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Submitted on 9 Nov 2010
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Formal verification of numerical programs: from C
annotated programs to Coq proofs
Sylvie Boldo
To cite this version:
Sylvie Boldo. Formal verification of numerical programs: from C annotated programs to Coq proofs.
NSV-3: Third International Workshop on Numerical Software Verification, Jul 2010, Edinburgh,
Scot-land, United Kingdom. �inria-00534400�
❋♦r♠❛❧ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s✿
❢r♦♠ ❈ ❛♥♥♦t❛t❡❞ ♣r♦❣r❛♠s t♦ ❈♦q ♣r♦♦❢s
⋆ ❙②❧✈✐❡ ❇♦❧❞♦ ■◆❘■❆ ❙❛❝❧❛② ✲ ❰❧❡✲❞❡✲❋r❛♥❝❡✱ Pr♦❱❛❧✱ ❖rs❛②✱ ❋✲✾✶✽✾✸ ▲❘■✱ ❯♥✐✈ P❛r✐s✲❙✉❞✱ ❈◆❘❙✱ ❖rs❛②✱ ❋✲✾✶✹✵✺ ❙②❧✈✐❡✳❇♦❧❞♦❅✐♥r✐❛✳❢r ❆❜str❛❝t✳ ◆✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s ♠❛② r❡q✉✐r❡ ❛ ❤✐❣❤ ❧❡✈❡❧ ♦❢ ❣✉❛r❛♥t❡❡✳ ❚❤✐s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ❛♣♣❧②✐♥❣ ❢♦r♠❛❧ ♠❡t❤♦❞s✱ s✉❝❤ ❛s ♠❛❝❤✐♥❡✲ ❝❤❡❝❦❡❞ ♣r♦♦❢s✳ ❇✉t t❤❡s❡ t♦♦❧s ❤❛♥❞❧❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r❡♠s ✇❤✐❧❡ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❈ ❝♦❞❡✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s ❤✐❣❤ ❧❡✈❡❧ ♦❢ ❝♦♥✜❞❡♥❝❡ ♦♥ ❈ ♣r♦❣r❛♠s✱ ✇❡ ✉s❡ ❛ ❝❤❛✐♥ ♦❢ t♦♦❧s✿ ❋r❛♠❛✲❈✱ ✐ts ❏❡ss✐❡ ♣❧✉❣✐♥✱ ❲❤② ❛♥❞ ❈♦q✳ ❚❤✐s r❡q✉✐r❡s t❤❡ ❈ ♣r♦❣r❛♠ t♦ ❜❡ ❛♥♥♦t❛t❡❞✿ t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ❢✉♥❝t✐♦♥ ♠✉st ❜❡ ♣r❡❝✐s❡❧② s♣❡❝✐✜❡❞✱ ❛♥❞ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❡ ❝♦rr❡❝t♥❡ss ♦❢ t❤❡ ♣r♦❣r❛♠ ❜② ♣r♦✈✐♥❣ ❜♦t❤ t❤❛t ✐t ♠❡❡ts ✐ts s♣❡❝✐✜❝❛t✐♦♥s ❛♥❞ t❤❛t ✐t ❞♦❡s ♥♦t ❢❛✐❧✳ ❊①❛♠♣❧❡s ✇✐❧❧ ❜❡ ❣✐✈❡♥ t♦ ✐❧❧✉str❛t❡ t❤❡ ❢❡❛t✉r❡s ♦❢ t❤✐s ❛♣♣r♦❛❝❤✳✶ ■♥tr♦❞✉❝t✐♦♥
●✐✈❡♥ ❛ ♣r♦❣r❛♠ ✉s✐♥❣ ✢♦❛t✐♥❣✲♣♦✐♥t ❛r✐t❤♠❡t✐❝✱ ✐t ✐s ♣r❡tt② ❤❛r❞ t♦ ❦♥♦✇ t❤❡ ✜✲ ♥❛❧ r♦✉♥❞✐♥❣ ❡rr♦r ♦❢ t❤❡ r❡s✉❧t✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♣r♦✈✐♥❣ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s ✇✐t❤ ❛ ✈❡r② ❤✐❣❤ ❧❡✈❡❧ ♦❢ ❣✉❛r❛♥t❡❡ ❜② ✉s✐♥❣ ❢♦r♠❛❧ ♠❡t❤♦❞s✳ ❊❛❝❤ ✢♦❛t✐♥❣✲♣♦✐♥t r❡s✉❧t ✐s ❛ ❝♦rr❡❝t r♦✉♥❞✐♥❣ ♦❢ t❤❡ ❡①❛❝t r❡❛❧ ✈❛❧✉❡ ❢♦r ❛❧❧ ❜❛s✐❝ ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❞✐✈✐s✐♦♥ ❛♥❞ sq✉❛r❡ r♦♦t✮✳ ❚❤✐s ♣r♦♣❡rt② ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ■❊❊❊✲✼✺✹ st❛♥❞❛r❞ ❬✶❪ ❛♥❞ ❛❧❧ ♠♦❞❡r♥ ♣r♦❝❡ss♦rs ❝♦♠♣❧② ✇✐t❤ ✐t✳ ◆❡✈❡rt❤❡❧❡ss✱ ❡✈❡♥ ✐❢ ❡❛❝❤ ❝♦♠♣✉t❛t✐♦♥ ✐s ❝♦rr❡❝t✱ ✐✳❡✳ t❤❡ ❜❡st ♣♦ss✐❜❧❡✱ t❤❡r❡ ✐s ♥♦ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡ ✜♥❛❧ r❡s✉❧t ❛❢t❡r ♠❛♥② s✉❝❤ ❝♦♠♣✉t❛t✐♦♥s ✐s st✐❧❧ ❛❝❝✉r❛t❡✳ ❙t❛t✐❝ ❛♥❛❧②s✐s ✐s ❛♥ ❛♣♣r♦❛❝❤ ❢♦r ❝❤❡❝❦✐♥❣ ❛ ♣r♦❣r❛♠ ✇✐t❤♦✉t r✉♥♥✐♥❣ ✐t✳ ❉❡❞✉❝t✐✈❡ ✈❡r✐✜❝❛t✐♦♥ t❡❝❤♥✐q✉❡s ✇❤✐❝❤ ♣❡r❢♦r♠ st❛t✐❝ ❛♥❛❧②s✐s ♦❢ ❝♦❞❡✱ r❡❧② ♦♥ t❤❡ ❛❜✐❧✐t② ♦❢ t❤❡♦r❡♠ ♣r♦✈❡rs t♦ ❝❤❡❝❦ ✈❛❧✐❞✐t② ♦❢ ❢♦r♠✉❧❛s ✐♥ ✜rst✲♦r❞❡r ❧♦❣✐❝ ♦r ❡✈❡♥ ♠♦r❡ ❡①♣r❡ss✐✈❡ ❧♦❣✐❝s✳ ❚❤❡② ✉s✉❛❧❧② ❝♦♠❡ ✇✐t❤ ❡①♣r❡ss✐✈❡ s♣❡❝✐✜❝❛t✐♦♥ ❧❛♥❣✉❛❣❡s s✉❝❤ ❛s ❏▼▲ ❬✷✱ ✸❪ ❢♦r ❏❛✈❛✱ ❆❈❙▲ ❬✹❪ ❢♦r ❈✱ ❙♣❡❝★ ❬✺❪ ❢♦r ❈★✱ ❡t❝✳ t♦ s♣❡❝✐❢② t❤❡ r❡q✉✐r❡♠❡♥ts✳ ❋♦r ❛✉t♦♠❛t✐❝ ❛♥❛❧②s✐s ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ❝♦❞❡s✱ t❤❡r❡ ❡①✐st s❡✈❡r❛❧ ♠❡t❤♦❞s ❢♦r ❜♦✉♥❞✐♥❣ t❤❡ ✜♥❛❧ ❡rr♦r ♦❢ ❛ ♣r♦❣r❛♠✱ ✐♥❝❧✉❞✐♥❣ ✐♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝✱ ❢♦r✇❛r❞ ❛♥❞ ❜❛❝❦✇❛r❞ ❛♥❛❧②s✐s ❬✻✱ ✼❪✳ ❆♥♦t❤❡r ❛♣♣r♦❛❝❤ ✐s ❛❜str❛❝t ✐♥t❡r♣r❡t❛t✐♦♥ ❜❛s❡❞ st❛t✐❝ ❛♥❛❧②s✐s✱ t❤❛t ✐♥❝❧✉❞❡s ❆stré❡ ❬✽✱ ✾❪ ❛♥❞ ❋❧✉❝t✉❛t ❬✶✵✱ ✶✶❪✳ ⋆❚❤✐s ✇♦r❦ ✇❛s ❢✉♥❞❡❞ ❜② t❤❡ ❋H st ✭❆◆❘✲✵✽✲❇▲❆◆✲✵✷✹✻✲✵✶✮ ♣r♦❥❡❝t ♦❢ t❤❡ ❋r❡♥❝❤ ♥❛t✐♦♥❛❧ r❡s❡❛r❝❤ ♦r❣❛♥✐③❛t✐♦♥ ✭❆◆❘✮✳✷ ❙②❧✈✐❡ ❇♦❧❞♦ ❋❧♦❛t✐♥❣✲♣♦✐♥t ❛r✐t❤♠❡t✐❝ ❤❛s ❜❡❡♥ ❢♦r♠❛❧✐③❡❞ ✉s✐♥❣ ❞❡❞✉❝t✐✈❡ ❢♦r♠❛❧ ♠❡t❤✲ ♦❞s s✐♥❝❡ ✶✾✽✾ ✐♥ ♦r❞❡r t♦ ❢♦r♠❛❧❧② ♣r♦✈❡ ❤❛r❞✇❛r❡ ❝♦♠♣♦♥❡♥ts ♦r ❛❧❣♦r✐t❤♠s ❬✶✷✕ ✶✹❪✳ ❲❡ ♠❛❝❤✐♥❡✲❝❤❡❝❦ ❛❧❧ ♣r♦♦❢s ✉s✐♥❣ t❤❡ ❈♦q ♣r♦♦❢ ❝❤❡❝❦❡r ❬✶✺❪✳ ❲❡ ✉s❡ ❛ ❤✐❣❤✲❧❡✈❡❧ ❢♦r♠❛❧✐③❛t✐♦♥ ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ❬✶✻✱ ✶✼❪✳ ❚❤❡r❡ ❡①✐st ❢❡✇ ✇♦r❦s ♦♥ s♣❡❝✐❢②✐♥❣ ❛♥❞ ♣r♦✈✐♥❣ ❜❡❤❛✈✐♦r❛❧ ♣r♦♣❡rt✐❡s ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♣r♦❣r❛♠s ✐♥ ❞❡❞✉❝t✐✈❡ ✈❡r✐✜❝❛t✐♦♥ s②st❡♠s✳ ❆ ✇♦r❦ ♦♥ ✢♦❛t✐♥❣✲ ♣♦✐♥t ✐♥ ❏▼▲ ❢♦r ❏❛✈❛ ✐s ♣r❡s❡♥t❡❞ ✐♥ ✷✵✵✻ ❜② ▲❡❛✈❡♥s ❬✶✽❪✳ ❆♥♦t❤❡r ♣r♦♣♦s❛❧ ❤❛s ❜❡❡♥ ♠❛❞❡ ✐♥ ✷✵✵✼ ❜② ❇♦❧❞♦ ❛♥❞ ❋✐❧❧✐âtr❡ ❬✶✾❪✳ ❆②❛❞ ❛♥❞ ▼❛r❝❤é ❡①t❡♥❞❡❞ t❤✐s t♦ ✐♥❝r❡❛s❡ ❣❡♥❡r✐❝✐t② ❛♥❞ ❤❛♥❞❧❡ ❡①❝❡♣t✐♦♥❛❧ ❜❡❤❛✈✐♦rs ❬✷✵❪✳ ❇♦❧❞♦ ❛♥❞ ◆❣✉②❡♥ ❡①t❡♥❞❡❞ t❤✐s t♦ ❤❛♥❞❧❡ ♠✉❧t✐♣❧❡ ❛r❝❤✐t❡❝t✉r❡s ❛♥❞ ❝♦♠♣✐❧❡rs ❬✷✶❪✳
✷ ❚♦♦❧s ❈❤❛✐♥
❲❡ st❛rt ❢r♦♠ ❛♥♥♦t❛t❡❞ ❈ ♣r♦❣r❛♠s ✉s✐♥❣ ❆❈❙▲ ❬✹❪✳ ❊❛❝❤ ❢✉♥❝t✐♦♥ ✐s ❛♥♥♦✲ t❛t❡❞ ✇✐t❤ ♣r❡✲❝♦♥❞✐t✐♦♥s ✭✇❤❛t t❤❡ ❢✉♥❝t✐♦♥ r❡q✉✐r❡s ❢r♦♠ t❤❡ ✐♥♣✉ts✮ ❛♥❞ ♣♦st✲❝♦♥❞✐t✐♦♥s ✭✇❤❛t t❤❡ ❢✉♥❝t✐♦♥ ❡♥s✉r❡s ❛t ✐ts ❡♥❞✮✳ ❚❤❡ ❛♥♥♦t❛t✐♦♥s ❛♥❞ r❡✲ q✉✐r❡♠❡♥ts ✭♣♦✐♥t❡r ❞❡r❡❢❡r❡♥❝✐♥❣ ❢♦r ❡①❛♠♣❧❡✮ ❛r❡ t❤❡♥ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ♣r♦♦❢ ♦❜❧✐❣❛t✐♦♥s t❤❛t ❤❛✈❡ t♦ ❜❡ s♦❧✈❡❞ ❜② ♣r♦♦❢ ❛ss✐st❛♥ts ♦r ❞❡❝✐s✐♦♥ ♣r♦❝❡❞✉r❡s✳ ❚❤❡ ❋r❛♠❛✲❈ ❢r❛♠❡✇♦r❦ ❤❛s ✢♦❛t✐♥❣✲♣♦✐♥t ❛♥♥♦t❛t✐♦♥s ❜❛s❡❞ ♦♥ ❬✶✾❪ t❤❛t ❛❧❧♦✇ t♦ s♣❡❝✐❢② ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❡❛❝❤ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠✲ ❜❡r ❤❛s ❛ ❣❤♦st ✈❛❧✉❡ ❝❛❧❧❡❞ ❡①❛❝t ✇❤✐❝❤ ❞♦❡s ♥♦t s✉✛❡r ❢r♦♠ r♦✉♥❞✐♥❣✳ ❚❤✐s r❡❛❧ ✈❛❧✉❡ ✐s t❤❡♥ ❝♦♠♣✉t❡❞ ✇✐t❤ t❤❡ s❛♠❡ ♦♣❡r❛t✐♦♥s ❛s t❤❡ ✢♦❛t ✈❛❧✉❡ ❡①❝❡♣t t❤❛t t❤❡ ❣❤♦st ♦♣❡r❛t✐♦♥ ✐s ❡①❛❝t✳ ❚❤❡ ♠❛❝r♦ r♦✉♥❞❴❡rr♦r(f) ✐s t❤❡♥ ✉s❡❞ ❢♦r ❆❈❙▲✲❛♥♥♦t❛t❡❞ ❈ ♣r♦❣r❛♠ ❋r❛♠❛✲❈✴❏❡ss✐❡ ♣❧✉❣✲✐♥ ❲❍❨ ✈❡r✐✜❝❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❣❡♥❡r❛t♦r ❱❡r✐✜❝❛t✐♦♥ ❝♦♥❞✐t✐♦♥s ❆✉t♦♠❛t✐❝ ♣r♦✈❡rs ✭❆❧t✲❊r❣♦✱●❛♣♣❛✱❈❱❈✸✱❡t❝✳✮ ■♥t❡r❛❝t✐✈❡ ♣r♦✈❡rs✭❈♦q✱P❱❙✱❡t❝✳✮ ❋✐❣✳ ✶✳ ❈❤❛✐♥ ♦❢ t♦♦❧s✿ ❢r♦♠ t❤❡ ❈ ♣r♦❣r❛♠ t♦ t❤❡ ♣r♦♦❢ ♦❜❧✐❣❛t✐♦♥s✳❋♦r♠❛❧ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s ✸ |f −❡①❛❝t(f)|✳ ❋♦r ❡①❛♠♣❧❡✱ t♦ ❝♦♠♣✉t❡ ❛ ♥❛✐✈❡ ❡①♣♦♥❡♥t✐❛❧ ❜② t❤❡ ♣♦❧②♥♦♠✐❛❧ ❡✈❛❧✉❛t✐♦♥ ♦❢ ✶✰①✰①✯①✴✷✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①❛❝t ✈❛❧✉❡ ✐s 1 + x + x2 2 ✭✇✐t❤ ♠❛t❤❡♠❛t✐❝❛❧ ♦♣❡r❛t✐♦♥s✮✳ ■♥s✐❞❡ t❤❡ ❛♥♥♦t❛t✐♦♥s✱ ❛❧❧ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❡①❛❝t✳ ❲❡ t❤❡♥ ✉s❡ t❤❡ ❋r❛♠❛✲❈ ♣❧❛t❢♦r♠✶ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❏❡ss✐❡ ♣❧✉❣✐♥ t❤❛t ✉s❡s ❲❤② ❬✷✷❪✳ ❚❤✐s ❝❤❛✐♥ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❋✐❣✉r❡ ✷✳ ❚❤❡ ❈ ❝♦❞❡ ✐s ❣✐✈❡♥ t♦ t❤❡ ❏❡ss✐❡ ♣❧✉❣✐♥ ♦❢ ❋r❛♠❛✲❈✱ ✐t ❝r❡❛t❡s ❛ ❲❤② ✜❧❡ ♦❢ ♣r♦♦❢ ♦❜❧✐❣❛t✐♦♥s ✭t❤❡♦r❡♠s t♦ ♣r♦✈❡✮ t❤❛t ❝❛♥ ❜❡ tr❛♥s❧❛t❡❞ ✐♥ ♦r❞❡r t♦ ❜❡ ❣✐✈❡♥ t♦ ❡✐t❤❡r ❛✉t♦♠❛t✐❝ ♦r ✐♥t❡r❛❝t✐✈❡ ♣r♦✈❡rs✳
✸ ❊①❛♠♣❧❡s
❚❤❡ ❢✉❧❧ ❝♦❞❡ ♦❢ ❛❧❧ t❤❡s❡ ❡①❛♠♣❧❡s ✭❛♥❞ ♠♦r❡✮ ❛♥❞ t❤❡✐r ♣r♦♦❢s ❛r❡ ❛✈❛✐❧❛❜❧❡ ♦♥ ❤tt♣✿✴✴✇✇✇✳❧r✐✳❢r✴⑦s❜♦❧❞♦✴r❡s❡❛r❝❤✳❤t♠❧✳ ✸✳✶ ❙t❡r❜❡♥③ s✉❜tr❛❝t✐♦♥ ❚❤✐s ❢✉♥❝t✐♦♥ ❝♦♠♣✉t❡s t❤❡ ❡①❛❝t s✉❜tr❛❝t✐♦♥ ✐❢ t❤❡ ✐♥♣✉ts ❛r❡ ♥❡❛r ❡♥♦✉❣❤ ♦♥❡ t♦ ❛♥♦t❤❡r ❬✷✸❪✳ ◆♦t❡ t❤❛t t❤❡ ❞✐✈✐s✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥s✐❞❡ t❤❡ ❛♥♥♦t❛t✐♦♥s ❛r❡ ❡①❛❝t✳ ✴∗❅ r❡q✉✐r❡s ② ✴ ✷ ✳ ❁❂ ① ❁❂ ✷ ✳ ∗ ② ❀ ❅ ❡♥s✉r❡s ❭ r❡s✉❧t ❂❂ ①−② ❀ ❅∗✴ ❢ ❧ ♦ ❛ t ❙t❡r❜❡♥③ ✭ ❢ ❧ ♦ ❛ t ① ✱ ❢ ❧ ♦ ❛ t ② ✮ ④ r❡t✉r♥ ①−② ❀ ⑥ ✸✳✷ ❱❡❧t❦❛♠♣✴❉❡❦❦❡r ❛❧❣♦r✐t❤♠ ❚❤✐s ❢✉♥❝t✐♦♥ ❝♦♠♣✉t❡s t❤❡ ❡①❛❝t ❡rr♦r ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❬✷✹✱ ✷✺❪ ✇✐t❤ ♦♥❧② ✢♦❛t✐♥❣✲♣♦✐♥t ♦♣❡r❛t✐♦♥s ✭❛♥❞ ♥♦ ❋▼❆✮✳ ❚❤❡r❡ ❛r❡ ❛❧s♦ ✉♥❞❡r✢♦✇ r❡str✐❝t✐♦♥s ❛♥❞ ♦✈❡r✢♦✇ r❡str✐❝t✐♦♥s s♦ t❤❛t ♥♦ ✐♥✜♥✐t② ✇✐❧❧ ❜❡ ❝r❡❛t❡❞ ❬✷✻❪✳ ✴∗❅ r❡q✉✐r❡s ①② ❂❂ ❭r♦✉♥❞❴❞♦✉❜❧❡✭❭ ◆❡❛r❡st❊✈❡♥ ✱ ①∗② ✮ ✫✫ ❅ ❭❛❜s ✭ ① ✮ ❁❂ ✵①✶ ✳ ♣✾✾✺ ✫✫ ❅ ❭❛❜s ✭ ② ✮ ❁❂ ✵①✶ ✳ ♣✾✾✺ ✫✫ ❅ ❭❛❜s ✭ ①∗② ✮ ❁❂ ✵①✶ ✳ ♣✶✵✷✶ ❀ ❅ ❡♥s✉r❡s ✭ ✭ ①∗② ❂❂ ✵ ⑤ ⑤ ✵①✶ ✳ ♣−✾✻✾ ❁❂ ❭❛❜s ✭ ①∗② ✮ ✮ ❅ ❂❂❃ ①∗② ❂❂ ①②✰❭r❡s✉❧t ✮ ❀ ❅∗✴ ❞♦✉❜❧❡ ❉❡❦❦❡r ✭ ❞♦✉❜❧❡ ① ✱ ❞♦✉❜❧❡ ② ✱ ❞♦✉❜❧❡ ①② ✮ ④ ❞♦✉❜❧❡ ❈✱ ♣① ✱ q① ✱ ❤① ✱ ♣② ✱ q② ✱ ❤② ✱ t① ✱ t② ✱ r✷ ❀ ✐♥t ✐ ❀ ❈❂✶❀ ✴∗❅ ❧♦♦♣ ✐♥✈❛r✐❛♥t ❈❂❂ ❭♣♦✇✭ ✷ ✳ ✱ ✐ ✮ ✫✫ ✵ ❁❂ ✐ ❁❂ ✷ ✼ ❀ ❅ ❧♦♦♣ ✈❛r✐❛♥t ✭✷✼− ✐ ✮ ❀ ∗✴ ❢ ♦ r ✭ ✐ ❂✵❀ ✐ ❁✷✼❀ ✐ ✰✰✮ ❈∗❂✷❀ ❈✰✰❀ ✴∗❅ ❛ss❡rt ❈ ❂❂ ❭♣♦✇✭ ✷ ✳ ✱ ✐ ✮ ✰ ✶ ✳ ✫✫ ✐ ❂❂✷✼❀ ∗✴ ✶ ❤tt♣✿✴✴❢r❛♠❛✲❝✳❝❡❛✳❢r✴✹ ❙②❧✈✐❡ ❇♦❧❞♦ ♣①❂①∗❈❀ q①❂①−♣① ❀ ❤①❂♣①✰q① ❀ t①❂①−❤① ❀ ♣②❂②∗❈❀ q②❂②−♣② ❀ ❤②❂♣②✰q② ❀ t②❂②−❤② ❀ r✷❂−①②✰❤①∗❤② ❀ r✷✰❂❤①∗ t② ❀ r✷✰❂❤②∗ t① ❀ r✷✰❂t① ∗ t② ❀ r❡t✉r♥ r✷ ❀ ⑥ ✸✳✸ ❑❛❤❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❛♥ ❛❝❝✉r❛t❡ ❞✐s❝r✐♠✐♥❛♥t ❚❤✐s ❢✉♥❝t✐♦♥ ❝♦♠♣✉t❡s ❛♥ ❛❝❝✉r❛t❡ ❞✐s❝r✐♠✐♥❛♥t ✉s✐♥❣ ❑❛❤❛♥✬s ❛❧❣♦r✐t❤♠ ❬✷✼❪✳ ❚❤❡ r❡s✉❧t ✐s ♣r♦✈❡❞ ❝♦rr❡❝t ✇✐t❤✐♥ ✷ ✉❧♣s✳ ❖✈❡r✢♦✇ ❛♥❞ ✉♥❞❡r✢♦✇ r❡str✐❝t✐♦♥s ❛r❡ ❣✐✈❡♥ ❬✷✽❪✳ ❲❡ ♥❡❡❞❡❞ ❛♥ ❛①✐♦♠❛t✐❝ t♦ ❡♥s✉r❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✉❧♣ ✐s t❤❡ ♣r♦♣❡r ♦♥❡ t❤❛t ✐s ♦♠✐tt❡❞ ❤❡r❡✳ ✴∗❅ r❡q✉✐r❡s ❅ ✭ ❜❂❂✵✳ ⑤ ⑤ ✵①✶ ✳ ♣−✾✶✻ ❁❂ ❭❛❜s ✭ ❜∗❜ ✮ ✮ ✫✫ ❅ ✭ ❛∗ ❝❂❂✵✳ ⑤ ⑤ ✵①✶ ✳ ♣−✾✶✻ ❁❂ ❭❛❜s ✭ ❛∗ ❝ ✮ ✮ ✫✫ ❅ ❭❛❜s ✭ ❜ ✮ ❁❂ ✵①✶ ✳ ♣✺✶✵ ✫✫ ❭❛❜s ✭ ❛ ✮ ❁❂ ✵①✶ ✳ ♣✾✾✺ ✫✫ ❭❛❜s ✭ ❝ ✮ ❁❂ ✵①✶ ✳ ♣✾✾✺ ✫✫ ❅ ❭❛❜s ✭ ❛∗ ❝ ✮ ❁❂ ✵①✶ ✳ ♣✶✵✷✶ ❀ ❅ ❡♥s✉r❡s ❭ r❡s✉❧t ❂❂✵✳ ⑤ ⑤ ❭❛❜s ✭❭ r❡s✉❧t −✭❜∗❜−❛∗ ❝ ✮ ✮ ❁❂ ✷ ✳ ∗ ✉❧♣ ✭❭ r❡s✉❧t ✮ ❀ ❅ ∗✴ ❞♦✉❜❧❡ ❞ ✐ s ❝ r ✐ ♠ ✐ ♥ ❛ ♥ t ✭ ❞♦✉❜❧❡ ❛ ✱ ❞♦✉❜❧❡ ❜ ✱ ❞♦✉❜❧❡ ❝ ✮ ④ ❞♦✉❜❧❡ ♣ ✱ q ✱ ❞ ✱ ❞♣ ✱ ❞q ❀ ♣❂❜∗❜ ❀ q❂❛∗ ❝ ❀ ✐ ❢ ✭ ♣✰q ❁❂ ✸∗ ❢ ❛ ❜ s ✭ ♣−q ✮ ✮ ❞❂♣−q ❀ ❡ ❧ s ❡ ④ ❞♣❂❉❡❦❦❡r ✭ ❜ ✱ ❜ ✱ ♣ ✮ ❀ ❞q❂❉❡❦❦❡r ✭ ❛ ✱ ❝ ✱ q ✮ ❀ ❞❂✭♣−q✮✰✭❞♣−❞q ✮ ❀ ⑥ r❡t✉r♥ ❞ ❀ ⑥ ✸✳✹ ❲❛✈❡ ❊q✉❛t✐♦♥ ❘❡s♦❧✉t✐♦♥ ❙❝❤❡♠❡ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s ❛ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡ ❢♦r t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ♦♥❡✲ ❞✐♠❡♥s✐♦♥❛❧ ❛❝♦✉st✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ ❬✷✾✱ ✸✵❪✳ ■ts r♦✉♥❞✐♥❣ ❡rr♦r ❜♦✉♥❞ r❡q✉✐r❡s ❛ ❤✐❣❤✲❧❡✈❡❧ ♣r❡❞✐❝❛t❡ ❞❡✜♥❡❞ ✐♥ ❈♦q ❛s ❝♦♥t❛✐♥✐♥❣ ❛ ♣r♦♣❡rt② ✏t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ Z t♦ R s✉❝❤ t❤❛t✳ ✳ ✳ ✑✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✐♥✐t✐❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ ♦♠✐tt❡❞✳ ◆♦t❡ t❤❛t t❤❡s❡ ♣r♦♦❢s ❛r❡ ♥♦t ❢✉❧❧② ❞♦♥❡ ❛s t❤❡② r❡q✉✐r❡ t❤❡ s❝❤❡♠❡ ♥♦t t♦ ❞✐✈❡r❣❡✳
❋♦r♠❛❧ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s ✺ ✴∗❅ ❛①✐♦♠❛t✐❝ ❞ ✐ r ✐ ❝ ❤ ❧ ❡ t ④ ❅ ♣r❡❞✐❝❛t❡ ❛ ♥ ❛ ❧ ② t ✐ ❝ ❴ ❡ r r ♦ r ④▲⑥ ❅ ✭ ❞♦✉❜❧❡ ∗∗♣ ✱ ✐♥t❡❣❡r ♥✐ ✱ ✐♥t❡❣❡r ✐ ✱ ✐♥t❡❣❡r ❦ ✱ ❞♦✉❜❧❡ ❛ ✮ ❅ r❡❛❞s ♣ ❬ ✳ ✳ ❪ ❬ ✳ ✳ ❪ ❀ ⑥ ∗✴ ✴∗❅ r❡q✉✐r❡s ♥ ✐ ❃❂ ✷ ✫✫ ♥❦ ❃❂ ✷ ❅ ✫✫ ❧ ✦❂ ✵ ❅ ✫✫ ❞① ❃ ✵ ✳ ✫✫ ❞t ❃ ✵ ✳ ✫✫ ✈ ❃ ✵ ✳ ❅ ✫✫ ❭❡①❛❝t ✭ ❞① ✮ ❃ ✵ ✳ ✫✫ ❭❡①❛❝t ✭ ❞t ✮ ❃ ✵ ✳ ❅ ✫✫ ❭❡①❛❝t ✭ ✈✮❂❂✈ ❅ ✫✫ ❭❛❜s ✭❭ ❡①❛❝t ✭ ❞① ✮ − ❞① ✮ ✴ ❞① ❁❂ ✵①✶ ✳ ♣−✺✸ ❅ ✫✫ ❭❛❜s ✭❭ ❡①❛❝t ✭ ❞t ✮ − ❞t ✮ ✴ ❞t ❁❂ ✵①✶ ✳ ♣−✺✶ ❅ ✫✫ ✸ ✳ ✴ ✺ ✳ ❁❂ ❭❡①❛❝t ✭ ❞t ✮✴❭ ❡①❛❝t ✭ ❞① ✮ ∗ ❭❡①❛❝t ✭ ✈ ✮ ❁❂ ✶−✵①✶ ✳ ♣−✺✵ ❅ ✫✫ ✵①✶ ✳ ♣−✶✵✵✵ ❁❂ ✈ ❁❂ ✵①✶ ✳ ♣✶✵✵✵ ❅ ✫✫ ♥ ✐ ❁❂ ✵①✶ ✳ ♣✻✹ ✫✫ ♥❦ ❁❂ ✹✶✾✹✸✵✹ ❅ ✫✫ ❭❡①❛❝t ✭ ❞① ✮ ❁❂ ✶ ❀ ❅ ❅ ❡♥s✉r❡s ❭ ❢ ♦ r ❛ ❧ ❧ ✐♥t ✐ ❀ ❭ ❢ ♦ r ❛ ❧ ❧ ✐♥t ❦ ❀ ❅ ✵ ❁❂ ✐ ❁❂ ♥ ✐ ❂❂❃ ✵ ❁❂ ❦ ❁❂ ♥❦ ❂❂❃ ❅ ❭r♦✉♥❞❴❡rr♦r ✭❭ r❡s✉❧t ❬ ✐ ❪ ❬ ❦ ❪ ✮ ❁❂ ✽✺✳✴✷∗✵ ①✶ ✳ ♣−✺✷∗✭❦✰✶✮∗✭❦ ✰✷✮❀ ∗✴ ❞♦✉❜❧❡ ∗∗ ❢♦r✇❛r❞❴♣r♦♣ ✭ ✐♥t ♥✐ ✱ ✐♥t ♥❦ ✱ ❞♦✉❜❧❡ ❞① ✱ ❞♦✉❜❧❡ ❞t ✱ ❞♦✉❜❧❡ ✈ ✱ ❞♦✉❜❧❡ ①s ✱ ❞♦✉❜❧❡ ❧ ✮ ④ ❞♦✉❜❧❡ ∗∗♣ ❀ ✐♥t ✐ ✱ ❦ ❀ ❞♦✉❜❧❡ ❛✶ ✱ ❛ ✱ ❞♣ ❀ ❛✶ ❂ ❞t ✴❞①∗✈ ❀ ❛ ❂ ❛✶∗❛✶ ❀ ✴∗❅ ❛ss❡rt ✶ ✳ ✴ ✹ ❁❂ ❛ ❁❂ ✶ ✫✫ ✵ ❁ ❭❡①❛❝t ✭ ❛ ✮ ❁❂ ✶ ✫✫ ❅ ❭r♦✉♥❞❴❡rr♦r ✭ ❛ ✮ ❁❂ ✵①✶ ✳ ♣−✹✾❀ ∗✴ ♣ ❂ ❛ r r ❛ ② ✷ ❞ ❴ ❛ ❧ ❧ ♦ ❝ ✭ ♥ ✐ ✰✶✱ ♥❦ ✰✶✮❀ ♣ ❬ ✵ ❪ ❬ ✵ ❪ ❂ ✵ ✳ ❀ ✴∗❅ ❧♦♦♣ ✐♥✈❛r✐❛♥t ✶ ❁❂ ✐ ❁❂ ♥ ✐ ✫✫ ❛ ♥ ❛ ❧ ② t ✐ ❝ ❴ ❡ r r ♦ r ✭ ♣ ✱ ♥✐ ✱ ✐ −✶ ✱✵ ✱ ❛ ✮ ❀ ❅ ❧♦♦♣ ✈❛r✐❛♥t ♥✐−✐ ❀ ∗✴ ❢ ♦ r ✭ ✐ ❂✶❀ ✐ ❁♥ ✐ ❀ ✐ ✰✰✮ ④ ♣ ❬ ✐ ❪ ❬ ✵ ❪ ❂ ♣❴③❡r♦ ✭ ①s ✱ ❧ ✱ ✐ ∗❞① ✮ ❀ ⑥ ♣ ❬ ♥ ✐ ❪ ❬ ✵ ❪ ❂✵✳❀ ♣ ❬ ✵ ❪ ❬ ✶ ❪ ❂ ✵ ✳ ❀ ✴∗❅ ❧♦♦♣ ✐♥✈❛r✐❛♥t ✶ ❁❂ ✐ ❁❂ ♥ ✐ ✫✫ ❛ ♥ ❛ ❧ ② t ✐ ❝ ❴ ❡ r r ♦ r ✭ ♣ ✱ ♥✐ ✱ ✐ −✶ ✱✶ ✱ ❛ ✮ ❀ ❅ ❧♦♦♣ ✈❛r✐❛♥t ♥✐−✐ ❀ ∗✴ ❢ ♦ r ✭ ✐ ❂✶❀ ✐ ❁♥ ✐ ❀ ✐ ✰✰✮ ④ ❞♣ ❂ ♣ ❬ ✐ ✰ ✶ ❪ ❬ ✵ ❪ − ✷ ✳ ∗ ♣ ❬ ✐ ❪ ❬ ✵ ❪ ✰ ♣ ❬ ✐ − ✶ ❪ ❬ ✵ ❪ ❀ ♣ ❬ ✐ ❪ ❬ ✶ ❪ ❂ ♣ ❬ ✐ ❪ ❬ ✵ ❪ ✰ ✵ ✳ ✺ ∗ ❛∗❞♣ ❀ ⑥ ♣ ❬ ♥ ✐ ❪ ❬ ✶ ❪ ❂ ✵ ✳ ❀ ✴∗ ♣ r♦ ♣ ❛ ❣ ❛ t✐♦ ♥ ❂ t✐♠❡ ❧♦♦♣ ∗✴ ✴∗❅ ❧♦♦♣ ✐♥✈❛r✐❛♥t ✶ ❁❂ ❦ ❁❂ ♥❦ ✫✫ ❛ ♥ ❛ ❧ ② t ✐ ❝ ❴ ❡ r r ♦ r ✭ ♣ ✱ ♥✐ ✱ ♥✐ ✱ ❦ ✱ ❛ ✮ ❀ ❅ ❧♦♦♣ ✈❛r✐❛♥t ♥❦−❦ ❀ ∗✴ ❢ ♦ r ✭ ❦❂✶❀ ❦❁♥❦ ❀ ❦✰✰✮ ④ ♣ ❬ ✵ ❪ ❬ ❦✰✶❪ ❂ ✵ ✳ ❀ ✴∗ t✐♠❡ ✐ t ❡ r ❛ t ✐ ♦ ♥ ❂ s♣❛❝❡ ❧♦♦♣ ∗✴ ✴∗❅ ❧♦♦♣ ✐♥✈❛r✐❛♥t ✶ ❁❂ ✐ ❁❂ ♥ ✐ ✫✫ ❛ ♥ ❛ ❧ ② t ✐ ❝ ❴ ❡ r r ♦ r ✭ ♣ ✱ ♥✐ ✱ ✐ −✶✱❦✰✶✱❛ ✮ ❀ ❅ ❧♦♦♣ ✈❛r✐❛♥t ♥✐−✐ ❀ ∗✴ ❢ ♦ r ✭ ✐ ❂✶❀ ✐ ❁♥ ✐ ❀ ✐ ✰✰✮ ④ ❞♣ ❂ ♣ ❬ ✐ ✰✶❪❬ ❦ ❪ − ✷ ✳ ∗ ♣ ❬ ✐ ❪ ❬ ❦ ❪ ✰ ♣ ❬ ✐ −✶❪❬ ❦ ❪ ❀ ♣ ❬ ✐ ❪ ❬ ❦✰✶❪ ❂ ✷ ✳ ∗ ♣ ❬ ✐ ❪ ❬ ❦ ❪ − ♣ ❬ ✐ ❪ ❬ ❦−✶❪ ✰ ❛∗❞♣ ❀ ⑥ ♣ ❬ ♥ ✐ ❪ ❬ ❦✰✶❪ ❂ ✵ ✳ ❀ ⑥ r❡t✉r♥ ♣ ❀ ⑥
✻ ❙②❧✈✐❡ ❇♦❧❞♦
✹ ❈♦♥❝❧✉s✐♦♥
❲❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❛t ❛ ✈❡r② ❤✐❣❤ ❣✉❛r❛♥t❡❡ ♦♥ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s ✐s ❛❝❤✐❡✈❛❜❧❡✳ ❆ ❣✐✈❡♥ s♣❡❝✐✜❡❞ ♣r♦❣r❛♠ ❝❛♥ ❜❡ ♣r♦✈❡❞ ✐♥ ✐ts ❞❡❡♣❡r ❞❡t❛✐❧s ✉s✐♥❣ ❈♦q ❛♥❞ t❤❡r❡❢♦r❡ ❛s❝❡rt❛✐♥✐♥❣ ✐ts ❝♦rr❡❝t♥❡ss✳ ❲❡ ❤❛✈❡ ❛ ✉s❛❜❧❡ ❝❤❛✐♥ ❢r♦♠ t❤❡ ❛♥♥♦✲ t❛t❡❞ ♣r♦❣r❛♠ t♦ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r❡♠s t❤❛t ✐s ❛❜❧❡ t♦ s❡❝✉r❡ ❛ ♣r♦❣r❛♠ ❜♦t❤ ❢r♦♠ ✐ts r♦✉♥❞✐♥❣ ❡rr♦rs ❛♥❞ ❢r♦♠ ✐ts ♦t❤❡r ♣♦ss✐❜❧❡ ❢❛✐❧✉r❡s ✭♣♦✐♥t❡r ❞❡r❡❢✲ ❡r❡♥❝✐♥❣✱ ♦✉t✲♦❢✲❜♦✉♥❞ ❛rr❛② ❛❝❝❡ss❡s✳ ✳ ✳ ✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡r❡ ❛r❡ ✽✹ s❛❢❡t② ♣r♦♦❢ ♦❜❧✐❣❛t✐♦♥s ❢♦r t❤❡ ❡①❛♠♣❧❡ ♦❢ ❙❡❝t✐♦♥ ✸✳✹ t❤❛t ❤❛✈❡ t♦ ❞♦ ✇✐t❤ ♠❡♠♦r② ❛❝❝❡ss✱ ✈❛r✐❛♥t ❞❡❝r❡❛s❡✱ ♦✈❡r✢♦✇✱ ❛♥❞ ♣r❡❝♦♥❞✐t✐♦♥ ❢♦r ❢✉♥❝t✐♦♥ ❝❛❧❧✳ ❚❤✐s ❛♣♣r♦❛❝❤ s✉✛❡rs ❢r♦♠ s❡✈❡r❛❧ ❞r❛✇❜❛❝❦s✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❛t t❤❡ s♣❡❝✲ ✐✜❝❛t✐♦♥s ♠✉st ❜❡ ❣✐✈❡♥✳ ❲❡ ♣r♦✈❡ t❤❡ s♣❡❝✐✜❝❛t✐♦♥s ❜✉t ✇❡ ❞♦ ♥♦t ✐♥❢❡r t❤❡♠ ❛t ❛❧❧✳ ❚❤✐s ❝♦✉❧❞ ❜❡ s♦❧✈❡❞ ❜② ✉s✐♥❣ ❛❜str❛❝t ✐♥t❡r♣r❡t❛t✐♦♥ ♦r ❜② ✉s✐♥❣ ❛♥② ♦t❤❡r t♦♦❧s t♦ ✐♥❢❡r s♣❡❝✐✜❝❛t✐♦♥s ✭❢♦r ❡①❛♠♣❧❡ t✉r♥❡❞ ✐♥t♦ ❛♥♦t❤❡r ❋r❛♠❛✲❈ ♣❧✉❣✐♥✮✳ ❲❡ t❤❡♥ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❡♠ t♦ ❡♥s✉r❡ t❤❡✐r ❝♦rr❡❝t♥❡ss✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❞♦ ♥♦t r❡❧② ♦♥ t❤❡ ❝♦rr❡❝t♥❡ss ♦❢ t❤❡ ❡①t❡r♥❛❧ t♦♦❧✳ ❚❤❡ s❡❝♦♥❞ ❞r❛✇❜❛❝❦ ✐s t❤❛t ❛❧❧ t❤♦s❡ ❡①❛♠♣❧❡s ♥❡❡❞ ✐♥t❡r❛❝t✐✈❡ ♣r♦♦❢s ✐♥ ❈♦q✳ ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ♦♥ s♠❛❧❧ ♣r♦❣r❛♠s✱ ❜✉t ❛✉t♦♠❛t✐③❛t✐♦♥s ✐s ❡ss❡♥t✐❛❧ t♦ s♣r❡❛❞ t❤♦s❡ t❡❝❤♥✐q✉❡s✳ ❆ ✜rst ✇❛② ✐s t♦ ✉s❡ ●❛♣♣❛ ❬✸✶✱ ✸✷❪ ❛s ❛♥ ❛✉t♦♠❛t✐❝ ♣r♦✈❡r ♦✉t♣✉t ♦❢ ❲❤②✳ ❚❤✐s ✐s q✉✐t❡ ❝♦♥✈❡♥✐❡♥t ❜✉t ✐t ♠❛② ♠❡❛♥ ❛ ❞r♦♣ ♦❢ ❣✉❛r❛♥t❡❡✳ ❍♦♣❡❢✉❧❧②✱ ●❛♣♣❛ ✐s ❛❜❧❡ t♦ ♣r♦❞✉❝❡ ❛ ❈♦q ♣r♦♦❢ ❛♥❞ ✇❡ ♠❛② ✉s❡ ❛ ●❛♣♣❛ t❛❝t✐❝ ✐♥s✐❞❡ ❈♦q t♦ ❜❡♥❡✜t ❢r♦♠ ●❛♣♣❛✬s ❛✉t♦♠❛t✐♦♥s ✐♥s✐❞❡ ✐♥t❡r❛❝t✐✈❡ ♣r♦♦❢s ❬✸✸❪✳ ❆ ♥❡✈❡r✲❡♥❞✐♥❣ ♣❡rs♣❡❝t✐✈❡ ✐s t♦ ✜♥❞ ❝✉♥♥✐♥❣ t❡❝❤♥✐q✉❡s t♦ ❜❡tt❡r ❜♦✉♥❞ t❤❡ r♦✉♥❞✐♥❣ ❡rr♦rs✱ ❡s♣❡❝✐❛❧❧② ✇❤❡♥ t❤❡② ❝♦♠♣❡♥s❛t❡✳ ❆ t❡❝❤♥✐q✉❡ t❤❛t st❛t❡s t❤❡ ❛♥❛❧②t✐❝❛❧ ❡rr♦r ❤❛s ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ✐♥ ❬✷✽❪ ❢♦r t❤❡ ❡①❛♠♣❧❡ ♦❢ ❙❡❝t✐♦♥ ✸✳✹ ✇❤❡r❡ ✉s✉❛❧ ♠❡t❤♦❞s ❣❛✈❡ ❛♥ ❡rr♦r ♣r♦♣♦rt✐♦♥❛❧ t♦ 2k t❤❛t ✇❛s ❝✉t ❞♦✇♥ t♦ k2✳ ❚❤✐s t❡❝❤♥✐q✉❡ ♦❢ t❤❡ ❛♥❛❧②t✐❝❛❧ ❡rr♦r ❛♥❞ ♣r❡❝✐s❡ ✢♦❛t✐♥❣✲♣♦✐♥t ❡rr♦r ❝❛♥❝❡❧❧❛t✐♦♥ ❝♦♠✐♥❣ ✇✐t❤ ✐ts ❢♦r♠❛❧ ♣r♦♦❢ ✐s ♥❡✇✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ✐t r❡q✉✐r❡s ✈❡r② ❣❡♥❡r✐❝ s♣❡❝✐✜❝❛t✐♦♥s ❛s t❤❡ ❧♦♦♣ ✐♥✈❛r✐❛♥t ♥❡❡❞s t♦ ❜❡ ❧♦❣✐❝❛❧❧② ❞❡✜♥❡❞✿ ✐t st❛t❡s t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ t❤❛t ❤❛s s✉❝❤ ❛♥❞ s✉❝❤ ♣r♦♣❡rt②✳ ❆♥❞ ❆❈❙▲ ❛❧❧♦✇s ✉s t♦ ❡①♣r❡ss s✉❝❤ ❛ ❤✐❣❤✲❧❡✈❡❧ ♣r♦♣❡rt② ♦♥ ❛ ❈ ♣r♦❣r❛♠✳ ❲❡ t❤❡♥ ✉s❡ ❈♦q ❛s ❛ ❜❛❝❦✲❡♥❞ t♦ ❢♦r♠❛❧❧② ❝❤❡❝❦ t❤❡ s♣❡❝✐✜❝❛t✐♦♥s✳ ❚❤✐s ❣❡♥❡r✐❝✐t② ✐s ❛♥ ❛❞✈❛♥t❛❣❡ ❝♦♠♣❛r❡❞ t♦ ❛✉t♦♠❛t✐❝ ♠❡t❤♦❞s t❤❛t ❝❛♥♥♦t ❡①♣r❡ss ♦✉r ❧♦♦♣ ✐♥✈❛r✐❛♥t✳❘❡❢❡r❡♥❝❡s
✶✳ ■❊❊❊✿ ■❊❊❊ ❙t❛♥❞❛r❞ ❢♦r ❋❧♦❛t✐♥❣✲P♦✐♥t ❆r✐t❤♠❡t✐❝✳ ■❊❊❊ ❙t❞✳ ✼✺✹✲✷✵✵✽ ✭✷✵✵✽✮ ✷✳ ✿ ❏▼▲✲❏❛✈❛ ▼♦❞❡❧✐♥❣ ▲❛♥❣✉❛❣❡ ✇✇✇✳❥♠❧s♣❡❝s✳♦r❣✳ ✸✳ ❇✉r❞②✱ ▲✳✱ ❈❤❡♦♥✱ ❨✳✱ ❈♦❦✱ ❉✳❘✳✱ ❊r♥st✱ ▼✳❉✳✱ ❑✐♥✐r②✱ ❏✳❘✳✱ ▲❡❛✈❡♥s✱ ●✳❚✳✱ ▲❡✐♥♦✱ ❑✳❘✳▼✳✱ P♦❧❧✱ ❊✳✿ ❆♥ ♦✈❡r✈✐❡✇ ♦❢ ❏▼▲ t♦♦❧s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦♥ ❙♦❢t✇❛r❡ ❚♦♦❧s ❢♦r ❚❡❝❤♥♦❧♦❣② ❚r❛♥s❢❡r ✭❙❚❚❚✮ ✼✭✸✮ ✭❏✉♥❡ ✷✵✵✺✮ ✷✶✷✕✷✸✷ ✹✳ ❇❛✉❞✐♥✱ P✳✱ ❋✐❧❧✐âtr❡✱ ❏✳❈✳✱ ▼❛r❝❤é✱ ❈✳✱ ▼♦♥❛t❡✱ ❇✳✱ ▼♦②✱ ❨✳✱ Pr❡✈♦st♦✱ ❱✳✿ ❆❈❙▲✿ ❆◆❙■✴■❙❖ ❈ ❙♣❡❝✐✜❝❛t✐♦♥ ▲❛♥❣✉❛❣❡✳ ✭✷✵✵✽✮ ❤tt♣✿✴✴❢r❛♠❛✲❝✳❝❡❛✳❢r✴❛❝s❧✳❤t♠❧✳ ✺✳ ❇❛r♥❡tt✱ ▼✳✱ ▲❡✐♥♦✱ ❑✳❘✳▼✳✱ ❘✉st❛♥✱ ❑✳✱ ▲❡✐♥♦✱ ▼✳✱ ❙❝❤✉❧t❡✱ ❲✳✿ ❚❤❡ ❙♣❡❝★ Pr♦❣r❛♠♠✐♥❣ ❙②st❡♠✿ ❆♥ ❖✈❡r✈✐❡✇✱ ❙♣r✐♥❣❡r ✭✷✵✵✹✮ ✹✾✕✻✾❋♦r♠❛❧ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s ✼ ✻✳ ❲✐❧❦✐♥s♦♥✱ ❏✳❍✳✿ ❘♦✉♥❞✐♥❣ ❊rr♦rs ✐♥ ❆❧❣❡❜r❛✐❝ Pr♦❝❡ss❡s✳ Pr❡♥t✐❝❡✲❍❛❧❧✱ ❯♣♣❡r ❙❛❞❞❧❡ ❘✐✈❡r✱ ◆❏ ✵✼✹✺✽✱ ❯❙❆ ✭✶✾✻✸✮ ✼✳ ❍✐❣❤❛♠✱ ◆✳❏✳✿ ❆❝❝✉r❛❝② ❛♥❞ st❛❜✐❧✐t② ♦❢ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s✳ ❙■❆▼ ✭✷✵✵✷✮ ✽✳ ❈♦✉s♦t✱ P✳✱ ❈♦✉s♦t✱ ❘✳✱ ❋❡r❡t✱ ❏✳✱ ▼❛✉❜♦r❣♥❡✱ ▲✳✱ ▼✐♥é✱ ❆✳✱ ▼♦♥♥✐❛✉①✱ ❉✳✱ ❘✐✈❛❧✱ ❳✳✿ ❚❤❡ ❆❙❚❘➱❊ ❛♥❛❧②③❡r✳ ■♥✿ ❊❙❖P✳ ◆✉♠❜❡r ✸✹✹✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✭✷✵✵✺✮ ✷✶✕✸✵ ✾✳ ▼♦♥♥✐❛✉①✱ ❉✳✿ ❆♥❛❧②s❡ st❛t✐q✉❡ ✿ ❞❡ ❧❛ t❤é♦r✐❡ à ❧❛ ♣r❛t✐q✉❡✳ ❍❛❜✐❧✐t❛t✐♦♥ t♦ ❞✐r❡❝t r❡s❡❛r❝❤✱ ❯♥✐✈❡rs✐té ❏♦s❡♣❤ ❋♦✉r✐❡r✱ ●r❡♥♦❜❧❡✱ ❋r❛♥❝❡ ✭❏✉♥❡ ✷✵✵✾✮ ✶✵✳ ●♦✉❜❛✉❧t✱ ❊✳✱ P✉t♦t✱ ❙✳✿ ❙t❛t✐❝ ❛♥❛❧②s✐s ♦❢ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s✳ ■♥ ❨✐✱ ❑✳✱ ❡❞✳✿ ❙❆❙✳ ❱♦❧✉♠❡ ✹✶✸✹ ♦❢ ▲◆❈❙✳✱ ❙♣r✐♥❣❡r ✭✷✵✵✻✮ ✶✽✕✸✹ ✶✶✳ ❉❡❧♠❛s✱ ❉✳✱ ●♦✉❜❛✉❧t✱ ❊✳✱ P✉t♦t✱ ❙✳✱ ❙♦✉②r✐s✱ ❏✳✱ ❚❡❦❦❛❧✱ ❑✳✱ ❱➹➞❞r✐♥❡✱ ❋✳✿ ❚♦✇❛r❞s ❛♥ ✐♥❞✉str✐❛❧ ✉s❡ ♦❢ ✢✉❝t✉❛t ♦♥ s❛❢❡t②✲❝r✐t✐❝❛❧ ❛✈✐♦♥✐❝s s♦❢t✇❛r❡✳ ■♥✿ ❋▼■❈❙✳ ❱♦❧✉♠❡ ✺✽✷✺ ♦❢ ▲◆❈❙✳✱ ❙♣r✐♥❣❡r ✭✷✵✵✾✮ ✺✸✕✻✾ ✶✷✳ ❈❛rr❡ñ♦✱ ❱✳❆✳✱ ▼✐♥❡r✱ P✳❙✳✿ ❙♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ■❊❊❊✲✽✺✹ ✢♦❛t✐♥❣✲♣♦✐♥t st❛♥❞❛r❞ ✐♥ ❍❖▲ ❛♥❞ P❱❙✳ ■♥✿ ❍❖▲✾✺✿ ✽t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❲♦r❦s❤♦♣ ♦♥ ❍✐❣❤❡r✲❖r❞❡r ▲♦❣✐❝ ❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❆s♣❡♥ ●r♦✈❡✱ ❯❚ ✭❙❡♣t❡♠❜❡r ✶✾✾✺✮ ✶✸✳ ❘✉ss✐♥♦✛✱ ❉✳▼✳✿ ❆ ♠❡❝❤❛♥✐❝❛❧❧② ❝❤❡❝❦❡❞ ♣r♦♦❢ ♦❢ ■❊❊❊ ❝♦♠♣❧✐❛♥❝❡ ♦❢ t❤❡ ✢♦❛t✐♥❣ ♣♦✐♥t ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❞✐✈✐s✐♦♥ ❛♥❞ sq✉❛r❡ r♦♦t ❛❧❣♦r✐t❤♠s ♦❢ t❤❡ ❆▼❉✲❑✼ ♣r♦❝❡ss♦r✳ ▲▼❙ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥ ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✶ ✭✶✾✾✽✮ ✶✹✽✕✷✵✵ ✶✹✳ ❍❛rr✐s♦♥✱ ❏✳✿ ❋♦r♠❛❧ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ✢♦❛t✐♥❣ ♣♦✐♥t tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳ ■♥✿ Pr♦✲ ❝❡❡❞✐♥❣s ♦❢ t❤❡ ❚❤✐r❞ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❋♦r♠❛❧ ▼❡t❤♦❞s ✐♥ ❈♦♠♣✉t❡r✲ ❆✐❞❡❞ ❉❡s✐❣♥✱ ❆✉st✐♥✱ ❚❡①❛s ✭✷✵✵✵✮ ✷✶✼✕✷✸✸ ✶✺✳ ❇❡rt♦t✱ ❨✳✱ ❈❛stér❛♥✱ P✳✿ ■♥t❡r❛❝t✐✈❡ ❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❛♥❞ Pr♦❣r❛♠ ❉❡✈❡❧♦♣♠❡♥t✳ ❈♦q✬❆rt✿ ❚❤❡ ❈❛❧❝✉❧✉s ♦❢ ■♥❞✉❝t✐✈❡ ❈♦♥str✉❝t✐♦♥s✳ ❙♣r✐♥❣❡r ✭✷✵✵✹✮ ✶✻✳ ❉❛✉♠❛s✱ ▼✳✱ ❘✐❞❡❛✉✱ ▲✳✱ ❚❤ér②✱ ▲✳✿ ❆ ❣❡♥❡r✐❝ ❧✐❜r❛r② ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①❛❝t ❝♦♠♣✉t✐♥❣✳ ■♥✿ ✶✹t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❚❤❡♦r❡♠ Pr♦✈✐♥❣ ✐♥ ❍✐❣❤❡r ❖r❞❡r ▲♦❣✐❝s✱ ❊❞✐♥❜✉r❣❤✱ ❙❝♦t❧❛♥❞ ✭✷✵✵✶✮ ✶✻✾✕✶✽✹ ✶✼✳ ❇♦❧❞♦✱ ❙✳✿ Pr❡✉✈❡s ❢♦r♠❡❧❧❡s ❡♥ ❛r✐t❤♠ét✐q✉❡s à ✈✐r❣✉❧❡ ✢♦tt❛♥t❡✳ P❤❉ t❤❡s✐s✱ ➱❝♦❧❡ ◆♦r♠❛❧❡ ❙✉♣ér✐❡✉r❡ ❞❡ ▲②♦♥ ✭✷✵✵✹✮ ✶✽✳ ▲❡❛✈❡♥s✱ ●✳❚✳✿ ◆♦t ❛ ♥✉♠❜❡r ♦❢ ✢♦❛t✐♥❣ ♣♦✐♥t ♣r♦❜❧❡♠s✳ ❏♦✉r♥❛❧ ♦❢ ❖❜❥❡❝t ❚❡❝❤✲ ♥♦❧♦❣② ✺✭✷✮ ✭✷✵✵✻✮ ✼✺✕✽✸ ✶✾✳ ❇♦❧❞♦✱ ❙✳✱ ❋✐❧❧✐âtr❡✱ ❏✳❈✳✿ ❋♦r♠❛❧ ❱❡r✐✜❝❛t✐♦♥ ♦❢ ❋❧♦❛t✐♥❣✲P♦✐♥t Pr♦❣r❛♠s✳ ■♥✿ ✶✽t❤ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ❈♦♠♣✉t❡r ❆r✐t❤♠❡t✐❝✱ ▼♦♥t♣❡❧❧✐❡r✱ ❋r❛♥❝❡ ✭❏✉♥❡ ✷✵✵✼✮ ✶✽✼✕✶✾✹ ✷✵✳ ❆②❛❞✱ ❆✳✱ ▼❛r❝❤é✱ ❈✳✿ ▼✉❧t✐✲♣r♦✈❡r ✈❡r✐✜❝❛t✐♦♥ ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♣r♦❣r❛♠s✳ ■♥ ●✐❡s❧✱ ❏✳✱ ❍ä❤♥❧❡✱ ❘✳✱ ❡❞s✳✿ ❋✐❢t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡ ♦♥ ❆✉t♦♠❛t❡❞ ❘❡❛s♦♥✐♥❣✳ ▲◆❆■✱ ❊❞✐♥❜✉r❣❤✱ ❙❝♦t❧❛♥❞✱ ❙♣r✐♥❣❡r ✭❏✉❧② ✷✵✶✵✮ ✷✶✳ ❇♦❧❞♦✱ ❙✳✱ ◆❣✉②❡♥✱ ❚✳▼✳❚✳✿ ❍❛r❞✇❛r❡✲✐♥❞❡♣❡♥❞❡♥t ♣r♦♦❢s ♦❢ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠s✳ ■♥ ▼✉ñ♦③✱ ❈✳✱ ❡❞✳✿ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙❡❝♦♥❞ ◆❆❙❆ ❋♦r♠❛❧ ▼❡t❤♦❞s ❙②♠♣♦s✐✉♠✳ ◆✉♠❜❡r ◆❆❙❆✴❈P✲✷✵✶✵✲✷✶✻✷✶✺ ✐♥ ◆❆❙❆ ❈♦♥❢❡r❡♥❝❡ P✉❜❧✐❝❛t✐♦♥✱ ❲❛s❤✐♥❣t♦♥ ❉✳❈✳✱ ❯❙❆ ✭❆♣r✐❧ ✷✵✶✵✮ ✶✹✕✷✸ ✷✷✳ ❋✐❧❧✐âtr❡✱ ❏✳❈✳✱ ▼❛r❝❤é✱ ❈✳✿ ❚❤❡ ❲❤②✴❑r❛❦❛t♦❛✴❈❛❞✉❝❡✉s ♣❧❛t❢♦r♠ ❢♦r ❞❡❞✉❝t✐✈❡ ♣r♦❣r❛♠ ✈❡r✐✜❝❛t✐♦♥✳ ■♥✿ ❈♦♠♣✉t❡r ❆✐❞❡❞ ❱❡r✐✜❝❛t✐♦♥ ✭❈❆❱✮✳ ❱♦❧✉♠❡ ✹✺✾✵ ♦❢ ▲◆❈❙✳✱ ❙♣r✐♥❣❡r ✭❏✉❧② ✷✵✵✼✮ ✷✸✳ ❙t❡r❜❡♥③✱ P✳❍✳✿ ❋❧♦❛t✐♥❣ ♣♦✐♥t ❝♦♠♣✉t❛t✐♦♥✳ Pr❡♥t✐❝❡ ❍❛❧❧ ✭✶✾✼✹✮ ✷✹✳ ❱❡❧t❦❛♠♣✱ ●✳❲✳✿ ❆❧❣♦❧♣r♦❝❡❞✉r❡s ✈♦♦r ❤❡t ❜❡r❡❦❡♥❡♥ ✈❛♥ ❡❡♥ ✐♥✇❡♥❞✐❣ ♣r♦❞✉❝t ✐♥ ❞✉❜❜❡❧❡ ♣r❡❝✐s✐❡✳ ❘❈✲■♥❢♦r♠❛t✐❡ ✷✷✱ ❚❡❝❤♥✐s❝❤❡ ❍♦❣❡s❝❤♦♦❧ ❊✐♥❞❤♦✈❡♥ ✭✶✾✻✽✮ ✷✺✳ ❉❡❦❦❡r✱ ❚✳❏✳✿ ❆ ✢♦❛t✐♥❣ ♣♦✐♥t t❡❝❤♥✐q✉❡ ❢♦r ❡①t❡♥❞✐♥❣ t❤❡ ❛✈❛✐❧❛❜❧❡ ♣r❡❝✐s✐♦♥✳ ◆✉♠❡r✐s❝❤❡ ▼❛t❤❡♠❛t✐❦ ✶✽✭✸✮ ✭✶✾✼✶✮ ✷✷✹✕✷✹✷
✽ ❙②❧✈✐❡ ❇♦❧❞♦ ✷✻✳ ❇♦❧❞♦✱ ❙✳✿ P✐t❢❛❧❧s ♦❢ ❛ ❢✉❧❧ ✢♦❛t✐♥❣✲♣♦✐♥t ♣r♦♦❢✿ ❊①❛♠♣❧❡ ♦♥ t❤❡ ❢♦r♠❛❧ ♣r♦♦❢ ♦❢ t❤❡ ✈❡❧t❦❛♠♣✴❞❡❦❦❡r ❛❧❣♦r✐t❤♠s✳ ■♥✿ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ t❤✐r❞ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡ ♦♥ ❆✉t♦♠❛t❡❞ ❘❡❛s♦♥✐♥❣ ✭■❏❈❆❘✮✱ ❙❡❛tt❧❡✱ ❯❙❆ ✭❆✉❣✉st ✷✵✵✻✮ ✺✷✕✻✻ ✷✼✳ ❑❛❤❛♥✱ ❲✳✿ ❖♥ t❤❡ ❈♦st ♦❢ ❋❧♦❛t✐♥❣✲P♦✐♥t ❈♦♠♣✉t❛t✐♦♥ ❲✐t❤♦✉t ❊①tr❛✲Pr❡❝✐s❡ ❆r✐t❤♠❡t✐❝✳ ❲♦r❧❞✲❲✐❞❡ ❲❡❜ ❞♦❝✉♠❡♥t ✭◆♦✈❡♠❜❡r ✷✵✵✹✮ ✷✽✳ ❇♦❧❞♦✱ ❙✳✿ ❑❛❤❛♥✬s ❛❧❣♦r✐t❤♠ ❢♦r ❛ ❝♦rr❡❝t ❞✐s❝r✐♠✐♥❛♥t ❝♦♠♣✉t❛t✐♦♥ ❛t ❧❛st ❢♦r✲ ♠❛❧❧② ♣r♦✈❡♥✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❡rs ✺✽✭✷✮ ✭❋❡❜r✉❛r② ✷✵✵✾✮ ✷✷✵✕✷✷✺ ✷✾✳ ❇♦❧❞♦✱ ❙✳✿ ❋❧♦❛ts ✫ ❘♦♣❡s✿ ❛ ❝❛s❡ st✉❞② ❢♦r ❢♦r♠❛❧ ♥✉♠❡r✐❝❛❧ ♣r♦❣r❛♠ ✈❡r✐✜❝❛t✐♦♥✳ ■♥✿ ✸✻t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦❧❧♦q✉✐✉♠ ♦♥ ❆✉t♦♠❛t❛✱ ▲❛♥❣✉❛❣❡s ❛♥❞ Pr♦❣r❛♠♠✐♥❣✳ ❱♦❧✉♠❡ ✺✺✺✻ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✲ ❆❘❈♦❙❙✳✱ ❘❤♦❞♦s✱ ●r❡❡❝❡✱ ❙♣r✐♥❣❡r ✭❏✉❧② ✷✵✵✾✮ ✾✶✕✶✵✷ ✸✵✳ ❇♦❧❞♦✱ ❙✳✱ ❈❧é♠❡♥t✱ ❋✳✱ ❋✐❧❧✐âtr❡✱ ❏✳❈✳✱ ▼❛②❡r♦✱ ▼✳✱ ▼❡❧q✉✐♦♥❞✱ ●✳✱ ❲❡✐s✱ P✳✿ ❋♦r♠❛❧ Pr♦♦❢ ♦❢ ❛ ❲❛✈❡ ❊q✉❛t✐♦♥ ❘❡s♦❧✉t✐♦♥ ❙❝❤❡♠❡✿ t❤❡ ▼❡t❤♦❞ ❊rr♦r✳ ■♥✿ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✜rst ■♥t❡r❛❝t✐✈❡ ❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❈♦♥❢❡r❡♥❝❡✳ ▲◆❈❙✱ ❊❞✐♥❜✉r❣❤✱ ❙❝♦t❧❛♥❞✱ ❙♣r✐♥❣❡r ✭❏✉❧② ✷✵✶✵✮ ✸✶✳ ❞❡ ❉✐♥❡❝❤✐♥✱ ❋✳✱ ▲❛✉t❡r✱ ❈✳✱ ▼❡❧q✉✐♦♥❞✱ ●✳✿ ❆ss✐st❡❞ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ✉s✐♥❣ ●❛♣♣❛✳ ■♥✿ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✵✵✻ ❆❈▼ ❙②♠♣♦s✐✉♠ ♦♥ ❆♣♣❧✐❡❞ ❈♦♠♣✉t✐♥❣✱ ❉✐❥♦♥✱ ❋r❛♥❝❡ ✭✷✵✵✻✮ ✶✸✶✽✕✶✸✷✷ ✸✷✳ ❉❛✉♠❛s✱ ▼✳✱ ▼❡❧q✉✐♦♥❞✱ ●✳✿ ❈❡rt✐✜❝❛t✐♦♥ ♦❢ ❜♦✉♥❞s ♦♥ ❡①♣r❡ss✐♦♥s ✐♥✈♦❧✈✐♥❣ r♦✉♥❞❡❞ ♦♣❡r❛t♦rs✳ ❚r❛♥s❛❝t✐♦♥s ♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❢t✇❛r❡ ✸✼✭✶✮ ✭✷✵✵✾✮ ✸✸✳ ❇♦❧❞♦✱ ❙✳✱ ❋✐❧❧✐âtr❡✱ ❏✳❈✳✱ ▼❡❧q✉✐♦♥❞✱ ●✳✿ ❈♦♠❜✐♥✐♥❣ ❈♦q ❛♥❞ ●❛♣♣❛ ❢♦r ❈❡rt✐❢②✲ ✐♥❣ ❋❧♦❛t✐♥❣✲P♦✐♥t Pr♦❣r❛♠s✳ ■♥✿ ✶✻t❤ ❙②♠♣♦s✐✉♠ ♦♥ t❤❡ ■♥t❡❣r❛t✐♦♥ ♦❢ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t❛t✐♦♥ ❛♥❞ ▼❡❝❤❛♥✐s❡❞ ❘❡❛s♦♥✐♥❣✳ ❱♦❧✉♠❡ ✺✻✷✺ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❆rt✐✜✲ ❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡✳✱ ●r❛♥❞ ❇❡♥❞✱ ❈❛♥❛❞❛✱ ❙♣r✐♥❣❡r ✭❏✉❧② ✷✵✵✾✮ ✺✾✕✼✹
