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Apr 08, 2023 315 likes •761 views

rt ts r t r srs s r t

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SLIDE 1
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❢♦r ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧

s❡♠✐❣r♦✉♣s ♦❢ ❛ ❣✐✈❡♥ ❣❡♥✉s

■▼◆❙ ✷✵✶✵ ✲ ■❜❡r✐❛♥ ▼❡❡t✐♥❣ ♦♥ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s

  • r❛♥❛❞❛✱ ❋❡❜r✉❛r② ✷✵✶✵

❱í❝t♦r ❇❧❛♥❝♦

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ➪❧❣❡❜r❛ ❯♥✐✈❡rs✐❞❛❞ ❞❡ ●r❛♥❛❞❛ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ P✳❆✳ ●❛r❝í❛✲❙á♥❝❤❡③ ❛♥❞ ❏✳ P✉❡rt♦✮

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SLIDE 2

❖✉t❧✐♥❡

✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ❈♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s✿ ❚❤❡♦r❡t✐❝❛❧ ❘❡s✉❧ts ✸ ●❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐♥ ♣r❛❝t✐❝❡ ✹ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ✹

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SLIDE 3

❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s

❙ = ♥✶, . . . , ♥♣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣✱ ❣❝❞(♥✐, ♥❥) = ✶✳ ♠✉❧t✐♣❧✐❝✐t②✿ ♠(❙) = ♠✐♥{♥ ∈ N : ♥ ∈ ❙\{✵}} ❣❡♥✉s✿ ❣(❙) = ★(N\❙) ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✿ ❋(❙) = ♠❛①{♥ ∈ N : ♥ ∈ ❙} S❣ = {❙ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ❣(❙) = ❣}, ♥❣ = ★S❣

▼✉❧t✐♣❧✐❝✐t② ✸✿ ✭❘♦s❛❧❡s✱ ✷✵✵✺✮✳ ❇♦✉♥❞s ✉s✐♥❣ ❉✐❝❦ ♣❛t❤s ✭❇r❛s✲❆♠♦rós ❛♥❞ ❉❡ ▼✐❡r✱ ✷✵✵✼✮ ❋✐❜♦♥❛❝❝✐ ❈♦♥❥❡❝t✉r❡✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✽✮✳ ❇♦✉♥❞s ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣s tr❡❡ ♠❡t❤♦❞✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✾✮ ■♠♣r♦✈❡❞ ❇♦✉♥❞s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜② ❣❡♥❡r❛t✐♥❣ tr❡❡s✿ ✭❊❧✐③❛❧❞❡✱ ✷✵✵✾✮ ■♠♣r♦✈❡❞ ❛s②♠♣t♦t✐❝ ❧♦✇❡r ❜♦✉♥❞✿ ✭❩❤❛♦✱ ✷✵✵✾✮

❙♦❢t✇❛r❡✿ ♥✉♠❡r✐❝❛❧s❣♣s ✐♥ ●❆P ✭❉❡❧❣❛❞♦ ❡t ❛❧✱ ✷✵✵✽✮

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SLIDE 4

❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s

❙ = ♥✶, . . . , ♥♣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣✱ ❣❝❞(♥✐, ♥❥) = ✶✳ ♠✉❧t✐♣❧✐❝✐t②✿ ♠(❙) = ♠✐♥{♥ ∈ N : ♥ ∈ ❙\{✵}} ❣❡♥✉s✿ ❣(❙) = ★(N\❙) ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✿ ❋(❙) = ♠❛①{♥ ∈ N : ♥ ∈ ❙} S❣ = {❙ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ❣(❙) = ❣}, ♥❣ = ★S❣

  • ▼✉❧t✐♣❧✐❝✐t② ✸✿ ✭❘♦s❛❧❡s✱ ✷✵✵✺✮✳
  • ❇♦✉♥❞s ✉s✐♥❣ ❉✐❝❦ ♣❛t❤s ✭❇r❛s✲❆♠♦rós ❛♥❞ ❉❡ ▼✐❡r✱ ✷✵✵✼✮
  • ❋✐❜♦♥❛❝❝✐ ❈♦♥❥❡❝t✉r❡✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✽✮✳
  • ❇♦✉♥❞s ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣s tr❡❡ ♠❡t❤♦❞✿ ✭❇r❛s✲❆♠♦rós✱

✷✵✵✾✮

  • ■♠♣r♦✈❡❞ ❇♦✉♥❞s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜② ❣❡♥❡r❛t✐♥❣ tr❡❡s✿

✭❊❧✐③❛❧❞❡✱ ✷✵✵✾✮

  • ■♠♣r♦✈❡❞ ❛s②♠♣t♦t✐❝ ❧♦✇❡r ❜♦✉♥❞✿ ✭❩❤❛♦✱ ✷✵✵✾✮

❙♦❢t✇❛r❡✿ ♥✉♠❡r✐❝❛❧s❣♣s ✐♥ ●❆P ✭❉❡❧❣❛❞♦ ❡t ❛❧✱ ✷✵✵✽✮

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SLIDE 5

❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s

❙ = ♥✶, . . . , ♥♣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣✱ ❣❝❞(♥✐, ♥❥) = ✶✳ ♠✉❧t✐♣❧✐❝✐t②✿ ♠(❙) = ♠✐♥{♥ ∈ N : ♥ ∈ ❙\{✵}} ❣❡♥✉s✿ ❣(❙) = ★(N\❙) ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✿ ❋(❙) = ♠❛①{♥ ∈ N : ♥ ∈ ❙} S❣ = {❙ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ❣(❙) = ❣}, ♥❣ = ★S❣

  • ▼✉❧t✐♣❧✐❝✐t② ✸✿ ✭❘♦s❛❧❡s✱ ✷✵✵✺✮✳
  • ❇♦✉♥❞s ✉s✐♥❣ ❉✐❝❦ ♣❛t❤s ✭❇r❛s✲❆♠♦rós ❛♥❞ ❉❡ ▼✐❡r✱ ✷✵✵✼✮
  • ❋✐❜♦♥❛❝❝✐ ❈♦♥❥❡❝t✉r❡✿ ✭❇r❛s✲❆♠♦rós✱ ✷✵✵✽✮✳
  • ❇♦✉♥❞s ✉s✐♥❣ t❤❡ s❡♠✐❣r♦✉♣s tr❡❡ ♠❡t❤♦❞✿ ✭❇r❛s✲❆♠♦rós✱

✷✵✵✾✮

  • ■♠♣r♦✈❡❞ ❇♦✉♥❞s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜② ❣❡♥❡r❛t✐♥❣ tr❡❡s✿

✭❊❧✐③❛❧❞❡✱ ✷✵✵✾✮

  • ■♠♣r♦✈❡❞ ❛s②♠♣t♦t✐❝ ❧♦✇❡r ❜♦✉♥❞✿ ✭❩❤❛♦✱ ✷✵✵✾✮

❙♦❢t✇❛r❡✿ ♥✉♠❡r✐❝❛❧s❣♣s ✐♥ ●❆P ✭❉❡❧❣❛❞♦ ❡t ❛❧✱ ✷✵✵✽✮

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SLIDE 6
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ◗ ⊆ Z♥✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❢ (P; ③) =

  • α∈◗

③α ◗ ✵ ◆ ✿ ❢ ◗ ③

◆ ✐ ✵ ③✐

✶ ③ ③✷ ③◆ ✶ ③◆

✶

✶ ③

❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮

❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P

♥✱

t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ P

♥ ③

❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ❢ P

♥ ③ ✐ ■ ✐

③✉✐

♥ ❥ ✶

✶ ③✈✐❥ ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ✶ ✶ ❛♥❞ ✉✐ ✈✐❥

♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳

❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆●

slide-7
SLIDE 7
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ◗ ⊆ Z♥✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❢ (P; ③) =

  • α∈◗

③α ◗ = {✵, . . . , ◆}✿ ❢ (◗, ③) = ◆

✐=✵ ③✐ = ✶ + ③ + ③✷ + · · · + ③◆

✶ ③◆

✶

✶ ③

❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮

❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P

♥✱

t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ P

♥ ③

❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ❢ P

♥ ③ ✐ ■ ✐

③✉✐

♥ ❥ ✶

✶ ③✈✐❥ ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ✶ ✶ ❛♥❞ ✉✐ ✈✐❥

♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳

❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆●

slide-8
SLIDE 8
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ◗ ⊆ Z♥✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❢ (P; ③) =

  • α∈◗

③α ◗ = {✵, . . . , ◆}✿ ❢ (◗, ③) = ◆

✐=✵ ③✐ = ✶ + ③ + ③✷ + · · · + ③◆= ✶ − ③◆+✶

✶ − ③

❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮

❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P

♥✱

t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ P

♥ ③

❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ❢ P

♥ ③ ✐ ■ ✐

③✉✐

♥ ❥ ✶

✶ ③✈✐❥ ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ✶ ✶ ❛♥❞ ✉✐ ✈✐❥

♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳

❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆●

slide-9
SLIDE 9
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ◗ ⊆ Z♥✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❢ (P; ③) =

  • α∈◗

③α ◗ ✵ ◆ ✿ ❢ ◗ ③

◆ ✐ ✵ ③✐

✶ ③ ③✷ ③◆ ✶ ③◆

✶

✶ ③

❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮

❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P ⊂ R♥✱ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ (P ∩ Z♥

+; ③) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧

t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ❢ (P ∩ Z♥

+; ③) =

  • ✐∈■

ε✐ ③✉✐

♥

  • ❥=✶

(✶ − ③✈✐❥ ) ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ε ∈ {✶, −✶} ❛♥❞ ✉✐, ✈✐❥ ∈ Z♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳

❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆●

slide-10
SLIDE 10
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ◗ ⊆ Z♥✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❢ (P; ③) =

  • α∈◗

③α ◗ ✵ ◆ ✿ ❢ ◗ ③

◆ ✐ ✵ ③✐

✶ ③ ③✷ ③◆ ✶ ③◆

✶

✶ ③

❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮

❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P ⊂ R♥✱ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ (P ∩ Z♥

+; ③) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧

t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ❢ (P ∩ Z♥

+; ③) =

  • ✐∈■

ε✐ ③✉✐

♥

  • ❥=✶

(✶ − ③✈✐❥ ) ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ε ∈ {✶, −✶} ❛♥❞ ✉✐, ✈✐❥ ∈ Z♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳

❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆●

slide-11
SLIDE 11
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ◗ ⊆ Z♥✱ ✐ts ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❢ (P; ③) =

  • α∈◗

③α ◗ ✵ ◆ ✿ ❢ ◗ ③

◆ ✐ ✵ ③✐

✶ ③ ③✷ ③◆ ✶ ③◆

✶

✶ ③

❚❤❡♦r❡♠ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮

❆ss✉♠❡ ♥✱ t❤❡ ❞✐♠❡♥s✐♦♥✱ ✐s ✜①❡❞✳ ●✐✈❡♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ P ⊂ R♥✱ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢ (P ∩ Z♥

+; ③) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧

t✐♠❡ ✐♥ t❤❡ ❢♦r♠ ❢ (P ∩ Z♥

+; ③) =

  • ✐∈■

ε✐ ③✉✐

♥

  • ❥=✶

(✶ − ③✈✐❥ ) ✇❤❡r❡ ■ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✲s✐③❡ ✐♥❞❡①✐♥❣ s❡t✱ ❛♥❞ ✇❤❡r❡ ε ∈ {✶, −✶} ❛♥❞ ✉✐, ✈✐❥ ∈ Z♥ ❢♦r ❛❧❧ ✐ ❛♥❞ ❥✳

❙♦❢t✇❛r❡✿ ▲❛tt❊ ✭❉❡▲♦❡r❛ ❡t ❛❧✳ ✷✵✵✺✮✱ ❜❛r✈✐♥♦❦ ✭❱❡r❞♦♦❧❛❡❣❡✱ ✷✵✵✽✮ ❆♣❧✐❝❛t✐♦♥s✿ ❙♦❧✈✐♥❣ ■P Pr♦❜❧❡♠s✱ ❈♦♠♣❧❡①✐t② r❡s✉❧ts ✐♥ ●❛♠❡ ❚❤❡♦r②✱ ▼✉❧t✐♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥✱ ❋✉③③② Pr♦❣r❛♠♠✐♥❣✱ ✳✳✳ ❛♥❞ ❈❖❯◆❚■◆●

slide-12
SLIDE 12

❈♦✉♥t✐♥❣ ✇✐t❤ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t P ⊆ R♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡✳ ❢ (P ∩ Z♥; ③) =

  • α∈P∩Z♥

③α =

  • ✐∈■

ε✐ ③✉✐

♥

  • ❥=✶

(✶ − ③✈✐❥ ) ★ P

♥

❢ P ✶ ❧✐♠

③ ✶ ❢ P ③

❊✛❡❝t✐✈❡ ❧❛tt✐❝❡ ♣♦✐♥t ❝♦✉♥t✐♥❣ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❣❡♥✳ ❢✉♥✳✿ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮✱ ✭❉❡ ▲♦❡r❛ ❡t✳ ❛❧✱ ✷✵✵✸✮✱ ✭▲❛ss❡rr❡ ❛♥❞ ❩❡r♦♥✱ ✷✵✵✸✱ ✷✵✵✼✮✱ ✳✳✳

slide-13
SLIDE 13

❈♦✉♥t✐♥❣ ✇✐t❤ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t P ⊆ R♥ ❛ r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡✳ ❢ (P ∩ Z♥; ③) =

  • α∈P∩Z♥

③α =

  • ✐∈■

ε✐ ③✉✐

♥

  • ❥=✶

(✶ − ③✈✐❥ ) ★(P ∩ Z♥) = ❢ (P; ✶) = ❧✐♠

③→✶ ❢ (P; ③)

❊✛❡❝t✐✈❡ ❧❛tt✐❝❡ ♣♦✐♥t ❝♦✉♥t✐♥❣ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❣❡♥✳ ❢✉♥✳✿ ✭❇❛r✈✐♥♦❦✱ ✶✾✾✹✮✱ ✭❉❡ ▲♦❡r❛ ❡t✳ ❛❧✱ ✷✵✵✸✮✱ ✭▲❛ss❡rr❡ ❛♥❞ ❩❡r♦♥✱ ✷✵✵✸✱ ✷✵✵✼✮✱ ✳✳✳

slide-14
SLIDE 14

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ❛♥❞ ■♥t❡❣❡r ♣♦✐♥ts ✐♥s✐❞❡ P♦❧②t♦♣❡s

❚❤❡♦r❡♠ ✭❑✉♥③✱ ✶✾✽✼❀ ❘♦s❛❧❡s ❡t ❛❧✳ ✷✵✵✷✮

▲❡t ♠ ❜❡ ❛♥ ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ ✶✳ ❚❤❡r❡ ❡①✐sts ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♠❛♣ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡❣❡r ♣♦✐♥ts ✭✐♥ Z♠−✶✮ ✐♥s✐❞❡ t❤❡ ♣♦❧②❤❡❞r♦♥✱ P♠✱ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s ①✐ ✶ ❢♦r ❛❧❧ ✐ ∈ {✶, . . . , ♠ − ✶}✱ ①✐ + ①❥ − ①✐+❥ ✵ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ ♠ − ✶✱ ①✐ + ①❥ − ①✐+❥−♠ −✶ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ > ♠ ❬❙❡❧♠❡r ✶✾✼✼❪

♠ ✶ ✐ ✶

①✐ ❣ ❛♥❞ t❤❡ s❡t ♦❢ ♥✉♠✳ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ♠✳ ①∗ ∈ P♠ ∩ Z♠−✶ ⇔ ❆♣(❙①∗, ♠) = {✵} ∪ {①∗

✐ ♠ + ✐ : ✐ = ✶, . . . , ♠ − ✶}✳

❆♣(❙, ♥) = {s ∈ ❙ : s − ♥ ∈ ❙}✱ ❙ = ❆♣(❙, ♥) ∪ {♥} ① P♠ ❣

♠ ✶

❆♣ ❙① ♠ ✵ ①✐ ♠ ✐ ✐ ✶ ♠ ✶ ✳

❣ ❙

✶ ♠ ✇ ❆♣ ❙ ♠

✇ ♠ ✶ ✷

slide-15
SLIDE 15

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ❛♥❞ ■♥t❡❣❡r ♣♦✐♥ts ✐♥s✐❞❡ P♦❧②t♦♣❡s

❚❤❡♦r❡♠ ✭❑✉♥③✱ ✶✾✽✼❀ ❘♦s❛❧❡s ❡t ❛❧✳ ✷✵✵✷✮

▲❡t ♠ ❜❡ ❛♥ ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ ✶✳ ❚❤❡r❡ ❡①✐sts ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♠❛♣ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡❣❡r ♣♦✐♥ts ✭✐♥ Z♠−✶✮ ✐♥s✐❞❡ t❤❡ ♣♦❧②t♦♣❡✱ P♠,❣✱ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s ①✐ ✶ ❢♦r ❛❧❧ ✐ ∈ {✶, . . . , ♠ − ✶}✱ ①✐ + ①❥ − ①✐+❥ ✵ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ ♠ − ✶✱ ①✐ + ①❥ − ①✐+❥−♠ −✶ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ > ♠ ❬❙❡❧♠❡r ✶✾✼✼❪ :

♠−✶

  • ✐=✶

①✐ = ❣ ❛♥❞ t❤❡ s❡t ♦❢ ♥✉♠✳ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ♠ ❛♥❞ ❣❡♥✉s ❣✳ ① P♠

♠ ✶

❆♣ ❙① ♠ ✵ ①✐ ♠ ✐ ✐ ✶ ♠ ✶ ✳ ❆♣ ❙ ♥ s ❙ s ♥ ❙ ✱ ❙ ❆♣ ❙ ♥ ♥ ①∗ ∈ P♠,❣ ∩ Z♠−✶ ⇔ ❆♣(❙①∗, ♠) = {✵} ∪ {①∗

✐ ♠ + ✐ : ✐ = ✶, . . . , ♠ − ✶}✳

❣(❙) =

✶ ♠

  • ✇∈❆♣(❙,♠)

✇ + ♠ − ✶ ✷

slide-16
SLIDE 16

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❣❡♥✉s

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ♠ ❛♥❞ ❣❡♥✉s ❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ♥❣

❣ ✶ ♠ ✶

♥♠ ❣

❚❤❡♦r❡♠

▲❡t ❣ ❜❡ ❛ ❣✐✈❡♥ ❣❡♥✉s✱ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

slide-17
SLIDE 17

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❣❡♥✉s

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ♠ ❛♥❞ ❣❡♥✉s ❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ♥❣ =

❣+✶

  • ♠=✶

♥♠,❣

❚❤❡♦r❡♠

▲❡t ❣ ❜❡ ❛ ❣✐✈❡♥ ❣❡♥✉s✱ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

slide-18
SLIDE 18

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❣❡♥✉s

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ♠ ❛♥❞ ❣❡♥✉s ❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ♥❣ =

❣+✶

  • ♠=✶

♥♠,❣

❚❤❡♦r❡♠

▲❡t ❣ ❜❡ ❛ ❣✐✈❡♥ ❣❡♥✉s✱ ❝♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

slide-19
SLIDE 19

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❚❤❡♦r❡♠ ✭❑✉♥③✱ ✶✾✽✼❀ ❘♦s❛❧❡s ❡t ❛❧✳ ✷✵✵✷✮

▲❡t ♠ ❜❡ ❛♥ ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ ✶✳ ❚❤❡r❡ ❡①✐sts ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♠❛♣ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡❣❡r ♣♦✐♥ts ✭✐♥ Z♠−✶✮ ✐♥s✐❞❡ t❤❡ ♣♦❧②❤❡❞r♦♥✱ P♠✱ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s ①✐ ✶ ❢♦r ❛❧❧ ✐ ∈ {✶, . . . , ♠ − ✶}✱ ①✐ + ①❥ − ①✐+❥ ✵ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ ♠ − ✶✱ ①✐ + ①❥ − ①✐+❥−♠ −✶ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ > ♠✱ ❬❙❡❧♠❡r ✶✾✼✼❪ ♠ ①✐ ✐ ❋ ♠ ❢♦r ❛❧❧ ✐ ✶ ♠ ✶ ✱ ♠ ①❦ ❦ ❋ ♠ ✇❤❡r❡ ❦ ❋ ♠♦❞ ♠ ❛♥❞ t❤❡ s❡t ♦❢ ♥✉♠✳ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ♠ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋✳ ①∗ ∈ P♠ ∩ Z♠−✶ ⇔ ❆♣(❙①∗, ♠) = {✵} ∪ {①∗

✐ ♠ + ✐ : ✐ = ✶, . . . , ♠ − ✶}✳

① P♠ ❣

♠ ✶

❆♣ ❙① ♠ ✵ ①✐ ♠ ✐ ✐ ✶ ♠ ✶ ✳ ❋ ❙ ♠❛① ❆♣ ❙ ♠ ♠

slide-20
SLIDE 20

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❚❤❡♦r❡♠ ✭❑✉♥③✱ ✶✾✽✼❀ ❘♦s❛❧❡s ❡t ❛❧✳ ✷✵✵✷✮

▲❡t ♠ ❜❡ ❛♥ ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ ✶✳ ❚❤❡r❡ ❡①✐sts ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♠❛♣ ❜❡t✇❡❡♥ t❤❡ ✐♥t❡❣❡r ♣♦✐♥ts ✭✐♥ Z♠−✶✮ ✐♥s✐❞❡ t❤❡ ♣♦❧②❤❡❞r♦♥✱ P♠✱ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t✐❡s ①✐ ✶ ❢♦r ❛❧❧ ✐ ∈ {✶, . . . , ♠ − ✶}✱ ①✐ + ①❥ − ①✐+❥ ✵ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ ♠ − ✶✱ ①✐ + ①❥ − ①✐+❥−♠ −✶ ❢♦r ❛❧❧ ✶ ✐ ❥ ♠ − ✶✱ ✐ + ❥ > ♠✱ ❬❙❡❧♠❡r ✶✾✼✼❪ : ♠ ①✐ + ✐ ≤ ❋ + ♠ ❢♦r ❛❧❧ ✐ ∈ {✶, . . . , ♠ − ✶}✱ ♠ ①❦∗ + ❦∗ = ❋ + ♠ ✇❤❡r❡ ❦∗ = ❋ (♠♦❞ ♠), ❛♥❞ t❤❡ s❡t ♦❢ ♥✉♠✳ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ♠ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋✳ ① P♠

♠ ✶

❆♣ ❙① ♠ ✵ ①✐ ♠ ✐ ✐ ✶ ♠ ✶ ✳ ①∗ ∈ P♠,❣ ∩ Z♠−✶ ⇔ ❆♣(❙①∗, ♠) = {✵} ∪ {①∗

✐ ♠ + ✐ : ✐ = ✶, . . . , ♠ − ✶}✳

❋(❙) = ♠❛①(❆♣(❙, ♠)) − ♠

slide-21
SLIDE 21

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❢♦r ❛♥② ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✐s ❞♦❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❆♥❛❣♦✉s❧②✳✳✳

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ t❤❡ ♠❛①✐♠❛❧ ❡♠❜❡❞❞✐♥❣ ❞✐♠❡♥s✐♦♥ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❢♦r ❛♥② ❣❡♥✉s ❣ ✭r❡s♣✳ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋✮ ✐s ❞♦❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

slide-22
SLIDE 22

❈♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ✜①❡❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❢♦r ❛♥② ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✐s ❞♦❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❆♥❛❣♦✉s❧②✳✳✳

❚❤❡♦r❡♠

❋♦r ✜①❡❞ ♠✉❧t✐♣❧✐❝✐t② ♠✱ ❝♦✉♥t✐♥❣ t❤❡ ♠❛①✐♠❛❧ ❡♠❜❡❞❞✐♥❣ ❞✐♠❡♥s✐♦♥ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ❢♦r ❛♥② ❣❡♥✉s ❣ ✭r❡s♣✳ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋✮ ✐s ❞♦❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

slide-23
SLIDE 23
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐♥ ♣r❛❝t✐❝❡

PPPP P

❣ ♠ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ♥❣ ✷ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✹ ✹ ✶ ✷ ✸ ✶ ✼ ✺ ✶ ✷ ✹ ✹ ✶ ✶✷ ✻ ✶ ✸ ✻ ✼ ✺ ✶ ✸✸ ✼ ✶ ✸ ✼ ✶✵ ✶✶ ✻ ✶ ✸✾ ✽ ✶ ✸ ✾ ✶✸ ✶✼ ✶✻ ✼ ✶ ✻✼ ✾ ✶ ✹ ✶✶ ✶✻ ✷✼ ✷✽ ✷✷ ✽ ✶ ✶✶✽ ✶✵ ✶ ✹ ✶✸ ✷✷ ✸✼ ✹✹ ✹✹ ✷✾ ✾ ✶ ✷✵✹ ✶✶ ✶ ✹ ✶✺ ✷✹ ✹✾ ✻✹ ✼✷ ✻✻ ✸✼ ✶✵ ✶ ✸✹✸ ✶✷ ✶ ✺ ✶✽ ✸✷ ✻✻ ✽✺ ✶✶✻ ✶✶✻ ✾✺ ✹✻ ✶✶ ✶ ✺✾✷ ✶✸ ✶ ✺ ✷✵ ✸✺ ✽✺ ✶✶✷ ✶✼✷ ✶✽✽ ✶✽✷ ✶✸✷ ✺✻ ✶✷ ✶ ✶✵✵✶ ✶✹ ✶ ✺ ✷✸ ✹✸ ✶✵✻ ✶✹✽ ✷✸✾ ✷✽✽ ✸✵✹ ✷✼✼ ✶✼✽ ✻✼ ✶✸ ✶ ✶✻✾✸ ✶✺ ✶ ✻ ✷✻ ✺✶ ✶✸✸ ✶✾✶ ✸✷✺ ✹✵✾ ✹✾✷ ✹✽✻ ✹✵✾ ✷✸✹ ✼✾ ✶✹ ✶ ✷✽✺✼

❚❛❜❧❡✿ ♥♠,❣ ❛♥❞ ♥❣✳

♥✾ ✶✷ ✽✸ ✽✺ s❡❝✳✭❜❛r✈✐♥♦❦✮✳✳✳ ✇❤✐❧❡ ♥✶✺ ✵ ✵✽ s❡❝✳ ✭♥✉♠❡r✐❝❛❧s❣♣s ✲●❆P✲✮

slide-24
SLIDE 24
  • ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✐♥ ♣r❛❝t✐❝❡

PPPP P

❣ ♠ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ♥❣ ✷ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✹ ✹ ✶ ✷ ✸ ✶ ✼ ✺ ✶ ✷ ✹ ✹ ✶ ✶✷ ✻ ✶ ✸ ✻ ✼ ✺ ✶ ✸✸ ✼ ✶ ✸ ✼ ✶✵ ✶✶ ✻ ✶ ✸✾ ✽ ✶ ✸ ✾ ✶✸ ✶✼ ✶✻ ✼ ✶ ✻✼ ✾ ✶ ✹ ✶✶ ✶✻ ✷✼ ✷✽ ✷✷ ✽ ✶ ✶✶✽ ✶✵ ✶ ✹ ✶✸ ✷✷ ✸✼ ✹✹ ✹✹ ✷✾ ✾ ✶ ✷✵✹ ✶✶ ✶ ✹ ✶✺ ✷✹ ✹✾ ✻✹ ✼✷ ✻✻ ✸✼ ✶✵ ✶ ✸✹✸ ✶✷ ✶ ✺ ✶✽ ✸✷ ✻✻ ✽✺ ✶✶✻ ✶✶✻ ✾✺ ✹✻ ✶✶ ✶ ✺✾✷ ✶✸ ✶ ✺ ✷✵ ✸✺ ✽✺ ✶✶✷ ✶✼✷ ✶✽✽ ✶✽✷ ✶✸✷ ✺✻ ✶✷ ✶ ✶✵✵✶ ✶✹ ✶ ✺ ✷✸ ✹✸ ✶✵✻ ✶✹✽ ✷✸✾ ✷✽✽ ✸✵✹ ✷✼✼ ✶✼✽ ✻✼ ✶✸ ✶ ✶✻✾✸ ✶✺ ✶ ✻ ✷✻ ✺✶ ✶✸✸ ✶✾✶ ✸✷✺ ✹✵✾ ✹✾✷ ✹✽✻ ✹✵✾ ✷✸✹ ✼✾ ✶✹ ✶ ✷✽✺✼

❚❛❜❧❡✿ ♥♠,❣ ❛♥❞ ♥❣✳

♥✾,✶✷ → ✽✸.✽✺ s❡❝✳✭❜❛r✈✐♥♦❦✮✳✳✳ ✇❤✐❧❡ ♥✶✺ → ✵.✵✽ s❡❝✳ ✭♥✉♠❡r✐❝❛❧s❣♣s ✲●❆P✲✮

slide-25
SLIDE 25

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❣ = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ① + ② = ❣, ① ≥ ✶, ② ≥ ✶} = {(①, ❣ − ①) : ❣ ≤ ✸ ① ≤ ✷❣ + ✶ ≤, ✶ ≤ ① ≤ ❣ − ✶} = {(①, ❣ − ①) : ① ∈

  • ♠❛①{✶, ❣

✸ }, ♠✐♥{❣ − ✶, ✷❣ + ✶ ✸ }

  • }

★ P✸ ❣

✷

★ ①

❣ ✸

①

✷❣ ✶ ✸ ✷ ✷❣ ✶ ✸ ❣ ✸ ❣ ✶ ✸

❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ ✶ ✸ ✳

❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ ✶ ✸ ✳

slide-26
SLIDE 26

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❣ = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ① + ② = ❣, ① ≥ ✶, ② ≥ ✶} = {(①, ❣ − ①) : ❣ ≤ ✸ ① ≤ ✷❣ + ✶ ≤, ✶ ≤ ① ≤ ❣ − ✶} = {(①, ❣ − ①) : ① ∈

  • ♠❛①{✶, ❣

✸ }, ♠✐♥{❣ − ✶, ✷❣ + ✶ ✸ }

  • }

★(P✸,❣ ∩ Z✷

+) = ★({① : ❣ ✸ ≤ ① ≤ ✷❣+✶ ✸

} ∩ Z✷

+) =

✷❣+✶

✸

  • −

❣

✸

  • =

❣+✶

✸

  • ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ ✶ ✸ ✳

❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ ✶ ✸ ✳

slide-27
SLIDE 27

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❣ = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ① + ② = ❣, ① ≥ ✶, ② ≥ ✶} = {(①, ❣ − ①) : ❣ ≤ ✸ ① ≤ ✷❣ + ✶ ≤, ✶ ≤ ① ≤ ❣ − ✶} = {(①, ❣ − ①) : ① ∈

  • ♠❛①{✶, ❣

✸ }, ♠✐♥{❣ − ✶, ✷❣ + ✶ ✸ }

  • }

★(P✸,❣ ∩ Z✷

+) = ★({① : ❣ ✸ ≤ ① ≤ ✷❣+✶ ✸

} ∩ Z✷

+) =

✷❣+✶

✸

  • −

❣

✸

  • =

❣+✶

✸

  • ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ + ✶ ✸

  • ✳

❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ ✶ ✸ ✳

slide-28
SLIDE 28

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❣ = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ① + ② = ❣, ① ≥ ✶, ② ≥ ✶} = {(①, ❣ − ①) : ❣ ≤ ✸ ① ≤ ✷❣ + ✶ ≤, ✶ ≤ ① ≤ ❣ − ✶} = {(①, ❣ − ①) : ① ∈

  • ♠❛①{✶, ❣

✸ }, ♠✐♥{❣ − ✶, ✷❣ + ✶ ✸ }

  • }

★(P✸,❣ ∩ Z✷

+) = ★({① : ❣ ✸ ≤ ① ≤ ✷❣+✶ ✸

} ∩ Z✷

+) =

✷❣+✶

✸

  • −

❣

✸

  • =

❣+✶

✸

  • ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ + ✶ ✸

  • ✳

❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❣❡♥✉s ❣ ✐s ❣ − ✶ ✸

  • ✳
slide-29
SLIDE 29

▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

▲❡t S✸,❣ ❜❡ t❤❡ s❡t ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ S♠❡

✸,❣ t❤❡ s❡t ♦❢ ▼❊❉✲s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸✿ ✶ ❚❤❡ s❡t S✸,❣ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ s❡t S♠❡ ✸,❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢

❣ ≡ ✷ (♠♦❞ ✸)✳

✷ ■❢ ❣

✷ ♠♦❞ ✸ ✱ t❤❡♥✱ ★

✸ ❣

★

♠❡ ✸ ❣

✶✱ ✐✳❡✳✱ t❤❡r❡ ❡①✐sts ♦♥❧② ♦♥❡ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸ t❤❛t ❞♦❡s ♥♦t ❤❛✈❡ ♠❛①✐♠❛❧ ❡♠❜❡❞❞✐♥❣ ❞✐♠❡♥s✐♦♥✳ ❙ ✸ ♥✷ ❢♦r s♦♠❡ ♥ ✸✳ ❣ ❙

✶ ✷ ✸

✶ ♥✷ ✶ ♥✷ ✶ ✭❙②❧✈❡st❡r✱ ✶✽✽✹✮ ❙ ✸ ❣ ✶ ✳

slide-30
SLIDE 30

▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

▲❡t S✸,❣ ❜❡ t❤❡ s❡t ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ S♠❡

✸,❣ t❤❡ s❡t ♦❢ ▼❊❉✲s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸✿ ✶ ❚❤❡ s❡t S✸,❣ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ s❡t S♠❡ ✸,❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢

❣ ≡ ✷ (♠♦❞ ✸)✳

✷ ■❢ ❣ ≡ ✷ (♠♦❞ ✸)✱ t❤❡♥✱ ★S✸,❣ = ★S♠❡ ✸,❣ + ✶✱ ✐✳❡✳✱ t❤❡r❡

❡①✐sts ♦♥❧② ♦♥❡ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸ t❤❛t ❞♦❡s ♥♦t ❤❛✈❡ ♠❛①✐♠❛❧ ❡♠❜❡❞❞✐♥❣ ❞✐♠❡♥s✐♦♥✳ ❙ ✸ ♥✷ ❢♦r s♦♠❡ ♥ ✸✳ ❣ ❙

✶ ✷ ✸

✶ ♥✷ ✶ ♥✷ ✶ ✭❙②❧✈❡st❡r✱ ✶✽✽✹✮ ❙ ✸ ❣ ✶ ✳

slide-31
SLIDE 31

▼❊❉ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

▲❡t S✸,❣ ❜❡ t❤❡ s❡t ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ S♠❡

✸,❣ t❤❡ s❡t ♦❢ ▼❊❉✲s❡♠✐❣r♦✉♣s ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸✿ ✶ ❚❤❡ s❡t S✸,❣ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ s❡t S♠❡ ✸,❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢

❣ ≡ ✷ (♠♦❞ ✸)✳

✷ ■❢ ❣ ≡ ✷ (♠♦❞ ✸)✱ t❤❡♥✱ ★S✸,❣ = ★S♠❡ ✸,❣ + ✶✱ ✐✳❡✳✱ t❤❡r❡

❡①✐sts ♦♥❧② ♦♥❡ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ♦❢ ❣❡♥✉s ❣ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ✸ t❤❛t ❞♦❡s ♥♦t ❤❛✈❡ ♠❛①✐♠❛❧ ❡♠❜❡❞❞✐♥❣ ❞✐♠❡♥s✐♦♥✳ ❙ = ✸, ♥✷ ❢♦r s♦♠❡ ♥ > ✸✳ ❣(❙) = ✶

✷ (✸ − ✶)(♥✷ − ✶) = ♥✷ − ✶

✭❙②❧✈❡st❡r✱ ✶✽✽✹✮ ⇒ ❙ = ✸, ❣ + ✶✳

slide-32
SLIDE 32

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❋≡✶ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ = ❋ + ✸, ✸① + ✷ ≤ ❋ + ✸, ①, ② ≥ ✶}✳ P✸,❋≡✷ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ ≤ ❋ + ✸, ✸① + ✷ = ❋ + ✸, ①, ② ≥ ✶}✳ ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s

❋ ✶ ✸ ❋ ✶ ✻

✶✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s

❋ ✶ ✸ ❋ ✷ ✻

✶✳

❋✐①✐♥❣ ❣❡♥✉s ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❆ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ✐s ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ❣❡♥✉s ❛♥❞ ✐ts ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✳ ❆❝t✉❛❧❧②✱ t❤❡ ✉♥✐q✉❡ ❧❛tt✐❝❡ ♣♦✐♥t ✐♥s✐❞❡ t❤❛t ♣♦❧②t♦♣❡ ✐s

❋ ✷ ✸ ✸❣ ❋ ✷ ✸

✐❢ ❋ ✶ ♠♦❞ ✸ ♦r

✸❣ ❋ ✶ ✸ ❋ ✶ ✸

✐❢ ❋ ✷ ♠♦❞ ✸ ✳ ❆♣ ❙ ♠ ✵ ❋ ✸ ✸❣ ❋ ✱ ❛♥❞ ❙ ❆♣ ❙ ♠ ♠ ✸ ❋ ✸ ✸❣ ❋ ✳

slide-33
SLIDE 33

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❋≡✶ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ = ❋ + ✸, ✸① + ✷ ≤ ❋ + ✸, ①, ② ≥ ✶}✳ P✸,❋≡✷ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ ≤ ❋ + ✸, ✸① + ✷ = ❋ + ✸, ①, ② ≥ ✶}✳ ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s ⌊ ❋+✶

✸ ⌋ − ⌈ ❋−✶ ✻ ⌉ + ✶✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s

❋ ✶ ✸ ❋ ✷ ✻

✶✳

❋✐①✐♥❣ ❣❡♥✉s ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❆ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ✐s ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ❣❡♥✉s ❛♥❞ ✐ts ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✳ ❆❝t✉❛❧❧②✱ t❤❡ ✉♥✐q✉❡ ❧❛tt✐❝❡ ♣♦✐♥t ✐♥s✐❞❡ t❤❛t ♣♦❧②t♦♣❡ ✐s

❋ ✷ ✸ ✸❣ ❋ ✷ ✸

✐❢ ❋ ✶ ♠♦❞ ✸ ♦r

✸❣ ❋ ✶ ✸ ❋ ✶ ✸

✐❢ ❋ ✷ ♠♦❞ ✸ ✳ ❆♣ ❙ ♠ ✵ ❋ ✸ ✸❣ ❋ ✱ ❛♥❞ ❙ ❆♣ ❙ ♠ ♠ ✸ ❋ ✸ ✸❣ ❋ ✳

slide-34
SLIDE 34

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❋≡✶ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ = ❋ + ✸, ✸① + ✷ ≤ ❋ + ✸, ①, ② ≥ ✶}✳ P✸,❋≡✷ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ ≤ ❋ + ✸, ✸① + ✷ = ❋ + ✸, ①, ② ≥ ✶}✳ ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s ⌊ ❋+✶

✸ ⌋ − ⌈ ❋−✶ ✻ ⌉ + ✶✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s ⌊ ❋+✶

✸ ⌋ − ⌈ ❋+✷ ✻ ⌉ + ✶✳

❋✐①✐♥❣ ❣❡♥✉s ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❆ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ✐s ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ❣❡♥✉s ❛♥❞ ✐ts ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✳ ❆❝t✉❛❧❧②✱ t❤❡ ✉♥✐q✉❡ ❧❛tt✐❝❡ ♣♦✐♥t ✐♥s✐❞❡ t❤❛t ♣♦❧②t♦♣❡ ✐s

❋ ✷ ✸ ✸❣ ❋ ✷ ✸

✐❢ ❋ ✶ ♠♦❞ ✸ ♦r

✸❣ ❋ ✶ ✸ ❋ ✶ ✸

✐❢ ❋ ✷ ♠♦❞ ✸ ✳ ❆♣ ❙ ♠ ✵ ❋ ✸ ✸❣ ❋ ✱ ❛♥❞ ❙ ❆♣ ❙ ♠ ♠ ✸ ❋ ✸ ✸❣ ❋ ✳

slide-35
SLIDE 35

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸

P✸,❋≡✶ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ = ❋ + ✸, ✸① + ✷ ≤ ❋ + ✸, ①, ② ≥ ✶}✳ P✸,❋≡✷ (♠♦❞ ✸) = {(①, ②) : ✷① − ② ≥ ✵, −① + ✷② ≥ −✶, ✸① + ✶ ≤ ❋ + ✸, ✸① + ✷ = ❋ + ✸, ①, ② ≥ ✶}✳ ❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s ⌊ ❋+✶

✸ ⌋ − ⌈ ❋−✶ ✻ ⌉ + ✶✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ ▼❊❉ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r ❋ ✭t❤❛t ✐s ♥♦t ❛ ♠✉❧t✐♣❧❡ ♦❢ ✸✮ ✐s ⌊ ❋+✶

✸ ⌋ − ⌈ ❋+✷ ✻ ⌉ + ✶✳

❋✐①✐♥❣ ❣❡♥✉s ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

❆ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣ ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✸ ✐s ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② ✐ts ❣❡♥✉s ❛♥❞ ✐ts ❋r♦❜❡♥✐✉s ♥✉♠❜❡r✳ ❆❝t✉❛❧❧②✱ t❤❡ ✉♥✐q✉❡ ❧❛tt✐❝❡ ♣♦✐♥t ✐♥s✐❞❡ t❤❛t ♣♦❧②t♦♣❡ ✐s ( ❋+✷

✸ , ✸❣−❋−✷ ✸

) ✐❢ ❋ ≡ ✶ (♠♦❞ ✸) ♦r ( ✸❣−❋−✶

✸

, ❋+✶

✸ ) ✐❢

❋ ≡ ✷ (♠♦❞ ✸)✳ ❆♣(❙, ♠) = {✵, ❋ + ✸, ✸❣ − ❋}✱ ❛♥❞ ❙ = ❆♣(❙, ♠), ♠ = ✸, ❋ + ✸, ✸❣ − ❋✳

slide-36
SLIDE 36

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹

Pr♦♣♦s✐t✐♦♥

▲❡t P✹,❣ = {(①, ②, ③) : ✷① − ② ≥ ✵, ① + ② − ③ ≥ ✵, −① + ② + ③ ≥ −✶, −② + ✷③ ≥ −✶, ① + ② + ③ = ❣, ① ≥ ✶, ② ≥ ✶, ③ ≥ ✶}✳

  • ■❢ ❣ ∈ [✸, ✽] t❤❡♥ ★(P✹,❣ ∩ Z✸) =

               ✶ ✐❢ ❣ = ✸, ✸ ✐❢ ❣ = ✹, ✹ ✐❢ ❣ = ✺, ✻ ✐❢ ❣ = ✻, ✼ ✐❢ ❣ = ✼, ✾ ✐❢ ❣ = ✽.

  • ■❢ ❣ ≥ ✾✱ t❤❡♥ ★(P✹,❣ ∩ Z✸) = ★(❚❆(❣) ∩ Z✷) +

★(❚❇(❣) ∩ Z✷) + ★(❘(❣) ∩ Z✷) − ★(❚❈(❣) ∩ Z✷) ✇❤❡r❡

  • ❚❆(❣) = {(①, ②) ∈ R✷ : ✸① + ② ≥ ❣, ① ≤ ✷❣+✶

✽

, ② ≤ ❣

✷ }✱

  • ❚❇(❣) = {(①, ②) ∈ R✷ : ① + ✸② ≥ ❣ − ✶, ① ≤ ❣+✶

✷ , ② ≤ ✷❣−✸ ✽

}✱

  • ❘(❣) = {(①, ②) ∈ R✷ : ✷❣+✶

✽

≤ ① ≤ ❣+✶

✷ , ✷❣−✸ ✽

≤ ② ≤

❣ ✷ }✱ ❛♥❞

  • ❚❈(❣) = {(①, ②) ∈ R✷ : ① + ② ≥ ❣, ① ≤ ❣+✶

✷ , ② ≤ ❣ ✷ }✳

slide-37
SLIDE 37

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹

T (g) B C g 6 (2g+1) 8 g-2 2 g+1 2 g-3 6 2g-3 8 g 2 g-3 2 T (g) T (g) R(g) A

slide-38
SLIDE 38

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹

❘✉♥♥✐♥❣ ❜❛r✈✐♥♦❦ ✭❢♦r ♣❛r❛♠❡tr✐❝ ♣♦❧②t♦♣❡s❀ ★(P(❣) ∩ Z♥) = q✉❛s②♣♦❧②♥♦♠✐❛❧✭❣✮✮✿ ✭❱❡r❞♦♦❧❛❡❣❡ ❡t ❛❧✳✱ ✷✵✵✼✮

★(❚❆(❣) ∩ Z✷) = ✶

✷

`¨ ❣+✺

✻

˝ − ¨ ❣

✹

˝ − ✶ ´ ` ✷❣ − ✸ ¨ ❣

✹

˝ − ✷ ¨ ❣

✷

˝ − ✷ − ✸ ¨ ❣+✺

✻

˝´ , ★(❚❇(❣) ∩ Z✷) = ✶

✷

`¨ ❣+✷

✻

˝ − ¨ ❣+✷

✹

˝´ ` −✸ ¨ ❣+✷

✹

˝ + ✷ ¨ ❣

✷

˝ − ✸ ¨ ❣+✷

✻

˝ − ✶ ´ , ★(❚❈(❣) ∩ Z✷) = ✶ ★(❘(❣) ∩ Z✷) = `¨ ❣+✶

✷

˝ − ˚ ✷❣+✶

✽

ˇ + ✶ ´ `¨ ❣

✷

˝ − ˚ ✷❣−✸

✽

ˇ + ✶ ´ .

slide-39
SLIDE 39

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹

❚❤❡♦r❡♠

▲❡t S✹,❣ ❜❡ t❤❡ s❡t ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ♠✉❧t✐♣❧✐❝✐t② ✹ ❛♥❞ ❣❡♥✉s ❣ ≥ ✾✳ ❚❤❡♥ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ S✹,❣ ✐s −❣ + ✺

✷⌊ ❣ ✹ ⌋ + +⌊ ❣ ✷ ⌋ + ✶ ✷⌊ ❣+✺ ✻ ⌋ + ⌊ ❣+✺ ✻ ⌋❣ − ⌊ ❣+✺ ✻ ⌋⌊ ❣ ✷ ⌋ − ✸ ✷⌊ ❣+✺ ✻ ⌋✷ −

⌊ ❣

✹ ⌋❣ + ✸ ✷⌊ ❣ ✹ ⌋✷ + ⌊ ❣ ✹ ⌋⌊ ❣ ✷ ⌋ − ✶ ✷⌊ ❣+✷ ✻ ⌋ + ⌊ ❣+✷ ✻ ⌋⌊ ❣ ✷ ⌋ − ✸ ✷⌊ ❣+✷ ✻ ⌋✷ + ✶ ✷⌊ ❣+✷ ✹ ⌋ + ✸ ✷⌊ ❣+✷ ✹ ⌋✷ − ⌊ ❣+✷ ✹ ⌋⌊ ❣ ✷ ⌋ + ⌊ ❣+✶ ✷ ⌋⌊ ❣ ✷ ⌋ − ⌊ ❣+✶ ✷ ⌋⌈ ✷❣−✸ ✽

⌉ + ⌊ ❣+✶

✷ ⌋ −

⌈ ✷❣−✼

✽

⌉⌊ ❣

✷ ⌋ + ⌈ ✷❣−✼ ✽

⌉⌈ ✷❣−✸

✽

⌉ − ⌈ ✷❣−✼

✽

⌉✳

slide-40
SLIDE 40

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹

❚❤❡♦r❡♠

❚❤❡ ♥✉♠❜❡r ♦❢ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹ ❛♥❞ ❋r♦❜❡♥✐♦✉s ♥✉♠❜❡r ❋ ✐s

                                                       “

❋−✶ ✹

”✷ ✐❢ ❋ ≡ ✶ (♠♦❞ ✹) ❛♥❞ ✺ ≤ ❋ ≤ ✾,

❋ ✷−✶✹❋+✶✹✶ ✶✻

✐❢ ❋ ≡ ✶ (♠♦❞ ✹) ❛♥❞ ✶✸ ≤ ❋ ≤ ✶✼,

−✸ ✷

j

❋+✶ ✶✷

k✷ + ✶

✹

j

❋+✶ ✶✷

k ❋ − ✸

✹

j

❋+✶ ✶✷

k + j

❋+✺ ✽

k✷ − ✶

✹

j

❋+✺ ✽

k ❋ + ✶

✹

j

❋+✺ ✽

k +

✺❋ ✻ − ✶✶ ✸✷ + ❋ ✷ ✸✷

✐❢ ❋ ≡ ✶ (♠♦❞ ✹) ❛♥❞ ❋ ≥ ✷✶, “

❋ ✹ −

j

❋+✶ ✽

k + ✶

✷

” “

❋ ✹ −

j

❋+✺ ✽

k + ✶

✷

” ✐❢ ❋ ≡ ✷ (♠♦❞ ✹),

❋ ✷ ✸✷ + ✼❋ ✶✻ − ✶✾ ✸✷ + ✸ ✹

j

❋+✶ ✽

k + j

❋+✶ ✽

k✷ +

✶ ✹

j

❋ ✶✷

k ❋ − ✾

✹

j

❋ ✶✷

k − ✸

✷

j

❋ ✶✷

k✷ − ✶

✹

j

❋+✶ ✽

k ❋ ✐❢ ❋ ≡ ✸ (♠♦❞ ✹), ✵ ♦t❤❡r✇✐s❡.

slide-41
SLIDE 41

◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ✹

❋✐①✐♥❣ ❣❡♥✉s ❛♥❞ ❋r♦❜❡♥✐✉s ♥✉♠❜❡r

♥✹,❋,❣ =                                                                                     

❋+✸ ✷

− j

✷❣+❋+✺ ✻

k ✐❢ ❋ ≡ ✶ (♠♦❞ ✹)✱ ✺❋ − ✽❣ ≥ ✺✱ ❛♥❞ ✷❣ − ❋ ≥ ✺, ❣ − j

✷❣+❋+✺ ✻

k ✐❢ ❋ ≡ ✶ (♠♦❞ ✹)✱ ✹❣ − ❋ ≥ ✷✸✱ ✷❣ − ❋ ≥ ✶ ❛♥❞ ✷❣ − ❋ ≤ ✸,

✸❋−✹❣+✺ ✹

✐❢ ❋ ≡ ✶ (♠♦❞ ✹)✱ ✺❋ − ✽❣ ≤ ✶✱ ✹❣ − ✸❋ ≤ ✶ ❛♥❞ ✷❣ − ❋ ≥ ✺,

✹❣−❋−✼ ✹

✐❢ ❋ ≡ ✶ (♠♦❞ ✹)✱ ✹❣ − ❋ ≥ ✶✶✱ ✹❣ − ❋ ≤ ✶✾✱ ✷❣ − ❋ ≥ ✶❛♥❞ ✷❣ − ❋ ≤ ✸,

✷❣−❋ ✷

✐❢ ❋ ≡ ✷ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≤ ✷✱ ❋ ≥ ✶✹✱ ❛♥❞ ✷❣ − ❋ ≥ ✷,

✸❋−✹❣+✻ ✹

✐❢ ❋ ≡ ✷ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≥ ✻✱ ✹❣ − ✸❋ ≤ ✷ ❛♥❞ ✷❣ − ❋ ≥ ✻,

❋−✷ ✹

✐❢ ❋ ≡ ✷ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≥ ✻✱ ❋ ≥ ✻✱ ❛♥❞ ✷❣ − ❋ ≤ ✹,

✹❣−❋−✷ ✹

− ¨ ❋+✻

✽

˝ ✐❢ ❋ ≡ ✷ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≤ ✷✱ ❋ ≤ ✶✵✱ ❛♥❞ ✷❣ − ❋ ≥ ✷✱

✸❋−✹❣+✼ ✹

✐❢ ❋ ≡ ✸ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≥ ✾✱ ✹❣ − ✸❋ ≤ ✸ ❛♥❞ ✷❣ − ❋ ≥ ✺✱

❋+✸ ✷

− j

✷❣+❋+✺ ✻

k ✐❢ ❋ ≡ ✸ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≤ ✺✱ ❛♥❞ ✷❣ − ❋ ≥ ✺✱ ❣ − j

✷❣+❋+✺ ✻

k ✐❢ ❋ ≡ ✸ (♠♦❞ ✹)✱ ✽❣ − ✺❋ ≤ ✺✱ ✹❣ − ❋ ≥ ✾✱ ✷❣ − ❋ ≥ ✶✱ ❛♥❞ ✷❣ − ❋ ≤ ✸

slide-42
SLIDE 42

❈♦♥❝❧✉s✐♦♥s

  • ❈♦✉♥t✐♥❣ ♥✉♠❡r✐❝❛❧ s❡♠✐❣r♦✉♣s ♦❢ ❛ ❣✐✈❡♥ ❣❡♥✉s ✭♦r

❋r♦❜❡♥✐✉s ♥✉♠❜❡r✮ ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

  • ❈♦♠♣❧❡①✐t② r❡s✉❧ts ❢♦r t❤❡s❡ t❛s❦s ❛r❡ st❛t❡❞✿ ♣♦❧②✲t✐♠❡

❢♦r ✜①❡❞ ❞✐♠❡♥s✐♦♥✳

  • ❙♦❢t✇❛r❡ ❢♦r ❣❡♥❡r❛❧ ❝♦♠♣✉t❛t✐♦♥s✿ ▲❛tt❊ ❛♥❞

❜❛r✈✐♥♦❦✳✳✳ ❜✉t s♦♠❡t❤✐♥❣ ❜❡tt❡r ✐s ♣♦ss✐❜❧❡ ❡①♣❧♦✐t✐♥❣ t❤❡ ❑✉♥③✬s ♣♦❧②t♦♣❡✳

  • ❙♣❡❝✐❛❧ ❝❛s❡s ✭♠✉❧t✐♣❧✐❝✐t② ✸ ❛♥❞ ✹✳✳✳✮ s♦❧✈❡❞ ✇✐t❤ s♦♠❡

❣❡♦♠❡tr② ❛♥❞ ❣❡♥✳ ❢✉♥✳ ❢♦r ♣❛r❛♠❡tr✐❝ ♣♦❧②t♦♣❡s✳

  • ❲❤❛t ❛❜♦✉t ❞♦✐♥❣ s♦♠❡ ■♥t❡❣❡r Pr♦❣r❛♠♠✐♥❣ ♦✈❡r t❤❡

❑✉♥③ P♦❧②t♦♣❡❄

slide-43
SLIDE 43

❚❤❛♥❦ ②♦✉

❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✵✾✵✶✳✶✷✷✽ ✈❜❧❛♥❝♦❅✉❣r✳❡s