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sst strt tt s t ts t r t

  1. ❆ ❈❧❛ss✐☞❝❛t✐♦♥ ♦❢ ❘❡str✐❝t❡❞ ▲❛tt✐❝❡ ❲❛❧❦s ✇✐t❤ ❙♠❛❧❧ ❙t❡♣s ❙t❡♣❤❡♥ ▼❡❧❝③❡r ✭❙❋❯✱ ■◆❘■❆✲▼❙❘✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼✐r❡✐❧❧❡ ❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✭❯♥✐✈❡rs✐t✓ ❡ ❇♦r❞❡❛✉① ✶✮✱ ❆❧✐♥ ❇♦st❛♥ ✭■◆❘■❆✲▼❙❘✮✱ ▼❛♥✉❡❧ ❑❛✉❡rs ✭❘■❙❈✮✱ ❛♥❞ ▼❛r♥✐ ▼✐s❤♥❛ ✭❙❋❯✮

  2. ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ t❤❡ st❡♣ s❡t ❙ ❂ ❢ ◆❊❀ ❙❊❀ ◆❲❀ ❙❲ ❣ t❤❡r❡ ❛r❡ ✾ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ✸✿ ▲❛tt✐❝❡ ❲❛❧❦s ✐♥ t❤❡ ◗✉❛rt❡r P❧❛♥❡ Given: ❆ s❡t ♦❢ ❞✐r❡❝t✐♦♥s Count: ◆✉♠❜❡r ♦❢ ✐♥t❡❣❡r ❧❛tt✐❝❡ ✇❛❧❦s ✐♥ t❤❡ ☞rst q✉❛❞r❛♥t ✉s✐♥❣ t❤❡s❡ st❡♣s✳

  3. ▲❛tt✐❝❡ ❲❛❧❦s ✐♥ t❤❡ ◗✉❛rt❡r P❧❛♥❡ Given: ❆ s❡t ♦❢ ❞✐r❡❝t✐♦♥s Count: ◆✉♠❜❡r ♦❢ ✐♥t❡❣❡r ❧❛tt✐❝❡ ✇❛❧❦s ✐♥ t❤❡ ☞rst q✉❛❞r❛♥t ✉s✐♥❣ t❤❡s❡ st❡♣s✳ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ t❤❡ st❡♣ s❡t ❙ ❂ ❢ ◆❊❀ ❙❊❀ ◆❲❀ ❙❲ ❣ t❤❡r❡ ❛r❡ ✾ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ✸✿

  4. ■❢ ✇❡ ❝❛♥ ❝❧❛ss✐❢② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❛s ❛❧❣❡❜r❛✐❝ ♦r ❉✲❋✐♥✐t❡ ✭s❛t✐s☞❡s ❛ ❧✐♥❡❛r ❖❉❊ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✍❝✐❡♥ts✮ t❤❡♥ ✇❡ ❦♥♦✇ t❤❡ ❢♦r♠ ♦❢ ✐ts ❣r♦✇t❤✳ ●r♦✇t❤ ♦❢ ❬ t ♥ ❪ ❋ ✭ t ✮ ◆❛t✉r❡ ♦❢ ❋ ✭ t ✮ ❝☞ ♥ ♥ s ❆❧❣❡❜r❛✐❝ �✭ s ✰✶✮ ❆☞ ♥ ♥ s ❧♦❣✭ ♥ ✮ r ❉✲❋✐♥✐t❡ ◆❡✐t❤❡r ❄ ❘❡❛❧✐st✐❝ ●♦❛❧s ❋✐♥❞✐♥❣ ❛♥ ❡①❛❝t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ✐s t♦♦ ❤❛r❞ ✐♥ ❣❡♥❡r❛❧ ✲ ✇❡ s❡❡❦ asymptotic estimates ✳

  5. ❘❡❛❧✐st✐❝ ●♦❛❧s ❋✐♥❞✐♥❣ ❛♥ ❡①❛❝t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ✐s t♦♦ ❤❛r❞ ✐♥ ❣❡♥❡r❛❧ ✲ ✇❡ s❡❡❦ asymptotic estimates ✳ ■❢ ✇❡ ❝❛♥ ❝❧❛ss✐❢② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❛s ❛❧❣❡❜r❛✐❝ ♦r ❉✲❋✐♥✐t❡ ✭s❛t✐s☞❡s ❛ ❧✐♥❡❛r ❖❉❊ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✍❝✐❡♥ts✮ t❤❡♥ ✇❡ ❦♥♦✇ t❤❡ ❢♦r♠ ♦❢ ✐ts ❣r♦✇t❤✳ ●r♦✇t❤ ♦❢ ❬ t ♥ ❪ ❋ ✭ t ✮ ◆❛t✉r❡ ♦❢ ❋ ✭ t ✮ ❝☞ ♥ ♥ s ❆❧❣❡❜r❛✐❝ �✭ s ✰✶✮ ❆☞ ♥ ♥ s ❧♦❣✭ ♥ ✮ r ❉✲❋✐♥✐t❡ ◆❡✐t❤❡r ❄

  6. ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s ❡✍❝✐❡♥t❧②❀ ❛♥s✇❡r q✉❡st✐♦♥s ❛❜♦✉t r❡❧❛t❡❞ ♣❤②s✐❝❛❧ s②st❡♠s ✭❧✐♠✐t✐♥❣ ❢r❡❡ ❡♥❡r❣②✮✳ ❉✲❋✐♥✐t❡ ❋✉♥❝t✐♦♥s ■❢ ✇❡ ❦♥♦✇ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❉✲❋✐♥✐t❡✱ ✇❡ ❝❛♥✿ ❞❡t❡r♠✐♥❡ ✐ts ❛s②♠♣t♦t✐❝s ✭❢r♦♠ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✮❀

  7. ❛♥s✇❡r q✉❡st✐♦♥s ❛❜♦✉t r❡❧❛t❡❞ ♣❤②s✐❝❛❧ s②st❡♠s ✭❧✐♠✐t✐♥❣ ❢r❡❡ ❡♥❡r❣②✮✳ ❉✲❋✐♥✐t❡ ❋✉♥❝t✐♦♥s ■❢ ✇❡ ❦♥♦✇ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❉✲❋✐♥✐t❡✱ ✇❡ ❝❛♥✿ ❞❡t❡r♠✐♥❡ ✐ts ❛s②♠♣t♦t✐❝s ✭❢r♦♠ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✮❀ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s ❡✍❝✐❡♥t❧②❀

  8. ❉✲❋✐♥✐t❡ ❋✉♥❝t✐♦♥s ■❢ ✇❡ ❦♥♦✇ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❉✲❋✐♥✐t❡✱ ✇❡ ❝❛♥✿ ❞❡t❡r♠✐♥❡ ✐ts ❛s②♠♣t♦t✐❝s ✭❢r♦♠ ❞✐☛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✮❀ ❝❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ✇❛❧❦s ❡✍❝✐❡♥t❧②❀ ❛♥s✇❡r q✉❡st✐♦♥s ❛❜♦✉t r❡❧❛t❡❞ ♣❤②s✐❝❛❧ s②st❡♠s ✭❧✐♠✐t✐♥❣ ❢r❡❡ ❡♥❡r❣②✮✳

  9. ❆s②♠♣t♦t✐❝s✿ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ✷ ♥ ✿ ■s ✐t ❉✲❋✐♥✐t❡❄

  10. ■s ✐t ❉✲❋✐♥✐t❡❄ YES - it is rational! ❆s②♠♣t♦t✐❝s✿ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ✷ ♥ ✿

  11. ❆s②♠♣t♦t✐❝s✿ ♣ ✑ ✁ ✹ ♥ ✹ ✸ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ♥ ✷ ❂ ✸ ✿ ✏ ✶ ✸� ✸ ■s ✐t ❉✲❋✐♥✐t❡❄

  12. ■s ✐t ❉✲❋✐♥✐t❡❄ Yes - it is algebraic! ❆s②♠♣t♦t✐❝s✿ ♣ ✑ ✁ ✹ ♥ ✹ ✸ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ♥ ✷ ❂ ✸ ✿ ✏ ✶ ✸� ✸

  13. ❆s②♠♣t♦t✐❝s✿ ✙ ✁ ✹ ♥ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ✽ ♥ ✷ ✿ ■s ✐t ❉✲❋✐♥✐t❡❄

  14. ■s ✐t ❉✲❋✐♥✐t❡❄ Yes! (But not algebraic) ❆s②♠♣t♦t✐❝s✿ ✙ ✁ ✹ ♥ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ✽ ♥ ✷ ✿

  15. ❆s②♠♣t♦t✐❝s✿ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ✜ ✁ ✹ ♥ ❀ ✜ ✷ ✿ ■s ✐t ❉✲❋✐♥✐t❡❄

  16. ■s ✐t ❉✲❋✐♥✐t❡❄ NO! ❆s②♠♣t♦t✐❝s✿ ❬ t ♥ ❪ ❋ ✭ t ✮ ✘ ✜ ✁ ✹ ♥ ❀ ✜ ✷ R ✿

  17. ●✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳ ❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮ ❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s

  18. ❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮ ❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s ●✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳

  19. ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ♥♦♥ ❉✲❋✐♥✐t❡♥❡ss✿ ■t❡r❛t❡❞ ❑❡r♥❡❧ ▼❡t❤♦❞ ❇♦✉♥❞❛r② ❱❛❧✉❡ ▼❡t❤♦❞ ❆s②♠♣t♦t✐❝ ❋♦r♠ ✭r❡s✉❧ts ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ●✲❋✉♥❝t✐♦♥s✮ ❉✲❋✐♥✐t❡♥❡ss ✐s ❛ ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt②✳ ✳ ✳ ❇✉t ❤♦✇ ❞♦ ✇❡ ☞♥❞ ✐t❄ ❚♦♦❧s ❢♦r ♣r♦✈✐♥❣ ❉✲❋✐♥✐t❡♥❡ss✿ ❈❧♦s✉r❡ ♣r♦♣❡rt✐❡s ♦❢ ❉✲❋✐♥✐t❡ ❢✉♥❝t✐♦♥s ●✉❡ss ❛♥❞ ❈❤❡❝❦ ▼❡t❤♦❞s ✭❇♦st❛♥ ✫ ❑❛✉❡rs✮✳ ❙❤♦✇ t❤❡ ●❋ ✐s t❤❡ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✭❇♦✉sq✉❡t✲▼✓ ❡❧♦✉ ✫ ▼✐s❤♥❛✮✳

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