Advanced Algebra II: Activities and Homework by KennyFelder

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Algebra IX: Finite Groups of Lie Type Finite-Dimensional Division Algebras

The finite teams of Lie sort are of crucial mathematical significance and the matter of knowing their irreducible representations is of significant curiosity. The illustration idea of those teams over an algebraically closed box of attribute 0 was once constructed by means of P. Deligne and G. Lusztig in 1976 and as a result in a sequence of papers via Lusztig culminating in his booklet in 1984.

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P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✵ ❜❛rs ✢♦❛t t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❧❛♥❞ ♦♥ t❤❡ t❡❛❝❤❡r✬s ❞❡s❦✳ ❆♥❞✱ ❛s q✉✐❝❦❧② ❛s s❤❡ ❛♣♣❡❛r❡❞✱ ❙❛❧❧② ✐s ❣♦♥❡ t♦ ❞♦ ♠♦r❡ ❣♦♦❞ ✐♥ t❤❡ ✇♦r❧❞✳ ▲❡t s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss✱ ❛♥❞ c r❡♣r❡s❡♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❛♥❞② ❜❛rs ❞✐str✐❜✉t❡❞✳ ❚✇♦ ❢♦r ❡❛❝❤ st✉❞❡♥t✱ ❛♥❞ ✜✈❡ ❢♦r t❤❡ t❡❛❝❤❡r✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t♦ s❤♦✇ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ❙❛❧❧② ❣❛✈❡ ♦✉t✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts✳ c (s) =❴❴❴❴❴❴ ❜✳ ❯s❡ t❤❛t ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ t❤❡r❡ ✇❡r❡ ✷✵ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ✇❡r❡ ❞✐str✐❜✉t❡❞❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❝✳ ◆♦✇ ✉s❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ ❙❛❧❧② ❞✐str✐❜✉t❡❞ ✸✺ ❝❛♥❞② ❜❛rs✱ ❤♦✇ ♠❛♥② st✉❞❡♥ts ✇❡r❡ ✐♥ t❤❡ ❝❧❛ss❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✵ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = ✐s ✏❙✉❜tr❛❝t t❤r❡❡✱ t❤❡♥ t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ f (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✶ ❚❤❡ ❢✉♥❝t✐♦♥ y (x) ✐s ✏●✐✈❡♥ ❛♥② ♥✉♠❜❡r✱ r❡t✉r♥ ✻✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ y (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✷ z (x) = x2 − 6x + 9 ❛✳ z (−1) =❴❴❴❴❴❴ ❜✳ z (0) = ❴❴❴❴❴❴ ❝✳ z (1) =❴❴❴❴❴❴ ❞✳ z (3) =❴❴❴❴❴❴ ❡✳ z (x + 2) =❴❴❴❴❴❴ ❢✳ z (z (x)) =❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✸ ❖❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts ♦❢ ♥✉♠❜❡rs✱ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ♦♥❡s ❝♦✉❧❞ ♣♦ss✐❜❧② ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❆❧❧ ■ ♥❡❡❞ ✐s ❛ ✏❨❡s✑ ♦r ✏◆♦✑✖②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ t❡❧❧ ♠❡ t❤❡ ❢✉♥❝t✐♦♥✦ ✭❇✉t ❣♦ ❛❤❡❛❞ ❛♥❞ ❞♦✱ ✐❢ ②♦✉ ✇❛♥t t♦.

T ✐s♥✬t ❛❧r❡❛❞② ✐♥ t❤❛t ❢♦r♠✮ ■❞❡♥t✐❢② t❤❡ s❧♦♣❡ ■❞❡♥t✐❢② t❤❡ y ✲✐♥t❡r❝❡♣t✱ ❛♥❞ ❣r❛♣❤ ✐t ❯s❡ t❤❡ s❧♦♣❡ t♦ ✜♥❞ ♦♥❡ ♣♦✐♥t ♦t❤❡r t❤❛♥ t❤❡ ●r❛♣❤ t❤❡ ❧✐♥❡ ❊①❡r❝✐s❡ ✶✳✺✺ y = 3x − 2 ❙❧♦♣❡✿❴❴❴❴❴❴❴❴❴❴❴ y ✲✐♥t❡r❝❡♣t✿❴❴❴❴❴❴❴❴❴❴❴ ❖t❤❡r ♣♦✐♥t✿❴❴❴❴❴❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✺✻ 2y − x = 4 ❊q✉❛t✐♦♥ ✐♥ y = ♠① + b y ✲✐♥t❡r❝❡♣t ♦♥ t❤❡ ❧✐♥❡ ✷✾ ❙❧♦♣❡✿❴❴❴❴❴❴❴❴❴❴❴ n✲✐♥t❡r❝❡♣t✿❴❴❴❴❴❴❴❴❴❴❴ ❖t❤❡r ♣♦✐♥t✿❴❴❴❴❴❴❴❴❴❴❴ ✶✸ ✶✳✶✸ ❍♦♠❡✇♦r❦✿ ●r❛♣❤✐♥❣ ▲✐♥❡s ❊①❡r❝✐s❡ ✶✳✺✼ 2y + 7x + 3 = 0 ❛✳ ❜✳ ❝✳ ❞✳ ❡✳ ✐s t❤❡ ❡q✉❛t✐♦♥ ❢♦r ❛ ❧✐♥❡✳ P✉t t❤✐s ❡q✉❛t✐♦♥ ✐♥t♦ t❤❡ ✏s❧♦♣❡✲✐♥t❡r❝❡♣t✑ ❢♦r♠ y = ♠① + b s❧♦♣❡ ❂ ❴❴❴❴❴❴❴❴❴❴❴ ②✲✐♥t❡r❝❡♣t ❂ ❴❴❴❴❴❴❴❴❴❴❴ ①✲✐♥t❡r❝❡♣t ❂ ❴❴❴❴❴❴❴❴❴❴❴ ●r❛♣❤ ✐t✳ ❊①❡r❝✐s❡ ✶✳✺✽ ❚❤❡ ♣♦✐♥ts (5, 2) ❛♥❞ (7, 8) ❧✐❡ ♦♥ ❛ ❧✐♥❡✳ ❛✳ ❋✐♥❞ t❤❡ s❧♦♣❡ ♦❢ t❤✐s ❧✐♥❡ ❜✳ ❋✐♥❞ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤✐s ❧✐♥❡ ❊①❡r❝✐s❡ ✶✳✺✾ ❲❤❡♥ ②♦✉✬r❡ ❜✉✐❧❞✐♥❣ ❛ r♦♦❢✱ ②♦✉ ♦❢t❡♥ t❛❧❦ ❛❜♦✉t t❤❡ ✏♣✐t❝❤✑ ♦❢ t❤❡ r♦♦❢✖✇❤✐❝❤ ✐s ❛ ❢❛♥❝② ✇♦r❞ t❤❛t ♠❡❛♥s ✐ts s❧♦♣❡✳ ❨♦✉ ❛r❡ ❜✉✐❧❞✐♥❣ ❛ r♦♦❢ s❤❛♣❡❞ ❧✐❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ r♦♦❢ ✐s ♣❡r❢❡❝t❧② s②♠♠❡tr✐❝❛❧✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ✳ ■♥ t❤❡ ❞r❛✇✐♥❣ ❜❡❧♦✇✱ t❤❡ r♦♦❢ ✐s t❤❡ t✇♦ t❤✐❝❦ ❜❧❛❝❦ ❧✐♥❡s✖t❤❡ ❝❡✐❧✐♥❣ ♦❢ t❤❡ ❤♦✉s❡ ✐s t❤❡ ❞♦tt❡❞ ❧✐♥❡ ✻✵✬ ❧♦♥❣✳ ❋✐❣✉r❡ ✶✳✷✸ ❛✳ ❲❤❛t ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ r♦♦❢ ❄ ❜✳ ❍♦✇ ❤✐❣❤ ✐s t❤❡ r♦♦❢ ❄ ❚❤❛t ✐s✱ ✇❤❛t ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡✐❧✐♥❣ ♦❢ t❤❡ ❤♦✉s❡✱ str❛✐❣❤t ✉♣ t♦ t❤❡ ♣♦✐♥t ❛t t❤❡ t♦♣ ♦❢ t❤❡ r♦♦❢ ❄ ❝✳ ❍♦✇ ❧♦♥❣ ✐s t❤❡ r♦♦❢ ❄ ❚❤❛t ✐s✱ ✇❤❛t ✐s t❤❡ ❝♦♠❜✐♥❡❞ ❧❡♥❣t❤ ♦❢ t❤❡ t✇♦ t❤✐❝❦ ❜❧❛❝❦ ❧✐♥❡s ✐♥ t❤❡ ❞r❛✇✐♥❣ ❛❜♦✈❡❄ ❊①❡r❝✐s❡ ✶✳✻✵ y = 3x✱ ❡①♣❧❛✐♥ ✇❤② ✸ ✐s t❤❡ s❧♦♣❡✳ ✭❉♦♥✬t ❥✉st s❛② ✏❜❡❝❛✉s❡ ✐t✬s t❤❡ m + b✳✑ ❊①♣❧❛✐♥ ✇❤② ∆y ∆x ✇✐❧❧ ❜❡ ✸ ❢♦r ❛♥② t✇♦ ♣♦✐♥ts ♦♥ t❤✐s ❧✐♥❡✱ ❥✉st ❧✐❦❡ ✇❡ ❡①♣❧❛✐♥❡❞ ✇❤② b ✐s t❤❡ ②✲✐♥t❡r❝❡♣t✳✮ ■♥ t❤❡ ❡q✉❛t✐♦♥ y= ♠① ❝❧❛ss ✶✸ ❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✶✾✶✶✽✴✶✳✷✴❃✳ ✐♥ ✐♥ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✸✵ ❊①❡r❝✐s❡ ✶✳✻✶ ❍♦✇ ❞♦ ②♦✉ ♠❡❛s✉r❡ t❤❡ ❤❡✐❣❤t ♦❢ ❛ ✈❡r② t❛❧❧ ♠♦✉♥t❛✐♥❄ ❨♦✉ ❝❛♥✬t ❥✉st s✐♥❦ ❛ r✉❧❡r ❞♦✇♥ ❢r♦♠ t❤❡ t♦♣ t♦ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ ♠♦✉♥t❛✐♥✦ ❙♦ ❤❡r❡✬s ♦♥❡ ✇❛② ②♦✉ ❝♦✉❧❞ ❞♦ ✐t✳ ❨♦✉ st❛♥❞ ❜❡❤✐♥❞ ❛ tr❡❡✱ ❛♥❞ ②♦✉ ♠♦✈❡ ❜❛❝❦ ✉♥t✐❧ ②♦✉ ❝❛♥ ❧♦♦❦ str❛✐❣❤t ♦✈❡r t❤❡ t♦♣ ♦❢ t❤❡ tr❡❡✱ t♦ t❤❡ t♦♣ ♦❢ t❤❡ ♠♦✉♥t❛✐♥✳ ❚❤❡♥ ②♦✉ ♠❡❛s✉r❡ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ tr❡❡✱ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ②♦✉ t♦ t❤❡ ♠♦✉♥t❛✐♥✱ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ ②♦✉ t♦ t❤❡ tr❡❡✳ ❙♦ ②♦✉ ♠✐❣❤t ❣❡t r❡s✉❧ts ❧✐❦❡ t❤✐s✳ ❋✐❣✉r❡ ✶✳✷✹ ❍♦✇ ❤✐❣❤ ✐s t❤❡ ♠♦✉♥t❛✐♥❄ ❊①❡r❝✐s❡ ✶✳✻✷ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ ✭❛ ✏r❡❧❛t✐♦♥✱✑ r❡♠❡♠❜❡r t❤♦s❡❄✮ s❤♦✇s ❤♦✇ ♠✉❝❤ ♠♦♥❡② ❙❝r♦♦❣❡ ▼❝❉✉❝❦ ❤❛s ❜❡❡♥ ✇♦rt❤ ❡✈❡r② ②❡❛r s✐♥❝❡ ✶✾✾✾✳ ❨❡❛r ✶✾✾✾ ✷✵✵✵ ✷✵✵✶ ✷✵✵✷ ✷✵✵✸ ✷✵✵✹ ◆❡t ❲♦rt❤ ✩✸ ❚r✐❧❧✐♦♥ ✩✹✳✺ ❚r✐❧❧✐♦♥ ✩✻ ❚r✐❧❧✐♦♥ ✩✼✳✺ ❚r✐❧❧✐♦♥ ✩✾ ❚r✐❧❧✐♦♥ ✩✶✵✳✺ ❚r✐❧❧✐♦♥ ❚❛❜❧❡ ✶✳✽ ❛✳ ❜✳ ❝✳ ❞✳ ❍♦✇ ♠✉❝❤ ✐s ❛ tr✐❧❧✐♦♥✱ ❛♥②✇❛②❄ ●r❛♣❤ t❤✐s r❡❧❛t✐♦♥✳ ❲❤❛t ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❣r❛♣❤❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❝❛♥ ▼r✳ ▼❝❉✉❝❦ ❡❛r♥ ✐♥ ✷✵ ②❡❛rs ❛t t❤✐s r❛t❡❄ ❊①❡r❝✐s❡ ✶✳✻✸ ▼❛❦❡ ✉♣ ❛♥❞ s♦❧✈❡ ②♦✉r ♦✇♥ ✇♦r❞ ♣r♦❜❧❡♠ ✉s✐♥❣ s❧♦♣❡✳ ✸✶ ✶✳✶✹ ❈♦♠♣♦s✐t❡ ❋✉♥❝t✐♦♥s ✶✹ ❊①❡r❝✐s❡ ✶✳✻✹ ❨♦✉ ❛r❡ t❤❡ ❢♦r❡♠❛♥ ❛t t❤❡ ❙❡s❛♠❡ ❙tr❡❡t ◆✉♠❜❡r ❋❛❝t♦r②✳ ❆ ❤✉❣❡ ❝♦♥✈❡②♦r ❜❡❧t r♦❧❧s ❛❧♦♥❣✱ ❝♦✈❡r❡❞ ✇✐t❤ ❜✐❣ ♣❧❛st✐❝ ♥✉♠❜❡rs ❢♦r ♦✉r ❝✉st♦♠❡rs✳ ❨♦✉r t✇♦ ❜❡st ❡♠♣❧♦②❡❡s ❛r❡ ❑❛t✐❡ ❛♥❞ ◆✐❝♦❧❛s✳ ❇♦t❤ ♦❢ t❤❡♠ st❛♥❞ ❛t t❤❡✐r st❛t✐♦♥s ❜② t❤❡ ❝♦♥✈❡②♦r ❜❡❧t✳ ◆✐❝♦❧❛s✬s ❥♦❜ ✐s✿ ✇❤❛t❡✈❡r ♥✉♠❜❡r ❝♦♠❡s t♦ ②♦✉r st❛t✐♦♥✱ ❛❞❞ ✷ ❛♥❞ t❤❡♥ ♠✉❧t✐♣❧② ❜② ✺✱ ❛♥❞ s❡♥❞ ♦✉t t❤❡ r❡s✉❧t✐♥❣ ♥✉♠❜❡r✳ ❑❛t✐❡ ✐s s✉❜tr❛❝t ✶✵✱ ❛♥❞ s❡♥❞ t❤❡ r❡s✉❧t ♥❡①t ♦♥ t❤❡ ❧✐♥❡✳ ❍❡r ❥♦❜ ✐s✿ ✇❤❛t❡✈❡r ♥✉♠❜❡r ❝♦♠❡s t♦ ②♦✉✱ ❞♦✇♥ t❤❡ ❧✐♥❡ t♦ ❙❡s❛♠❡ ❙tr❡❡t✳ ❛✳ ❋✐❧❧ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✳ ❚❤✐s ♥✉♠❜❡r ❝♦♠❡s ❞♦✇♥ t❤❡ ❧✐♥❡ ✲✺ ✲✸ ✲✶ ✷ ✹ ✻ ✶✵ x 2x ◆✐❝♦❧❛s ❝♦♠❡s ✉♣ ✇✐t❤ t❤✐s ♥✉♠✲ ❜❡r✱ ❛♥❞ s❡♥❞s ✐t ❞♦✇♥ t❤❡ ❧✐♥❡ t♦ ❑❛t✐❡ ❑❛t✐❡ t❤❡♥ s♣✐ts ♦✉t t❤✐s ♥✉♠❜❡r ❚❛❜❧❡ ✶✳✾ ❜✳ ■♥ ❛ ♠❛ss✐✈❡ ❞♦✇♥s✐③✐♥❣ ❡✛♦rt✱ ②♦✉ ❛r❡ ❣♦✐♥❣ t♦ ✜r❡ ◆✐❝♦❧❛s✳ ❑❛t✐❡ ✐s ❣♦✐♥❣ t♦ t❛❦❡ ♦✈❡r ❜♦t❤ ❢✉♥❝t✐♦♥s ✭◆✐❝♦❧❛s✬s ❛♥❞ ❤❡r ♦✇♥✮✳ ❙♦ ②♦✉ ✇❛♥t t♦ ❣✐✈❡ ❑❛t✐❡ ❛ ♥✉♠❜❡r✱ ❛♥❞ s❤❡ ✜rst ❞♦❡s ◆✐❝♦❧❛s✬s ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡♥ ❤❡r ♦✇♥✳ ❇✉t ♥♦✇ ❑❛t✐❡ ✐s ♦✈❡r✇♦r❦❡❞✱ s♦ s❤❡ ❝♦♠❡s ✉♣ ✇✐t❤ ❛ s❤♦rt❝✉t✿ ♦♥❡ ❢✉♥❝t✐♦♥ s❤❡ ❝❛♥ ❞♦✱ t❤❛t ❝♦✈❡rs ❜♦t❤ ◆✐❝♦❧❛s✬s ❥♦❜ ❛♥❞ ❤❡r ♦✇♥✳ ❲❤❛t ❞♦❡s ❑❛t✐❡ ❞♦ t♦ ❡❛❝❤ ♥✉♠❜❡r ②♦✉ ❣✐✈❡ ❤❡r❄ ✭❆♥s✇❡r ✐♥ ✇♦r❞s✳✮ ❊①❡r❝✐s❡ ✶✳✻✺ ❚❛②❧♦r ✐s ❞r✐✈✐♥❣ ❛ ♠♦t♦r❝②❝❧❡ ❛❝r♦ss t❤❡ ❝♦✉♥tr②✳ ❊❛❝❤ ❞❛② ❤❡ ❝♦✈❡rs ✺✵✵ ♠✐❧❡s✳ ❆ ♣♦❧✐❝❡♠❛♥ st❛rt❡❞ t❤❡ s❛♠❡ ♣❧❛❝❡ ❚❛②❧♦r ❞✐❞✱ ✇❛✐t❡❞ ❛ ✇❤✐❧❡✱ ❛♥❞ t❤❡♥ t♦♦❦ ♦✛✱ ❤♦♣✐♥❣ t♦ ❝❛t❝❤ s♦♠❡ ✐❧❧❡❣❛❧ ❛❝t✐✈✐t②✳ ❚❤❡ ♣♦❧✐❝❡♠❛♥ st♦♣s ❡❛❝❤ ❞❛② ❡①❛❝t❧② ✜✈❡ ♠✐❧❡s ❜❡❤✐♥❞ ❚❛②❧♦r✳ ▲❡t d ❡q✉❛❧ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s t❤❡② ❤❛✈❡ ❜❡❡♥ ❞r✐✈✐♥❣✳ ✭❙♦ ❛❢t❡r t❤❡ ✜rst ❞❛②✱ d = 1✳✮ ▲❡t T ❜❡ p ❡q✉❛❧ t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s t❤❡ ♣♦❧✐❝❡♠❛♥ ❤❛s ❞r✐✈❡♥✳ t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s ❚❛②❧♦r ❤❛s ❞r✐✈❡♥✳ ▲❡t ❛✳ ❆❢t❡r t❤r❡❡ ❞❛②s✱ ❤♦✇ ❢❛r ❤❛s ❚❛②❧♦r ❣♦♥❡❄ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❜✳ ❍♦✇ ❢❛r ❤❛s t❤❡ ♣♦❧✐❝❡♠❛♥ ❣♦♥❡❄ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❝✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ T (d) t❤❛t ❣✐✈❡s t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s ❚❛②❧♦r ❤❛s tr❛✈❡❧❡❞✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❤♦✇ ♠❛♥② ❞❛②s ❤❡ ❤❛s ❜❡❡♥ tr❛✈❡❧✐♥❣✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❞✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ p (T ) t❤❛t ❣✐✈❡s t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡ t❤❡ ♣♦❧✐❝❡♠❛♥ ❤❛s tr❛✈❡❧❡❞✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❞✐st❛♥❝❡ t❤❛t ❚❛②❧♦r ❤❛s tr❛✈❡❧❡❞✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❡✳ ◆♦✇ ✇r✐t❡ t❤❡ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥ p (T (d)) t❤❛t ❣✐✈❡s t❤❡ ♥✉♠❜❡r ♦❢ ♠✐❧❡s t❤❡ ♣♦❧✐❝❡✲ ♠❛♥ ❤❛s tr❛✈❡❧❡❞✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ❤❡ ❤❛s ❜❡❡♥ tr❛✈❡❧✐♥❣✳ ❊①❡r❝✐s❡ ✶✳✻✻ ❘❛s❤♠✐ ✐s ❛ ❤♦♥♦r st✉❞❡♥t ❜② ❞❛②❀ ❜✉t ❜② ♥✐❣❤t✱ s❤❡ ✇♦r❦s ❛s ❛ ❤✐t ♠❛♥ ❢♦r t❤❡ ♠♦❜✳ ❊❛❝❤ ♠♦♥t❤ s❤❡ ❣❡ts ♣❛✐❞ ✩✶✵✵✵ ❜❛s❡✱ ♣❧✉s ❛♥ ❡①tr❛ ✩✶✵✵ ❢♦r ❡❛❝❤ ♣❡rs♦♥ s❤❡ ❦✐❧❧s✳ ❖❢ ❝♦✉rs❡✱ s❤❡ ❣❡ts ♣❛✐❞ ✐♥ ❝❛s❤✖❛❧❧ ✩✷✵ ❜✐❧❧s✳ ▲❡t k ❡q✉❛❧ t❤❡ ♥✉♠❜❡r ♦❢ ♣❡♦♣❧❡ ❘❛s❤♠✐ ❦✐❧❧s ✐♥ ❛ ❣✐✈❡♥ ♠♦♥t❤✳ ▲❡t ♠ ❜❡ t❤❡ ❛♠♦✉♥t ♦❢ ♠♦♥❡② s❤❡ ✐s ♣❛✐❞ t❤❛t ♠♦♥t❤✱ ✐♥ ❞♦❧❧❛rs✳ ▲❡t ✶✹ ❚❤✐s b ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ✩✷✵ ❜✐❧❧s s❤❡ ❣❡ts✳ ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✶✾✶✵✾✴✶✳✶✴❃✳ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✸✷ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ m (k) t❤❛t t❡❧❧s ❤♦✇ ♠✉❝❤ ♠♦♥❡② ❘❛s❤♠✐ ♠❛❦❡s✱ ✐♥ ❛ ❣✐✈❡♥ ♠♦♥t❤✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣❡♦♣❧❡ s❤❡ ❦✐❧❧s✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❜✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ b (m) t❤❛t t❡❧❧s ❤♦✇ ♠❛♥② ❜✐❧❧s ❘❛s❤♠✐ ❣❡ts✱ ✐♥ ❛ ❣✐✈❡♥ ♠♦♥t❤✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❞♦❧❧❛rs s❤❡ ♠❛❦❡s✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❝✳ ❲r✐t❡ ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥ b (m (k))❜ t❤❛t ❣✐✈❡s t❤❡ ♥✉♠❜❡r ♦❢ ❜✐❧❧s ❘❛s❤♠✐ ❣❡ts✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣❡♦♣❧❡ s❤❡ ❦✐❧❧s✳ ❞✳ ■❢ ❘❛s❤♠✐ ❦✐❧❧s ✺ ♠❡♥ ✐♥ ❛ ♠♦♥t❤✱ ❤♦✇ ♠❛♥② ✩✷✵ ❜✐❧❧s ❞♦❡s s❤❡ ❡❛r♥❄ ❋✐rst✱ tr❛♥s❧❛t❡ t❤✐s q✉❡st✐♦♥ ✐♥t♦ ❢✉♥❝t✐♦♥ ♥♦t❛t✐♦♥✖t❤❡♥ s♦❧✈❡ ✐t ❢♦r ❛ ♥✉♠❜❡r✳ ❡✳ ■❢ ❘❛s❤♠✐ ❡❛r♥s ✶✵✵ ✩✷✵ ❜✐❧❧s ✐♥ ❛ ♠♦♥t❤✱ ❤♦✇ ♠❛♥② ♠❡♥ ❞✐❞ s❤❡ ❦✐❧❧❄ ❋✐rst✱ tr❛♥s❧❛t❡ t❤✐s q✉❡st✐♦♥ ✐♥t♦ ❢✉♥❝t✐♦♥ ♥♦t❛t✐♦♥✖t❤❡♥ s♦❧✈❡ ✐t ❢♦r ❛ ♥✉♠❜❡r✳ ❊①❡r❝✐s❡ ✶✳✻✼ ▼❛❦❡ ✉♣ ❛ ♣r♦❜❧❡♠ ❧✐❦❡ ❡①❡r❝✐s❡s ★✷ ❛♥❞ ★✸✳ ❇❡ s✉r❡ t♦ t❛❦❡ ❛❧❧ t❤❡ r✐❣❤t st❡♣s✿ ❞❡✜♥❡ t❤❡ s❝❡♥❛r✐♦✱ ❞❡✜♥❡ ②♦✉r ✈❛r✐❛❜❧❡s ❝❧❡❛r❧②✱ ❛♥❞ t❤❡♥ s❤♦✇ t❤❡ ❢✉♥❝t✐♦♥s t❤❛t r❡❧❛t❡ t❤❡ ✈❛r✐❛❜❧❡s✳ ❚❤✐s ✐s ❥✉st ❧✐❦❡ t❤❡ ♣r♦❜❧❡♠s ✇❡ ❞✐❞ ❧❛st ✇❡❡❦✱ ❡①❝❡♣t t❤❛t ②♦✉ ❤❛✈❡ t♦ ✉s❡ t❤r❡❡ ✈❛r✐❛❜❧❡s✱ r❡❧❛t❡❞ ❜② ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥✳ ❊①❡r❝✐s❡ ✶✳✻✽ f (x) = √ x+2 ✳ g (x) = x2 + x ✳ ❛✳ f (7) = ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❜✳ g (7) = ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❝✳ f (g (x)) =❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❞✳ f (f (x)) = ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❡✳ g (f (x)) = ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❢✳ g (g (x)) = ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❣✳ f (g (3)) =❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✻✾ h (x) = x − 5✳ h (i (x)) = x✳ ❈❛♥ ②♦✉ ✜♥❞ ✇❤❛t ❢✉♥❝t✐♦♥ i (x) ✐s✱ t♦ ♠❛❦❡ t❤✐s ❤❛♣♣❡♥❄ ✶✺ ✶✳✶✺ ❍♦♠❡✇♦r❦✿ ❈♦♠♣♦s✐t❡ ❋✉♥❝t✐♦♥s ❊①❡r❝✐s❡ ✶✳✼✵ ❆♥ ✐♥❝❤✇♦r♠ ✭❡①❛❝t❧② ♦♥❡ ✐♥❝❤ ❧♦♥❣✱ ♦❢ ❝♦✉rs❡✮ ✐s ❝r❛✇❧✐♥❣ ✉♣ ❛ ②❛r❞st✐❝❦ ✭❣✉❡ss ❤♦✇ ❧♦♥❣ t❤❛t ✐s❄✮✳ ❆❢t❡r t❤❡ ✜rst ❞❛②✱ t❤❡ ✐♥❝❤✇♦r♠✬s ❤❡❛❞ ✭❧❡t✬s ❥✉st ❛ss✉♠❡ t❤❛t✬s ❛t t❤❡ ❢r♦♥t✮ ✐s ❛t t❤❡ ✸✧ ♠❛r❦✳ ❆❢t❡r t❤❡ s❡❝♦♥❞ ❞❛②✱ t❤❡ ✐♥❝❤✇♦r♠✬s ❤❡❛❞ ✐s ❛t t❤❡ ✻✧ ♠❛r❦✳ ❆❢t❡r t❤❡ t❤✐r❞ ❞❛②✱ t❤❡ ✐♥❝❤✇♦r♠✬s ❤❡❛❞ ✐s ❛t t❤❡ ✾✧ ♠❛r❦✳ ▲❡t h d ❡q✉❛❧ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s t❤❡ ✇♦r♠ ❤❛s ❜❡❡♥ ❝r❛✇❧✐♥❣✳ ✭❙♦ ❛❢t❡r t❤❡ ✜rst ❞❛②✱ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❝❤❡s t❤❡ ❤❡❛❞ ❤❛s ❣♦♥❡✳ ▲❡t t d = 1✳✮ ▲❡t ❜❡ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ✇♦r♠✬s t❛✐❧✳ ❛✳ ❆❢t❡r ✶✵ ❞❛②s✱ ✇❤❡r❡ ✐s t❤❡ ✐♥❝❤✇♦r♠✬s ❤❡❛❞❄ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❜✳ ■ts t❛✐❧❄ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❝✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ h (d) t❤❛t ❣✐✈❡s t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❝❤❡s t❤❡ ❤❡❛❞ ❤❛s tr❛✈❡❧❡❞✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❤♦✇ ♠❛♥② ❞❛②s t❤❡ ✇♦r♠ ❤❛s ❜❡❡♥ tr❛✈❡❧✐♥❣✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❞✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t (h) t❤❛t ❣✐✈❡s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ t❛✐❧✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❤❡❛❞✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❡✳ ◆♦✇ ✇r✐t❡ t❤❡ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥ t (h (d)) t❤❛t ❣✐✈❡s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ t❛✐❧✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s t❤❡ ✇♦r♠ ❤❛s ❜❡❡♥ tr❛✈❡❧✐♥❣✳ ✶✺ ❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✶✾✶✵✼✴✶✳✶✴❃✳ ✸✸ ❊①❡r❝✐s❡ ✶✳✼✶ ➣ ❚❤❡ ♣r✐❝❡ ♦❢ ❣❛s st❛rt❡❞ ♦✉t ❛t ✶✵✵ ✴❣❛❧❧♦♥ ♦♥ t❤❡ ✶st ♦❢ t❤❡ ♠♦♥t❤✳ ❊✈❡r② ❞❛② s✐♥❝❡ t❤❡♥✱ ✐t ❤❛s ➣ ❣♦♥❡ ✉♣ ✷ ✴❣❛❧❧♦♥✳ ▼② ❝❛r t❛❦❡s ✶✵ ❣❛❧❧♦♥s ♦❢ ❣❛s✳ ✭❆s ②♦✉ ♠✐❣❤t ❤❛✈❡ ❣✉❡ss❡❞✱ t❤❡s❡ ♥✉♠❜❡rs ❛r❡ ❛❧❧ ✜❝t✐♦♥❛❧✳✮ ▲❡t d ❡q✉❛❧ t❤❡ ❞❛t❡ ✭s♦ t❤❡ ✶st ♦❢ t❤❡ ♠♦♥t❤ ✐s ✶✱ ❛♥❞ s♦ ♦♥✮✳ ▲❡t ♦❢ ❣❛s✱ ✐♥ ❝❡♥ts✳ ▲❡t c g ❡q✉❛❧ t❤❡ ♣r✐❝❡ ♦❢ ❛ ❣❛❧❧♦♥ ❡q✉❛❧ t❤❡ t♦t❛❧ ♣r✐❝❡ r❡q✉✐r❡❞ t♦ ✜❧❧ ✉♣ ♠② ❝❛r✱ ✐♥ ❝❡♥ts✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ g (d) t❤❛t ❣✐✈❡s t❤❡ ♣r✐❝❡ ♦❢ ❣❛s ♦♥ ❛♥② ❣✐✈❡♥ ❞❛② ♦❢ t❤❡ ♠♦♥t❤✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❜✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ c (g) t❤❛t t❡❧❧s ❤♦✇ ♠✉❝❤ ♠♦♥❡② ✐t t❛❦❡s t♦ ✜❧❧ ✉♣ ♠② ❝❛r✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r✐❝❡ ♦❢ ❛ ❣❛❧❧♦♥ ♦❢ ❣❛s✳ ❴❴❴❴❴❴❴❴❴❴❴❴❴❴ ❝✳ ❲r✐t❡ ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥ c (g (d)) t❤❛t ❣✐✈❡s t❤❡ ❝♦st ♦❢ ✜❧❧✐♥❣ ✉♣ ♠② ❝❛r ♦♥ ❛♥② ❣✐✈❡♥ ❞❛② ♦❢ t❤❡ ♠♦♥t❤✳ ❞✳ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦❡s ✐t t❛❦❡ t♦ ✜❧❧ ✉♣ ♠② ❝❛r ♦♥ t❤❡ ✶✶t❤ ♦❢ t❤❡ ♠♦♥t❤❄ ❋✐rst✱ tr❛♥s❧❛t❡ t❤✐s q✉❡st✐♦♥ ✐♥t♦ ❢✉♥❝t✐♦♥ ♥♦t❛t✐♦♥✖t❤❡♥ s♦❧✈❡ ✐t ❢♦r ❛ ♥✉♠❜❡r✳ ➣ ❡✳ ❖♥ ✇❤❛t ❞❛② ❞♦❡s ✐t ❝♦st ✶✱✵✹✵ ✭♦t❤❡r✇✐s❡ ❦♥♦✇♥ ❛s ✩✶✵✳✹✵✮ t♦ ✜❧❧ ✉♣ ♠② ❝❛r❄ ❋✐rst✱ tr❛♥s❧❛t❡ t❤✐s q✉❡st✐♦♥ ✐♥t♦ ❢✉♥❝t✐♦♥ ♥♦t❛t✐♦♥✖t❤❡♥ s♦❧✈❡ ✐t ❢♦r ❛ ♥✉♠❜❡r✳ ❊①❡r❝✐s❡ ✶✳✼✷ ▼❛❦❡ ✉♣ ❛ ♣r♦❜❧❡♠ ❧✐❦❡ ♥✉♠❜❡rs ✶ ❛♥❞ ✷✳ ❇❡ s✉r❡ t♦ t❛❦❡ ❛❧❧ t❤❡ r✐❣❤t st❡♣s✿ ❞❡✜♥❡ t❤❡ s❝❡♥❛r✐♦✱ ❞❡✜♥❡ ②♦✉r ✈❛r✐❛❜❧❡s ❝❧❡❛r❧②✱ ❛♥❞ t❤❡♥ s❤♦✇ t❤❡ ✭❝♦♠♣♦s✐t❡✮ ❢✉♥❝t✐♦♥s t❤❛t r❡❧❛t❡ t❤❡ ✈❛r✐❛❜❧❡s✳ ❊①❡r❝✐s❡ ✶✳✼✸ f (x) = x x2 +3x+4 ✳ ❋✐♥❞ f (g (x)) ✐❢.

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