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❆ ◆❡✇ ❊✣❝✐❡♥t ❆❧❣♦r✐t❤♠ ❢♦r ❱❛❧✐❞❛t✐♥❣ ❈❤❡❜②s❤❡✈ ❆♣♣r♦①✐♠❛t✐♦♥s ♦❢ ▲❖❉❊ ❙♦❧✉t✐♦♥s

◆✐❝♦❧❛s ❇❘■❙❊❇❆❘❘❊✱ ▼✐♦❛r❛ ❏❖▲❉❊❙

✱ ❛♥❞ ❋❧♦r❡♥t ❇❘❊❍❆❘❉ ❊◆❙ ❞❡ ▲②♦♥ ❛♥❞ ▲❆❆❙✲❈◆❘❙

❏❛♥✉❛r② ✶✻✱ ✷✵✶✼ ❏◆❈❋ ✷✵✶✼✱ ▼❛rs❡✐❧❧❡✱ ❈■❘▼

✶✴✷✺

❚❤❡ ❙♣❛❝❡❝r❛❢t ❘❡♥❞❡③✲❱♦✉s Pr♦❜❧❡♠

ISS

shuttle

▲✐♥❡❛r✐③❡❞ ❊q✉❛t✐♦♥ ♦❢ t❤❡ ■♥✲P❧❛♥❡ ▼♦t✐♦♥ ✹ ✸ ✶ ❝♦s ❆♣♣r♦①✐♠❛t✐♥❣ s♦❧✉t✐♦♥s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧s✳ ❱❛❧✐❞❛t✐♥❣ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ✇✐t❤ ❝❡rt✐✜❡❞ ❡rr♦r ❜♦✉♥❞s✳

✷✴✷✺

❚❤❡ ❙♣❛❝❡❝r❛❢t ❘❡♥❞❡③✲❱♦✉s Pr♦❜❧❡♠

ISS

shuttle

▲✐♥❡❛r✐③❡❞ ❊q✉❛t✐♦♥ ♦❢ t❤❡ ■♥✲P❧❛♥❡ ▼♦t✐♦♥ z′′(t) +

• ✹ −

✸ ✶ + e ❝♦s t

• z(t) = c

❆♣♣r♦①✐♠❛t✐♥❣ s♦❧✉t✐♦♥s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧s✳ ❱❛❧✐❞❛t✐♥❣ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ✇✐t❤ ❝❡rt✐✜❡❞ ❡rr♦r ❜♦✉♥❞s✳

✷✴✷✺

❚❤❡ ❙♣❛❝❡❝r❛❢t ❘❡♥❞❡③✲❱♦✉s Pr♦❜❧❡♠

ISS

shuttle

▲✐♥❡❛r✐③❡❞ ❊q✉❛t✐♦♥ ♦❢ t❤❡ ■♥✲P❧❛♥❡ ▼♦t✐♦♥ z′′(t) +

• ✹ −

✸ ✶ + e ❝♦s t

• z(t) = c

❆♣♣r♦①✐♠❛t✐♥❣ s♦❧✉t✐♦♥s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧s✳ ❱❛❧✐❞❛t✐♥❣ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ✇✐t❤ ❝❡rt✐✜❡❞ ❡rr♦r ❜♦✉♥❞s✳

✷✴✷✺

❚❤❡ ❙♣❛❝❡❝r❛❢t ❘❡♥❞❡③✲❱♦✉s Pr♦❜❧❡♠

ISS

shuttle

▲✐♥❡❛r✐③❡❞ ❊q✉❛t✐♦♥ ♦❢ t❤❡ ■♥✲P❧❛♥❡ ▼♦t✐♦♥ z′′(t) +

• ✹ −

✸ ✶ + e ❝♦s t

• z(t) = c

❆♣♣r♦①✐♠❛t✐♥❣ s♦❧✉t✐♦♥s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧s✳ ❱❛❧✐❞❛t✐♥❣ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ✇✐t❤ ❝❡rt✐✜❡❞ ❡rr♦r ❜♦✉♥❞s✳

✷✴✷✺

✶

❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✐♥ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

✷

❆♣♣r♦①✐♠❛t✐♥❣ ❋✉♥❝t✐♦♥s ✇✐t❤ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

✸

❚❤❡ ❱❛❧✐❞❛t✐♦♥ ❆❧❣♦r✐t❤♠

✸✴✷✺

✶

❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✐♥ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

✷

❆♣♣r♦①✐♠❛t✐♥❣ ❋✉♥❝t✐♦♥s ✇✐t❤ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

✸

❚❤❡ ❱❛❧✐❞❛t✐♦♥ ❆❧❣♦r✐t❤♠

✹✴✷✺

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ♦✈❡r ❝♦♠♣❛❝t ✐♥t❡r✈❛❧ I✿ f (r)(t) + ar−✶(t)f (r−✶)(t) + · · · + a✶(t)f ′(t) + a✵(t)f (t) = g(t). ❚❤❡ ❢✉♥❝t✐♦♥s ai ❛♥❞ g ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❛t ❧❡❛st ❝♦♥t✐♥✉♦✉s✳ ■♥✐t✐❛❧ ❱❛❧✉❡ Pr♦❜❧❡♠ ✭■❱P✮ ❛t

✵

✿

✵ ✵ ✵ ✶ ✶ ✵ ✶

❚❤✐s ✐s ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ♣r♦❜❧❡♠✿ ▲

✶ ✶ ✶ ✵ ✵

❇ ✵

✵ ✵ ✶ ✵

▲ ❇ ✵

✵ ✶

❊①❛♠♣❧❡ ▲

✷

✹ ✸ ✶ ❝♦s ❇

✵ ✵

✺✴✷✺

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ♦✈❡r ❝♦♠♣❛❝t ✐♥t❡r✈❛❧ I✿ f (r)(t) + ar−✶(t)f (r−✶)(t) + · · · + a✶(t)f ′(t) + a✵(t)f (t) = g(t). ❚❤❡ ❢✉♥❝t✐♦♥s ai ❛♥❞ g ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❛t ❧❡❛st ❝♦♥t✐♥✉♦✉s✳ ■♥✐t✐❛❧ ❱❛❧✉❡ Pr♦❜❧❡♠ ✭■❱P✮ ❛t t✵ ∈ I✿ f (t✵) = v✵ f ′(t✵) = v✶ . . . f (r−✶)(t✵) = vr−✶. ❚❤✐s ✐s ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ♣r♦❜❧❡♠✿ ▲

✶ ✶ ✶ ✵ ✵

❇ ✵

✵ ✵ ✶ ✵

▲ ❇ ✵

✵ ✶

❊①❛♠♣❧❡ ▲

✷

✹ ✸ ✶ ❝♦s ❇

✵ ✵

✺✴✷✺

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ♦✈❡r ❝♦♠♣❛❝t ✐♥t❡r✈❛❧ I✿ f (r)(t) + ar−✶(t)f (r−✶)(t) + · · · + a✶(t)f ′(t) + a✵(t)f (t) = g(t). ❚❤❡ ❢✉♥❝t✐♦♥s ai ❛♥❞ g ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❛t ❧❡❛st ❝♦♥t✐♥✉♦✉s✳ ■♥✐t✐❛❧ ❱❛❧✉❡ Pr♦❜❧❡♠ ✭■❱P✮ ❛t t✵ ∈ I✿ f (t✵) = v✵ f ′(t✵) = v✶ . . . f (r−✶)(t✵) = vr−✶. ❚❤✐s ✐s ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ♣r♦❜❧❡♠✿ ▲ = ∂r + ar−✶∂r−✶ + · · · + a✶∂ + a✵ : Cr(I) → C✵(I), ❇t✵ : f →

• f (t✵), f ′(t✵), . . . , f (r−✶)(t✵)
• : Cr(I) → Rr.

▲ ❇ ✵

✵ ✶

❊①❛♠♣❧❡ ▲

✷

✹ ✸ ✶ ❝♦s ❇

✵ ✵

✺✴✷✺

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ♦✈❡r ❝♦♠♣❛❝t ✐♥t❡r✈❛❧ I✿ f (r)(t) + ar−✶(t)f (r−✶)(t) + · · · + a✶(t)f ′(t) + a✵(t)f (t) = g(t). ❚❤❡ ❢✉♥❝t✐♦♥s ai ❛♥❞ g ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❛t ❧❡❛st ❝♦♥t✐♥✉♦✉s✳ ■♥✐t✐❛❧ ❱❛❧✉❡ Pr♦❜❧❡♠ ✭■❱P✮ ❛t t✵ ∈ I✿ f (t✵) = v✵ f ′(t✵) = v✶ . . . f (r−✶)(t✵) = vr−✶. ❚❤✐s ✐s ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ♣r♦❜❧❡♠✿ ▲ = ∂r + ar−✶∂r−✶ + · · · + a✶∂ + a✵ : Cr(I) → C✵(I), ❇t✵ : f →

• f (t✵), f ′(t✵), . . . , f (r−✶)(t✵)
• : Cr(I) → Rr.
• ▲ · f

= g, ❇t✵ · f = (v✵, . . . , vr−✶). ❊①❛♠♣❧❡ ▲

✷

✹ ✸ ✶ ❝♦s ❇

✵ ✵

✺✴✷✺

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

▲✐♥❡❛r ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ♦✈❡r ❝♦♠♣❛❝t ✐♥t❡r✈❛❧ I✿ f (r)(t) + ar−✶(t)f (r−✶)(t) + · · · + a✶(t)f ′(t) + a✵(t)f (t) = g(t). ❚❤❡ ❢✉♥❝t✐♦♥s ai ❛♥❞ g ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ ❛t ❧❡❛st ❝♦♥t✐♥✉♦✉s✳ ■♥✐t✐❛❧ ❱❛❧✉❡ Pr♦❜❧❡♠ ✭■❱P✮ ❛t t✵ ∈ I✿ f (t✵) = v✵ f ′(t✵) = v✶ . . . f (r−✶)(t✵) = vr−✶. ❚❤✐s ✐s ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ♣r♦❜❧❡♠✿ ▲ = ∂r + ar−✶∂r−✶ + · · · + a✶∂ + a✵ : Cr(I) → C✵(I), ❇t✵ : f →

• f (t✵), f ′(t✵), . . . , f (r−✶)(t✵)
• : Cr(I) → Rr.
• ▲ · f

= g, ❇t✵ · f = (v✵, . . . , vr−✶). ❊①❛♠♣❧❡ ▲ = ∂✷ + ✹ − ✸ ✶ + e ❝♦s t ❇ · z = (z(t✵), z′(t✵))

✺✴✷✺

❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙♦❧✉t✐♦♥

❚❤❡♦r❡♠ ✭P✐❝❛r❞✲▲✐♥❞❡❧ö❢ ✕ ❧✐♥❡❛r ❝❛s❡✮ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ (▲, ❇t✵) : Cr(I) → C✵(I) × Rr, ✐s ❛ ✭❜✐❝♦♥t✐♥✉♦✉s✮ ✐s♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✿ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r♠ ❛ ✲❞✐♠❡♥s✐♦♥❛❧ ❛✣♥❡ s♣❛❝❡✳ ❋♦r ✜①❡❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛t

✵✱ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ s♦❧✉t✐♦♥✳

❙♦♠❡ ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s❄ ❍♦✇ t♦ ❜♦✉♥❞ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥❄

✻✴✷✺

❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙♦❧✉t✐♦♥

❚❤❡♦r❡♠ ✭P✐❝❛r❞✲▲✐♥❞❡❧ö❢ ✕ ❧✐♥❡❛r ❝❛s❡✮ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ (▲, ❇t✵) : Cr(I) → C✵(I) × Rr, ✐s ❛ ✭❜✐❝♦♥t✐♥✉♦✉s✮ ✐s♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✿ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r♠ ❛ r✲❞✐♠❡♥s✐♦♥❛❧ ❛✣♥❡ s♣❛❝❡✳ ❋♦r ✜①❡❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛t t✵✱ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ s♦❧✉t✐♦♥✳ ❙♦♠❡ ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s❄ ❍♦✇ t♦ ❜♦✉♥❞ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥❄

✻✴✷✺

❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❙♦❧✉t✐♦♥

❚❤❡♦r❡♠ ✭P✐❝❛r❞✲▲✐♥❞❡❧ö❢ ✕ ❧✐♥❡❛r ❝❛s❡✮ ❚❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r✿ (▲, ❇t✵) : Cr(I) → C✵(I) × Rr, ✐s ❛ ✭❜✐❝♦♥t✐♥✉♦✉s✮ ✐s♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t✿ ❚❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r♠ ❛ r✲❞✐♠❡♥s✐♦♥❛❧ ❛✣♥❡ s♣❛❝❡✳ ❋♦r ✜①❡❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛t t✵✱ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ s♦❧✉t✐♦♥✳ ❙♦♠❡ ♣r♦❜❧❡♠s✿ ❍♦✇ t♦ ✜♥❞ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s❄ ❍♦✇ t♦ ❜♦✉♥❞ t❤❡ ❡rr♦r ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥❄

✻✴✷✺

Pr❡✈✐♦✉s ❲♦r❦s

❙♣❡❝tr❛❧ ♠❡t❤♦❞s✿

❙❡♠✐♥❛❧ ✇♦r❦s ❜② ❖rs③❛❣✱ ❚r❡❢❡t❤❡♥ ❛♥❞ ♦t❤❡rs ❋❛st ❛❧❣♦r✐t❤♠ ❢♦r ▲❖❉❊s✿ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞

❘✐❣♦r♦✉s ❛r✐t❤♠❡t✐❝s✿

■♥t❡r✈❛❧ ❛♥❛❧②s✐s ✭▼♦♦r❡✮ ❘✐❣♦r♦✉s P♦❧②♥♦♠✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥s ✭❘P❆s✮✿ ✉❧tr❛✲❛r✐t❤♠❡t✐❝s ✭❊♣st❡✐♥✱ ▼✐r❛♥❦❡r✱ ❘✐✈❧✐♥✮✱ ❚❛②❧♦r ♠♦❞❡❧s ✭▼❛❦✐♥♦ ❛♥❞ ❇❡r③✮✱ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❇r✐s❡❜❛rr❡✱ ❏♦❧❞❡s

✱✮✳

❆ ♣♦st❡r✐♦r✐ ✈❛❧✐❞❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿

◗✉❛s✐✲◆❡✇t♦♥ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞s ✭❨❛♠❛♠♦t♦✱ ▲❡ss❛r❞✮ ❉✲✜♥✐t❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ✭❇❡♥♦✐t✱ ❏♦❧❞❡s

✱✱ ▼❡③③❛r♦❜❜❛✮

✼✴✷✺

Pr❡✈✐♦✉s ❲♦r❦s

❙♣❡❝tr❛❧ ♠❡t❤♦❞s✿

❙❡♠✐♥❛❧ ✇♦r❦s ❜② ❖rs③❛❣✱ ❚r❡❢❡t❤❡♥ ❛♥❞ ♦t❤❡rs ❋❛st ❛❧❣♦r✐t❤♠ ❢♦r ▲❖❉❊s✿ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞

❘✐❣♦r♦✉s ❛r✐t❤♠❡t✐❝s✿

■♥t❡r✈❛❧ ❛♥❛❧②s✐s ✭▼♦♦r❡✮ ❘✐❣♦r♦✉s P♦❧②♥♦♠✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥s ✭❘P❆s✮✿ ✉❧tr❛✲❛r✐t❤♠❡t✐❝s ✭❊♣st❡✐♥✱ ▼✐r❛♥❦❡r✱ ❘✐✈❧✐♥✮✱ ❚❛②❧♦r ♠♦❞❡❧s ✭▼❛❦✐♥♦ ❛♥❞ ❇❡r③✮✱ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❇r✐s❡❜❛rr❡✱ ❏♦❧❞❡s

✱✮✳

❆ ♣♦st❡r✐♦r✐ ✈❛❧✐❞❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿

◗✉❛s✐✲◆❡✇t♦♥ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞s ✭❨❛♠❛♠♦t♦✱ ▲❡ss❛r❞✮ ❉✲✜♥✐t❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ✭❇❡♥♦✐t✱ ❏♦❧❞❡s

✱✱ ▼❡③③❛r♦❜❜❛✮

✼✴✷✺

Pr❡✈✐♦✉s ❲♦r❦s

❙♣❡❝tr❛❧ ♠❡t❤♦❞s✿

❙❡♠✐♥❛❧ ✇♦r❦s ❜② ❖rs③❛❣✱ ❚r❡❢❡t❤❡♥ ❛♥❞ ♦t❤❡rs ❋❛st ❛❧❣♦r✐t❤♠ ❢♦r ▲❖❉❊s✿ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞

❘✐❣♦r♦✉s ❛r✐t❤♠❡t✐❝s✿

■♥t❡r✈❛❧ ❛♥❛❧②s✐s ✭▼♦♦r❡✮ ❘✐❣♦r♦✉s P♦❧②♥♦♠✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥s ✭❘P❆s✮✿ ✉❧tr❛✲❛r✐t❤♠❡t✐❝s ✭❊♣st❡✐♥✱ ▼✐r❛♥❦❡r✱ ❘✐✈❧✐♥✮✱ ❚❛②❧♦r ♠♦❞❡❧s ✭▼❛❦✐♥♦ ❛♥❞ ❇❡r③✮✱ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❇r✐s❡❜❛rr❡✱ ❏♦❧❞❡s

✱✮✳

❆ ♣♦st❡r✐♦r✐ ✈❛❧✐❞❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✿

◗✉❛s✐✲◆❡✇t♦♥ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞s ✭❨❛♠❛♠♦t♦✱ ▲❡ss❛r❞✮ ❉✲✜♥✐t❡ ❛♣♣r♦❛❝❤ ❛♥❞ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ✭❇❡♥♦✐t✱ ❏♦❧❞❡s

✱✱ ▼❡③③❛r♦❜❜❛✮

✼✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

s✶

t✵

. . . sr−✶−i

t✵

ϕ❞s✶ . . . ❞sr−i. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ❑ ✇❤❡r❡✿

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

s✶

t✵

. . . sr−✶−i

t✵

ϕ❞s✶ . . . ❞sr−i

• =

t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s . ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ❑ ✇❤❡r❡✿

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ❑ ✇❤❡r❡✿

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❑ · ϕ(t) =

r−✶

• i=✵

ai(t) t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ(s)❞s. ❑ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳ ✭s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✬s✮

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❑ · ϕ(t) =

r−✶

• i=✵

ai(t) t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ(s)❞s. ⇒ ❑ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳ ✭s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✬s✮

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❑ · ϕ(t) =

r−✶

• i=✵

ai(t) t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ(s)❞s. ⇒ ❑ ✐s ❛ ❝♦♠♣❛❝t ♦♣❡r❛t♦r✳ ψ(t) = g(t) + ✭s♦♠❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ vj✬s✮.

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❊①❛♠♣❧❡

❑ ✹ ✸ ✶ ❝♦s

✵

❞ ✹ ✸ ✶ ❝♦s

✵

❞ ✳

✵ ✵ ✵

✹ ✸ ✶ ❝♦s ✳

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❊①❛♠♣❧❡

❑ · ϕ = t

• ✹ −

✸ ✶ + e ❝♦s t t

t✵

ϕ(s)❞s −

• ✹ −

✸ ✶ + e ❝♦s t t

t✵

sϕ(s)❞s✳

✵ ✵ ✵

✹ ✸ ✶ ❝♦s ✳

✽✴✷✺

❘❡❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ■♥t❡❣r❛❧ ❖♣❡r❛t♦r

▲❡t ϕ = f (r) ∈ C✵(I) ✇✐t❤ f (t✵) = v✵ . . . f (r−✶)(t✵) = vr−✶✳ ❚❤❡♥ ❢♦r i ∈ [✵, r − ✶]✿ f (i)(t) =

r−✶

• j=i

(t − t✵)j−i (j − i)! vj + t

t✵

(t − s)r−✶−i (r − ✶ − i)! ϕ❞s. ❲❡ ❣❡t ❛♥ ✐♥t❡❣r❛❧ r❡❢♦r♠✉❧❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ, ✇❤❡r❡✿ ❊①❛♠♣❧❡

❑ · ϕ = t

• ✹ −

✸ ✶ + e ❝♦s t t

t✵

ϕ(s)❞s −

• ✹ −

✸ ✶ + e ❝♦s t t

t✵

sϕ(s)❞s✳

ψ(t) = c − (z(t✵) + (t − t✵)z′(t✵))

• ✹ −

✸ ✶ + e ❝♦s t

• ✳

✽✴✷✺

✶

❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✐♥ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

✷

❆♣♣r♦①✐♠❛t✐♥❣ ❋✉♥❝t✐♦♥s ✇✐t❤ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

✸

❚❤❡ ❱❛❧✐❞❛t✐♦♥ ❆❧❣♦r✐t❤♠

✾✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ T✵(X) = ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ T✵(X) = ✶ T✶(X) = X ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ T✵(X) = ✶ T✶(X) = X T✷(X) = ✷X ✷ − ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ T✵(X) = ✶ T✶(X) = X T✷(X) = ✷X ✷ − ✶ T✸(X) = ✹X ✸ − ✸X ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ T✵(X) = ✶ T✶(X) = X T✷(X) = ✷X ✷ − ✶ T✸(X) = ✹X ✸ − ✸X T✹(X) = ✽X ✹ − ✽X ✷ + ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ T✵(X) = ✶ T✶(X) = X T✷(X) = ✷X ✷ − ✶ T✸(X) = ✹X ✸ − ✸X T✹(X) = ✽X ✹ − ✽X ✷ + ✶ T✺(X) = ✶✻X ✺ − ✷✵X ✸ + ✺X ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ ❝♦s ❝♦s ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ Tn(❝♦s ϑ) = ❝♦s nϑ. ✶ ✶ ✶ ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ Tn(❝♦s ϑ) = ❝♦s nϑ. ⇒ ∀t ∈ [−✶, ✶], |Tn(t)| ≤ ✶. ❲❡ ❞❡✜♥❡ ❢♦r ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ Tn(❝♦s ϑ) = ❝♦s nϑ. ⇒ ∀t ∈ [−✶, ✶], |Tn(t)| ≤ ✶. ❲❡ ❞❡✜♥❡ T−n = Tn ❢♦r n ≥ ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ✶ ✷ ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ Tn(❝♦s ϑ) = ❝♦s nϑ. ⇒ ∀t ∈ [−✶, ✶], |Tn(t)| ≤ ✶. ❲❡ ❞❡✜♥❡ T−n = Tn ❢♦r n ≥ ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ TnTm = ✶ ✷(Tn+m + Tn−m). ■♥t❡❣r❛t✐♦♥✿ ✶ ✷

✶

✶

✶

✶

✶✵✴✷✺

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❚❤❡ ❈❤❡❜②s❤❡✈ ❢❛♠✐❧② ♦❢ ♣♦❧②♥♦♠✐❛❧s✿ T✵(X) = ✶, T✶(X) = X, Tn+✷(X) = ✷XTn+✶(X) − Tn(X). −✶ ✶ ❚r✐❣♦♥♦♠❡tr✐❝ r❡❧❛t✐♦♥✿ Tn(❝♦s ϑ) = ❝♦s nϑ. ⇒ ∀t ∈ [−✶, ✶], |Tn(t)| ≤ ✶. ❲❡ ❞❡✜♥❡ T−n = Tn ❢♦r n ≥ ✵✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ TnTm = ✶ ✷(Tn+m + Tn−m). ■♥t❡❣r❛t✐♦♥✿

• Tn = ✶

✷ Tn+✶ n + ✶ − Tn−✶ n − ✶

• .

✶✵✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t

✵

❝♦s ❝♦s ❞ ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ ✐❢ ✵

✷

✐❢ ✵ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ ✶

✵

❝♦s ❝♦s ❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t = π

✵

f (❝♦s ϑ)g(❝♦s ϑ)❞ϑ. ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ ✵ ✐❢ ✐❢ ✵

✷

✐❢ ✵ ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ ✶

✵

❝♦s ❝♦s ❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t = π

✵

f (❝♦s ϑ)g(❝♦s ϑ)❞ϑ. ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ Tn, Tm =    ✵ ✐❢ n = ±m, π ✐❢ n = m = ✵,

π ✷

✐❢ n = ±m = ✵. ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ ✶

✵

❝♦s ❝♦s ❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t = π

✵

f (❝♦s ϑ)g(❝♦s ϑ)❞ϑ. ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ Tn, Tm =    ✵ ✐❢ n = ±m, π ✐❢ n = m = ✵,

π ✷

✐❢ n = ±m = ✵. ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ an = ✶ π π

✵

f (❝♦s ϑ) ❝♦s nϑ❞ϑ, n ∈ Z. ❈❤❡❜②s❤❡✈ s❡r✐❡s✿ ✶ ✶ ▼❛✐♥ q✉❡st✐♦♥✿ ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t = π

✵

f (❝♦s ϑ)g(❝♦s ϑ)❞ϑ. ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ Tn, Tm =    ✵ ✐❢ n = ±m, π ✐❢ n = m = ✵,

π ✷

✐❢ n = ±m = ✵. ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ an = ✶ π π

✵

f (❝♦s ϑ) ❝♦s nϑ❞ϑ, n ∈ Z. ❈❤❡❜②s❤❡✈ s❡r✐❡s✿

• f (t) =
• n∈Z

anTn(t), t ∈ [−✶, ✶]. ▼❛✐♥ q✉❡st✐♦♥✿ ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t = π

✵

f (❝♦s ϑ)g(❝♦s ϑ)❞ϑ. ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ Tn, Tm =    ✵ ✐❢ n = ±m, π ✐❢ n = m = ✵,

π ✷

✐❢ n = ±m = ✵. ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ an = ✶ π π

✵

f (❝♦s ϑ) ❝♦s nϑ❞ϑ, n ∈ Z. ❈❤❡❜②s❤❡✈ s❡r✐❡s✿

• f (t) =
• n∈Z

anTn(t), t ∈ [−✶, ✶]. ▼❛✐♥ q✉❡st✐♦♥✿ f = f ? ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈❤❡❜②s❤❡✈ ❙❡r✐❡s

❙❝❛❧❛r ♣r♦❞✉❝t✿ f , g = ✶

−✶

f (t)g(t) √ ✶ − t✷ ❞t = π

✵

f (❝♦s ϑ)g(❝♦s ϑ)❞ϑ. ❖rt❤♦❣♦♥❛❧✐t② r❡❧❛t✐♦♥s✿ Tn, Tm =    ✵ ✐❢ n = ±m, π ✐❢ n = m = ✵,

π ✷

✐❢ n = ±m = ✵. ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts✿ an = ✶ π π

✵

f (❝♦s ϑ) ❝♦s nϑ❞ϑ, n ∈ Z. ❈❤❡❜②s❤❡✈ s❡r✐❡s✿

• f (t) =
• n∈Z

anTn(t), t ∈ [−✶, ✶]. ▼❛✐♥ q✉❡st✐♦♥✿ f = f ? ✭✐♥ ✇❤✐❝❤ s❡♥s❡❄✮

✶✶✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ ✐s ✐♥

✷ ✶

✶

✷ ♦✈❡r

✶ ✶ ✱ t❤❡♥ ❝♦♥✈❡r❣❡s t♦ ✐♥

✷ ✶

✶

✷ ✳

❚❤❡♦r❡♠ ■❢ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t ✶ ✶ ✱ t❤❡♥ ❛s ✳ ✶✲♥♦r♠ ✐♥ ×✶✿

×✶

×✶

×✶

×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t ✶ ✶ ✱ t❤❡♥ ❛s ✳ ✶✲♥♦r♠ ✐♥ ×✶✿

×✶

×✶

×✶

×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿

×✶

×✶

×✶

×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿ f ×✶ =

• n∈Z

|an| ×✶

×✶

×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿ f ×✶ =

• n∈Z

|an| ≥ f ∞ ×✶

×✶

×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿ f ×✶ =

• n∈Z

|an| ≥ f ∞ ×✶ = {f | f ×✶ < ∞} ×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿ f ×✶ =

• n∈Z

|an| ≥ f ∞ ×✶ = {f | f ×✶ < ∞} ×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳

✶

×✶

✵

❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿ f ×✶ =

• n∈Z

|an| ≥ f ∞ ×✶ = {f | f ×✶ < ∞} ×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ C✶ ⊂ ×✶ ⊂ C✵ ❚❤❡♦r❡♠ ■❢ ✐s ✭ ✶✮✱ t❤❡♥ ✳

✶✷✴✷✺

❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠s ❢♦r ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

▲❡t f [N] =

|n|≤N anTn✳

❚❤❡♦r❡♠ ■❢ f ✐s ✐♥ L✷(✶/ √ ✶ − t✷) ♦✈❡r [−✶, ✶]✱ t❤❡♥ f [N] ❝♦♥✈❡r❣❡s t♦ f ✐♥ L✷(✶/ √ ✶ − t✷)✳ ❚❤❡♦r❡♠ ■❢ f ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛❞♠✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❞❡r✐✈❛t✐✈❡s ❛t x ∈ [−✶, ✶]✱ t❤❡♥ f [N](x) → f (x) ❛s N → ∞✳ ✶✲♥♦r♠ ✐♥ ×✶✿ f ×✶ =

• n∈Z

|an| ≥ f ∞ ×✶ = {f | f ×✶ < ∞} ×✶ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ C✶ ⊂ ×✶ ⊂ C✵ ❚❤❡♦r❡♠ ■❢ f ✐s Cr ✭r ≥ ✶✮✱ t❤❡♥ an = O(n−r)✳

✶✷✴✷✺

❆♣♣r♦①✐♠❛t✐♥❣ ♦✉r ❊①❛♠♣❧❡

❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t → ✹ − ✸ ✶ + e ❝♦s t ♦✈❡r [−✶, ✶] ✭e = ✵.✺✮✿

−✶ ✶ ✶.✺ ✷

α(t)

−✶ ✶ ✶.✺ ✷

✶✸✴✷✺

❆♣♣r♦①✐♠❛t✐♥❣ ♦✉r ❊①❛♠♣❧❡

❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t → ✹ − ✸ ✶ + e ❝♦s t ♦✈❡r [−✶, ✶] ✭e = ✵.✺✮✿

−✶ ✶ ✶.✺ ✷

α(t)

−✶ ✶ ✶.✺ ✷

✶.✽✷ |α(t) − ✶.✽✷| ≤ ✵.✷

✶✸✴✷✺

❆♣♣r♦①✐♠❛t✐♥❣ ♦✉r ❊①❛♠♣❧❡

❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t → ✹ − ✸ ✶ + e ❝♦s t ♦✈❡r [−✶, ✶] ✭e = ✵.✺✮✿

−✶ ✶ ✶.✺ ✷

α(t)

−✶ ✶ ✶.✺ ✷

✶.✽✷−✵.✶✽T✷(t) |α(t) − (✶.✽✷ − ✵.✶✽T✷(t))| ≤ ✵.✵✵✼

✶✸✴✷✺

✶

❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✐♥ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

✷

❆♣♣r♦①✐♠❛t✐♥❣ ❋✉♥❝t✐♦♥s ✇✐t❤ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s

✸

❚❤❡ ❱❛❧✐❞❛t✐♦♥ ❆❧❣♦r✐t❤♠

✶✹✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s

✶ ✵

✵

❞

✵

✶

✵ ✵ ✵ ✶ ✶ ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

t✵

Tj(s)ϕ(s)❞s.

✵

✶

✵ ✵ ✵ ✶ ✶ ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

t✵

Tj(s)ϕ(s)❞s.

Ti

✵

✶

✵ i ✵ ✵ ✶ ✶ ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

t✵

Tj(s)ϕ(s)❞s.

Ti TjTi

✵

✶

✵ i ✵ i − j i + j ✵ ✶ ✶ ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

t✵

Tj(s)ϕ(s)❞s.

Ti TjTi t

t✵ TjTi ✶

✵ i ✵ i − j i + j ✵ ✶ i − j − ✶ i + j + ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

t✵

Tj(s)ϕ(s)❞s.

Ti TjTi t

t✵ TjTi ✶

✵ i ✵ i − j i + j ✵ ✶/i i − j − ✶ i + j + ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

t✵

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

t✵

Tj(s)ϕ(s)❞s.

Ti TjTi t

t✵ TjTi ✶

✵ i ✵ i − j i + j ✵ ✶/i ✶/i i − j − ✶ i + j + ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

−✶

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

−✶

Tj(s)ϕ(s)❞s.

Ti TjTi t

−✶ TjTi ✶

✵ i ✵ i − j i + j ✵ ✶/i✷ ✶/i i − j − ✶ i + j + ✶

×✶ ✷

❞❡❣ ❞❡❣

×✶

✶ ❞❡❣ ✶ ❞❡❣

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

−✶

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

−✶

Tj(s)ϕ(s)❞s.

Ti TjTi t

−✶ TjTi

βj t

−✶ TjTi

✵ i ✵ i − j i + j ✵ ✶/i✷ ✶/i i − j − ✶ i + j + ✶

×✶ ✷

−❞❡❣ βj ❞❡❣ βj

×✶

i − j − ✶ − ❞❡❣ βj i + j + ✶ + ❞❡❣ βj

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

−✶

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

−✶

Tj(s)ϕ(s)❞s.

Ti TjTi t

−✶ TjTi

βj t

−✶ TjTi

✵ i ✵ i − j i + j ✵ ✶/i✷ ✶/i i − j − ✶ i + j + ✶

×✶ ✷

−❞❡❣ βj ❞❡❣ βj βj×✶/i i − j − ✶ − ❞❡❣ βj i + j + ✶ + ❞❡❣ βj

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

−✶

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

−✶

Tj(s)ϕ(s)❞s.

Ti TjTi t

−✶ TjTi

βj t

−✶ TjTi

✵ i ✵ i − j i + j ✵ ✶/i✷ ✶/i i − j − ✶ i + j + ✶ βj×✶/i✷ −❞❡❣ βj ❞❡❣ βj βj×✶/i i − j − ✶ − ❞❡❣ βj i + j + ✶ + ❞❡❣ βj

♠❛①

✵

❞❡❣ ♠❛①

✵

✶ ❞❡❣ ❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

−✶

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

−✶

Tj(s)ϕ(s)❞s.

Ti TjTi t

−✶ TjTi

βj t

−✶ TjTi

✵ i ✵ i − j i + j ✵ ✶/i✷ ✶/i i − j − ✶ i + j + ✶ βj×✶/i✷ −❞❡❣ βj ❞❡❣ βj βj×✶/i i − j − ✶ − ❞❡❣ βj i + j + ✶ + ❞❡❣ βj

h = ♠❛①

✵≤j<r ❞❡❣ βj,

d = ♠❛①

✵≤j<r(j + ✶ + ❞❡❣ βj).

❑ ✐s ❛♥ ✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

❑ · ϕ(t) =

r−✶

• j=✵

aj(t) t

−✶

(t − s)r−✶−j (r − ✶ − j)! ϕ(s)❞s =

r−✶

• j=✵

βj(t) t

−✶

Tj(s)ϕ(s)❞s.

Ti TjTi t

−✶ TjTi

βj t

−✶ TjTi

✵ i ✵ i − j i + j ✵ ✶/i✷ ✶/i i − j − ✶ i + j + ✶ βj×✶/i✷ −❞❡❣ βj ❞❡❣ βj βj×✶/i i − j − ✶ − ❞❡❣ βj i + j + ✶ + ❞❡❣ βj

h = ♠❛①

✵≤j<r ❞❡❣ βj,

d = ♠❛①

✵≤j<r(j + ✶ + ❞❡❣ βj).

⇒ ❑ ✐s ❛♥ (h, d)✲❛❧♠♦st ❜❛♥❞❡❞ ♦♣❡r❛t♦r✳

✶✺✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

j = ✵ ↓ i = ✵ → ❚❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ♦♣❡r❛t♦r ❑✳

✶✻✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑

j = ✵ ↓ i = ✵ → ❚❤❡ ✜♥❛❧✲❞✐♠❡♥s✐♦♥❛❧ tr✉♥❝❛t✐♦♥ ❑[N]✳

✶✻✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡

❑ · ϕ = t

• ✹ −

✸ ✶ + e ❝♦s t t

t✵

ϕ(s)❞s +

• −✹ +

✸ ✶ + e ❝♦s t t

t✵

sϕ(s)❞s

✶✼✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡

❑ · ϕ ≈ t(✶.✽✷ − ✵.✶✽T✷(t)) t

t✵

ϕ(s)❞s + (−✶.✽✷ + ✵.✶✽T✷(t)) t

t✵

sϕ(s)❞s

✶✼✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡

❑ · ϕ ≈ (✶.✼✸T✶(t) − ✵.✵✾T✸(t))

• β✵(t)

t

t✵

ϕ(s)❞s + (−✶.✽✷ + ✵.✶✽T✷(t))

• β✶(t)

t

t✵

sϕ(s)❞s

✶✼✴✷✺

❚❤❡ ❆❧♠♦st✲❇❛♥❞❡❞ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❖♣❡r❛t♦r ❑ ✲ ❊①❛♠♣❧❡

❑ · ϕ ≈ (✶.✼✸T✶(t) − ✵.✵✾T✸(t))

• β✵(t)

t

t✵

ϕ(s)❞s + (−✶.✽✷ + ✵.✶✽T✷(t))

• β✶(t)

t

t✵

sϕ(s)❞s

✶✼✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ❊q✉✐✈❛❧❡♥t t♦ ■ ❑

◆

✇❤❡r❡ ✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ ❑ ✳ ✵ ✽✷

✵

✶ ✼✸

✶

✵ ✶✽

✷

✵ ✵✾

✸✳

❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

✵ ✻

✵

✶ ✶✾

✶

✵ ✻✷

✷

✵ ✶✼

✸

✵ ✵✺

✹

✵ ✵✶

✺

✷ ✶ ✶✵

✸ ✻

✸ ✷ ✶✵

✸ ✼

✺ ✽ ✶✵

✺ ✽

✼ ✻ ✶✵

✻ ✾

✶ ✷ ✶✵

✻ ✶✵

✶ ✹ ✶✵

✼ ✶✶

✶ ✾ ✶✵

✽ ✶✷

✷ ✵ ✶✵

✾ ✶✸

✷ ✻ ✶✵

✶✵ ✶✹

✷ ✺ ✶✵

✶✶ ✶✺

✸ ✵ ✶✵

✶✷ ✶✻

✷ ✻ ✶✵

✶✸ ✶✼

✸ ✵ ✶✵

✶✹ ✶✽

✷ ✺ ✶✵

✶✺ ✶✾

✷ ✻ ✶✵

✶✻ ✷✵

✶✽✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ❊q✉✐✈❛❧❡♥t t♦

• ■ + ❑

◆

• · ϕ = ψ ✇❤❡r❡ ϕ = z′′✳

❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ ❑ ✳ ✵ ✽✷

✵

✶ ✼✸

✶

✵ ✶✽

✷

✵ ✵✾

✸✳

❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

✵ ✻

✵

✶ ✶✾

✶

✵ ✻✷

✷

✵ ✶✼

✸

✵ ✵✺

✹

✵ ✵✶

✺

✷ ✶ ✶✵

✸ ✻

✸ ✷ ✶✵

✸ ✼

✺ ✽ ✶✵

✺ ✽

✼ ✻ ✶✵

✻ ✾

✶ ✷ ✶✵

✻ ✶✵

✶ ✹ ✶✵

✼ ✶✶

✶ ✾ ✶✵

✽ ✶✷

✷ ✵ ✶✵

✾ ✶✸

✷ ✻ ✶✵

✶✵ ✶✹

✷ ✺ ✶✵

✶✶ ✶✺

✸ ✵ ✶✵

✶✷ ✶✻

✷ ✻ ✶✵

✶✸ ✶✼

✸ ✵ ✶✵

✶✹ ✶✽

✷ ✺ ✶✵

✶✺ ✶✾

✷ ✻ ✶✵

✶✻ ✷✵

✶✽✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ≈ ❊q✉✐✈❛❧❡♥t t♦

• ■ + ❑[◆]

· ϕ = ψ ✇❤❡r❡ ϕ = z′′✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ ❑ ✳ ✵ ✽✷

✵

✶ ✼✸

✶

✵ ✶✽

✷

✵ ✵✾

✸✳

❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

✵ ✻

✵

✶ ✶✾

✶

✵ ✻✷

✷

✵ ✶✼

✸

✵ ✵✺

✹

✵ ✵✶

✺

✷ ✶ ✶✵

✸ ✻

✸ ✷ ✶✵

✸ ✼

✺ ✽ ✶✵

✺ ✽

✼ ✻ ✶✵

✻ ✾

✶ ✷ ✶✵

✻ ✶✵

✶ ✹ ✶✵

✼ ✶✶

✶ ✾ ✶✵

✽ ✶✷

✷ ✵ ✶✵

✾ ✶✸

✷ ✻ ✶✵

✶✵ ✶✹

✷ ✺ ✶✵

✶✶ ✶✺

✸ ✵ ✶✵

✶✷ ✶✻

✷ ✻ ✶✵

✶✸ ✶✼

✸ ✵ ✶✵

✶✹ ✶✽

✷ ✺ ✶✵

✶✺ ✶✾

✷ ✻ ✶✵

✶✻ ✷✵

✶✽✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ≈ ❊q✉✐✈❛❧❡♥t t♦

• ■ + ❑[◆]

· ϕ = ψ ✇❤❡r❡ ϕ = z′′✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑[N]✳ ✵ ✽✷

✵

✶ ✼✸

✶

✵ ✶✽

✷

✵ ✵✾

✸✳

❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

✵ ✻

✵

✶ ✶✾

✶

✵ ✻✷

✷

✵ ✶✼

✸

✵ ✵✺

✹

✵ ✵✶

✺

✷ ✶ ✶✵

✸ ✻

✸ ✷ ✶✵

✸ ✼

✺ ✽ ✶✵

✺ ✽

✼ ✻ ✶✵

✻ ✾

✶ ✷ ✶✵

✻ ✶✵

✶ ✹ ✶✵

✼ ✶✶

✶ ✾ ✶✵

✽ ✶✷

✷ ✵ ✶✵

✾ ✶✸

✷ ✻ ✶✵

✶✵ ✶✹

✷ ✺ ✶✵

✶✶ ✶✺

✸ ✵ ✶✵

✶✷ ✶✻

✷ ✻ ✶✵

✶✸ ✶✼

✸ ✵ ✶✵

✶✹ ✶✽

✷ ✺ ✶✵

✶✺ ✶✾

✷ ✻ ✶✵

✶✻ ✷✵

✶✽✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ≈ ❊q✉✐✈❛❧❡♥t t♦

• ■ + ❑[◆]

· ϕ = ψ ✇❤❡r❡ ϕ = z′′✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑[N]✳ ψ ≈ −✵.✽✷T✵ − ✶.✼✸T✶ + ✵.✶✽T✷ + ✵.✵✾T✸✳ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

✵ ✻

✵

✶ ✶✾

✶

✵ ✻✷

✷

✵ ✶✼

✸

✵ ✵✺

✹

✵ ✵✶

✺

✷ ✶ ✶✵

✸ ✻

✸ ✷ ✶✵

✸ ✼

✺ ✽ ✶✵

✺ ✽

✼ ✻ ✶✵

✻ ✾

✶ ✷ ✶✵

✻ ✶✵

✶ ✹ ✶✵

✼ ✶✶

✶ ✾ ✶✵

✽ ✶✷

✷ ✵ ✶✵

✾ ✶✸

✷ ✻ ✶✵

✶✵ ✶✹

✷ ✺ ✶✵

✶✶ ✶✺

✸ ✵ ✶✵

✶✷ ✶✻

✷ ✻ ✶✵

✶✸ ✶✼

✸ ✵ ✶✵

✶✹ ✶✽

✷ ✺ ✶✵

✶✺ ✶✾

✷ ✻ ✶✵

✶✻ ✷✵

✶✽✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ≈ ❊q✉✐✈❛❧❡♥t t♦

• ■ + ❑[◆]

· ϕ = ψ ✇❤❡r❡ ϕ = z′′✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑[N]✳ ψ ≈ −✵.✽✷T✵ − ✶.✼✸T✶ + ✵.✶✽T✷ + ✵.✵✾T✸✳ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

✵ ✻

✵

✶ ✶✾

✶

✵ ✻✷

✷

✵ ✶✼

✸

✵ ✵✺

✹

✵ ✵✶

✺

✷ ✶ ✶✵

✸ ✻

✸ ✷ ✶✵

✸ ✼

✺ ✽ ✶✵

✺ ✽

✼ ✻ ✶✵

✻ ✾

✶ ✷ ✶✵

✻ ✶✵

✶ ✹ ✶✵

✼ ✶✶

✶ ✾ ✶✵

✽ ✶✷

✷ ✵ ✶✵

✾ ✶✸

✷ ✻ ✶✵

✶✵ ✶✹

✷ ✺ ✶✵

✶✶ ✶✺

✸ ✵ ✶✵

✶✷ ✶✻

✷ ✻ ✶✵

✶✸ ✶✼

✸ ✵ ✶✵

✶✹ ✶✽

✷ ✺ ✶✵

✶✺ ✶✾

✷ ✻ ✶✵

✶✻ ✷✵

✶✽✴✷✺

❆♣♣r♦①✐♠❛t❡ ❙♦❧✉t✐♦♥ ✲ ❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ s♦❧✈❡ z′′(t) +

• ✹ −

✸ ✶+✵.✺ ❝♦s t

• z(t) = c ✇✐t❤ z(−✶) = ✵✱

z′(−✶) = ✶ ❛♥❞ c = ✶✳ ≈ ❊q✉✐✈❛❧❡♥t t♦

• ■ + ❑[◆]

· ϕ = ψ ✇❤❡r❡ ϕ = z′′✳ ❲❡ ❤❛✈❡ ❛ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ■ + ❑[N]✳ ψ ≈ −✵.✽✷T✵ − ✶.✼✸T✶ + ✵.✶✽T✷ + ✵.✵✾T✸✳ ❍❡♥❝❡✱ ❜② ✐♥✈❡rt✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠✱ ✇❡ ❣❡t✿

• ϕ = −✵.✻T✵ − ✶.✶✾T✶ + ✵.✻✷T✷ + ✵.✶✼T✸ − ✵.✵✺T✹ − ✵.✵✶T✺

+ ✷.✶ · ✶✵−✸T✻ + ✸.✷ · ✶✵−✸T✼ − ✺.✽ · ✶✵−✺T✽ − ✼.✻ · ✶✵−✻T✾ + ✶.✷ · ✶✵−✻T✶✵ + ✶.✹ · ✶✵−✼T✶✶ − ✶.✾ · ✶✵−✽T✶✷ − ✷.✵ · ✶✵−✾T✶✸ + ✷.✻ · ✶✵−✶✵T✶✹ + ✷.✺ · ✶✵−✶✶T✶✺ − ✸.✵ · ✶✵−✶✷T✶✻ − ✷.✻ · ✶✵−✶✸T✶✼ + ✸.✵ · ✶✵−✶✹T✶✽ + ✷.✺ · ✶✵−✶✺T✶✾ − ✷.✻ · ✶✵−✶✻T✷✵

✶✽✴✷✺

• ❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s

❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ (■ + ❑) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ϕ✿

• ϕ − ϕ∗×✶.

❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ❑ ❚ ❚ ❆ ❑ ❆ ■ ❑

✶ ✐♥❥❡❝t✐✈❡

■❢ ❉❚ ×✶ ■ ❆ ■ ❑

×✶

✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚

×✶

✶

×✶

❚

×✶

✶

✶✾✴✷✺

• ❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s

❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ (■ + ❑) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ϕ✿

• ϕ − ϕ∗×✶.

◆❛✐✈❡ ♠❡t❤♦❞✿

• ϕ − ϕ∗×✶ ≤ (■ + ❑)−✶×✶

ϕ + ❑ · ϕ − ψ×✶. ❇✉t ♣r♦❜❧❡♠s✿

❍♦✇ t♦ ❝♦♠♣✉t❡ (■ + ❑)−✶×✶ r✐❣♦r♦✉s❧②❄ ❈♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ✐ss✉❡s✳ ❇✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s ❞✉❡ t♦ ✐♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝s✳ ❚✐❣❤t♥❡ss ♦❢ t❤❡ ❜♦✉♥❞❄

❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ❑ ❚ ❚ ❆ ❑ ❆ ■ ❑

✶ ✐♥❥❡❝t✐✈❡

■❢ ❉❚ ×✶ ■ ❆ ■ ❑

×✶

✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚

×✶

✶

×✶

❚

×✶

✶

✶✾✴✷✺

• ❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s

❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ (■ + ❑) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ϕ✿

• ϕ − ϕ∗×✶.

◆❛✐✈❡ ♠❡t❤♦❞✿

• ϕ − ϕ∗×✶ ≤ (■ + ❑)−✶×✶

ϕ + ❑ · ϕ − ψ×✶. ❇✉t ♣r♦❜❧❡♠s✿

❍♦✇ t♦ ❝♦♠♣✉t❡ (■ + ❑)−✶×✶ r✐❣♦r♦✉s❧②❄ ❈♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ✐ss✉❡s✳ ❇✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s ❞✉❡ t♦ ✐♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝s✳ ❚✐❣❤t♥❡ss ♦❢ t❤❡ ❜♦✉♥❞❄

❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ❑ ❚ ❚ ❆ ❑ ❆ ■ ❑

✶ ✐♥❥❡❝t✐✈❡

■❢ ❉❚ ×✶ ■ ❆ ■ ❑

×✶

✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚

×✶

✶

×✶

❚

×✶

✶

✶✾✴✷✺

• ❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s

❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ (■ + ❑) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ϕ✿

• ϕ − ϕ∗×✶.

❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ ⇔ ❚ · ϕ = ϕ, ❚ · ϕ = ϕ − ❆ · (ϕ + ❑ · ϕ − ψ) , ❆ ≈ (■ + ❑)−✶ ✐♥❥❡❝t✐✈❡. ■❢ ❉❚ ×✶ ■ ❆ ■ ❑

×✶

✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚

×✶

✶

×✶

❚

×✶

✶

✶✾✴✷✺

• ❡♥❡r❛❧ ■❞❡❛s ❢♦r ❱❛❧✐❞❛t✐♦♥ ♦❢ ▲✐♥❡❛r Pr♦❜❧❡♠s

❘❡❝❛❧❧✿ ❋♦r t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ϕ (■ + ❑) · ϕ = ψ, ✇❡ ✇❛♥t t♦ ✈❛❧✐❞❛t❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ϕ✿

• ϕ − ϕ∗×✶.

❘❡❢♦r♠✉❧❛t✐♦♥ ❛s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥✿ ϕ + ❑ · ϕ = ψ ⇔ ❚ · ϕ = ϕ, ❚ · ϕ = ϕ − ❆ · (ϕ + ❑ · ϕ − ψ) , ❆ ≈ (■ + ❑)−✶ ✐♥❥❡❝t✐✈❡. ■❢ ❉❚×✶ = ■ − ❆ (■ + ❑)×✶ = k < ✶✱ ❚ ✐s ❝♦♥tr❛❝t✐✈❡ ❛♥❞ ✇❡ ❣❡t ❛ t✐❣❤t ❡♥❝❧♦s✉r❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✿ ❚ · ϕ − ϕ×✶ ✶ + k ≤ ϕ − ϕ∗×✶ ≤ ❚ · ϕ − ϕ×✶ ✶ − k .

✶✾✴✷✺

❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐①

❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ (■ + ❑)−✶. ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿

✷

✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳

✷✵✴✷✺

❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐①

❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ (■ + ❑[N])−✶. ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿

✷

✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳

✷✵✴✷✺

❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐①

❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ (■ + ❑[N])−✶. ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿

✷

✳ ❈♦♠♣✉t✐♥❣ ❛♥ ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳

✷✵✴✷✺

❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐①

❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ (■ + ❑[N])−✶. ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿ O(n✷(h + d))✳

• ❈♦♠♣✉t✐♥❣ ❛♥

❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿ ✳

✷✵✴✷✺

❈♦♠♣✉t✐♥❣ ❛♥ ❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ▼❛tr✐①

❲❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡ ♠❛tr✐①✿ ❆ ≈ (■ + ❑[N])−✶. ❚✇♦ ♣♦ss✐❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈♦♠♣✉t✐♥❣ t❤❡ ✭❞❡♥s❡✮ ✐♥✈❡rs❡✱ ✉s✐♥❣ ❖❧✈❡r ❛♥❞ ❚♦✇♥s❡♥❞✬s ❛❧❣♦r✐t❤♠✿ O(n✷(h + d))✳

• ❈♦♠♣✉t✐♥❣ ❛♥ (h′, d′) ❛❧♠♦st✲❜❛♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐♥✈❡rs❡✿

O(n(h′ + d′)(h + d))✳

✷✵✴✷✺

❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ❢♦r ♦✉r ❊①❛♠♣❧❡

✷✶✴✷✺

❆♣♣r♦①✐♠❛t❡ ■♥✈❡rs❡ ❢♦r ♦✉r ❊①❛♠♣❧❡

✷✶✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶

• ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r

+ ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶

• ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r

+ ❆(❑ − ❑[N])×✶

• ❚r✉♥❝❛t✐♦♥ ❡rr♦r

. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ ✲❛❧♠♦st✲❜❛♥❞❡❞ ✿ ✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ (h′, d′)✲❛❧♠♦st✲❜❛♥❞❡❞ A✿ O(n(h′ + d′)(h + d))✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ ❆ ■ ❑

×✶

✶ ✺ ✶✵

✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✶✴✷✮

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r ♥♦r♠✿ ■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶. ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ tr✐✈✐❛❧❧② ❤❛♥❞❧❡❞✳ ❈♦♠♣✉t✐♥❣ ×✶✲♥♦r♠ ❂ ♠❛①✐♠✉♠ ♦❢ ✶✲♥♦r♠s ♦❢ t❤❡ ❝♦❧✉♠♥s✳ ❲✐t❤ (h′, d′)✲❛❧♠♦st✲❜❛♥❞❡❞ A✿ O(n(h′ + d′)(h + d))✳ ❊①❛♠♣❧❡ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐s✿ ■ − ❆

• ■ + ❑[N]

×✶ ≤ ✶.✺ · ✶✵−✸

✷✷✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❑ ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❑ ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❑ − ❑[N] ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿ ❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶ ✸ ✶✵

✸

✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶.✸ · ✶✵−✸ ✺ ✷ ✶✵

✸

✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶.✸ · ✶✵−✸ ✺.✷ · ✶✵−✸ ✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶.✸ · ✶✵−✸ ✺.✷ · ✶✵−✸ ✾.✹ · ✶✵−✸ + ✷.✼ · ✶✵−✸ ✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶.✸ · ✶✵−✸ ✺.✷ · ✶✵−✸ ✶.✷✶ · ✶✵−✷ ✶ ✷✶ ✶✵

✷

✷✸✴✷✺

❈♦♠♣✉t✐♥❣ t❤❡ ❖♣❡r❛t♦r ◆♦r♠ ✭✷✴✷✮

■ − ❆(■ + ❑)×✶ ≤ ■ − ❆(■ + ❑[N])×✶ + ❆(❑ − ❑[N])×✶.

❆ ■ ❆(❑ − ❑[N]) ❉✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❆♣♣❧② ❆ ❛♥❞ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥✳ ❇♦✉♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦❧✉♠♥s✿

❯s✐♥❣ t❤❡ ❜♦✉♥❞s ✐♥ ✶/i ❛♥❞ ✶/i✷✿ ♣♦ss✐❜❧② ❜✐❣ ♦✈❡r❡st✐♠❛t✐♦♥s✳ ❯s✐♥❣ ❛ ✜rst ♦r❞❡r ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✿ ❞✐✛❡r❡♥❝❡s ✐♥ ✶/i✷ ❛♥❞ ✶/i✹✳

❚r✉♥❝❛t✐♦♥ ❡rr♦r ♦❢ t❤❡ ❡①❛♠♣❧❡ ✶.✸ · ✶✵−✸ ✺.✷ · ✶✵−✸ ✶.✷✶ · ✶✵−✷ ⇒ ✶.✷✶ · ✶✵−✷

✷✸✴✷✺

❇r✐♥❣ ♦✉r ❊①❛♠♣❧❡ t♦ t❤❡ ❊♥❞

❦ ≤ ✶.✺ · ✶✵−✸ + ✶.✷✶ · ✶✵−✷✳ ❚

×✶

❆ ❑

×✶

✻ ✹✽ ✶✵

✶✻✳

❍❡♥❝❡✿ ✻ ✹✽ ✶✵

✶✻

✶ ❦

×✶

✻ ✹✽ ✶✵

✶✻

✶ ❦

✷✹✴✷✺

❇r✐♥❣ ♦✉r ❊①❛♠♣❧❡ t♦ t❤❡ ❊♥❞

❦ ≤ ✶.✸✻ · ✶✵−✷✳ ❚

×✶

❆ ❑

×✶

✻ ✹✽ ✶✵

✶✻✳

❍❡♥❝❡✿ ✻ ✹✽ ✶✵

✶✻

✶ ❦

×✶

✻ ✹✽ ✶✵

✶✻

✶ ❦

✷✹✴✷✺

❇r✐♥❣ ♦✉r ❊①❛♠♣❧❡ t♦ t❤❡ ❊♥❞

❦ ≤ ✶.✸✻ · ✶✵−✷✳ ❚ · ϕ − ϕ×✶ = ❆( ϕ + ❑ · ϕ − ψ)×✶ = ✻.✹✽ · ✶✵−✶✻✳ ❍❡♥❝❡✿ ✻ ✹✽ ✶✵

✶✻

✶ ❦

×✶

✻ ✹✽ ✶✵

✶✻

✶ ❦

✷✹✴✷✺

❇r✐♥❣ ♦✉r ❊①❛♠♣❧❡ t♦ t❤❡ ❊♥❞

❦ ≤ ✶.✸✻ · ✶✵−✷✳ ❚ · ϕ − ϕ×✶ = ❆( ϕ + ❑ · ϕ − ψ)×✶ = ✻.✹✽ · ✶✵−✶✻✳ ❍❡♥❝❡✿ ✻.✹✽ · ✶✵−✶✻ ✶ + ❦ ≤

• ϕ − ϕ∗×✶

≤ ✻.✹✽ · ✶✵−✶✻ ✶ − ❦

✷✹✴✷✺

❇r✐♥❣ ♦✉r ❊①❛♠♣❧❡ t♦ t❤❡ ❊♥❞

❦ ≤ ✶.✸✻ · ✶✵−✷✳ ❚ · ϕ − ϕ×✶ = ❆( ϕ + ❑ · ϕ − ψ)×✶ = ✻.✹✽ · ✶✵−✶✻✳ ❍❡♥❝❡✿ ✻.✸✾ · ✶✵−✶✻ ≤

• ϕ − ϕ∗×✶

≤ ✻.✺✼ · ✶✵−✶✻

✷✹✴✷✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P♦ss✐❜❧❡ ❊①t❡♥s✐♦♥s

❆ ❣❡♥❡r❛❧ s♦❢t✇❛r❡ t♦ ❝❡rt✐❢② ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ s♦❧✉t✐♦♥s ♦❢ ▲❖❉❊s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳ ❆ s✐♠♣❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ▲❖❉❊s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s✿

◆♦♥✲❧✐♥❡❛r ❖❉❊s✳ ❖t❤❡r ♦rt❤♦❣♦♥❛❧ ❢❛♠✐❧✐❡s ♦❢ ♣♦❧②♥♦♠✐❛❧s✳

❆ ❈❖◗ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t♦ ❝❡rt✐❢② t❤✐s ❛❧❣♦r✐t❤♠✳

✷✺✴✷✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P♦ss✐❜❧❡ ❊①t❡♥s✐♦♥s

❆ ❣❡♥❡r❛❧ s♦❢t✇❛r❡ t♦ ❝❡rt✐❢② ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ s♦❧✉t✐♦♥s ♦❢ ▲❖❉❊s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳ ❆ s✐♠♣❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ▲❖❉❊s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s✿

◆♦♥✲❧✐♥❡❛r ❖❉❊s✳ ❖t❤❡r ♦rt❤♦❣♦♥❛❧ ❢❛♠✐❧✐❡s ♦❢ ♣♦❧②♥♦♠✐❛❧s✳

❆ ❈❖◗ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t♦ ❝❡rt✐❢② t❤✐s ❛❧❣♦r✐t❤♠✳

✷✺✴✷✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P♦ss✐❜❧❡ ❊①t❡♥s✐♦♥s

❆ ❣❡♥❡r❛❧ s♦❢t✇❛r❡ t♦ ❝❡rt✐❢② ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ s♦❧✉t✐♦♥s ♦❢ ▲❖❉❊s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳ ❆ s✐♠♣❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ▲❖❉❊s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s✿

◆♦♥✲❧✐♥❡❛r ❖❉❊s✳ ❖t❤❡r ♦rt❤♦❣♦♥❛❧ ❢❛♠✐❧✐❡s ♦❢ ♣♦❧②♥♦♠✐❛❧s✳

❆ ❈❖◗ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t♦ ❝❡rt✐❢② t❤✐s ❛❧❣♦r✐t❤♠✳

✷✺✴✷✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P♦ss✐❜❧❡ ❊①t❡♥s✐♦♥s

❆ ❣❡♥❡r❛❧ s♦❢t✇❛r❡ t♦ ❝❡rt✐❢② ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ s♦❧✉t✐♦♥s ♦❢ ▲❖❉❊s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳ ❆ s✐♠♣❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ▲❖❉❊s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s✿

◆♦♥✲❧✐♥❡❛r ❖❉❊s✳ ❖t❤❡r ♦rt❤♦❣♦♥❛❧ ❢❛♠✐❧✐❡s ♦❢ ♣♦❧②♥♦♠✐❛❧s✳

❆ ❈❖◗ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t♦ ❝❡rt✐❢② t❤✐s ❛❧❣♦r✐t❤♠✳

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❈♦♥❝❧✉s✐♦♥ ❛♥❞ P♦ss✐❜❧❡ ❊①t❡♥s✐♦♥s

❆ ❣❡♥❡r❛❧ s♦❢t✇❛r❡ t♦ ❝❡rt✐❢② ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ s♦❧✉t✐♦♥s ♦❢ ▲❖❉❊s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳ ❆ s✐♠♣❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ▲❖❉❊s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s✿

◆♦♥✲❧✐♥❡❛r ❖❉❊s✳ ❖t❤❡r ♦rt❤♦❣♦♥❛❧ ❢❛♠✐❧✐❡s ♦❢ ♣♦❧②♥♦♠✐❛❧s✳

❆ ❈❖◗ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t♦ ❝❡rt✐❢② t❤✐s ❛❧❣♦r✐t❤♠✳

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❈♦♥❝❧✉s✐♦♥ ❛♥❞ P♦ss✐❜❧❡ ❊①t❡♥s✐♦♥s

❆ ❣❡♥❡r❛❧ s♦❢t✇❛r❡ t♦ ❝❡rt✐❢② ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ s♦❧✉t✐♦♥s ♦❢ ▲❖❉❊s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✳ ❆ s✐♠♣❧❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ▲❖❉❊s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❋✉t✉r❡ ❞✐r❡❝t✐♦♥s✿

◆♦♥✲❧✐♥❡❛r ❖❉❊s✳ ❖t❤❡r ♦rt❤♦❣♦♥❛❧ ❢❛♠✐❧✐❡s ♦❢ ♣♦❧②♥♦♠✐❛❧s✳

❆ ❈❖◗ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t♦ ❝❡rt✐❢② t❤✐s ❛❧❣♦r✐t❤♠✳

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