤❡ ✐❞❡❛❧ U ✭✇✐t❤ ✉✐,❥ ♦❢ ✇❡✐❣❤t ❉ − ✶✮ ✐s ✇❍❙U (t❉−✶)/(✶ − t)♥.

✭▼✐❧♥♦r✱ ❖❧✐♥✐❦✱ ❲❛❧r❡✐❝❤✱ s❡❡ ✏P♦✐♥❝❛ré s❡r✐❡s ♦❢ ❛ ❣r❛❞❡❞ ❛❧❣❡❜r❛✑✱ ❇♦✉r❜❛❦✐✮

■♥ ❳ ❯ ✇❍❙ t ❞❡t ❆ t❉

✶

t ❉

✶

♣ ✶ ✷

✶ t❉

✶ ♥ ♣ ✶ ✶

t ♥

✶✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❜❤②❛♥❦❛r✬s ❢♦r♠✉❧❛ ✇✐t❤ ✇❡✐❣❤ts ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ U

Q[❯, ❳] = ⊕❞∈NQ[❯, ❳]❞ ✭✇✐t❤ ✇❡✐❣❤t ❞❡❣r❡❡s✮ ◗✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s ✐❞❡❛❧s ❛r❡ t❤♦s❡ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛t❡❞ ❜② ❛ s❡t ♦❢ q✉❛s✐✲❤♦♠♦❣❡♥❡♦✉s s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s✳ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ✇❍❙■(t) =

❞∈N ❞✐♠

• Q[❯,❳]❞

■❞

• t❞✳

❋✐rst Pr♦♣❡rt✐❡s ✭✇❡❧❧✲❦♥♦✇♥✮ ■♥ Q[❯]✱ t❤❡ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ♦❢ t❤❡ ✐❞❡❛❧ U ✭✇✐t❤ ✉✐,❥ ♦❢ ✇❡✐❣❤t ❉ − ✶✮ ✐s ✇❍❙U (t❉−✶) ■♥ Q[❳, ❯]✱ ❚❤❡ ✇❡✐❣❤t❡❞ ❍✐❧❜❡rt s❡r✐❡s ♦❢ t❤❡ ✐❞❡❛❧ U ✭✇✐t❤ ✉✐,❥ ♦❢ ✇❡✐❣❤t ❉ − ✶✮ ✐s ✇❍❙U (t❉−✶)/(✶ − t)♥.

✭▼✐❧♥♦r✱ ❖❧✐♥✐❦✱ ❲❛❧r❡✐❝❤✱ s❡❡ ✏P♦✐♥❝❛ré s❡r✐❡s ♦❢ ❛ ❣r❛❞❡❞ ❛❧❣❡❜r❛✑✱ ❇♦✉r❜❛❦✐✮

■♥ Q[❳, ❯], ✇❍❙U (t) = ❞❡t(❆(t❉−✶)) t(❉−✶)(♣−✶

✷ )(✶ − t❉−✶)♥(♣−✶)(✶ − t)♥ .

✶✹✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐♥❣ ❛ r❡❣✉❧❛r s❡q✉❡♥❝❡

I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼) . ({✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . ❢♣) ✐s ❛ Q[❳, ❯]/U ✲r❡❣✉❧❛r s❡q✉❡♥❝❡✿

✇❍❙

U +{✉✐,❥ − ∂❢✐

∂①❥ },❢✶,...,❢♣(t)

= (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣❍❙U (t❉−✶) ✶ t❉

✶ ♥ ✶ ♣ ✶

t❉ ♣ ❞❡t ❆ t❉

✶

✶ t❉

✶ ♥ ♣ ✶ ✶

t ♥t ❉

✶

♣ ✶ ✷

✶ t❉

✶ ♥ ♣ ✶

t❉ ♣ ❞❡t ❆ t❉

✶

✶ t ♥t ❉

✶

♣ ✶ ✷

❍❙ t

✶✺✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐♥❣ ❛ r❡❣✉❧❛r s❡q✉❡♥❝❡

I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼) . ({✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . ❢♣) ✐s ❛ Q[❳, ❯]/U ✲r❡❣✉❧❛r s❡q✉❡♥❝❡✿

✇❍❙

U +{✉✐,❥ − ∂❢✐

∂①❥ },❢✶,...,❢♣(t)

= (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣❍❙U (t❉−✶) = (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣ ❞❡t(❆(t❉−✶)) (✶ − t❉−✶)♥(♣−✶)(✶ − t)♥t(❉−✶)(♣−✶

✷ )

✶ t❉

✶ ♥ ♣ ✶

t❉ ♣ ❞❡t ❆ t❉

✶

✶ t ♥t ❉

✶

♣ ✶ ✷

❍❙ t

✶✺✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐♥❣ ❛ r❡❣✉❧❛r s❡q✉❡♥❝❡

I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼) . ({✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . ❢♣) ✐s ❛ Q[❳, ❯]/U ✲r❡❣✉❧❛r s❡q✉❡♥❝❡✿

✇❍❙

U +{✉✐,❥ − ∂❢✐

∂①❥ },❢✶,...,❢♣(t)

= (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣❍❙U (t❉−✶) = (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣ ❞❡t(❆(t❉−✶)) (✶ − t❉−✶)♥(♣−✶)(✶ − t)♥t(❉−✶)(♣−✶

✷ )

= (✶ − t❉−✶)♥−♣(✶ − t❉)♣ ❞❡t(❆(t❉−✶)) (✶ − t)♥t(❉−✶)(♣−✶

✷ )

❍❙ t

✶✺✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❆❞❞✐♥❣ ❛ r❡❣✉❧❛r s❡q✉❡♥❝❡

I = ❢✶, . . . , ❢♣, ▼❛①▼✐♥♦rs(▼) . ({✉✐,❥ − ∂❢✐

∂①❥ }, ❢✶, . . . ❢♣) ✐s ❛ Q[❳, ❯]/U ✲r❡❣✉❧❛r s❡q✉❡♥❝❡✿

✇❍❙

U +{✉✐,❥ − ∂❢✐

∂①❥ },❢✶,...,❢♣(t)

= (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣❍❙U (t❉−✶) = (✶ − t❉−✶)(♥−✶)♣(✶ − t❉)♣ ❞❡t(❆(t❉−✶)) (✶ − t❉−✶)♥(♣−✶)(✶ − t)♥t(❉−✶)(♣−✶

✷ )

= (✶ − t❉−✶)♥−♣(✶ − t❉)♣ ❞❡t(❆(t❉−✶)) (✶ − t)♥t(❉−✶)(♣−✶

✷ )

= ❍❙I(t).

✶✺✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

• ❡♥❡r✐❝✐t② ❄

Pr♦❜❧❡♠✿ ✉♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ✐s t❤❡ ❍✐❧❜❡rt s❡r✐❡s ❡q✉❛❧ t♦ ❍❙I(t) =

• ✶ − t❉−✶♥−♣

✶ − t❉♣ ❞❡t

• ❆
• t❉−✶

(✶ − t)♥t(❉−✶)(♣−✶

✷ )

? ❚❤❡♦r❡♠ ✭❊✐s❡♥❜✉❞✱❈♦♥❝❛✱❍❡r③♦❣✮ ▲❡t ❇ ❜❡ ❛ ♣ ♥ ✶ ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s ✐♥ ❳ ✿ ❇ ❢✶ ✶ ❢✶ ♥

✶

✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣ ✶ ❢♣ ♥

✶

■❢ ❝♦❞✐♠ ▼❛①▼✐♥♦rs ❇ ♥ ♣✱ t❤❡♥ ❍❙ ▼❛①▼✐♥♦rs ❇ t ❞❡t ❆ t t

♣ ✶ ✷

✶ t ❞✐♠

▼❛①▼✐♥♦rs ❇

❆♥s✇❡r✿ ❊q✉✐✈❛❧❡♥t t♦ ❞✐♠ ✵✳

✶✻✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

• ❡♥❡r✐❝✐t② ❄

Pr♦❜❧❡♠✿ ✉♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ✐s t❤❡ ❍✐❧❜❡rt s❡r✐❡s ❡q✉❛❧ t♦ ❍❙I(t) =

• ✶ − t❉−✶♥−♣

✶ − t❉♣ ❞❡t

• ❆
• t❉−✶

(✶ − t)♥t(❉−✶)(♣−✶

✷ )

? ❚❤❡♦r❡♠ ✭❊✐s❡♥❜✉❞✱❈♦♥❝❛✱❍❡r③♦❣✮ ▲❡t ❇ ❜❡ ❛ ♣ × (♥ − ✶) ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s ✐♥ K[❳]✿ ❇ =    ❢✶,✶ . . . ❢✶,♥−✶ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,♥−✶    ■❢ ❝♦❞✐♠(▼❛①▼✐♥♦rs(❇)) = ♥ − ♣✱ t❤❡♥ ❍❙▼❛①▼✐♥♦rs(❇)(t) = ❞❡t(❆(t)) t(♣−✶

✷ )(✶ − t)❞✐♠(▼❛①▼✐♥♦rs(❇)) .

❆♥s✇❡r✿ ❊q✉✐✈❛❧❡♥t t♦ ❞✐♠ ✵✳

✶✻✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

• ❡♥❡r✐❝✐t② ❄

Pr♦❜❧❡♠✿ ✉♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ✐s t❤❡ ❍✐❧❜❡rt s❡r✐❡s ❡q✉❛❧ t♦ ❍❙I(t) =

• ✶ − t❉−✶♥−♣

✶ − t❉♣ ❞❡t

• ❆
• t❉−✶

(✶ − t)♥t(❉−✶)(♣−✶

✷ )

? ❚❤❡♦r❡♠ ✭❊✐s❡♥❜✉❞✱❈♦♥❝❛✱❍❡r③♦❣✮ ▲❡t ❇ ❜❡ ❛ ♣ × (♥ − ✶) ♠❛tr✐① ✇❤♦s❡ ❡♥tr✐❡s ❛r❡ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s ✐♥ K[❳]✿ ❇ =    ❢✶,✶ . . . ❢✶,♥−✶ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ❢♣,✶ . . . ❢♣,♥−✶    ■❢ ❝♦❞✐♠(▼❛①▼✐♥♦rs(❇)) = ♥ − ♣✱ t❤❡♥ ❍❙▼❛①▼✐♥♦rs(❇)(t) = ❞❡t(❆(t)) t(♣−✶

✷ )(✶ − t)❞✐♠(▼❛①▼✐♥♦rs(❇)) .

❆♥s✇❡r✿ ❊q✉✐✈❛❧❡♥t t♦ ❞✐♠(I) = ✵✳

✶✻✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t②

❍❙I(t) =

• ✶ − t❉−✶♥−♣

✶ − t❉♣ ❞❡t

• ❆
• t❉−✶

(✶ − t)♥t(❉−✶)(♣−✶

✷ )

. ❍♦♠♦❣❡♥❡♦✉s Dr❡❣ = ❞❡❣(❍❙I(t)) + ✶ = ❉(♣ − ✶) + (❉ − ✷)♥ + ✷. ❉❊●(I) = ❍❙I(✶) = ♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣.

❇❛❝❦ t♦ t❤❡ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❝❛s❡ ❝♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡✿ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s

r❡❣

❉ ♣ ✶ ❉ ✷ ♥ ✷ ❉❊●

♥ ✶ ♣ ✶ ❉♣ ❉

✶ ♥

♣

❇♦✉♥❞s ❛r❡ r❡❛❝❤❡❞ ✐♥ ♣r❛❝t✐❝❡ ✦

r❡❣ ✐♥❞❡♣❡♥❞❛♥t ♦❢ ♥ ✇❤❡♥ ❉

✷✳

✶✼✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t②

❍❙I(t) =

• ✶ − t❉−✶♥−♣

✶ − t❉♣ ❞❡t

• ❆
• t❉−✶

(✶ − t)♥t(❉−✶)(♣−✶

✷ )

. ❍♦♠♦❣❡♥❡♦✉s Dr❡❣ = ❞❡❣(❍❙I(t)) + ✶ = ❉(♣ − ✶) + (❉ − ✷)♥ + ✷. ❉❊●(I) = ❍❙I(✶) = ♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣.

❇❛❝❦ t♦ t❤❡ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❝❛s❡ ❝♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡✿ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s

r❡❣

❉ ♣ ✶ ❉ ✷ ♥ ✷ ❉❊●

♥ ✶ ♣ ✶ ❉♣ ❉

✶ ♥

♣

❇♦✉♥❞s ❛r❡ r❡❛❝❤❡❞ ✐♥ ♣r❛❝t✐❝❡ ✦

r❡❣ ✐♥❞❡♣❡♥❞❛♥t ♦❢ ♥ ✇❤❡♥ ❉

✷✳

✶✼✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❉❡❣r❡❡ ❛♥❞ r❡❣✉❧❛r✐t②

❍❙I(t) =

• ✶ − t❉−✶♥−♣

✶ − t❉♣ ❞❡t

• ❆
• t❉−✶

(✶ − t)♥t(❉−✶)(♣−✶

✷ )

. ❍♦♠♦❣❡♥❡♦✉s Dr❡❣ = ❞❡❣(❍❙I(t)) + ✶ = ❉(♣ − ✶) + (❉ − ✷)♥ + ✷. ❉❊●(I) = ❍❙I(✶) = ♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣.

❇❛❝❦ t♦ t❤❡ ♥♦♥✲❤♦♠♦❣❡♥❡♦✉s ❝❛s❡ ❝♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ♣❛rt ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡✿ ◆♦♥✲❤♦♠♦❣❡♥❡♦✉s Dr❡❣ ≤ ❉(♣ − ✶) + (❉ − ✷)♥ + ✷. ❉❊●(I) ≤ ♥−✶

♣−✶

• ❉♣(❉ − ✶)♥−♣.

❇♦✉♥❞s ❛r❡ r❡❛❝❤❡❞ ✐♥ ♣r❛❝t✐❝❡ ✦ Dr❡❣ ✐♥❞❡♣❡♥❞❛♥t ♦❢ ♥ ✇❤❡♥ ❉ = ✷✳

✶✼✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♠♣❧❡①✐t②

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − →

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ♥ + Dr❡❣ Dr❡❣ ω ❖ (♥ · ❉❊●ω)

❋✐rst ❛♥❛❧②s✐s✿ ❤♦✇ ❞♦❡s t❤❡ ❝♦♠♣❧❡①✐t② ❜❡❤❛✈❡ ✇❤❡♥ ♥ ❣r♦✇s ❄ ❈♦♠♣❧❡①✐t② ♦❢ ❣r❡✈❧❡① ●❇ ❢❡❛s✐❜❧❡ ❡①♣♦♥❡♥t ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ r♦✇ ❡❝❤❡❧♦♥ ❢♦r♠✳ ✐❢ ❉ ✷ ❖ ♥✷♣ ✱ ♣♦❧②♥♦♠✐❛❧ ✐♥ ♥✳ ✐❢ ❉ ✷ ❖ ❉ ✶ ❡ ♥ ❈♦♠♣❧❡①✐t② ♦❢ ❝❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣ ❖ ♥ ♥ ✶ ♣ ✶ ❉♣ ❉ ✶

♥ ♣

✉♣♣❡r ❜♦✉♥❞❡❞ ❜② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❣r❡✈❧❡① ●❇

✶✽✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♠♣❧❡①✐t②

❙②st❡♠ − → ❣r❡✈❧❡① ●❇ − →

❈❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣❧❡① ●❇.

❈♦♠♣❧❡①✐t② ❖ ♥ + Dr❡❣ Dr❡❣ ω ❖ (♥ · ❉❊●ω)

❋✐rst ❛♥❛❧②s✐s✿ ❤♦✇ ❞♦❡s t❤❡ ❝♦♠♣❧❡①✐t② ❜❡❤❛✈❡ ✇❤❡♥ ♥ ❣r♦✇s ❄ ❈♦♠♣❧❡①✐t② ♦❢ ❣r❡✈❧❡① ●❇ ω ❢❡❛s✐❜❧❡ ❡①♣♦♥❡♥t ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ r♦✇ ❡❝❤❡❧♦♥ ❢♦r♠✳ ✐❢ ❉ = ✷ ❖

• ♥✷♣ω

✱ ♣♦❧②♥♦♠✐❛❧ ✐♥ ♥✳ ✐❢ ❉ > ✷ ❖ ((❉ − ✶)❡)♥ω) ❈♦♠♣❧❡①✐t② ♦❢ ❝❤❛♥❣❡ ♦❢ ♦r❞❡r✐♥❣ ❖

• ♥
• ♥ − ✶

♣ − ✶ ω ❉♣ω(❉ − ✶)(♥−♣)ω

• ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❣r❡✈❧❡① ●❇

✶✽✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♠♣❧❡①✐t② ✭■■✮

❚❤❡ ♥✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❢♦r ❝♦♠♣✉t✐♥❣ ❛ ❧❡① ●❇ ♦❢ I ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖

• ❉❊●(I)

❧♦❣(❉)+❧♦❣(✷)+✶ ❧♦❣(❉−✶)

ω

• .

❧♦❣ ❉ ❧♦❣ ✷ ✶ ❧♦❣ ❉ ✶ ✹ ✵✸ ❢♦r ❉ ✷

❝♦♠♣❧❡①✐t② ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

❧✐♠

❉

❧♦❣ ❉ ❧♦❣ ✷ ✶ ❧♦❣ ❉ ✶ ✶ ◆✉♠❡r✐❝❛❧ ❡✈❛❧✉❛t✐♦♥s✿ ❧♦❣

♥

r❡❣

♥

❧♦❣ ❉❊● ✶ ✸

✶✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♠♣❧❡①✐t② ✭■■✮

❚❤❡ ♥✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❢♦r ❝♦♠♣✉t✐♥❣ ❛ ❧❡① ●❇ ♦❢ I ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖

• ❉❊●(I)

❧♦❣(❉)+❧♦❣(✷)+✶ ❧♦❣(❉−✶)

ω

• .

❧♦❣(❉) + ❧♦❣(✷) + ✶ ❧♦❣(❉ − ✶) < ✹.✵✸ ❢♦r ❉ > ✷,

❝♦♠♣❧❡①✐t② ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

❧✐♠

❉

❧♦❣ ❉ ❧♦❣ ✷ ✶ ❧♦❣ ❉ ✶ ✶ ◆✉♠❡r✐❝❛❧ ❡✈❛❧✉❛t✐♦♥s✿ ❧♦❣

♥

r❡❣

♥

❧♦❣ ❉❊● ✶ ✸

✶✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♠♣❧❡①✐t② ✭■■✮

❚❤❡ ♥✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❢♦r ❝♦♠♣✉t✐♥❣ ❛ ❧❡① ●❇ ♦❢ I ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖

• ❉❊●(I)

❧♦❣(❉)+❧♦❣(✷)+✶ ❧♦❣(❉−✶)

ω

• .

❧♦❣(❉) + ❧♦❣(✷) + ✶ ❧♦❣(❉ − ✶) < ✹.✵✸ ❢♦r ❉ > ✷,

❝♦♠♣❧❡①✐t② ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

❧✐♠

❉

❧♦❣ ❉ ❧♦❣ ✷ ✶ ❧♦❣ ❉ ✶ ✶ ◆✉♠❡r✐❝❛❧ ❡✈❛❧✉❛t✐♦♥s✿ ❧♦❣

♥

r❡❣

♥

❧♦❣ ❉❊● ✶ ✸

✶✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♠♣❧❡①✐t② ✭■■✮

❚❤❡ ♥✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❢♦r ❝♦♠♣✉t✐♥❣ ❛ ❧❡① ●❇ ♦❢ I ✐s ✉♣♣❡r ❜♦✉♥❞❡❞ ❜② ❖

• ❉❊●(I)

❧♦❣(❉)+❧♦❣(✷)+✶ ❧♦❣(❉−✶)

ω

• .

❧♦❣(❉) + ❧♦❣(✷) + ✶ ❧♦❣(❉ − ✶) < ✹.✵✸ ❢♦r ❉ > ✷,

❝♦♠♣❧❡①✐t② ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝r✐t✐❝❛❧ ♣♦✐♥ts✳

❧✐♠

❉→∞

❧♦❣(❉) + ❧♦❣(✷) + ✶ ❧♦❣(❉ − ✶) = ✶. ◆✉♠❡r✐❝❛❧ ❡✈❛❧✉❛t✐♦♥s✿ ❧♦❣ ♥+Dr❡❣

♥

• ❧♦❣ (❉❊●(I)) ≈ ✶.✸

✶✾✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♥❝❧✉s✐♦♥✴P❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥ ◆❡✇ ❝♦♠♣❧❡①✐t② ❡st✐♠❛t❡s✳ ❯♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❜❡❤❛✈✐♦✉r✳ ❆ ♥❡✇ ❢❛♠✐❧② ♦❢ str✉❝t✉r❡❞ s②st❡♠s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ ❛r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ✇✐t❤ ●rö❜♥❡r ❜❛s❡s✳ ❍✐❧❜❡rt s❡r✐❡s ✜rst st❡♣ t♦✇❛r❞s ❛ ❞❡❞✐❝❛t❡❞ ●❇ ❛❧❣♦r✐t❤♠✳ ❲♦r❦ ✐♥ ♣r♦❣r❡ss ▼✐①❡❞ ❞❡❣r❡❡s✳ ❋✺✴❘❡❞✉❝t✐♦♥s t♦ ③❡r♦✳ t♦♦❧s ❢r♦♠ ❤♦♠♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛✳

✷✵✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❈♦♥❝❧✉s✐♦♥✴P❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥ ◆❡✇ ❝♦♠♣❧❡①✐t② ❡st✐♠❛t❡s✳ ❯♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❜❡❤❛✈✐♦✉r✳ ❆ ♥❡✇ ❢❛♠✐❧② ♦❢ str✉❝t✉r❡❞ s②st❡♠s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ ❛r✐t❤♠❡t✐❝ ❝♦♠♣❧❡①✐t② ✇✐t❤ ●rö❜♥❡r ❜❛s❡s✳ ❍✐❧❜❡rt s❡r✐❡s ✜rst st❡♣ t♦✇❛r❞s ❛ ❞❡❞✐❝❛t❡❞ ●❇ ❛❧❣♦r✐t❤♠✳ ❲♦r❦ ✐♥ ♣r♦❣r❡ss ▼✐①❡❞ ❞❡❣r❡❡s✳ ❋✺✴❘❡❞✉❝t✐♦♥s t♦ ③❡r♦✳ t♦♦❧s ❢r♦♠ ❤♦♠♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛✳

✷✵✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r

❘❡❧❛t❡❞ Pr♦❜❧❡♠s

❈r②♣t♦❧♦❣② ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r② ❈r②♣t❛♥❛❧②s✐s ♦❢ ❍❋❊ ✭❑✐♣♥✐s✱ ❙❤❛♠✐r✱ ❇❡tt❛❧❡✱ P❡rr❡t✱ ❋❛✉❣èr❡✱ ✳✳✳✮✳ ❈♦✉rt♦✐s ❛✉t❤❡♥t✐✜❝❛t✐♦♥ s❝❤❡♠❡ ✭❈♦✉rt♦✐s✱ P❡rr❡t✱ ❋❛✉❣èr❡✱ ▲❡✈②✲❞✐t✲❱❡❤❡❧✱ ❙❛❢❡② ❊❧ ❉✐♥✱ ❙✳✱ ✳✳✳✮✳ ❘❛♥❦ ♠❡tr✐❝ ❞❡❝♦❞✐♥❣ ✭●❛❜✐❞✉❧✐♥✱ ▲♦✐❞r❡❛✉✱ ❖✉r✐✈s❦✐✱ ❏♦❤❛♥♥ss♦♥✱ ✳✳✳✮✳ ▼✉❧t✐❤♦♠♦❣❡♥❡♦✉s s②st❡♠s✳ ❊♥✉♠❡r❛t✐✈❡ ❣❡♦♠❡tr②✿ ❙❝❤✉❜❡rt ❝❛❧❝✉❧✉s✱ ❙❡❝❛♥t ❝♦♥❥❡❝t✉r❡✱ ✳ ✳ ✳ P♦❧❛r✴❇✐♣♦❧❛r ✈❛r✐❡t✐❡s✳

• ❧♦❜❛❧ ♦♣t✐♠✐③❛t✐♦♥ ✭❝❢✳ ❆✉r❡❧✐❡♥✬s t❛❧❦✮✱✳✳✳

✷✶✴✷✶ P❏ ❙♣❛❡♥❧❡❤❛✉❡r