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May 07, 2023 •113 likes •597 views

strt t t s rsts ss t

  1. ❉✐str✐❜✉t❡❞ ❲❡✐❣❤t❡❞ ▼❛t❝❤✐♥❣ ✈✐❛ ❘❛♥❞♦♠✐③❡❞ ❈♦♠♣♦s❛❜❧❡ ❈♦r❡s❡ts ▼♦❤❛♠♠❛❞❍♦ss❡✐♥ ❇❛t❡♥✐ ●♦♦❣❧❡ ❘❡s❡❛r❝❤ ◆❡✇ ❨♦r❦✱ ❯❙❆ ❙❡♣❡❤r ❆ss❛❞✐ ❱❛❤❛❜ ▼✐rr♦❦♥✐ ✸✻t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ✶✷ ❏✉♥❡ ✷✵✶✾ ✖ ▲♦♥❣ ❇❡❛❝❤✱ ❈❛❧✐❢♦r♥✐❛✱ ❯❙❆ P♦st❡r ★✶✻✶✱ P❛❝✐✜❝ ❇❛❧❧r♦♦♠

  2. ▼❛t❝❤✐♥❣✿ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❡rt❡①✲❞✐s❥♦✐♥t ❡❞❣❡s ❈❧✉st❡r✐♥❣✱ ♣❛rt✐t✐♦♥✐♥❣ ❋✐♥❞✐♥❣ ♠♦t✐❢s ✐♥ ❜✐♦✐♥❢♦r♠❛t✐❝s ❚r❛❞❡ ♠❛r❦❡t✐♥❣✱ ♦♥❧✐♥❡ ❛❞✈❡rt✐s❡♠❡♥t ❑✐❞♥❡② ❡①❝❤❛♥❣❡ ▲✐♥❡❛r ❛❧❣❡❜r❛✱ ♠❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥ ▼❛ss✐✈❡ ❣r❛♣❤s ❊✈❡r②✇❤❡r❡✿ ✇❡❜ ❣r❛♣❤✱ s♦❝✐❛❧ ♥❡t✇♦r❦s✱ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s✱ ❡t❝✳

  3. ▼❛ss✐✈❡ ❣r❛♣❤s ❊✈❡r②✇❤❡r❡✿ ✇❡❜ ❣r❛♣❤✱ s♦❝✐❛❧ ♥❡t✇♦r❦s✱ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s✱ ❡t❝✳ ▼❛t❝❤✐♥❣✿ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❡rt❡①✲❞✐s❥♦✐♥t ❡❞❣❡s ◮ ❈❧✉st❡r✐♥❣✱ ♣❛rt✐t✐♦♥✐♥❣ ◮ ❋✐♥❞✐♥❣ ♠♦t✐❢s ✐♥ ❜✐♦✐♥❢♦r♠❛t✐❝s ◮ ❚r❛❞❡ ♠❛r❦❡t✐♥❣✱ ♦♥❧✐♥❡ ❛❞✈❡rt✐s❡♠❡♥t ◮ ❑✐❞♥❡② ❡①❝❤❛♥❣❡ ◮ ▲✐♥❡❛r ❛❧❣❡❜r❛✱ ♠❛tr✐① ❞❡❝♦♠♣♦s✐t✐♦♥

  4. ❙❡q✉❡♥t✐❛❧ r❡s✉❧ts ❱❛r✐❛♥ts✿ ◮ ❲❡✐❣❤t❡❞ ✈s ✉♥✇❡✐❣❤t❡❞ ◮ ❇✐♣❛rt✐t❡ ✈s ♥♦♥✲❜✐♣❛rt✐t❡ ❆❧❣♦r✐t❤♠✐❝ r❡s✉❧ts✿ ◮ ❋✐rst ♣♦❧②t✐♠❡ ❛❧❣♦r✐t❤♠✿ ✏❇❧♦ss♦♠✑ ❞❡❝♦♠♣♦s✐t✐♦♥ ❬❊❞♠✻✺❛❪ ◮ ❊①t❡♥❞❡❞ t♦ ✇❡✐❣❤t❡❞ ❝❛s❡ ❬❊❞♠✻✺❜❪ ❖ ( ♠ √ ♥ ) ❬●❛❜❚❛r✾✶❪ ◮ ❋❛st❡st ✐♥ t✐♠❡ ˜ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s✿ ◮ ●r❡❡❞② ❛❧❣♦r✐t❤♠✿ ✷✲❛♣♣r♦① ✐♥ ❖ ( ♠ log ♥ ) t✐♠❡ ◮ ✶ + ǫ ❛♣♣r♦① ✐♥ ˜ ❖ ( ♠ /ǫ ) t✐♠❡ ❬❉✉❛P❡t✶✹❪

  5. ❙❡q✉❡♥t✐❛❧ r❡s✉❧ts ❱❛r✐❛♥ts✿ ◮ ❲❡✐❣❤t❡❞ ✈s ✉♥✇❡✐❣❤t❡❞ ◮ ❇✐♣❛rt✐t❡ ✈s ♥♦♥✲❜✐♣❛rt✐t❡ ❆❧❣♦r✐t❤♠✐❝ r❡s✉❧ts✿ ◮ ❋✐rst ♣♦❧②t✐♠❡ ❛❧❣♦r✐t❤♠✿ ✏❇❧♦ss♦♠✑ ❞❡❝♦♠♣♦s✐t✐♦♥ ❬❊❞♠✻✺❛❪ ◮ ❊①t❡♥❞❡❞ t♦ ✇❡✐❣❤t❡❞ ❝❛s❡ ❬❊❞♠✻✺❜❪ ❖ ( ♠ √ ♥ ) ❬●❛❜❚❛r✾✶❪ ◮ ❋❛st❡st ✐♥ t✐♠❡ ˜ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s✿ ◮ ●r❡❡❞② ❛❧❣♦r✐t❤♠✿ ✷✲❛♣♣r♦① ✐♥ ❖ ( ♠ log ♥ ) t✐♠❡ ◮ ✶ + ǫ ❛♣♣r♦① ✐♥ ˜ ❖ ( ♠ /ǫ ) t✐♠❡ ❬❉✉❛P❡t✶✹❪

  6. ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t ✿ ✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ▲❛r❣❡ s❝❛❧❡ ❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖ ( ♠ ) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖ ( ♥ ) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦

  7. ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t ✿ ✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ▲❛r❣❡ s❝❛❧❡ ❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖ ( ♠ ) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖ ( ♥ ) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦

  8. ■♠♣♦rt❛♥t ✿ ✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ▲❛r❣❡ s❝❛❧❡ ❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖ ( ♠ ) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖ ( ♥ ) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦ ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ◮ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ◮ ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ◮ ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ◮ ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞

  9. ▲❛r❣❡ s❝❛❧❡ ❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠s ❞♦ ♥♦t ✇♦r❦ ◮ ❖ ( ♠ ) r✉♥t✐♠❡ ✐s ♣r♦❤✐❜✐t✐✈❡ ◮ ❖ ( ♥ ) ♠❡♠♦r② ♥♦t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ◮ ❙✐♠♣❧② r❡❛❞✐♥❣ ✐♥♣✉t ❞❛t❛ ♦♥❝❡ ♠❛② t❛❦❡ t♦♦ ❧♦♥❣✦ ▼❛♣❘❡❞✉❝❡✿ ❞❡ ❢❛❝t♦ ✐♥❞✉str② st❛♥❞❛r❞ ◮ ❙♣❧✐t ❞❛t❛ ❛❝r♦ss ♠❛♥② ♠❛❝❤✐♥❡s ◮ ❙♣❧✐t ❝♦♠♣✉t❛t✐♦♥ ✐♥t♦ s❡✈❡r❛❧ r♦✉♥❞s ◮ ❉❛t❛ ✐s s❡♥t t♦ ♠❛❝❤✐♥❡s ❜❛s❡❞ ♦♥ ❦❡②s ◮ ▼❛❝❤✐♥❡s ♣r♦❞✉❝❡ ♦✉t♣✉t ❢♦r ♥❡①t r♦✉♥❞ ■♠♣♦rt❛♥t ✿ ✶ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✷ ♥✉♠❜❡r ♦❢ ♠❛❝❤✐♥❡s ✸ ♠❡♠♦r② ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡ ✹ ❛♠♦✉♥t ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❡❛❝❤ ♠❛❝❤✐♥❡

  10. ▲❛r❣❡ s❝❛❧❡ ❙✐♠✉❧❛t✐♦♥ ✏●r❡❡❞② ❛❧❣♦r✐t❤♠s ❛r❡ ♣r❛❝t✐t✐♦♥❡rs✬ ❜❡st ❢r✐❡♥❞s✖t❤❡② ❛r❡ ✐♥t✉✐t✐✈❡✱ s✐♠♣❧❡ t♦ ✐♠♣❧❡♠❡♥t✱ ❛♥❞ ♦❢t❡♥ ❧❡❛❞ t♦ ✈❡r② ❣♦♦❞ s♦❧✉t✐♦♥s✳ ❍♦✇❡✈❡r✱ ✐♠♣❧❡♠❡♥t✐♥❣ ❣r❡❡❞② ❛❧❣♦r✐t❤♠s ✐♥ ❛ ❞✐str✐❜✉t❡❞ s❡tt✐♥❣ ✐s ❝❤❛❧❧❡♥❣✐♥❣ s✐♥❝❡ t❤❡ ❣r❡❡❞② ❝❤♦✐❝❡ ✐s ✐♥❤❡r❡♥t❧② s❡q✉❡♥t✐❛❧✱ ❛♥❞ ✐t ✐s ♥♦t ❝❧❡❛r ❤♦✇ t♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❡①tr❛ ♣r♦❝❡ss✐♥❣ ♣♦✇❡r✳✑ ✖ ❬❑✉♠▼♦s❱❛s❱❛t✶✸❪ ❚✇♦ t②♣✐❝❛❧ t②♣❡s ♦❢ r❡s✉❧ts✿ ✶ ❘❡❧❛t✐✈❡❧② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ r♦✉♥❞s t♦ ✏❢❛✐t❤❢✉❧❧②✑ s✐♠✉❧❛t❡ t❤❡ ❣r❡❡❞② ❛❧❣♦r✐t❤♠✳ ✷ ❱❡r② s♠❛❧❧ ♥✉♠❜❡r ♦❢ r♦✉♥❞s ✭♦♥❡ ♦r t✇♦✮ ❢♦r ✏✇❡❛❦✑ s✐♠✉❧❛t✐♦♥✱ ❖ ( ✶ ) ✇♦rs❡ t❤❛♥ t❤❡ ❣r❡❡❞② ❛❧❣♦r✐t❤♠✳

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