●rö❜♥❡r ❇❛s❡s ♦❢ ❙tr✉❝t✉r❡❞ ❙②st❡♠s ❛♥❞ t❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❈r②♣t♦❧♦❣② ❏❡❛♥✲❈❤❛r❧❡s ❋❛✉❣èr❡✱ ▼♦❤❛❜ ❙❛❢❡② ❊❧ ❉✐♥ P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r ❯P▼❈ ✕ ❈◆❘❙ ✕ ■◆❘■❆ P❛r✐s ✲ ❘♦❝q✉❡♥❝♦✉rt ▲■P✻ ✕ ❙❆▲❙❆ t❡❛♠ ❘✉❤r ❯♥✐✈❡rs✐tät ✕ ▼❆❨❆ ❲♦r❦s❤♦♣ ✷✵✶✶ ✷✵✶✶✱ ✵✼✴✷✽ ✶✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄ ❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ❈r②♣t♦ ♣r✐♠✐t✐✈❡ ✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄ ❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄ ❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❆❧❣❡❜r❛✐❝ ❈r②♣t❛♥❛❧②s✐s ❈r②♣t♦ ♣r✐♠✐t✐✈❡ ♠♦❞❡❧✐♥❣ ❙②st❡♠ s♦❧✈✐♥❣ ■ss✉❡s ❲❤✐❝❤ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧✐♥❣ ❄ ❚r❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥s✴♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ❄ ❙♦❧✈✐♥❣ t♦♦❧s✿ ●rö❜♥❡r ❜❛s❡s ❄ ❙❆❚✲s♦❧✈❡rs ❄ ✳✳✳ ❙tr✉❝t✉r❡ ❄ ✷✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
◆♦♥✲❧✐♥❡❛r✐t② ❙❡❝✉r✐t② ❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① ✶ ✵ ❢♦r ① ✵✮✳ ② ❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿ tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳ ❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳ ✳ ✳ ✳ ■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄ ▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣② ❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿ tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳ ❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳ ✳ ✳ ✳ ■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄ ▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣② ❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t② � ❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① � = ✵✮✳ ✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿ ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳ ✳ ✳ ✳ ■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄ ▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣② ❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t② � ❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① � = ✵✮✳ ❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿ � tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ � ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳ ✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄ ▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣② ❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t② � ❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① � = ✵✮✳ ❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿ � tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ � ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳ ❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿ � ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳ ✳ ✳ ✳ ✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
❈♦♠♣❧❡①✐t② ❄ ❉❡❞✐❝❛t❡❞ ❛❧❣♦r✐t❤♠s ❄ ▼♦t✐✈❛t✐♦♥s✿ ❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡s ✐♥ ❈r②♣t♦❧♦❣② ❲❤❡r❡ ❞♦❡s t❤❡ str✉❝t✉r❡ ❝♦♠❡ ❢r♦♠ ❄ ◆♦♥✲❧✐♥❡❛r✐t② → ❙❡❝✉r✐t② � ❙♦♠❡t✐♠❡s ❜✐✭♦r ♠✉❧t✐✮✲❧✐♥❡❛r ✭❡✳❣✳ ❆❊❙ ❙✲❜♦①❡s✿ ① · ② − ✶ = ✵ ❢♦r ① � = ✵✮✳ ❆s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✴s✐❣♥❛t✉r❡✿ � tr❛♣❞♦♦r ✭❡✳❣✳ ❍❋❊✱ ▼✉❧t✐✲❍❋❊✱ ▼❝❊❧✐❡❝❡✮✳ � ❘❡❞✉❝✐♥❣ t❤❡ ❦❡② s✐③❡s ✐s ❛ ❝♦♠♠♦♥ ✐ss✉❡ → ♣♦t❡♥t✐❛❧ ✇❡❛❦♥❡ss❡s ❞✉❡ t♦ t❤❡ str✉❝t✉r❡✳ ❙②♠♠❡tr✐❡s✱ ✐♥✈❛r✐❛♥ts✿ � ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ✉♥❞❡r s♦♠❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❡✳❣✳ ▼✐♥❘❛♥❦✮✳ ✳ ✳ ✳ ■♠♣❛❝t ♦♥ t❤❡ s♦❧✈✐♥❣ ♣r♦❝❡ss ❄ ✸✴✷✽ P❏ ❙♣❛❡♥❧❡❤❛✉❡r
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