❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t② ♦❢ ❛ ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ t❤❡ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ❏❛♥ ❑♦❝❤❛♥♦✇s❦✐ ❯♥✐✈❡rs✐t② ✐♥ ❑✐❡❧❝❡✴❏❛❣✐❡❧❧♦♥✐❛♥ ❯♥✐✈❡rs✐t② ✐♥ ❑r❛❦ó✇✱ P♦❧❛♥❞ ❲❛rs❛✇✱ ❆♣r✐❧ ✷✵✶✸ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼♦t✐✈❛t✐♦♥s ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② ❲❡ ♣r❡s❡♥t ❛♥ ✐s♦♠❡tr✐❝ ✈❡rs✐♦♥ ♦❢ t❤❡ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ P ✇✐t❤ ❛ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡ s♣❛❝❡ P ✐s ✐s♦♠♦r♣❤✐❝ t♦ P❡➟❝③②➠s❦✐✬s s♣❛❝❡ ✇✐t❤ ❛ ✉♥✐✈❡rs❛❧ ❜❛s✐s ❛s ✇❡❧❧ ❛s t♦ ❑❛❞❡❝✬s ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ s♣❛❝❡ ✇✐t❤ t❤❡ ❜♦✉♥❞❡❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt②✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼♦t✐✈❛t✐♦♥s ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② P❡➟❝③②➠s❦✐ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❑❛❞❡❝ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r t❤❡ ❝❧❛ss ♦❢ s♣❛❝❡s ✇✐t❤ t❤❡ ❇❆P✳ P❡➟❝③②➠s❦✐ s❤♦✇❡❞ t❤❛t ❡✈❡r② ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❇❆P ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ s♣❛❝❡ ✇✐t❤ ❛ ❜❛s✐s✳ P❡➟❝③②➠s❦✐ ✫ ❲♦❥t❛s③❝③②❦ ❝♦♥str✉❝t❡❞ ❛♥ ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r ❛❧❧ s♣❛❝❡s ✇✐t❤ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼♦t✐✈❛t✐♦♥s ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② P❡➟❝③②➠s❦✐ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❑❛❞❡❝ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r t❤❡ ❝❧❛ss ♦❢ s♣❛❝❡s ✇✐t❤ t❤❡ ❇❆P✳ P❡➟❝③②➠s❦✐ s❤♦✇❡❞ t❤❛t ❡✈❡r② ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❇❆P ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ s♣❛❝❡ ✇✐t❤ ❛ ❜❛s✐s✳ P❡➟❝③②➠s❦✐ ✫ ❲♦❥t❛s③❝③②❦ ❝♦♥str✉❝t❡❞ ❛♥ ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r ❛❧❧ s♣❛❝❡s ✇✐t❤ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼♦t✐✈❛t✐♦♥s ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② P❡➟❝③②➠s❦✐ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❑❛❞❡❝ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r t❤❡ ❝❧❛ss ♦❢ s♣❛❝❡s ✇✐t❤ t❤❡ ❇❆P✳ P❡➟❝③②➠s❦✐ s❤♦✇❡❞ t❤❛t ❡✈❡r② ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❇❆P ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ s♣❛❝❡ ✇✐t❤ ❛ ❜❛s✐s✳ P❡➟❝③②➠s❦✐ ✫ ❲♦❥t❛s③❝③②❦ ❝♦♥str✉❝t❡❞ ❛♥ ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r ❛❧❧ s♣❛❝❡s ✇✐t❤ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼♦t✐✈❛t✐♦♥s ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② P❡➟❝③②➠s❦✐ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❑❛❞❡❝ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r t❤❡ ❝❧❛ss ♦❢ s♣❛❝❡s ✇✐t❤ t❤❡ ❇❆P✳ P❡➟❝③②➠s❦✐ s❤♦✇❡❞ t❤❛t ❡✈❡r② ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❇❆P ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ s♣❛❝❡ ✇✐t❤ ❛ ❜❛s✐s✳ P❡➟❝③②➠s❦✐ ✫ ❲♦❥t❛s③❝③②❦ ❝♦♥str✉❝t❡❞ ❛♥ ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r ❛❧❧ s♣❛❝❡s ✇✐t❤ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ▼♦t✐✈❛t✐♦♥s ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② P❡➟❝③②➠s❦✐ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❑❛❞❡❝ ❝♦♥str✉❝t❡❞ ❛ ❝♦♠♣❧❡♠❡♥t❛❜❧② ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r t❤❡ ❝❧❛ss ♦❢ s♣❛❝❡s ✇✐t❤ t❤❡ ❇❆P✳ P❡➟❝③②➠s❦✐ s❤♦✇❡❞ t❤❛t ❡✈❡r② ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❇❆P ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ s♣❛❝❡ ✇✐t❤ ❛ ❜❛s✐s✳ P❡➟❝③②➠s❦✐ ✫ ❲♦❥t❛s③❝③②❦ ❝♦♥str✉❝t❡❞ ❛♥ ✉♥✐✈❡rs❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ❢♦r ❛❧❧ s♣❛❝❡s ✇✐t❤ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
❘❡s✉❧ts ❉❡✜♥✐t✐♦♥ ❈❛t❡❣♦r② t❤❡♦r② ε ✲✐s♦♠❡tr✐❡s ▼❛✐♥ r❡s✉❧ts ❋❉❉ ❚❤❡♦r❡♠s ❇✐❜❧✐♦❣r❛♣❤② ▲❡t ❳ ✱ ❨ ❜❡ ❇❛♥❛❝❤ s♣❛❝❡s✱ ε > ✵✳ ❢ : ❳ → ❨ ✐s ❛♥ ε ✲✐s♦♠❡tr② ✐❢ ( ✶ + ε ) − ✶ · � ① � ≤ � ❢ ( ① ) � ≤ ( ✶ + ε ) · � ① � ∀ ① ∈ ❳ ✳ ❆♥ ✐s♦♠❡tr② ❢ : ❳ → ❨ t❤❛t ✐s ❛♥ ε ✲✐s♦♠❡tr② ❢♦r ❡✈❡r② ε > ✵✱ ✐✳❡✳ � ❢ ( ① ) � = � ① � ∀ ① ∈ ❳ ✳ ❆ ❇❛♥❛❝❤ s♣❛❝❡ ❨ ✐s ( ✶ + ε ) ✲❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❳ ✐❢ ❨ ⊆ ❳ ∃ ❚ : ❳ → ❨ s✉❝❤ t❤❛t � ❚② − ② � ≤ ε � ② � ∀ ② ∈ ❨ ✳ ❏♦❛♥♥❛ ●❛r❜✉❧✐➠s❦❛ ■s♦♠❡tr✐❝ ✉♥✐q✉❡♥❡ss ❛♥❞ ❤♦♠♦❣❡♥❡✐t②
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