SLIDE 1
❙t❛r✜s❤ ♦♥ str✐❦❡
❉❛♥✐❡❧ ❏✳ ❇❡r♥st❡✐♥✱ P❡t❡r ❇✐r❦♥❡r✱ ❚❛♥❥❛ ▲❛♥❣❡ ✷✵✶✵✳✵✽✳✵✾
SLIDE 2 ❊❞✇❛r❞s ❝✉r✈❡s
❆♥ ❊❞✇❛r❞s ❝✉r✈❡ ♦✈❡r ❛ ♥♦♥✲❜✐♥❛r② ✜❡❧❞ ❦ ✐s ❛ ❝✉r✈❡ ①✷ + ② ✷ = ✶ + ❞①✷② ✷ ✇✐t❤ ❞ / ∈ {✵, ✶}✳ ❊❞✇❛r❞s ❛❞❞✐t✐♦♥ ❧❛✇✿ (①✶, ②✶) + (①✷, ②✷) = (①✸, ②✸) ✇✐t❤ ①✸ = ①✶②✷ + ②✶①✷ ✶ + ❞①✶①✷②✶②✷ , ②✸ = ②✶②✷ − ①✶①✷ ✶ − ❞①✶①✷②✶②✷ . ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ (✵, ✶)✳ ◆❡❣❛t✐♦♥✿ −(①✶, ②✶) = (−①✶, ②✶)✳ ❯s❡ ♣r♦❥❡❝t✐✈❡ r❡♣r❡s❡♥t❛t✐♦♥ t♦ ❛✈♦✐❞ ❞✐✈✐s✐♦♥s✿
◮ (❳✶ : ❨✶ : ❩✶) r❡♣r❡s❡♥ts (❳✶/❩✶, ❨✶/❩✶)✳ ◮ ❆❞❞✐t✐♦♥ ❝♦sts ✶✵▼ + ✶❙ + ✶▼❞✳ ◮ ❉♦✉❜❧✐♥❣ ❝♦sts ✸▼ + ✹❙✳
SLIDE 3 ❊①❛♠♣❧❡✿ ①✷ + ② ✷ = ✶ − ✸✵①✷② ✷
② ①
- ♥❡✉tr❛❧ = (✵, ✶)
- (✵, −✶) ♦❢ ♦r❞❡r ✷ •
(−✶, ✵) ♦❢ ♦r❞❡r ✹ • P✶ = (①✶, ②✶)
- P✷ = (①✷, ②✷)
- P✸ = (①✸, ②✸)
- ❈♦♠♣❛r❡ t♦ st❛♥❞❛r❞ ❏❛❝♦❜✐❛♥ ❱ ✷ = ❯✸ − ✸❯❲ ✹ + ❜❲ ✻✿
◮ ❆❞❞✐t✐♦♥ ✶✶▼ + ✺❙✳ ❊❞✇❛r❞s s❛✈❡s ✹❙ + ✶▼ − ✶▼❞✳ ◮ ❉♦✉❜❧✐♥❣ ✸▼ + ✺❙✳ ❊❞✇❛r❞s s❛✈❡s ✶❙✳
SLIDE 4 ❊①❛♠♣❧❡✿ ①✷ + ② ✷ = ✶ − ✸✵①✷② ✷
② ① ♥❡✉tr❛❧ ✵ ✶ ✵ ✶ ♦❢ ♦r❞❡r ✷ ✶ ✵ ♦❢ ♦r❞❡r ✹ P✶ ①✶ ②✶ P✷ ①✷ ②✷ P✸ ①✸ ②✸ ❈♦♠♣❛r❡ t♦ st❛♥❞❛r❞ ❏❛❝♦❜✐❛♥ ❱ ✷ = ❯✸ − ✸❯❲ ✹ + ❜❲ ✻✿
◮ ❆❞❞✐t✐♦♥ ✶✶▼ + ✺❙✳ ❊❞✇❛r❞s s❛✈❡s ✹❙ + ✶▼ − ✶▼❞✳ ◮ ❉♦✉❜❧✐♥❣ ✸▼ + ✺❙✳ ❊❞✇❛r❞s s❛✈❡s ✶❙✳
SLIDE 5 ❚✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s
✷✵✵✽ ❇❡r♥st❡✐♥✕❇✐r❦♥❡r✕❏♦②❡✕▲❛♥❣❡✕P❡t❡rs✿
- ❡♥❡r❛❧✐③❡ t♦ ✏t✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s✑
❛①✷ + ② ✷ = ✶ + ❞①✷② ✷ ✇✐t❤ ❛ = ✵, ❞ = ✵, ❛ = ❞✳ ❆❞❞✐t✐♦♥ ❧❛✇✿ (①✶, ②✶) + (①✷, ②✷) = (①✸, ②✸) ✇✐t❤ ①✸ = ①✶②✷ + ②✶①✷ ✶ + ❞①✶①✷②✶②✷ , ②✸ = ②✶②✷ − ❛①✶①✷ ✶ − ❞①✶①✷②✶②✷ . ❆❞✈❛♥t❛❣❡s✿
◮ ▼♦r❡ ✢❡①✐❜❧❡✿ ♥♦t ♥❡❝❡ss❛r✐❧② ❛ ♣♦✐♥t ♦❢ ♦r❞❡r ✹✳ ◮ ❈♦✈❡rs ❛❧❧ ▼♦♥t❣♦♠❡r② ❝✉r✈❡s✳ ◮ ❈♦✈❡rs ❡✈❡♥ ♠♦r❡ ❝✉r✈❡s ✇✐t❤ ❛ ✷✲✐s♦❣❡♥②✳ ◮ ❙❛✈❡s t✐♠❡ ✇❤❡♥ ❞ ✐s ❛ r❛t✐♦ ♦❢ s♠❛❧❧ ✐♥t❡❣❡rs✳
SLIDE 6
❙♣❡❡❞ ♦❢ t✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s
Pr♦❥❡❝t✐✈❡ ❛❞❞✐t✐♦♥✿ ✶✵▼ + ✶❙ + ✶▼❞ + ✶▼❛✳ Pr♦❥❡❝t✐✈❡ ❞♦✉❜❧✐♥❣✿ ✸▼ + ✹❙ + ✶▼❛✳ ✷✵✵✽ ❍✐s✐❧✕❲♦♥❣✕❈❛rt❡r✕❉❛✇s♦♥✿ ❞✉❛❧ ❛❞❞✐t✐♦♥ ❧❛✇ ①✶ ②✶ ①✷ ②✷ ①✶②✶ ①✷②✷ ❛①✶①✷ ②✶②✷ ①✶②✶ ①✷②✷ ①✶②✷ ②✶①✷ ❡①t❡♥❞❡❞ ❝♦♦r❞✐♥❛t❡s ❳ ❨ ❩ ❚ ✇✐t❤ ❚ ❳❨ ❩❀ ❜♦✉♥❝✐♥❣ ❜❡t✇❡❡♥ ♣r♦❥❡❝t✐✈❡ ❛♥❞ ❡①t❡♥❞❡❞ ❝♦♦r❞✐♥❛t❡s✳ ❆❞❞✐t✐♦♥✿ ✾▼ ✶▼❛✳ ❖♥❧② ✽▼ ❢♦r ❛ ✶✳ ❉♦✉❜❧✐♥❣✿ ✸▼ ✹❙ ✶▼❛✳ ◆♦t❡ t❤❡ ❛❞❞✐t✐♦♥ s♣❡❡❞✉♣ ❢♦r ❛ ✶✳
SLIDE 7 ❙♣❡❡❞ ♦❢ t✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s
Pr♦❥❡❝t✐✈❡ ❛❞❞✐t✐♦♥✿ ✶✵▼ + ✶❙ + ✶▼❞ + ✶▼❛✳ Pr♦❥❡❝t✐✈❡ ❞♦✉❜❧✐♥❣✿ ✸▼ + ✹❙ + ✶▼❛✳ ✷✵✵✽ ❍✐s✐❧✕❲♦♥❣✕❈❛rt❡r✕❉❛✇s♦♥✿ ❞✉❛❧ ❛❞❞✐t✐♦♥ ❧❛✇ (①✶, ②✶) + (①✷, ②✷) = ①✶②✶ + ①✷②✷ ❛①✶①✷ + ②✶②✷ , ①✶②✶ − ①✷②✷ ①✶②✷ − ②✶①✷
❡①t❡♥❞❡❞ ❝♦♦r❞✐♥❛t❡s (❳ : ❨ : ❩ : ❚) ✇✐t❤ ❚ = ❳❨ /❩❀ ❜♦✉♥❝✐♥❣ ❜❡t✇❡❡♥ ♣r♦❥❡❝t✐✈❡ ❛♥❞ ❡①t❡♥❞❡❞ ❝♦♦r❞✐♥❛t❡s✳ ❆❞❞✐t✐♦♥✿ ✾▼ + ✶▼❛✳ ❖♥❧② ✽▼ ❢♦r ❛ = −✶✳ ❉♦✉❜❧✐♥❣✿ ✸▼ + ✹❙ + ✶▼❛✳ ◆♦t❡ t❤❡ ❛❞❞✐t✐♦♥ s♣❡❡❞✉♣ ❢♦r ❛ = −✶✳
SLIDE 8 ❚❤❡ ♣ − ✶ ♠❡t❤♦❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥
✷✷✸✷✼✾✷✺✻✵ − ✶ ❤❛s ♣r✐♠❡ ❞✐✈✐s♦rs ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ✶✼✱ ✶✾✱ ✷✸✱ ✷✾✱ ✸✶✱ ✸✼✱ ✹✶✱ ✹✸✱ ✺✸✱ ✻✶✱ ✻✼✱ ✼✶✱ ✼✸✱ ✼✾✱ ✽✾✱ ✾✼✱ ✶✵✸✱ ✶✵✾✱ ✶✶✸✱ ✶✷✼✱ ✶✸✶✱ ✶✸✼✱ ✶✺✶✱ ✶✺✼✱ ✶✽✶✱ ✶✾✶✱ ✶✾✾✱ ❡t❝✳ ❆♥ ♦❞❞ ♣r✐♠❡ ♣ ❞✐✈✐❞❡s ✷✷✸✷✼✾✷✺✻✵ − ✶ ✐✛ ♦r❞❡r ♦❢ ✷ ✐♥ ❋∗
♣ ❞✐✈✐❞❡s ✷✸✷✼✾✷✺✻✵✳
▼❛♥② ✇❛②s ❢♦r t❤✐s t♦ ❤❛♣♣❡♥✿ ✷✸✷✼✾✷✺✻✵ ❤❛s ✾✻✵ ❞✐✈✐s♦rs✳ ❲❤② s♦ ♠❛♥②❄ ✷✸✷✼✾✷✺✻✵ = ✷✹ · ✸✷ · ✺ · ✼ · ✶✶ · ✶✸ · ✶✼ · ✶✾✳ ❈♦♠♣✉t❡ ❣❝❞{✷✷✸✷✼✾✷✺✻✵ − ✶, ♥} t♦ ♦❜t❛✐♥ t❤❡ ♣r♦❞✉❝t ♦❢ ❛❧❧ ♣r✐♠❡s ♣ ❞✐✈✐❞✐♥❣ ♥ s✉❝❤ t❤❛t t❤❡ ♦r❞❡r ♦❢ ✷ ✐♥ ❋∗
♣ ❞✐✈✐❞❡s ✷✸✷✼✾✷✺✻✵✳
✭❆ss✉♠✐♥❣ ♥ sq✉❛r❡❢r❡❡ ❢♦r s✐♠♣❧✐❝✐t②✳✮
SLIDE 9
❊❈▼✿ t❤❡ ❡❧❧✐♣t✐❝✲❝✉r✈❡ ♠❡t❤♦❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥
❚❛❦❡ ❛ ❝✉r✈❡ ♦✈❡r ◗ ✇✐t❤ ❛ ♣♦✐♥t ●✳ ❊✳❣✳ ❚❛❦❡ ❛♥② ●✱ ❝♦♠♣✉t❡ ❊❞✇❛r❞s ❞ ❢r♦♠ ●✳ ❘❡❞✉❝❡ t❤✐s ❝✉r✈❡ ♠♦❞✉❧♦ ♥✳ ✭❈❛♥ r❡❝②❝❧❡ ❝✉r✈❡ ❢♦r ♠❛♥② ❞✐✛❡r❡♥t ♥❀ ❊❈▼ ✐s ❣r❡❡♥✳✮ ❈♦♠♣✉t❡ [s]● ♠♦❞✉❧♦ ♥ ❢♦r s♦♠❡ ✈❡r② s♠♦♦t❤ s✳ ■❢ t❤❡ ♦r❞❡r ♦❢ ● ✐♥ ❊(❋♣) ❞✐✈✐❞❡s s t❤❡♥ t❤❡ ♣♦✐♥t [s]● ✐s t❤❡ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ♠♦❞✉❧♦ ♣✳ ❉❡t❡❝t ❜② ❛ s✉✐t❛❜❧❡ ❣❝❞ ❝♦♠♣✉t❛t✐♦♥✳ ❊ ♠♦❞✉❧♦ ♣ ❤❛s ♦r❞❡r ✐♥ [♣ + ✶ − ✷√♣, ♣ + ✶ + ✷√♣]❀ t❤✐s ♠❛② ♦r ♠❛② ♥♦t ❜❡ s♠♦♦t❤ ❜✉t ✇❡ ❝❛♥ ✈❛r② ❊✳ ❈✉r✈❡ ♦♣❡r❛t✐♦♥s ♠♦r❡ ❡①♣❡♥s✐✈❡ t❤❛♥ ✐♥ ♣ − ✶ ♠❡t❤♦❞❀ ❜✉t ✈❛r②✐♥❣ ❝✉r✈❡s ♠❛❦❡s ♠❡t❤♦❞ ♠✉❝❤ ❢❛st❡r ♦✈❡r❛❧❧✳
SLIDE 10 ❚♦rs✐♦♥ ♣♦✐♥ts
❈✉r✈❡ ♦✈❡r ◗ ❤❛s s♦♠❡ t♦rs✐♦♥ ♣♦✐♥ts✿ ♣♦✐♥ts ♦❢ ✜♥✐t❡ ♦r❞❡r✳ ❆❧❧ ♣♦ss✐❜❧❡ t♦rs✐♦♥ ❣r♦✉♣s ✭▼❛③✉r✬s t❤❡♦r❡♠✮✿
◮ ❩/♠ ❢♦r ♠ ∈ {✶, ✷, ✸, ✹, ✺, ✻, ✼, ✽, ✾, ✶✵, ✶✷}✱ ◮ ❩/✷ × ❩/✷♠ ❢♦r ♠ ∈ {✶, ✷, ✸, ✹}✳
■❢ ❛ ♣♦✐♥t ❤❛s ✜♥✐t❡ ♦r❞❡r ♦♥ t❤❡ ❝✉r✈❡ ♦✈❡r ◗ t❤❡♥ t❤❡ ♣♦✐♥t ❤❛s t❤❡ s❛♠❡ ✜♥✐t❡ ♦r❞❡r ♦✈❡r ❩/♥ ❛♥❞ ♦✈❡r ❋♣✳ ❉♦♥✬t ❝❤♦♦s❡ ● ❛s ❛ t♦rs✐♦♥ ♣♦✐♥t✳ ▼✐♥✐♠✐③❡ tr♦✉❜❧❡ ❜② ❝❤♦♦s✐♥❣ ❝✉r✈❡ ✇✐t❤ t♦rs✐♦♥ ❩/✶❄ ◆♦✿ ♣❡♦♣❧❡ tr② t♦ ✉s❡ ❝✉r✈❡s ✇✐t❤ ♠❛♥② t♦rs✐♦♥ ♣♦✐♥ts✳ ❡✳❣✳ ✷✵✵✽✕✷✵✶✵ ❇❡r♥st❡✐♥✕❇✐r❦♥❡r✕▲❛♥❣❡✕P❡t❡rs ✏❊❈▼ ✉s✐♥❣ ❊❞✇❛r❞s ❝✉r✈❡s✑ ✭s♦❢t✇❛r❡✿ ✏❊❊❈▼✲▼P❋◗✑✮ s❛✈❡ t✐♠❡ ✐♥ ❊❈▼ ❜② ✉s✐♥❣ ❊❞✇❛r❞s ❝✉r✈❡s❀ ❝♦♥str✉❝t ❢❛♠✐❧✐❡s ♦❢ ❊❞✇❛r❞s ❝✉r✈❡s ✇✐t❤ t♦rs✐♦♥ ❩/✶✷ ♦r ❩/✷ × ❩/✽✳
SLIDE 11 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪ Pr❬♣r✐♠❡ ♣ ✶ ❘ ❤❛s s♠♦♦t❤
❪ Pr❬♣r✐♠❡ ♣ ✶ ❘ ❤❛s s♠♦♦t❤ ❊ ❋♣ ❪ Pr❬✐♥t❡❣❡r ✶ ❘ ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ✶ ❘ ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪ Pr❬✐♥t❡❣❡r t❩ ✶ ❘ ✐s s♠♦♦t❤❪ Pr❬✐♥t❡❣❡r ❩ ✶ ❘ t ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 12 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪ Pr❬♣r✐♠❡ ♣ ✶ ❘ ❤❛s s♠♦♦t❤ ❊ ❋♣ ❪ Pr❬✐♥t❡❣❡r ✶ ❘ ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ✶ ❘ ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪ Pr❬✐♥t❡❣❡r t❩ ✶ ❘ ✐s s♠♦♦t❤❪ Pr❬✐♥t❡❣❡r ❩ ✶ ❘ t ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 13 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #❊(❋♣)❪ Pr❬✐♥t❡❣❡r ✶ ❘ ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ✶ ❘ ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪ Pr❬✐♥t❡❣❡r t❩ ✶ ❘ ✐s s♠♦♦t❤❪ Pr❬✐♥t❡❣❡r ❩ ✶ ❘ t ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 14 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #❊(❋♣)❪
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≈ Pr❬✐♥t❡❣❡r ∈ [✶, ❘] ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ✶ ❘ ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪ Pr❬✐♥t❡❣❡r t❩ ✶ ❘ ✐s s♠♦♦t❤❪ Pr❬✐♥t❡❣❡r ❩ ✶ ❘ t ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 15 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪
?
≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #❊(❋♣)❪
?
≈ Pr❬✐♥t❡❣❡r ∈ [✶, ❘] ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪ Pr❬✐♥t❡❣❡r t❩ ✶ ❘ ✐s s♠♦♦t❤❪ Pr❬✐♥t❡❣❡r ❩ ✶ ❘ t ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 16 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #❊(❋♣)❪
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≈ Pr❬✐♥t❡❣❡r ∈ [✶, ❘] ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬✐♥t❡❣❡r ∈ t❩ ∩ [✶, ❘] ✐s s♠♦♦t❤❪ Pr❬✐♥t❡❣❡r ❩ ✶ ❘ t ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 17 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #❊(❋♣)❪
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≈ Pr❬✐♥t❡❣❡r ∈ [✶, ❘] ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬✐♥t❡❣❡r ∈ t❩ ∩ [✶, ❘] ✐s s♠♦♦t❤❪
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≈ Pr❬✐♥t❡❣❡r ∈ ❩ ∩ [✶, ❘/t] ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t s♠❛❧❧❡r ❘ t ❧❛r❣❡r Pr✳
SLIDE 18 ❲❤② ♣❡♦♣❧❡ ✇❛♥t ❜✐❣ t♦rs✐♦♥
❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ✏r❛♥❞♦♠✑ ❡❧❧✐♣t✐❝ ❝✉r✈❡✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #● ✐♥ ❊(❋♣)❪
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≈ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ❤❛s s♠♦♦t❤ #❊(❋♣)❪
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≈ Pr❬✐♥t❡❣❡r ∈ [✶, ❘] ✐s s♠♦♦t❤❪✳ ❙t❛♥❞❛r❞ s❡r✐❡s ♦❢ ❤❡✉r✐st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s ✇❤❡♥ ❊❈▼ ✉s❡s ❛ ❝✉r✈❡ ❦♥♦✇♥ t♦ ❤❛✈❡ t t♦rs✐♦♥ ♣♦✐♥ts✿ Pr❬♣r✐♠❡ ♣ ∈ [✶, ❘] ✐s ❢♦✉♥❞ ❜② t❤✐s ❝✉r✈❡❪
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≈ Pr❬✐♥t❡❣❡r ∈ t❩ ∩ [✶, ❘] ✐s s♠♦♦t❤❪
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≈ Pr❬✐♥t❡❣❡r ∈ ❩ ∩ [✶, ❘/t] ✐s s♠♦♦t❤❪✳ ▲❛r❣❡r t ⇒ s♠❛❧❧❡r ❘/t ⇒ ❧❛r❣❡r Pr✳
SLIDE 19
❙t❛r✜s❤ ♦♥ str✐❦❡
❲♦✉❧❞ ❧✐❦❡ t♦ ❝♦♠❜✐♥❡ ❛ = −✶ s♣❡❡❞✉♣ ✇✐t❤ ❧❛r❣❡ t♦rs✐♦♥✳ ▲❡t✬s ❧♦♦❦ ❝❧♦s❡r ❛t −①✷ + ② ✷ = ✶ − ✸✵①✷② ✷✿ ❙✐♥❣✉❧❛r✐t② ❛t ✐♥✜♥✐t② ❜❧♦✇s ✉♣ t♦ t✇♦ ♣♦✐♥ts ♦❢ ♦r❞❡r ✷✳ ❊❊❈▼ ♣❛♣❡r ♣r♦✈❡❞✿ ❛r❜✐tr❛r② ❞ ✇✐t❤ ❛ = −✶ ❝❛♥♥♦t ❛❝❤✐❡✈❡ ❤✐❣❤❡st t♦rs✐♦♥ s✉❝❤ ❛s ❩/✶✷ ❛♥❞ ❩/✷ × ❩/✽✳ ❚❤✐s ♣❛♣❡r✿ ❉♦❡s ❛ ✶ s♣❡❡❞✉♣ ♦✉t✇❡✐❣❤ s♠❛❧❧❡r t♦rs✐♦♥❄
SLIDE 20
❙t❛r✜s❤ ♦♥ str✐❦❡
❲♦✉❧❞ ❧✐❦❡ t♦ ❝♦♠❜✐♥❡ ❛ = −✶ s♣❡❡❞✉♣ ✇✐t❤ ❧❛r❣❡ t♦rs✐♦♥✳ ▲❡t✬s ❧♦♦❦ ❝❧♦s❡r ❛t −①✷ + ② ✷ = ✶ − ✸✵①✷② ✷✿ ❙✐♥❣✉❧❛r✐t② ❛t ✐♥✜♥✐t② ❜❧♦✇s ✉♣ t♦ t✇♦ ♣♦✐♥ts ♦❢ ♦r❞❡r ✷✳ ❊❊❈▼ ♣❛♣❡r ♣r♦✈❡❞✿ ❛r❜✐tr❛r② ❞ ✇✐t❤ ❛ = −✶ ❝❛♥♥♦t ❛❝❤✐❡✈❡ ❤✐❣❤❡st t♦rs✐♦♥ s✉❝❤ ❛s ❩/✶✷ ❛♥❞ ❩/✷ × ❩/✽✳ ❚❤✐s ♣❛♣❡r✿ ❉♦❡s ❛ ✶ s♣❡❡❞✉♣ ♦✉t✇❡✐❣❤ s♠❛❧❧❡r t♦rs✐♦♥❄
SLIDE 21
❙t❛r✜s❤ ♦♥ str✐❦❡
❲♦✉❧❞ ❧✐❦❡ t♦ ❝♦♠❜✐♥❡ ❛ = −✶ s♣❡❡❞✉♣ ✇✐t❤ ❧❛r❣❡ t♦rs✐♦♥✳ ▲❡t✬s ❧♦♦❦ ❝❧♦s❡r ❛t −①✷ + ② ✷ = ✶ − ✸✵①✷② ✷✿ ❙✐♥❣✉❧❛r✐t② ❛t ✐♥✜♥✐t② ❜❧♦✇s ✉♣ t♦ t✇♦ ♣♦✐♥ts ♦❢ ♦r❞❡r ✷✳ ❊❊❈▼ ♣❛♣❡r ♣r♦✈❡❞✿ ❛r❜✐tr❛r② ❞ ✇✐t❤ ❛ = −✶ ❝❛♥♥♦t ❛❝❤✐❡✈❡ ❤✐❣❤❡st t♦rs✐♦♥ s✉❝❤ ❛s ❩/✶✷ ❛♥❞ ❩/✷ × ❩/✽✳ ❚❤✐s ♣❛♣❡r✿ ❉♦❡s ❛ = −✶ s♣❡❡❞✉♣ ♦✉t✇❡✐❣❤ s♠❛❧❧❡r t♦rs✐♦♥❄
SLIDE 22
❊①❛♠♣❧❡✿ ❈♦♥str✉❝t✐♥❣ ❩/✽ ✇✐t❤ ❛ = −✶
❚✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡ ❤❛s ✶ ❛✣♥❡ ♣♦✐♥t ♦❢ ♦r❞❡r ✷ ❛t (✵, −✶)✳ P♦✐♥ts ♦❢ ♦r❞❡r ✹ ❞♦✉❜❧✐♥❣ t♦ (✵, −✶) ❡①✐st ✐✛ ❛ = ♦r ❞ = ✳ ❋♦r ❛ = −✶ ♦♥❧② ♣♦ss✐❜✐❧✐t② ✐s ❞ = ✳ ▼❛♥② ♠♦r❡ r❡str✐❝t✐♦♥s ♦♥ ❞ t♦ ❝♦♥str✉❝t ♣♦✐♥ts ♦❢ ♦r❞❡r ✽ ❛♥❞ ♥♦♥✲t♦rs✐♦♥ ♣♦✐♥t ●✳ ❘❡s✉❧t✐♥❣ t❤❡♦r❡♠✿ ▲❡t (r, s) ❜❡ ❛ r❛t✐♦♥❛❧ ♣♦✐♥t ✇✐t❤ r, s = ✵ ❛♥❞ s = ±✹r ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❙ ✷ = ❘✸ + ✹✽❘ ♦✈❡r ◗✳ ❉❡✜♥❡ ✉ = ✷r/s✱ ✈ = (✷r ✸ − s✷)/s✷✱ ❞ = (✶✻✉✹)/(✹✉✹ − ✶)✷✳ ❚❤❡♥ t❤❡ t✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡ −①✷ + ② ✷ = ✶ + ❞①✷② ✷ ❤❛s t♦rs✐♦♥ ❣r♦✉♣ ❩/✽ ❛♥❞ ♥♦♥✲t♦rs✐♦♥ ♣♦✐♥t (✷✉✷, (✹✉✹ − ✶)/✈)✳ ❈❛♥ t❛❦❡ (✹, −✶✻) ❛♥❞ ✐ts ♠✉❧t✐♣❧❡s ❛s ✈❛❧✉❡s ❢♦r (r, s)✳ P❛♣❡r ❤❛s s✐♠✐❧❛r ❝♦♥str✉❝t✐♦♥s ❢♦r ❩/✷ × ❩/✹ ❛♥❞ ❩/✻❀ ♥♦t❡ t❤❛t t♦rs✐♦♥ ❣r♦✉♣s ❛r❡ ❤❛❧❢✲s✐③❡ ❝♦♠♣❛r❡❞ t♦ ❊❊❈▼ ♣❛♣❡r✳
SLIDE 23 ❊①♣❡r✐♠❡♥ts ❛♥❞ ❝♦♥❝❧✉s✐♦♥s
❲❡ ♠♦❞✐✜❡❞ ❊❊❈▼✲▼P❋◗ t♦ s✉♣♣♦rt ♦✉r ♥❡✇ ❝✉r✈❡s✳ ❊❊❈▼ ♣❛♣❡r ♦♣t✐♠✐③❡❞ ❊❊❈▼✲▼P❋◗ s ❡t❝✳ ❢♦r ♦❧❞ ❝✉r✈❡s✳ ❲❡ tr✐❡❞ t❤♦s❡ ♣❛r❛♠❡t❡rs ✇✐t❤ t❤♦✉s❛♥❞s ♦❢ ♦✉r ❝✉r✈❡s ❢♦r ❛❧❧ ❜✲❜✐t ♣r✐♠❡s ❢♦r ❡❛❝❤ ❜ ∈ {✶✺, ✶✻, . . . , ✷✻}✳ ❍❛♣♣② ♦❜s❡r✈❛t✐♦♥✿ ❖✉r ♥❡✇ ❩ ✻ ❛ ✶ ❝✉r✈❡s ❛r❡ t❤❡ ♥❡✇ ♣r✐❝❡✴♣❡r❢♦r♠❛♥❝❡ ❧❡❛❞❡rs ❢♦r ❊❈▼✦ ❊❊❈▼✲▼P❋◗ ♥♦✇ ✉s❡s t❤❡s❡ ❝✉r✈❡s ❜② ❞❡❢❛✉❧t✳
♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ❝✉r✈❡ ♦✉t✇❡✐❣❤s ❧♦ss ✐♥ ♣r✐♠❡s ❢♦✉♥❞ ♣❡r ❝✉r✈❡✳ ❙✉r♣r✐s✐♥❣ ♦❜s❡r✈❛t✐♦♥✿ ■t✬s ❛ ❣❛✐♥✱ ♥♦t ❛ ❧♦ss✦ ❙✇✐t❝❤✐♥❣ ❢r♦♠ ❩ ✷ ❩ ✽ t♦ ♦✉r ♥❡✇ ❝✉r✈❡s ❞❡❝r❡❛s❡s t♦rs✐♦♥ ❜✉t ✐♥❝r❡❛s❡s ♣r✐♠❡s ❢♦✉♥❞ ♣❡r ❝✉r✈❡✳ ❙♦♠❡ ♣❛rt✐❝✉❧❛r ❝✉r✈❡s ❛r❡ ❡✈❡♥ ♠♦r❡ ❡✛❡❝t✐✈❡✳
SLIDE 24 ❊①♣❡r✐♠❡♥ts ❛♥❞ ❝♦♥❝❧✉s✐♦♥s
❲❡ ♠♦❞✐✜❡❞ ❊❊❈▼✲▼P❋◗ t♦ s✉♣♣♦rt ♦✉r ♥❡✇ ❝✉r✈❡s✳ ❊❊❈▼ ♣❛♣❡r ♦♣t✐♠✐③❡❞ ❊❊❈▼✲▼P❋◗ s ❡t❝✳ ❢♦r ♦❧❞ ❝✉r✈❡s✳ ❲❡ tr✐❡❞ t❤♦s❡ ♣❛r❛♠❡t❡rs ✇✐t❤ t❤♦✉s❛♥❞s ♦❢ ♦✉r ❝✉r✈❡s ❢♦r ❛❧❧ ❜✲❜✐t ♣r✐♠❡s ❢♦r ❡❛❝❤ ❜ ∈ {✶✺, ✶✻, . . . , ✷✻}✳ ❍❛♣♣② ♦❜s❡r✈❛t✐♦♥✿ ❖✉r ♥❡✇ ❩/✻ ❛ = −✶ ❝✉r✈❡s ❛r❡ t❤❡ ♥❡✇ ♣r✐❝❡✴♣❡r❢♦r♠❛♥❝❡ ❧❡❛❞❡rs ❢♦r ❊❈▼✦ ❊❊❈▼✲▼P❋◗ ♥♦✇ ✉s❡s t❤❡s❡ ❝✉r✈❡s ❜② ❞❡❢❛✉❧t✳
- ❛✐♥ ✐♥ # ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ❝✉r✈❡
♦✉t✇❡✐❣❤s ❧♦ss ✐♥ # ♣r✐♠❡s ❢♦✉♥❞ ♣❡r ❝✉r✈❡✳ ❙✉r♣r✐s✐♥❣ ♦❜s❡r✈❛t✐♦♥✿ ■t✬s ❛ ❣❛✐♥✱ ♥♦t ❛ ❧♦ss✦ ❙✇✐t❝❤✐♥❣ ❢r♦♠ ❩ ✷ ❩ ✽ t♦ ♦✉r ♥❡✇ ❝✉r✈❡s ❞❡❝r❡❛s❡s t♦rs✐♦♥ ❜✉t ✐♥❝r❡❛s❡s ♣r✐♠❡s ❢♦✉♥❞ ♣❡r ❝✉r✈❡✳ ❙♦♠❡ ♣❛rt✐❝✉❧❛r ❝✉r✈❡s ❛r❡ ❡✈❡♥ ♠♦r❡ ❡✛❡❝t✐✈❡✳
SLIDE 25 ❊①♣❡r✐♠❡♥ts ❛♥❞ ❝♦♥❝❧✉s✐♦♥s
❲❡ ♠♦❞✐✜❡❞ ❊❊❈▼✲▼P❋◗ t♦ s✉♣♣♦rt ♦✉r ♥❡✇ ❝✉r✈❡s✳ ❊❊❈▼ ♣❛♣❡r ♦♣t✐♠✐③❡❞ ❊❊❈▼✲▼P❋◗ s ❡t❝✳ ❢♦r ♦❧❞ ❝✉r✈❡s✳ ❲❡ tr✐❡❞ t❤♦s❡ ♣❛r❛♠❡t❡rs ✇✐t❤ t❤♦✉s❛♥❞s ♦❢ ♦✉r ❝✉r✈❡s ❢♦r ❛❧❧ ❜✲❜✐t ♣r✐♠❡s ❢♦r ❡❛❝❤ ❜ ∈ {✶✺, ✶✻, . . . , ✷✻}✳ ❍❛♣♣② ♦❜s❡r✈❛t✐♦♥✿ ❖✉r ♥❡✇ ❩/✻ ❛ = −✶ ❝✉r✈❡s ❛r❡ t❤❡ ♥❡✇ ♣r✐❝❡✴♣❡r❢♦r♠❛♥❝❡ ❧❡❛❞❡rs ❢♦r ❊❈▼✦ ❊❊❈▼✲▼P❋◗ ♥♦✇ ✉s❡s t❤❡s❡ ❝✉r✈❡s ❜② ❞❡❢❛✉❧t✳
- ❛✐♥ ✐♥ # ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ❝✉r✈❡
♦✉t✇❡✐❣❤s ❧♦ss ✐♥ # ♣r✐♠❡s ❢♦✉♥❞ ♣❡r ❝✉r✈❡✳ ❙✉r♣r✐s✐♥❣ ♦❜s❡r✈❛t✐♦♥✿ ■t✬s ❛ ❣❛✐♥✱ ♥♦t ❛ ❧♦ss✦ ❙✇✐t❝❤✐♥❣ ❢r♦♠ ❩/✷ × ❩/✽ t♦ ♦✉r ♥❡✇ ❝✉r✈❡s ❞❡❝r❡❛s❡s t♦rs✐♦♥ ❜✉t ✐♥❝r❡❛s❡s # ♣r✐♠❡s ❢♦✉♥❞ ♣❡r ❝✉r✈❡✳ ❙♦♠❡ ♣❛rt✐❝✉❧❛r ❝✉r✈❡s ❛r❡ ❡✈❡♥ ♠♦r❡ ❡✛❡❝t✐✈❡✳
SLIDE 26 ◆✉♠❜❡r ♦❢ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ♣r✐♠❡ ❢♦✉♥❞
✶✺✲❜✐t ♣r✐♠❡s
620 640 660 680 700 720 740 760 780 800 1 250 500 750 1000
✶✻✲❜✐t ♣r✐♠❡s
750 800 850 900 950 1000 1050 1 250 500 750 1000
✶✼✲❜✐t ♣r✐♠❡s ✶✽✲❜✐t ♣r✐♠❡s ✶✾✲❜✐t ♣r✐♠❡s ✷✵✲❜✐t ♣r✐♠❡s ✷✶✲❜✐t ♣r✐♠❡s ✷✷✲❜✐t ♣r✐♠❡s ✷✸✲❜✐t ♣r✐♠❡s ✷✹✲❜✐t ♣r✐♠❡s ✷✺✲❜✐t ♣r✐♠❡s ✷✻✲❜✐t ♣r✐♠❡s
SLIDE 27 ◆✉♠❜❡r ♦❢ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ♣r✐♠❡ ❢♦✉♥❞
✶✺✲❜✐t ♣r✐♠❡s ✶✻✲❜✐t ♣r✐♠❡s ✶✼✲❜✐t ♣r✐♠❡s
1000 1050 1100 1150 1200 1250 1300 1350 1 250 500 750 1000
✶✽✲❜✐t ♣r✐♠❡s
1350 1400 1450 1500 1550 1600 1650 1700 1750 1 250 500 750 1000
✶✾✲❜✐t ♣r✐♠❡s ✷✵✲❜✐t ♣r✐♠❡s ✷✶✲❜✐t ♣r✐♠❡s ✷✷✲❜✐t ♣r✐♠❡s ✷✸✲❜✐t ♣r✐♠❡s ✷✹✲❜✐t ♣r✐♠❡s ✷✺✲❜✐t ♣r✐♠❡s ✷✻✲❜✐t ♣r✐♠❡s
SLIDE 28 ◆✉♠❜❡r ♦❢ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ♣r✐♠❡ ❢♦✉♥❞
✶✺✲❜✐t ♣r✐♠❡s ✶✻✲❜✐t ♣r✐♠❡s ✶✼✲❜✐t ♣r✐♠❡s ✶✽✲❜✐t ♣r✐♠❡s ✶✾✲❜✐t ♣r✐♠❡s
1700 1750 1800 1850 1900 1950 2000 2050 2100 1 250 500 750 1000
✷✵✲❜✐t ♣r✐♠❡s
2150 2200 2250 2300 2350 2400 2450 2500 2550 2600 2650 1 250 500 750 1000
✷✶✲❜✐t ♣r✐♠❡s ✷✷✲❜✐t ♣r✐♠❡s ✷✸✲❜✐t ♣r✐♠❡s ✷✹✲❜✐t ♣r✐♠❡s ✷✺✲❜✐t ♣r✐♠❡s ✷✻✲❜✐t ♣r✐♠❡s
SLIDE 29 ◆✉♠❜❡r ♦❢ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ♣r✐♠❡ ❢♦✉♥❞
✶✺✲❜✐t ♣r✐♠❡s ✶✻✲❜✐t ♣r✐♠❡s ✶✼✲❜✐t ♣r✐♠❡s ✶✽✲❜✐t ♣r✐♠❡s ✶✾✲❜✐t ♣r✐♠❡s ✷✵✲❜✐t ♣r✐♠❡s ✷✶✲❜✐t ♣r✐♠❡s
2700 2800 2900 3000 3100 3200 3300 3400 3500 1 250 500 750 1000
✷✷✲❜✐t ♣r✐♠❡s
3600 3700 3800 3900 4000 4100 4200 4300 4400 1 250 500 750 1000
✷✸✲❜✐t ♣r✐♠❡s ✷✹✲❜✐t ♣r✐♠❡s ✷✺✲❜✐t ♣r✐♠❡s ✷✻✲❜✐t ♣r✐♠❡s
SLIDE 30 ◆✉♠❜❡r ♦❢ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ♣r✐♠❡ ❢♦✉♥❞
✶✺✲❜✐t ♣r✐♠❡s ✶✻✲❜✐t ♣r✐♠❡s ✶✼✲❜✐t ♣r✐♠❡s ✶✽✲❜✐t ♣r✐♠❡s ✶✾✲❜✐t ♣r✐♠❡s ✷✵✲❜✐t ♣r✐♠❡s ✷✶✲❜✐t ♣r✐♠❡s ✷✷✲❜✐t ♣r✐♠❡s ✷✸✲❜✐t ♣r✐♠❡s
4600 4700 4800 4900 5000 5100 5200 5300 5400 1 250 500 750 1000
✷✹✲❜✐t ♣r✐♠❡s
5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 1 250 500 750 1000
✷✺✲❜✐t ♣r✐♠❡s ✷✻✲❜✐t ♣r✐♠❡s
SLIDE 31 ◆✉♠❜❡r ♦❢ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ♣r✐♠❡ ❢♦✉♥❞
✶✺✲❜✐t ♣r✐♠❡s ✶✻✲❜✐t ♣r✐♠❡s ✶✼✲❜✐t ♣r✐♠❡s ✶✽✲❜✐t ♣r✐♠❡s ✶✾✲❜✐t ♣r✐♠❡s ✷✵✲❜✐t ♣r✐♠❡s ✷✶✲❜✐t ♣r✐♠❡s ✷✷✲❜✐t ♣r✐♠❡s ✷✸✲❜✐t ♣r✐♠❡s ✷✹✲❜✐t ♣r✐♠❡s ✷✺✲❜✐t ♣r✐♠❡s
7200 7400 7600 7800 8000 8200 8400 8600 1 250 500 750 1000
✷✻✲❜✐t ♣r✐♠❡s
9000 9200 9400 9600 9800 10000 10200 10400 10600 1 250 500 750 1000
SLIDE 32
❚❤❡ ❡♥❞
SLIDE 33
❚❤❛♥❦s✦
