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t rt rr rsr Z Z Ps s sss rrs P t

  1. ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ▼P▲❆✱ ◆❛t✐♦♥❛❧ ❛♥❞ ❑❛♣♦❞✐str✐❛♥ ❯♥✐✈❡rs✐t② ♦❢ ❆t❤❡♥s ❆t❤❡♥s ❖❝t♦❜❡r ✷✵✵✾ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✶ ✴ ✺✷

  2. ❖✉t❧✐♥❡ ❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ✷ ❚❤❡ ❉✐s❝r❡t❡ ❋r❋❚ ●❛❧♦✐s P❤❛s❡ ❙♣❛❝❡s ❚❤❡ ❇❛❧✐❛♥✲■t③②❦s♦♥ ◗✉❛♥t✉♠ ❖s❝✐❧❧❛t♦r ✸ ❚❤❡ ◗✉❛♥t✉♠ ❋r❋❚ ❊❧❡♠❡♥ts ♦❢ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ❚❤❡ ◗❋❚ ❆❧❣♦r✐t❤♠ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❉❋r❋❚ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✷ ✴ ✺✷

  3. ❖✉t❧✐♥❡ ❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ✷ ❚❤❡ ❉✐s❝r❡t❡ ❋r❋❚ ●❛❧♦✐s P❤❛s❡ ❙♣❛❝❡s ❚❤❡ ❇❛❧✐❛♥✲■t③②❦s♦♥ ◗✉❛♥t✉♠ ❖s❝✐❧❧❛t♦r ✸ ❚❤❡ ◗✉❛♥t✉♠ ❋r❋❚ ❊❧❡♠❡♥ts ♦❢ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ❚❤❡ ◗❋❚ ❆❧❣♦r✐t❤♠ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❉❋r❋❚ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✷ ✴ ✺✷

  4. ❖✉t❧✐♥❡ ❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ✷ ❚❤❡ ❉✐s❝r❡t❡ ❋r❋❚ ●❛❧♦✐s P❤❛s❡ ❙♣❛❝❡s ❚❤❡ ❇❛❧✐❛♥✲■t③②❦s♦♥ ◗✉❛♥t✉♠ ❖s❝✐❧❧❛t♦r ✸ ❚❤❡ ◗✉❛♥t✉♠ ❋r❋❚ ❊❧❡♠❡♥ts ♦❢ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ❚❤❡ ◗❋❚ ❆❧❣♦r✐t❤♠ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❉❋r❋❚ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✷ ✴ ✺✷

  5. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❖✉t❧✐♥❡ ✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ✷ ❚❤❡ ❉✐s❝r❡t❡ ❋r❋❚ ●❛❧♦✐s P❤❛s❡ ❙♣❛❝❡s ❚❤❡ ❇❛❧✐❛♥✲■t③②❦s♦♥ ◗✉❛♥t✉♠ ❖s❝✐❧❧❛t♦r ✸ ❚❤❡ ◗✉❛♥t✉♠ ❋r❋❚ ❊❧❡♠❡♥ts ♦❢ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ❚❤❡ ◗❋❚ ❆❧❣♦r✐t❤♠ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❉❋r❋❚ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✸ ✴ ✺✷

  6. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❙♦♠❡ ❤✐st♦r② ✶✾✷✾✿ ❋✐rst ❛♣♣❡❛r❛♥❝❡ ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ❜✐❜❧✐♦❣r❛♣❤② ✭◆✳ ❲✐❡♥❡r✮ ❬❍❡r♠✐t✐❛♥ ♣♦❧②♥♦♠✐❛❧s❪ ❬✷✸❪ ✶✾✽✵✿ ❯s❡❞ ✐♥ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s ✭❱✳ ◆❛♠✐❛s✮ ❬●r❡❡♥ ❢✉♥❝t✐♦♥s ♦❢ ❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r❪ ❬✶✻❪ ✶✾✾✸✿ ❊①♣❧❛✐♥s ❋r❡s♥❡❧ ❞✐☛r❛❝t✐♦♥ ✭❍✳▼✳ ❖③❛❦t❛s✱ ❉✳ ▼❡♥❞❧♦✈✐❝✮ ❛♥❞ ✐s ❡st❛❜❧✐s❤❡❞ ✐♥ t❤❡ ☞❡❧❞ ♦❢ ♦♣t✐❝s✳ ❬✷✹❪ ✶✾✾✹✿ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ t❤❡ ☞❡❧❞ ♦❢ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ✭▲✳❇✳ ❆❧♠❡✐❞❛✮✳ ❬✶❪ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✹ ✴ ✺✷

  7. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❉❡☞♥✐t✐♦♥ ♦❢ t❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✭❋r❋❚✮ ❉❡☞♥✐t✐♦♥ ✷✳✶ ✭❋r❋❚✮ ❋♦r ❡❛❝❤ ❛ ∈ R ✱ ✇❡ ❞❡☞♥❡ t❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❛s Z ∞ F ❛ ( ❢ )( ù ) = ❑ ❛ ( ù❀ t ) ❢ ( t ) ❞t ✭✶✮ −∞ Z ∞ r ✶ − ✐ ❝♦t ( ❛ ) ❡ ✐ ❝♦t ( ❛ ) ù ✷ ❂ ✷ ❡ ✐ ❝s❝ ( ❛ ) ù t + ✐ ❝♦t ( ❛ ) t ✷ ❂ ✷ ❢ ( t ) ❞t = ✷ ð −∞ ✇❤❡r❡ ❑ ❛ ( ù❀ t ) = ❈ ❛ ❡①♣ [ − ✐ ð ( ✷ ù t s✐♥ ❛ − ( t ✷ + ù ✷ ) ❝♦t ❛ )] ✭✷✮ ✐s t❤❡ tr❛♥s❢♦r♠✬s ❦❡r♥❡❧ ❛♥❞ ❈ ❛ = √ ✶ − ✐ ❝♦t ❛✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ✷ ❛ ❂ð = ♥ ∈ Z ✱ t❤❡♥ F ❛ ( ❢ ) = F ✷ ❛ ❂ð ( ❢ ) ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✺ ✴ ✺✷

  8. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❋r❋❚ ♣r♦♣❡rt✐❡s ❈♦r♦❧❧❛r② ✷✳✶ ✭❋r❋❚ ♣r♦♣❡rt✐❡s✮ ∀ ❛ ❀ ❜ ∈ R ✶ F ❛ + ❜ ( ❢ ) = F ❛ ( F ❜ ( ❢ )) = F ❜ ( F ❛ ( ❢ )) ✷ ■❢ ❛ = ð❂ ✷ ✱ t❤❡♥ F ❛ ≡ F  ❢ ( t ) ♥ = ✷ ❦ ♥ = ✷ ❦ + ✶ ✱ ∵ F ✷ ( ❢ ) = ❢ ( − t ) ✸ ■❢ ❛ ❂ð = ♥ ∈ Z ✱ F ❛ ( ❢ ) = ❢ ( − t ) ✹ F ❛ ( ä ( t − ã )) = ❑ ❛ ( ù❀ ã ) ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✻ ✴ ✺✷

  9. ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❋r❋❚ ❛s ❛ ❝♦♥✈♦❧✉t✐♦♥ ❉❡☞♥✐t✐♦♥ ✷✳✷ ✭❈❤✐r♣ ❋✉♥❝t✐♦♥✮ ❆ ❝❤✐r♣ ❢✉♥❝t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t s✇❡❡♣s ❛♥ ✐♥t❡r✈❛❧ [ ù ✵ ❀ ù ✶ ] ♦❢ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ✐♥ ❛ s♣❡❝✐☞❝ t✐♠❡ ✐♥t❡r✈❛❧ [ t ✵ ❀ t ✶ ] ✳ ■❢ t❤❡ s✇❡❡♣ ✐s ❧✐♥❡❛r✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦❢ t❤❡ ❢♦r♠ ❡ ✐ ð ( ÷ t + ã ) t ✱ ✇❤❡r❡ ÷ ✐s t❤❡ s✇❡❡♣ r❛t❡✳ ❘❡♠❛r❦ ✷✳✶ ❋r❋❚✬s ❦❡r♥❡❧ ✐s ❛ ❝❤✐r♣ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛ s✇❡❡♣ r❛t❡ ♦❢ ❝♦t ❛ ✭❡q✳ ✭✷✮✱ ❞❡❢✳ ✷✳✷✮✳ t ✷ ❝♦t ❛ − ✷ t ù ❝s❝ ❛ + ù ✷ ❝♦t ❛ = t ✷ ( ❝♦t ❛ − ❝s❝ ❛ ) + ( t − ù ) ✷ ❝s❝ ❛ + ù ✷ ( ❝♦t ❛ − ❝s❝ ❛ ) ❍❡♥❝❡ t❤❡ ❋r❋❚ ♠❛② ❜❡ ✈✐❡✇❡❞ ❛s ❛♣♣❧②✐♥❣ ❛ ❝❤✐r♣ ❝♦♥✈♦❧✉t✐♦♥ ✇✐t❤ ❛ s✇❡❡♣ r❛t❡ ♦❢ ❝s❝ ❛ ✐♥ ❜❡t✇❡❡♥ t✇♦ ❝❤✐r♣ ♠✉❧t✐♣❧✐❝❛t✐♦♥s ✇✐t❤ ❛ s✇❡❡♣ r❛t❡ ♦❢ ❝♦t ❛ ❬✻❪ ❲✐❧❧✐❛♠ ❱❛ss✐❧✐s ❑❛r❛❣❡♦r❣♦s ✭❯♦❆✮ ◗✉❛♥t✉♠ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥ Z ♣ × Z ♣ P❤❛s❡ ❙♣❛❝❡s ✼ ✴ ✺✷

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