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slide-1
SLIDE 1

❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✶ ❆❧❡❥❛♥❞r♦ ●á♠❡③✷ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦✶

✶❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡

❯♥✐✈❡rs✐t② ♦❢ ❈á❞✐③ ✭❙P❆■◆✮

✷■❊❙ ▼❛r ▼❡❞✐t❡rrá♥❡♦

✭❙P❆■◆✮

slide-2
SLIDE 2

■♥tr♦❞✉❝t✐♦♥

◮ ■t ✐s ❡st✐♠❛t❡❞ t❤❛t ♦♥❡ ✐♥ ❡✐❣❤t ✇♦♠❡♥ ✇✐❧❧ s✉✛❡r ❜r❡❛st

❝❛♥❝❡r✱ ❜❡✐♥❣ ❛♣♣r♦①✐♠❛t❡❧② ✽✵ ✪ ♦❢ t❤❡♠ ❞✉❝t❛❧ ❝❛r❝✐♥♦♠❛s✳

◮ ❈♦♠♣✉t❡r s✐♠✉❧❛t✐♦♥ ❝❛♥ ❜❡ ❛♥ ❡①❝❡❧❧❡♥t t♦♦❧ t♦ ✐♥✈❡st✐❣❛t❡ ✐t✳ ◮ ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✭❈P▼✮ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ s✐♠✉❧❛t❡

❜✐♦❧♦❣✐❝❛❧ s②st❡♠s✱ ✐♥ ❛ ✇✐❞❡ s❝❛❧❡ r❛♥❣❡✳

◮ ❚❤✐s ✇♦r❦ ♣r♦♣♦s❡s ❛ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈P▼ ✉s✐♥❣

❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②✳

◮ P❛r❛❧❧❡❧ ♠♦❞❡❧ ✐s ❛♣♣❧✐❡❞ t♦ ♠♦❞❡❧ ❜r❡❛st ❝❛♥❝❡r✱ ❛ ❦✐♥❞ ♦❢

❉✉❝t❛❧ ❈❛r❝✐♥♦♠❛ ■♥ ❙✐t✉ ✭❉❈■❙✮✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-3
SLIDE 3

❇✐♦❧♦❣② ♦❢ ❆❞❡♥♦❝❛r❝✐♥♦♠❛s ✐♥ ❙✐t✉✿ ◆❛t✉r❛❧ ❍✐st♦r②

◮ ❉✉❝ts ❛r❡ ❝♦♠♣♦s❡❞ ♦❢ t✇♦ ❧❛②❡rs ♦❢ ❝❡❧❧s✿ t❤❡ ✐♥♥❡r♠♦st ❧❛②❡r

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

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-4
SLIDE 4

❇✐♦❧♦❣② ♦❢ ❆❞❡♥♦❝❛r❝✐♥♦♠❛s ✐♥ ❙✐t✉✿ ◆❛t✉r❛❧ ❍✐st♦r②

◮ ❇r❡❛st ❞✉❝t❛❧ ❛❞❡♥♦❝❛r❝✐♥♦♠❛s ❜❡❣✐♥ ✇✐t❤ ♦♥❡ ✭♦r s❡✈❡r❛❧

♠✉t❛t✐♦♥s✮ ✐♥ t❤❡ ❣❡♥♦♠❡s ♦❢ t❤❡s❡ ❝❡❧❧s✳

◮ ❚❤❡ ❧✉♠✐♥❛❧ ❝❡❧❧s ✐♥✈❛❞❡ t❤❡ ❧✐❣❤t ♦❢ t❤❡ ❞✉❝t✱ ❜r❡❛❦✐♥❣ t❤❡

♥♦r♠❛❧ ❞♦✉❜❧❡✲❧❛②❡r ❛r❝❤✐t❡❝t✉r❡ ✭❤②♣❡r♣❧❛s✐❛✮✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-5
SLIDE 5

❇✐♦❧♦❣② ♦❢ ❆❞❡♥♦❝❛r❝✐♥♦♠❛s ✐♥ ❙✐t✉✿ ◆❛t✉r❛❧ ❍✐st♦r②

◮ ❇r❡❛st ❞✉❝t❛❧ ❛❞❡♥♦❝❛r❝✐♥♦♠❛s ❜❡❣✐♥ ✇✐t❤ ♦♥❡ ✭♦r s❡✈❡r❛❧

♠✉t❛t✐♦♥s✮ ✐♥ t❤❡ ❣❡♥♦♠❡s ♦❢ t❤❡s❡ ❝❡❧❧s✳

◮ ❚❤❡ ❧✉♠✐♥❛❧ ❝❡❧❧s ✐♥✈❛❞❡ t❤❡ ❧✐❣❤t ♦❢ t❤❡ ❞✉❝t✱ ❜r❡❛❦✐♥❣ t❤❡

♥♦r♠❛❧ ❞♦✉❜❧❡✲❧❛②❡r ❛r❝❤✐t❡❝t✉r❡ ✭❤②♣❡r♣❧❛s✐❛✮

◮ ❚❤❡♥ ❣♦ ❜② ✜❧❧✐♥❣ ♦✉t t❤❡ ❧✐❣❤t ♦❢ t❤❡ ❞✉❝t ✭❉❈■❙✮✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-6
SLIDE 6

❇✐♦❧♦❣② ♦❢ ❆❞❡♥♦❝❛r❝✐♥♦♠❛s ✐♥ ❙✐t✉✿ ◆❛t✉r❛❧ ❍✐st♦r②

◮ ❆♥❞ ✜♥❛❧❧② ❜r❡❛❦✐♥❣ t❤❡ ❜❛s❡♠❡♥t ♠❡♠❜r❛♥❡ ❛♥❞ ✐♥✈❛❞✐♥❣ t❤❡

❣❧❛♥❞✉❧❛r ♣❛r❡♥❝❤②♠❛✳

◮ ❆t t❤✐s ♣♦✐♥t t❤❡ ❞✐s❡❛s❡ ❛❝q✉✐r❡s ❛♥ ✐♥✜❧tr❛t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝✱

❛♣♣❡❛rs t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ♠❡t❛st❛t✐❝ ♣r♦❝❡ss❡s ❛♥❞ tr❡❛t♠❡♥t ❝♦sts r✐s❡ ❝♦♥s✐❞❡r❛❜❧②✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-7
SLIDE 7

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧

◮ ❚❤✐s ♠♦❞❡❧ ✇♦r❦s ✇✐t❤ ❛ ✷❉ ❣r✐❞ ζ ♦❢ ♥♦❞❡s ✇✐t❤ ♥✉❧❧

❜♦r❞❡r❧✐♥❡ ❝♦♥❞✐t✐♦♥s✳

◮ ❊❛❝❤ ♥♦❞❡ ♦❢ ❤❛✈❡ ❝♦♦r❞✐♥❛t❡s (x, y)✱ ❛♥❞ ❛ s②♠❜♦❧ k ∈ Σ✳ ◮ ❆ ❝❡❧❧ ✐s ❛ s✉❜s❡t S = {(i, j, k) : k is constant} ♦❢ ζ✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-8
SLIDE 8

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧

◮ ❖♥❡ ♥♦❞❡ ✭❜❡❧♦♥❣✐♥❣ t♦ ❛ ❝❡❧❧ ♦r t♦ ❛♥ ❡♠♣t② s♣❛❝❡✮ ✐s

r❛♥❞♦♠❧② s❡❧❡❝t❡❞ ✐♥ ❡❛❝❤ t✐♠❡ st❡♣✳

◮ ❚❤✐s ❝❡❧❧ tr✐❡s t♦ ❝❤❛♥❣❡ ✐ts ❧♦❝❛t✐♦♥✱ s✐③❡✱ ❡t❝✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-9
SLIDE 9

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧

◮ ❖♥❡ ♥♦❞❡ ✭❜❡❧♦♥❣✐♥❣ t♦ ❛ ❝❡❧❧ ♦r t♦ ❛♥ ❡♠♣t② s♣❛❝❡✮ ✐s

r❛♥❞♦♠❧② s❡❧❡❝t❡❞ ✐♥ ❡❛❝❤ t✐♠❡ st❡♣✳

◮ ❆ t❛r❣❡t ♥♦❞❡ ✐s r❛♥❞♦♠❧② s❡❧❡❝t❡❞ ✐♥ ❡❛❝❤ t✐♠❡ st❡♣✳ ◮ ❚❤✐s ❝❡❧❧ tr✐❡s t♦ ❝❤❛♥❣❡ ✐ts ❧♦❝❛t✐♦♥✱ s✐③❡✱ ❡t❝✳ ◮ ❙✉rr♦✉♥❞✐♥❣ ❝❡❧❧s ✭♥❡✐❣❤❜♦rs✮ tr✐❡s t♦ ♦❝❝✉♣② t❤❛t ♥♦❞❡✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-10
SLIDE 10

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧✿ Pr♦❜❛❜✐❧✐t② ❋♦r ❈❤❛♥❣❡

P

  • α (i, j, k) → α
  • i

′, j ′, k ′

=

  • e− ∆H

Tm

if ∆H > ✵ ✶ if ∆H ≤ ✵ ✇❤❡r❡

◮ t❤❡ tr✐♣❧❡t (i, j, k) s♣❡❝✐✜❡s ❛ ♥♦❞❡ ♦❢ t❤❡ ❣r✐❞✳✳ ◮ ❛♥❞ Tm ✐s t❤❡ t❡♠♣❡r❛t✉r❡ ♣❛r❛♠❡t❡r✳ ◮ ❈❤❛♥❣❡ ✐s ❛❝❝❡♣t❡❞ ♦r r❡❥❡❝t❡❞ ❜❛s❡❞ ♦♥ ❛♥ ❡♥❡r❣② ❢✉♥❝t✐♦♥

❝❛❧❧❡❞ t❤❡ ❤❛♠✐❧t♦♥✐❛♥ H✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-11
SLIDE 11

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧✿ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ H

H =

  • (i,j,k),(i′,j′,k′)

Jτ(α(i,j,k)),τ(α(i′,j′,k′))

  • ✶ − δα(i,j,k),α(i′,j′,k′)
  • +
  • α

λV (τ)(V (α) − Vt(α))✷ +

  • α

λS(τ)(S (α) − St(α))✷ ✇❤❡r❡

◮ τ r❡♣r❡s❡♥ts t❤❡ t②♣❡ ♦❢ ❛❣❡♥t ✭❧✉♠✐♥❛❧✱ ❡①tr❛❝❡❧❧✉❧❛r ♦r

♠②♦❡♣✐t❤❡❧✐❛❧ ❝❡❧❧s✮✳

◮ ❚❤❡ ✜rst t❡r♠ ❞❡s❝r✐❜❡s t❤❡ ❡♥❡r❣② ♦❢ ❛❞❤❡s✐♦♥ ❜❡t✇❡❡♥ ❛ ❝❡❧❧

❛♥❞ ✐ts ♥❡✐❣❤❜♦rs✳

◮ ❚❤❡ s❡❝♦♥❞ t❡r♠ ❞❡✜♥❡s t❤❡ ✈♦❧✉♠❡ ❛♥❞ t❤❡ ❞❡❣r❡❡ ♦❢

❝♦♠♣r❡ss✐❜✐❧✐t② ♦❢ t❤❡ ❝❡❧❧✳

◮ ❚❤❡ t❤✐r❞ t❡r♠ ♠♦❞❡❧s t❤❡ ❡❧❛st✐❝✐t② ♦❢ t❤❡ ❝❡❧❧✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-12
SLIDE 12

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧✿ ❉❛t❛ ❙tr✉❝t✉r❡s

◮ ❆ ✷❉ ❛rr❛② t♦ s✐♠✉❧❛t❡ t❤❡ ❣r✐❞ ζ✳ ◮ ❆ ❧✐st Ξ ♦❢ ♥♦❞❡s t❤❛s ❤❛✈❡ ❜❡❡♥ ♣r♦❝❡ss❡❞✳ ◮ ❚❤❡ ❧✐st Ξ ✇✐❧ ❜❡ ♣r♦❝❡ss❡❞ ✉♥❞❡r tr❛♥s❛❝t✐♦♥s✳ ◮ P❛r❛❧❧❡❧ t❛s❦ ❝❤♦♦s❡ ❛ ♥♦❞❡✱ ❡✈❛❧✉❛t❡ H ❛♥❞ ❛♣♣❧② δ✳ ◮ ❉❛t❛ str✉❝t✉r❡s ❛r❡ ❛❝❝❡ss❡❞ ✇✐t❤✐♥ tr❛♥s❛❝t✐♦♥s✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-13
SLIDE 13

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧✿ ❆❧❣♦r✐t❤♠ ■

❆❧❣♦r✐t❤♠ ❊✈♦❧✈❡✭③❡t❛ ✱ ❥✐✮④

❢♦r✭✐❂✶✱ ✐❁♥✐t❡r❛t✐♦♥s ✴♥t❛s❦s ✱ ✐✰✰✮④

①❂r❛♥❞♦♠✭①♠❛①✮❀

②❂r❛♥❞♦♠✭②♠❛①✮✿

r✉♥■♥❚r❛♥s❛❝t✐♦♥ ✭✮④✐❢ ✦✭✭①✱②✮ ✐♥ ❥✐✮④

❝❡❧❧❂ ③❡t❛❬①❪❬②❪❀

❥✐✳❛❞❞✭✭①✱②✱❦✮✮❀⑥

❡❧s❡ ❣♦t♦ ✶✳✶

✶✵

♥❡✐❣❤❜♦✉r❳ ❂ ❣❡t❘❛♥❞♦♠✭r❛♥❣❳▼✐♥ ✱r❛♥❣❳▼❛①✮❀

✶✶

♥❡✐❣❤❜♦✉r❨ ❂ ❣❡t❘❛♥❞♦♠✭r❛♥❣❨▼✐♥ ✱r❛♥❣❨▼❛①✮❀

✶✷

❝❡❧❧◆❡✐❣❤❜♦✉r ❂ ③❡t❛ ❬♥❡✐❣❤❜♦✉r❳ ❪❬ ♥❡✐❣❤❜♦✉r❨ ❪❀

✶✸

❏❂ ❣❡t❊♥❡r❣②❆❞❤❡s✐♦♥❋♦r◆❡✐❣❤❜♦rs ✭❝❡❧❧✮❀

✶✹

❱❂❧❛♠❜❞❛❱✯ ❣❡t❱♦❧✉♠❡❋♦r❈❡❧❧ ✭❝❡❧❧✮❀

✶✺

❙❂❧❛♠❜❞❛❙✯ ❣❡t❙✉r❢❛❝❡❈❡❧❧ ✭❝❡❧❧✮❀

✶✻

❞❡❧t❛❍❂❏✰❱✰❙❀

✶✼

✐❢✭❞❡❧t❛❍ ❃✵✮

✶✽

P❂▼❛t❤✳❡①♣✭✲❞❡❧t❛❍✴♣❛r❛♠s✳ ❣❡t❚❡♠♣❡r❛t✉r❡ ✭✮✮❀

✶✾

❡❧s❡ ✐❢✭❞❡❧t❛❍ ❁❂✵✮P❂✶❀

✷✵

✐❢✭♣❃❂ r❛♥❞♦♠ ✭✮✮ ③❡t❛❬①❪❬②❪❂ ❝❡❧❧◆❡✐❣❤❜♦✉r ✭✮❀

✷✶

⑥⑥

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-14
SLIDE 14

❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧✿ ❍♦✇ ■t ❲♦r❦s✳✳✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-15
SLIDE 15

❚❤❡ ❊①♣❡r✐♠❡♥t

◮ ■♠♣❧❡♠❡♥t❛t✐♦♥ ✇✐t❤ ❏❛✈❛ ❛♥❞ ❈❧♦❥✉r❡ ✭❢♦r ❙❚▼✮✳ ◮ ❋r♦♠ ❛ ♥♦r♠❛❧ ❞✉❝t t♦ ❛ ❉❈■❙ ❞✉❝t✳ ◮ ❙✐♠✉❧❛t✐♦♥ ❝❛rr✐❡❞ ♦✉t ♦♥ ✽ ♥♦❞❡s ♦❢ ♦✉r ❝❧✉st❡r ♦❢ ♣r♦❝❡ss♦rs ◮ ❋♦r ❡❛❝❤ ♥♦❞❡✿ t✇♦ ■♥t❡❧ ❳❡♦♥ ❊✺ ♣r♦❝❡ss♦rs ✭✶✻ ❝♦r❡s✮✱ ✷✳✻

  • ❍❩ ❛♥❞ ✶✷✽●❇ ❘❆▼ ②✐❡❧❞✐♥❣ ✷✵✳✽ ●❋▲❖P❙✳

◮ ●r✐❞ ♦❢ ✾✵✵ × ✾✵✵ ❛♥❞ ✶✵✾ ✐t❡r❛t✐♦♥s✳ ◮ ◆✉♠❜❡r ♦❢ P❛r❛❧❧❡❧ ❚❛s❦✿ ✷ t♦ ✶✻✳ ◮ ▼❡❛♥ ❡①❡❝✉t✐♦♥ t✐♠❡s ❛♥❞ s♣❡❡❞✉♣s ❛r❡ ❝❛❧❝✉❧❛t❡❞✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-16
SLIDE 16

❚❤❡ ❊①♣❡r✐♠❡♥t✿ ❊①❡❝✉t✐♦♥ ❚✐♠❡s

◮ ❙❡q✉❡♥t✐❛❧✿ ✷✹✳✶✸ s❡❝♦♥❞s✳ ◮ ▲♦❝❦s✲❜❛s❡❞ t❡❝❤♥✐q✉❡✿ ✻✳✸✷ s❡❝♦♥❞s ✭✶✷ t❛s❦s✮✳ ◮ ❙❚▼ t❡❝❤♥✐q✉❡✿ ✷✳✷✽ s❡❝♦♥❞s ✭✶✷ t❛s❦s✮✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-17
SLIDE 17

❚❤❡ ❊①♣❡r✐♠❡♥t✿ ❙♣❡❡❞✉♣s

◮ ▼❛①✐♠✉♠ s♣❡❡❞✉♣✿ ✶✵✳✼✻ ✭✶✷ t❤r❡❛❞s✮✳ ◮ ◆♦ ✐♠♣r♦✈❡♠❡♥t ❢♦r ✶✻ t❤r❡❛❞s✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-18
SLIDE 18

❈♦♥❝❧✉ss✐♦♥s ❛♥❞ ❋✉t✉r❡ ❲♦r❦

◮ ❚❤❡ ✇♦r❦ ♣r♦♣♦s❡s ❛ ❣❡♥❡r❛❧ ♣r♦❝❡❞✉r❡ t♦ ❞♦ ♣❛r❛❧❧❡❧

s✐♠✉❧❛t✐♦♥s ❢♦r ❈P▼ ♠♦❞❡❧✳

◮ ❲❡ ❛♣♣❧② t❤❡ ♣r♦❝❡❞✉r❡ t♦ ❇r❡❛st ❆❞❡♥♦❝❛r❝✐♥♦♠❛ ✐♥ s✐t✉

✭❉❈■❙✮✳

◮ ❲❡ ♣r♦t❡❝t ❞❛t❛ str✉❝t✉r❡s ✇✐t❤✐♥ tr❛♥s❛❝t✐♦♥s✱ ❛♥❞ ❞✐✈✐❞❡ t❤❡

✇♦r❦ ❜❡t✇❡❡♥ ♣❛r❛❧❧❡❧ t❛s❦✳

◮ ❲❡ ♦❜t❛✐♥ ❛ ♠❛①✐♠✉♥ s♣❡❡❞✉♣ ❢♦r ✶✷ ♣❛r❛❧❧❡❧ t❛s❦s✳ ◮ ❲❡ ❝❤❡❝❦ t❤❛t s✉❜s❡q✉❡♥t ✐♥❝r❡❛s❡s ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ t❛s❦s ❞♦

♥♦t ♦✛❡r ♣❡r❢♦r♠❛♥❝❡ ✐♠♣r♦✈❡♠❡♥ts✳

◮ ●♦♦❞ s❝❛❧❛❜✐❧✐t② ❢♦r ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s✳ ◮ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦✉r ❢✉t✉r❡ ✇♦r❦ ✐♥✿

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

s✐t✉✳

◮ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❞❛t❛ ♣❛rt✐t✐♦♥ s❝❤❡♠❡ ❢♦r ●P❯

❛r❝❤✐t❡❝t✉r❡s✳

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②

slide-19
SLIDE 19

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳✳✳ ◗✉❡st✐♦♥s❄

❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②