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SLIDE 1

❖♥ ❛ ❝❧❛ss ♦❢ 3✲❛❧❣❡❜r❛s

❆♥❞r❡❛ ❙❛♥t✐

❯♥✐✈❡rs✐t② ♦❢ ❇♦❧♦❣♥❛

❲✐♥t❡r ❙❝❤♦♦❧ ✐♥ ●❡✐❧♦✱ ◆♦r✇❛②✿

  • ❡♦♠❡tr②✱ ❆♥❛❧②s✐s ❛♥❞ P❤②s✐❝s✱ ✺t❤✲✾t❤ ▼❛r❝❤ ✷✵✶✽

❇❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✳ ❈❛♥t❛r✐♥✐ ❛♥❞ ❆✳ ❘✐❝❝✐❛r❞♦

❛r❳✐✈✿✶✼✶✵✳✵✺✸✼✺

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SLIDE 2

P❧❛♥ ♦❢ t❤❡ t❛❧❦✿ ✕ ❏♦r❞❛♥ ❛❧❣❡❜r❛s ❛♥❞ ❏♦r❞❛♥ tr✐♣❧❡ s②st❡♠s ✕ ❚r✐♣❧❡ s②st❡♠s ✐♥ ●❡♦♠❡tr② ❛♥❞ ❙✉♣❡r❣r❛✈✐t② ✕ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s ✕ ❚❑❑ ❝♦♥str✉❝t✐♦♥ ✕ ■♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s ✕ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s ✕ ❊①❝❡♣t✐♦♥❛❧ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s

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SLIDE 3

❏♦r❞❛♥ ❛❧❣❡❜r❛s ❉❡❢✳ ❆ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ A s❛t✐s❢②✐♥❣ a ✆ ♣b ✆ a2q ✏ ♣a ✆ bq ✆ a2 ❢♦r ❛❧❧ a, b P A✳ ❏♦r❞❛♥✱ ✈♦♥ ◆❡✉♠❛♥♥ ❛♥❞ ❲✐❣♥❡r ❝❧❛ss✐✜❡❞ ✐♥ ✬✸✹ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ ❢♦r♠❛❧❧② r❡❛❧ ❏♦r❞❛♥ ❛❧❣❡❜r❛s✿ ✌ t❤❡ ❛❧❣❡❜r❛s ✱ ✱ ♦❢ ❍❡r♠✐t✐❛♥ r❡❛❧✱ ❝♦♠♣❧❡① ❛♥❞ q✉❛t❡r♥✐♦♥✐❝ ✂ ♠❛tr✐❝❡s ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✆ ✏ ♣

  • q

✌ t❤❡ s♣✐♥ ❢❛❝t♦rs ♣ q ✏ ❵ ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♣ q ✆ ♣ q ✏ ♣

② q ✌ t❤❡ ❆❧❜❡rt ❛❧❣❡❜r❛ ♦❢ ✂ ❍❡r♠✐t✐❛♥ ♠❛tr✐❝❡s ♦✈❡r ♦❝t♦♥✐♦♥s

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SLIDE 4

❏♦r❞❛♥ ❛❧❣❡❜r❛s ❉❡❢✳ ❆ ❏♦r❞❛♥ ❛❧❣❡❜r❛ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ A s❛t✐s❢②✐♥❣ a ✆ ♣b ✆ a2q ✏ ♣a ✆ bq ✆ a2 ❢♦r ❛❧❧ a, b P A✳ ❏♦r❞❛♥✱ ✈♦♥ ◆❡✉♠❛♥♥ ❛♥❞ ❲✐❣♥❡r ❝❧❛ss✐✜❡❞ ✐♥ ✬✸✹ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ ❢♦r♠❛❧❧② r❡❛❧ ❏♦r❞❛♥ ❛❧❣❡❜r❛s✿ ✌ t❤❡ ❛❧❣❡❜r❛s hR

n✱ hC n✱ hH n ♦❢ ❍❡r♠✐t✐❛♥ r❡❛❧✱ ❝♦♠♣❧❡① ❛♥❞ q✉❛t❡r♥✐♦♥✐❝

n ✂ n ♠❛tr✐❝❡s ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ a ✆ b ✏ 1

2♣ab baq

✌ t❤❡ s♣✐♥ ❢❛❝t♦rs J♣Rnq ✏ Rn ❵ R ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♣v, αq ✆ ♣w, βq ✏ ♣αw βv, ①v, w② αβq ✌ t❤❡ ❆❧❜❡rt ❛❧❣❡❜r❛ hO

3 ♦❢ 3 ✂ 3 ❍❡r♠✐t✐❛♥ ♠❛tr✐❝❡s ♦✈❡r ♦❝t♦♥✐♦♥s

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SLIDE 5

❏♦r❞❛♥ tr✐♣❧❡ s②st❡♠s

  • ✐✈❡♥ ❛♥② ❏♦r❞❛♥ ❛❧❣❡❜r❛ ♣A, ✆q ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❛ tr✐♣❧❡ ♣r♦❞✉❝t ♦♥ A ❜②

♣xyzq :✏ ♣x ✆ yq ✆ z ♣z ✆ yq ✆ x ✁ ♣x ✆ zq ✆ y ✇❤❡r❡ x, y, z P A✳ ■t ❤♦❧❞s✿ ✶✳ ♣uv♣xyzqq ✏ ♣♣uvxqyzq ✁ ♣x♣vuyqzq ♣xy♣uvzqq ✷✳ ♣xzyq ✏ ♣yzxq ❢♦r ❛❧❧ u, v, x, y, z✳ ❉❡❢✳ ❬▼❡②❜❡r❣ ✬✼✵❪ ❆ ❏♦r❞❛♥ tr✐♣❧❡ s②st❡♠ ✭s❤♦rt❧②✱ ❏❚❙✮ ✐s ❛ ✈✳s✳ ✇✐t❤ ❛ tr✐♣❧❡ ♣r♦❞✉❝t s❛t✐s❢②✐♥❣ ✶✳ ❛♥❞ ✷✳ ❛❜♦✈❡✳

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SLIDE 6

❏♦r❞❛♥ tr✐♣❧❡ s②st❡♠s ❊①❛♠♣❧❡s ♦❢ ❏❚❙✳ ✌ t❤❡ s♣❛❝❡ Mat♣p, q; Kq ♦❢ p ✂ q ♠❛tr✐❝❡s ♦✈❡r K ✏ R, C, H ✇✐t❤ ♣xyzq ✏ xy✝z zy✝x ❛♥❞ s✉❜❛❧❣❡❜r❛s ✭❡✳❣✳ ❍❡r♠✐t✐❛♥ ♠❛tr✐❝❡s hK

p ✇❤❡♥ p ✏ q✮

✌ t❤❡ s♣✐♥ ❢❛❝t♦rs J♣Rnq ✇✐t❤ t❤❡ tr✐♣❧❡ ♣r♦❞✉❝t ♥❛t✉r❛❧❧② ✐♥❞✉❝❡❞ ❜② ❏♦r❞❛♥ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✌ t❤❡ ❆❧❜❡rt ❛❧❣❡❜r❛ hO

3 ✇✐t❤ t❤❡ ✐♥❞✉❝❡❞ tr✐♣❧❡ ♣r♦❞✉❝t ❛♥❞ ✐ts

s✉❜❛❧❣❡❜r❛ M♣1, 2; Oq ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ r❡❛❧ ❏❚❙ ✐s ❞✉❡ t♦ ▲♦♦s ✐♥ ✬✼✼ ❛♥❞ ◆❡❤❡r ✐♥ ✬✽✵✳

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SLIDE 7

❚r✐♣❧❡ s②st❡♠s ✐♥ ●❡♦♠❡tr② ❛♥❞ ❙✉♣❡r❣r❛✈✐t②

✕ ❏♦r❞❛♥ ❛❧❣❡❜r❛s ❛♥❞ ❏♦r❞❛♥ tr✐♣❧❡ s②st❡♠s ❛r❡ ✉s❡❞ ✐♥ t❤❡♦r② ♦❢ ❍❡r♠✐t✐❛♥ s②♠♠❡tr✐❝ s♣❛❝❡s ❬❙❛t❛❦❡✱ ❇❡rtr❛♠❪ ❛♥❞ ✐♥ ❈❘ ❣❡♦♠❡tr② ❬❑❛✉♣✱ ❩❛✐ts❡✈❪ ✕ ❚r✐♣❧❡ s②st❡♠s s❛t✐s❢②✐♥❣ s✐♠✐❧❛r ✭❜✉t ❞✐✛❡r❡♥t✮ ❛①✐♦♠s ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ q✉❛t❡r♥✐♦♥✐❝ ❑ä❤❧❡r s②♠♠❡tr✐❝ s♣❛❝❡s ❬❲♦❧❢ ✬✻✺❀ ❆❧❡❦s❡❡✈s❦②✱ ❈♦rtés ✬✵✺❪ ✕ ❏♦r❞❛♥ ❛❧❣❡❜r❛s ✇❡r❡ ✉s❡❞ t♦ ❝♦♥str✉❝t N ✏ 2 ▼❛①✇❡❧❧✲❊✐♥st❡✐♥ D ✏ 5 s✉♣❡r❣r❛✈✐t② t❤❡♦r✐❡s ❬●ü♥❛②❞✐♥✱ ❙✐❡rr❛✱ ❚♦✇♥s❡♥❞ ✬✽✹❪ ✕ N ✏ 5 3✲❛❧❣❡❜r❛s ❛♣♣❡❛r ✐♥ D ✏ 3 s✉♣❡r❝♦♥❢♦r♠❛❧ ❈❤❡r♥✲❙✐♠♦♥s t❤❡♦r✐❡s ❞❡s❝r✐❜✐♥❣ ❞②♥❛♠✐❝s ♦❢ ❜r❛♥❡s ✐♥ s✉♣❡r❣r❛✈✐t② ❬❆❤❛r♦♥②✱ ❇❡r❣♠❛♥✱ ❏❛✛❡r✐s✬ ✵✽❪✱ ❬❞❡ ▼❡❞❡✐r♦s✱ ❋✐❣✉❡r♦❛✲❖✬❋❛rr✐❧❧✱ ▼é♥❞❡③✲❊s❝♦❜❛r✱ ❘✐tt❡r ✬✵✾❪

slide-8
SLIDE 8

❑❛♥t♦r tr✐♣❧❡ s②st❡♠s ■✳ ▲✳ ❑❛♥t♦r ✐♥tr♦❞✉❝❡❞ ✐♥ ✬✼✷ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❏♦r❞❛♥ tr✐♣❧❡ s②st❡♠s✳ ❉❡❢✳ ❆ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠ ✭s❤♦rt❧②✱ ❑❚❙✮ ✐s ❛ ✈✳s✳ V ✇✐t❤ ❛ tr✐♣❧❡ ♣r♦❞✉❝t s❛t✐s❢②✐♥❣ ✶✳ ♣uv♣xyzqq ✏ ♣♣uvxqyzq ✁ ♣x♣vuyqzq ♣xy♣uvzqq ✷✳ KKuv♣xqy ✏ K♣yxuqv ✁ K♣yxvqu ❢♦r ❛❧❧ u, v, x, y, z P V ✳ ❚❤❡ ✏❑❛♥t♦r t❡♥s♦r✑ Kxy♣zq ✏ ♣xzyq ✁ ♣yzxq ♠❡❛s✉r❡s t❤❡ ❢❛✐❧✉r❡ ♦❢ t❤❡ ♣r♦❞✉❝t t♦ ❜❡ s②♠♠❡tr✐❝ ✐♥ ♦✉t❡r ❛r❣✉♠❡♥ts✳ ■❢ Kxy ✏ 0 ❢♦r ❛❧❧ x, y P V t❤❡♥ ♦♥❡ ❣❡ts ❛ ❏❚❙✳

slide-9
SLIDE 9

❑❛♥t♦r tr✐♣❧❡ s②st❡♠s

❊①❛♠♣❧❡s ♦❢ ❑❚❙✳ ✌ ❚❤❡ s♣❛❝❡ Mat♣p, q; Kq ♦❢ p ✂ q ♠❛tr✐❝❡s ♦✈❡r K ✏ R, C, H ✇✐t❤ ♣xyzq ✏ xy✝z zy✝x ✁ zx ✝y ✌ ❚❤❡ s♣❛❝❡ Mat♣p, q; Kq ✂ Mat♣q, r; Kq ✇✐t❤ ♠✐①❡❞ tr✐♣❧❡ ♣r♦❞✉❝t ✄✄ x1 x2 ☛ ✄ y1 y2 ☛ ✄ z1 z2 ☛☛ ✏ ✄ x1y✝

1 z1 z1y✝ 1 x1 ✁ z1x2y✝ 2

x2y✝

2 z2 z2y✝ 2 x2 ✁ y✝ 1 x1z2

☛ ✇❤❡r❡ x ✏ ♣x1, x2q✱ y ✏ ♣y1, y2q ❛♥❞ z ✏ ♣z1, z2q✳ ❙✐♠♣❧❡ r❡❛❧ ❑❚❙ ♦❢ ❝❧❛ss✐❝❛❧ t②♣❡ t❤❛t ❛r❡ ♥♦t ❏❚❙ ❛r❡ ❣✐✈❡♥ ❜② ✭♠♦❞✐✜❝❛t✐♦♥s ♦❢✮ t❤❡s❡ ❡①❛♠♣❧❡s ❬❆s❛♥♦✱ ❑❛♥❡②✉❦✐ ✬✾✶❪✳ ❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ❡①❝❡♣t✐♦♥❛❧ t②♣❡ ❤❛✈❡ ❜❡❡♥ ❝♦♥s✐❞❡r❡❞ ❜② ▼♦♥❞♦❝ ✐♥ ✬✵✼✳

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SLIDE 10

❈♦♥str✉❝t✐♦♥ ♦❢ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s

▲❡t g ✏ g✁d ❵ ☎ ☎ ☎ ❵ gd ❜❡ ❛ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ Z✲❣r❛❞✐♥❣ ✇✐t❤ ❡q✉❛❧ ❞❡♣t❤ ❛♥❞ ❤❡✐❣❤t✳ ▲❡t σ : g Ñ g ❜❡ ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥ ♦❢ g✳ ❙❡t V ✏ g✁1 ❛♥❞ ♣xyzq :✏ rrx, σ♣yqs, zs ❢♦r ❛❧❧ x, y, z P V ✳

slide-11
SLIDE 11

❈♦♥str✉❝t✐♦♥ ♦❢ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s

▲❡t g ✏ g✁d ❵ ☎ ☎ ☎ ❵ gd ❜❡ ❛ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ Z✲❣r❛❞✐♥❣ ✇✐t❤ ❡q✉❛❧ ❞❡♣t❤ ❛♥❞ ❤❡✐❣❤t✳ ▲❡t σ : g Ñ g ❜❡ ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥ ♦❢ g✳ ❙❡t V ✏ g✁1 ❛♥❞ ♣xyzq :✏ rrx, σ♣yqs, zs ❢♦r ❛❧❧ x, y, z P V ✳ ❚❤❡♥ ♣uv♣xyzqq ✏rru, σ♣vqs, rrx, σ♣yqs, zss ✏rrrru, σ♣vqs, xs, σ♣yqs, zs rrx, rru, σ♣vqs, σ♣yqss, zs rrx, σ♣yqs, rru, σ♣vqs, zss ✏♣♣uvxqyzq rrx, σrrσ♣uq, vs, yss, zs ♣xy♣uvzqq ✏♣♣uvxqyzq ✁ ♣x♣vuyqzq ♣xy♣uvzqq , ❢♦r ❛❧❧ u, v, x, y, z P V ✳ ■♥ ♦t❤❡r ✇♦r❞s t❤❡ ♣r✐♥❝✐♣❛❧ ✐❞❡♥t✐t② ♦❢ ❑❚❙ ❤♦❧❞s✳

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SLIDE 12

❈♦♥str✉❝t✐♦♥ ♦❢ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s

▲❡t g ✏ g✁d ❵ ☎ ☎ ☎ ❵ gd ❜❡ ❛ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ Z✲❣r❛❞✐♥❣ ✇✐t❤ ❡q✉❛❧ ❞❡♣t❤ ❛♥❞ ❤❡✐❣❤t✳ ▲❡t σ : g Ñ g ❜❡ ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥ ♦❢ g✳ ❙❡t V ✏ g✁1 ❛♥❞ ♣xyzq :✏ rrx, σ♣yqs, zs ❢♦r ❛❧❧ x, y, z P V ✳ ✌ ■❢ ❞❡♣t❤ d ✏ 1 t❤❡♥ Kxy♣zq ✏ ♣xzyq ✁ ♣yzxq ✏ rrx, σ♣zqs, ys ✁ rry, σ♣zqs, xs ✏ rrx, ys, σ♣zqs ✏ 0 t❤❛t ✐s t❤❡ ❑❛♥t♦r t❡♥s♦rs ❛❧❧ ✈❛♥✐s❤✳ ❚❤✐s ✐s t❤❡ ❝❛s❡ ♦❢ ❏❚❙✳ ✌ ❑❚❙ ❝♦rr❡s♣♦♥❞ t♦ ❞❡♣t❤ d ✏ 2✳

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SLIDE 13

❚❑❑ ❝♦♥str✉❝t✐♦♥ r❡✈✐s✐t❡❞

▲❡t V ❜❡ ❛ ❝❡♥t❡r❧❡ss ❑❚❙ ❛♥❞ ❞❡✜♥❡ t❤❡ ♦♣❡r❛t♦rs✱ ❢♦r x, y, z P V ✿ Lxy♣zq :✏ ♣xyzq , ϕx♣zq :✏ Lzx , Dxy♣zq :✏ ✁ϕKxy♣zq . g ✏ g♣V q ✏ ①Kxy② ❵ V ❵ ①Lxy② ❵ ①ϕx② ❵ ①Dxy② ✁2 ✁1 1 2 ❚❤❡ ▲✐❡ ❜r❛❝❦❡t ✐s ❞❡✜♥❡❞ ❜② rx, ys :✏ Kxy , rA, zs :✏ A♣zq ❢♦r A ❡✐t❤❡r Lxy✱ ϕx ♦r Dxy ❛♥❞ ❡①t❡♥❞❡❞ ✉s✐♥❣ ✌ tr❛♥s✐t✐✈✐t② ✭✐❢ A P gi✱ i ➙ 0✱ ❛♥❞ rA, zs ✏ 0 ❢♦r ❛❧❧ z P g✁1 t❤❡♥ A ✏ 0✮ ✌ ❏❛❝♦❜✐ ✐❞❡♥t✐t②✱ s✐♥❝❡ g ✐s ❢✉♥❞❛♠❡♥t❛❧ ✭g✁1 ❣❡♥❡r❛t❡s g✁2✮✳

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SLIDE 14

❚❑❑ ❝♦♥str✉❝t✐♦♥ r❡✈✐s✐t❡❞

▲❡t V ❜❡ ❛ ❝❡♥t❡r❧❡ss ❑❚❙ ❛♥❞ ❞❡✜♥❡ t❤❡ ♦♣❡r❛t♦rs✱ ❢♦r x, y, z P V ✿ Lxy♣zq :✏ ♣xyzq , ϕx♣zq :✏ Lzx , Dxy♣zq :✏ ✁ϕKxy♣zq . g ✏ g♣V q ✏ ①Kxy② ❵ V ❵ ①Lxy② ❵ ①ϕx② ❵ ①Dxy② ✁2 ✁1 1 2 ✌ g♣V q ✐s ❛ s✉❜❛❧❣❡❜r❛ ♦❢ t❤❡ ❚❛♥❛❦❛ ♣r♦❧♦♥❣❛t✐♦♥ g✽ ♦❢ m ✏ g✁2 ❵ g✁1❀ ✌ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♣ ✐s ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥ ♦❢ g♣V q σ : Kxy Ð Ñ Dxy , x Ð Ñ ✁ϕx , Lxy Ð Ñ ✁Lyx

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SLIDE 15

❚❑❑ ❝♦♥str✉❝t✐♦♥ r❡✈✐s✐t❡❞ ❚❤♠ ■✳ ❚❤❡r❡ ✐s ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ s✐♠♣❧❡ ❑❚❙ V ❛♥❞ ♣❛✐rs ♣g, σq✱ ✇❤❡r❡ g ✏ g✁2❵☎ ☎ ☎❵g2 ✐s s✐♠♣❧❡ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ 5✲❣r❛❞✐♥❣ ❛♥❞ σ ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥✳ ❋✉rt❤❡r♠♦r❡ g ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ✭r❡s♣✳ ❧✐♥❡❛r❧② ❝♦♠♣❛❝t✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ V ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ✭r❡s♣✳ ❧✐♥❡❛r❧② ❝♦♠♣❛❝t✮✳

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SLIDE 16

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❑❛♥t♦r tr✐♣❧❡ s②st❡♠s

❲❡ ❝❧❛ss✐✜❡❞ t❤❡ s✐♠♣❧❡ ❑❚❙ ♦✈❡r C✳ ▲❡t ✉s st❛rt ✇✐t❤ ❧✐♥❡❛r❧② ❝♦♠♣❛❝t ❝❛s❡✳ ❇② ❛ ✇❡❧❧✲❦♥♦✇♥ t❤❡♦r❡♠ ♦❢ ❊✳ ❈❛rt❛♥✱ s✐♠♣❧❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛s ♦❢ ✈❡❝t♦r ✜❡❧❞s ♦♥ ❛♥ m✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ ❛r❡ ✭❧♦❝❛❧❧②✱ ❢♦r♠❛❧❧②✮ ✐s♦♠♦r♣❤✐❝ t♦✿ ✌ Wm ✏ ✦ ➦m

i✏1 Pi ❇ ❇xi ⑤ Pi P Crrx1, . . . , xmss

✮ ✌ Sm ✏ ✦ X P Wm ⑤ div♣Xq ✏ 0 ✮ ✌ Hm ✏ ✦ X P Wm ⑤ LXω ✏ 0 ✮ ♣m ✏ 2kq✱ ✇❤❡r❡ ω ✏ ➦k

i✏1 dxi ❫ dxki ✐s

s②♠♣❧❡❝t✐❝ ❢♦r♠ ✌ Km ✏ ✦ X P Wm ⑤ LXα ✏ fα ❢♦r f P Crrx1, . . . , xmss ✮ ♣m ✏ 2k 1q ✇❤❡r❡ α ✏ dxm ➦k

i✏1 xidxki ✐s ❝♦♥t❛❝t ❢♦r♠

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SLIDE 17

■♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❑❚❙

❚❤♠ ■■✳ ❚❤❡r❡ ❛r❡ ♥♦ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ ❧✐♥❡❛r❧② ❝♦♠♣❛❝t ❑❚❙✳ Pr♦♦❢✳ ❆♥② Z✲❣r❛❞✐♥❣ ♦❢ Wm ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛ss✐❣♥✐♥❣ ❞❡❣r❡❡s deg♣xiq ✏ ki , deg♣ ❇ ❇xi q ✏ ✁ki , ❢♦r ❛ m✲t✉♣❧❡ ♦❢ ✐♥t❡❣❡rs ♣k1, . . . , kmq✳ ■♥ ♣❛rt✐❝✉❧❛r deg♣xn

i

❇ ❇xj q ✏ nki ✁ kj ❛♥❞ Wm ❞♦❡s ♥♦t ❛❞♠✐t ❛♥② 5✲❣r❛❞✐♥❣ ✭gi ✘ 0 ❢♦r ✐♥✜♥✐t❡❧② ♠❛♥② i✮✳ ❙✐♠✐❧❛r❧② ❢♦r Sm✱ Hm ❛♥❞ Km✳

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SLIDE 18

❙tr✉❝t✉r❡ t❤❡♦r② ❢♦r ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❑❚❙

Pr♦❜❧❡♠✿ ❍♦✇ t♦ ❝❧❛ss✐❢② t❤❡ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ 5✲❣r❛❞❡❞ ▲✐❡ ❛❧❣❡❜r❛s❄ ❚❤♠ ■■■✳ ❚❤❡r❡ ✐s ❛ ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ♣g, σq ❛♥❞ ✐s♦♠♦r✲ ♣❤✐s♠ ❝❧❛ss❡s ♦❢ ♣go, θq✱ ✇❤❡r❡ g ✐s ❛ s✐♠♣❧❡ ❝♦♠♣❧❡① Z✲❣r❛❞❡❞ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ✐♥✈♦❧✉t✐♦♥ σ ❛♥❞ go ✐s ❛ r❡❛❧ ❛❜s♦❧✉t❡❧② s✐♠♣❧❡ Z✲❣r❛❞❡❞ ▲✐❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ ❣r❛❞❡✲r❡✈❡rs✐♥❣ ❈❛rt❛♥ ✐♥✈♦❧✉t✐♦♥✳ ❙♦❧✉t✐♦♥✿ ❋✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s✐♠♣❧❡ ❑❚❙ ♦✈❡r C ❛r❡ ✐♥ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ 5✲❣r❛❞✐♥❣s ♦❢ r❡❛❧ ❢♦r♠s ♦❢ s✐♠♣❧❡ ❝♦♠♣❧❡① ▲✐❡ ❛❧❣❡❜r❛s✳

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SLIDE 19

❚❤❡ ❛❧❣❡❜r❛ ♦❢ ❞❡r✐✈❛t✐♦♥s ❚❤♠ ■❱✳ ▲❡t V ❜❡ ❛ s✐♠♣❧❡ ❑❚❙ ✇✐t❤ ❛ss♦❝✐❛t❡❞ r❡❛❧ ❢♦r♠ ♣go, θq✳ ▲❡t ♣go

0qss ❜❡ t❤❡ s❡♠✐s✐♠♣❧❡ ♣❛rt ♦❢ go 0 ❛♥❞ go 0 ✏ k ❵ p ✐ts ❈❛rt❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥✳

❚❤❡♥ t❤❡ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ❞❡r✐✈❛t✐♦♥s ♦❢ V ✐s der♣V q ✏ k ❜ C . Pr♦♦❢✳ ❚❤❡ s♣❛❝❡ der♣V q ❝♦♥s✐sts ♦❢ ✭t❤❡ r❡str✐❝t✐♦♥ t♦ V ✏ g✁1 ♦❢✮ t❤❡ 0✲❞❡❣r❡❡ ❞❡r✐✈❛t✐♦♥s ♦❢ g ❝♦♠♠✉t✐♥❣ ✇✐t❤ σ✳ ❆♥② ❞❡r✐✈❛t✐♦♥ ✐s ✐♥♥❡r ❛♥❞ ✐t ✐s ❦♥♦✇♥ t❤❛t t❤❡ r❡str✐❝t✐♦♥ t♦ ♣go

0qss ♦❢ t❤❡ ❈❛rt❛♥ ✐♥✈♦❧✉t✐♦♥ ♦❢ go ✐s

st✐❧❧ ❛ ❈❛rt❛♥ ✐♥✈♦❧✉t✐♦♥✳

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SLIDE 20

❋✐♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❑❚❙ ❚❤♠ ❱✳ ❯♣ t♦ ✐s♦♠♦r♣❤✐s♠ t❤❡r❡ ❛r❡ 8 ✐♥✜♥✐t❡ s❡r✐❡s ♦❢ ❝❧❛ss✐❝❛❧ s✐♠♣❧❡ ❑❚❙ ❛♥❞ 23 ❡①❝❡♣t✐♦♥❛❧ ❝❛s❡s✳ ❚❤❡ ❡①❝❡♣t✐♦♥❛❧ ❑❚❙ ❝❛♥ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ t❤r❡❡ ♠❛✐♥ ❝❧❛ss❡s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❣r❛❞✐♥❣ ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ ❛❧❣❡❜r❛✿ ✭✐✮ ♦❢ ❝♦♥t❛❝t t②♣❡ ✐❢ dim g✁2 ✏ 1❀ ✭✐✐✮ ♦❢ ❡①t❡♥❞❡❞ P♦✐♥❝❛ré t②♣❡ ✐❢ g✁2 ✏ U ❛♥❞ g0 ⑩ so♣Uq❀ ✭✐✐✐✮ ♦❢ s♣❡❝✐❛❧ t②♣❡ ♦t❤❡r✇✐s❡✳

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SLIDE 21

❊①❝❡♣t✐♦♥❛❧ ❑❚❙ ❛ss♦❝✐❛t❡❞ t♦ E7 ❈♦♥t❛❝t t②♣❡

go ❙❛t❛❦❡ ❉✐❛❣r❛♠ ❑❚❙ E V ✂ V ✏ S der♣V q ✏ so♣6, Cq ❵ so♣6, Cq E V I ✂ V ✏ S der♣V q ✏ gl♣6, Cq E V II ✂ V ✏ S der♣V q ✏ so♣2, Cq ❵ so♣10, Cq

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SLIDE 22

❊①❝❡♣t✐♦♥❛❧ ❑❚❙ ❛ss♦❝✐❛t❡❞ t♦ E7 P♦✐♥❝❛ré t②♣❡

go ❙❛t❛❦❡ ❉✐❛❣r❛♠ ❑❚❙ E V ✂ V ✏ S ❜ C2 der♣V q ✏ so♣5, Cq ❵ so♣5, Cq ❵ so♣2, Cq E V I ✂ V ✏ S ❜ C2 der♣V q ✏ so♣3, Cq ❵ so♣7, Cq ❵ sl♣2, Cq E V II ✂ V ✏ S ❜ C2 der♣V q ✏ so♣9, Cq ❵ so♣2, Cq

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SLIDE 23

❊①❝❡♣t✐♦♥❛❧ ❑❚❙ ❛ss♦❝✐❛t❡❞ t♦ E7 ❙♣❡❝✐❛❧ t②♣❡

go ❙❛t❛❦❡ ❉✐❛❣r❛♠ ❑❚❙ E V

V ✏ Λ3♣C7q✝ der♣V q ✏ so♣7, Cq E V I ◆♦ ❑❚❙ E V II ◆♦ ❑❚❙

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SLIDE 24

❊①❝❡♣t✐♦♥❛❧ ❑❚❙ ♦❢ s♣❡❝✐❛❧ t②♣❡ g ✏ E7

▲❡t η ❜❡ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ C7 ❛♥❞ ✼ t❤❡ ❛ss♦❝✐❛t❡❞ ♠✉s✐❝❛❧ ✐s♦♠♦r♣❤✐s♠ t❤❛t s❡♥❞s Λ3♣C7q✝ t♦ Λ3C7✳ ▲❡t ✌ ❜❡ t❤❡ ♥❛t✉r❛❧ ♣r♦❥❡❝t✐♦♥ ♦❢ Λ3C7 ❜ Λ3♣C7q✝ t♦ sl♣7, Cq✳ ❚❤♠ ❱■✳ ❚❤❡ ✈❡❝t♦r s♣❛❝❡ Λ3♣C7q✝ ✇✐t❤ tr✐♣❧❡ ♣r♦❞✉❝t ♣αβγq ✏ 2 7η♣α, βqγ ✁ ♣β✼ ✌ αq ☎ γ ✐s ❛ s✐♠♣❧❡ ❑❚❙ ✇✐t❤ ❛ss♦❝✐❛t❡❞ ▲✐❡ ❛❧❣❡❜r❛ g ✏ E7 ❛♥❞ ❞❡r✐✈❛t✐♦♥ ❛❧❣❡❜r❛ so♣7, Cq✳

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SLIDE 25

❚❤❛♥❦s✦