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Dec 05, 2023 278 likes •1.26k views

rs Ptr

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SLIDE 1

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♥✲❊♥❣❡❧ ❣r♦✉♣s

P❛tr✐③✐❛ ▲❖◆●❖❇❆❘❉■

❯◆■❱❊❘❙■❚➚ ❉❊●▲■ ❙❚❯❉■ ❉■ ❙❆▲❊❘◆❖

❆❞❞✐t✐✈❡ ❈♦♠❜✐♥❛t♦r✐❝s ✴ ❈♦♠❜✐♥❛t♦✐r❡ ❛❞❞✐t✐✈❡ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡ ✲ ▼❛rs❡✐❧❧❡ ❍❨❇❘■❉ ❈❖◆❋❊❘❊◆❈❊ ✵✼ ✲ ✶✶ ❙❡♣t❡♠❜❡r ✷✵✷✵

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 2

❆ ♥❡✇ ♣❛♣❡r

▼❡r❝❡❞❡ ▼❛❥ ✭❙❛❧❡r♥♦✮

P✳ ▲✳✱ ▼✳ ▼❛❥✱

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♠✲❊♥❣❡❧ ❣r♦✉♣s✱ ■♥t✳ ❏✳ ●r♦✉♣ ❚❤❡♦r②✱ ✾ ✭✷✵✷✵✮✱ ♥♦✳ ✹✱ ✷✼✼✲✷✾✶✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 3

❇❛s✐❝ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥s ■❢ ❙, ❚ ❛r❡ s✉❜s❡ts ♦❢ ❛ ❣r♦✉♣ ●✱ t❤❡♥ ✇❡ ✇r✐t❡ ❙❚ = {❛❜ | ❛ ∈ ❙, ❜ ∈ ❚} ❛♥❞ ❙✷ = {❛❜ | ❛, ❜ ∈ ❙} . ❙✷ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ sq✉❛r❡ ♦❢ ❙✳ ■❢ ● ✐s ❛♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ ❙ + ❚ = {❛ + ❜ | ❛ ∈ ❙, ❜ ∈ ❚} ❛♥❞ ✷❙ = {❛ + ❜ | ❛, ❜ ∈ ❙} . ❙ + ❚ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✭▼✐♥❦♦✇s❦✐✮ s✉♠s❡t ♦❢ ❙ ❛♥❞ ❚✱ ✷❙ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❞♦✉❜❧❡ ♦❢ ❙✳ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ✇❡ ❞❡♥♦t❡ ❜② |❙| t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❙✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 4

❇❛s✐❝ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥s ■❢ ❙, ❚ ❛r❡ s✉❜s❡ts ♦❢ ❛ ❣r♦✉♣ ●✱ t❤❡♥ ✇❡ ✇r✐t❡ ❙❚ = {❛❜ | ❛ ∈ ❙, ❜ ∈ ❚} ❛♥❞ ❙✷ = {❛❜ | ❛, ❜ ∈ ❙} . ❙✷ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ sq✉❛r❡ ♦❢ ❙✳ ■❢ ● ✐s ❛♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ ❙ + ❚ = {❛ + ❜ | ❛ ∈ ❙, ❜ ∈ ❚} ❛♥❞ ✷❙ = {❛ + ❜ | ❛, ❜ ∈ ❙} . ❙ + ❚ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✭▼✐♥❦♦✇s❦✐✮ s✉♠s❡t ♦❢ ❙ ❛♥❞ ❚✱ ✷❙ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❞♦✉❜❧❡ ♦❢ ❙✳ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ✇❡ ❞❡♥♦t❡ ❜② |❙| t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❙✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 5

❇❛s✐❝ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥s ■❢ ❙, ❚ ❛r❡ s✉❜s❡ts ♦❢ ❛ ❣r♦✉♣ ●✱ t❤❡♥ ✇❡ ✇r✐t❡ ❙❚ = {❛❜ | ❛ ∈ ❙, ❜ ∈ ❚} ❛♥❞ ❙✷ = {❛❜ | ❛, ❜ ∈ ❙} . ❙✷ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ sq✉❛r❡ ♦❢ ❙✳ ■❢ ● ✐s ❛♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ ❙ + ❚ = {❛ + ❜ | ❛ ∈ ❙, ❜ ∈ ❚} ❛♥❞ ✷❙ = {❛ + ❜ | ❛, ❜ ∈ ❙} . ❙ + ❚ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ✭▼✐♥❦♦✇s❦✐✮ s✉♠s❡t ♦❢ ❙ ❛♥❞ ❚✱ ✷❙ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❞♦✉❜❧❡ ♦❢ ❙✳ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ✇❡ ❞❡♥♦t❡ ❜② |❙| t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❙✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 6

Pr♦❜❧❡♠s

▲❡t ● ❜❡ ❛ ❣r♦✉♣✱ ❙, ❚ ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✳ ❉✐r❡❝t ♣r♦❜❧❡♠s✿ t♦ ✜♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t |❙❚| ✐♥ t❡r♠s ♦❢ |❙| ❛♥❞ |❚| ✳ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✿ t♦ ✜♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ♦❢ ❙ ❛♥❞ ❚ ✐❢ t❤❡r❡ ✐s ❛ ❜♦✉♥❞ ❢♦r |❙❚| ✳ ❇❛s✐❝ ♣r♦❜❧❡♠s ✐♥ ❆❞❞✐t✐✈❡ ❈♦♠❜✐♥❛t♦r✐❝s✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-7
SLIDE 7

Pr♦❜❧❡♠s

▲❡t ● ❜❡ ❛ ❣r♦✉♣✱ ❙, ❚ ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✳ ❉✐r❡❝t ♣r♦❜❧❡♠s✿ t♦ ✜♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t |❙❚| ✐♥ t❡r♠s ♦❢ |❙| ❛♥❞ |❚| ✳ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✿ t♦ ✜♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ♦❢ ❙ ❛♥❞ ❚ ✐❢ t❤❡r❡ ✐s ❛ ❜♦✉♥❞ ❢♦r |❙❚| ✳ ❇❛s✐❝ ♣r♦❜❧❡♠s ✐♥ ❆❞❞✐t✐✈❡ ❈♦♠❜✐♥❛t♦r✐❝s✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-8
SLIDE 8

Pr♦❜❧❡♠s

▲❡t ● ❜❡ ❛ ❣r♦✉♣✱ ❙, ❚ ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✳ ❉✐r❡❝t ♣r♦❜❧❡♠s✿ t♦ ✜♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t |❙❚| ✐♥ t❡r♠s ♦❢ |❙| ❛♥❞ |❚| ✳ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✿ t♦ ✜♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ str✉❝t✉r❡ ♦❢ ❙ ❛♥❞ ❚ ✐❢ t❤❡r❡ ✐s ❛ ❜♦✉♥❞ ❢♦r |❙❚| ✳ ❇❛s✐❝ ♣r♦❜❧❡♠s ✐♥ ❆❞❞✐t✐✈❡ ❈♦♠❜✐♥❛t♦r✐❝s✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 9

Pr♦❜❧❡♠s

▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ●✳ Pr♦❜❧❡♠ ✶ ❋✐♥❞ ✐♥❢♦r♠❛t✐♦♥ ♦♥ |❙✷|✳ Pr♦❜❧❡♠ ✷ ❉❡t❡r♠✐♥❡ t❤❡ str✉❝t✉r❡ ♦❢ ❙ ✐❢ |❙✷| s❛t✐s✜❡s |❙✷| ≤ α|❙| + β ❢♦r s♦♠❡ s♠❛❧❧ α ≥ ✶ ❛♥❞ s♠❛❧❧ |β|✳ Pr♦❜❧❡♠s ♦❢ ❦✐♥❞ ✭✷✮ ❛r❡ ❝❛❧❧❡❞ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ♦❢ s♠❛❧❧ ❞♦✉❜❧✐♥❣ t②♣❡✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-10
SLIDE 10

Pr♦❜❧❡♠s

▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ●✳ Pr♦❜❧❡♠ ✶ ❋✐♥❞ ✐♥❢♦r♠❛t✐♦♥ ♦♥ |❙✷|✳ Pr♦❜❧❡♠ ✷ ❉❡t❡r♠✐♥❡ t❤❡ str✉❝t✉r❡ ♦❢ ❙ ✐❢ |❙✷| s❛t✐s✜❡s |❙✷| ≤ α|❙| + β ❢♦r s♦♠❡ s♠❛❧❧ α ≥ ✶ ❛♥❞ s♠❛❧❧ |β|✳ Pr♦❜❧❡♠s ♦❢ ❦✐♥❞ ✭✷✮ ❛r❡ ❝❛❧❧❡❞ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ♦❢ s♠❛❧❧ ❞♦✉❜❧✐♥❣ t②♣❡✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 11

Pr♦❜❧❡♠s

▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ●✳ Pr♦❜❧❡♠ ✶ ❋✐♥❞ ✐♥❢♦r♠❛t✐♦♥ ♦♥ |❙✷|✳ Pr♦❜❧❡♠ ✷ ❉❡t❡r♠✐♥❡ t❤❡ str✉❝t✉r❡ ♦❢ ❙ ✐❢ |❙✷| s❛t✐s✜❡s |❙✷| ≤ α|❙| + β ❢♦r s♦♠❡ s♠❛❧❧ α ≥ ✶ ❛♥❞ s♠❛❧❧ |β|✳ Pr♦❜❧❡♠s ♦❢ ❦✐♥❞ ✭✷✮ ❛r❡ ❝❛❧❧❡❞ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ♦❢ s♠❛❧❧ ❞♦✉❜❧✐♥❣ t②♣❡✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 12

❇❛❝❦❣r♦✉♥❞

❘❡♠❛r❦ ✶ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ ❣r♦✉♣ ●✱ ♦❜✈✐♦✉s❧②

|❙| ≤ |❙✷| ≤ |❙|✷.

❚❤❡ ❜♦✉♥❞s ❛r❡ s❤❛r♣✳ ❘❡♠❛r❦ ✷ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ❡❧❡♠❡♥ts✱ ❦ > ✵✳ ❚❤❡♥

|✷❙| ≥ ✷❦ − ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 13

❇❛❝❦❣r♦✉♥❞

❘❡♠❛r❦ ✶ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ ❣r♦✉♣ ●✱ ♦❜✈✐♦✉s❧②

|❙| ≤ |❙✷| ≤ |❙|✷.

❚❤❡ ❜♦✉♥❞s ❛r❡ s❤❛r♣✳ ❘❡♠❛r❦ ✷ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ❡❧❡♠❡♥ts✱ ❦ > ✵✳ ❚❤❡♥

|✷❙| ≥ ✷❦ − ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 14

❇❛❝❦❣r♦✉♥❞

❘❡♠❛r❦ ✸ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ❡❧❡♠❡♥ts✱ ❦ > ✵✳ ■❢ ❙ ✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✿

❙ = {❛, ❛ + r, ❛ + ✷r, · · · , ❛ + (❦ − ✶)r},

t❤❡♥

|✷❙| = ✷❦ − ✶.

❘❡♠❛r❦ ✹ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ❡❧❡♠❡♥ts✳ ■❢

|✷❙| = ✷❦ − ✶,

t❤❡♥ ❙ ✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 15

❇❛❝❦❣r♦✉♥❞

❘❡♠❛r❦ ✸ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ❡❧❡♠❡♥ts✱ ❦ > ✵✳ ■❢ ❙ ✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✿

❙ = {❛, ❛ + r, ❛ + ✷r, · · · , ❛ + (❦ − ✶)r},

t❤❡♥

|✷❙| = ✷❦ − ✶.

❘❡♠❛r❦ ✹ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ❡❧❡♠❡♥ts✳ ■❢

|✷❙| = ✷❦ − ✶,

t❤❡♥ ❙ ✐s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 16

❇❛❝❦❣r♦✉♥❞

❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ✶✾✺✾✮ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ≥ ✸ ❡❧❡♠❡♥ts ❛♥❞ s✉♣♣♦s❡ t❤❛t |✷❙| ≤ ✷|❙| − ✶ + ❜, ✇❤❡r❡ ✵ ≤ ❜ ≤ ❦ − ✸. ❚❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❦ + ❜ ✳ ■♥ ♣❛rt✐❝✉❧❛r ❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ✶✾✺✾✮ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ≥ ✸ ❡❧❡♠❡♥ts ❛♥❞ s✉♣♣♦s❡ t❤❛t |✷❙| ≤ ✸|❙| − ✹. ❚❤❡♥ t❤❡r❡ ❡①✐st ✐♥t❡❣❡rs ❛ ❛♥❞ q s✉❝❤ t❤❛t q > ✵ ❛♥❞ ❙ ⊆ {❛, ❛ + q, ❛ + ✷q, . . . , ❛ + (✷❦ − ✹)q} .

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 17

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ r❡❝❡♥t❧② ❜② ♠❛♥② ❛✉t❤♦rs✿ ❊✳ ❇r❡✉✐❧❧❛r❞✱ ❇✳ ●r❡❡♥✱ ■✳❩✳ ❘✉③s❛✱ ❚✳ ❚❛♦✱ ✳ ✳ ✳ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ✐♥ ❛❜❡❧✐❛♥ ❣r♦✉♣s ❜② ♠❛♥② ❛✉t❤♦rs✿ ❨✳ ❖✳ ❍❛♠✐❞♦✉♥❡✱ ❇✳ ●r❡❡♥✱ ▼✳ ❑♥❡s❡r✱ ❆✳❙✳ ▲❧❛❞ó✱ ❆✳ P❧❛❣♥❡✱ P✳P✳ P❛❧❢②✱ ■✳❩✳ ❘✉③s❛✱ ❖✳ ❙❡rr❛✱ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ✳ ✳ ✳ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ✐♥ t♦rs✐♦♥✲❢r❡❡ ♥♦♥ ❛❜❡❧✐❛♥ ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❜② ▲✳❱✳ ❇r❛✐❧♦✈s❦②✱ ●✳❆✳ ❋r❡✐♠❛♥✱ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛✱ ❚✳❈✳ ❚❛♦✱ ❑✳❏✳ ❇örö❝③❦②✱ P✳P✳ Pá❧❢②✱ ❖✳ ❙❡rr❛✱ ✳ ✳ ✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 18

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ r❡❝❡♥t❧② ❜② ♠❛♥② ❛✉t❤♦rs✿ ❊✳ ❇r❡✉✐❧❧❛r❞✱ ❇✳ ●r❡❡♥✱ ■✳❩✳ ❘✉③s❛✱ ❚✳ ❚❛♦✱ ✳ ✳ ✳ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ✐♥ ❛❜❡❧✐❛♥ ❣r♦✉♣s ❜② ♠❛♥② ❛✉t❤♦rs✿ ❨✳ ❖✳ ❍❛♠✐❞♦✉♥❡✱ ❇✳ ●r❡❡♥✱ ▼✳ ❑♥❡s❡r✱ ❆✳❙✳ ▲❧❛❞ó✱ ❆✳ P❧❛❣♥❡✱ P✳P✳ P❛❧❢②✱ ■✳❩✳ ❘✉③s❛✱ ❖✳ ❙❡rr❛✱ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ✳ ✳ ✳ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ✐♥ t♦rs✐♦♥✲❢r❡❡ ♥♦♥ ❛❜❡❧✐❛♥ ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❜② ▲✳❱✳ ❇r❛✐❧♦✈s❦②✱ ●✳❆✳ ❋r❡✐♠❛♥✱ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛✱ ❚✳❈✳ ❚❛♦✱ ❑✳❏✳ ❇örö❝③❦②✱ P✳P✳ Pá❧❢②✱ ❖✳ ❙❡rr❛✱ ✳ ✳ ✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 19

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦❜❧❡♠s ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

Pr♦♣♦s✐t✐♦♥ ■❢ ❙ ✐s ❛ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡t ♦❢ t❤❡ ❣r♦✉♣ ♦❢ t❤❡ ✐♥t❡❣❡rs✱ t❤❡♥ ✇❡ ❤❛✈❡

|✷❙| ≥ ✷|❙| − ✶.

▼♦r❡ ❣❡♥❡r❛❧❧②✿ ❚❤❡♦r❡♠ ✭❏✳❍✳❇✳ ❑❡♠♣❡r♠❛♥✱ ✶✾✺✻✮ ■❢ ❙ ✐s ❛ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❤❛✈❡

|❙✷| ≥ ✷|❙| − ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 20

❇❛s✐❝ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥ ■❢ ❛, r = ✶ ❛r❡ ❡❧❡♠❡♥ts ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣r♦✉♣ ●✱ ❛ ❣❡♦♠❡tr✐❝ ❧❡❢t

✭r✐❣t❤✮ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r ❛♥❞ ❧❡♥❣t❤ ♥ ✐s t❤❡ s✉❜s❡t ♦❢ ● {❛, ❛r, ❛r ✷, · · · , ❛r ♥−✶} ({❛, r❛, r ✷❛, · · · , r ♥−✶❛}).

■❢ ● ✐s ❛♥ ❛❞❞✐t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣✱ t❤❡♥

{❛, ❛ + r, ❛ + ✷r, · · · , ❛ + (♥ − ✶)r}

✐s ❝❛❧❧❡❞ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ❞✐✛❡r❡♥❝❡ r ❛♥❞ ❧❡♥❣t❤ ♥✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 21

❇❛s✐❝ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥ ■❢ ❛, r = ✶ ❛r❡ ❡❧❡♠❡♥ts ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣r♦✉♣ ●✱ ❛ ❣❡♦♠❡tr✐❝ ❧❡❢t

✭r✐❣t❤✮ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ r❛t✐♦ r ❛♥❞ ❧❡♥❣t❤ ♥ ✐s t❤❡ s✉❜s❡t ♦❢ ● {❛, ❛r, ❛r ✷, · · · , ❛r ♥−✶} ({❛, r❛, r ✷❛, · · · , r ♥−✶❛}).

■❢ ● ✐s ❛♥ ❛❞❞✐t✐✈❡ ❛❜❡❧✐❛♥ ❣r♦✉♣✱ t❤❡♥

{❛, ❛ + r, ❛ + ✷r, · · · , ❛ + (♥ − ✶)r}

✐s ❝❛❧❧❡❞ ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ❞✐✛❡r❡♥❝❡ r ❛♥❞ ❧❡♥❣t❤ ♥✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 22

❆♥ ❡①❛♠♣❧❡

❊①❛♠♣❧❡ ■❢ ❙ = {❛, ❛r, ❛r ✷, · · · , ❛r ♥−✶} ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✐♥ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ❛♥❞ ❛r = r❛✱ t❤❡♥

❙✷ = {❛✷, ❛✷r, ❛✷r ✷, · · · , ❛✷r ✷♥−✷} ❤❛s ♦r❞❡r ✷|❙| − ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 23

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ❇✳▼✳ ❙❝❤❡✐♥✱ ✶✾✾✶✮ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ |❙| = ❦ ≥ ✷✱

|❙✷| = ✷|❙| − ✶

✐❢ ❛♥❞ ♦♥❧② ✐❢

❙ = {❛, ❛q, · · · , ❛q❦−✶}, ❛♥❞ ❡✐t❤❡r ❛q = q❛ ♦r ❛q❛−✶ = q−✶.

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ |❙✷| = ✷|❙| − ✶✱ t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❧❡❢t ❝♦s❡t ♦❢ ❛ ❝②❝❧✐❝ s✉❜❣r♦✉♣ ♦❢ ●✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 24

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛✱ ✶✾✾✽✮ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ |❙| = ❦ ≥ ✹✱ t❤❡♥

|❙✷| ≤ ✷|❙|

✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐st ❛, q ∈ ●, ❛q = q❛ s✉❝❤ t❤❛t

❙ = {❛, ❛q, · · · , ❛q❦} \ {❝}, ✇✐t❤ ❝ ∈ {❛, ❛q}.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 25

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ✶✾✺✾✮ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡❣❡rs ✇✐t❤ ❦ ≥ ✸ ❡❧❡♠❡♥ts ❛♥❞ s✉♣♣♦s❡ t❤❛t |✷❙| ≤ ✸|❙| − ✹✳ ❚❤❡♥ t❤❡r❡ ❡①✐st ✐♥t❡❣❡rs ❛ ❛♥❞ q s✉❝❤ t❤❛t q > ✵ ❛♥❞ ❙ ⊆ {❛, ❛ + q, ❛ + ✷q, . . . , ❛ + (✷❦ − ✹)q} . ❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ✶✾✼✸✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❛❜❡❧✐❛♥ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✸|❙| − ✹,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ✷|❙| − ✸✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 26

❋r❡✐♠❛♥✬s ❈♦♥❥❡❝t✉r❡

❈♦♥❥❡❝t✉r❡ ✭●✳❆✳ ❋r❡✐♠❛♥✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✸|❙| − ✹,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ✷|❙| − ✸✳ ❈♦♥❥❡❝t✉r❡ ✭●✳❆✳ ❋r❡✐♠❛♥✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✷|❙| − ✶ + ❜, ❢♦r ✵ ≤ ❜ ≤ |❙| − ✸,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st |❙| + ❜✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 27

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❉❡✜♥✐t✐♦♥s ▲❡t ● ❜❡ ❛ ❣r♦✉♣ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ t♦t❛❧ ♦r❞❡r r❡❧❛t✐♦♥ ≤ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t ●✳ ❲❡ s❛② t❤❛t (●, ≤) ✐s ❛♥ ♦r❞❡r❡❞ ✭r✐❣❤t✲♦r❞❡r❡❞✮ ❣r♦✉♣ ✐❢ ❢♦r ❛❧❧ ❛, ❜, ①, ② ∈ ●✱ t❤❡ ✐♥❡q✉❛❧✐t②

❛ ≤ ❜ ✐♠♣❧✐❡s t❤❛t ①❛② ≤ ①❜② ✭❛ ≤ ❜ ✐♠♣❧✐❡s t❤❛t ❛① ≤ ❜①✮ ✳

❆ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✭r✐❣❤t✲♦r❞❡r❛❜❧❡✮ ✐❢ t❤❡r❡ ❡①✐sts ❛ t♦t❛❧ ♦r❞❡r r❡❧❛t✐♦♥ ≤ ♦♥ t❤❡ s❡t ●✱ s✉❝❤ t❤❛t (●, ≤) ✐s ❛♥ ♦r❞❡r❡❞ ✭❛ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 28

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❉❡✜♥✐t✐♦♥s ▲❡t ● ❜❡ ❛ ❣r♦✉♣ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❛ t♦t❛❧ ♦r❞❡r r❡❧❛t✐♦♥ ≤ ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t ●✳ ❲❡ s❛② t❤❛t (●, ≤) ✐s ❛♥ ♦r❞❡r❡❞ ✭r✐❣❤t✲♦r❞❡r❡❞✮ ❣r♦✉♣ ✐❢ ❢♦r ❛❧❧ ❛, ❜, ①, ② ∈ ●✱ t❤❡ ✐♥❡q✉❛❧✐t②

❛ ≤ ❜ ✐♠♣❧✐❡s t❤❛t ①❛② ≤ ①❜② ✭❛ ≤ ❜ ✐♠♣❧✐❡s t❤❛t ❛① ≤ ❜①✮ ✳

❆ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✭r✐❣❤t✲♦r❞❡r❛❜❧❡✮ ✐❢ t❤❡r❡ ❡①✐sts ❛ t♦t❛❧ ♦r❞❡r r❡❧❛t✐♦♥ ≤ ♦♥ t❤❡ s❡t ●✱ s✉❝❤ t❤❛t (●, ≤) ✐s ❛♥ ♦r❞❡r❡❞ ✭❛ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 29

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❆♥② r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ❆♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s t♦rs✐♦♥✲❢r❡❡ ✭❋✳❲✳ ▲❡✈✐✮✳ ❚❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ❣r♦✉♣s ❝♦♥t❛✐♥s t❤❡ ❝❧❛ss ♦❢ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ✭❑✳ ■✇❛s❛✇❛✱ ❆✳■✳ ▼❛❧✟ ❝❡✈✱ ❇✳❍✳ ◆❡✇♠❛♥✮✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ♦r❞❡r❛❜❧❡✳ ❚❤❡ ❣r♦✉♣ ①, ❝ | ❝① = ❝−✶ ✐s ♥♦t ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 30

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❆♥② r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ❆♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s t♦rs✐♦♥✲❢r❡❡ ✭❋✳❲✳ ▲❡✈✐✮✳ ❚❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ❣r♦✉♣s ❝♦♥t❛✐♥s t❤❡ ❝❧❛ss ♦❢ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ✭❑✳ ■✇❛s❛✇❛✱ ❆✳■✳ ▼❛❧✟ ❝❡✈✱ ❇✳❍✳ ◆❡✇♠❛♥✮✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ♦r❞❡r❛❜❧❡✳ ❚❤❡ ❣r♦✉♣ ①, ❝ | ❝① = ❝−✶ ✐s ♥♦t ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 31

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❆♥② r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ❆♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s t♦rs✐♦♥✲❢r❡❡ ✭❋✳❲✳ ▲❡✈✐✮✳ ❚❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ❣r♦✉♣s ❝♦♥t❛✐♥s t❤❡ ❝❧❛ss ♦❢ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ✭❑✳ ■✇❛s❛✇❛✱ ❆✳■✳ ▼❛❧✟ ❝❡✈✱ ❇✳❍✳ ◆❡✇♠❛♥✮✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ♦r❞❡r❛❜❧❡✳ ❚❤❡ ❣r♦✉♣ ①, ❝ | ❝① = ❝−✶ ✐s ♥♦t ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 32

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❆♥② r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ❆♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s t♦rs✐♦♥✲❢r❡❡ ✭❋✳❲✳ ▲❡✈✐✮✳ ❚❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ❣r♦✉♣s ❝♦♥t❛✐♥s t❤❡ ❝❧❛ss ♦❢ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ✭❑✳ ■✇❛s❛✇❛✱ ❆✳■✳ ▼❛❧✟ ❝❡✈✱ ❇✳❍✳ ◆❡✇♠❛♥✮✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ♦r❞❡r❛❜❧❡✳ ❚❤❡ ❣r♦✉♣ ①, ❝ | ❝① = ❝−✶ ✐s ♥♦t ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 33

❖r❞❡r❡❞ ❛♥❞ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s

❆♥② r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ❆♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ● ✐s ♦r❞❡r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s t♦rs✐♦♥✲❢r❡❡ ✭❋✳❲✳ ▲❡✈✐✮✳ ❚❤❡ ❝❧❛ss ♦❢ ♦r❞❡r❡❞ ❣r♦✉♣s ❝♦♥t❛✐♥s t❤❡ ❝❧❛ss ♦❢ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ✭❑✳ ■✇❛s❛✇❛✱ ❆✳■✳ ▼❛❧✟ ❝❡✈✱ ❇✳❍✳ ◆❡✇♠❛♥✮✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ♦r❞❡r❛❜❧❡✳ ❚❤❡ ❣r♦✉♣ ①, ❝ | ❝① = ❝−✶ ✐s ♥♦t ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 34

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✶✹✮ ▲❡t (●, ≤) ❜❡ ❛♥ ♦r❞❡r❡❞ ❣r♦✉♣ ❛♥❞ ❧❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ♦❢ ♦r❞❡r ❦ ≥ ✸✳ ❆ss✉♠❡ t❤❛t

|❙✷| ≤ ✸|❙| − ✹.

❚❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❡①✐st ❛, q ∈ ●✱ s✉❝❤ t❤❛t q❛ = ❛q ❛♥❞ ❙ ✐s ❛ s✉❜s❡t ♦❢

{❛, ❛q, ❛q✷, · · · , ❛q✷❦−✹}.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 35

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✶✹✮ ▲❡t (●, ≤) ❜❡ ❛♥ ♦r❞❡r❡❞ ❣r♦✉♣ ❛♥❞ ❧❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ♦❢ ♦r❞❡r ❦ ≥ ✸✳ ❆ss✉♠❡ t❤❛t

|❙✷| ≤ ✸|❙| − ✹.

❚❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❡①✐st ❛, q ∈ ●✱ s✉❝❤ t❤❛t q❛ = ❛q ❛♥❞ ❙ ✐s ❛ s✉❜s❡t ♦❢

{❛, ❛q, ❛q✷, · · · , ❛q✷❦−✹}.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 36

❖r❞❡r❡❞ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭●✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✶✹✮ ▲❡t (●, ≤) ❜❡ ❛♥ ♦r❞❡r❡❞ ❣r♦✉♣ ❛♥❞ ❧❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✸. ❆ss✉♠❡ t❤❛t |❙✷| ≤ ✸|❙| − ✸. ❚❤❡♥ < ❙ > ✐s ❛❜❡❧✐❛♥✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 37

❋r❡✐♠❛♥✬s ❈♦♥❥❡❝t✉r❡

❈♦♥❥❡❝t✉r❡ ✭●✳❆✳ ❋r❡✐♠❛♥✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✸|❙| − ✹,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ✷|❙| − ✸✳ ❈♦♥❥❡❝t✉r❡ ✭●✳❆✳ ❋r❡✐♠❛♥✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✷|❙| − ✶ + ❜, ❢♦r ✵ ≤ ❜ ≤ |❙| − ✸,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st |❙| + ❜✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 38

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❉❡✜♥✐t✐♦♥

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❲❡ s❛② t❤❛t ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ✐❢ ❢♦r ❡✈❡r② ❆✱ ❇ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✱ t❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t ❣ ∈ ❆❇ t❤❛t ❝❛♥ ❜❡ ✉♥✐q✉❡❧② ❡①♣r❡ss❡❞ ❛s ❛ ♣r♦❞✉❝t ♦❢ ❛♥ ❡❧❡♠❡♥t ♦❢ ❆ ❛♥❞ ❛♥ ❡❧❡♠❡♥t ♦❢ ❇✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 39

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❉❡✜♥✐t✐♦♥

▲❡t ● ❜❡ ❛ ❣r♦✉♣✳ ❲❡ s❛② t❤❛t ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ✐❢ ❢♦r ❡✈❡r② ❆✱ ❇ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✱ t❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t ❣ ∈ ❆❇ t❤❛t ❝❛♥ ❜❡ ✉♥✐q✉❡❧② ❡①♣r❡ss❡❞ ❛s ❛ ♣r♦❞✉❝t ♦❢ ❛♥ ❡❧❡♠❡♥t ♦❢ ❆ ❛♥❞ ❛♥ ❡❧❡♠❡♥t ♦❢ ❇✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 40

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❆♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ◆♦t ❛❧❧ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ✭❊✳ ❘✐♣s ❛♥❞ ❨✳ ❙❡❣❡✈✮✳ ❆ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s✳ ◆♦t ❛❧❧ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ❛r❡ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s ✭◆✳ ❉✉♥✜❡❧❞✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 41

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❆♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ◆♦t ❛❧❧ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ✭❊✳ ❘✐♣s ❛♥❞ ❨✳ ❙❡❣❡✈✮✳ ❆ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s✳ ◆♦t ❛❧❧ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ❛r❡ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s ✭◆✳ ❉✉♥✜❡❧❞✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 42

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❆♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ◆♦t ❛❧❧ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ✭❊✳ ❘✐♣s ❛♥❞ ❨✳ ❙❡❣❡✈✮✳ ❆ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s✳ ◆♦t ❛❧❧ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ❛r❡ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s ✭◆✳ ❉✉♥✜❡❧❞✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 43

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❆♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ◆♦t ❛❧❧ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ✭❊✳ ❘✐♣s ❛♥❞ ❨✳ ❙❡❣❡✈✮✳ ❆ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s✳ ◆♦t ❛❧❧ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ❛r❡ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s ✭◆✳ ❉✉♥✜❡❧❞✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 44

❯♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❆♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ● ✐s t♦rs✐♦♥✲❢r❡❡✳ ◆♦t ❛❧❧ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ✭❊✳ ❘✐♣s ❛♥❞ ❨✳ ❙❡❣❡✈✮✳ ❆ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✳ ❋r❡❡ ❣r♦✉♣s ❛r❡ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s✳ ◆♦t ❛❧❧ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s ❛r❡ r✐❣❤t✲♦r❞❡r❡❞ ❣r♦✉♣s ✭◆✳ ❉✉♥✜❡❧❞✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 45

❋r❡✐♠❛♥✬s ❈♦♥❥❡❝t✉r❡

❈♦♥❥❡❝t✉r❡ ✭●✳❆✳ ❋r❡✐♠❛♥✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✸|❙| − ✹,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ✷|❙| − ✸✳ ❈♦♥❥❡❝t✉r❡ ✭●✳❆✳ ❋r❡✐♠❛♥✮ ■❢ ● ✐s ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✹✱ ❛♥❞

|❙✷| ≤ ✷|❙| − ✶ + ❜, ❢♦r ✵ ≤ ❜ ≤ |❙| − ✸,

t❤❡♥ ❙ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st |❙| + ❜✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 46

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛✱ ✶✾✾✽✮ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ |❙| = ❦ ≥ ✹ ❛♥❞

|❙✷| ≤ ✷|❙|,

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ✐❢

❜ = ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 47

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛✱ ✶✾✾✽✮ ■❢ ❙ ✐s ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ |❙| = ❦ ≥ ✹ ❛♥❞

|❙✷| ≤ ✷|❙|,

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ✐❢

❜ = ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 48

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❆✳ ❆❜❞♦❧❧❛❤✐✱ ❋✳ ❏❛❢❛r✐✱ ✷✵✶✾✮ ▲❡t ● ❜❡ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ❛♥❞ ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ❝♦♥t❛✐♥✐♥❣ t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✇✐t❤ |❙| ≥ ✼✳ ■❢

|❙✷| ≤ ✷|❙| + ✶,

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ✐❢

❜ = ✷ ❛♥❞ |❙| ≥ ✼.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 49

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❑✳❏✳ ❇örö❝③❦②✱ P✳P✳ Pá❧❢②✱ ❖✳ ❙❡rr❛✱ ✷✵✶✶✮ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ❛♥❞ |❙| = ❦ ≥ ✻✻✳ ■❢

|❙✷| ≤ ✷|❙| − ✶ + ❜, ✇❤❡r❡ ✵ ≤ ❜ ≤ ✶ ✷|❙|

✶ ✻ − ✷,

t❤❡♥ ❙ ✐s ❛ s✉❜s❡t ♦❢ ❛ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ |❙| + ❜✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ✐❢

❜ ≤ ✶ ✷|❙|

✶ ✻ − ✷

❛♥❞ |❙| ≥ ✻✻.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 50

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❑✳❏✳ ❇örö❝③❦②✱ P✳P✳ Pá❧❢②✱ ❖✳ ❙❡rr❛✱ ✷✵✶✶✮ ▲❡t ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ❛♥❞

|❙| = ❦ ≥ ✻✸✳

■❢

|❙✷| ≤ ✷|❙| − ✶ + ❜, ✇❤❡r❡ ✵ ≤ ❜ ≤ ✶ ✷|❙|

✶ ✸ − ✶

✷

t❤❡♥ ❙ ✐s ❛ s✉❜s❡t ♦❢ ❛ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ |❙| + ❜✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ✐❢

❜ ≤ ✶ ✷|❙|

✶ ✸ − ✶

✷ ❛♥❞ |❙| ≥ ✻✸.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 51

♥✲❊♥❣❡❧ ❣r♦✉♣s

❉❡✜♥✐t✐♦♥

▲❡t ● ❜❡ ❛ ❣r♦✉♣ ❛♥❞ ♥ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳

❲❡ s❛② t❤❛t ● ✐s ❛♥ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐❢ [①,♥ ②] = ✶✱ ❢♦r ❛♥② ①, ② ∈ ● ✱ ✇❤❡r❡ [①,♥ ②] ✐s ❞❡✜♥❡❞ ❜② ♣✉tt✐♥❣

[①,✶ ②] = [①, ②] = ①−✶② −✶①② ✱

❛♥❞✱ ❜② ✐♥❞✉❝t✐♦♥✱ ✐❢ ✐ > ✶✱ [①,✐ ②] = [[①,✐−✶ ②], ②] ✳ ❲❡ ❛❧s♦ ♣✉t [①,✵ ②] = ①✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 52

♥✲❊♥❣❡❧ ❣r♦✉♣s

❉❡✜♥✐t✐♦♥

▲❡t ● ❜❡ ❛ ❣r♦✉♣ ❛♥❞ ♥ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳

❲❡ s❛② t❤❛t ● ✐s ❛♥ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐❢ [①,♥ ②] = ✶✱ ❢♦r ❛♥② ①, ② ∈ ● ✱ ✇❤❡r❡ [①,♥ ②] ✐s ❞❡✜♥❡❞ ❜② ♣✉tt✐♥❣

[①,✶ ②] = [①, ②] = ①−✶② −✶①② ✱

❛♥❞✱ ❜② ✐♥❞✉❝t✐♦♥✱ ✐❢ ✐ > ✶✱ [①,✐ ②] = [[①,✐−✶ ②], ②] ✳ ❲❡ ❛❧s♦ ♣✉t [①,✵ ②] = ①✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 53

♥✲❊♥❣❡❧ ❣r♦✉♣s

❆❜❡❧✐❛♥ ❣r♦✉♣s ❛r❡ ❡①❛❝t❧② t❤❡ ✶✲❊♥❣❡❧ ❣r♦✉♣s✳ ◆✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ ❝❧❛ss ≤ ♥ ❛r❡ ♥✲❊♥❣❡❧ ❣r♦✉♣s✳ ❆ ✜♥✐t❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭▼✳ ❩♦r♥✮✳ ❆ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s♦❧✉❜❧❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭❑✳ ●r✉❡♥❜❡r❣✮✳ ❚❤❡r❡ ❡①✐sts ❛ ♥♦♥✲♥✐❧♣♦t❡♥t ✸✲❊♥❣❡❧ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 54

♥✲❊♥❣❡❧ ❣r♦✉♣s

❆❜❡❧✐❛♥ ❣r♦✉♣s ❛r❡ ❡①❛❝t❧② t❤❡ ✶✲❊♥❣❡❧ ❣r♦✉♣s✳ ◆✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ ❝❧❛ss ≤ ♥ ❛r❡ ♥✲❊♥❣❡❧ ❣r♦✉♣s✳ ❆ ✜♥✐t❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭▼✳ ❩♦r♥✮✳ ❆ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s♦❧✉❜❧❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭❑✳ ●r✉❡♥❜❡r❣✮✳ ❚❤❡r❡ ❡①✐sts ❛ ♥♦♥✲♥✐❧♣♦t❡♥t ✸✲❊♥❣❡❧ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 55

♥✲❊♥❣❡❧ ❣r♦✉♣s

❆❜❡❧✐❛♥ ❣r♦✉♣s ❛r❡ ❡①❛❝t❧② t❤❡ ✶✲❊♥❣❡❧ ❣r♦✉♣s✳ ◆✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ ❝❧❛ss ≤ ♥ ❛r❡ ♥✲❊♥❣❡❧ ❣r♦✉♣s✳ ❆ ✜♥✐t❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭▼✳ ❩♦r♥✮✳ ❆ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s♦❧✉❜❧❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭❑✳ ●r✉❡♥❜❡r❣✮✳ ❚❤❡r❡ ❡①✐sts ❛ ♥♦♥✲♥✐❧♣♦t❡♥t ✸✲❊♥❣❡❧ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 56

♥✲❊♥❣❡❧ ❣r♦✉♣s

❆❜❡❧✐❛♥ ❣r♦✉♣s ❛r❡ ❡①❛❝t❧② t❤❡ ✶✲❊♥❣❡❧ ❣r♦✉♣s✳ ◆✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ ❝❧❛ss ≤ ♥ ❛r❡ ♥✲❊♥❣❡❧ ❣r♦✉♣s✳ ❆ ✜♥✐t❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭▼✳ ❩♦r♥✮✳ ❆ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s♦❧✉❜❧❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭❑✳ ●r✉❡♥❜❡r❣✮✳ ❚❤❡r❡ ❡①✐sts ❛ ♥♦♥✲♥✐❧♣♦t❡♥t ✸✲❊♥❣❡❧ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 57

♥✲❊♥❣❡❧ ❣r♦✉♣s

❆❜❡❧✐❛♥ ❣r♦✉♣s ❛r❡ ❡①❛❝t❧② t❤❡ ✶✲❊♥❣❡❧ ❣r♦✉♣s✳ ◆✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ ❝❧❛ss ≤ ♥ ❛r❡ ♥✲❊♥❣❡❧ ❣r♦✉♣s✳ ❆ ✜♥✐t❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭▼✳ ❩♦r♥✮✳ ❆ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s♦❧✉❜❧❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s ♥✐❧♣♦t❡♥t ✭❑✳ ●r✉❡♥❜❡r❣✮✳ ❚❤❡r❡ ❡①✐sts ❛ ♥♦♥✲♥✐❧♣♦t❡♥t ✸✲❊♥❣❡❧ ❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 58

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ✉♥✐q✉❡ ♣r♦❞✉❝t ♥✲❊♥❣❡❧ ❣r♦✉♣s

❚❤❡♦r❡♠ ❆ ✭P✳ ▲✳✱ ▼✳▼❛❥✱ ✷✵✷✵✮ ▲❡t ● ❜❡ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛♥❞ ❙ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| = ❦ ≥ ✻✷✳ ■❢

|❙✷| ≤ ✷|❙| − ✶ + ❜, ✇❤❡r❡ ✵ ≤ ❜ ≤ ✶ ✷|❙|

✶ ✷

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥ ❛♥❞ ❙ ✐s ❛ s✉❜s❡t ♦❢ ❛ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ |❙| + ❜✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐❢

❜ ≤ ✶ ✷|❙|

✶ ✷

❛♥❞ |❙| ≥ ✻✷.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 59

◗✉❡st✐♦♥s

◗✉❡st✐♦♥

■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈❡ t❤❛t ✐♥ ❚❤❡♦r❡♠ ❆ t❤❡ ❜♦✉♥❞ ❢♦r ❜ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ |❙| ✐s |❙|

✶ ✷❄

◗✉❡st✐♦♥

■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈❡ t❤❛t ✐♥ ❚❤❡♦r❡♠ ❆ t❤❡ ❜♦✉♥❞ ❢♦r ❜ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ |❙| ✐s ❧✐♥❡❛r ❄

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 60

◗✉❡st✐♦♥s

◗✉❡st✐♦♥

■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈❡ t❤❛t ✐♥ ❚❤❡♦r❡♠ ❆ t❤❡ ❜♦✉♥❞ ❢♦r ❜ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ |❙| ✐s |❙|

✶ ✷❄

◗✉❡st✐♦♥

■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈❡ t❤❛t ✐♥ ❚❤❡♦r❡♠ ❆ t❤❡ ❜♦✉♥❞ ❢♦r ❜ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ |❙| ✐s ❧✐♥❡❛r ❄

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 61

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❆✳ ❆❜❞♦❧❧❛❤✐✱ ❋✳ ❏❛❢❛r✐✱ ✷✵✶✾✮ ▲❡t ● ❜❡ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ❛♥❞ ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ❝♦♥t❛✐♥✐♥❣ t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✇✐t❤ |❙| ≥ ✼✳ ■❢

|❙✷| ≤ ✷|❙| + ✶

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ✐❢

❜ = ✷ ❛♥❞ |❙| ≥ ✼.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 62

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣s

❚❤❡♦r❡♠ ✭❆✳ ❆❜❞♦❧❧❛❤✐✱ ❋✳ ❏❛❢❛r✐✱ ✷✵✶✾✮ ▲❡t ● ❜❡ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ❛♥❞ ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ● ❝♦♥t❛✐♥✐♥❣ t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✇✐t❤ |❙| ≥ ✼✳ ■❢

|❙✷| ≤ ✷|❙| + ✶

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ✐❢

❜ = ✷ ❛♥❞ |❙| ≥ ✼.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 63

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♥✲❊♥❣❡❧ ❣r♦✉♣s

❚❤❡♦r❡♠ ❇ ✭P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✷✵✮ ▲❡t ● ❜❡ ❛ r✐❣t❤✲♦r❞❡r❡❞ ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛♥❞ ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| > ✷✵✳ ■❢

|❙✷| ≤ ✷|❙| + ✷

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② r✐❣❤t✲♦r❞❡r❡❞ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐❢

❜ = ✸ ❛♥❞ |❙| > ✷✵.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 64

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♥✲❊♥❣❡❧ ❣r♦✉♣s

❚❤❡♦r❡♠ ❇ ✭P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✷✵✮ ▲❡t ● ❜❡ ❛ r✐❣t❤✲♦r❞❡r❡❞ ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛♥❞ ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| > ✷✵✳ ■❢

|❙✷| ≤ ✷|❙| + ✷

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② r✐❣❤t✲♦r❞❡r❡❞ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐❢

❜ = ✸ ❛♥❞ |❙| > ✷✵.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 65

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♥✲❊♥❣❡❧ ❣r♦✉♣s

❚❤❡♦r❡♠ ❈ ✭P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✷✵✮ ▲❡t ● ❜❡ ❛ r✐❣t❤✲♦r❞❡r❡❞ ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛♥❞ ❙ ❜❡ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ |❙| ≥ ✸✻. ■❢

|❙✷| ≤ ✷|❙| + ✸

t❤❡♥ ❙ ✐s ❛❜❡❧✐❛♥✳ ❍❡♥❝❡ ❋r❡✐♠❛♥✬s ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❛♥② r✐❣❤t✲♦r❞❡r❡❞ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐❢

❜ = ✹ ❛♥❞ |❙| ≥ ✻✷.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 66

◗✉❡st✐♦♥s

◗✉❡st✐♦♥ ✭❲❡❧❧✲❦♥♦✇♥✮

■s ❡✈❡r② r✐❣❤t✲♦r❞❡r❛❜❧❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛ ♥✐❧♣♦t❡♥t ❣r♦✉♣❄

❚❤❡ ❛♥s✇❡r ✐s ❨❊❙ ✐❢ ♥ = ✷, ✸, ✹✳ ❚❤❡ ❛♥s✇❡r ✐s ❨❊❙ ✐❢ ● ✐s ♦r❞❡r❛❜❧❡ ✭❨✳❑✳ ❑✐♠ ❛♥❞ ❆✳❍✳ ❘❤❡♠t✉❧❧❛✮✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 67

◗✉❡st✐♦♥s

◗✉❡st✐♦♥ ✭❆✳■✳ ❑♦❦♦r✐♥✮

■s ❡✈❡r② t♦rs✐♦♥✲❢r❡❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣❄

◗✉❡st✐♦♥

■s ❡✈❡r② t♦rs✐♦♥✲❢r❡❡ ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛ r✐❣❤t✲♦r❞❡r❛❜❧❡ ❣r♦✉♣❄

❚❤❡ ❛♥s✇❡r ✐s ❨❊❙ ✐❢ ♥ = ✷, ✸, ✹✱ s✐♥❝❡ ✐♥ t❤❡s❡ ❝❛s❡s ● ✐s ♥✐❧♣♦t❡♥t✱ ❛♥❞ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ❛r❡ ♦r❞❡r❛❜❧❡✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 68

Pr♦♦❢s

■♥ ♦r❞❡r t♦ ♣r♦✈❡ ❚❤❡♦r❡♠s ❆✱ ❇✱ ❈ ✇❡ ✉s❡ r❡s✉❧ts ❛❜♦✉t

♥✲❊♥❣❡❧ ❣r♦✉♣s✳

▲❡♠♠❛ ✶ ▲❡t ● ❜❡ ❛ t♦rs✐♦♥✲❢r❡❡ ♥✲❊♥❣❡❧ ❣r♦✉♣✱ ①, ② ∈ ●✳ ■❢ [①♠, ②] = ✶, ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ♠✱ t❤❡♥ [①, ②] = ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 69

Pr♦♦❢s

▲❡♠♠❛ ✷ ✭❨✳❑✳ ❑✐♠✱ ❆✳❍✳ ❘❤❡♠t✉❧❧❛✮ ▲❡t ● ❜❡ ❛♥ ♥✲❊♥❣❡❧ ❣r♦✉♣✱ ①, ② ∈ ●✳ ❚❤❡♥ ①② = ①, ①②, ①② ✷, · · · , ①② ♥−✶. ❆♥ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s str♦♥❣❧② r❡str❛✐♥❡❞✳ ▲❡♠♠❛ ✸ ▲❡t ● ❜❡ ❛ t♦rs✐♦♥✲❢r❡❡ ♥✲❊♥❣❡❧ ❣r♦✉♣✱ ①, ② ∈ ●✳ ■❢ ② −✶①α② ∈ ①, ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r α✱ t❤❡♥ ①, ② ✐s ❛❜❡❧✐❛♥.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 70

Pr♦♦❢s

▲❡♠♠❛ ✷ ✭❨✳❑✳ ❑✐♠✱ ❆✳❍✳ ❘❤❡♠t✉❧❧❛✮ ▲❡t ● ❜❡ ❛♥ ♥✲❊♥❣❡❧ ❣r♦✉♣✱ ①, ② ∈ ●✳ ❚❤❡♥ ①② = ①, ①②, ①② ✷, · · · , ①② ♥−✶. ❆♥ ♥✲❊♥❣❡❧ ❣r♦✉♣ ✐s str♦♥❣❧② r❡str❛✐♥❡❞✳ ▲❡♠♠❛ ✸ ▲❡t ● ❜❡ ❛ t♦rs✐♦♥✲❢r❡❡ ♥✲❊♥❣❡❧ ❣r♦✉♣✱ ①, ② ∈ ●✳ ■❢ ② −✶①α② ∈ ①, ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r α✱ t❤❡♥ ①, ② ✐s ❛❜❡❧✐❛♥.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 71

❚❤❡ ✐s♦♣❡r✐♠❡tr✐❝ ♠❡t❤♦❞

■♥ ♦r❞❡r t♦ ♣r♦✈❡ ❚❤❡♦r❡♠ ❆ ✇❡ ❛❧s♦ ✉s❡ ♠❡t❤♦❞s ❢r♦♠ ❆❞❞✐t✐✈❡ ❈♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t ● ❜❡ ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❈ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ ●✱ ♠ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❚❤❡ ♠✲✐s♦♣❡r✐♠❡tr✐❝ ♥✉♠❜❡r ❦♠(❈) ♦❢ ❈ ✐s ❞❡✜♥❡❞ ❜② ❦♠(❈) = ♠✐♥{|❳❈| − |❳| | ❳ ⊆ ●, |❳| ≥ ♠}. ❆ ✜♥✐t❡ s✉❜s❡t ❆ ♦❢ ● ✐s ❛♥ ♠✲❢r❛❣♠❡♥t ❢♦r ❈ ✐❢ |❆| ≥ ♠ ❛♥❞ |❆❈| − |❆| = ❦♠(❈)✳ ❆♥ ♠✲❛t♦♠ ❢♦r ❈ ✐s ❛♥ ♠✲❢r❛❣♠❡♥t ♦❢ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t②✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 72

❚❤❡ ✐s♦♣❡r✐♠❡tr✐❝ ♠❡t❤♦❞

❘❡♠❛r❦ ■❢ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ t❤❡♥ ❛♥② ♠✲❛t♦♠ ❤❛s ❝❛r❞✐♥❛❧✐t② ♠✳ ❈♦♥❥❡❝t✉r❡ ✭❨✳❖✳ ❍❛♠✐❞♦✉♥❡✮ ❆♥② ♠✲❛t♦♠ ✐♥ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ❤❛s ❝❛r❞✐♥❛❧✐t② ♠✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 73

❚❤❡ ✐s♦♣❡r✐♠❡tr✐❝ ♠❡t❤♦❞

❘❡♠❛r❦ ■❢ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ t❤❡♥ ❛♥② ♠✲❛t♦♠ ❤❛s ❝❛r❞✐♥❛❧✐t② ♠✳ ❈♦♥❥❡❝t✉r❡ ✭❨✳❖✳ ❍❛♠✐❞♦✉♥❡✮ ❆♥② ♠✲❛t♦♠ ✐♥ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ❤❛s ❝❛r❞✐♥❛❧✐t② ♠✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 74

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❘❡♠❛r❦ ■❢ ❙, ❚ ❛r❡ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❛❜❡❧✐❛♥ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❤❛✈❡

|❙❚| ≥ |❙| + |❚| − ✶.

❲❡ ❤❛✈❡ t❤❡ ❡q✉❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❙, ❚ ❛r❡ ♣r♦❣r❡ss✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ r❛t✐♦✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 75

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❘❡♠❛r❦ ■❢ ❙, ❚ ❛r❡ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❛❜❡❧✐❛♥ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❤❛✈❡

|❙❚| ≥ |❙| + |❚| − ✶.

❲❡ ❤❛✈❡ t❤❡ ❡q✉❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❙, ❚ ❛r❡ ♣r♦❣r❡ss✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ r❛t✐♦✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 76

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❚❤❡♦r❡♠ ✭❏✳❍✳❇✳ ❑❡♠♣❡r♠❛♥✱ ✶✾✺✻✮ ■❢ ❙, ❚ ❛r❡ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ t❤❡♥ ✇❡ ❤❛✈❡

|❙❚| ≥ |❙| + |❚| − ✶.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-77
SLIDE 77

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❚❤❡♦r❡♠ ✭▲✳ ❇r❛✐❧♦✈s❦②✱ ●✳❆✳ ❋r❡✐♠❛♥✱ ✶✾✾✵✮

■❢ ❙, ❚ ❛r❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ |❙|, |❚| ≥ ✷✱ t❤❡♥ |❙❚| ≥ |❙| + |❚| ✉♥❧❡ss t❤❡r❡ ❛r❡ ❜ ∈ ❙−✶, ❝ ∈ ❚ −✶ s✉❝❤ t❤❛t ❜❙ ❛♥❞ ❚❝ ❛r❡ ♣r♦❣r❡ss✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ r❛t✐♦✳

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ |❙❚| = |❙| + |❚| − ✶✱ t❤❡♥ ❙ ❛♥❞ ❚ ❧✐❡ ✐♥ ❛ ❧❡❢t ❛♥❞ ❛ r✐❣❤t ❝♦s❡t ♦❢ ❛ ❝②❝❧✐❝ s✉❜❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 78

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❚❤❡♦r❡♠ ✭❏✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛✱ ✶✾✾✽✮ ■❢ ❙, ❚ ❛r❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ |❚| ≥ ✹✱ t❤❡♥

|❙❚| ≥ |❙| + |❚| + ✶

✐❢ ❙ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ s♦♠❡ ❧❡❢t ❝♦s❡t ♦❢ ❛ ❝②❝❧✐❝ s✉❜❣r♦✉♣✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 79

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❚❤❡♦r❡♠ ✭❆✳ ❆❜❞♦❧❧❛❤✐✱ ❋✳ ❏❛❢❛r✐✱ ✷✵✶✾✮ ▲❡t ● ❜❡ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣ ❛♥❞ ❧❡t ❙, ❚ ❜❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✱ ✶ ∈ ❙✱ ❙ ♥♦♥✲❛❜❡❧✐❛♥✳ ❚❤❡♥ ❢♦r ❛❧❧ s✉❜s❡ts ❚ ♦❢ ● ✇✐t❤ |❚| ≥ ✼✱

|❙❚| ≥ |❙| + |❚| + ✷.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 80

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❚❤❡♦r❡♠ ✭❑✳❏✳ ❇örö❝③❦②✱ P✳P✳ Pá❧❢②✱ ❖✳ ❙❡rr❛✱ ✷✵✶✶✮ ▲❡t ❙, ❚ ❜❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ ❛♥② t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣ ●✳ ■❢ ❙ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ s♦♠❡ ❧❡❢t ❝♦s❡t ♦❢ ❛ ❝②❝❧✐❝ s✉❜❣r♦✉♣ ♦❢ ●✱ t❤❡♥ ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❦ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❝(❦) s✉❝❤ t❤❛t ✐❢

|❚| > ❝(❦),

t❤❡♥

|❙❚| > |❙| + |❚| + ❦.

❋✉rt❤❡r♠♦r❡

❝(❦) = ✸✷(❦ + ✸)✻.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 81

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❘❡♠❛r❦ ■❢ ● ✐s ❛❜❡❧✐❛♥✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡

❝(❦) = ❦✷ + ✷❦ + ✸.

■❢ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡

❝(❦) = ✹(✷❦ + ✸)✸.

❝(❦) ✐s ❛t ❧❡❛st q✉❛❞r❛t✐❝✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 82

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❘❡♠❛r❦ ■❢ ● ✐s ❛❜❡❧✐❛♥✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡

❝(❦) = ❦✷ + ✷❦ + ✸.

■❢ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡

❝(❦) = ✹(✷❦ + ✸)✸.

❝(❦) ✐s ❛t ❧❡❛st q✉❛❞r❛t✐❝✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 83

❖r❞❡r ♦❢ t❤❡ ♣r♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts

❘❡♠❛r❦ ■❢ ● ✐s ❛❜❡❧✐❛♥✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡

❝(❦) = ❦✷ + ✷❦ + ✸.

■❢ ● ✐s ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ❣r♦✉♣✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡

❝(❦) = ✹(✷❦ + ✸)✸.

❝(❦) ✐s ❛t ❧❡❛st q✉❛❞r❛t✐❝✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 84

Pr♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts ✐♥ ❊♥❣❡❧ ❣r♦✉♣s

❚❤❡♦r❡♠ ❉ ✭P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✷✵✮ ▲❡t ● ❜❡ ❛ t♦rs✐♦♥✲❢r❡❡ ♥✲❊♥❣❡❧ ❛♥❞ ❧❡t ❙, ❚ ❜❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✳ ■❢ ❙ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ s♦♠❡ ❧❡❢t ❝♦s❡t ♦❢ ❛ ❝②❝❧✐❝ s✉❜❣r♦✉♣ ♦❢ ●✱ t❤❡♥ ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❦ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❝(❦) s✉❝❤ t❤❛t ✐❢

|❚| > ❝(❦),

t❤❡♥

|❙❚| > |❙| + |❚| + ❦.

❋✉rt❤❡r♠♦r❡

❝(❦) = ✹(❦ + ✸)✷.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 85

Pr♦❞✉❝t ♦❢ t✇♦ ✜♥✐t❡ s✉❜s❡ts ✐♥ ❊♥❣❡❧ ❣r♦✉♣s

❚❤❡♦r❡♠ ❊ ✭P✳ ▲✳✱ ▼✳ ▼❛❥✱ ✷✵✷✵✮ ▲❡t ● ❜❡ ❛ ✉♥✐q✉❡ ♣r♦❞✉❝t ♥✲❊♥❣❡❧ ❣r♦✉♣ ❛♥❞ ❧❡t ❙, ❚ ❜❡ ✜♥✐t❡ s✉❜s❡ts ♦❢ ●✳ ■❢ ❙ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ s♦♠❡ ❧❡❢t ❝♦s❡t ♦❢ ❛ ❝②❝❧✐❝ s✉❜❣r♦✉♣ ♦❢ ●✱ t❤❡♥ ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❦ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❝(❦) s✉❝❤ t❤❛t ✐❢

|❚| > ❝(❦),

t❤❡♥

|❙❚| > |❙| + |❚| + ❦.

❋✉rt❤❡r♠♦r❡

❝(❦) = ✹❦✷.

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 86

❚❤❛♥❦ ②♦✉ ❢♦r t❤❡ ❛tt❡♥t✐♦♥ !

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 87

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ❉✐♣❛rt✐♠❡♥t♦ ❞✐ ▼❛t❡♠❛t✐❝❛ ❯♥✐✈❡rs✐tà ❞✐ ❙❛❧❡r♥♦ ✈✐❛ ●✐♦✈❛♥♥✐ P❛♦❧♦ ■■✱ ✶✸✷✱ ✽✹✵✽✹ ❋✐s❝✐❛♥♦ ✭❙❛❧❡r♥♦✮✱ ■t❛❧② ❊✲♠❛✐❧ ❛❞❞r❡ss ✿ ♣❧♦♥❣♦❜❛r❞✐❅✉♥✐s❛✳✐t

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 88

❇✐❜❧✐♦❣r❛♣❤②

❆✳❆❜❞♦❧❧❛❤✐✱ ❋✳ ❏❛❢❛r✐✱ ❈❛r❞✐♥❛❧✐t② ♦❢ ♣r♦❞✉❝t s❡ts ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❣r♦✉♣ ❛❧❣❡❜r❛s✱ ❛r❳✐✈✿✶✽✵✽✳✵✽✼✵✽✳ ❨✳ ❇✐❧✉✱ ❙tr✉❝t✉r❡ ♦❢ s❡ts ✇✐t❤ s♠❛❧❧ s✉♠s❡ts✱ ❆stér✐sq✉❡✱ ✷✺✽ ✭✶✾✾✾✮✱ ✼✼✲✶✵✽✳ ❑✳❏✳ ❇örö❝③❦②✱ P✳P✳ Pá❧❢②✱ ❖✳ ❙❡rr❛✱ ❖♥ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ s✉♠s❡ts ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s✱ ❇✉❧❧✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✹✹ ✭✷✵✶✷✮✱ ✶✵✸✹✲✶✵✹✶✳ ▲✳❱✳ ❇r❛✐❧♦✈s❦②✱ ●✳❆✳ ❋r❡✐♠❛♥✱ ❖♥ ❛ ♣r♦❞✉❝t ♦❢ ✜♥✐t❡ s✉❜s❡ts ✐♥ ❛ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣✱ ❏✳ ❆❧❣❡❜r❛✱ ✶✸✵ ✭✶✾✾✵✮✱ ✹✻✷✲✹✼✻✳ ❊✳ ❇r❡✉✐❧❧❛r❞✱ ❇✳ ●r❡❡♥✱ ❚✳ ❚❛♦✱ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ❣r♦✉♣s✱ ❊r❞➤s ❈❡♥t❡♥♥✐❛❧✱ ❇♦❧②❛✐ ❙♦❝✳ ▼❛t❤✳ ❙t✉❞✐❡s✱ ✷✺✱ ✷✵✶✸✳ ❊✳ ❇r❡✉✐❧❧❛r❞✱ ❇✳ ●r❡❡♥ ❆♣♣r♦①✐♠❛t❡ ❣r♦✉♣s ■✿ t❤❡ t♦rs✐♦♥✲❢r❡❡ ♥✐❧♣♦t❡♥t ❝❛s❡✱ ❏✳ ■♥st✳ ▼❛t❤✳ ❏✉ss✐❡✉✱ ✶✵ ✭✷✵✶✶✮✱ ✸✼❄✺✼✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 89

❇✐❜❧✐♦❣r❛♣❤②

❏✳✲▼✳ ❉❡s❤♦✉✐❧❧❡rs✱ ❋✳ ❍❡♥♥❡❝❛rt✱ ❆✳ P❧❛❣♥❡✱ ❖♥ s♠❛❧❧ s✉♠s❡ts ✐♥ (Z/✷Z)♥✱ ❈♦♠❜✐♥❛t♦r✐❝❛✱ ✷✹ ✭✷✵✵✹✮✱ ✺✸✲✻✽✳ ▼✳ ❈❤❛♥❣✱ ❆ ♣♦❧②♥♦♠✐❛❧ ❜♦✉♥❞ ✐♥ ❋r❡✐♠❛♥✬s t❤❡♦r❡♠✱ ❉✉❦❡ ▼❛t❤✳ ❏✳ ✶✶✸✭✸✮ ✭✷✵✵✷✮✱ ✸✾✾✲✹✶✾✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❛ str✉❝t✉r❛❧ t❤❡♦r② ♦❢ s❡t ❛❞❞✐t✐♦♥✳

❚r❛♥s❧❛t✐♦♥s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦♥♦❣r❛♣❤s✱ ✸✼ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘❤♦❞❡ ■s❧❛♥❞✱ ✶✾✼✸✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ❲❤❛t ✐s t❤❡ str✉❝t✉r❡ ♦❢ ❑ ✐❢ ❑ + ❑ ✐s s♠❛❧❧❄

◆✉♠❜❡r ❚❤❡♦r② ✭◆❡✇ ❨♦r❦✱ ✶✾✽✹✲✶✾✽✺✮✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✶✷✹✵ ✭✶✾✽✼✮✱ ✶✵✾✲✶✸✹✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ❖♥ ✜♥✐t❡ s✉❜s❡ts ♦❢ ♥♦♥❛❜❡❧✐❛♥ ❣r♦✉♣s ✇✐t❤ s♠❛❧❧

❞♦✉❜❧✐♥❣✱ Pr♦❝ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✹✵✭✾✮ ✭✷✵✶✷✮✱ ✷✾✾✼✲✸✵✵✷✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 90

❇✐❜❧✐♦❣r❛♣❤②

  • ✳❆✳ ❋r❡✐♠❛♥✱ ❖♥ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ ✜♥✐t❡ s❡ts✳ ■✳✱ ■③✈✳ ❱②ss✳ ❯❝❡❜♥✳

❩❛✈❡❞✳ ▼❛t❡♠❛t✐❦❛✱ ✻ ✭✶✸✮ ✭✶✾✺✾✮✱ ✷✵✷✲✷✶✸✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ■♥✈❡rs❡ ♣r♦❜❧❡♠s ♦❢ ❛❞❞✐t✐✈❡ ♥✉♠❜❡r t❤❡♦r②✳ ■❱✿ ❖♥

t❤❡ ❛❞❞✐t✐♦♥ ♦❢ ✜♥✐t❡ s❡ts✳ ■■✱ ❊❧❛❜✉ˇ ③✳ ●♦s✳ P❡❞✳ ■♥st✳ ❯ˇ ❝❡♥✳ ❩❛♣✳✱ ✽ ✭✶✾✻✵✮✱ ✼✷✲✶✶✻✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ❍✳ ❍❛❧❜❡rst❛♠✱ ■✳❩✳ ❘✉③s❛✱ ■♥t❡❣❡r s✉♠ s❡ts

❝♦♥t❛✐♥✐♥❣ ❧♦♥❣ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✹✻✭✷✮ ✭✶✾✾✷✮✱ ✶✾✸✲✷✵✶✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥

♦r❞❡r❡❞ ❣r♦✉♣s✱ ❏✳ ❆✉str❛❧✐❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✾✻ ✭✷✵✶✹✮✱ ♥♦✳ ✸✱ ✸✶✻✲✸✷✺✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱

❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♦r❞❡r❡❞ ♥✐❧♣♦t❡♥t ❣r♦✉♣s ♦❢ ❝❧❛ss ✷✱ ❊✉r♦♣❡❛♥ ❏✳ ❈♦♠❜✳✱ ✻✼✱ ✽✼✲✾✺✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 91

❇✐❜❧✐♦❣r❛♣❤②

  • ✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱

❉✐r❡❝t ❛♥❞ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ✐♥ ❛❞❞✐t✐✈❡ ♥✉♠❜❡r t❤❡♦r② ❛♥❞ ✐♥ ♥♦♥✲❛❜❡❧✐❛♥ ❣r♦✉♣ t❤❡♦r②✱ ❊✉r♦♣❡❛♥ ❏✳ ❈♦♠❜✐♥✳✱ ✹✵ ✭✷✵✶✹✮✱ ✹✷✲✺✹✳

  • ✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ❆

s♠❛❧❧ ❞♦✉❜❧✐♥❣ str✉❝t✉r❡ t❤❡♦r❡♠ ✐♥ ❛ ❇❛✉♠s❧❛❣✲❙♦❧✐t❛r ❣r♦✉♣✱ ❊✉r♦♣❡❛♥ ❏✳ ❈♦♠❜✐♥✳✱ ✹✹ ✭✷✵✶✺✮✱ ✶✵✻✲✶✷✹✳

  • ✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❆✳ P❧❛❣♥❡✱ ❉✳❏✳❙✳

❘♦❜✐♥s♦♥✱ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ❖♥ t❤❡ str✉❝t✉r❡ ♦❢ s✉❜s❡ts ♦❢ ❛♥ ♦r❞❡r❛❜❧❡ ❣r♦✉♣ ✇✐t❤ s♦♠❡ s♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦♣❡rt✐❡s✱ ❏✳ ❆❧❣❡❜r❛ ✹✹✺✭✶✮ ✭✷✵✶✻✮✱ ✸✵✼✲✸✷✻✳

  • ✳❆✳ ❋r❡✐♠❛♥✱ ▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❆✳ P❧❛❣♥❡✱ ❨✳❱✳

❙t❛♥❝❤❡s❝✉✱ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♦r❞❡r❡❞ ❣r♦✉♣s✿ ●❡♥❡r❛t♦rs ❛♥❞ str✉❝t✉r❡✱ ●r♦✉♣s ●❡♦♠✳ ❉②♥✳ ✶✶✭✷✮ ✭✷✵✶✼✮✱ ✺✽✺✲✻✶✷✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

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SLIDE 92

❇✐❜❧✐♦❣r❛♣❤②

  • ✳❆✳ ❋r❡✐♠❛♥✱ ❇✳▼✳ ❙❝❤❡✐♥✱ ●r♦✉♣ ❛♥❞ s❡♠✐❣r♦✉♣ t❤❡♦r❡t✐❝

❝♦♥s✐❞❡r❛t✐♦♥s ✐♥s♣✐r❡❞ ❜② ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ♦❢ t❤❡ ❛❞❞✐t✐✈❡ ♥✉♠❜❡r t❤❡♦r②✱ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✶✸✷✵✱ ♣♣✳ ✶✷✶✲✶✹✵ ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✱ ✶✾✽✽✮✳ ❘✳❏✳ ●❛r❞✐♥❡r✱ P✳ ●r♦♥❝❤✐✱ ❆ ❇r✉♥♥✲▼✐♥❦♦✇s❦✐ ✐♥❡q✉❛❧✐t② ❢♦r t❤❡ ✐♥t❡❣❡r ❧❛tt✐❝❡✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✺✸ ✭✷✵✵✶✮✱ ✸✾✾✺✲✹✵✷✹✳ ❇✳ ●r❡❡♥✱ ❲❤❛t ✐s ✳✳✳ ❛♥ ❛♣♣r♦①✐♠❛t❡ ❣r♦✉♣❄✱ ◆♦t✐❝❡s ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✺✾ ✭✷✵✶✷✮✱ ♥♦✳ ✺✱ ✻✺✺✲✻✺✻✳ ❇✳❏✳ ●r❡❡♥✱ ■✳❩✳ ❘✉③s❛✱ ❋r❡✐♠❛♥✬s t❤❡♦r❡♠ ✐♥ ❛♥ ❛r❜✐tr❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣✱ ❏✳ ▲♦♥❞✳ ▼❛t❤✳ ❙♦❝✳ ✭✷✮ ✼✺✭✶✮ ✭✷✵✵✼✮✱ ✶✻✸✲✶✼✺✳ ❇✳❏✳ ●r❡❡♥✱ ❚✳❈✳ ❚❛♦✱ ❋r❡✐♠❛♥✬s t❤❡♦r❡♠ ✐♥ ✜♥✐t❡ ✜❡❧❞s ✈✐❛ ❡①tr❡♠❛❧ s❡t t❤❡♦r②✱ ❈♦♠❜✐♥✳ Pr♦❜❛❜✳ ❈♦♠♣✉t✳ ✶✽ ✭✷✵✵✾✮✱ ✸✸✺✲✸✺✺✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-93
SLIDE 93

❇✐❜❧✐♦❣r❛♣❤②

❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❝♦♥♥❡❝t✐✈✐t② t❤❡♦r② ✐♥ ❣r❛♣❤s t♦ ❢❛❝t♦r✐③❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ✐♥ ❣r♦✉♣s✱ ❊✉r♦♣❡❛♥ ❏✳ ❈♦♠❜✐♥✳ ✷ ✭✶✾✽✶✮✱ ✸✹✾✲✸✺✺✳ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❚❤❡ ✐s♦♣❡r✐♠❡tr✐❝ ♠❡t❤♦❞✱ ❈♦♠❜✐♥❛t♦r✐❛❧ ♥✉♠❜❡r t❤❡♦r② ❛❞❞✐t✐✈❡ ❣r♦✉♣ t❤❡♦r②✱ ❆❞✈✳ ❈♦✉rs❡s ▼❛t❤✳ ❈❘▼ ❇❛r❝❡❧♦♥❛✱ ❇✐r❦❤ä✉s❡r ❱❡r❧❛❣✱ ❇❛s❡❧ ✷✵✵✾✱ ✷✹✶✲✷✺✷✳ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❚✇♦ ✐♥✈❡rs❡ r❡s✉❧ts✱ ❈♦♠❜✐♥❛t♦r✐❝❛ ✸✸ ✭✷✵✶✸✮✱ ✷✶✼✲✷✸✵✳ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳❙✳ ▲❧❛❞ó✱ ❖✳ ❙❡rr❛ ❖♥ s✉❜s❡ts ✇✐t❤ s♠❛❧❧ ♣r♦❞✉❝t ✐♥ t♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s✱ ❈♦♠❜✐♥❛t♦r✐❝❛ ✶✽ ✭✶✾✾✽✮✱ ✺✷✾✲✺✹✵✳ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳ P❧❛❣♥❡✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❋r❡✐♠❛♥✬s ✸❦ − ✸ ❚❤❡♦r❡♠✱ ❆❝t❛ ❆r✐t❤✳ ✶✵✸ ✭✷✵✵✷✮✱ ✶✹✼✲✶✺✻✳ ❨✳❖✳ ❍❛♠✐❞♦✉♥❡✱ ❆✳ P❧❛❣♥❡✱ ❆ ♠✉❧t✐♣❧❡ s❡t ✈❡rs✐♦♥ ♦❢ t❤❡ ✸❦ − ✸ t❤❡♦r❡♠✱ ❘❡✈✳ ▼❛t✳ ■❜❡r♦❛♠✳ ✷✶ ✭✷✵✵✺✮✱ ✶✸✸✲✶✻✶✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-94
SLIDE 94

❇✐❜❧✐♦❣r❛♣❤②

▼✳ ❍❡r③♦❣✱ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❙♦♠❡ r❡s✉❧ts ♦♥ ♣r♦❞✉❝ts ♦❢ ✜♥✐t❡ s✉❜s❡ts ✐♥ ❣r♦✉♣s✱ ❈✳▼✳ ❈❛♠♣❜❡❧❧✱ ▼✳❘✳ ◗✉✐❝❦✱ ❊✳❋✳ ❘♦❜❡rts♦♥✱ ❈✳▼✳ ❘♦♥❡②✲❉♦✉❣❛❧✱ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✐❡s ✹✷✷ ✭✷✵✶✺✮ ✲ ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✸✱ ✷✽✻✲✸✵✺✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ■❙❇◆✿✾✼✽✶✶✵✼✺✶✹✺✹✻✳ ❏✳❍✳❇✳ ❑❡♠♣❡r♠❛♥✱ ❖♥ ❝♦♠♣❧❡①❡s ✐♥ ❛ s❡♠✐❣r♦✉♣✱ ■♥❞❛❣✳ ▼❛t❤✳ ✶✽ ✭✶✾✺✻✮✱ ✷✹✼✲✷✺✹✳ ❨✳❑✳ ❑✐♠✱ ❆✳❍✳ ❘❤❡♠t✉❧❧❛✱ ❲❡❛❦ ♠✐♥✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ❛♥❞ ♣♦❧②❝②❝❧✐❝ ❣r♦✉♣s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✷✸ ✭✶✾✾✺✮✱ ✼✶✶✲✼✶✹✳ ❙✳ ❑✐♦♥❦❡✱ ❏✳ ❘❛✐♠❜❛✉❧t✱ ❖♥ ❣❡♦♠❡tr✐❝ ❛s♣❡❝ts ♦❢ ❞✐✛✉s❡ ❣r♦✉♣s✳ ❲✐t❤ ❛♥ ❛♣♣❡♥❞✐① ❜② ◆❛t❤❛♥ ❉✉♥✜❡❧❞✱ ❉♦❝✳ ▼❛t❤✳ ✷✶ ✭✷✵✶✻✮✱ ✽✼✸✲✾✶✺✳ ▼✳ ❑♥❡s❡r✱ ❆❜s❝❤ät③✉♥❣ ❞❡r ❛s②♠♣t♦t✐s❝❤❡♥ ❉✐❝❤t❡ ✈♦♥ ❙✉♠♠❡♥♠❡♥❣❡♥✱ ▼❛t❤✳ ❩✳✱ ✺✽ ✭✶✾✺✸✮✱ ✹✺✾✲✹✽✹✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-95
SLIDE 95

❇✐❜❧✐♦❣r❛♣❤②

❱✳❋✳ ▲❡✈✱ P✳❨✳ ❙♠❡❧✐❛♥s❦②✱ ❖♥ ❛❞❞✐t✐♦♥ ♦❢ t✇♦ ❞✐st✐♥❝t s❡ts ♦❢ ✐♥t❡❣❡rs✱ ❆❝t❛ ❆r✐t❤✳✱ ✼✵ ✭✶✾✾✺✮✱ ♥♦✳ ✶✱ ✽✺✲✾✶✳ P✳ ▲♦♥❣♦❜❛r❞✐✱ ▼✳ ▼❛❥✱ ❙♠❛❧❧ ❞♦✉❜❧✐♥❣ ✐♥ ♠✲❊♥❣❡❧ ❣r♦✉♣s✱ ■♥t✳ ❏✳

  • r♦✉♣ ❚❤❡♦r②✱ ✾ ✭✷✵✷✵✮✱ ♥♦✳ ✹✱ ✷✼✼✲✷✾✶✳

▼✳❇✳ ◆❛t❤❛♥s♦♥✱ ❆❞❞✐t✐✈❡ ◆✉♠❜❡r ❚❤❡♦r②✿ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ t❤❡ ●❡♦♠❡tr② ♦❢ ❙✉♠s❡ts✱ ❙♣r✐♥❣❡r✱ ✶✾✾✻✳ ❊✳ ❘✐♣s✱ ❨✳ ❙❡❣❡✈✱ ❚♦rs✐♦♥✲❢r❡❡ ❣r♦✉♣s ✇✐t❤♦✉t ✉♥✐q✉❡ ♣r♦❞✉❝t ♣r♦♣❡rt②✱ ❏✳ ❆❧❣❡❜r❛ ✶✵✽ ✭✶✾✽✼✮✱ ✶✶✻✲✶✷✻✳ ❉✳❏✳❙✳ ❘♦❜✐♥s♦♥✱ ❆ ❝♦✉rs❡ ✐♥ t❤❡♦r② ♦❢ ❣r♦✉♣s✱ ✷♥❞ ❡❞✳✱ ❙♣r✐♥❣❡r✱ ✶✾✾✻✳ ■✳ ❩✳ ❘✉③s❛✱ ●❡♥❡r❛❧✐③❡❞ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s ❛♥❞ s✉♠s❡ts✱ ❆❝t❛ ▼❛t❤✳ ❍✉♥❣❛r✳ ✻✺✭✹✮ ✭✶✾✾✹✮✱ ✸✼✾✲✸✽✽✳ ■✳❩✳ ❘✉③s❛✱ ❆♥ ❛♥❛❧♦❣ ♦❢ ❋r❡✐♠❛♥✬s t❤❡♦r❡♠ ✐♥ ❣r♦✉♣s✱ ❆stér✐sq✉❡✱ ✷✺✽ ✭✶✾✾✾✮✱ ✸✷✸✲✸✷✻✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-96
SLIDE 96

❇✐❜❧✐♦❣r❛♣❤②

❚✳ ❙❛♥❞❡rs✱ ❆ ♥♦t❡ ♦♥ ❋r❡✐♠❛♥✬s t❤❡♦r❡♠ ✐♥ ✈❡❝t♦r s♣❛❝❡s✱ ❈♦♠❜✐♥✳ Pr♦❜❛❜✳ ❈♦♠♣✉t✳✶✼✭✷✮ ✭✷✵✵✽✮✱ ✷✾✼✲✸✵✺✳ ❚✳ ❙❛♥❞❡rs✱ ❚❤❡ str✉❝t✉r❡ t❤❡♦r② ♦❢ s❡t ❛❞❞✐t✐♦♥ r❡✈✐s✐t❡❞✱ ❇✉❧❧✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✺✵✶ ✭✷✵✶✸✮✱ ♥♦✳ ✶✱ ✾✸✕✶✷✼✳ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ❖♥ ❛❞❞✐t✐♦♥ ♦❢ t✇♦ ❞✐st✐♥❝t s❡ts ♦❢ ✐♥t❡❣❡rs✱ ❆❝t❛ ❆r✐t❤✳ ✼✺ ✭✶✾✾✻✮✱ ♥♦✳ ✷✱ ♣♣✳ ✶✾✶✲✶✾✹✳ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ❖♥ t❤❡ str✉❝t✉r❡ ♦❢ s❡ts ✇✐t❤ s♠❛❧❧ ❞♦✉❜❧✐♥❣ ♣r♦♣❡rt② ♦♥ t❤❡ ♣❧❛♥❡✳ ■✱ ❆❝t❛ ❆r✐t❤✳ ✽✸ ✭✶✾✾✽✮✱ ♥♦✳ ✷✱ ♣♣✳ ✶✷✼✲✶✹✶✳ ❨✳❱✳ ❙t❛♥❝❤❡s❝✉✱ ❚❤❡ str✉❝t✉r❡ ♦❢ ❞✲❞✐♠❡♥s✐♦♥❛❧ s❡ts ✇✐t❤ s♠❛❧❧ s✉♠s❡t✱ ❏✳ ◆✉♠❜❡r ❚❤❡♦r② ✶✸✵ ✭✷✵✶✵✮✱ ♥♦✳ ✷✱ ♣♣✳ ✷✽✾✲✸✵✸✳ ❚✳ ❚❛♦✱ ❱❛♥ ❍✳ ❱✉✱ ❆❞❞✐t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❈❛♠❜r✐❞❣❡ st✉❞✐❡s ✐♥ ❛❞✈❛♥❝❡❞ ♠❛t❤❡♠❛t✐❝s ✶✵✺✱ ✷✵✵✻✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙

slide-97
SLIDE 97

❇✐❜❧✐♦❣r❛♣❤②

❚✳ ❚❛♦✱ Pr♦❞✉❝t s❡t ❡st✐♠❛t❡s ❢♦r ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❣r♦✉♣s✱ ❈♦♠❜✐♥❛t♦r✐❝❛ ✷✽ ✭✷✵✵✽✮✱ ♥♦✳ ✺✱ ♣♣✳ ✺✹✼✲✺✾✹✳ ❚✳❈✳ ❚❛♦✱ Pr♦❞✉❝t s❡t ❡st✐♠❛t❡s ❢♦r ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❣r♦✉♣s✱ ❈♦♠❜✐♥❛t♦r✐❝❛ ✷✽✭✺✮ ✭✷✵✵✽✮✱ ✺✹✼✲✺✾✹✳ ❚✳❈✳ ❚❛♦✱ ❆♥ ❡❧❡♠❡♥t❛r② ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❋r❡✐♠❛♥ t❤❡♦r❡♠✱ ❤tt♣✿✴✴t❡rr②t❛♦✳✇♦r❞♣r❡ss✳❝♦♠✴✷✵✵✾✴✶✶✴✶✵✴❛♥✲❡❧❡♠❡♥t❛r②✲♥♦♥✲ ❝♦♠♠✉t❛t✐✈❡ ✲❢r❡✐♠❛♥✲t❤❡♦r❡♠ ✭✷✵✵✾✮✱ ✶✲✻✳ ❚✳❈✳ ❚❛♦✱ ❍❛♠✐❞♦✉♥❡✬s ❋r❡✐♠❛♥✲❑♥❡ss❡r t❤❡♦r❡♠ ❢♦r ♥♦♥❛❜❡❧✐❛♥ ❣r♦✉♣s✱ ❤tt♣✿✴✴t❡rr②t❛♦✳✇♦r❞♣r❡ss✳❝♦♠✴✷✵✶✶✴✵✸✴✶✷✴❤❛♠✐❞♦✉♥❡s✲ ❢r❡✐♠❛♥✲❦♥❡s❡✲t❤❡♦r❡♠✲❢♦r✲♥♦♥❛❜❡❧✐❛♥✲❣r♦✉♣s ✭✷✵✶✶✮✱ ✶✲✽✳

P❛tr✐③✐❛ ▲♦♥❣♦❜❛r❞✐ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧❡r♥♦ ❙▼❆▲▲ ❉❖❯❇▲■◆● ■◆ ♥✲❊◆●❊▲ ●❘❖❯P❙