r trs s rsts t - - PowerPoint PPT Presentation
r trs s rsts t tr sttsts r rss tr r ss
◆❡✉r❛❧ ♥❡t✇♦r❦s✿ s♦♠❡ r❡s✉❧ts ❛❜♦✉t
✭✐✮ ❙♣✐❦❡ tr❛✐♥ st❛t✐st✐❝s ❀ ✭✐✐✮ ▲✐♥❡❛r r❡s♣♦♥s❡ t❤❡♦r②✳ ❇r✉♥♦ ❈❡ss❛❝
◆❡✉r♦▼❛t❤❈♦♠♣ ❚❡❛♠✱■◆❘■❆ ❙♦♣❤✐❛ ❆♥t✐♣♦❧✐s✱❋r❛♥❝❡✳
✵✺✲✵✾✲✷✵✶✶
❇r✉♥♦ ❈❡ss❛❝ ✭■◆❘■❆✮ ◆❡✉r❛❧ ♥❡t✇♦r❦s✿ s♦♠❡ r❡s✉❧ts ❛❜♦✉t ✵✺✲✵✾✲✷✵✶✶ ✶ ✴ ✶✾
❚❛❜❧❡ ♦❢ ❝♦♥t❡♥ts
✶
❙♣✐❦❡ tr❛✐♥ st❛t✐st✐❝s ❛♥❞ ●✐❜❜s ❞✐str✐❜✉t✐♦♥s
✷
▲✐♥❡❛r r❡s♣♦♥s❡ ✐♥ ❝❤❛♦t✐❝ ♥❡✉r❛❧ ♥❡t✇♦r❦s
❇r✉♥♦ ❈❡ss❛❝ ✭■◆❘■❆✮ ◆❡✉r❛❧ ♥❡t✇♦r❦s✿ s♦♠❡ r❡s✉❧ts ❛❜♦✉t ✵✺✲✵✾✲✷✵✶✶ ✷ ✴ ✶✾
❙♣✐❦❡ tr❛✐♥ st❛t✐st✐❝s ❛♥❞ ●✐❜❜s ❞✐str✐❜✉t✐♦♥s
❈❤❛r❛❝t❡r✐③✐♥❣ s♣✐❦❡ tr❛✐♥s st❛t✐st✐❝s
❋✐❣✉r❡✿ ❘❛st❡r ♣❧♦t✴s♣✐❦❡ tr❛✐♥✳
❈❤❛r❛❝t❡r✐③✐♥❣ s♣✐❦❡ tr❛✐♥s st❛t✐st✐❝s
❋✐❣✉r❡✿ ❘❛st❡r ♣❧♦t✴s♣✐❦❡ tr❛✐♥✳
❆ss✉♠❡ t❤❛t s♣✐❦❡ tr❛✐♥s st❛t✐st✐❝s ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛♥ ❤✐❞❞❡♥ ♣r♦❜❛❜✐❧✐t② µ✳
❈❤❛r❛❝t❡r✐③✐♥❣ s♣✐❦❡ tr❛✐♥s st❛t✐st✐❝s
❋✐❣✉r❡✿ ❘❛st❡r ♣❧♦t✴s♣✐❦❡ tr❛✐♥✳
❆ss✉♠❡ t❤❛t s♣✐❦❡ tr❛✐♥s st❛t✐st✐❝s ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛♥ ❤✐❞❞❡♥ ♣r♦❜❛❜✐❧✐t② µ✳ ❈❛♥ ♦♥❡ ❤❛✈❡ ❛ r❡❛s♦♥❛❜❧❡ ✐❞❡❛ ♦❢ ✇❤❛t µ ✐s ✐♥ ❛ ♥❡✉r❛❧ ♥❡t✇♦r❦ ♠♦❞❡❧ ❄
❘❛st❡r ♣❧♦t
✏s♣✐❦✐♥❣ st❛t❡✑ ω❦(♥) = ✶ ✐❢ ∃t ∈]♥ − ✶, ♥] s✉❝❤ t❤❛t ❱❦(t) ≥ θ; ✵ ♦t❤❡r✇✐s❡✳
❋✐❣✉r❡✿ ❘❛st❡r ♣❧♦t✳
❙♣✐❦❡ ♣❛tt❡r♥ ω(♥) = (ω❦(♥))◆
❦=✶
❙♣✐❦❡ ❜❧♦❝❦ ω♥
♠ = { ω(♠) ω(♠ + ✶) . . . ω(♥) }
❘❛st❡r ♣❧♦t ω ❞❡❢ = ω+∞
−∞
❚❤❡ ●❡♥❡r❛❧✐③❡❞ ■♥t❡❣r❛t❡ ❛♥❞ ❋✐r❡ ▼♦❞❡❧ ❘✉❞♦❧♣❤✲❉❡st❡①❤❡✱✷✵✵✻
❈❦ ❞❱❦ ❞t = −❣▲,❦ (❱❦ − ❊▲) −
◆
❣❦❥(t, ω) (❱❦ − ❊❥) + ■❦(t), ❣❦❥(t, ω) = ●❦❥α❦❥ (t, ω) α❦❥ (t, ω) =
α❦❥ (t − r) ω❥(r)
❋✐❣✉r❡✿ P♦st❙②♥❛♣t✐❝ P♦t❡♥t✐❛❧✳ ❋r♦♠ ❋✳ ●r❛♠♠♦♥t✱ ▲❡❝t✉r❡ ✐♥ ▲❡s ❍♦✉❝❤❡s✱ ✷✵✵✾✳
❚❤❡ ●❡♥❡r❛❧✐③❡❞ ■♥t❡❣r❛t❡ ❛♥❞ ❋✐r❡ ▼♦❞❡❧ ❘✉❞♦❧♣❤✲❉❡st❡①❤❡✱✷✵✵✻
❈❦ ❞❱❦ ❞t = −❣▲,❦ (❱❦ − ❊▲) −
◆
❣❦❥(t, ω) (❱❦ − ❊❥) + ■❦(t), ❣❦❥(t, ω) = ●❦❥α❦❥ (t, ω) α❦❥ (t, ω) =
α❦❥ (t − r) ω❥(r) ❙②♥❛♣t✐❝ r❡s♣♦♥s❡ α❦❥ (t) = t τ❦❥ ❡
− t
τ❦❥ ❍(t)
❚❤❡ ●❡♥❡r❛❧✐③❡❞ ■♥t❡❣r❛t❡ ❛♥❞ ❋✐r❡ ▼♦❞❡❧ ❘✉❞♦❧♣❤✲❉❡st❡①❤❡✱✷✵✵✻
❈❦ ❞❱❦ ❞t = −❣▲,❦ (❱❦ − ❊▲) −
◆
❣❦❥(t, ω) (❱❦ − ❊❥) + ✐(❡①t)
❦
(t) + σ❇ξ❦(t), ❣❦❥(t, ω) = ●❦❥α❦❥ (t, ω) ❈❛♥♦♥✐❝❛❧ ❡q✉❛t✐♦♥s ❈❦ ❞❱❦ ❞t + ❣❦ (t, ω) ❱❦ = ✐❦(t, ω), ✐❦(t, ω) = ❣▲,❦ ❊▲ +
◆
❲❦❥ α❦❥ (t, ω) + ✐(❡①t)
❦
(t) + σ❇ξ❦(t), ❲❦❥ = ❊❥ ●❦❥
❋❧♦✇ ❣✐✈❡♥ ❛ r❛st❡r
❈❦ ❞❱❦ ❞t + ❣❦ (t, ω) ❱❦ = ✐❦(t, ω), Γ❦(t✶, t✷, ω) = ❡
− ✶
❈❦
R t✷
t✶ ❣❦(✉,ω) ❞✉.
❱❦(t, ω) = Γ❦(s, t, ω) ❱❦(s) + ✶ ❈❦ t
s
Γ❦(t✶, t, ω) ✐❦(t✶, ω) ❞t✶.
▲❛st r❡s❡t t✐♠❡
■❢ ❱❦(t) ≥ θ✱ ♥❡✉r♦♥ ❦ ✜r❡s✳
❋✐❣✉r❡✿ ❋r♦♠ ❈❡ss❛❝✱ ❏✳ ▼❛t❤✳ ◆❡✉r♦✳ ✷✵✶✶✳
❉❡❧❛②❡❞ r❡s❡t t♦ ❛ r❛♥❞♦♠ ✈❛❧✉❡ ❱r❡s❡t✳ ❙♣✐❦❡s ❛r❡ r❡❣✐st❡r❡❞ ❛t ✐♥t❡❣❡r t✐♠❡s ✭✐♥ ❛ t✐♠❡ ✉♥✐t t❤❛t ❝❛♥ ❜❡ ❛r❜✐tr❛r② s♠❛❧❧✮✳ ❱❦(t, ω) = Γ❦(τ❦(t, ω), t, ω) ❱r❡s❡t + ✶ ❈❦ t
τ❦(t,ω)
Γ❦(t✶, t, ω) ✐❦(t✶, ω) ❞t✶.
❊①♣❧✐❝✐t ❢♦r♠ ♦❢ t❤❡ ♠❡♠❜r❛♥❡ ♣♦t❡♥t✐❛❧ ❣✐✈❡♥ ❛ r❛st❡r
❱❦(t, ω) = Γ❦(τ❦(t, ω), t, ω) ❱r❡s❡t + ✶ ❈❦ t
τ❦(t,ω)
Γ❦(t✶, t, ω) ✐❦(t✶, ω) ❞t✶. ✐❦(t, ω) = ❣▲,❦ ❊▲ +
◆
❲❦❥ α❦❥ (t, ω) + ✐(❡①t)
❦
(t) + σ❇ξ❦(t). ❱❦(t, ω) = ❱ (❞❡t)
❦
(t, ω) + ❱ (♥♦✐s❡)
❦
(t, ω).
❉❡t❡r♠✐♥✐st✐❝ ♣❛rt
❱ (❞❡t)
❦
(t, ω) = ❱ (s②♥)
❦
(t, ω) + ❱ (❡①t)
❦
(t, ω) ❙②♥❛♣t✐❝ ❝♦♥tr✐❜✉t✐♦♥ ❱ (s②♥)
❦
(t, ω) = ✶ ❈❦
◆
❲❦❥ t
τ❦(t,ω)
Γ❦(t✶, t, ω)α❦❥ (t✶, ω) ❞t✶, ❊①t❡r♥❛❧ ✰ ❧❡❛❦ ❝♦♥tr✐❜✉t✐♦♥
❱ (❡①t)
❦
(t, ω) = ❊▲ τ▲,❦ t
τ❦(t,ω)
Γ❦(t✶, t, ω)❞t✶ + ✶ ❈❦ t
τ❦(t,ω)
✐(❡①t)
❦
(t✶)Γ❦(t✶, t, ω)❞t✶, ✇❤❡r❡ τ▲,❦
❞❡❢
=
❈❦ ❣▲,❦ .
❙t♦❝❤❛st✐❝ ♣❛rt
❱ (♥♦✐s❡)
❦
(τ❦(t, ω), t, ω) = Γ❦(τ❦(t, ω), t, ω)❱r❡s❡t + ❱ (❇)
❦
(τ❦(t, ω), t, ω) ✇✐t❤ ❱ (❇)
❦
(t, ω) = σ❇ ❈❦ t
τ❦(t,ω)
Γ❦(t✶, t, ω)❞❇❦(t✶).
σ✷
❦(t, ω) = Γ✷ ❦(τ❦(t, ω), t, ω) σ✷ ❘ +
σ❇ ❈❦ ✷ t
τ❦(t,ω)
Γ✷
❦(t✶, t, ω) ❞t✶.
❈♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②
Pr♦♣♦s✐t✐♦♥ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ω(♥) ❝♦♥❞✐t✐♦♥❛❧❧② t♦ ω♥−✶
−∞ ✐s ❣✐✈❡♥ ❜②✿
P♥
−∞
◆
P♥
−∞
✇✐t❤ P♥
−∞
ω❦(♥) π (❳❦(♥ − ✶, ω)) + (✶ − ω❦(♥)) (✶ − π (❳❦(♥ − ✶, ω))) , ✇❤❡r❡ ❳❦(♥ − ✶, ω) = θ − ❱ (❞❡t)
❦
(♥ − ✶, ω) σ❦(♥ − ✶, ω) , ❛♥❞ π(①) = ✶ √ ✷π +∞
①
❡− ✉✷
✷ ❞✉.
❙❡t✿ φ (♥, ω) =
◆
φ❦ (♥, ω) φ❦ (♥, ω) = ω❦(♥) ❧♦❣ π (❳❦(♥ − ✶, ω)) + (✶ − ω❦(♥)) ❧♦❣ (✶ − π (❳❦(♥ − ✶, ω))) , s♦ t❤❛t P♥
−∞
❚❤❡♥✿ ❈♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❧♦❝❦s ❣✐✈❡♥ t❤❡ ♣❛st P♥
♠
−∞
P♥
❧=♠ φ(❧,ω).
❚❤❡♦r❡♠✱ ❈❡ss❛❝ ✷✵✶✶✱ ❏✳ ▼❛t❤✳ ◆❡✉r♦✳ ❋♦r ❡❛❝❤ ❝❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs t❤❡ ❣■❋ ♠♦❞❡❧ ❤❛s ❛ ✉♥✐q✉❡ ●✐❜❜s ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣♦t❡♥t✐❛❧ φ✳ ❊①♣❧✐❝✐t ●✐❜❜s ♣♦t❡♥t✐❛❧✳ ❊①♣❧✐❝✐t ❞❡♣❡♥❞❡♥❝❡ ✐♥ ♣❛r❛♠❡t❡rs✳ ❍♦❧❞s ❢♦r ❛ t✐♠❡✲❞❡♣❡♥❞❡♥t st✐♠✉❧✉s ✭♥♦♥ st❛t✐♦♥❛r✐t②✮✳
▼❛r❦♦✈✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥s✳
❚❤❡ ●✐❜❜s ♣♦t❡♥t✐❛❧ ❤❛s ✐♥✜♥✐t❡ r❛♥❣❡ ✭♥♦♥ ▼❛r❦♦✈✐❛♥✮✳ ▼❛r❦♦✈✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐t❤ ♠❡♠♦r② ❞❡♣t❤ ❉ ❛♣♣r♦❛❝❤❡s t❤❡ ❡①❛❝t st❛t✐st✐❝s ✇✐t❤ ❛ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❞✐✈❡r❣❡♥❝❡ ❝♦♥✈❡r❣✐♥❣ ❡①♣♦♥❡♥t✐❛❧❧② ❢❛st t♦ ✵ ❛s ❉ → ∞✳
▼❛r❦♦✈✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥s✳
P♦❧②♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥✳ φ(❉)(ω♥
♥−❉) = ▲
λ❧(♥) φ❧(ω♥
♥−❉),
✇❤❡r❡✿ φ❧(ω♥
♥−❉) = ω✐✶(t✶) . . . ω✐♥(t♥),
✐❧ ∈ { ✶, . . . , ◆ } , t❧ ∈ { ♥ − ❉, . . . , ♥ } .
❚❤❡ ♠❛①✐♠❛❧ ❡♥tr♦♣② ♣r✐♥❝✐♣❧❡
❈♦♥s✐❞❡r t❤❡ st❛t✐♦♥❛r② ❝❛s❡✳ φ(❉)(ω✵
−❉) = ▲
λ❧ φ❧(ω✵
−❉),
❱❛r✐❛t✐♦♥❛❧ ♣r✐♥❝✐♣❧❡ ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ♣r❡ss✉r❡✳ P
= s✉♣
ν∈M✐♥✈
▲
λ❧ ν
−❉)
❚❤❡r❡❢♦r❡✱ t❤❡ ♠♦♥♦♠✐❛❧s φ❧ ❝♦♥st✐t✉t❡ ❛ ❝❛♥♦♥✐❝❛❧ ❜❛s✐s ❢♦r ❝♦♥str❛✐♥ts ✇❤✐❧❡ t❤❡ λ❧✬s ❛r❡ ❝♦♥❥✉❣❛t❡❞ ♣❛r❛♠❡t❡rs✳ ❚❤❡ λ❧✬s ❞❡♣❡♥❞ ❡①♣❧✐❝✐t❧② ♦♥ ♥❡t✇♦r❦ ♣❛r❛♠❡t❡rs ✭s②♥❛♣t✐❝ ✇❡✐❣❤ts✱ st✐♠✉❧✉s✮✳
❙t❛t✐st✐❝❛❧ ▼♦❞❡❧s ❤✐❡r❛r❝❤②
❇❡r♥♦✉❧❧✐ ❉ = ✵✳ ▼❡♠♦r②❧❡ss✳ ◆❡✉r♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ✏■s✐♥❣✑ ✭❙❝❤♥❡✐❞♠❛♥ ❡t ❛❧✱ ♥❛t✉r❡ ✷✵✵✻✮ ❉ = ✵✳ ▼❡♠♦r②❧❡ss✳ ◆❡✉r♦♥s ❛r❡ s♣❛t✐❛❧❧② ❝♦rr❡❧❛t❡❞ ❜✉t t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t✳ P♦❧②♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥ ✭▼❛rr❡ ❡t ❛❧✱ ✷✵✵✾❀ ❱❛sq✉❡③ ❡t ❛❧ ✷✵✶✶✮ ■♥✜♥✐t❡ r❛♥❣❡
▲✐♥❡❛r r❡s♣♦♥s❡ ✐♥ ❝❤❛♦t✐❝ ♥❡✉r❛❧ ♥❡t✇♦r❦s✳
✭❈❡ss❛❝✲❙❡♣✉❧❝❤r❡✱ ✷✵✵✹✱ ✷✵✵✻✮
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