❆ 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❆■◆✮
Pr tt r r - - PowerPoint PPT Presentation
Pr tt r r Ptts t tr rst r t
✶❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡
❯♥✐✈❡rs✐t② ♦❢ ❈á❞✐③ ✭❙P❆■◆✮
✷■❊❙ ▼❛r ▼❡❞✐t❡rrá♥❡♦
✭❙P❆■◆✮
◮ ■t ✐s ❡st✐♠❛t❡❞ t❤❛t ♦♥❡ ✐♥ ❡✐❣❤t ✇♦♠❡♥ ✇✐❧❧ s✉✛❡r ❜r❡❛st
◮ ❈♦♠♣✉t❡r s✐♠✉❧❛t✐♦♥ ❝❛♥ ❜❡ ❛♥ ❡①❝❡❧❧❡♥t t♦♦❧ t♦ ✐♥✈❡st✐❣❛t❡ ✐t✳ ◮ ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✭❈P▼✮ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ s✐♠✉❧❛t❡
◮ ❚❤✐s ✇♦r❦ ♣r♦♣♦s❡s ❛ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈P▼ ✉s✐♥❣
◮ P❛r❛❧❧❡❧ ♠♦❞❡❧ ✐s ❛♣♣❧✐❡❞ t♦ ♠♦❞❡❧ ❜r❡❛st ❝❛♥❝❡r✱ ❛ ❦✐♥❞ ♦❢
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
◮ ❉✉❝ts ❛r❡ ❝♦♠♣♦s❡❞ ♦❢ t✇♦ ❧❛②❡rs ♦❢ ❝❡❧❧s✿ t❤❡ ✐♥♥❡r♠♦st ❧❛②❡r
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
◮ ❇r❡❛st ❞✉❝t❛❧ ❛❞❡♥♦❝❛r❝✐♥♦♠❛s ❜❡❣✐♥ ✇✐t❤ ♦♥❡ ✭♦r s❡✈❡r❛❧
◮ ❚❤❡ ❧✉♠✐♥❛❧ ❝❡❧❧s ✐♥✈❛❞❡ t❤❡ ❧✐❣❤t ♦❢ t❤❡ ❞✉❝t✱ ❜r❡❛❦✐♥❣ t❤❡
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
◮ ❇r❡❛st ❞✉❝t❛❧ ❛❞❡♥♦❝❛r❝✐♥♦♠❛s ❜❡❣✐♥ ✇✐t❤ ♦♥❡ ✭♦r s❡✈❡r❛❧
◮ ❚❤❡ ❧✉♠✐♥❛❧ ❝❡❧❧s ✐♥✈❛❞❡ t❤❡ ❧✐❣❤t ♦❢ t❤❡ ❞✉❝t✱ ❜r❡❛❦✐♥❣ t❤❡
◮ ❚❤❡♥ ❣♦ ❜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
◮ ❆♥❞ ✜♥❛❧❧② ❜r❡❛❦✐♥❣ t❤❡ ❜❛s❡♠❡♥t ♠❡♠❜r❛♥❡ ❛♥❞ ✐♥✈❛❞✐♥❣ t❤❡
◮ ❆t t❤✐s ♣♦✐♥t t❤❡ ❞✐s❡❛s❡ ❛❝q✉✐r❡s ❛♥ ✐♥✜❧tr❛t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝✱
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
◮ ❚❤✐s ♠♦❞❡❧ ✇♦r❦s ✇✐t❤ ❛ ✷❉ ❣r✐❞ ζ ♦❢ ♥♦❞❡s ✇✐t❤ ♥✉❧❧
◮ ❊❛❝❤ ♥♦❞❡ ♦❢ ❤❛✈❡ ❝♦♦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
◮ ❖♥❡ ♥♦❞❡ ✭❜❡❧♦♥❣✐♥❣ t♦ ❛ ❝❡❧❧ ♦r t♦ ❛♥ ❡♠♣t② s♣❛❝❡✮ ✐s
◮ ❚❤✐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
◮ ❖♥❡ ♥♦❞❡ ✭❜❡❧♦♥❣✐♥❣ t♦ ❛ ❝❡❧❧ ♦r t♦ ❛♥ ❡♠♣t② s♣❛❝❡✮ ✐s
◮ ❆ 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
′, j ′, k ′
Tm
◮ 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♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
◮ τ r❡♣r❡s❡♥ts t❤❡ t②♣❡ ♦❢ ❛❣❡♥t ✭❧✉♠✐♥❛❧✱ ❡①tr❛❝❡❧❧✉❧❛r ♦r
◮ ❚❤❡ ✜rst t❡r♠ ❞❡s❝r✐❜❡s t❤❡ ❡♥❡r❣② ♦❢ ❛❞❤❡s✐♦♥ ❜❡t✇❡❡♥ ❛ ❝❡❧❧
◮ ❚❤❡ s❡❝♦♥❞ t❡r♠ ❞❡✜♥❡s t❤❡ ✈♦❧✉♠❡ ❛♥❞ t❤❡ ❞❡❣r❡❡ ♦❢
◮ ❚❤❡ 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
◮ ❆ ✷❉ ❛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
✶
❆❧❣♦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♦ ✶✳✶
✾
⑥
✶✵
♥❡✐❣❤❜♦✉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
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
◮ ■♠♣❧❡♠❡♥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✮✱ ✷✳✻
◮ ●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
◮ ❙❡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
◮ ▼❛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
◮ ❚❤❡ ✇♦r❦ ♣r♦♣♦s❡s ❛ ❣❡♥❡r❛❧ ♣r♦❝❡❞✉r❡ t♦ ❞♦ ♣❛r❛❧❧❡❧
◮ ❲❡ ❛♣♣❧② t❤❡ ♣r♦❝❡❞✉r❡ t♦ ❇r❡❛st ❆❞❡♥♦❝❛r❝✐♥♦♠❛ ✐♥ s✐t✉
◮ ❲❡ ♣r♦t❡❝t ❞❛t❛ str✉❝t✉r❡s ✇✐t❤✐♥ tr❛♥s❛❝t✐♦♥s✱ ❛♥❞ ❞✐✈✐❞❡ t❤❡
◮ ❲❡ ♦❜t❛✐♥ ❛ ♠❛①✐♠✉♥ s♣❡❡❞✉♣ ❢♦r ✶✷ ♣❛r❛❧❧❡❧ t❛s❦s✳ ◮ ❲❡ ❝❤❡❝❦ t❤❛t s✉❜s❡q✉❡♥t ✐♥❝r❡❛s❡s ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ t❛s❦s ❞♦
◮ ●♦♦❞ s❝❛❧❛❜✐❧✐t② ❢♦r ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s✳ ◮ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦✉r ❢✉t✉r❡ ✇♦r❦ ✐♥✿
◮ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ t♦ ♦t❤❡r ❣❧❛♥❞✉❧❛r ♥❡♦♣❧❛♠s ✐♥
◮ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ❞❛t❛ ♣❛rt✐t✐♦♥ s❝❤❡♠❡ ❢♦r ●P❯
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②
❆♥t♦♥✐♦ ❏✳ ❚♦♠❡✉✱ ❆❧❡❥❛♥❞r♦ ●á♠❡③✱ ❆❧❜❡rt♦ ●✳ ❙❛❧❣✉❡r♦ ❆ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r ❈❡❧❧✉❧❛r P♦tts ▼♦❞❡❧ ✇✐t❤ ❙♦❢t✇❛r❡ ❚r❛♥s❛❝t✐♦♥❛❧ ▼❡♠♦r②