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  1. ❘❛♥❞♦♠ ❲❛❧❦s ✐♥ ❉❡♣❡♥❞❡♥t ❘❛♥❞♦♠ ❊♥✈✐r♦♥♠❡♥ts ❲❡r♥❡r ❙❝❤❡✐♥❤❛r❞t ❉✐r❦ ❑r♦❡s❡ ❲♦r❦s❤♦♣ ♦♥ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ❛t t❤❡ ♦❝❝❛s✐♦♥ ♦❢ r❡t✐r❡♠❡♥t ❊r✐❦ ✈❛♥ ❉♦♦r♥ ❙❡♣t❡♠❜❡r ✷✻✱ ✷✵✶✹ ✶ ✴ ✷✸

  2. ❖✉t❧✐♥❡ ■ ■♥tr♦❞✉❝t✐♦♥ ■ ●❡♥❡r❛❧ ❜❡❤❛✈✐♦r ■ ❊✈❛❧✉❛t✐♥❣ t❤❡ ❞r✐❢t ❢♦r ✏s✇❛♣ ♠♦❞❡❧✑ ■ ✐✐❞ ❡♥✈✐r♦♥♠❡♥t✿ tr❛♣♣✐♥❣ ♣❤❡♥♦♠❡♥♦♥ ■ ▼❛r❦♦✈ ❡♥✈✐r♦♥♠❡♥t ■ ✷✲❞❡♣❡♥❞❡♥t ❡♥✈✐r♦♥♠❡♥t ■ ▼♦✈✐♥❣ ❛✈❡r❛❣❡ ❡♥✈✐r♦♥♠❡♥t ■ ❈♦♥❝❧✉s✐♦♥s ✷ ✴ ✷✸

  3. ■♥tr♦❞✉❝t✐♦♥ ✭✶✲❞✐♠❡♥s✐♦♥❛❧✮ ❘❛♥❞♦♠ ❲❛❧❦ ✐♥ ❘❛♥❞♦♠ ❊♥✈✐r♦♥♠❡♥t ✭❘❲❘❊✮✿ ✶✳ ✏❯♥❞❡r❧②✐♥❣✑ ❡♥✈✐r♦♥♠❡♥t ❯ ✐s r❛♥❞♦♠ ❜✉t ✜①❡❞ ✭❆ss✉♠❡ ❯ st❛t✐♦♥❛r② ❛♥❞ ❡r❣♦❞✐❝ ✇✳r✳t✳ ❧♦❝❛t✐♦♥✮ ✷✳ ❘❛♥❞♦♠ ✇❛❧❦ ❢ ❳ ♥ ❀ ♥ = ✵ ❀ ✶ ❀ ✿ ✿ ✿ ❣ ✇✐t❤ ❝♦♥❞✐t✐♦♥❛❧ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s✱ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❡♥✈✐r♦♥♠❡♥t✿ P ( ❳ ♥ + ✶ = ✐ + ✶ ❥ ❳ ♥ = ✐ ❀ ❯ = ✉ ) = ☛ ✐ ( ✉ ) P ( ❳ ♥ + ✶ = ✐ � ✶ ❥ ❳ ♥ = ✐ ❀ ❯ = ✉ ) = ☞ ✐ ( ✉ ) = ✶ � ☛ ✐ ( ✉ ) ✿ ❘❲❘❊s ❡①❤✐❜✐t ✐♥t❡r❡st✐♥❣ ❛♥❞ ✉♥✉s✉❛❧ ❜❡❤❛✈✐♦r✱ ♥♦t s❡❡♥ ✐♥ ♦r❞✐♥❛r② r❛♥❞♦♠ ✇❛❧❦s✳ ✸ ✴ ✷✸

  4. ■♥tr♦❞✉❝t✐♦♥✖▼♦t✐✈❛t✐♦♥ ❘❲❘❊s ✭ ❞ ✲❞✐♠❡♥s✐♦♥❛❧✮ ✉s❡❞ ✐♥ ♣❤②s✐❝s t♦ ♠♦❞❡❧ ♠♦t✐♦♥ t❤r♦✉❣❤ ❞✐s♦r❣❛♥✐③❡❞ ✭r❛♥❞♦♠✮ ♠❡❞✐❛✳ ✹ ✴ ✷✸

  5. ■♥tr♦❞✉❝t✐♦♥✖●♦❛❧ ❚❤❡♦r❡t✐❝❛❧ ❜❡❤❛✈✐♦r ♦❢ ❞r✐❢t ✐♥ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❘❲❘❊ ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ ❜✉t ■ ♥✉♠❡r✐❝❛❧ ❡✈❛❧✉❛t✐♦♥❄ ■ ✐♥✢✉❡♥❝❡ ♦❢ ❞❡♣❡♥❞❡♥t ❡♥✈✐r♦♥♠❡♥ts❄ ■ ❞ ✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡❄❄ ❖✉r ❣♦❛❧✿ ❞❡✈❡❧♦♣ t❤❡♦r② ❛♥❞ ♠❡t❤♦❞s t♦ ❝♦♠♣✉t❡ t❤❡ ❞r✐❢t ♦❢ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ❢♦r ❞❡♣❡♥❞❡♥t ❡♥✈✐r♦♥♠❡♥ts✳ ❈✉rr❡♥t r❡s✉❧ts✿ ❢♦r ✶✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ✺ ✴ ✷✸

  6. ●❡♥❡r❛❧ ❜❡❤❛✈✐♦r ❉r✐❢t ❜❡❤❛✈✐♦r ♦❢ ❢ ❳ ♥ ❣ ❢♦❧❧♦✇s✱ ✐♥ ♣r✐♥❝✐♣❧❡✱ ❢r♦♠ ❙♦❧♦♠♦♥ ✶✾✼✺✱ ❆❧✐❧✐ ✶✾✾✾ ❑❡② q✉❛♥t✐t✐❡s✿ ✛ ✐ = ✛ ✐ ( ✉ ) = ☞ ✐ ( ✉ ) ☛ ✐ ( ✉ ) ❙ = ✶ + ✛ ✶ + ✛ ✶ ✛ ✷ + ✛ ✶ ✛ ✷ ✛ ✸ + ✁ ✁ ✁ ❛♥❞ ✶ ✶ ✶ ❋ = ✶ + + + + ✁ ✁ ✁ ✛ � ✶ ✛ � ✶ ✛ � ✷ ✛ � ✶ ✛ � ✷ ✛ � ✸ ✻ ✴ ✷✸

  7. ●❡♥❡r❛❧ ❜❡❤❛✈✐♦r✖❚r❛♥s✐❡♥❝❡✴❘❡❝✉rr❡♥❝❡ ❲✐❧❧ ❢ ❳ ♥ ❣ ✉❧t✐♠❛t❡❧② ♠♦✈❡ t♦ + ✶ ♦r �✶ ✱ ♦r ♥♦t❄ ❚❤❡♦r❡♠ ✶ ✭❙♦❧♦♠♦♥✕❆❧✐❧✐✮ ✶✳ ■❢ ❊ [ ❧♥ ✛ ✵ ] ❁ ✵ ✱ t❤❡♥ ❛❧♠♦st s✉r❡❧② ❧✐♠ ♥ ✦✶ ❳ ♥ = ✶ ✿ ✷✳ ■❢ ❊ [ ❧♥ ✛ ✵ ] ❃ ✵ ✱ t❤❡♥ ❛❧♠♦st s✉r❡❧② ❧✐♠ ♥ ✦✶ ❳ ♥ = �✶ ✿ ✸✳ ■❢ ❊ [ ❧♥ ✛ ✵ ] = ✵ ✱ t❤❡♥ ❛❧♠♦st s✉r❡❧② ❧✐♠ ✐♥❢ ♥ ✦✶ ❳ ♥ = �✶ ❛♥❞ ❧✐♠ s✉♣ ♥ ✦✶ ❳ ♥ = ✶ ✿ ✼ ✴ ✷✸

  8. ●❡♥❡r❛❧ ❜❡❤❛✈✐♦r✖▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ■❢ ❢ ❳ ♥ ❣ ✉❧t✐♠❛t❡❧② ♠♦✈❡s t♦ + ✶ ♦r �✶ ✱ ❤♦✇ ❢❛st❄ ❚❤❡♦r❡♠ ✷ ✭❙♦❧♦♠♦♥✕❆❧✐❧✐✮ ❳ ♥ ✶ ✶✳ ■❢ ❊ [ ❙ ] ❁ ✶ ✱ t❤❡♥ ❛❧♠♦st s✉r❡❧② ❧✐♠ ♥ = ✷ ❊ [ ❙ ] � ✶ ✿ ♥ ✦✶ ❳ ♥ � ✶ ✷✳ ■❢ ❊ [ ❋ ] ❁ ✶ ✱ t❤❡♥ ❛❧♠♦st s✉r❡❧② ❧✐♠ ♥ = ✷ ❊ [ ❋ ] � ✶ ✿ ♥ ✦✶ ❳ ♥ ✸✳ ■❢ ❊ [ ❙ ] = ✶ ❛♥❞ ❊ [ ❋ ] = ✶ ✱ t❤❡♥ ❛❧♠♦st s✉r❡❧② ❧✐♠ ♥ = ✵ ✳ ♥ ✦✶ ❧✐♠ ♥ ✦✶ ❳ ♥ ❂ ♥ ✐s t❤❡ ❞r✐❢t ♦❢ t❤❡ ♣r♦❝❡ss ❢ ❳ ♥ ❣ ✱ ❞❡♥♦t❡ ❜② ❱ ✳ ✽ ✴ ✷✸

  9. ❙✇❛♣ ♠♦❞❡❧ ■ ❙✐♠♣❧❡ ❜✉t ✈❡rs❛t✐❧❡ ♠♦❞❡❧ ✇✐t❤ r✐❝❤ ❜❡❤❛✈✐♦r ■ ☛ ✐ ( ✉ ) ❛♥❞ ☞ ✐ ( ✉ ) ♦♥❧② ❞❡♣❡♥❞ ♦♥ ✉ ✈✐❛ ✉ ✐ ■ ❊❛❝❤ ❯ ✐ ❝❛♥ ♦♥❧② t❛❦❡ ✈❛❧✉❡ � ✶ ✭s✇❛♣✮ ♦r + ✶ ✭♥♦ s✇❛♣✮✳ ■ ❋♦r s♦♠❡ ✜①❡❞ ✈❛❧✉❡ ♣ ✐♥ ( ✵ ❀ ✶ ) ✱ ❧❡t ☞ ✐ ( ✉ ) = ✶ � ☛ ✐ ( ✉ ) ❛♥❞ � ✐❢ ✉ ✐ = ✶ ♣ ☛ ✐ ( ✉ ) = ✶ � ♣ ✐❢ ✉ ✐ = � ✶ ■ ❆s ❛ r❡s✉❧t ✇❡ ❤❛✈❡ � ❯ ✐ ✶ � ♣ P ( ❯ ✐ = � ✶ ) + ✶ � ♣ ♣ � ✶ � ♣ ✛ ✐ = P ( ❯ ✐ = ✶ ) = ♣ ♣ ❘❲✿ ❘❲❘❊✿ ✾ ✴ ✷✸

  10. ❉❡♣❡♥❞❡♥❝② str✉❝t✉r❡ ❙t❛♥❞❛r❞ ✐✐❞ ❝❛s❡ ✭❙✐♥❛✐ ✶✾✽✸✮✿ ■ ❆ss✉♠❡s t❤❡ ❢ ❯ ✐ ❣ ❛r❡ ✐✐❞ ✇✐t❤✱ ❢♦r s♦♠❡ ✵ ❁ ☛ ❁ ✶✱ P ( ❯ ✐ = ✶ ) = ☛❀ P ( ❯ ✐ = � ✶ ) = ✶ � ☛✿ ❖t❤❡r ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤❡ ❡♥✈✐r♦♥♠❡♥t st❛t❡s ❢ ❯ ✐ ❣ ✿ ■ ▼❛r❦♦✈ ❡♥✈✐r♦♥♠❡♥t ■ ❦ ✲❞❡♣❡♥❞❡♥t ♣r♦❝❡ss ■ ▼♦✈✐♥❣ ❛✈❡r❛❣❡ ♦❢ ❛♥ ✐✐❞ ❡♥✈✐r♦♥♠❡♥t✿ ✶✵ ✴ ✷✸

  11. ■✐❞ ❝❛s❡ P❛r❛♠❡t❡rs ☛ ❛♥❞ ♣ ✳ � � ✶ � ♣ ❊ [ ❧♥ ✛ ✵ ] = ( ✶ � ✷ ☛ ) ❧♥ ♣ ❍❡♥❝❡ ❳ ♥ ✦ + ✶ ❛✳s✳ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r ☛ ❃ ✶ ❂ ✷ ❛♥❞ ♣ ❃ ✶ ❂ ✷✱ ♦r ☛ ❁ ✶ ❂ ✷ ❛♥❞ ♣ ❁ ✶ ❂ ✷✳ ✶ ✶ � ❯ ✶ �� ♥ � �� ✶ � � ✶ � ♣ ❊ [ ❙ ] = ❊ [ ✛ ✶ ✁ ✁ ✁ ✛ ♥ ] = = ❊ ♣ �� � ❯ ✶ � ✶ � ♣ ♥ = ✵ ♥ = ✵ ✶ � ❊ ♣ �� � ❯ ✶ � ✶ � ♣ ✶ � ♣ ( ✶ � ☛ ) + ✶ � ♣ ♣ ✐✛ = ♣ ☛ ❁ ✶ ✿ ❊ ♣ ❍❡♥❝❡ ✇❤❡♥ ☛ ❃ ✶ ❂ ✷ ❛♥❞ ♣ ✷ ( ✶ ❂ ✷ ❀ ☛ ) ✱ ♦r ☛ ❁ ✶ ❂ ✷ ❛♥❞ ♣ ✷ ( ☛❀ ✶ ❂ ✷ ) ✱ t❤❡ ❞r✐❢t ✐s ✶ ☛ � ♣ ❱ = ✷ ❊ [ ❙ ] � ✶ = ( ✷ ♣ � ✶ ) ☛ ( ✶ � ♣ ) + ( ✶ � ☛ ) ♣ ❃ ✵ ✿ ✶✶ ✴ ✷✸

  12. ■✐❞ ❝❛s❡ 1 V = 0 V = 0 ↑ X → −∞ X → ∞ p V < 0 V > 0 X → −∞ X → ∞ 1 2 V > 0 V < 0 X → −∞ X → ∞ V = 0 V = 0 X → −∞ X → ∞ 1 α → 0 1 2 ❋✐❣✉r❡ ✶✿ ❉r✐❢t r❡❣✐♠❡s ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♣ ❛♥❞ ☛ ✳ ●r❡② ❛r❡❛s✿ ♣r♦❝❡ss tr❛♥s✐❡♥t✱ ❜✉t ❞r✐❢t ✐s ③❡r♦✦ ✶✷ ✴ ✷✸

  13. ■ ❈♦♥s✐❞❡r ☛ ✸ ❂ ✹✱ ✐✳❡✳ ✷✺✪ ♦❢ ❧♦❝❛t✐♦♥s ❛r❡ ❜❧❛❝❦ ✭s✇❛♣♣❡❞✮ ■ ❲❤❡♥ ✶ ❂ ✷ ❁ ♣ ❁ ✸ ❂ ✹✱ ❞r✐❢t t♦ t❤❡ r✐❣❤t ✐s ✬♦❜✈✐♦✉s✬ ■ ❲❤❡♥ ✸ ❂ ✹ ❁ ♣ ❁ ✶✱ ■ st✐❧❧ ✭str♦♥❣✮ ♣✉s❤ t♦ t❤❡ r✐❣❤t ✐♥ ♠❛❥♦r✐t② ♦❢ ❧♦❝❛t✐♦♥s ✭❤❡♥❝❡ ❳ ♥ ✦ ✶ ✮ ■ ❜✉t ✐♥ r❡♠❛✐♥✐♥❣ ✷✺✪ ♦❢ ❧♦❝❛t✐♦♥s str♦♥❣ ✭✦✮ ♣✉s❤ t♦ t❤❡ ❧❡❢t ■ ❍❡♥❝❡ ❱ ✵ ✐♥ ❧❛tt❡r ❝❛s❡✿ ❞✐✣❝✉❧t t♦ ❝r♦ss ❜❛rr✐❡r ♦❢ ♠❛♥② ❛❞❥❛❝❡♥t ❜❧❛❝❦ ❧♦❝❛t✐♦♥s ✦ ■ ❆♣♣❛r❡♥t❧② t❤❡ ✏❝✉t✲♦✛✑ ✈❛❧✉❡ ❢♦r ♣ ✐s ❡q✉❛❧ t♦ ☛ ✳✳✳ ■ ✳✳✳ ✐♥ t❤❡ ✐✐❞ ❝❛s❡✦ ■✐❞ ❝❛s❡✖❚r❛♣♣✐♥❣ ♣❤❡♥♦♠❡♥♦♥ ❍♦✇ ❝❛♥ ❜♦t❤ ❳ ♥ ✦ ✶ ❛♥❞ ❱ = ✵ ❜❡ tr✉❡❄ ✭❡✳❣✳ ✇❤❡♥ ♣ ❃ ☛ ❃ ✶ ❂ ✷✮ ✶✸ ✴ ✷✸

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