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❍✐❣❤✲♦r❞❡r ❛❝❝✉r❛❝②✱ ❡♥tr♦♣② st❛❜✐❧✐t② ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞s ❢♦r ❤②♣❡r❜♦❧✐❝ ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❯❧r✐❦ ❙❦r❡ ❋❥♦r❞❤♦❧♠ ✇✐t❤ ❙✐❞❞❤❛rt❤ ▼✐s❤r❛ ❛♥❞ ❊✐t❛♥ ❚❛❞♠♦r ❊❚❍ ❩ür✐❝❤ ❏✉♥❡ ✷✻✱ ✷✵✶✷

❙❡q✉❡♥❝❡s ✉ ❛♣♣r♦①✐♠❛t✐♥❣ ✭✶✮ ♠✐❣❤t s❛t✐s❢② ✉ q ✉ ❙❖▼❊❚❍■◆● t ① ❲❡ ✇✐❧❧ st✉❞② ❙❖▼❊❚❍■◆● t♦ ❞❡r✐✈❡ st❛❜✐❧✐t② ❜♦✉♥❞s ❢♦r ❤✐❣❤✲♦r❞❡r ❛❝❝✉r❛t❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡s✳ ❊♥tr♦♣② ❝♦♥❞✐t✐♦♥s ❈♦♥s✐❞❡r t❤❡ s❝❛❧❛r ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇ ✉ t + ❢ ( ✉ ) ① = ✵ . ✭✶✮ ❉❡✜♥✐t✐♦♥ ✭❊♥tr♦♣② ♣❛✐r✮ ❆ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ( η ( ✉ ) , q ( ✉ )) s✉❝❤ t❤❛t q ′ ( ✉ ) = η ′ ( ✉ ) ❢ ′ ( ✉ ) ✳ ▼✉❧t✐♣❧② ✭✶✮ ❜② ❡♥tr♦♣② ✈❛r✐❛❜❧❡s ✈ := η ′ ( ✉ ) ✿ η ( ✉ ) t + q ( ✉ ) ① = ✵ ✭❡♥tr♦♣② ✐s ❝♦♥s❡r✈❡❞✮✳ ❍♦✇❡✈❡r✱ ❡♥tr♦♣② s❤♦✉❧❞ ❜❡ ❞✐ss✐♣❛t❡❞ ❛t s❤♦❝❦s✳ ❊♥tr♦♣② ❝♦♥❞✐t✐♦♥✿ ❋♦r ❛❧❧ ❝♦♥✈❡① ❡♥tr♦♣② ♣❛✐rs ( η, q ) ✱ η ( ✉ ) t + q ( ✉ ) ① � ✵ . ■♠♣❧✐❡s ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ❢♦r s❝❛❧❛r ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❛♥❞ ❧✐♥❡❛r s②st❡♠s✳ ❯✳ ❙✳ ❋❥♦r❞❤♦❧♠ ✭❊❚❍ ❩ür✐❝❤✮ ❍✐❣❤✲♦r❞❡r ❡♥tr♦♣② st❛❜✐❧✐t② ❏✉♥❡ ✷✻✱ ✷✵✶✷ ✷ ✴ ✶✼

❊♥tr♦♣② ❝♦♥❞✐t✐♦♥s ❈♦♥s✐❞❡r t❤❡ s❝❛❧❛r ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇ ✉ t + ❢ ( ✉ ) ① = ✵ . ✭✶✮ ❉❡✜♥✐t✐♦♥ ✭❊♥tr♦♣② ♣❛✐r✮ ❆ ♣❛✐r ♦❢ ❢✉♥❝t✐♦♥s ( η ( ✉ ) , q ( ✉ )) s✉❝❤ t❤❛t q ′ ( ✉ ) = η ′ ( ✉ ) ❢ ′ ( ✉ ) ✳ ▼✉❧t✐♣❧② ✭✶✮ ❜② ❡♥tr♦♣② ✈❛r✐❛❜❧❡s ✈ := η ′ ( ✉ ) ✿ η ( ✉ ) t + q ( ✉ ) ① = ✵ ✭❡♥tr♦♣② ✐s ❝♦♥s❡r✈❡❞✮✳ ❍♦✇❡✈❡r✱ ❡♥tr♦♣② s❤♦✉❧❞ ❜❡ ❞✐ss✐♣❛t❡❞ ❛t s❤♦❝❦s✳ ❊♥tr♦♣② ❝♦♥❞✐t✐♦♥✿ ❋♦r ❛❧❧ ❝♦♥✈❡① ❡♥tr♦♣② ♣❛✐rs ( η, q ) ✱ η ( ✉ ) t + q ( ✉ ) ① � ✵ . ■♠♣❧✐❡s ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ❢♦r s❝❛❧❛r ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❛♥❞ ❧✐♥❡❛r s②st❡♠s✳ ❙❡q✉❡♥❝❡s ✉ ε ❛♣♣r♦①✐♠❛t✐♥❣ ✭✶✮ ♠✐❣❤t s❛t✐s❢② η ( ✉ ε ) t + q ( ✉ ε ) ① = ❙❖▼❊❚❍■◆● . ❲❡ ✇✐❧❧ st✉❞② ❙❖▼❊❚❍■◆● t♦ ❞❡r✐✈❡ st❛❜✐❧✐t② ❜♦✉♥❞s ❢♦r ❤✐❣❤✲♦r❞❡r ❛❝❝✉r❛t❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡s✳ ❯✳ ❙✳ ❋❥♦r❞❤♦❧♠ ✭❊❚❍ ❩ür✐❝❤✮ ❍✐❣❤✲♦r❞❡r ❡♥tr♦♣② st❛❜✐❧✐t② ❏✉♥❡ ✷✻✱ ✷✵✶✷ ✷ ✴ ✶✼

❋r♦♠ ✭✸✮ ✇❡ ❣❡t t❤❛t ❢♦r ❡✈❡r② ❝♦♥✈❡① ✉ ✇✐t❤ ✉ ❞ ✵✱ ✷ ✉ ❚ ✉ ✵ ❈ ✉ ① ❚ ✵ ▲ ✶ ▲ ✶ ▲ ✷ ✵ ❚ ❚❤✐s ❣✐✈❡s t❤❡ ♦s❝✐❧❧❛t✐♦♥ ❜♦✉♥❞ ✶ ✉ ① ▲ ✷ ✵ ❚ ❚❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ✭✷✮ ❣✐✈❡s ✉ ✉ ✵ ▲ ✵ ❚ ▲ ❯s✐♥❣ ❝♦♠♣❡♥s❛t❡❞ ❝♦♠♣❛❝t♥❡ss✱ t❤❡s❡ st❛❜✐❧✐t② ❡st✐♠❛t❡s ❛r❡ ❡♥♦✉❣❤ t♦ s❤♦✇ t❤❛t ✉ ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ❡♥tr♦♣② s♦❧✉t✐♦♥ ♦❢ ✭✶✮✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✈✐s❝♦✉s ❛♣♣r♦①✐♠❛t✐♦♥s ❈♦♥s✐❞❡r t❤❡ ✈✐s❝♦✉s ❛♣♣r♦①✐♠❛t✐♦♥ ✉ ε t + ❢ ( ✉ ε ) ① = ε ( ❉ ( ✉ ε ) ✉ ε ① ) ① ✭✷✮ ❢♦r s♦♠❡ ❉ ( ✉ ε ) � ❝ > ✵✳ ▼✉❧t✐♣❧② ✭✷✮ ❜② ✈ = η ′ ( ✉ ) ✿ � � η ( ✉ ε ) t + q ( ✉ ε ) ① = ε ❉ ( ✉ ε ) η ′ ( ✉ ε ) ✉ ε ① − εη ′′ ( ✉ ε ) ❉ ( ✉ ε )( ✉ ε ① ) ✷ ✭✸✮ ① ❯✳ ❙✳ ❋❥♦r❞❤♦❧♠ ✭❊❚❍ ❩ür✐❝❤✮ ❍✐❣❤✲♦r❞❡r ❡♥tr♦♣② st❛❜✐❧✐t② ❏✉♥❡ ✷✻✱ ✷✵✶✷ ✸ ✴ ✶✼

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✈✐s❝♦✉s ❛♣♣r♦①✐♠❛t✐♦♥s ❈♦♥s✐❞❡r t❤❡ ✈✐s❝♦✉s ❛♣♣r♦①✐♠❛t✐♦♥ ✉ ε t + ❢ ( ✉ ε ) ① = ε ( ❉ ( ✉ ε ) ✉ ε ① ) ① ✭✷✮ ❢♦r s♦♠❡ ❉ ( ✉ ε ) � ❝ > ✵✳ ▼✉❧t✐♣❧② ✭✷✮ ❜② ✈ = η ′ ( ✉ ) ✿ � � η ( ✉ ε ) t + q ( ✉ ε ) ① = ε ❉ ( ✉ ε ) η ′ ( ✉ ε ) ✉ ε ① − εη ′′ ( ✉ ε ) ❉ ( ✉ ε )( ✉ ε ① ) ✷ ✭✸✮ ① ❋r♦♠ ✭✸✮ ✇❡ ❣❡t t❤❛t ❢♦r ❡✈❡r② ❝♦♥✈❡① η ( ✉ ) ✇✐t❤ η ′′ ( ✉ ) � ❞ > ✵✱ � η ( ✉ ε ( ❚ )) � ▲ ✶ ( R ) − � η ( ✉ ε ( ✵ )) � ▲ ✶ ( R ) � − ❈ ε � ✉ ε ① � ✷ ∀ ❚ > ✵ . ▲ ✷ ( R × [ ✵ , ❚ ]) ❚❤✐s ❣✐✈❡s t❤❡ ♦s❝✐❧❧❛t✐♦♥ ❜♦✉♥❞ ✶ � ✉ ε ① � ▲ ✷ ( R × [ ✵ , ❚ ]) � √ ε ❚❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ✭✷✮ ❣✐✈❡s � ✉ ε � ▲ ∞ ( R × [ ✵ , ❚ ]) � � ✉ ε ( ✵ ) � ▲ ∞ ( R ) . ❯s✐♥❣ ❝♦♠♣❡♥s❛t❡❞ ❝♦♠♣❛❝t♥❡ss✱ t❤❡s❡ st❛❜✐❧✐t② ❡st✐♠❛t❡s ❛r❡ ❡♥♦✉❣❤ t♦ s❤♦✇ t❤❛t ✉ ε ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ❡♥tr♦♣② s♦❧✉t✐♦♥ ♦❢ ✭✶✮✳ ❯✳ ❙✳ ❋❥♦r❞❤♦❧♠ ✭❊❚❍ ❩ür✐❝❤✮ ❍✐❣❤✲♦r❞❡r ❡♥tr♦♣② st❛❜✐❧✐t② ❏✉♥❡ ✷✻✱ ✷✵✶✷ ✸ ✴ ✶✼

❋✐rst✲♦r❞❡r ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞s ❲❡ ❝♦♥s✐❞❡r t❤❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ❞ ✶ � � ❞t ✉ ✐ + ❋ ✐ + ✶ / ✷ − ❋ ✐ − ✶ / ✷ = ✵ . ∆ ① ❋✐rst✲♦r❞❡r s❝❤❡♠❡s ❛r❡ ❣❡♥❡r❛❧❧② ♦❢ t❤❡ ❢♦r♠ ❢ ✐ + ✶ / ✷ − ❢ ✐ + ✶ / ✷ ❉ ✐ + ✶ / ✷ ( ✉ ✐ + ✶ − ✉ ✐ ) − ❉ ✐ − ✶ / ✷ ( ✉ ✐ − ✉ ✐ − ✶ ) ❞ ❞t ✉ ✐ + = ∆ ① ∆ ① � �� � � �� � � �� � ≈ ✉ t ≈ ❢ ( ✉ ) ① ≈ ∆ ① ( ❉ ( ✉ ) ✉ ① ) ① ✭ ❢ ✐ + ✶ / ✷ = ( ❢ ( ✉ ✐ ) + ❢ ( ✉ ✐ + ✶ )) / ✷✮ ❢♦r s♦♠❡ ❉ ✐ + ✶ / ✷ � ❝ > ✵✳ ❯✳ ❙✳ ❋❥♦r❞❤♦❧♠ ✭❊❚❍ ❩ür✐❝❤✮ ❍✐❣❤✲♦r❞❡r ❡♥tr♦♣② st❛❜✐❧✐t② ❏✉♥❡ ✷✻✱ ✷✵✶✷ ✹ ✴ ✶✼

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✜rst✲♦r❞❡r s❝❤❡♠❡s ❬▲❛① ✶✾✼✶❪ s❤♦✇❡❞ t❤❛t t❤❡ ✭❢✉❧❧② ❞✐s❝r❡t❡✮ ▲❛①✲❋r✐❡❞r✐❝❤s s❝❤❡♠❡ s❛t✐s✜❡s � � � � η ( ✉ ♥ | ✉ ♠ ✐ + ✶ − ✉ ♠ η ( ✉ ✵ ✐ | ✷ ∆ t ✐ )∆ ① − ✐ )∆ ① � − ❈ ∀ ♥ ∈ N ✐ ✐ ♠ � ♥ ✐ ✭❣❡♥❡r❛❧✐③❡❞ t♦ ❊✲s❝❤❡♠❡s ❜② ❈♦q✉❡❧ ❛♥❞ ▲❡❋❧♦❝❤✱ ✶✾✾✶✮✳ ❚❤✐s ❣✐✈❡s t❤❡ ♦s❝✐❧❧❛t✐♦♥ ❜♦✉♥❞ � � | ✉ ♠ ✐ + ✶ − ✉ ♠ ✐ | ✷ ∆ t � ❈ , ♠ � ♥ ✐ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ❜♦✉♥❞ � ✉ ε ✶ ① � ▲ ✷ � √ ε ✳ ❚❤❡ s❝❤❡♠❡ ✐s ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞✱ � ✉ ♥ � ▲ ∞ ( R ) � � ✉ ✵ � ▲ ∞ ( R ) . ❯s✐♥❣ t❤❡s❡ ❡st✐♠❛t❡s✱ ❬❉✐P❡r♥❛ ✶✾✽✸❪ ♣r♦✈❡❞ ❝♦♥✈❡r❣❡♥❝❡ t♦ t❤❡ ❡♥tr♦♣② s♦❧✉t✐♦♥ ♦❢ ✭✶✮ ❢♦r ❛❧❧ ✉ ✵ ∈ ▲ ∞ ( R ) ✱ ✉s✐♥❣ ❝♦♠♣❡♥s❛t❡❞ ❝♦♠♣❛❝t♥❡ss✳ ❯✳ ❙✳ ❋❥♦r❞❤♦❧♠ ✭❊❚❍ ❩ür✐❝❤✮ ❍✐❣❤✲♦r❞❡r ❡♥tr♦♣② st❛❜✐❧✐t② ❏✉♥❡ ✷✻✱ ✷✵✶✷ ✺ ✴ ✶✼