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  1. ❈♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥s✱ ❇♦❝❤♥❡r✬s t❤❡♦r❡♠✱ ❛♥❞ ♠♦r❡ ▼✐❦②♦✉♥❣ ❏✉♥ ❚❡①❛s ❆✫▼ ❯♥✐✈❡rs✐t② P❆❙■ ✷✵✶✹ ❋✐rst ♣❛rt ❜❛s❡❞ ♦♥ t❤❡ ❜♦♦❦ ✏■♥t❡r♣♦❧❛t✐♦♥ ♦❢ ❙♣❛t✐❛❧ ❉❛t❛✿ ❙♦♠❡ ❚❤❡♦r② ❢♦r ❑r✐❣✐♥❣✑ ❜② ▼✳ ▲✳ ❙t❡✐♥ ✭✶✾✾✾✮ ✶✴✷✹

  2. ❖✉t❧✐♥❡ ▼❡❛♥ sq✉❛r❡ ♣r♦♣❡rt✐❡s✱ ❇♦❝❤♥❡r✬s t❤❡♦r❡♠✱ ✳✳✳ ✶ ✷ ❋r❡q✉❡♥t✐st✬ ❡st✐♠❛t✐♦♥✱ s♦♠❡ ♣r❛❝t✐❝❛❧ ❝♦♠♠❡♥ts ✸ ❈♦✈❛r✐❛♥❝❡ ♠♦❞❡❧s ♦♥ ❛ s♣❤❡r❡ ✷✴✷✹

  3. ❖✉t❧✐♥❡ ▼❡❛♥ sq✉❛r❡ ♣r♦♣❡rt✐❡s✱ ❇♦❝❤♥❡r✬s t❤❡♦r❡♠✱ ✳✳✳ ✶ ✷ ❋r❡q✉❡♥t✐st✬ ❡st✐♠❛t✐♦♥✱ s♦♠❡ ♣r❛❝t✐❝❛❧ ❝♦♠♠❡♥ts ✸ ❈♦✈❛r✐❛♥❝❡ ♠♦❞❡❧s ♦♥ ❛ s♣❤❡r❡ ✸✴✷✹

  4. ▼❡❛♥ sq✉❛r❡ ♣r♦♣❡rt✐❡s ❘❡❧❛t❡ ❝♦✈❛r✐❛♥❝❡ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ✜❡❧❞ ❛♥❞ t❤❡ s♠♦♦t❤♥❡ss ♦❢ ✐ts r❡❛❧✐③❛t✐♦♥ ❩ ✱ ❛ r❛♥❞♦♠ ✜❡❧❞ ♦♥ R ❞ ✱ ✐s ♠❡❛♥ sq✉❛r❡ ❝♦♥t✐♥✉♦✉s ❛t ① ✐❢ ❊ ❢ ❩ ( ② ) � ❩ ( ① ) ❣ ✷ ✦ ✵ ❛s ② ✦ ① ❋♦r ✇❡❛❦❧② st❛t✐♦♥❛r② r❛♥❞♦♠ ✜❡❧❞✱ ♠❡❛♥ sq✉❛r❡ ❝♦♥t✐♥✉✐t② ✐✛ ✐ts ❛✉t♦❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ❛t t❤❡ ♦r✐❣✐♥ ▼❡❛♥ sq✉❛r❡ ❝♦♥t✐♥✉✐t② ❞♦❡s ◆❖❚ ✐♠♣❧② ❝♦♥t✐♥✉✐t② ♦❢ ✐ts r❡❛❧✐③❛t✐♦♥s ✹✴✷✹

  5. ▼❡❛♥ sq✉❛r❡ ♣r♦♣❡rt✐❡s ❩ ♦♥ R ✐s ♠❡❛♥ sq✉❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t ✐❢ t❤❡r❡ ❡①✐st ✵ ( t ) = ❧✐♠ ♥ ✦✶ ❢ ❩ ( t + ❤ ♥ ) � ❩ ( t ) ❣ ❂ ❤ ♥ ✭✐♥ ▲ ✷ ✮ ❢♦r ❤ ♥ ✦ ✵ ❩ ❊①t❡♥s✐♦♥ t♦ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥ ❛♥❞ ❤✐❣❤❡r ♦r❞❡r ✐s str❛✐❣❤t❢♦r✇❛r❞ ❚❤❡r❡ ❝❛♥ ❜❡ ❛♥❛❧②t✐❝ r❡❛❧✐③❛t✐♦♥s ♦❢ r❛♥❞♦♠ ✜❡❧❞✱ ♥♦t ♠❡❛♥ sq✉❛r❡ ❞✐✛❡r❡♥t✐❛❜❧❡✿ ❩ ( t ) = ❝♦s ( ❳t + ❨ ) ✱ ❳ ❛♥❞ ❨ ✐♥❞❡♣❡♥❞❡♥t✱ ❳ ❢♦❧❧♦✇✐♥❣ st❛♥❞❛r❞ ❈❛✉❝❤②✱ ❨ ❢♦❧❧♦✇✐♥❣ ✉♥✐❢♦r♠ ♦♥ [ ✵ ❀ ✷ ✙ ] ❆❧❧ r❡❛❧✐③❛t✐♦♥s ❛r❡ ♦❜✈✐♦✉s❧② ❛♥❛❧②t✐❝ ❈♦✈ ❢ ❩ ( s ) ❀ ❩ ( t ) ❣ = ❡①♣ ( �❥ s � t ❥ ❂ ✷ ) ✺✴✷✹

  6. ▼❡❛♥ sq✉❛r❡ s♠♦♦t❤♥❡ss not smooth smooth 1.0 1.0 0.8 0.8 2 2 1 0.6 0.6 1 0 0 0.4 0.4 −1 −1 0.2 0.2 −2 −2 0.0 0.0 0.0 0.4 0.8 0.0 0.4 0.8 ✻✴✷✹

  7. ❙♣❡❝tr❛❧ ♠❡t❤♦❞s ❉❡✜♥❡ ❝♦♠♣❧❡① r❛♥❞♦♠ ✜❡❧❞s✿ ❩ ( ① ) = ❯ ( ① )+ ✐❱ ( ① ) ❩ ( ① ) = � ♥ ❦ = ✶ ❩ ❦ ❡①♣ ( ✐ ✇ ❚ ❦ ① ) ✱ ✇ ❦ ✷ R ❞ ✱ ❩ ✶ ❀✿✿✿❀ ❩ ♥ ❛r❡ ♠❡❛♥ ✵ ❝♦♠♣❧❡① r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ❊ ( ❩ ✐ ❩ ❥ ) = ✵ ✭ ✐ � ❥ ✮✱ ❊ ❥ ❩ ✐ ❥ ✷ ❁ ✶ ■ts ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❑ ( ① ) = � ♥ ❦ = ✶ ❢ ❦ ❡①♣ ( ✐ ✇ ❚ ❦ ① ) ❈♦♥s✐❞❡r ▲ ✷ ❧✐♠✐t ♦❢ t❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ✼✴✷✹

  8. ❇♦❝❤♥❡r✬s ❚❤❡♦r❡♠ ❆ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ❑ ♦♥ R ❞ ✐s t❤❡ ❛✉t♦❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ❛ ✇❡❛❦❧② st❛t✐♦♥❛r② ♠❡❛♥ sq✉❛r❡ ❝♦♥t✐♥✉♦✉s ❝♦♠♣❧❡①✲✈❛❧✉❡❞ r❛♥❞♦♠ ✜❡❧❞ ♦♥ R ❞ ✐✛ ✐t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s � R ❞ ❡①♣ ( ✐ ✇ ❚ ① ) ❋ ( ❞✇ ) ❑ ( ① ) = ✇✐t❤ ❋ ❛ ♣♦s✐t✐✈❡ ✜♥✐t❡ ♠❡❛s✉r❡✳ ❲❡♥ ❋ ❤❛s ❛ ❞❡♥s✐t② ✇rt ▲❡❜❡sq✉❡ ♠❡❛s✉r❡✱ ✇❡ ❤❛✈❡ t❤❡ s♣❡❝tr❛❧ ❞❡♥s✐t② ❢ ❛♥❞ ✶ � R ❞ ❡①♣ ( � ✐ ✇ ❚ ① ) ❑ ( ① ) ❞① ❢ ( ✇ ) = ( ✷ ✙ ) ❞ ✽✴✷✹

  9. ❋♦r ✐s♦tr♦♣✐❝ r❛♥❞♦♠ ✜❡❧❞s ❇② ❇♦❝❤♥❡r✬s ❚❤❡♦r❡♠✱ ❢♦r ❛ ✇❡❛❦❧② ✐s♦tr♦♣✐❝ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ r❛♥❞♦♠ ✜❡❧❞ ❩ ♦♥ R ❞ ✱ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ✜♥✐t❡ ♠❡❛s✉r❡ ❋ s✉❝❤ t❤❛t � R ❞ ❡①♣ ( ✐ ✇ ❚ ① ) ❋ ( ❞✇ ) ❑ ( ❥ ① ❥ ) = � ◆♦t❡ ❑ ( r ) = ❅ ❜ ❞ ❑ ( r ❥ ① ❥ ) ❯ ( ❞① ) ❢♦r ❯ ✉♥✐❢♦r♠ ♠❡❛s✉r❡ ♦♥ ❅ ❜ ❞ ✮ � � R ❞ ❡①♣ ( ✐r ✇ ❚ ① ) ❋ ( ❞✇ ) ❣ ❯ ( ❞① ) ❑ ( r ) = ❢ ❅ ❜ ❞ � � ❝♦s ( r ✇ ❚ ① ) ❯ ( ❞① ) ❣ ❋ ( ❞✇ ) = R ❞ ❢ ❅ ❜ ❞ ✾✴✷✹

  10. ❋♦r ✐s♦tr♦♣✐❝ r❛♥❞♦♠ ✜❡❧❞s ❝♦♥t✬❞ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ � ❝♦s ( r ✇ ❚ ① ) ❯ ( ❞① ) = � ( ❞ ❂ ✷ )( ✷ r ❥ ✇ ❥ ) ( ❞ � ✷ ) ❂ ✷ ❏ ( ❞ � ✷ ) ❂ ✷ ( r ❥ ✇ ❥ ) ❅ ❜ ❞ ✭ ❏ ✗ ✿ t❤❡ ♦r❞✐♥❛r② ❇❡ss❡❧ ❢✉♥❝t✐♦♥✮ ✮ � ✶ ❑ ( r ) = ✷ ( ❞ � ✷ ) ❂ ✷ � ( ❞ ❂ ✷ ) ( r✉ ) � ( ❞ � ✷ ) ❂ ✷ ❏ ( ❞ � ✷ ) ❂ ✷ ( r✉ ) ❞● ( ✉ ) ✵ ❢♦r ● ♥♦♥❞❡❝r❡❛s✐♥❣✱ ❜♦✉♥❞❡❞ ♦♥ [ ✵ ❀ ✶ ) ✱ ● ( ✵ ) = ✵ ✿ ❍❛♥❦❡❧ tr❛♥s❢♦r♠ ♦❢ ♦r❞❡r ( ❞ � ✶ ) ❂ ✷ ♦❢ ● ■♥ ❢❛❝t✱ t❤✐s ✐s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✶✵✴✷✹

  11. ❋♦r ✐s♦tr♦♣✐❝ r❛♥❞♦♠ ✜❡❧❞s ■t ✐s ❞✐✣❝✉❧t t♦ ✈❡r✐❢② ✐❢ ❛♥② ❣✐✈❡♥ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ t❤❡ ❍❛♥❦❡❧ tr❛♥s❢♦r♠ ♦❢ ❛ ❢✉♥❝t✐♦♥ ● s❛t✐s❢②✐♥❣ t❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥ ❆ ❝♦♥t✐♥✉♦✉s ✐s♦tr♦♣✐❝ ❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ � ✶ ✵ ( r✉ ) � ✶ s✐♥ ( r✉ ) ❞● ( ✉ ) ✐♥ R ✸ ✿ ❑ ( r ) = � ✶ ✵ ❡①♣ ( � r ✷ ✉ ✷ ) ❞● ( ✉ ) ✐♥ ✏❡✈❡r②✑ ❞✐♠❡♥s✐♦♥✿ ❑ ( r ) = � ✶ ✵ r ❞ � ✶ ❥ ❑ ( r ) ❥ ❞r ❁ ✶ ✱ t❤❡r❡ ❡①✐sts ❛ ■♥✈❡rs✐♦♥ ❢♦r♠✉❧❛ ✿ ✐❢ � ✶ ✵ ✉ ❞ � ✶ ❢ ( ✉ ) ❞✉ ❁ ✶ ✿ ♥♦♥♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ ❢ ✇✐t❤ � ✶ ❢ ( ✉ ) = ( ✷ ✙ ) � ❞ ❂ ✷ ( ✉r ) � ( ❞ � ✷ ) ❂ ✷ ❏ ( ❞ � ✷ ) ❂ ✷ ( ✉r ) r ❞ � ✶ ❑ ( r ) ❞r ✵ ✶✶✴✷✹

  12. ❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ❝♦♥t✐♥✉♦✉s ✐s♦tr♦♣✐❝ ❛✉t♦❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ▼❛tér♥ ❝❧❛ss✿ ❢ ( ✉ ) = ✷ ✗ � ✶ ✣ � ( ✗ + ❞ ✷ ) ☛ ✷ ✗ ✙ ❞ ❂ ✷ ( ☛ ✷ + ✉ ✷ ) ✗ + ❞ ❂ ✷ ✐♥ t❤❡ ■♥✈❡rs✐♦♥ ❢♦r♠✉❧❛ � ✗ ☛ � ❞ � ❞ � ❑ ( ❞ ) = ❑ ✗ ❀ ❞ = ❥ s ✶ � s ✷ ❥ ✷ ✗ � ✶ � ( ✗ ) ☞ ☞ ❑ ✗ ✐s t❤❡ ♠♦❞✐✜❡❞ ❇❡ss❡❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ♦❢ ♦r❞❡r ✗ � ▼❛tér♥ ❝❧❛ss ✐s ✈❛❧✐❞ ✐♥ R ♣ ❢♦r ❛❧❧ ♣ ✷ N ✶✷✴✷✹

  13. ❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ❝♦♥t✐♥✉♦✉s ✐s♦tr♦♣✐❝ ❛✉t♦❝♦✈❛r✐❛♥❝❡ ❢✉♥❝t✐♦♥ ❙♣❤❡r✐❝❛❧ ♠♦❞❡❧✿ ❑ ( r ) = ❝ ( ✶ � ✸ ✶ ✷ ✚ ✸ r ✸ ) ❢♦r r ✔ ✚ ❛♥❞ ✷ ✚ r + ③❡r♦ ❡✈❡r②✇❤❡r❡ ❡❧s❡ ❱❛❧✐❞ ✉♣ t♦ ✸ ❞✐♠❡♥s✐♦♥ ❙♣❡❝tr❛❧ ❞❡♥s✐t② ❤❛s ♦s❝✐❧❧❛t✐♦♥ ❛t ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s ❙♣❡❝tr❛❧ ❞❡♥s✐t② ♦❢ ❑ ( r ) = ✛ ( ☛ �❥ r ❥ ) + ✿ ❢ ( ✇ ) = ✛ ✶ � ❝♦s ( ☛ ✇ ) ✙ ✇ ✷ ✶✸✴✷✹

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