Power series analysis as a major breakthrough to improve the efficiency of Asymptotic Numerical Method in the vicinity of bifurcations

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Preprint submitted on 12 Jun 2012

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Power series analysis as a major breakthrough to improve the efficiency of Asymptotic Numerical Method

in the vicinity of bifurcations

Bruno Cochelin, Médale Marc

To cite this version:

Bruno Cochelin, Médale Marc. Power series analysis as a major breakthrough to improve the efficiency

of Asymptotic Numerical Method in the vicinity of bifurcations. 2012. �hal-00707513�

P♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s ❛s ❛ ♠❛❥♦r ❜r❡❛❦t❤r♦✉❣❤ t♦

✐♠♣r♦✈❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❆s②♠♣t♦t✐❝ ◆✉♠❡r✐❝❛❧

▼❡t❤♦❞ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❜✐❢✉r❝❛t✐♦♥s

❇r✉♥♦ ❈❖❈❍❊▲■◆

▼❛r❝ ▼❊❉❆▲❊

1

▲▼❆✱ ❯P❘ ✼✵✺✶ ❈◆❘❙

✸✶ ❝❤❡♠✐♥ ❏♦s❡♣❤ ❆✐❣✉✐❡r

✶✸✹✵✷ ▼❆❘❙❊■▲▲❊ ❈❡❞❡① ✷✵✱ ❋r❛♥❝❡

2

❈❡♥tr❛❧❡ ▼❛rs❡✐❧❧❡✱ ❚❡❝❤♥♦♣♦❧❡ ❞❡ ❈❤❛t❡❛✉✲●♦♠❜❡rt✱

✶✸✹✺✶ ▼❆❘❙❊■▲▲❊ ❈❡❞❡① ✷✵✱ ❋r❛♥❝❡

3

■❯❙❚■✱ ❯▼❘ ✼✸✹✸ ❈◆❘❙✲❯♥✐✈❡rs✐té ❆✐①✲▼❛rs❡✐❧❧❡

✺ r✉❡ ❊♥r✐❝♦ ❋❡r♠✐✱ ❚❡❝❤♥♦♣♦❧❡ ❞❡ ❈❤❛t❡❛✉✲●♦♠❜❡rt✱

✶✸✹✺✸ ▼❛rs❡✐❧❧❡ ❈❡❞❡① ✶✸✱ ❋r❛♥❝❡

❏✉♥❡ ✶✷✱ ✷✵✶✷

❆❜str❛❝t ✿ ❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts t❤❡ ♦✉t❝♦♠❡ ♦❢ ♣♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s ✐♥ t❤❡

❢r❛♠❡✇♦r❦ ♦❢ t❤❡ ❆s②♠♣t♦t✐❝ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞✳ ❲❡ t❤❡♦r❡t✐❝❛❧❧② ❞❡♠♦♥✲

str❛t❡ ❛♥❞ ♥✉♠❡r✐❝❛❧❧② ❡✈✐❞❡♥❝❡ t❤❛t t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ❣❡♦♠❡tr✐❝ ♣♦✇❡r s❡✲

r✐❡s ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts ✐s ❛ ❣❡♥❡r✐❝ ❜❡❤❛✈✐♦r✳ ❙♦ ✇❡

♣r♦♣♦s❡ t♦ ✉s❡ t❤✐s ❤❛❧❧♠❛r❦ ❛s ❛ ❜✐❢✉r❝❛t✐♦♥ ✐♥❞✐❝❛t♦r t♦ ❧♦❝❛t❡ ❛♥❞ ❝♦♠✲

♣✉t❡ ✈❡r② ❡✣❝✐❡♥t❧② ❛♥② s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❋✐♥❛❧❧②✱ ❛ ♣♦✇❡r s❡r✐❡s t❤❛t r❡❝♦✈❡rs ❛♥ ♦♣t✐♠❛❧ st❡♣ ❧❡♥❣t❤ ✐s ❜✉✐❧❞ ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❜✐❢✉r❝❛✲

t✐♦♥ ♣♦✐♥ts✳ ❚❤❡ r❡❧✐❛❜✐❧✐t② ❛♥❞ r♦❜✉st♥❡ss ♦❢ t❤✐s ♣♦✇❡r❢✉❧ ❛♣♣r♦❛❝❤ ✐s t❤❡♥

❞❡♠♦♥str❛t❡❞ ♦♥ t✇♦ ❛♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡s ❢r♦♠ str✉❝t✉r❛❧ ♠❡❝❤❛♥✐❝s ❛♥❞

❤②❞r♦❞②♥❛♠✐❝s✳

❑❡②✇♦r❞s ✿ P❛t❤✲❢♦❧❧♦✇✐♥❣✱ ❝♦♥t✐♥✉❛t✐♦♥✱ ❜✐❢✉r❝❛t✐♦♥✱ ♣♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s✱

❆s②♠♣t♦t✐❝ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞✳

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ♠❛✐♥ ♣✉r♣♦s❡ ♦❢ ❝♦♥t✐♥✉❛t✐♦♥ ♦r ♣❛t❤✲❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠s ✐s t♦ ♠❛♣

st❡❛❞② st❛t❡ s♦❧✉t✐♦♥s ✐♥ t❤❡ ♣❛r❛♠❡t❡r s♣❛❝❡ ❜② ❝♦♠♣✉t✐♥❣ ❜r❛♥❝❤❡s ♦❢ s♦✲

❧✉t✐♦♥s ❢♦r ❛ ❣✐✈❡♥ r❛♥❣❡ ♦❢ ❝♦♥tr♦❧ ♣❛r❛♠❡t❡rs ❬✶✽✱ ✷✸❪✳ ■♥ ❛ ♠♦r❡ ❞❡t❛✐❧❡❞

✶

✇❛②✱ ♦♥❡ ❧♦♦❦s ❢♦r s✐♥❣❧❡ ♦r ♠✉❧t✐♣❧❡ s♦❧✉t✐♦♥ r❛♥❣❡s✱ ❝♦♠♣✉t❡ ❜✐❢✉r❝❛t✐♦♥

❞✐❛❣r❛♠s ❛♥❞ ✜♥❛❧❧② ♣❡r❢♦r♠ st❛❜✐❧✐t② ❛♥❛❧②s❡s t♦ ❞❡t❡r♠✐♥❡ r❡❣✐♦♥s ♦❢ st❛✲

❜❧❡ ❛♥❞ ✉♥st❛❜❧❡ s♦❧✉t✐♦♥s ❬✶✺✱ ✷❪✳ ❋✐rst ♦r❞❡r ♣r❡❞✐❝t♦r✲❝♦rr❡❝t♦r ❛❧❣♦r✐t❤♠s

✭❊✉❧❡r ♣r❡❞✐❝t♦r✱ ◆❡✇t♦♥✲❘❛♣❤s♦♥ ❜❛s❡❞ ❝♦rr❡❝t♦r✮ ✇✐t❤ ♣s❡✉❞♦✲❛r❝✲❧❡♥❣t❤

♣❛r❛♠❡t❡r✐③❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ✇✐❞❡❧② ✉s❡❞ ❢♦r ❞❡❝❛❞❡s ❬✶✽✱ ✶✺✱ ✷✸❪✳ ◆❡✈❡rt❤❡✲

❧❡ss✱ st❡♣✲❧❡♥❣❤ ❛❞❛♣t✐✈✐t② ♠❛② ❜❡ ✐♥ tr♦✉❜❧❡ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❜✐❢✉r❝❛t✐♦♥

♣♦✐♥ts ❧❡❛❞✐♥❣ t♦ ❛ ✇❡❛❦ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✣❝✐❡♥❝② ❛♥❞ s♦♠❡t✐♠❡s ❧❛❝❦ ♦❢

❝♦♥✈❡r❣❡♥❝❡✳

❆♥ ❛❧t❡r♥❛t❡ ✇❛② t♦ ✜rst ♦r❞❡r ♣r❡❞✐❝t♦r ❛❧❣♦r✐t❤♠s st❛♥❞s ✐♥ ❤✐❣❤✲♦r❞❡r

♣r❡❞✐❝t♦rs t❤❛t ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ✐♥ ❝♦♥t✐♥✉❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ t❤❡

❆s②♠♣t♦t✐❝ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞ ✭❆◆▼✮✳ ❚❤✐s ♠❡t❤♦❞ ❝♦♥s✐sts ✐♥ ❛ ❝♦♠❜✐♥❛✲

t✐♦♥ ♦❢ ❛ ❤✐❣❤✲♦r❞❡r ♣❡rt✉r❜❛t✐♦♥ ♠❡t❤♦❞✱ ❛ ❞✐s❝r❡t✐③❛t✐♦♥ t❡❝❤♥✐q✉❡ ❛♥❞ ❛

♣❛r❛♠❡t❡r✐③❛t✐♦♥ str❛t❡❣② ❬✶✸✱ ✸✱ ✾✱ ✶✻❪✳ ■t ✐s ❛ ❣❡♥❡r❛❧ ❛♥❞ ❡✣❝✐❡♥t ♥♦♥✲❧✐♥❡❛r s♦❧✉t✐♦♥ ♠❡t❤♦❞ ❬✶✷❪✱ ✇❤✐❝❤ ❤❛s ❜❡❡♥ s✉❝❝❡ss❢✉❧❧② ❛♣♣❧✐❡❞ ✐♥ ♠❛✐♥❧② s♦❧✐❞ ❛♥❞

str✉❝t✉r❛❧ ♠❡❝❤❛♥✐❝s✱ ❜✉t ❛❧s♦ ✐♥ ❢❡✇ ❤②❞r♦❞②♥❛♠✐❝s ♣r♦❜❧❡♠s ❬✽✱ ✶✱ ✼✱ ✶✼❪✳

■♥ t❤❡ ❆◆▼ t❤❡ st❡♣ ❧❡♥❣t❤ ✐s ❞❡t❡r♠✐♥❡❞ ❛✲♣♦st❡r✐♦r✐ ❢r♦♠ t❤❡ ❝♦♥✈❡r❣❡♥❝❡

♣r♦♣❡rt✐❡s ♦❢ t❤❡ s❡r✐❡s ❛t t❤❡ ❝✉rr❡♥t st❡♣ ✿ ✐t s❤♦rt❡♥s ❛s t❤❡ ♥♦♥ ❧✐♥❡❛r✐✲

t✐❡s ✐♥❝r❡❛s❡ ❛♥❞ ✇✐❞❡♥s ❛s s♦♦♥ ❛s t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s s♦❢t❡r✱ ♣r♦✈✐❞✐♥❣ ❛ s❡❧❢ ❛❞❛♣t✐✈✐t② ❛❧❧ ❛❧♦♥❣ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ❛s t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ❛♣✲

♣r♦❛❝❤❡s t♦ ❛ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t t❤❡ s❡❧❢ ❛❞❛♣t✐✈❡ st❡♣ ❧❡♥❣t❤ ❝❛♥ ❜❡❝♦♠❡ ✈❡r② s♠❛❧❧ ❛♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ❆◆▼ ♣r♦❝❡❞✉r❡ ❜❡❝♦♠❡s s✐❣✲

♥✐✜❝❛♥t❧② ♣❡♥❛❧✐s❡❞✳ ❘❡♣❧❛❝✐♥❣ ♥❛t✉r❛❧ ♣♦✇❡r s❡r✐❡s ❜② r❛t✐♦♥❛❧ ♦♥❡s ✭P❛❞é

❛♣♣r♦①✐♠❛t❡s✮ ✐♠♣r♦✈❡s t❤❡ ❣❧♦❜❛❧ ❝♦♥t✐♥✉❛t✐♦♥ ❡✣❝✐❡♥❝② ❛s ✐t ❡♥❛❜❧❡s t♦

❣❡t st❡♣ ❧❡♥❣t❤s r♦✉❣❤❧② t✇✐❝❡ ❛s ❧♦♥❣ ❛s ✐♥ t❤❡ ❝❧❛ss✐❝❛❧ ♣♦✇❡r s❡r✐❡s r❡♣✲

r❡s❡♥t❛t✐♦♥ ❬✶✶✱ ✺✱ ✶✻❪✱ ❜✉t ❡✈❡♥ ✐♥ t❤❛t ❝❛s❡ t❤❡ st❡♣ ❧❡♥❣t❤ ♠❛② ❜❡ ❢❛r

❢r♦♠ ✐ts ♦♣t✐♠❛❧ ✈❛❧✉❡✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts ❛♥❞ ❡♠❛♥❛t✲

✐♥❣ ❜r❛♥❝❤❡s ❛r❡ ♦t❤❡r ♣❛rt✐❝✉❧❛r❧② ✐♠♣♦rt❛♥t ✐ss✉❡s✳ ❚❤❡② ❤❛✈❡ ❜❡❡♥ t❛❝❦❧❡❞

✐♥ t❤❡ ❆◆▼ ❢r❛♠❡✇♦r❦ ❜② ❝♦♠♣✉t✐♥❣ ❛❧♦♥❣ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ❡✐t❤❡r t❤❡ ③❡r♦s

♦❢ P❛❞é ❛♣♣r♦①✐♠❛t❡s ♦r ❛ s❝❛❧❛r ❜✐❢✉r❝❛t✐♦♥ ✐♥❞✐❝❛t♦r ♦❜t❛✐♥❡❞ ❢r♦♠ ❛♥ ❛❞✲

❞✐t✐♦♥❛❧ ♣❡rt✉r❜❡❞ ♣r♦❜❧❡♠ s♦❧✉t✐♦♥ ❬✷✹✱ ✷✽✱ ✻✱ ✼✱ ✶✼❪✳ ◆♦t ♦♥❧② t❤❡s❡ ♠❡t❤♦❞s

❤❛✈❡ ❛♥ ❡①tr❛ ❝♦st ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ✉♣ t♦ t✇✐❝❡ ❛s t❤❡ ♦r✐❣✐♥❛❧ ❆◆▼ ♣r♦❜❧❡♠✱

❜✉t ❛❧s♦ t❤❡② ❝♦✉❧❞ ❜❡ s♦♠❡t✐♠❡s ❧❛❝❦✐♥❣ s♦♠❡ r♦❜✉st♥❡ss✳

❆♥ ❛❧t❡r♥❛t❡ ♣r♦♠✐s✐♥❣ ❞✐r❡❝t✐♦♥ ✇❛s ♣✐♦♥❡❡r❡❞ ✐♥ t❤❡ ♠✐❞ s❡✈❡♥t✐❡s ❜②

❱❛♥ ❉②❦❡ ✇❤♦ ✐♥✐t✐❛t❡❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❤✐❣❤ ♦r❞❡r ♣❡rt✉r❜❛t✐♦♥ s❡r✐❡s

❛♥❞ t❤❡✐r ❛♥❛❧②s✐s ✐♥ ♦r❞❡r t♦ r❡✈❡❛❧ t❤❡✐r ❛♥❛❧②t✐❝❛❧ str✉❝t✉r❡ ❬✷✺❪✳ ❍❡ ♣r♦✲

♣♦s❡❞ s❡✈❡r❛❧ s❡r✐❡s ✐♠♣r♦✈❡♠❡♥ts ❛♥❞ ❝♦♠♣❧❡t✐♦♥ ❡✐t❤❡r t♦ ❝♦♠♣✉t❡ t❤❡♠

♠✉❝❤ ♠♦r❡ ❡✣❝✐❡♥t❧② ♦r t♦ ❡①t❡♥❞ t❤❡✐r r❛♥❣❡ ♦❢ ✉t✐❧✐t②✱ ✐♥ ❛♥ ✉❧t✐♠❛t❡ ❣♦❛❧

t♦ r❡✈❡❛❧ t❤❡✐r ✉♥❞❡r❧②✐♥❣ ♠❛t❤❡♠❛t✐❝❛❧ ♦r ♣❤②s✐❝❛❧ ♠❡❛♥✐♥❣ ❬✷✻✱ ✷✼❪✳ ❍❡

♠❛❞❡ ✐t ❝❧❡❛r t❤❛t ♣♦✇❡r s❡r✐❡s ✉♥❞♦✉❜t❡❞❧② ❝♦♥t❛✐♥ ♥✉♠❡r♦✉s r❡❧❡✈❛♥t ✐♥✲

❢♦r♠❛t✐♦♥ ❛♥❞ ✇❛②s ❝❛♥ ❜❡ ❢♦✉♥❞ t♦ ❡①tr❛❝t t❤❡♠✳ ❇✉t s✉r♣r✐s✐♥❣❧②✱ ♥♦♥❡ ♦❢

❱❛♥ ❉②❦❡✬s ♣♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s ❤❛s ❜❡♥❡✜t❡❞ t♦ t❤❡ ❆◆▼ ②❡t✳

❙♦✱ ✐♥ t❤❡ ♣r❡s❡♥t ✇♦r❦ ✇❡ ❤❛✈❡ ❛♥❛❧②s❡❞ t❤❡ ♣♦✇❡r s❡r✐❡s t❤❛t ❛r✐s❡ ❛t

❡✈❡r② st❡♣ ♦❢ t❤❡ ❆◆▼ ❛❧❣♦r✐t❤♠ ❛♥❞ ✐t t✉r♥s ♦✉t t❤❛t ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢

s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts ❛ ❣❡♦♠❡tr✐❝ ♣♦✇❡r s❡r✐❡s ❛❧✇❛②s ❡♠❡r❣❡s✳ ▼♦r❡♦✈❡r

✷

t❤✐s ❜❡❤❛✈✐♦r ✐s ❣❡♥❡r✐❝ ❛s ✐t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r♦❜❧❡♠ t♦ ❜❡ s♦❧✈❡❞✳

❚❤❡r❡❢♦r❡ ✇❡ ♣r♦♣♦s❡ t♦ ✉s❡ t❤✐s ❤❛❧❧♠❛r❦ ❛s ❛ r❡❧✐❛❜❧❡ ❜✐❢✉r❝❛t✐♦♥ ✐♥❞✐❝❛t♦r t♦ ❛❝❝✉r❛t❡❧② ❛♥❞ ❝❤❡❛♣❧② ❝♦♠♣✉t❡ ❛♥② s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❋✐♥❛❧❧②✱

❛s s♦♦♥ ❛s ✐t ✐s ❞❡t❡❝t❡❞✱ ❛ ♥❡✇ ♣♦✇❡r s❡r✐❡s t❤❛t r❡❝♦✈❡rs ❛♥ ♦♣t✐♠❛❧ st❡♣

❧❡♥❣t❤ ✐s ❜✉✐❧❞ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts✱ ✇❤❡r❡ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ✐s

❝r✐t✐❝❛❧❧② ♣❡♥❛❧✐s❡❞ ✐♥ t❤❡ ❝❧❛ss✐❝❛❧ ❆◆▼✳

❚❤❡ ♣r❡s❡♥t ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ t❤❡ t❤❡♦r❡t✐❝❛❧

❜❛❝❦❣r♦✉♥❞ r❡❧❛t❡❞ t♦ ❜✐❢✉r❝❛t✐♦♥ ❛♥❛❧②s✐s ✐s ✜rst r❡❝❛❧❧❡❞✳ ❋♦r♠❛❧ ❞❡r✐✈❛✲

t✐♦♥s ♦❢ ❆▼◆ ♣♦✇❡r s❡r✐❡s ❛r❡ ❝❛rr✐❡❞ ♦✉t ✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♣❡rt✉r❜❡❞

❜✐❢✉r❝❛t✐♦♥ ❛♥❛❧②s✐s✳ ❚❤❡ ❜❛s✐❝ ❢❡❛t✉r❡s ♦❢ ❆◆▼ ♣♦✇❡r s❡r✐❡s ❛r❡ t❤❡♥ ❤✐❣❤✲

❧✐❣❤t❡❞ ❛♥❞ ❞✐s❝✉ss❡❞✳ ❙❡❝t✐♦♥ ✸ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ✇❛② ✇❡ ❤❛✈❡ ✐♠♣❧❡♠❡♥t❡❞

❣❡♦♠❡tr✐❝❛❧ ♣♦✇❡r s❡r✐❡s ❡①tr❛❝t✐♦♥ ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✇❡r s❡r✐❡s✳ ❚❤❡ ❝♦♠✲

♣✉t❛t✐♦♥ ♦❢ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts ❛♥❞ ❡♠❛♥❛t✐♥❣ ❜✐❢✉r❝❛t✐♦♥ ❜r❛♥❝❤❡s ❛r❡ t❤❡♥

❡♠♣❤❛s✐③❡❞✳ ❚❤❡ r❡❧✐❛❜✐❧✐t②✱ r♦❜✉st♥❡ss ❛♥❞ ❡✣❝✐❡♥❝② ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ✐s ♣r❡✲

s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✹ t❤r♦✉❣❤♦✉t t✇♦ ❛♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡s ✿ t❤❡ ❜✉❝❦❧✐♥❣ ♦❢ ❛♥

✐♥❡①t❡♥s✐❜❧❡ ❡❧❛st✐❝ ❜❡❛♠ ❛♥❞ t❤❡ ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞ ✢♦✇ ✐♥ ❛ s②♠♠❡tr✐❝

s✉❞❞❡♥ ❡①♣❛♥s✐♦♥✳ ❋✐♥❛❧❧②✱ ❝♦♥❝❧✉s✐♦♥s ❛♥❞ ❢✉t✉r❡ ✇♦r❦ ❞✐r❡❝t✐♦♥s ❛r❡ ❣✐✈❡♥

✐♥ ❙❡❝t✐♦♥ ✺✳

✷ ❋♦r♠❛❧ ❞❡r✐✈❛t✐♦♥s ♦❢ ❆▼◆ ♣♦✇❡r s❡r✐❡s ✐♥ t❤❡

✈✐❝✐♥✐t② ♦❢ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✜rst ✐♥tr♦❞✉❝❡ t❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞❡❞ t❤❡♦r❡t✐❝❛❧ ❜❛❝❦✲

❣r♦✉♥❞ ❢♦r ❛♥❛❧②s✐♥❣ ❆◆▼ ♣♦✇❡r s❡r✐❡s ❝♦♠♣✉t❡❞ ♥❡❛r s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥

♣♦✐♥t✳ ▲❡t R(u, λ) = 0 ❜❡ ❛♥ ❛❧❣❡❜r❛✐❝ s②st❡♠ ♦❢ n ♥♦♥❧✐♥❡❛r s♠♦♦t❤ ❡q✉❛✲

t✐♦♥s ✇❤❡r❡ u ∈ R

✐s ❛ ✈❡❝t♦r ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s ❛♥❞ λ ∈ R ❛ s✐♥❣❧❡ ❝♦♥tr♦❧

♦r ❜✐❢✉r❝❛t✐♦♥ ♣❛r❛♠❡t❡r✳ ❚❤❡ ❡①t❡♥❞❡❞ st❛t❡ ✈❡❝t♦r U = u

λ

∈ R

✐s

✐♥tr♦❞✉❝❡❞ ❢♦r ❝♦♠♣❛❝t♥❡ss s✐♥❝❡ ✐t ✐♥❝❧✉❞❡s t❤❡ ♣❛r❛♠❡t❡r λ ❛s ✐ts ❧❛st st❛t❡

✈❛r✐❛❜❧❡✱ s♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s②st❡♠ r❡❛❞s ✿

R(U) = 0 ✭✶✮

✇✐t❤♦✉t ❞✐st✐♥❣✉✐s❤✐♥❣ ❜❡t✇❡❡♥ st❛t❡ ✈❛r✐❛❜❧❡s ❛♥❞ ♣❛r❛♠❡t❡r✳ ●❡♥❡r✐❝ s♦❧✉✲

t✐♦♥s ♦❢ ✭✶✮ ❛r❡ ❜r❛♥❝❤❡s ♦❢ s♦❧✉t✐♦♥s✱ ✐❡✱ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t✐♥✉❛ ♦❢ s♦❧✉t✐♦♥

♣♦✐♥ts✳ ■♥ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❜❛s❡❞ ♦♥ t❤❡ ❆◆▼✱ t❤❡s❡ ❜r❛♥❝❤❡s

❛r❡ r❡♣r❡s❡♥t❡❞ ❛t t❤❡ ❝✉rr❡♥t st❡♣ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦✇❡r s❡r✐❡s ❡①♣❛♥✲

s✐♦♥ ❬✶✸✱ ✾✱ ✶✵✱ ✶✷❪ ✿

U (a) = U

+ a U

+ a

U

+ · · · + a

U

✭✷✮

✇✐t❤ a t❤❡ ♣❛t❤ ♣❛r❛♠❡t❡r ❞❡✜♥❡❞ ✐♥ t❤❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ❡q✉❛t✐♦♥✳ ■♥ t❤❡

❢♦❧❧♦✇✐♥❣✱ ✇❡ ❛♥❛❧②s❡ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ st❛rt✐♥❣ ♣♦✐♥t U

✐s ❛ s✐♠♣❧❡ ❜✐❢✉r✲

❝❛t✐♦♥ ♣♦✐♥t✱ t❤❡♥✱ ❛ r❡❣✉❧❛r ♣♦✐♥t ❝❧♦s❡❞ t♦ ❛ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥✳

✸

✷✳✶ ❇r❛♥❝❤❡s ♦❢ s♦❧✉t✐♦♥ ❛t ❛ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t

▲❡t U

❜❡ ❛ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ✇❤❡r❡ t✇♦ ❜r❛♥❝❤❡s ♦❢ s♦❧✉t✐♦♥s

❝r♦ss tr❛♥s✈❡rs❛❧❧②✱ ✐❡✱ ✇✐t❤ t✇♦ ❞✐st✐♥❝t t❛♥❣❡♥ts U

❛♥❞ U

✳ ❚❤❡ ❦❡r♥❡❧ ♦❢

t❤❡ ❥❛❝♦❜✐❛♥ ♠❛tr✐① R

❡✈❛❧✉❛t❡❞ ❛t U

✐s t✇♦ ❞✐♠❡♥s✐♦♥❛❧✱

N (R

) = ❙♣❛♥ {U

, U

}, N (R

) = ❙♣❛♥ {ψ}. ✭✸✮

✇❤✐❧❡ t❤❡ ❦❡r♥❡❧ ♦❢ t❤❡ tr❛♥s♣♦s❡❞ ❥❛❝♦❜✐❛♥ ♠❛tr✐① ✐s ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ❬✶✺❪✳

❚❤❡ t✇♦ ✭♥♦r♠❛❧✐③❡❞✮ t❛♥❣❡♥t ✈❡❝t♦rs U

(i = 1, 2) s❛t✐s❢② t❤❡ ✜rst ♦r❞❡r

❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✭✶✮ ❛♥❞ t❤❡ s♦✲❝❛❧❧❡❞ ✧❛❧❣❡❜r❛✐❝ ❜✐❢✉r❝❛t✐♦♥ ❡q✉❛t✐♦♥✧ ❬✶✽❪✱

✇❤✐❝❤ ✐s t❤❡ s❡❝♦♥❞ ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✭✶✮ ♣r♦❥❡❝t❡❞ ♦♥ ❧❡❢t ♥✉❧❧✲✈❡❝t♦r ψ ♦❢ R

✳

R

U

= 0

ψ

R

U

U

= 0 ✭✹✮

❲❡ ❛❧s♦ ❞❡✜♥❡ t❤❡ ✭♥♦r♠❛❧✐③❡❞✮ ✈❡❝t♦r U

❛s ❜❡✐♥❣ ✐♥s✐❞❡ N (R

) ❛♥❞ ♦r✲

t❤♦❣♦♥❛❧ t♦ U

✭s❛♠❡ ❢♦r U

✮✳ ❚❤❡ ❜r❛♥❝❤ ♦❢ s♦❧✉t✐♦♥s ❡♠❛♥❛t✐♥❣ ❢r♦♠ U

❛❧♦♥❣ t❤❡ t❛♥❣❡♥t U

❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② ✿

U (a) = U

+ a U

+ a

U

+ a

U

+ · · · ✭✺✮

✇❤❡r❡ a = (U (a) − U

)

U

✐s t❤❡ ❝❧❛ss✐❝❛❧ ♣s❡✉❞♦ ❛r❝✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✳

❯s✐♥❣ ❛ ▲②❛♣✉♥♦✈✲❙❝❤♠✐❞t ❞❡❝♦♠♣♦s✐t✐♦♥✱ t❤❡ ✈❡❝t♦r U

✱ p ≤ 2 ✱ r❡❛❞s ✿ U

= β

U

| {z }

∈N(R^c_,U)

+ V

| {z }

∈N^T(R^c_,U)

✭✻✮

✇❤❡r❡ V

✐s s♦❧✉t✐♦♥ ♦❢ ❛♥ ❛✉❣♠❡♥t❡❞ ✐♥✈❡rt✐❜❧❡ ❧✐♥❡❛r s②st❡♠ ✭s❡❡ ❛♣♣❡♥❞✐①

❢♦r ❞❡t❛✐❧❧❡❞ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ s❝❛❧❛r β

❛♥❞ t❤❡ ✈❡❝t♦r V

✮✳ ❲❡ r❡❢❡r t♦ ❬✸❪

❛♥❞ ❬✶✼❪ ❢♦r ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❜r❛♥❝❤❡s ♦❢ s♦❧✉t✐♦♥s ❛t ❛ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ✐♥

s♦❧✐❞ ♦r ✢✉✐❞ ♠❡❝❤❛♥✐❝s ♣r♦❜❧❡♠s✳

✷✳✷ P❡rt✉r❜❡❞ ❜✐❢✉r❝❛t✐♦♥ ❛♥❛❧②s✐s ❛t s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ✿

❧♦✇ ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥

❙♠❛❧❧ ✐♠♣❡r❢❡❝t✐♦♥s ❛r❡ ❛❧✇❛②s ♣r❡s❡♥t ✐♥ ❡①♣❡r✐♠❡♥ts✱ ❜✉t ❛❧s♦ ✐♥ ❝♦♠✲

♣✉t❛t✐♦♥s ♦✇✐♥❣ t♦ ✜♥✐t❡ ♣r❡❝✐s✐♦♥ ♦❢ ❝♦♠♣✉t❡rs ❛♥❞ t❤❡✐r ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳

❚❤✉s✱ ✇❡ ♥♦✇ ♣❛② ❛tt❡♥t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❡rt✉r❜❡❞ ♣r♦❜❧❡♠ ✿

R

(U, µ) = R(U ) + µP = 0 ✭✼✮

✇❤❡r❡ P ✐s ❛ ♥♦r♠❛❧✐③❡❞ ♣❡rt✉r❜❛t✐♦♥ ✈❡❝t♦r ❛♥❞ µ ❛ ❢r❡❡ ♣❡rt✉r❜❛t✐♦♥ ❛♠✲

♣❧✐t✉❞❡✳ ■♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ψ

P 6= 0 ✱ ✐❡✱ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ✈❡❝t♦r P ✐s ♥♦♥✲

❞❡❣❡♥❡r❛t❡ ♦r ❜✐❢✉r❝❛t✐♦♥ ❞❡str♦②✐♥❣✱ ❛s ♦♣♣♦s❡❞ t♦ ❜✐❢✉r❝❛t✐♦♥ ♣r❡s❡r✈✐♥❣

✹

❢♦r t❤❡ ❡①❝❡♣t✐♦♥❛❧ ❝❛s❡ ✇❤❡r❡ ψ

P = 0 ❬✷✷❪✳ ❍❡♥❝❡✱ ✇✐t❤♦✉t ❛♥② ❧♦ss ♦❢

❣❡♥❡r❛❧❧② ♦♥❡ ❝❛♥ s❡t ✿

P = ψ + P

✇✐t❤ ψ

P

= 0 ✭✽✮

❲✐t❤ t❤❡ ❛❞❞✐t✐♦♥♥❛❧ ❢r❡❡ ♣❛r❛♠❡t❡r µ s♦❧✉t✐♦♥s ♦❢ ✭✼✮ ❛r❡ ♥♦ ♠♦r❡

❜r❛♥❝❤❡s ❜✉t t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t✐♥✉❛ ♦❢ s♦❧✉t✐♦♥ ♣♦✐♥ts✳ ❆t t❤❡ ❜✐❢✉r❝❛t✐♦♥

♣♦✐♥t U

✱ t❤❡ ✷❉ ♠❛♥✐❢♦❧❞ ❝❛♥ ❜❡ ❢♦✉♥❞ ✉♥❞❡r t❤❡ ❢♦r♠ ♦❢ ❛ t✇♦✲♣❛r❛♠❡t❡r

❡①♣❛♥s✐♦♥ ✿

U = U

+ b

U

+ b

U

| {z }

∈N(R^c_,U)

+ V (b

, b

)

| {z }

∈N^T(R^c_,U)

µ = µ(b

, b

)

✭✾✮

✇❤❡r❡ b

✱ b

❛r❡ t❤❡ ❛♠♣❧✐t✉❞❡s ♦❢ U − U

♦♥ t❤❡ ❜❛s❡ ✈❡❝t♦rs U

❛♥❞ U

♦❢ N (R

) ✳ ❆❝❝♦r❞✐♥❣ t♦ ❜✐❢✉r❝❛t✐♦♥ t❤❡♦r② ❡①♣❛♥s✐♦♥ ♦❢ V ❛♥❞ µ s❤♦✉❧❞

❜❡❣✐♥ ❛t ♦r❞❡r t✇♦ ✐♥ b

, b

❛♥❞ t❤❡ s❡❝♦♥❞ ♦r❞❡r ♦❢ µ s❤♦✉❧❞ s♦❧✈❡ ❢♦r t❤❡

✧❛❧❣❡❜r❛✐❝ ❜✐❢✉r❝❛t✐♦♥ ❡q✉❛t✐♦♥✧ ✿

µ(b

, b

) = b

b

ψ

R

U

U

| {z }

µ1

✭✶✵✮

✷✳✷✳✶ P❡rt✉r❜❡❞ ❜r❛♥❝❤❡s

❲❤❡♥ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ❛♠♣❧✐t✉❞❡ ✐s ✜①❡❞ t♦ ❛ ❣✐✈❡♥ ✈❛❧✉❡ µ

✱ ❣❡♥❡r✐❝ s♦✲

❧✉t✐♦♥s ♦❢ ✭✼✮ ❣❡t ❜❛❝❦ t♦ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t✐♥✉❛ ❝❛❧❧❡❞ ♣❡rt✉r❜❡❞ ❜r❛♥❝❤❡s✳

❚❤❡② ❧♦♦❦ ❧✐❦❡ t❤❡ t✇♦ ❤②♣❡r❜♦❧❛❡ s❤♦✇♥ ✐♥ ✜❣✉r❡✭✶✮ ❛♥❞ t❤❡② ❛r❡ ❣✐✈❡♥ ❛t

❧♦✇ ♦r❞❡r ❜② t❤❡ ♣❛r❛♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ✿

U(b

, b

) = U

+ b

U

+ b

U

✇✐t❤ b

b

=

. ✭✶✶✮

Ut1 Ut2

Uc

U0

d Ut1

❋✐❣✉r❡ ✶ ✕ P❡rt✉❜❡❞ ❜✐❢✉r❝❛t✐♦♥ ❞✐❛❣r❛♠ ❛t ❧♦✇❡st ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥✳

✺

✷✳✸ P❡rt✉r❜❡❞ ❜r❛♥❝❤ ❡①♣r❡ss✐♦♥ ❢r♦♠ ❛ r❡❣✉❧❛r ♣♦✐♥t ❧♦✲

❝❛t❡❞ ❝❧♦s❡ t♦ ❛ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t

▲❡t d ❜❡ ❛ ❣✐✈❡♥ ❞✐st❛♥❝❡✱ ♦♥❡ s❡ts b

= d ❛♥❞ b

=

1d

✐♥t♦ ✭✶✶✮ t♦ ❞❡✜♥❡

❛ r❡❣✉❧❛r ♣♦✐♥t ✿

U

= U

+ d U

+ µ

µ

d U

✭✶✷✮

❛t ❛ ❞✐st❛♥❝❡ d ❛❧♦♥❣ U

❢r♦♠ t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❲❡ ♥♦✇ ❞❡✜♥❡ t❤❡ ♥❡✇

♣❛t❤ ♣❛r❛♠❡t❡r a = d − b

❛♥❞ r❡✇r✐t❡ ✭✶✶✮ ❛s ✿ U (a) = U

+ a (−U

) + µ

µ

d

a d

(1 −

) U

✭✶✸✮

❚❤✐s ❧❛st ❡q✉❛t✐♦♥ ❣✐✈❡s ❛ ♣❛r❛♠❡tr✐❝ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ♣❡rt✉r❜❛t❡❞ ❜r❛♥❝❤

❢r♦♠ t❤❡ r❡❣✉❧❛r s♦❧✉t✐♦♥ ♣♦✐♥t U

✭a = 0✮ t♦✇❛r❞ t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t U

✭ a > 0 ✮✳ ❚❤❡ r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥ t❤❛t ❛♣♣❡❛rs ✐♥ U

✬s ❛♠♣❧✐t✉❞❡ ❤❛s ❛ ♣♦❧❡ ❢♦r a = d ✱ ✇❤✐❝❤ ♠❛❦❡s t❤❡ ♣❡rt✉r❜❡❞ ❜r❛♥❝❤ t✉r♥ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ U

♥❡❛r t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ◆♦t✐❝❡ t❤❛t ✇❤❡♥ µ

✐s ✈❡r② s♠❛❧❧ t❤❡ ❛♠♣❧✐t✉❞❡ ❢♦r U

✐s ❛❧s♦ ✈❡r② s♠❛❧❧ ❡①❝❡♣t ✇❤❡♥ a ✐s ✈❡r② ❝❧♦s❡ t♦ d ✳

❚❤✐s s✐♠♣❧❡ r❡❛s♦♥✐♥❣ t❤❛t ❧❡❛❞s t♦ ✭✶✸✮ ❛❧r❡❛❞② s❤♦✇s t❤❡ ❦❡② ♣♦✐♥t ♦❢ t❤❡

♣r❡s❡♥t ❛♥❛❧②s✐s ✿ ❢♦r ❛♥② ♥♦♥ ③❡r♦ ✐♠♣❡r❢❡❝t✐♦♥ t❤❡ ✭✈❡❝t♦r✮ ❢✉♥❝t✐♦♥ U (a)

❤❛s ❛ s✐♥❣✉❧❛r✐t② ❢♦r a = d ✱ ✐❡✱ ❢♦r ❛ ♣♦s✐t✐✈❡ r❡❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡r a

❛♥❞ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t② ✐s ♦❢ t❤❡ ❦✐♥❞ (1−

)

✳ ❚❤❡ ❚❛②❧♦r s❡r✐❡s ♦❢

s✉❝❤ ❛ r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥ ✐s ✿

+ (

)

+ (

)

+ · · · ✳ ■t r❡♣r❡s❡♥ts ❛ ❣❡♦♠❡tr✐❝❛❧

♣r♦❣r❡ss✐♦♥ ✇❤♦s❡ ❝♦♠♠♦♥ r❛t✐♦ ✐s

❛♥❞ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ d ✳ ❚❤❡s❡

r❡s✉❧ts ❛r❡ ❣❧♦❜❛❧❧② ✐♥ ❧✐♥❡ ✇✐t❤ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❆◆▼ s❡r✐❡s ✇❤❡♥ t❤❡

❡①♣❛♥s✐♦♥ ♣♦✐♥t ✐s ❝❧♦s❡ t♦ ❛ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❍♦✇❡✈❡r✱ t❤❡ ♣✐❝t✉r❡ ✐s ♥♦t

❝♦♠♣❧❡t❡ ❜❡❝❛✉s❡ ✭✶✸✮ ✐s ♦♥❧② ❛ ✜rst ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥✳

✷✳✹ ❊①t❡♥t✐♦♥ t♦ ❤✐❣❤❡r ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥

❚♦ ❡①t❡♥❞ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧ts ❛t ❤✐❣❤ ♦r❞❡rs ✇❡ ❜❡❣✐♥ ✇✐t❤ t❤❡ ❞❡r✐✈❛t✐♦♥

♦❢ ❛ t✇♦ ♣❛r❛♠❡t❡r ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ✷❉ ♠❛♥✐❢♦❧❞ ✉♥❞❡r t❤❡ ❢♦r♠ ✿ U (a

, a

) = U

+ a

U

+ a

U

+a

V

+ a

a

V

+ a

V

+a

V

+ a

a

V

+ a

a

V

+ a

V

+ · · ·

✭✶✹✮

µ(a

, a

) = a

µ

+ a

a

µ

+ a

µ

+a

µ

+ a

a

µ

+ a

a

µ

+ a

µ

+ · · ·

✭✶✺✮

✇❤❡r❡ a

❛♥❞ a

❛r❡ ❞❡✜♥❡❞ ❛s t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ U − U

♦♥ t❤❡ ❜❛s❡ ✈❡❝t♦r U

❛♥❞ U

♦❢ N (R

) ✳ ■♥ t❤❡ ♣❡rs♣❡❝t✐✈❡ t♦ ❛♥❛❧②s❡ ❆◆▼ ♣♦✇❡r s❡r✐❡s✱

t❤❡ ❝❤♦✐❝❡ ♦❢ ✭ a

✱ a

✮ ♣❛r❛♠❡t❡rs ✐s ❜❡tt❡r t❤❛♥ ✭ b

✱ b

✮✱ ✇❤✐❝❤ ✇❡r❡ t❤❡

❜❡st ♦♥❡ ❢♦r ❛ ✜rst ♦r❞❡r ❛♥❛❧②s✐s✳ ❆❧❧ t❤❡ ✈❡❝t♦rs V

✬s ❛r❡ ♦rt❤♦❣♦♥❛❧ t♦ U

✻

❛♥❞ U

✱ t❤❡✐r ❡①♣r❡ss✐♦♥ ❤❛✈❡ ❜❡❡♥ r❡♣♦rt❡❞ ✐♥ ❛♣♣❡♥❞✐① ❢♦r ❜r❡✈✐t② ✐♥ t❤❡

♣r❡s❡♥t❛t✐♦♥✳

❚❤❡ ♥❡①t st❡♣ ✐s t♦ r❡✇r✐t❡ t❤❡ ✷❉ ♠❛♥✐❢♦❧❞ ❜② t❛❦✐♥❣ a

❛♥❞ µ ❛s ♣❛✲

r❛♠❡t❡rs ✐♥st❡❛❞ ♦❢ a

❛♥❞ a

✳ ❋♦r t❤✐s ✇❡ ❢♦❧❧♦✇ t❤❡ ♣r♦❝❡❞✉r❡ ♣r♦♣♦s❡❞ ❜②

❉❛♠✐❧ ❛♥❞ P♦t✐❡r✲❋❡rr② ✐♥ ❬✶✸❪ t♦ ❣❡t ✿

U (a

, µ) = U

+ a

U

+ a

U

+ a

U

+ · · · +

1

a1

U

+ U

+ a

U

+ a

U

+ · · · +

11

a³₁

U

+

U

+ +

1

U

+ · · · +

11

a⁵1

U

+ · · ·

✳✳✳

✭✶✻✮

❚❤❡ ✜rst ❧✐♥❡ ♦❢ ✭✶✻✮ ❡①❛❝t❧② ❝♦rr❡s♣♦♥❞s t♦ ✭✺✮✱ t❤❡ ❝❛s❡ ✇✐t❤ ♥♦ ✐♠♣❡r❢❡❝✲

t✐♦♥✳ ◆♦t✐❝❡ ❛❧s♦ t❤❛t s✐♥❣✉❧❛r t❡r♠s ♦❢ t❤❡ ❦✐♥❞

1

❛♣♣❡❛r ✐♥ s✉❜s❡q✉❡♥t

❧✐♥❡s ♦❢ t❤✐s ❡①♣r❡ss✐♦♥✳

✷✳✹✳✶ ❋✐rst ♦r❞❡r tr✉♥❝❛t✐♦♥ ✐♥ µ

■♥ ♦✉r ❆◆▼ ❝♦♥t✐♥✉❛t✐♦♥ ♣r♦❝❡ss ✐♠♣❡r❢❡❝t✐♦♥s ❛♣♣❡❛r ❜❡❝❛✉s❡ t❤❡ s♦✲

❧✉t✐♦♥ ♣♦✐♥ts ✇❤❡r❡ t❤❡ s❡r✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ❞♦ ♥♦t ❡①❛❝t❧② s❛t✐s❢② ✭✶✮ ❜✉t

♦♥❧② ✉♣ t♦ ❛ r❡q✉✐r❡❞ ✉s❡r✲❞❡✜♥❡❞ ♣r❡❝✐s✐♦♥ ε

✱ ✐❡✱ U

✐s ❛ s♦❧✉t✐♦♥ ♣♦✐♥t ✐❢

||R(U

)|| < ε

✳ ❚❤❡ ❆◆▼ s❡r✐❡s ❡✈❛❧✉❛t❡❞ ❛t U

❛r❡ ✐♥❞❡❡❞ t❤❡ ❡①❛❝t ❢♦r♠❛❧

s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣❡rt✉r❜❡❞ ♣r♦❜❧❡♠ R(U ) − R(U

) = 0 ✳ ■♥ ❛♣♣❧✐❝❛t✐♦♥s✱ ε

✐s ❣❡♥❡r❛❧❧② s❡t t♦ ❛ ✈❡r② s♠❛❧❧ ✈❛❧✉❡✱ ❤❡♥❝❡✱ t❤❡ ✐♠♣❡r❢❡❝t✐♦♥ ❛♠♣❧✐t✉❞❡ µ

❝♦♥s✐❞❡r❡❞ ❤❡r❛❢t❡r ✇✐❧❧ ❛❧s♦ ❜❡ ✈❡r② s♠❛❧❧✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ✇❡ ❜❡❣✐♥ t❤✐s

❛♥❛❧②s✐s ✐♥ ❧✐♠✐t✐♥❣ t♦ ❛ ✜rst ♦r❞❡r tr✉♥❝❛t✉r❡ ✐♥ µ ✳

✷✳✹✳✷ P❡rt✉r❜❡❞ ❜r❛♥❝❤ ❡①♣r❡ss✐♦♥ ❢♦r♠ ❛ r❡❣✉❧❛r ♣♦✐♥t U

❆s ❛❜♦✈❡ ✇❡ s❡t t❤❡ ✐♠♣❡r❢❡❝t✐♦♥ ❛♠♣❧✐t✉❞❡ µ t♦ ❛ ✜①❡❞ ✈❛❧✉❡ µ

t♦

❞❡✜♥❡ ♣❡rt✉r❜❡❞ ❜r❛♥❝❤❡s✳ ❚❤❡♥✱ ✇❡ ❞❡♣♦rt t❤❡ ❡①♣❛♥s✐♦♥ ♣♦✐♥t ❢r♦♠ U

t♦

U

❜② t❛❦✐♥❣ a = d − a

t♦ ❣❡t ✿

U (a) = U

+ a U

+ a

U

+ a

U

+ · · · +

11

d(d−a)

U

+ aU

+ a

U

+ · · · ✭✶✼✮

❚❤❡ ♥❡✇ ✈❡❝t♦r U

❛♥❞ U

✭❛t U

✮ ❝❛♥ ❜❡ ❡❛s✐❧② ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ U

❛♥❞

U

✭❛t U

✮ ♦❢ ✭✶✻✮ ❛♥❞ d✳

▲❡t r ❜❡ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✜rst ❧✐♥❡ ♦❢ ✭✶✼✮✱ ✐❡✱ t❤❡ s❡r✐❡s

❢♦r t❤❡ ♣❡r❢❡❝t ❜r❛♥❝❤✳ ▲❡t ✉s s❝❛❧❡ t❤❡ ♣❛t❤ ♣❛r❛♠❡t❡r a ❜② r ✱ s♦ t❤❛t t❤❡

r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣❡r❢❡❝t ❜r❛♥❝❤ ❜❡❝♦♠❡s ✉♥✐t② ❛♥❞ t❤❡ ♥♦r♠ ♦❢

t❤❡ s❡r✐❡s ❝♦❡✣❝✐❡♥t ❜❡❝♦♠❡ ♦❢ ♦r❞❡r ♦❢ ✉♥✐t②✳ ❉❡♥♦t✐♥❣ ¯ a =

✱ ✇❡ ❣❡t t❤❡

✼

✜♥❛❧ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♣❡rt✉r❜❡❞ ❜r❛♥❝❤❡s ✿

U (¯ a) = U

+ ¯ a (r U

) + ¯ a

(r

U

) + ¯ a

(r

U

) + · · · +

d

d(1−¯a^r_d)

U

+ ¯ a(r U

) + ¯ a

(r

U

) + · · · ✭✶✽✮

❚❤❡ ❆◆▼ ♠❡t❤♦❞ ❞♦❡s ♥♦t ❣✐✈❡ ❛❝❝❡ss t♦ ✭✶✽✮ ❜✉t ♦♥❧② t♦ ✐ts ♣♦✇❡rs s❡r✐❡s✳ ❊①♣❛♥❞✐♥❣ t❤❡ r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥ ♦❢ ✭✶✽✮ ♦♥❡ ❝❛♥ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①✲

♣r❡ss✐♦♥ ❢♦r t❤❡ s❡r✐❡s ❝♦❡✣❝✐❡♥t U ¯

❛t ♦r❞❡r p ♦❢ t❤❡ r❡s❝❛❧❡❞ ❆◆▼ ❡①♣❛♥s✐♦♥

❡✈❛❧✉❛t❡❞ ♥❡❛r ❛ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ✿ U ¯

= r

U

+ µ

µ

r

U

+ µ

µ

d ( r

d )

U

✭✶✾✮

❚❤❡ ✜rst t❡r♠ r

U

❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣❡r❢❡❝t ❝❛s❡ s✐t✉❛t✐♦♥ µ

= 0 ✳ ❇❡❝❛✉s❡

♦❢ t❤❡ r❡s❝❛❧❧✐♥❣✱ ✐ts ♥♦r♠ ✐s ♦❢ ♦r❞❡r ♦❢ ✉♥✐t②✳ ❚❤❡ s❡❝♦♥❞ t❡r♠

r

U

✐s

❛ t❤❡ ✜rst ♦r❞❡r ❝♦rr❡❝t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❞✉❡ t♦ t❤❡ ✐♠♣❡r❢❡❝t✐♦♥ ❛♥❞

✐ts ♥♦r♠ ✐s ❛❧s♦ ♦❢ ♦r❞❡r ♦❢ ✉♥✐t②✳ ❚❤❡ t❤✐r❞ t❡r♠

(

)

U

✐s ❛ss♦❝✐❛t❡❞

✇✐t❤ t❤❡ s✐♥❣✉❧❛r✐t② ❜r♦✉❣❤t ❜② t❤❡ ♣❡rt✉r❜❡❞ ❜✐❢✉r❝❛t✐♦♥✳ ■ts ❛♠♣❧✐t✉❞❡ ✐s

❛ ♣r♦❞✉❝t ♦❢ t❤❡ s♠❛❧❧ ✐♠♣❡r❢❡❝t✐♦♥ µ

✇✐t❤ t❤❡ p ♣♦✇❡r ♦❢

✱ ✇❤✐❝❤ ❝❛♥ ❜❡

✈❡r② ❧❛r❣❡ ✇❤❡♥ d << r ✳

■t ❢♦❧❧♦✇s t❤❡ ✐♠♣♦rt❛♥t r❡s✉❧ts ✿ ✇❤❡♥ t❤❡ ❞✐st❛♥❝❡ d ❜❡t✇❡❡♥ U

❛♥❞ U

✐s s♠❛❧❧❡r t❤❛♥ r ✱ ✐❡✱ t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ✐s ✐♥s✐❞❡ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡

♦❢ t❤❡ ✑♣❡r❢❡❝t✑ s❡r✐❡s✱ t❤❡ ❧❛st t❡r♠ ♦❢ ✭✶✾✮ ❞♦♠✐♥❛t❡s t❤❡ s❡r✐❡s ❛t ❤✐❣❤ ♦r❞❡r

✇❤❛t❡✈❡r t❤❡ ✐♠♣❡r❢❡❝t✐♦♥ ♠❛❣♥✐t✉❞❡ µ

✳ ▼♦r❡♦✈❡r✱ ❛t ❤✐❣❤ ♦r❞❡r t❤❡ s❡r✐❡s

❜❡❤❛✈❡s ❧✐❦❡ ❛ ❣❡♦♠❡tr✐❝❛❧ ♣r♦❣r❡ss✐♦♥ ✇✐t❤ ❛ ❝♦♠♠♦♥ r❛t✐♦

✭♦r

❢♦r t❤❡

♥♦♥ s❝❛❧❡❞ s❡r✐❡s✮✱ ❛♥❞ t❤❡ s❡r✐❡s t❡r♠ U ¯

✭♦r U

❢♦r t❤❡ ♥♦♥s❝❛❧❡❞ s❡r✐❡s✮

❜❡❝♦♠❡s ❝♦❧✐♥❡❛r t♦ t❤❡ ✈❡❝t♦r U

✳

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡♥ d < r ❛♥❞ µ

6= 0✱ t❤❡ ♣♦❧❡ ♦❢ t❤❡ r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥

✐♥ ✭✶✽✮ ✭♦r ✭✶✼✮✮ ❜❡❝♦♠❡s t❤❡ ♥❡❛r❡st s✐♥❣✉❧❛r✐t② ♦❢ U (¯ a) ❛♥❞ s❡t t❤❡ r❛❞✐✉s ♦❢

❝♦♥✈❡r❣❡♥❝❡ t♦ d ✱ ✐♥st❡❛❞ ♦❢ r ❢♦r t❤❡ ♣❡r❢❡❝t ❝❛s❡ µ

= 0 ✳ ❚❤✐s ✐s ♣❡r❢❡❝t❡❧②

✐♥ ❧✐♥❡ ✇✐t❤ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♠❛❞❡ ♦♥ ❆◆▼ s❡r✐❡s ❛t ❡①♣❛♥s✐♦♥ ♣♦✐♥t ❝❧♦s❡

t♦ ❛ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✱ ❛s ✐t ✇✐❧❧ ❜❡ s❤♦✇♥ ♦♥ ❡①❛♠♣❧❡s ❜❡❧♦✇✳

✷✳✹✳✸ ❘❡t✉r♥ t♦ ❤✐❣❤ ♦r❞❡r t❡r♠s ✐♥ µ

▲❡t ✉s ♥♦✇ ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ ❤✐❣❤ ♦r❞❡r µ t❡r♠ ✐♥ ✭✶✻✮ t❤❛t ❤❛✈❡ ❜❡❡♥

s✉♣♣r❡ss❡❞ ❛❜♦✈❡✱ ❝❧❛✐♠✐♥❣ t❤❛t µ ✐s ✈❡r② s♠❛❧❧ ✐♥ ♦✉r ❛♣♣❧✐❝❛t✐♦♥s✳ ❖♥❡

q✉❡st✐♦♥ st✐❧❧ r❡♠❛✐♥s ✿ ✇❤❛t ✐s t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ ♠♦r❡ s✐♥❣✉❧❛r t❡r♠s s✉❝❤ ❛s

µ²0

µ²₁₁ 1

a³₁

❛s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❧❡ss s✐♥❣✉❧❛r t❡r♠

r❡t❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s

❛♥❛❧②s✐s ❄✳ ❚♦ ❛♥s✇❡r ✐t✱ ❧❡t ✉s ❛♥❛❧②s❡ t❤❡ ❚❛②❧♦r s❡r✐❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t♦②

♠♦❞❡❧ ✿

f(x) = 1 1 − x

| {z }

f1(x)

+ 10

1 (1 − x)

| {z }

f2(x)

. ✭✷✵✮

✽

❆❝❝♦r❞✐♥❣ t♦ ❱❛♥ ❉②❦❡ ❬✷✺❪ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✱ s✐❣♥ ♣❛tt❡r♥ ✐♥ t❤❡

❚❛②❧♦r s❡r✐❡s ❝♦❡✣❝✐❡♥ts ❛♥❞ ✈❛❧✉❡s ♦❢ ❝♦❡✣❝✐❡♥ts ❛t ❤✐❣❤ ♦r❞❡r ❛r❡ ❜♦t❤

❣♦✈❡r♥❡❞ ❜② t❤❡ ♥❡❛r❡st s✐♥❣✉❧❛r✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❲❤❡♥ t✇♦ s✐♥❣✉❧❛r✐t✐❡s

❛r❡ ❛t t❤❡ s❛♠❡ ❞✐st❛♥❝❡✱ ❛s ✐♥ t❤✐s ❡①❛♠♣❧❡✱ t❤❡ ♠♦st s✐♥❣✉❧❛r ♦♥❡s ✇✐❧❧

✐♠♣♦s❡ ✐ts❡❧❢ ✭❤❡r❡ f

✮✱ r❛t❤❡r t❤❛♥ t❤❡ ♦♥❡ ✇✐t❤ t❤❡ ❧❛r❣❡st ❛♠♣❧✐t✉❞❡ ✭❤❡r❡

f

✮✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r♦♠✐♥❡♥❝❡ ♦❢ f

♦♥❧② ♦❝❝✉rs ❛t ✈❡r② ❤✐❣❤ ♦r❞❡r ❛s ✐t ✐s r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✶✱ ❜❡❝❛✉s❡ ♦❢ ✐ts s♠❛❧❧ ❛♠♣❧✐t✉❞❡ 10

✳

♦r❞❡r ✶ ✷ ✸ ✳ ✳ ✳ ✹✸ ✹✹ ✳ ✳ ✳ ✾✾ ✶✵✵

f

✶ ✶ ✶ ✳ ✳ ✳ ✶ ✶ ✳ ✳ ✳ ✶ ✶

f

✵✳✵✵✸ ✵✳✵✵✻ ✵✳✵✶ ✳ ✳ ✳ ✵✳✾✾ ✶✳✵✸✺ ✳ ✳ ✳ ✺✳✵✺✵ ✺✳✶✺✶

❚❛❜❧❡ ✶ ✕ ❚❛②❧♦r ❝♦❡✣❝✐❡♥ts ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f

=

❛♥❞ f

= 10

✳

■♥ ✭✶✻✮✱ t❤❡ ♠♦st s✐♥❣✉❧❛r t❡r♠s ❤❛✈❡ ❛ ♠✉❝❤ s♠❛❧❧❡r ❛♠♣❧✐t✉❞❡ t❤❛♥ t❤❡

❧❡ss s✐♥❣✉❧❛r ♦♥❡s ✇❤❡♥ µ

✐s ✈❡r② s♠❛❧❧✳ Pr❛❝t✐❝❛❧❧②✱ ✇✐t❤ µ

= 10

❛♥❞ ❛ tr✉♥❝❛t✉r❡ ❛r♦✉♥❞ ♦r❞❡r 20 ✱ t❤❡s❡ ♠♦r❡ s✐♥❣✉❧❛r t❡r♠s ❞♦ ♥♦t s❤♦✇ ✉♣ ✐♥

♦✉r ❛♥❛❧②s✐s✳

✸ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s ✐♥ t❤❡

❆s②♠♣t♦t✐❝ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞

■t ❤❛s ❜❡❡♥ ♣r❡✈✐♦✉s❧② ♣♦✐♥t❡❞ ♦✉t t❤❛t ♣♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s ✐s t✇♦❢♦❧❞

✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤❡ ❆s②♠♣t♦t✐❝ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞ ✿ ✭✐✮ ✐t ♣r♦✈✐❞❡s ❛♥

❛❝❝✉r❛t❡ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t ✇❛② t♦ ❞❡t❡❝t ❛♥❞ ❝♦♠♣✉t❡ ❜✐❢✉r❝❛t✐♦♥

♣♦✐♥ts ✭s♦❧✉t✐♦♥ ❛t ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts✮ ❀ ✭✐✐✮ ✐t ❡♥❛❜❧❡s ✉s t♦ ❝♦♥str✉❝t ❛♥

❡♥❤❛♥❝❡❞ ♣♦✇❡r s❡r✐❡s t❤❛t r❡❝♦✈❡rs ❛♥ ♦♣t✐♠❛❧ st❡♣ ❧❡♥❣t❤ ✐♥ t❤❡ ✈✐❝✐♥✐t② ♦❢

❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❝❧❛ss✐❝❛❧ ❆◆▼ ✐s ♦t❤❡r✇✐s❡ ❞r❛st✐❝❛❧❧② ♣❡♥❛❧✐s❡❞✳

❙♦ t❤❡ ✇❛② ✇❡ ❤❛✈❡ ✐♠♣❧❡♠❡♥t❡❞ t❤❡s❡ ❦❡② ♣♦✐♥ts ❛r❡ ♥♦✇ ♣r❡s❡♥t❡❞✳

✸✳✶ ❉❡t❡❝t✐♦♥ ♦❢ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✐♥ t❤❡ ❆◆▼

❆♥② ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t✇♦ q✉❛♥t✐t✐❡s ✿ ✐ts

❝♦♠♠♦♥ r❛t✐♦ ❛♥❞ ✐ts s❝❛❧❡ ❢❛❝t♦r✱ ✇❤✐❝❤ ✐s ❤❡r❡ ❛ ✜❡❧❞ ✈❡❝t♦r✳ ❙♦✱ t❤❡ ♣r♦✲

♣♦s❡❞ ❆◆▼ ❛❧❣♦r✐t❤♠ ✐♥❝❧✉❞❡s ❛t ❡✈❡r② ❝♦♥t✐♥✉❛t✐♦♥ st❡♣ ❛♥ ❛♥❛❧②s✐s ♦❢ t❤❡

❧❛st ✈❡❝t♦r s❡q✉❡♥❝❡ t♦ ❝❤❡❝❦ ✐❢ t❤❡② ♠❛❦❡ ✉♣ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ♦r ♥♦t✳

▲❡t U

, U

, U

, U

❜❡ t❤❡ ❧❛st ❢♦✉r ✜❡❧❞ ✈❡❝t♦r s❡q✉❡♥❝❡ ✐♥ t❤❡ ❆◆▼

♣♦✇❡r s❡r✐❡s ✭✷✮ ❝♦♠♣✉t❡❞ ✉♣ t♦ ❛♥ ♦r❞❡r ♦❢ tr♦♥❝❛t✉r❡ n✳ ❚❤❡♥✱ t❤❡ ❞❡t❡❝✲

t✐♦♥ ♦❢ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❝❤❡❝❦s ❢♦r ❝♦❧❧✐♥❡❛r✐t② ❛♥❞ ♣r♦♣♦rt✐♦♥❛❧✐t② ♦❢

✾

t❤✐s ✈❡❝t♦r s❡q✉❡♥❝❡✳ ❚❤✐s t❡st r❡❛❞s ❛s ❢♦❧❧♦✇s ✿

❢♦r n − 3 ≤ p ≤ n − 1 : α

= (U

· U

)/(U

· U

) ❛♥❞ U

= U

− α

U

✐❢

n−2

X

p=n−3

(α

− α

)/α

< ε

❛♥❞

n−1

X

p=n−3

kU

k/kU

k < ε

t❤❡♥ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ✐s ❞❡t❡❝t❡❞✱ s♦ ♦♥❡ ❝♦♠♣✉t❡s ✿

✕ ✐ts ❝♦♠♠♦♥ r❛t✐♦ ✿

=

n−1

✕ ✐ts s❝❛❧❡ ❢❛❝t♦r ✿ U

α

✭✷✶✮

❆❝❝♦r❞✐♥❣ t♦ s❡❝t✐♦♥ ✷✱ α ✐s t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ❝❧♦s❡st ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ❛♥❞

t❤❡ ♥♦r♠❛❧✐s❡❞ s❝❛❧❡ ❢❛❝t♦r ✭

✮ ✐s t❤❡ ✈❡❝t♦r U

♣r❡✈✐♦✉s❧② ❞❡✜♥❡❞✳

■♥ ♦✉r ❝❛❧❝✉❧❛t✐♦♥s s❡tt✐♥❣ t❤❡ t✇♦ ♥✉♠❡r✐❝❛❧ t❤r❡s❤♦❧❞ ✈❛❧✉❡s t♦ ε

≈ 10

❛♥❞ ε

≈ 10

❣❛✈❡ ✉s r❡❧✐❛❜❧❡ ❛♥❞ r♦❜✉st ❜✐❢✉r❝❛t✐♦♥ ❞❡t❡❝t✐♦♥ t❡st✳

✸✳✷ ❊♥❤❛♥❝❡❞ ♣♦✇❡r s❡r✐❡s t♦ r❡❝♦✈❡r ♦♣t✐♠❛❧ st❡♣ ❧❡♥❣t❤

▲❡t ✉s ♥♦✇ s✉♣♣♦s❡ ❛ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥ ❤❛s ❜❡❡♥ ❥✉st ❞❡t❡❝t❡❞ ❛t t❤❡ ❤✐❣❤❡st ♦r❞❡rs ♦❢ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♠❡t❤♦❞ ✇✐t❤ t❤❡ ♣r♦♣♦s❡❞ t❡st ✭✷✶✮✳

❚❤❡♥✱ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✇❡r s❡r✐❡s ✭✷✮ ❝❛♥ ❜❡ s♣❧✐tt❡❞ ✐♥ t✇♦ ♣❛rts ✿ ❛ ❣❡♦♠❡tr✐❝

♣♦✇❡r s❡r✐❡s ❛♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ♣❛rt ❢r♦♠ t❤❡ ♦r✐❣♥❛❧ ♦♥❡✱ ❛s ❢♦❧❧♦✇s ✿ U (a) = U

+ a U ˆ

+ a

U ˆ

+ · · · + a

U ˆ

+

h a α + ( a

α )

+ · · · + ( a

α )

+ ( a α )

i

U

α

✭✷✷✮

✇❤❡r❡ U ˆ

= U

− α

U

❢♦r 0 < p ≤ n − 1 ✳ ■♥tr♦❞✉❝✐♥❣ t❤❡ ❢♦r♠❛❧

❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ♣♦✇❡r s❡r✐❡s t♦❣❡t❤❡r ✇✐t❤ ❛ ❝♦♠♣❧❡t✐♦♥ ❛t

✐♥✜♥✐t②✱ ❛♥❞ ✉s✐♥❣

= U

✱ ♦♥❡ ✜♥❛❧❧② ❣❡ts ✿ U (a) = U

+ a U ˆ

+ a

U ˆ

+ · · · + a

U ˆ

+

a α

1 −

||U

||α

U

✭✷✸✮

❚❤❡ ♣♦✇❡r s❡r✐❡s ❛♥❛❧②s✐s ✐♥tr♦❞✉❝❡❞ ✐♥ s❡❝t✐♦♥ ✷ ❡♥❛❜❧❡s ✉s t♦ ✉♥❞❡rst❛♥❞ t❤❡

✉♥❞❡r❧②✐♥❣ ✐ss✉❡s r❡❧❛t❡❞ t♦ t❤❡s❡ t✇♦ ♣❛rts✳ ❚❤❡ ✜rst ♣❛rt ♦❢ ✭✷✸✮ ✐s t❤❡ ♣♦✇❡r s❡r✐❡s t❤❛t ❛❝t✉❛❧❧② ❞❡s❝r✐❜❡s t❤❡ ♣❡r❢❡❝t ❜r❛♥❝❤✱ ✇❤❡r❡❛s t❤❡ s❡❝♦♥❞ ♣❛rt✱

✐✳❡✳✱ t❤❡ ❣❡♦♠❡tr✐❝ ♣♦✇❡r s❡r✐❡s ❛❝❝♦✉♥ts ❢♦r t❤❡ ✐♥t❡r❛❝t✐♦♥ ❛♥❞ ❛♠♣❧✐✜❝❛t✐♦♥

♦❢ ❛♥② ✐♠♣❡r❢❡❝t✐♦♥ ✇✐t❤ t❤❡ ♦t❤❡r ❜r❛♥❝❤ t❤❛t ❝r♦ss❡s t❤❡ ♣r❡s❡♥t ♦♥❡ ❛t t❤❡ ♥❡①t ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳ ❚❤❡♥✱ t❤❡ ❦❡② ♣♦✐♥t t♦ r❡❝♦✈❡r ❛♥ ♦♣t✐♠❛❧ st❡♣

❧❡♥❣t❤ ♦✈❡r t❤✐s ❝♦♥t✐♥✉❛t✐♦♥ st❡♣ ✐s t♦ ♦♥❧② ❝♦♥s✐❞❡r t❤❡ ❢♦r♠❡r ♣♦✇❡r s❡r✐❡s t♦ ❛❝❝✉r❛t❡❧② r❡♣r❡s❡♥t t❤❡ ❝✉rr❡♥t ❜r❛♥❝❤✳ ❙♦ t❤❡ ❡♥❤❛♥❝❡❞ ♣♦✇❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ r❡❛❞s ✿

U (a) = U

+ a U ˆ

+ a

U ˆ

+ · · · + a

U ˆ

✭✷✹✮

✇❤✐❝❤ r❡s✉❧ts ❢r♦♠ ❛♥ ❛❞❞✐t✐✈❡ s✐♥❣✉❧❛r✐t② ❡①tr❛❝t✐♦♥ ♦❢ ✭✷✮ ✐♥ t❤❡ s❛♠❡ ✇❛②

❛s ❢♦r♠❡r❧② ♣r♦♣♦s❡❞ ✐♥ ❬✷✺❪✳

✶✵

✸✳✸ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ✈❡❝t♦r ❛t ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts

❖♥❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❡♥❤❛♥❝❡❞ ♣♦✇❡r s❡r✐❡s ✭✷✹✮✱ ♦♥❡ ❝❛♥ ❛❝❝✉r❛t❡❧②

❧♦❝❛t❡ t❤❡ ♥❡❛r❡st s✐♥❣✉❧❛r✐t②✱ ✐✳❡✳✱ s✐♠♣❧❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ❛♥❞ ❝♦♠♣✉t❡

❜♦t❤ t❤❡ s♦❧✉t✐♦♥ ✈❡❝t♦r ❛♥❞ ✐ts t✇♦ t❛♥❣❡♥t ✈❡❝t♦rs ❛t t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t✳

❘❡❝❛❧❧✐♥❣ t❤❛t ✐t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣♦✇❡r s❡r✐❡s✱

✇❤✐❝❤ ✐♥ t❤❡ ♣r❡s❡♥t ❝❛s❡ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❝♦♠♠♦♥ r❛t✐♦ ♦❢ t❤❡ ❣❡♦♠❡tr✐❝

♣r♦❣r❡ss✐♦♥✱ t❤❡ s♦❧✉t✐♦♥ ✈❡❝t♦r ❛t ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ✐s t❤❡r❡❢♦r❡ ❝♦♠♣✉t❡❞

❜② ✐♥s❡rt✐♥❣ a = α ✐♥ ✭✷✹✮ ✿

U

= U (a = α) = U

+ α U ˆ

+ α

U ˆ

+ · · · + α

U ˆ

✭✷✺✮

❋✉rt❤❡r♠♦r❡✱ t❤❡ t❛♥❣❡♥t ✈❡❝t♦r t♦ t❤❡ ❝✉rr❡♥t ❜r❛♥❝❤ ✐s ❝♦♠♣✉t❡❞ ❛t t❤❡

❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ❛s ❢♦❧❧♦✇s ✿ U

= ∂U

∂a (a = α) = ˆ U

+ 2α U ˆ

+ · · · + (n − 1)α

U ˆ

✭✷✻✮

❋✐♥❛❧❧②✱ t❤❡ t❛♥❣❡♥t ✈❡❝t♦r t♦ t❤❡ ♦t❤❡r ❜r❛♥❝❤ U

✐s ❡❛s✐❧② ❛♥❞ ❝❤❡❛♣❧② ❝♦♠✲

♣✉t❡❞ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❦♥♦✇♥ ✈❡❝t♦rs U

❛♥❞ U

✱ ♠❡❛♥✇❤✐❧❡

s❛t✐s❢②✐♥❣ t❤❡ ❛❧❣❡❜r❛✐❝ ❜✐❢✉r❝❛t✐♦♥ ❡q✉❛t✐♦♥ ✭✹✲❜✮✳

U

= β U

+ γ U

ψ

R

U

U

= 0 ✭✷✼✮

❆s ❢❛r ❛s t❤❡ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥t ❛♥❞ t❤❡ t✇♦ t❛♥❣❡♥t ✈❡❝t♦rs ❛ss♦❝✐❛t❡❞ t♦

t❤❡ ❝r♦ss✐♥❣ ❜r❛♥❝❤❡s ❛r❡ ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞✱ ❜r❛♥❝❤ s✇✐t❝❤✐♥❣ ❝❛♥ ❜❡

♣❡r❢♦r♠❡❞ ❛t ✇✐s❤ ✐♥ t❤❡ ❝♦✉rs❡ ♦❢ ❝♦♥t✐♥✉❛t✐♦♥✳

✹ ❆♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡s

❋♦r ✐❧❧✉str❛t✐♥❣ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠✱ ✇❡ ✜rst ❝♦♥✲

s✐❞❡r t❤❡ ❜✉❝❦❧✐♥❣ ♣r♦❜❧❡♠ ♦❢ ❛♥ ✐♥❡①t❡♥s✐❜❧❡ ❡❧❛st✐❝ ❜❡❛♠✳ ■t ✐s ❛ s♠❛❧❧ s✐③❡

❞✐s❝r❡t❡ ♣r♦❜❧❡♠ ❝♦♠✐♥❣ ❢r♦♠ s♦❧✐❞ ♠❡❝❤❛♥✐❝s✱ ✇❤✐❝❤ ♣r❡s❡♥ts ❛♥ ❡❧❛❜♦r❛t❡

❜✐❢✉r❝❛t✐♦♥ ❞✐❛❣r❛♠ ✇✐t❤ ♠❛♥② ✐♥t❡r❝♦♥♥❡❝t❡❞ ❜r❛♥❝❤❡s ♦❢ s♦❧✉t✐♦♥s✳ ❖♥ t❤❡

♦t❤❡r ❤❛♥❞ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❤❛s ❛❧s♦ ❜❡❡♥ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ ❤✐❣❤ ♣❡r✲

❢♦r♠❛♥❝❡ ❢r❛♠❡✇♦r❦ ❬✷✶❪ t♦ ❞❡❛❧ ✇✐t❤ ❧❛r❣❡ s✐③❡ ❞✐s❝r❡t❡ ♣r♦❜❧❡♠s s✉❝❤ t❤♦s❡

t❤❛t ❛r✐s❡ ✐♥ ♠❛♥② ✢✉✐❞ ♠❡❝❤❛♥✐❝s ♦r ❝♦✉♣❧❡❞ ✢✉✐❞ ✢♦✇ ❛♥❞ ❤❡❛t tr❛♥s❢❡r

♣r♦❜❧❡♠s✳ ❲❡ ❤❛✈❡ ♣❡r❢♦r♠❡❞ s❡✈❡r❛❧ ✈❛❧✐❞❛t✐♦♥ t❡sts ❛♥❞ ♣r❡s❡♥t ❜❡❧♦✇ t❤❡

❛❧❣♦r✐t❤♠ ❛❝❝✉r❛❝② ❛♥❞ ❡✣❝✐❡♥❝② ❢♦r ❛♥ ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞ ✢♦✇ ♣r♦❜❧❡♠

✐♥ ❛ s②♠♠❡tr✐❝ s✉❞❞❡♥ ❡①♣❛♥s✐♦♥✳

✹✳✶ ❇✉❝❦❧✐♥❣ ❛♥❛❧②s✐s ♦❢ ❛♥ ❡❧❛st✐❝❛ ❜❡❛♠

❲❡ ❝♦♥s✐❞❡r t❤❡ ❛①✐❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❛♥ ✐♥❡①t❡♥s✐❜❧❡ ❡❧❛st✐❝ ❜❡❛♠✱ ♦❢t❡♥

r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❊✉❧❡r ❜❡❛♠ ♦r ❊❧❛st✐❝❛✱ ❢♦r ✇❤✐❝❤ ✇❡ ❛❞♦♣t ❝❧❛♠♣❡❞✲

❝❧❛♠♣❡❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ▲❡t s ∈ [0, 1] ❜❡ t❤❡ ♥♦♥❞✐♠❡♥s✐♦♥❛❧ ❝✉r✈✐✲

❧✐♥❡❛r ❝♦♦r❞✐♥❛t❡✱ (x(s), y(s)) t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❜❡❛♠ ❝❡♥t❡r❧✐♥❡✱ θ(s) t❤❡

✶✶

r♦t❛t✐♦♥ ♦❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥ ❛♥❞ m(s) t❤❡ ❜❡♥❞✐♥❣ ♠♦♠❡♥t ✭❛❧❧ ♥♦♥❞✐♠❡♥✲

t✐♦♥❛❧✮✳ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣♦s✐t✐♦♥s ♦❢ t❤❡ ❜❡❛♠ ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣

❖❉❊ s②st❡♠ ✿

x

= cos(θ) y

= sin(θ) θ

= m

m

= −4π

(P sin(θ) + T cos(θ))

✭✷✽✮

✇✐t❤ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s x(0) = 0 ✱ y(0) = 0 ✱ θ(0) = 0 ✱ m(0) = 2πC

✭❛t ❧❡❢t✮ ❛♥❞ y(1) = 0 ✱ θ(1) = 0 ✭❛t r✐❣❤t✮✳ ❚❤❡ ❛①✐❛❧ ❛♣♣❧✐❡❞ ❢♦r❝❡ P ✐s

❤❡r❡ t❤❡ ❝♦♥tr♦❧ ♣❛r❛♠❡t❡r✱ T ❛♥❞ C ❛r❡ t❤❡ ❧❛t❡r❛❧ ❛♥❞ t♦rq✉❡ r❡❛❝t✐♦♥s ❛t t❤❡ ❜♦✉♥❞❛r②✳ ❚❤✐s ❖❉❊ s②st❡♠ ✐s tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❛ ❞✐s❝r❡t❡ ♣r♦❜❧❡♠ ✭✷✼✻

❞✳♦✳❢✮ ❜② ❛♣♣❧②✐♥❣ ❛♥ ♦rt❤♦❣♦♥❛❧ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ✇✐t❤ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧

✐♥t❡r♣♦❧❛t✐♦♥ ❬✶✹❪✳ ❚❤❡ ❜✐❢✉r❝❛t✐♦♥ ❞✐❛❣r❛♠ ✐s ♠❛❞❡ ♦❢ ❛ tr✐✈✐❛❧ ❝♦♠♣r❡ss✐✈❡

s♦❧✉t✐♦♥ ✭ x = s ✱ y = 0 ✱ θ = 0 ✱ m = 0 ✮ ❢r♦♠ ✇❤✐❝❤ ❜✐❢✉r❝❛t❡❞ ❜r❛♥❝❤❡s

❡♠❛♥❛t❡ ❛t ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts P = {1 ; 2.04 ; 4 ; ...}✳ ❆❞❞✐t✐♦♥❛❧ ❜r❛♥❝❤❡s ♦❢

s♦❧✉t✐♦♥s ✐♥t❡r❝♦♥♥❡❝t t❤❡s❡ ❜✐❢✉r❝❛t❡❞ ❜r❛♥❝❤❡s ❛t ✧s❡❝♦♥❞❛r②✧ ❜✐❢✉r❝❛t✐♦♥

♣♦✐♥ts✳

✹✳✶✳✶ ❉❡t❛✐❧❧❡❞ ❛♥❛❧②s✐s ♦❢ ❝♦♥t✐♥✉❛t✐♦♥ ❛♥❞ ❜✐❢✉r❝❛t✐♦♥ ❞❡t❡❝t✐♦♥

❙t❛rt✐♥❣ ❢r♦♠ ❛ ❦♥♦✇♥ ♣♦✐♥t U

♦♥ t❤❡ ✜rst ❜✐❢✉r❝❛t❡❞ ❜r❛♥❝❤✱ ✇❡ ♣❡r✲

❢♦r♠ ✹ st❡♣s ♦❢ ❆◆▼ ❝♦♥t✐♥✉❛t✐♦♥ ✇✐t❤ ❛ tr♦♥❝❛t✉r❡ ♦r❞❡r ♦❢ 20 ❛♥❞ ❛

✉s❡r✲r❡q✉✐r❡❞ ♣r❡❝✐s✐♦♥ ε

= 10

✳ ❚✇♦ ❣❡♦♠❡tr✐❝❛❧ s❡r✐❡s ❛r❡ ❞❡t❡❝t❡❞ ❜② t❤❡ t❡st ✭✷✶✮ ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ❜✐❢✉r❝❛t✐♦♥ ♣♦✐♥ts ❛r❡ ♠❛r❦❡❞ ✇✐t❤ ❝✐r❝❧❡s ♦♥

t❤❡ ❜✐❢✉r❝❛t✐♦♥ ❞✐❛❣r❛♠ r❡♣♦rt❡❞ ✐♥ ✜❣✉r❡ ✷✳ ❚❤❡ ❜✐❢✉r❝❛t✐♦♥ ✭■✮ ❞❡t❡❝t❡❞ ❛t t❤❡ ✜rst ❝♦♥t✐♥✉❛t✐♦♥ st❡♣ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ✜rst ❜✉❦❧✐♥❣

❜r❛♥❝❤ ✇✐t❤ t❤❡ tr✐✈✐❛❧ ❝♦♠♣r❡ss✐✈❡ s♦❧✉t✐♦♥✳ ❚❤❡ ✐♠♣❡r❢❡❝t✐♦♥ ❛♠♣❧✐t✉❞❡ µ

✐s ❤❡r❡ ✈❡r② s♠❛❧❧ ✭ µ

= 2.1423 10

✮ ❜❡❝❛✉s❡ t❤❡ st❛rt ♣♦✐♥t U

✐s ✈❡r②

❛❝❝✉r❛t❡ ✭ R(U

) = 1.4615 10

❛♥❞ µ

= −0.0698 ✮✳ ❊✈❡♥ t❤♦✉❣❤✱ s✐♥❝❡

t❤❡ r❛t✐♦

✐s ❧❛r❣❡ ✭❛❜♦✉t 17 ✐♥ t❤❛t ❝❛s❡✮ t❤❡ ❣❡♦♠❡tr✐❝❛❧ s❡r✐❡s ❡♥❞ ✉♣ ✐♠✲

♣♦s✐♥❣ ✐ts❡❧❢ ❛t ♦r❞❡rs ❤✐❣❤❡r t❤❛♥ 13 ✳ ❚❤❡ ❡♥❞ ♣♦✐♥t ♦❢ t❤❡ ✜rst st❡♣ U

✐s

❝♦♠♣✉t❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ ✭✷✹✮ t♦ r❡❝♦✈❡r ❛♥ ♦♣t✐♠❛❧ st❡♣ ❧❡♥❣t❤✳ ❚❤❡ ♦r✐❣✐♥❛❧

s❡r✐❡s ✭✷✮ ✇♦✉❧❞ ❤❛✈❡ ❞❡❧✐✈❡r ❛ ♠✉❝❤ s❤♦rt❡r st❡♣ ❧❡♥❣t❤ ❛s ✐t ✐s r❡♣♦rt❡❞ ✐♥

✜❣✉r❡ ✷✳ ❲❡ ♥♦t❡ ✐♥ ♣❛ss✐♥❣ t❤❛t t❤✐s ❡①❡♠♣❧❡ ✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ✐♥ ❬✷✵❪

✇❤❡r❡ t❤❡ ❛✉t❤♦r s❛②s t❤❛t ❛ ❥✉❞✐❝✐♦✉s r❡✲♦r❞❡r✐♥❣ ♦❢ t❤❡ s❡r✐❡s t❡r♠s ❧❡❛❞s t♦ s✉♣r✐s✐♥❣ ✐♠♣r♦✈❡♠❡♥ts✳

❋♦r t❤❡ s❡❝♦♥❞ ❝♦♥t✐♥✉❛t✐♦♥ st❡♣✱ ✐❡✱ ✇✐t❤ t❤❡ s❡r✐❡s ✐ss✉❡❞ ❢r♦♠ U

✱ ♥♦

❣❡♦♠❡tr✐❝❛❧ s❡r✐❡ ✇❛s ❞❡t❡❝t❡❞ ❜② t❤❡ ❛❧❣♦r✐t❤♠✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡ ❜✐❢✉r❝❛t✐♦♥

♣♦✐♥t ✭■■✮ ✐s ❧♦❝❛t❡❞ ✐♥s✐❞❡ t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳ ❚❤❡ r❡❛s♦♥ ✐♥ t❤❡

♣r❡s❡♥t ❝❛s❡ ✐s t❤❛t t❤❡ r❛t✐♦

✐s ♥♦t ❧❛r❣❡ ❡♥♦✉❣❤ s♦ t❤❛t t❤❡ t❡r♠ (

)

❝♦♠♣❡♥s❛t❡s ❛ s♦ t✐♥② µ

✈❛❧✉❡✳ ■♥ ♦t❤❡r ✇♦r❞s t❤❡ s✐♥❣✉❧❛r✐t② ❜r♦✉❣❤t ❜② t❤❡ ❜✐❢✉r❝❛t✐♦♥ ✭■■✮ ✐s ❛❝t✉❛❧❧② t❤❡ ♥❡❛r❡st ♦♥❡ ❜✉t ❛s ✐ts ❛♠♣❧✐t✉❞❡ ✐s s♠❛❧❧

❛♥❞ t❤❡ ♦t❤❡r s✐♥❣✉❧❛r✐t✐❡s ❛r❡ ♥♦t s♦ ❢❛r t❤❡ ❣❡♦♠❡tr✐❝❛❧ s❡r✐❡s ❞♦❡s ♥♦t s✉❝❝❡❡❞ t♦ ✐♠♣♦s❡ ✐ts❡❧❢✳ ❍❡♥❝❡✱ ❛ st❛♥❞❛r❞ ❆◆▼ st❡♣ ✐s ♣❡r❢♦r♠❡❞ ❢r♦♠

✶✷