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◆✐❦♦❧❛♦s ❋♦✉♥t♦✉❧❛❦✐s✶ ❑♦♥st❛♥t✐♥♦s P❛♥❛❣✐♦t♦✉✶

✸✲❝♦♥♥❡❝t❡❞ ❝♦r❡s ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

✶▼❛①✲P❧❛♥❝❦✲■♥st✐t✉t ❢ür ■♥❢♦r♠❛t✐❦✱ ●❡r♠❛♥②

▲❛❜♦r❛t♦✐r❡ ❞✬■♥❢♦r♠❛t✐q✉❡ ❞❡ P❛r✐s✲◆♦r❞✱ ✶✻ ◆♦✈❡♠❜❡r ✷✵✶✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s

❊r❞➤s ❛♥❞ ❘é♥②✐ ♣r♦✈❡❞ t❤❛t ❛r♦✉♥❞ ♣❝r = ✶

♥ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡

str✉❝t✉r❡ ♦❢ ●(♥, ♣) ♦❝❝✉rs✿

✐❢ ♣ < ✶−ε

♥ ✱ t❤❡♥ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ●♥,♣ ❝♦♥t❛✐♥ ❖(❧♦❣ ♥) ✈❡rt✐❝❡s❀

✐❢ ♣ > ✶+ε

♥ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t ✇✐t❤ Θ(♥)

✈❡rt✐❝❡s✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ❖(❧♦❣ ♥) ✈❡rt✐❝❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s

❊r❞➤s ❛♥❞ ❘é♥②✐ ♣r♦✈❡❞ t❤❛t ❛r♦✉♥❞ ♣❝r = ✶

♥ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡

str✉❝t✉r❡ ♦❢ ●(♥, ♣) ♦❝❝✉rs✿

✐❢ ♣ < ✶−ε

♥ ✱ t❤❡♥ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ●♥,♣ ❝♦♥t❛✐♥ ❖(❧♦❣ ♥) ✈❡rt✐❝❡s❀

✐❢ ♣ > ✶+ε

♥ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t ✇✐t❤ Θ(♥)

✈❡rt✐❝❡s✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ❖(❧♦❣ ♥) ✈❡rt✐❝❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❣r❛♣❤s

❊r❞➤s ❛♥❞ ❘é♥②✐ ♣r♦✈❡❞ t❤❛t ❛r♦✉♥❞ ♣❝r = ✶

♥ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡

str✉❝t✉r❡ ♦❢ ●(♥, ♣) ♦❝❝✉rs✿

✐❢ ♣ < ✶−ε

♥ ✱ t❤❡♥ ❛❧❧ ❝♦♠♣♦♥❡♥ts ♦❢ ●♥,♣ ❝♦♥t❛✐♥ ❖(❧♦❣ ♥) ✈❡rt✐❝❡s❀

✐❢ ♣ > ✶+ε

♥ ✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t ✇✐t❤ Θ(♥)

✈❡rt✐❝❡s✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ❖(❧♦❣ ♥) ✈❡rt✐❝❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s

▲❡t P♥ ❜❡ t❤❡ s❡t ♦❢ ❧❛❜❡❧❡❞ ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ❛♥❞ ❧❡t P♥ ❞❡♥♦t❡ ❛ ❣r❛♣❤ t❛❦❡♥ ❛t r❛♥❞♦♠ ❢r♦♠ P♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/|P♥|✳ ❍♦✇ ♠❛♥② ❡❞❣❡s ❞♦❡s P♥ t②♣✐❝❛❧❧② ❤❛✈❡❄

• ✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ s❤♦✇❡❞ t❤❛t

❡(P♥) − κ♥ √ ✷πλ♥

❞

→ ◆(✵, ✶), ✇❤❡r❡ κ ≈ ✷.✷✶✸✷✻ ❛♥❞ λ ≈ ✵.✹✸✵✸✹ ❛r❡ ❝♦♥st❛♥ts✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s

▲❡t P♥ ❜❡ t❤❡ s❡t ♦❢ ❧❛❜❡❧❡❞ ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ❛♥❞ ❧❡t P♥ ❞❡♥♦t❡ ❛ ❣r❛♣❤ t❛❦❡♥ ❛t r❛♥❞♦♠ ❢r♦♠ P♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/|P♥|✳ ❍♦✇ ♠❛♥② ❡❞❣❡s ❞♦❡s P♥ t②♣✐❝❛❧❧② ❤❛✈❡❄

• ✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ s❤♦✇❡❞ t❤❛t

❡(P♥) − κ♥ √ ✷πλ♥

❞

→ ◆(✵, ✶), ✇❤❡r❡ κ ≈ ✷.✷✶✸✷✻ ❛♥❞ λ ≈ ✵.✹✸✵✸✹ ❛r❡ ❝♦♥st❛♥ts✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s

▲❡t P♥ ❜❡ t❤❡ s❡t ♦❢ ❧❛❜❡❧❡❞ ❛❧❧ ♣❧❛♥❛r ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ❛♥❞ ❧❡t P♥ ❞❡♥♦t❡ ❛ ❣r❛♣❤ t❛❦❡♥ ❛t r❛♥❞♦♠ ❢r♦♠ P♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶/|P♥|✳ ❍♦✇ ♠❛♥② ❡❞❣❡s ❞♦❡s P♥ t②♣✐❝❛❧❧② ❤❛✈❡❄

• ✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ s❤♦✇❡❞ t❤❛t

❡(P♥) − κ♥ √ ✷πλ♥

❞

→ ◆(✵, ✶), ✇❤❡r❡ κ ≈ ✷.✷✶✸✷✻ ❛♥❞ λ ≈ ✵.✹✸✵✸✹ ❛r❡ ❝♦♥st❛♥ts✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s

❲❤❛t ✐s t❤❡ t②♣✐❝❛❧ ❝♦♠♣♦♥❡♥t str✉❝t✉r❡ ♦❢ P♥❄

• ✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ ♣r♦✈❡❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞

❝♦♠♣♦♥❡♥ts ♦❢ P♥ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ✶ + ❳, ✇❤❡r❡ ❳ L = P♦(ν)✱ ❛♥❞ ν ≈ ✵.✵✸✼✹✸✾✳ ❚②♣✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❣✐❛♥t ❝♦♠♣♦♥❡♥t ❝♦♥t❛✐♥✐♥❣ ♥ − ❖❈(✶) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s t❤❡ r❡♠❛✐♥✐♥❣ ❝♦♠♣♦♥❡♥ts ❛r❡ ✏s♠❛❧❧✧✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s

❲❤❛t ✐s t❤❡ t②♣✐❝❛❧ ❝♦♠♣♦♥❡♥t str✉❝t✉r❡ ♦❢ P♥❄

• ✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ ♣r♦✈❡❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞

❝♦♠♣♦♥❡♥ts ♦❢ P♥ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ✶ + ❳, ✇❤❡r❡ ❳ L = P♦(ν)✱ ❛♥❞ ν ≈ ✵.✵✸✼✹✸✾✳ ❚②♣✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❣✐❛♥t ❝♦♠♣♦♥❡♥t ❝♦♥t❛✐♥✐♥❣ ♥ − ❖❈(✶) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s t❤❡ r❡♠❛✐♥✐♥❣ ❝♦♠♣♦♥❡♥ts ❛r❡ ✏s♠❛❧❧✧✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤s

❲❤❛t ✐s t❤❡ t②♣✐❝❛❧ ❝♦♠♣♦♥❡♥t str✉❝t✉r❡ ♦❢ P♥❄

• ✐♠é♥❡③ ❛♥❞ ◆♦② ✭✷✵✵✾✮ ♣r♦✈❡❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞

❝♦♠♣♦♥❡♥ts ♦❢ P♥ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ✶ + ❳, ✇❤❡r❡ ❳ L = P♦(ν)✱ ❛♥❞ ν ≈ ✵.✵✸✼✹✸✾✳ ❚②♣✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ♣❧❛♥❛r ❣r❛♣❤ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❣✐❛♥t ❝♦♠♣♦♥❡♥t ❝♦♥t❛✐♥✐♥❣ ♥ − ❖❈(✶) ✈❡rt✐❝❡s✱ ✇❤❡r❡❛s t❤❡ r❡♠❛✐♥✐♥❣ ❝♦♠♣♦♥❡♥ts ❛r❡ ✏s♠❛❧❧✧✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❆ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❝♦♥s✐sts ♦❢ ✐ts ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✿ t❤❡ ♠❛①✐♠❛❧ s✉❜❣r❛♣❤s ♦❢ ❝♦♥♥❡❝t✐✈✐t② ❛t ❧❡❛st ✷✳

A connected graph

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✺

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❆ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❝♦♥s✐sts ♦❢ ✐ts ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✿ t❤❡ ♠❛①✐♠❛❧ s✉❜❣r❛♣❤s ♦❢ ❝♦♥♥❡❝t✐✈✐t② ❛t ❧❡❛st ✷✳

A connected graph

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✺

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❆ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ❝♦♥s✐sts ♦❢ ✐ts ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✿ t❤❡ ♠❛①✐♠❛❧ s✉❜❣r❛♣❤s ♦❢ ❝♦♥♥❡❝t✐✈✐t② ❛t ❧❡❛st ✷✳

A connected graph

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✺

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❆ss✉♠❡ t❤❛t ✇❡ s❛♠♣❧❡ ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✉♥✐❢♦r♠❧② ❢r♦♠ t❤❡ ❢❛♠✐❧② ♦❢ ❣r❛♣❤s ✇❤♦s❡ ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ❜❡❧♦♥❣ t♦ t❤❡ ❢❛♠✐❧② B✳ ▲❡t ❇(①) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ B✿ ❇(①) =

∞

• ♥=✶

|B♥| ♥! ①♥, ❛♥❞ ❧❡t ρB ❜❡ ✐ts r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✻

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❆ss✉♠❡ t❤❛t ✇❡ s❛♠♣❧❡ ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✉♥✐❢♦r♠❧② ❢r♦♠ t❤❡ ❢❛♠✐❧② ♦❢ ❣r❛♣❤s ✇❤♦s❡ ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ❜❡❧♦♥❣ t♦ t❤❡ ❢❛♠✐❧② B✳ ▲❡t ❇(①) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ B✿ ❇(①) =

∞

• ♥=✶

|B♥| ♥! ①♥, ❛♥❞ ❧❡t ρB ❜❡ ✐ts r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✻

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❆ss✉♠❡ t❤❛t ✇❡ s❛♠♣❧❡ ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✉♥✐❢♦r♠❧② ❢r♦♠ t❤❡ ❢❛♠✐❧② ♦❢ ❣r❛♣❤s ✇❤♦s❡ ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ❜❡❧♦♥❣ t♦ t❤❡ ❢❛♠✐❧② B✳ ▲❡t ❇(①) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ B✿ ❇(①) =

∞

• ♥=✶

|B♥| ♥! ①♥, ❛♥❞ ❧❡t ρB ❜❡ ✐ts r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✻

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

■t t✉r♥s ♦✉t t❤❛t ✇❤❛t ❞❡t❡r♠✐♥❡s t❤❡ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ s❛♠♣❧❡❞ ❢r♦♠ t❤✐s ❢❛♠✐❧② ✐s ρB❇′′(ρB)

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

■t t✉r♥s ♦✉t t❤❛t ✇❤❛t ❞❡t❡r♠✐♥❡s t❤❡ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ s❛♠♣❧❡❞ ❢r♦♠ t❤✐s ❢❛♠✐❧② ✐s ρB❇′′(ρB)

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

❚❤❡♦r❡♠ ❬P❛♥❛❣✐♦t♦✉ ❛♥❞ ❙t❡❣❡r ✭✷✵✵✾✮❪ ▲❡t ❈♥ ❜❡ ❛ r❛♥❞♦♠ ❣r❛♣❤ s❛♠♣❧❡❞ ✉♥✐❢♦r♠❧② ❢r♦♠ t❤❡ ❢❛♠✐❧② ♦❢ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ♦♥ ♥ ✈❡rt✐❝❡s ✇✐t❤ ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ✐♥ B✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ − ♦(✶) ✐❢ ρB❇′′(ρB) > ✶✱ t❤❡♥ ❛❧❧ ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ❤❛✈❡ s✐③❡ ❖(❧♦❣ ♥)❀ ✐❢ ρB❇′′(ρB) < ✶✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❜✐❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ ♦r❞❡r Θ(♥)✱ ❜✉t ❡✈❡r② ♦t❤❡r ❝♦♠♣♦♥❡♥t ❤❛s ♦(♥) ✈❡rt✐❝❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✽

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s

◗✉❡st✐♦♥ ■❢ ❈♥ ✐s ❛ r❛♥❞♦♠ ❣r❛♣❤ ♦♥ ♥ ✈❡rt✐❝❡s s❛♠♣❧❡❞ ❢r♦♠ ❛ ❝❡rt❛✐♥ ❝❧❛ss ♦❢ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✱ ✇❤❛t ✐s t❤❡ t②♣✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ✐ts ✸✲❝♦♥♥❡❝t❡❞ ❜✉✐❧❞✐♥❣s ❜❧♦❝❦s❄ ❲❤❛t ✐s ❛ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ♦❢ ❛ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤❄

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✾

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦s ✭❚r❛❦❤t❡♥❜r♦t✕❚✉tt❡✮

❉❡✜♥✐t✐♦♥ ✲ ◆❡t✇♦r❦s ❆ ♥❡t✇♦r❦ ✐s ❛ ❣r❛♣❤ ✇✐t❤ t✇♦ ❞✐st✐♥❣✉✐s❤❡❞ ✈❡rt✐❝❡s ✇❤✐❝❤ ✇❡ ❝❛❧❧ ♣♦❧❡s✱ s♦ t❤❛t ✐❢ ✇❡ ❛❞❞ ❛♥ ❡❞❣❡ ❜❡t✇❡❡♥ t❤❡♠✱ t❤❡♥ t❤❡ r❡s✉❧t✐♥❣ ✭♠✉❧t✐✮❣r❛♣❤ ❜❡❧♦♥❣s t♦ t❤❡ ❝❡rt❛✐♥ ❝❧❛ss ♦❢ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦s ✭❚r❛❦❤t❡♥❜r♦t✕❚✉tt❡✮

❉❡✜♥✐t✐♦♥ ✲ ◆❡t✇♦r❦s ❆ ♥❡t✇♦r❦ ✐s ❛ ❣r❛♣❤ ✇✐t❤ t✇♦ ❞✐st✐♥❣✉✐s❤❡❞ ✈❡rt✐❝❡s ✇❤✐❝❤ ✇❡ ❝❛❧❧ ♣♦❧❡s✱ s♦ t❤❛t ✐❢ ✇❡ ❛❞❞ ❛♥ ❡❞❣❡ ❜❡t✇❡❡♥ t❤❡♠✱ t❤❡♥ t❤❡ r❡s✉❧t✐♥❣ ✭♠✉❧t✐✮❣r❛♣❤ ❜❡❧♦♥❣s t♦ t❤❡ ❝❡rt❛✐♥ ❝❧❛ss ♦❢ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳

+

• A network

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦s ✭❚r❛❦❤t❡♥❜r♦t✕❚✉tt❡✮

❉❡✜♥✐t✐♦♥ ✲ ◆❡t✇♦r❦s ❆ ♥❡t✇♦r❦ ✐s ❛ ❣r❛♣❤ ✇✐t❤ t✇♦ ❞✐st✐♥❣✉✐s❤❡❞ ✈❡rt✐❝❡s ✇❤✐❝❤ ✇❡ ❝❛❧❧ ♣♦❧❡s✱ s♦ t❤❛t ✐❢ ✇❡ ❛❞❞ ❛♥ ❡❞❣❡ ❜❡t✇❡❡♥ t❤❡♠✱ t❤❡♥ t❤❡ r❡s✉❧t✐♥❣ ✭♠✉❧t✐✮❣r❛♣❤ ❜❡❧♦♥❣s t♦ t❤❡ ❝❡rt❛✐♥ ❝❧❛ss ♦❢ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳

+

• A network

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦ ❞❡❝♦♠♣♦s✐t✐♦♥

❆ ♥❡t✇♦r❦ ✐s✿ ❛♥ ❡❞❣❡❀ ❛ s❡r✐❡s ♥❡t✇♦r❦ ✭t②♣❡ ❙✮❀ ❛ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✭t②♣❡ P✮❀ ❛ ❝♦r❡ ♥❡t✇♦r❦ ✭t②♣❡ ❍✮✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦ ❞❡❝♦♠♣♦s✐t✐♦♥

❆ ♥❡t✇♦r❦ ✐s✿ ❛♥ ❡❞❣❡❀ ❛ s❡r✐❡s ♥❡t✇♦r❦ ✭t②♣❡ ❙✮❀ ❛ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✭t②♣❡ P✮❀ ❛ ❝♦r❡ ♥❡t✇♦r❦ ✭t②♣❡ ❍✮✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦ ❞❡❝♦♠♣♦s✐t✐♦♥

❆ ♥❡t✇♦r❦ ✐s✿ ❛♥ ❡❞❣❡❀ ❛ s❡r✐❡s ♥❡t✇♦r❦ ✭t②♣❡ ❙✮❀ ❛ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✭t②♣❡ P✮❀ ❛ ❝♦r❡ ♥❡t✇♦r❦ ✭t②♣❡ ❍✮✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦ ❞❡❝♦♠♣♦s✐t✐♦♥

❆ ♥❡t✇♦r❦ ✐s✿ ❛♥ ❡❞❣❡❀ ❛ s❡r✐❡s ♥❡t✇♦r❦ ✭t②♣❡ ❙✮❀ ❛ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✭t②♣❡ P✮❀ ❛ ❝♦r❡ ♥❡t✇♦r❦ ✭t②♣❡ ❍✮✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

◆❡t✇♦r❦ ❞❡❝♦♠♣♦s✐t✐♦♥

❆ ♥❡t✇♦r❦ ✐s✿ ❛♥ ❡❞❣❡❀ ❛ s❡r✐❡s ♥❡t✇♦r❦ ✭t②♣❡ ❙✮❀ ❛ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✭t②♣❡ P✮❀ ❛ ❝♦r❡ ♥❡t✇♦r❦ ✭t②♣❡ ❍✮✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❙❡r✐❡s ♥❡t✇♦r❦s

❆ s❡r✐❡s ♥❡t✇♦r❦ ✐s✿

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❙❡r✐❡s ♥❡t✇♦r❦s

❆ s❡r✐❡s ♥❡t✇♦r❦ ✐s✿

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❙❡r✐❡s ♥❡t✇♦r❦s

❆ s❡r✐❡s ♥❡t✇♦r❦ ✐s✿

A series network

P + H + e N

+

• ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉

❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

P❛r❛❧❧❡❧ ♥❡t✇♦r❦s

❆ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✐s✿

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

P❛r❛❧❧❡❧ ♥❡t✇♦r❦s

❆ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✐s✿

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

P❛r❛❧❧❡❧ ♥❡t✇♦r❦s

❆ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦ ✐s✿

e

N N N

+ − − + The two types of parallel networks ❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❈♦r❡ ♥❡t✇♦r❦s

❆ ❝♦r❡ ♥❡t✇♦r❦ ✐s✿

+

• ❚❤❡ ✉♥❞❡r❧②✐♥❣ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ❛ ❝♦r❡✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❈♦r❡ ♥❡t✇♦r❦s

❆ ❝♦r❡ ♥❡t✇♦r❦ ✐s✿

N N N N N +

• ❚❤❡ ✉♥❞❡r❧②✐♥❣ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ❛ ❝♦r❡✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

• ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ✲ ◆(①, ②) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❝❧❛ss ♦❢ ♥❡t✇♦r❦s❀ ✲ ❚(①, ②) ❜❡ t❤❡ ❡✳❣✳❢✳ ♦❢ t❤❡ ❝❧❛ss ♦❢ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❢r♦♠ ✇❤✐❝❤ ✇❡ ❝❤♦♦s❡ t❤❡ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✺

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

• ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

▲❡t ✲ ◆(①, ②) ❜❡ t❤❡ ❡♥✉♠❡r❛t✐♥❣ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❝❧❛ss ♦❢ ♥❡t✇♦r❦s❀ ✲ ❚(①, ②) ❜❡ t❤❡ ❡✳❣✳❢✳ ♦❢ t❤❡ ❝❧❛ss ♦❢ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❢r♦♠ ✇❤✐❝❤ ✇❡ ❝❤♦♦s❡ t❤❡ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✺

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

• ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s

❚❤❡s❡ ❢✉♥❝t✐♦♥s s❛t✐s❢②✿ Φ(①, ②, ◆(①, ②)) = ✵, ✇❤❡r❡ Φ(①, ②, ③) = ❚(①, ③) − ❧♦❣ ✶ + ③ ✶ + ②

• +

①③✷ ✶ + ①③ .

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✻

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❆✐♠ ❲❡ st✉❞② r❛♥❞♦♠ ♥❡t✇♦r❦s ♦♥ ♥ ✈❡rt✐❝❡s✱ ✇❤❡r❡ t❤❡ ❝♦r❡s ❛r❡ s❛♠♣❧❡❞ ❢r♦♠ ❛ ❣✐✈❡♥ ❝❧❛ss ♦❢ ✸✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❆ ❝❡rt❛✐♥ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♥❡t✇♦r❦s ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❝❧❛ss ♦❢ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ②✐❡❧❞s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❘♦✉❣❤ ❡q✉✐✈❛❧❡♥❝❡ ■❢ ❛ ♣r♦♣❡rt② ❤♦❧❞s ❛✳❛✳s✳ ❢♦r s✉❝❤ ❛ ❝❧❛ss ♦❢ ♥❡t✇♦r❦s✱ t❤❡♥ ✐t ❛❧s♦ ❤♦❧❞s ❛✳❛✳s✳ ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛ss ♦❢ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✽

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❲❤❛t ❞❡t❡r♠✐♥❡s t❤❡ t②♣✐❝❛❧ str✉❝t✉r❡ ♦❢ ❛ r❛♥❞♦♠ ♥❡t✇♦r❦ ♦♥ ♥ ✈❡rt✐❝❡s ✐s t❤❡ s✐❣♥ ♦❢ Φ③(ρ◆(✶), ✶, ◆(ρ◆(✶), ✶))✱ ✇❤❡r❡ ρ◆(✶) ✐s t❤❡ r❛❞✐✉s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ◆(①, ✶)✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✶✾

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❚❤❡♦r❡♠ ❬❋✳ ❛♥❞ P❛♥❛❣✐♦t♦✉❪ ▲❡t ◆♥ ❜❡ ❛ r❛♥❞♦♠ ♥❡t✇♦r❦ ♦♥ ♥ ✈❡rt✐❝❡s✳ ■❢ Φ③(ρ◆(✶), ✶, ◆(ρ◆(✶), ✶)) > ✵✱ t❤❡♥ ❛❧❧ ❝♦r❡s ♦❢ ◆♥ ❤❛✈❡ ❖❈(❧♦❣ ♥) ✈❡rt✐❝❡s✳ ■❢ Φ③(ρ◆(✶), ✶, ◆(ρ◆(✶), ✶)) < ✵✱ t❤❡♥ ❢♦r s♦♠❡ γ❈ > ✵ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❝♦r❡ ✇✐t❤ γ❈♥ + ♦♣(♥) ✈❡rt✐❝❡s✱ ❜✉t ❡✈❡r② ♦t❤❡r ❝♦r❡ ❤❛s ♦♣(♥) ✈❡rt✐❝❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❲❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ✲ ❛s②♠♣t♦t✐❝ ❝♦✉♥ts ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ s♠❛❧❧ ❝♦r❡s❀ ✲ t❤❡ ♦r❞❡r ♦❢ t❤❡ ✏❣✐❛♥t✧ ❝♦r❡❀ ✲ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ♦❢ ◆♥ ❛s ♥ → ∞✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❲❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ✲ ❛s②♠♣t♦t✐❝ ❝♦✉♥ts ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ s♠❛❧❧ ❝♦r❡s❀ ✲ t❤❡ ♦r❞❡r ♦❢ t❤❡ ✏❣✐❛♥t✧ ❝♦r❡❀ ✲ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ♦❢ ◆♥ ❛s ♥ → ∞✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❲❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ✲ ❛s②♠♣t♦t✐❝ ❝♦✉♥ts ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ s♠❛❧❧ ❝♦r❡s❀ ✲ t❤❡ ♦r❞❡r ♦❢ t❤❡ ✏❣✐❛♥t✧ ❝♦r❡❀ ✲ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ♦❢ ◆♥ ❛s ♥ → ∞✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❘❛♥❞♦♠ ♥❡t✇♦r❦s

❲❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ ✲ ❛s②♠♣t♦t✐❝ ❝♦✉♥ts ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ s♠❛❧❧ ❝♦r❡s❀ ✲ t❤❡ ♦r❞❡r ♦❢ t❤❡ ✏❣✐❛♥t✧ ❝♦r❡❀ ✲ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ♦❢ ◆♥ ❛s ♥ → ∞✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❊①❛♠♣❧❡ ✲ ❇✐❝♦♥♥❡❝t❡❞ ❘❛♥❞♦♠ P❧❛♥❛r ●r❛♣❤s

■❢ ✇❡ s❛♠♣❧❡ t❤❡ ❝♦r❡s ❢r♦♠ t❤❡ ❝❧❛ss ♦❢ ✸✲❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤s✱ t❤❡ r❡s✉❧t✐♥❣ ♥❡t✇♦r❦ ❝♦rr❡s♣♦♥❞s t♦ r❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤s❀ ■t t✉r♥s ♦✉t t❤❛t t❤✐s ❝❧❛ss ♦❢ ♥❡t✇♦r❦s ❢❛❧❧s ✐♥t♦ t❤❡ s❡❝♦♥❞ ✏❝❛t❡❣♦r②✧✿ ❚❤❡♦r❡♠ ❆ r❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤ ♦♥ ♥ ✈❡rt✐❝❡s ❤❛s ❛ ✉♥✐q✉❡ ❝♦r❡ ♦❢ ♦r❞❡r ❝♣♥ + ♦(♥)✱ ✇❤❡r❡ ❝♣ = ✵.✼✻✺...✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦r❡ ❤❛s ❖(♥✷/✸) ✈❡rt✐❝❡s✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ − ♦(✶)✳ ❚❤✐s ✇❛s ❛❧s♦ s❤♦✇♥ r❡❝❡♥t❧② ❜② ●✐♠é♥❡③✱ ◆♦② ❛♥❞ ❘✉é✱ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❛♥❛❧②t✐❝ ♠❡t❤♦❞s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❊①❛♠♣❧❡ ✲ ❇✐❝♦♥♥❡❝t❡❞ ❘❛♥❞♦♠ P❧❛♥❛r ●r❛♣❤s

■❢ ✇❡ s❛♠♣❧❡ t❤❡ ❝♦r❡s ❢r♦♠ t❤❡ ❝❧❛ss ♦❢ ✸✲❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤s✱ t❤❡ r❡s✉❧t✐♥❣ ♥❡t✇♦r❦ ❝♦rr❡s♣♦♥❞s t♦ r❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤s❀ ■t t✉r♥s ♦✉t t❤❛t t❤✐s ❝❧❛ss ♦❢ ♥❡t✇♦r❦s ❢❛❧❧s ✐♥t♦ t❤❡ s❡❝♦♥❞ ✏❝❛t❡❣♦r②✧✿ ❚❤❡♦r❡♠ ❆ r❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤ ♦♥ ♥ ✈❡rt✐❝❡s ❤❛s ❛ ✉♥✐q✉❡ ❝♦r❡ ♦❢ ♦r❞❡r ❝♣♥ + ♦(♥)✱ ✇❤❡r❡ ❝♣ = ✵.✼✻✺...✱ ✇❤❡r❡❛s ❡✈❡r② ♦t❤❡r ❝♦r❡ ❤❛s ❖(♥✷/✸) ✈❡rt✐❝❡s✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ − ♦(✶)✳ ❚❤✐s ✇❛s ❛❧s♦ s❤♦✇♥ r❡❝❡♥t❧② ❜② ●✐♠é♥❡③✱ ◆♦② ❛♥❞ ❘✉é✱ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❛♥❛❧②t✐❝ ♠❡t❤♦❞s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✷

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❊①❛♠♣❧❡ ✲ ❇✐❝♦♥♥❡❝t❡❞ ❘❛♥❞♦♠ P❧❛♥❛r ●r❛♣❤s

❋♦r ❡✈❡r② ✹ ≤ ℓ = ❖

• ♥

❧♦❣ ♥

✷/✺ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦r❡s ✇✐t❤ ℓ ✈❡rt✐❝❡s ✐s ❢♦r ❛♥② ε > ✵ ❝ℓ♥(✶ ± ε), ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ − ♦(✶)✱ ✇❤❡r❡ ❝ℓ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❝❧❛ss ♦❢ ✸✲❝♦♥♥❡❝t❡❞ ♣❧❛♥❛r ❣r❛♣❤s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✸

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛♥❛❧②s❡ t❤❡ ♦✉t♣✉t ♦❢ ❇♦❧t③♠❛♥♥ s❛♠♣❧❡rs ✇❤✐❝❤ ❛r❡ r❛♥❞♦♠✐s❡❞ ❛❧❣♦r✐t❤♠s t❤❛t ❣❡♥❡r❛t❡ ♥❡t✇♦r❦s❀ ✲ ■♥ ♦✉r ❝❛s❡✱ t❤❡s❡ ❛r❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠s t❤❛t ❝❛❧❧ ❡❛❝❤ ♦t❤❡r r❡❝✉rs✐✈❡❧②✱ r❡✢❡❝t✐♥❣ t❤❡ r❡❝✉rs✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ♥❡t✇♦r❦✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛♥❛❧②s❡ t❤❡ ♦✉t♣✉t ♦❢ ❇♦❧t③♠❛♥♥ s❛♠♣❧❡rs ✇❤✐❝❤ ❛r❡ r❛♥❞♦♠✐s❡❞ ❛❧❣♦r✐t❤♠s t❤❛t ❣❡♥❡r❛t❡ ♥❡t✇♦r❦s❀ ✲ ■♥ ♦✉r ❝❛s❡✱ t❤❡s❡ ❛r❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠s t❤❛t ❝❛❧❧ ❡❛❝❤ ♦t❤❡r r❡❝✉rs✐✈❡❧②✱ r❡✢❡❝t✐♥❣ t❤❡ r❡❝✉rs✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ♥❡t✇♦r❦✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✹

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

❊①❛♠♣❧❡✿ ❇♦❧t③♠❛♥♥ s❛♠♣❧❡r ❢♦r ◆❡t✇♦r❦s Γ◆(①, ②)✿ Γ◆ ← ❡ ✇✳♣✳

② ◆(①,②)❀

Γ◆ ← Γ❙(①, ②) ✇✳♣✳

❙(①,②) ◆(①,②)❀

Γ◆ ← ΓP(①, ②) ✇✳♣✳ P(①,②)

◆(①,②)❀

Γ◆ ← Γ❍(①, ②) ✇✳♣✳ ❍(①,②)

◆(①,②)❀

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✺

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

❊①❛♠♣❧❡✿ ❇♦❧t③♠❛♥♥ s❛♠♣❧❡r ❢♦r ❈♦r❡ ◆❡t✇♦r❦s Γ❍(①, ②) ✿ ❚ ← Γ❚(①, ◆(①, ②)) ❢♦r ❡❛❝❤ ❡❞❣❡ ❡ ♦❢ ❚ γ❡ ← Γ◆(①, ②) r❡♣❧❛❝❡ ❡✈❡r② ❡ ✐♥ ❚ ❜② γ❡ ❘❡t✉r♥ ❚✱ r❡❧❛❜❡❧✐♥❣ r❛♥❞♦♠❧② ✐ts ✈❡rt✐❝❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✻

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ ❡❛❝❤ r♦✉t✐♥❡❀ ✲ ▲❡t ✲ ❆◆❡t ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♥❡t✇♦r❦ r♦✉t✐♥❡❀ ✲ ❆❙❡r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ s❡r✐❡s ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❆P❛r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❱T ❛♥❞ ❊T ❞❡♥♦t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✐♥ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ ❡❛❝❤ r♦✉t✐♥❡❀ ✲ ▲❡t ✲ ❆◆❡t ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♥❡t✇♦r❦ r♦✉t✐♥❡❀ ✲ ❆❙❡r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ s❡r✐❡s ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❆P❛r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❱T ❛♥❞ ❊T ❞❡♥♦t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✐♥ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ ❡❛❝❤ r♦✉t✐♥❡❀ ✲ ▲❡t ✲ ❆◆❡t ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♥❡t✇♦r❦ r♦✉t✐♥❡❀ ✲ ❆❙❡r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ s❡r✐❡s ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❆P❛r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❱T ❛♥❞ ❊T ❞❡♥♦t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✐♥ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ ❡❛❝❤ r♦✉t✐♥❡❀ ✲ ▲❡t ✲ ❆◆❡t ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♥❡t✇♦r❦ r♦✉t✐♥❡❀ ✲ ❆❙❡r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ s❡r✐❡s ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❆P❛r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❱T ❛♥❞ ❊T ❞❡♥♦t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✐♥ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ ❡❛❝❤ r♦✉t✐♥❡❀ ✲ ▲❡t ✲ ❆◆❡t ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♥❡t✇♦r❦ r♦✉t✐♥❡❀ ✲ ❆❙❡r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ s❡r✐❡s ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❆P❛r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❱T ❛♥❞ ❊T ❞❡♥♦t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✐♥ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ ❛r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ ❡❛❝❤ r♦✉t✐♥❡❀ ✲ ▲❡t ✲ ❆◆❡t ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♥❡t✇♦r❦ r♦✉t✐♥❡❀ ✲ ❆❙❡r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ s❡r✐❡s ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❆P❛r ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♥❡t✇♦r❦s r♦✉t✐♥❡❀ ✲ ❱T ❛♥❞ ❊T ❞❡♥♦t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ❛♥❞ ❡❞❣❡s ✐♥ ❝♦r❡s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✼

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ ❚❡❝❤♥✐q✉❡s

❲❡ s❤♦✇ t❤❛t ❢♦r ❡❛❝❤ ❩ ∈ {❆◆❡t, ❆❙❡r, ❆P❛r, ❱T , ❊T } ✇❡ ❤❛✈❡ P (|❩ − ③♥| < ε♥) > ✶ − ❡−❈ε✷♥, ✇❤❡r❡ ③ ∈ {❛◆❡t, ❛❙❡r, ❛P❛r, ✈T , ❡T } ❛♥❞ t❤❡ ✈❡❝t♦r ❛ := [❛◆❡t, ❛❙❡r, ❛P❛r, ✈T , ❡T ]❚ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ▼❛ = r✱ ✇❤❡r❡ ▼ =        

✶ ◆(①,②) ρ◆◆(①,②) ❙(①,②) ◆(①,②)−✶ ✷P(①,②)

✵ ✵ ✵ ✶ ✵ ✶ ✵

❙(①,②) ◆(①,②)

−✶

❙(①,②)◆(①,②) P(①,②)

✵ ✵

P(①,②) ◆(①,②) ρ◆P(①,②)◆(①,②) ❙(①,②)

−✶ ✵ ✵ −✶ ✶ ✵ ✵ ✶         , r =       µ ✶ ✵ ✵ ✵       ✭✶✮ ❛♥❞ µ = −ρ′

◆(✶)

ρ◆(✶)✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✽

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

✲ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ❆❍ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ❈♦r❡ ◆❡t✇♦r❦s r♦✉t✐♥❡✱ t❤❡♥ ❆❍ = α❍♥ + ♦♣(♥), ✇❤❡r❡ α❍ = ✷µ❍(①, ②).

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✾

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

✲ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ❆❍ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝❛❧❧s ♦❢ t❤❡ ❈♦r❡ ◆❡t✇♦r❦s r♦✉t✐♥❡✱ t❤❡♥ ❆❍ = α❍♥ + ♦♣(♥), ✇❤❡r❡ α❍ = ✷µ❍(①, ②).

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✷✾

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ tr❡❛t Γ❍(①, ②) ❛s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ t❤❛t r❡❛❞s ✐ts ✐♥♣✉ts ❢r♦♠ ❛ ❧✐st (❚✶, ❚✷, . . .), ✇❤❡r❡ t❤❡ {❚✐}✐≥✶ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t s❛♠♣❧❡s ❢r♦♠ t❤❡ ❝❧❛ss ♦❢ ❝♦r❡s✱ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❇♦❧t③♠❛♥♥ ❞✐str✐❜✉t✐♦♥✳ ✲ ■❢ ❈❦(♥) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦r❡s ♦❢ s✐③❡ ❦ ✐♥ ❛ ♥❡t✇♦r❦ ✇✐t❤ ♥ ✈❡rt✐❝❡s✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❜♦✉♥❞ ✐t ❜② ❧♦♦❦✐♥❣ ✐♥s✐❞❡ (❚✶, . . . , ❚⌈α❍♥+ε♥⌉) ❛♥❞ ✉s❡ ❈❤❡r♥♦✛ ❜♦✉♥❞s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

Pr♦♦❢ t❡❝❤♥✐q✉❡s

✲ ❲❡ tr❡❛t Γ❍(①, ②) ❛s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ t❤❛t r❡❛❞s ✐ts ✐♥♣✉ts ❢r♦♠ ❛ ❧✐st (❚✶, ❚✷, . . .), ✇❤❡r❡ t❤❡ {❚✐}✐≥✶ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t s❛♠♣❧❡s ❢r♦♠ t❤❡ ❝❧❛ss ♦❢ ❝♦r❡s✱ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❇♦❧t③♠❛♥♥ ❞✐str✐❜✉t✐♦♥✳ ✲ ■❢ ❈❦(♥) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦r❡s ♦❢ s✐③❡ ❦ ✐♥ ❛ ♥❡t✇♦r❦ ✇✐t❤ ♥ ✈❡rt✐❝❡s✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❜♦✉♥❞ ✐t ❜② ❧♦♦❦✐♥❣ ✐♥s✐❞❡ (❚✶, . . . , ❚⌈α❍♥+ε♥⌉) ❛♥❞ ✉s❡ ❈❤❡r♥♦✛ ❜♦✉♥❞s✳

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸✵

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❖✉t❧♦♦❦

❯s❡ t❤❡s❡ r❡s✉❧ts ❢♦r ❛❧❣♦r✐t❤♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s❀ ❞❡t❡r♠✐♥❡ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ❛♥❞ t❤❡ s✐③❡ ♦❢ t❤❡ ❣✐❛♥t ❝♦r❡❀ ❤✐❣❤❡r ❝♦♥♥❡❝t✐✈✐t✐❡s❄❄❄

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❖✉t❧♦♦❦

❯s❡ t❤❡s❡ r❡s✉❧ts ❢♦r ❛❧❣♦r✐t❤♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s❀ ❞❡t❡r♠✐♥❡ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ❛♥❞ t❤❡ s✐③❡ ♦❢ t❤❡ ❣✐❛♥t ❝♦r❡❀ ❤✐❣❤❡r ❝♦♥♥❡❝t✐✈✐t✐❡s❄❄❄

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❖✉t❧♦♦❦

❯s❡ t❤❡s❡ r❡s✉❧ts ❢♦r ❛❧❣♦r✐t❤♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s❀ ❞❡t❡r♠✐♥❡ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ❛♥❞ t❤❡ s✐③❡ ♦❢ t❤❡ ❣✐❛♥t ❝♦r❡❀ ❤✐❣❤❡r ❝♦♥♥❡❝t✐✈✐t✐❡s❄❄❄

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸✶

❈❧❛ss✐❝❛❧ r❛♥❞♦♠ ❣r❛♣❤s ❘❛♥❞♦♠ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❛♥❞♦♠ ❜✐❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ❘❡s✉❧ts ❋✉rt❤❡r ❞✐r❡❝t✐♦♥s

❚❤❛♥❦ ②♦✉✦

❋♦✉♥t♦✉❧❛❦✐s✱ P❛♥❛❣✐♦t♦✉ ❚❤❡ str✉❝t✉r❡ ♦❢ r❛♥❞♦♠ ✷✲❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✸✷