r t t t - - PowerPoint PPT Presentation

▶
r t t t r t s ts
SMART_READER_LITE
LIVE PREVIEW

r t t t - - PowerPoint PPT Presentation

Dec 18, 2022 328 likes •874 views

tt r tt ts r t t t rt sts

slide-1
SLIDE 1

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

▲❛❜✲❙❚■❈❈✱ ❊◆❙❚❆✲❇r❡t❛❣♥❡ ▼❘■❙✱ ❊◆❙❚❆✲P❛r✐s✱ ▼❛r❝❤ ✶✺✱ ✷✵✶✽

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-2
SLIDE 2

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

▼♦t✐✈❛t✐♦♥ ❬✺❪❬✹❪

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-3
SLIDE 3

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s ❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-4
SLIDE 4

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s ❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-5
SLIDE 5

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❈♦♥s✐❞❡r t❤❡ s②st❡♠ S : ˙ ①(t) = ❢(①(t)) ❉❡♥♦t❡ ❜② ϕ(t,①) t❤❡ ✢♦✇ ♠❛♣✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-6
SLIDE 6

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

P♦s✐t✐✈❡ ✐♥✈❛r✐❛♥t s❡t

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-7
SLIDE 7

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❆ s❡t A ✐s ♣♦s✐t✐✈❡ ✐♥✈❛r✐❛♥t ❬✶❪ ✐❢ ① ∈ A,t ≥ ✵ = ⇒ ϕ(t,①) ∈ A. ❖r ❡q✉✐✈❛❧❡♥t❧② ϕ([✵,∞],A) ⊂ A. ❚❤❡ s❡t ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ ✐♥✈❛r✐❛♥t s❡ts ✐s ❛ ❝♦♠♣❧❡t❡ ❧❛tt✐❝❡✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-8
SLIDE 8

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❑❧❡❡♥❡ ❛❧❣❡❜r❛

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-9
SLIDE 9

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

▲❛tt✐❝❡

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-10
SLIDE 10

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❆ ❧❛tt✐❝❡ (L ,≤) ✐s ❛ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ s❡t✱ ❝❧♦s❡❞ ✉♥❞❡r ❧❡❛st ✉♣♣❡r ❛♥❞ ❣r❡❛t❡st ❧♦✇❡r ❜♦✉♥❞s ❬✷❪✳ ❆ ♠❛❝❤✐♥❡ ❧❛tt✐❝❡ (L▼,≤) ♦❢ L ✐s ❝♦♠♣❧❡t❡ s✉❜❧❛tt✐❝❡ ♦❢ (L ,≤) ✇❤✐❝❤ ✐s ✜♥✐t❡✳ ▼♦r❡♦✈❡r ❜♦t❤ L ❛♥❞ L▼ ❤❛✈❡ t❤❡ s❛♠❡ t♦♣ ❛♥❞ ❜♦tt♦♠✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-11
SLIDE 11

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

▼❛❝❤✐♥❡ ❧❛tt✐❝❡

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-12
SLIDE 12

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❑❧❡❡♥❡ ❛❧❣❡❜r❛

❲❡ ❝♦♥s✐❞❡r ❛ s❡t F ♦❢ ❛✉t♦♠♦r♣❤✐s♠ ❢ ✿P (X) → P (X) s✉❝❤ t❤❛t ❢ (X) = X ❢ (A∩B) = ❢ (A)∩❢ (B) ◆♦t❡ t❤❛t ❢ ✐s ✐♥❝❧✉s✐♦♥ ♠♦♥♦t♦♥✐❝✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-13
SLIDE 13

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❑❧❡❡♥❡(+,·,∗)

❑❧❡❡♥❡(∩,◦,∗)

❆❞❞✐t✐♦♥ ❛+❜ ❢ ∩❣ Pr♦❞✉❝t ❛·❜ ❢ ◦❣ ❆ss♦❝✐❛t✐✈✐t② ❛+(❜ +❝) = (❛+❜)+❝ ❢ ∩(❣ ∩❤) = (❢ ∩❣)∩❤ ❛(❜❝) = (❛❜)❝ ❢ ◦(❣ ◦❤) = (❢ ◦❣)◦❤ ❈♦♠♠✉t❛t✐✈✐t② ❛+❜ = ❜ +❛ ❢ ∩❣ = ❣ ∩❢ ❉✐str✐❜✉t✐✈✐t② ❛(❜ +❝) = (❛❜)+(❛❝) ❢ ◦(❣ ∩❤) = (❢ ◦❣)∩(❢ ◦❤) (❜ +❝)❛ = (❜❛)+(❝❛) (❣ ∩❤)◦❢ = (❣ ◦❢ )∩(❤ ◦❢ ) ③❡r♦ ❛+✵ = ❛ ❢ ∩⊤ = ❢ ❖♥❡ ❛✶ = ✶❛ = ❛ ❢ ◦■❞ = ■❞ ◦❢ = ❢ ❆♥♥✐❤✐❧❛t✐♦♥ ❛✵ = ✵❛ = ✵ ❢ ◦⊤ = ⊤ ■❞❡♠♣♦t❡♥❝❡ ❛+❛ = ❛ ❢ ∩❢ = ❢ P❛rt✐❛❧ ♦r❞❡r ❛ ≤ ❜ ⇔ ❛+❜ = ❜ ❢ ⊃ ❣ ⇔ ❢ ∩❣ = ❣ ❑❧❡❡♥❡ st❛r ❛∗ = ✶+❛+❛❛+❛❛❛+... ❢ ∗ = ■❞ ∩❢ ∩❢ ✷ ∩❢ ✸ ∩...

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-14
SLIDE 14

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❘❡❞✉❝❡rs

❚♦ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ❢ ∈ F✱ ✇❡ ❝❛♥ ❛ss♦❝✐❛t❡ t❤❡ r❡❞✉❝❡r R = ■❞ ∩❢ ✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-15
SLIDE 15

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❲❡ ❤❛✈❡ A ⊂ B ⇒ R (A) ⊂ R (B) ♠♦♥♦t♦♥✐❝✐t② R (A) ⊂ A ❞❡❣r♦✇t❤

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-16
SLIDE 16

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠✳ ❲❡ ❤❛✈❡ (■❞ ∩❢ )∞ = ❢ ∗ Pr♦♦❢✳ ❙✐♥❝❡ ❢ ✐s s✉❝❤ t❤❛t ❢ (A∩B)) = ❢ (A)∩❢ (B), ✇❡ ❤❛✈❡ (■❞ ∩❢ )✷ (A) = (■❞ ∩❢ )(A∩❢ (A)) = = A∩❢ (A)∩❢ (A∩❢ (A)) = A∩❢ (A)∩❢ ✷ (A) ❛♥❞ (■❞ ∩❢ )∞ (A) = A∩❢ (A)∩❢ ✷ (A)∩❢ ✸ (A)∩··· = ❢ ∗ (A).

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-17
SLIDE 17

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❲❡ ❞❡✜♥❡ ❋✐①(❢ ∗) = {A|❢ ∗(A) = A} = ❋✐①(■❞ ∩❢ ) ❋r♦♠ t❤❡ ❑♥❛st❡r✕❚❛rs❦✐ t❤❡♦r❡♠✱ ✐t ✐s ❛ ❝♦♠♣❧❡t❡ s✉❜❧❛tt✐❝❡ ♦❢ L ✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-18
SLIDE 18

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

✭❛✮ ✿ ❘❡❞ ♥♦❞❡s ✿ A✱ ✭❜✮✿A∩❢ (A)✱ ✭❝✮✿A∩❢ (A)∩❢ ✷ (A)✱ ✭❞✮✿❢ ∗ (A).

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-19
SLIDE 19

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

  • ♦❛❧✳ ❈♦♠♣✉t❡ ✇✐t❤ ❝❧♦s✉r❡ s❡ts ❢ ∗

✐ ✱✐ ∈ {✶,✷,...}✱ ✐✳❡✳✱ ❝♦♠♣✉t❡

✇✐t❤ ❡①♣r❡ss✐♦♥s s✉❝❤ ❛s ❢ ∗ (A)∩(❣∗ (A)∪❤∗ (A))∗ ❲❡ ✇❛♥t t♦ ❢❛❝t♦r✐③❡ t❤❡ ✜①❡❞ ♣♦✐♥t ♦♣❡r❛t♦rs ❛s ♠✉❝❤ ❛s ♣♦ss✐❜❧❡✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-20
SLIDE 20

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❋❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt✐❡s ❬✸❪ ❢ ∗ ∩❢ ∗ = ❢ ∗ (❢ ∗)∗ = ❢ ∗ (❢ ∗ ∩❣∗)∗ = (❢ ∩❣)∗ ❢ ∗ ◦(❢ ◦❣∗)∗ = (❢ ∩❣)∗

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-21
SLIDE 21

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❉❡❛❧✐♥❣ st❛t❡ ❡q✉❛t✐♦♥s

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-22
SLIDE 22

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❉❡✜♥❡ − → ❢ (A) = ϕ([−✶,✵],A) ← − ❢ (A) = ϕ([✵,✶],A) ❲❡ ❤❛✈❡ − → ❢ (R♥) = R♥ − → ❢ (A∩B) = − → ❢ (A)∩− → ❢ (B) ← − ❢ (R♥) = R♥ ← − ❢ (A∩B) = ← − ❢ (A)∩← − ❢ (B)

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-23
SLIDE 23

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❚❤❡ s❡ts − → ❢ ∗ (A),← − ❢ ∗ (A) ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❧❛r❣❡st ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ✐♥✈❛r✐❛♥t s❡ts ✐♥❝❧✉❞❡❞ ✐♥ A✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-24
SLIDE 24

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❚❤❡ ❧❛r❣❡st ✐♥✈❛r✐❛♥t s❡t ✐♥❝❧✉❞❡❞ ✐♥ A ✐s − → ❢ ∩← − ❢ ∗ (A)

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-25
SLIDE 25

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

■❧❧✉str❛t✐♦♥

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-26
SLIDE 26

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s ❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-27
SLIDE 27

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s ❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-28
SLIDE 28

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❑❧❡❡♥❡ ✐♥t❡r✈❛❧s

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-29
SLIDE 29

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

  • ✐✈❡♥ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ❢ ✱ ✇❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢ ∗(❛) ✇❤❡r❡ ❛ ✐s ✐♥

(L ,≤) ✭❢♦r ✐♥st❛♥❝❡ (R♥,⊂)✮✳ ▼❛❝❤✐♥❡ s✉❜❧❛tt✐❝❡ L▼ ♦❢ L ✭♠❛③❡ ❢♦r ✐♥st❛♥❝❡✮✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-30
SLIDE 30

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

■♥t❡r✈❛❧ ❛✉t♦♠♦r♣❤✐s♠

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-31
SLIDE 31

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❆♥ ✐♥t❡r✈❛❧ ❛✉t♦♠♦r♣❤✐s♠ [❢ −,❢ +] ❝♦♥t❛✐♥✐♥❣ ❢ ✐s ❛ ♣❛✐r ♦❢ t✇♦ ♠❛❝❤✐♥❡ ❛✉t♦♠♦r♣❤✐s♠ ❢ −,❢ + ✇✐t❤ s✉❝❤ t❤❛t ❛ ∈ L▼ ⇒ ❢ − (❛) ≤ ❢ (❛) ≤ ❢ + (❛). ▲❡♠♠❛✳ ❲❡ ❤❛✈❡ ❋✐①

  • (❢ −)∗

⊂ L▼ ∩❋✐①(❢ ∗) ⊂ ❋✐①

  • (❢ +)∗

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-32
SLIDE 32

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❋✐①❡❞ ♣♦✐♥ts ❋✐①

  • (❢ −)∗

✐♥ ♠❛❣❡♥t❛✱ ❋✐①

  • (❢ +)∗

✐♥ ❜❧✉❡

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-33
SLIDE 33

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠✳ ■❢ ❛ ∈ [❛−,❛+]✱ ✇❤❡r❡ ❛−,❛+ ❜♦t❤ ❜❡❧♦♥❣ t♦ L▼✱ t❤❡♥ (✐) ❢ ∗ (❛) ∈

  • (❢ −)∗ (❛−),(❢ +)∗ (❛+)
  • (✐✐)

❢ ∗ ◦(❢ −)∗ (❛−) = (❢ −)∗ (❛−) (✐✐✐) ❢ ∗ (❛) ≤ (■❞ ∩❢ +)✐ (❛+), ∀✐ ≥ ✵

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-34
SLIDE 34

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❆❧❣♦r✐t❤♠

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-35
SLIDE 35

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❈♦♠♣✉t❛t✐♦♥ ♦❢ ❢ ∗ (❛),❛ ∈ [❛]

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-36
SLIDE 36

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❇♦♦❧❡❛♥ ❧❛tt✐❝❡

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-37
SLIDE 37

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❆ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ L ✐s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡✳ ❊✈❡r② ❡❧❡♠❡♥t ❛ ❤❛s ❛ ✉♥✐q✉❡ ❝♦♠♣❧❡♠❡♥t ❛✱ s❛t✐s❢②✐♥❣ ❛ ∨❛ = ⊤ ❛♥❞ ❛ ∧❛ = ⊥✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-38
SLIDE 38

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❲❡ ❤❛✈❡ ❛ ≤ ❜ ⇔ ❜ ≤ ❛ ❛ ∨❜ = ❜ ∧❛ ✭❉❡ ▼♦r❣❛♥✬s ❧❛✇s✮ ❛ ∧❜ = ❜ ∨❛

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-39
SLIDE 39

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-40
SLIDE 40

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

[❛−,❛+] =

  • ❛+,❛−
  • ❢ ([❛−,❛+])

= [❢ (❛−),❢ (❛+)] [❛−,❛+]∧[❜−,❜+] = [❛− ∧❜−,❛+ ∧❜+] [❛−,❛+]∨[❜−,❜+] = [❛− ∨❜−,❛+ ∨❜+]

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-41
SLIDE 41

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

▼♦♥♦t♦♥✐❝ ❝❛s❡

❈♦♠♣✉t❡ ① = ❢ ∗

✶ (❛)∨(❢ ∗ ✷ (❜)∧❢ ∗ ✸ (❝)). ❲❡ ❤❛✈❡

① ∈ [ ❢ ∗

✶ (❛−)∨(❢ ∗ ✷ (❜−)∧❢ ∗ ✸ (❝−))

  • ■❞ ∩❢ ✐

✶

  • (❛+)∨
  • ■❞ ∩❢ ✐

✷

  • (❜+)∧
  • ■❞ ∩❢ ✐

✸

  • (❝+)
  • ] .

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-42
SLIDE 42

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

◆♦♥ ♠♦♥♦t♦♥✐❝ ❝❛s❡

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-43
SLIDE 43

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ① = ❢ ∗

✶ (❛)∧❢ ∗ ✷

  • ❜
  • . ❆♣♣❧②✐♥❣ ✐♥t❡r✈❛❧

❛r✐t❤♠❡t✐❝ r✉❧❡s✱ ✇❡ ❣❡t ① ∈

  • ❢ ∗

✶

  • ❛−

∧❢ ∗

✷

  • ❜+
  • ,❢ ∗

✶

  • ❛+
  • ∧❢ ∗

✷

  • ❜−
  • ,

✐✳❡✳✱ ✇❡ ♥❡❡❞ t♦ ❣♦ ✉♣ t♦ t❤❡ ✜①❡❞ ♣♦✐♥t ❢♦r ❜♦t❤ ❜♦✉♥❞s✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-44
SLIDE 44

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❋♦r✇❛r❞ r❡❛❝❤ s❡t

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-45
SLIDE 45

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❋♦r✇❛r❞ r❡❛❝❤ s❡t ♦❢ A ❞❡✜♥❡❞ ❜② ❋♦r✇(❢,A) = {① | ∃t ≥ ✵,∃①✵ ∈ A,ϕ(t,①✵) = ①}. We get ❋♦r✇(❢,A) = ← − ❢ ∗ A

  • .

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-46
SLIDE 46

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

▼♦♥♦t♦♥✐❝ ♣❛t❤ ♣❧❛♥♥✐♥❣

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-47
SLIDE 47

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❚❤❡ s❡t ♦❢ ♣❛t❤s t❤❛t st❛rt ✐♥ t❤❡ s❡t A ❛♥❞ r❡❛❝❤ B ✐s ❣✐✈❡♥ ❜② P❛t❤(A,B) = ❋♦r✇(A)∩❇❛❝❦(B) = ← − ❢ ∗ A

  • ∩−

→ ❢ ∗ B

  • .

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-48
SLIDE 48

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❆ t♦ ❇ ♣r♦❜❧❡♠

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-49
SLIDE 49

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❈♦♥s✐❞❡r t✇♦ s❡ts A,B s✉❝❤ t❤❛t B ⊂ A✳ ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ s❡t X = ❈❛♣tA→B = {① | ∃t ≥ ✵,ϕ(t,①) ∈ B ❛♥❞ ∀t✶ ∈ [✵,t],ϕ(t✶,①) ∈ A}.

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-50
SLIDE 50

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s ❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-51
SLIDE 51

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

◆♦♥ ♠♦♥♦t♦♥✐❝ ♣❛t❤ ♣❧❛♥♥✐♥❣

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-52
SLIDE 52

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❋✐♥❞ t❤❡ s❡t X ♦❢ ❛❧❧ ♣❛t❤s t❤❛t st❛rt ✐♥ A✱ ❛✈♦✐❞ B ❛♥❞ r❡❛❝❤ C✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-53
SLIDE 53

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

❋r❛♥❝♦ ❇❧❛♥❝❤✐♥✐ ❛♥❞ ❙t❡❢❛♥♦ ▼✐❛♥✐✳ ❙❡t✲❚❤❡♦r❡t✐❝ ▼❡t❤♦❞s ✐♥ ❈♦♥tr♦❧✳ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐♥❡ss ▼❡❞✐❛✱ ❖❝t♦❜❡r ✷✵✵✼✳ ❇✳ ❆✳ ❉❛✈❡② ❛♥❞ ❍✳ ❆✳ Pr✐❡st❧❡②✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❛♥❞ ❖r❞❡r✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✭■❙❇◆ ✵✺✷✶✼✽✹✺✶✹✮✱ ✷✵✵✷✳ ❉❡①t❡r ❑♦③❡♥✳ ❆ ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ❢♦r ❦❧❡❡♥❡ ❛❧❣❡❜r❛s ❛♥❞ t❤❡ ❛❧❣❡❜r❛ ♦❢ r❡❣✉❧❛r ❡✈❡♥ts✳ ■♥ ●✐❧❡s ❑❛❤♥✱ ❡❞✐t♦r✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙✐①t❤ ❆♥♥✉❛❧ ■❊❊❊ ❙②♠♣✳ ♦♥ ▲♦❣✐❝ ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ▲■❈❙ ✶✾✾✶✱ ♣❛❣❡s ✷✶✹✕✷✷✺✳ ■❊❊❊ ❈♦♠♣✉t❡r ❙♦❝✐❡t② Pr❡ss✱ ❏✉❧② ✶✾✾✶✳ ❚✳ ▲❡ ▼é③♦✱ ▲✳ ❏❛✉❧✐♥✱ ❛♥❞ ❇✳ ❩❡rr✳ ■♥♥❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❝❛♣t✉r❡ ❜❛s✐♥ ♦❢ ❛ ❞②♥❛♠✐❝❛❧ s②st❡♠✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s

slide-54
SLIDE 54

▼♦t✐✈❛t✐♦♥ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ❇♦♦❧❡❛♥ ❧❛tt✐❝❡ ❆♣♣❧✐❝❛t✐♦♥s

■♥ ❆❜str❛❝ts ♦❢ t❤❡ ✾t❤ ❙✉♠♠❡r ❲♦r❦s❤♦♣ ♦♥ ■♥t❡r✈❛❧ ▼❡t❤♦❞s✳ ▲②♦♥✱ ❋r❛♥❝❡✱ ❏✉♥❡ ✶✾✲✷✷✱ ✷✵✶✻✳ ❚✳ ▲❡ ▼é③♦✱ ▲✳ ❏❛✉❧✐♥✱ ❛♥❞ ❇✳ ❩❡rr✳ ❆♥ ✐♥t❡r✈❛❧ ❛♣♣r♦❛❝❤ t♦ ❝♦♠♣✉t❡ ✐♥✈❛r✐❛♥t s❡ts✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥ ♦♥ ❆✉t♦♠❛t✐❝ ❈♦♥tr♦❧✱ ✻✷✿✹✷✸✻✕✹✷✹✸✱ ✷✵✶✼✳

❑❧❡❡♥❡ ❛❧❣❡❜r❛ t♦ ❝♦♠♣✉t❡ ✇✐t❤ ✐♥✈❛r✐❛♥t s❡ts ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s