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  1. ■◆❚❊◆❙❊ ❆❯❚❖▼❖❘P❍■❙▼❙ ❖❋ ●❘❖❯P❙ ▼✐♠❛ ❙t❛♥♦❥❦♦✈s❦✐ ❋❘❆❯❊◆ ■◆ ❉❊❘ ▼❆❚❍❊▼❆❚■❑ ❑♦♥st❛♥③✱ ✷✹ ❏❛♥✉❛r② ✷✵✶✼

  2. ❏♦✐♥t ✇♦r❦ ✇✐t❤

  3. ❍❡♥❞r✐❦ ▲❡♥str❛ ✭❯▲✮

  4. ❆♥❞r❡❛ ▲✉❝❝❤✐♥✐ ✭❯♥✐P❉✮

  5. ❏♦♥ ●♦♥③á❧❡③ ❙á♥❝❤❡③ ✭❊❍❯✮

  6. ●r♦✉♣s ❛♥❞ s②♠♠❡tr✐❡s

  7. ▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ❳ ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ ❙ ♥ ❉ ✷ ♥ ●▲ ❱ ●❛❧ ❦ ❆✉t ● ❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts

  8. ❉ ✷ ♥ ●▲ ❱ ●❛❧ ❦ ❆✉t ● ❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts ▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ( ❳ ) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ • ❙ ♥

  9. ●▲ ❱ ●❛❧ ❦ ❆✉t ● ❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts ▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ( ❳ ) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ • ❙ ♥ • ❉ ✷ ♥

  10. ●❛❧ ❦ ❆✉t ● ❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts ▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ( ❳ ) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ • ❙ ♥ • ❉ ✷ ♥ • ●▲ ( ❱ )

  11. ❆✉t ● ❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts ▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ( ❳ ) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ • ❙ ♥ • ❉ ✷ ♥ • ●▲ ( ❱ ) • ●❛❧ ( ℓ/ ❦ )

  12. ❙②♠♠❡tr✐❡s ♦❢ ♦❜❥❡❝ts ▲❡t ❳ ❜❡ ❛♥ ♦❜❥❡❝t ❛♥❞ ❧❡t ❙②♠ ( ❳ ) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ✐ts s②♠♠❡tr✐❡s✳ • ❙ ♥ • ❉ ✷ ♥ • ●▲ ( ❱ ) • ●❛❧ ( ℓ/ ❦ ) • ❆✉t ( ● )

  13. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  14. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  15. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  16. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  17. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  18. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  19. ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ❙ ✻ ✇❤✐❝❤ ❤❛s ✼✷✵ ❡❧❡♠❡♥ts✳ P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts

  20. ✇❤✐❝❤ ❤❛s ✼✷✵ ❡❧❡♠❡♥ts✳ P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = ❙ ✻

  21. P❡r♠✉t❛t✐♦♥ ♦❢ ♣♦✐♥ts ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = ❙ ✻ ✇❤✐❝❤ ❤❛s ✼✷✵ ❡❧❡♠❡♥ts✳

  22. ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

  23. ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

  24. ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

  25. ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

  26. ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ❉ ✶✷ ✇❤✐❝❤ ❤❛s ✶✷ ❡❧❡♠❡♥ts✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥

  27. ✇❤✐❝❤ ❤❛s ✶✷ ❡❧❡♠❡♥ts✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥ ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = ❉ ✶✷

  28. ❙②♠♠❡tr✐❡s ♦❢ ❛ r❡❣✉❧❛r ♣♦❧②❣♦♥ ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = ❉ ✶✷ ✇❤✐❝❤ ❤❛s ✶✷ ❡❧❡♠❡♥ts✳

  29. ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ✐❞ ❳ ✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

  30. ❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥ ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = { ✐❞ ❳ , ρ } ✳

  31. ❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥ ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = { ✐❞ ❳ , ρ } ✳

  32. ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ❳ ✐❞ ❳ ✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥

  33. ❙②♠♠❡tr✐❡s ♦❢ ❛ ❝♦❧♦✉r❡❞ ♣♦❧②❣♦♥ ❚❤❡ ❣r♦✉♣ ♦❢ s②♠♠❡tr✐❡s ✐s ❙②♠ ( ❳ ) = { ✐❞ ❳ } ✳

  34. ❚❤❡♥ ❙②♠ ❱ ●▲ ❦ ❱ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s ❱ ❱ s✉❝❤ t❤❛t✿ ❢♦r ❛❧❧ ✉ ✈ ❱ ✱ ♦♥❡ ❤❛s ✉ ✈ ✉ ✈ ❀ ❢♦r ❛❧❧ ❦ ❛♥❞ ✈ ❱ ✱ ♦♥❡ ❤❛s ✈ ✈ ❀ t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦ ✱ t❤❡♥ ●▲ ❦ ❱ ●▲ ♥ ❦ ✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ ▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦ ✳

  35. ❢♦r ❛❧❧ ❦ ❛♥❞ ✈ ❱ ✱ ♦♥❡ ❤❛s ✈ ✈ ❀ t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦ ✱ t❤❡♥ ●▲ ❦ ❱ ●▲ ♥ ❦ ✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ ▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦ ✳ ❚❤❡♥ ❙②♠ ( ❱ ) = ●▲ ❦ ( ❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿ • ❢♦r ❛❧❧ ✉ , ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( ✉ + ✈ ) = α ( ✉ ) + α ( ✈ ) ❀

  36. t❤❡ ♠❛♣ ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦ ✱ t❤❡♥ ●▲ ❦ ❱ ●▲ ♥ ❦ ✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ ▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦ ✳ ❚❤❡♥ ❙②♠ ( ❱ ) = ●▲ ❦ ( ❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿ • ❢♦r ❛❧❧ ✉ , ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( ✉ + ✈ ) = α ( ✉ ) + α ( ✈ ) ❀ • ❢♦r ❛❧❧ λ ∈ ❦ ❛♥❞ ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( λ ✈ ) = λα ( ✈ ) ❀

  37. ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦ ✱ t❤❡♥ ●▲ ❦ ❱ ●▲ ♥ ❦ ✳ ❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ ▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦ ✳ ❚❤❡♥ ❙②♠ ( ❱ ) = ●▲ ❦ ( ❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿ • ❢♦r ❛❧❧ ✉ , ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( ✉ + ✈ ) = α ( ✉ ) + α ( ✈ ) ❀ • ❢♦r ❛❧❧ λ ∈ ❦ ❛♥❞ ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( λ ✈ ) = λα ( ✈ ) ❀ • t❤❡ ♠❛♣ α ✐s ❜✐❥❡❝t✐✈❡✳

  38. ❙②♠♠❡tr✐❡s ♦❢ ❛ ✈❡❝t♦r s♣❛❝❡ ▲❡t ❦ ❜❡ ❛ ✜❡❧❞ ❛♥❞ ❧❡t ❱ ❜❡ ❛ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❦ ✳ ❚❤❡♥ ❙②♠ ( ❱ ) = ●▲ ❦ ( ❱ ) ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s α : ❱ → ❱ s✉❝❤ t❤❛t✿ • ❢♦r ❛❧❧ ✉ , ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( ✉ + ✈ ) = α ( ✉ ) + α ( ✈ ) ❀ • ❢♦r ❛❧❧ λ ∈ ❦ ❛♥❞ ✈ ∈ ❱ ✱ ♦♥❡ ❤❛s α ( λ ✈ ) = λα ( ✈ ) ❀ • t❤❡ ♠❛♣ α ✐s ❜✐❥❡❝t✐✈❡✳ ■❢ ❱ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥ ♥ ♦✈❡r ❦ ✱ t❤❡♥ ●▲ ❦ ( ❱ ) ∼ = ●▲ ♥ ( ❦ ) ✳

  39. ❚❤❡♥ ❙②♠ ❦ ❆✉t ❦ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛♣s s✉❝❤ t❤❛t✿ ●▲ ❦ ❀ ❢♦r ❛❧❧ ① ② ✱ ♦♥❡ ❤❛s ①② ① ② ✳ ✭ ❢♦r ❛❧❧ ① ❦ ✱ ♦♥❡ ❤❛s ① ① ✮ ■❢ t❤❡ ❡①t❡♥s✐♦♥ ❦ ✐s ●❛❧♦✐s✱ t❤❡♥ ●❛❧ ❦ ❆✉t ❦ ✳ ❙②♠♠❡tr✐❡s ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s ▲❡t ℓ/ ❦ ❜❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ✜❡❧❞s✳

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