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❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣ ♦❢ ❛ ❉✐s❦ ✇✐t❤ ❈♦♥❣r✉❡♥t ❙♠❛❧❧❡r ❉✐s❦s

❇❛❧á③s ❈s✐❦ós

❊öt✈ös ▲♦rá♥❞ ❯♥✐✈❡rs✐t② ❇✉❞❛♣❡st

❘❡tr♦s♣❡❝t✐✈❡ ❲♦r❦s❤♦♣ ♦♥ ❉✐s❝r❡t❡ ●❡♦♠❡tr②✱ ❖♣t✐♠✐③❛t✐♦♥✱ ❛♥❞ ❙②♠♠❡tr② ◆♦✈❡♠❜❡r ✷✺✲✷✾✱ ✷✵✶✸ ❋✐❡❧❞s ■♥st✐t✉t❡✱ ❚♦r♦♥t♦

Pr♦❜❧❡♠✿ ❋✐♥❞ rn

p = max{r : n ❞✐s❦s ♦❢ r❛❞✐✉s r ❝❛♥ ❜❡ ♣❛❝❦❡❞ ✐♥t♦ ❛ ✉♥✐t ❞✐s❦✳}

✭✶✲✹ tr✐✈✐❛❧❀ ✺✲✼ ●r❛❤❛♠ ✭✶✾✻✽✮❀ ✽✲✶✵ P✐r❧ ✭✶✾✻✾✮❀ ✶✶ ▼❡❧✐ss❡♥ ✭✶✾✾✹✮❀ ✶✷ ❋♦❞♦r ✭✷✵✵✵✮❀ ✶✸ ❋♦❞♦r ✭✷✵✵✸✮❀ ✶✹ ✉♥s♦❧✈❡❞✮

❈♦✈❡r✐♥❣ ♦❢ ❛ ❉✐s❦ ✇✐t❤ ❈♦♥❣r✉❡♥t ❉✐s❦s

Pr♦❜❧❡♠✿ ❋✐♥❞ rn

c = min{r : n ❞✐s❦s ♦❢ r❛❞✐✉s r ❝❛♥ ❝♦✈❡r ❛ ✉♥✐t ❞✐s❦✳}

❙♦❧✉t✐♦♥s ❢♦r n ≤ 8✿ ❋♦r n = 5 ❛♥❞ 6 t❤❡ ♦♣t✐♠❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❤❛s ♦♥❧② ❛ ♠✐rr♦r s②♠♠❡tr②✳

✭✶✲✹✱ ✼ tr✐✈✐❛❧❀ ✺✲✻ ❑✳ ❇❡③❞❡❦ ✭✶✾✼✾✱✶✾✽✸✮❀ ✽✲✾ ●✳ ❋❡❥❡s ❚ót❤ ✭✶✾✾✾✮❀ ✶✵ ❄✮

❘✳ ❈♦♥♥❡❧❧②✬s Pr♦❜❧❡♠✿

◮ ●✐✈❡♥ n ❛♥❞ rn p ≤ r ≤ rn c ✱ ✜♥❞ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ n ❞✐s❦s ♦❢ r❛❞✐✉s

r✱ t❤❛t ❝♦✈❡rs t❤❡ ♠♦st ♦❢ t❤❡ ❛r❡ ♦❢ ❛ ✉♥✐t ❞✐s❦✳

◮ ❯♥❞❡rst❛♥❞ ❤♦✇ t❤❡ r♦t❛t✐♦♥❛❧ s②♠♠❡tr② ♦❢ t❤❡ ♦♣t✐♠❛❧

❝♦♥✜❣✉r❛t✐♦♥ ❢♦r n = 5✱ r = r5

p ✐s ❧♦st ❛s r ❣r♦✇s ❝♦♥t✐♥✉♦✉s❧② ❢r♦♠

r5

p t♦ r5 c✳ ???

− →

???

− →

❉❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❱♦❧✉♠❡ ♦❢ ❋❧♦✇❡rs

❉❡✜♥✐t✐♦♥s

◮ ❆ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f(x1, . . . , xk) ✐s ❛ ❢♦r♠❛❧ ❡①♣r❡ss✐♦♥ ❜✉✐❧t ❢r♦♠

t❤❡ ✈❛r✐❛❜❧❡s x1, . . . , xn ❛♥❞ t❤❡ ❜✐♥❛r② ♦♣❡r❛t✐♦♥ s②♠❜♦❧s ∧ ❛♥❞ ∨✳ ❊①❛♠♣❧❡✿ x1 ∧ (x2 ∨ x3)✳

◮ ❆ ✢♦✇❡r ✐s ❛ ❜♦❞② ♦❜t❛✐♥❡❞ ❜② ❡✈❛❧✉❛t✐♥❣ ❛ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧

f(x1, . . . , xk) ♦♥ s♦♠❡ ❜❛❧❧s xi = Bi ✇✐t❤ ♦♣❡r❛t✐♦♥s ∨ = ∪✱ ∧ = ∩✳

◮ ❚❤❡ ♣♦✇❡r ♦❢ ❛ ♣♦✐♥t p ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❜❛❧❧ B = B(c, r) ✐s

PB(p) = p − c2 − r2✳

◮ ❚❤❡ ✭tr✉♥❝❛t❡❞✮ ❉✐r✐❝❤❧❡t✕❱♦r♦♥♦✐ ❝❡❧❧ ♦❢ t❤❡ ❜❛❧❧ Bi ✐♥ t❤❡ ✢♦✇❡r

f(B1, . . . , Bk) ✐s t❤❡ s❡t Ci = {x : f(PB1(x), . . . , f(PBk(x)) = f(PBi(x)}, ✇❤❡r❡ f ✐s ❡✈❛❧✉❛t❡❞ ♦♥ t❤❡ ♣♦✇❡rs ✇✐t❤ ♦♣❡r❛t✐♦♥s ∨ = min✱ ∧ = max✳

◮ ❚❤❡ ✇❛❧❧ Wij ❜❡t✇❡❡♥ t❤❡ ❉✐r✐❝❤❧❡t✕❱♦r♦♥♦✐ ❝❡❧❧s Ci ❛♥❞ Cj ✐❢

Wij = Ci ∩ Cj✳

❉❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❱♦❧✉♠❡ ♦❢ ❋❧♦✇❡rs

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t ❡❛❝❤ ✈❛r✐❛❜❧❡ xi ♦❝❝✉rs ✐♥ t❤❡ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f(x1, . . . , xk) ❡①❛❝t❧② ♦♥❝❡✳ ▲❡t ǫij ❜❡ 1 ✐❢ t❤❡ t❤❡ s❤♦rt❡st s✉❜❡①♣r❡ss✐♦♥ ♦❢ f t❤❛t ❝♦♥t❛✐♥s ❜♦t❤ xi ❛♥❞ xj ✐s ❛ ∨ ♦❢ s❤♦rt❡r ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧s✱ −1 ♦t❤❡r✇✐s❡✳ ■❢ t❤❡ ❜❛❧❧ Bi(t) = B(ci(t), ri) ♠♦✈❡ ✐♥ ❛ ❞✐✛❡r❡♥t✐❛❜❧❡ ✇❛②✱ t❤❡♥ t❤❡ ✈♦❧✉♠❡ V (t) ♦❢ t❤❡ ✢♦✇❡r f(B1(t), . . . , Bk(t)) ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ❡❛❝❤ t ❛t ✇❤✐❝❤ t❤❡ ❜❛❧❧s Bi(t) ❛r❡ ❞✐✛❡r❡♥t ❛♥❞ V ′ =

• 1≤i<j≤k

ǫijV oln−1(Wij)d′

ij,

✇❤❡r❡ dij = ci − cj✳

❖❜s❡r✈❛t✐♦♥ ✭❘✳ ❈♦♥♥❡❧❧②✮

■❢ ❛ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❜❛❧❧s ♠❛①✐♠✐③❡s t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ✢♦✇❡r✱ t❤❡♥ t❤❡ t❡♥s❡❣r✐t② ♦❜t❛✐♥❡❞ ❜② ❝♦♥♥❡❝t✐♥❣ ci ❛♥❞ cj ❜② ❛ ❝❛❜❧❡ ✐❢ ǫij = −1 ❛♥❞ ❛ str✉t ✐❢ ǫij = 1 ✐s r✐❣✐❞✳ ❚❤❡ ✈♦❧✉♠❡s ♦❢ t❤❡ ✇❛❧❧s ♣r♦✈✐❞❡ ❛ s❡❧❢ str❡ss ωij = ǫij

V oln−1(Wij) dij

✳

❋♦r♠✉❧❛❡ ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ P❧❛♥❡

◮ ❖r✐❡♥t t❤❡ ♣❧❛♥❡ ❛♥❞ ❛❧❧ t❤❡ ❝✐r❝❧❡s ✐♥ t❤❡ ♣♦s✐t✐✈❡ ❞✐r❡❝t✐♦♥✳ ◮ J ✕ r♦t❛t✐♦♥ ❜② + π 2 ✳ ◮ ❋♦r ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ k ❞✐s❦s Di = D(ci, ri) ✇✐t❤ ❜♦✉♥❞❛r② ❝✐r❝❧❡s

Ci = S(ci, ri)✱ ❞❡♥♦t❡ ❜② pij t❤❡ ♣♦✐♥t ✇❤❡r❡ Ci ❡♥t❡rs Cj✱ ♣r♦✈✐❞❡❞ t❤❛t t❤✐s ♣♦✐♥t ❡①✐sts✳

◮ ❋♦r ❛ ✢♦✇❡r f(D1, . . . , Dk)✱ ❞❡✜♥❡ t❤❡ ✈❡rt❡① s❡t V ❛s t❤❡ s❡t ♦❢

❝r♦ss✐♥❣s pij✱ ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✢♦✇❡r✳

❚❤❡♦r❡♠

❆ss✉♠❡ t❤❛t t❤❡ ❞✐s❦s Di = D(ci, ri) ♠♦✈❡ s♠♦♦t❤❧② ❛♥❞ t❤❡ s♣❡❡❞ ✈❡❝t♦rs ♦❢ t❤❡✐r ❝❡♥t❡rs ❛r❡ vi✱ i = 1, . . . , k✳ ■❢ ❛❧❧ t❤❡ ❞✐s❦s ❛r❡ ❞✐✛❡r❡♥t✱ t❤❡♥ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛r❡❛ V ♦❢ t❤❡ ✢♦✇❡r f(D1, . . . , Dk) ❡①✐sts ❛♥❞ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❢♦❧❧♦✇s V ′ =

k

• i=1
• vi, J

pij∈V

ǫijpij −

• pji∈V

ǫijpji

• =
• pij

ǫijvi − vj, Jpij.

❋♦r♠✉❧❛❡ ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ P❧❛♥❡

❈r✐t✐❝❛❧ ❈♦♥✜❣✉r❛t✐♦♥s ❛♥❞ t❤❡ ❍❡ss✐❛♥

◮ ●✐✈❡♥ ❛ ❧❛tt✐❝❡ ♣♦❧②♥♦♠✐❛❧ f(x1, . . . , xk)✱ ❛♥ ❛rr❛♥❣❡♠❡♥t ♦❢ ❞✐s❦s

D1, . . . , Dk ✐s ❝❛❧❧❡❞ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ✐❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❛r❡❛ ♦❢ f(D1, . . . , Dk) ✐s 0 ❢♦r ❛♥② s♠♦♦t❤ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❞✐s❦s✳

❈♦r♦❧❧❛r②

❆ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❞✐s❦s ✐s ❝r✐t✐❝❛❧ ❢♦r ❛ ❣✐✈❡♥ f✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∀i :

• pij∈V

ǫijpij −

• pji∈V

ǫijpji = 0.

❚❤❡♦r❡♠

■❢ ❢♦r ❛ ❣✐✈❡♥ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥✱ t❤❡r❡ ❛r❡ ♥♦ t❛♥❣❡♥t ♣❛✐r ♦❢ ❝✐r❝❧❡s✱ t❤❡ ❝♦♥t❛❝t ♣♦✐♥t ♦❢ ✇❤✐❝❤ ✐s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✢♦✇❡r✱ t❤❡♥ t❤❡ ❛r❡❛ ♦❢ t❤❡ ✢♦✇❡r ✐s t✇✐❝❡ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t t❤✐s ❝♦♥✜❣✉r❛t✐♦♥ ❛♥❞ ✐ts ❍❡ss✐❛♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ Hess(V, V) =

• pij∈V

ǫij rirj sin θij (vi − vj)T (ci − pij)(cj − pij)T (vi − vj), ✇❤❡r❡ ✳

❈♦✈❡r✐♥❣ t❤❡ ▼♦st ✇✐t❤ ✷ ❈♦♥❣r✉❡♥t ❉✐s❦s

❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣s ❜② ✸ ❉✐s❦s

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❇✳ ❙③❛❧❦❛✐

❋✐rst ❙t❡♣✳ ❈❧❛ss✐❢② ❛❧❧ ❝♦♠❜✐♥❛t♦r✐❛❧❧② ❞✐✛❡r❡♥t ❛rr❛♥❣❡♠❡♥ts t❤❛t ❝❛♥ ♣♦ss✐❜❧② ❣✐✈❡ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✳

◮ ❋✐♥❞ ❝r✐t❡r✐❛ t❤❛t r✉❧❡ ♦✉t ❣❡♦♠❡tr✐❝❛❧❧② ♥♦t r❡❛❧✐③❛❜❧❡ ❝♦♠❜✐♥❛t♦r✐❛❧

str✉❝t✉r❡s✳

◮ ❋✐♥❞ ❝r✐t❡r✐❛ t❤❛t ❛r❡ ♥❡❝❡ss❛r✐❧② s❛t✐s✜❡❞ ❜② ♦♣t✐♠❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✳ ◮ ❉❡✈❡❧♦♣ ❛ s♦❢t✇❛r❡ t❤❛t ❧✐sts ❛❧❧ t❤❡ r❡♠❛✐♥✐♥❣ ❝❛s❡s✳

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

❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣s ❜② ✸ ❉✐s❦s

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❇✳ ❙③❛❧❦❛✐

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

▲❡♠♠❛

• ✐✈❡♥ ❛ ❝✐r❝❧❡ C✱ ❛♥ ✐♥t❡rs❡❝t✐♥❣ str❛✐❣❤t ❧✐♥❡ e ❛♥❞ t✇♦ ♣♦✐♥ts P, Q ∈ e

♦♥ ❞✐✛❡r❡♥t s✐❞❡s ♦❢ C✱ ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❤❛❧❢ ♣❧❛♥❡s H ❜♦✉♥❞❡❞ ❜② e✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♣♦✐♥t R ∈ H s✉❝❤ t❤❛t a + b + c = 0✳

❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣s ❜② ✸ ❉✐s❦s

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❇✳ ❙③❛❧❦❛✐

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛ r❡q✉✐r❡s t♦ s❤♦✇ t❤❛t t✇♦ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡s ♦❢ ❞❡❣r❡❡ ❢♦✉r ❤❛✈❡ ❡①❛❝t❧② ♦♥❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥t ✐♥ H✳

❚❤❡♦r❡♠

❋♦r ❛♥② r3

p ≤ r ≤ r3 c✱ t❤❡ ♦♣✐♠❛❧ ❝♦✈❡r✐♥❣ ♦❢ ❛ ✉♥✐t ❞✐s❦ ✇✐t❤ t❤r❡❡

❝♦♥❣r✉❡♥t ❞✐s❦s ♦❢ r❛❞✐✉s r ✐s ❣✐✈❡♥ ❜② t❤❡ ✉♥✐q✉❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧ t②♣❡

❖♣t✐♠❛❧ ❈♦✈❡r✐♥❣s ❜② ✺ ❉✐s❦s

❈♦♠♣✉t❡r s❡❛r❝❤ ❢♦r ♦♣t✐♠❛❧ ❝♦✈❡r✐♥❣ ✇❛s ❞♦♥❡ ❜② ❚❛r♥❛✐✕●ás♣ár✕❍✐♥❝③✱ ✉s✐♥❣ ❛ ♠❡❝❤❛♥✐❝❛❧ ♠♦❞❡❧✳ ❚❤❡② ♦❜s❡r✈❡❞ t❤❛t

◮ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥❀ ◮ ❢♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ r ≥ r5 p✱ t❤❡ ✉♥✐q✉❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ ❝r✐t✐❝❛❧

❝♦♥✜❣✉r❛t✐♦♥ ✐s t❤❡ ❜❡st❀

◮ ❛s r ✐♥❝r❡❛s❡s ❛❜♦✈❡ ❛ ❝❡rt❛✐♥ ✈❛❧✉❡ r0✱ t✇♦ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥s

❣r♦✇ ♦✉t ♦❢ t❤❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ ♦♥❡✱ ❛♥ ❡❣❣ s❤❛♣❡❞ ❛♥❞ ❛ ♣✉♠♣❦✐♥ s❤❛♣❡❞✳

◗✉❡st✐♦♥

❍♦✇ ❝❛♥ ✇❡ ❝♦♠♣✉t❡ r0❄ ■♥ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥s s❤♦✉❧❞ ✇❡ ❞❡❢♦r♠ t❤❡ r♦t❛t✐♦♥❛❧❧② s②♠♠❡tr✐❝ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❛t r0 t♦ ♣✉s❤ t♦✇❛r❞ t❤❡ ♣✉♠♣❦✐♥ ♦r ❡❣❣ s❤❛♣❡❞ ❝r✐t✐❝❛❧ ❝♦♥✜❣✉r❛t✐♦♥s❄

◮ ❚❤❡ ❍❡ss✐❛♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✉s✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛❡ ♣r❡s❡♥t❡❞

❛❜♦✈❡✳

◮ ❋✐①✐♥❣ t❤❡ ❜✐❣ ❝✐r❝❧❡✱ t❤❡ ❍❡ss✐❛♥ ✐s ❞❡✜♥❡❞ ♦♥ ❛ ✶✵✲❞✐♠❡♥s✐♦♥❛❧

❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳

◮ ■♥✜♥✐t❡s✐♠❛❧ r♦t❛t✐♦♥s ❛❜♦✉t t❤❡ ♦r✐❣✐♥ ❛r❡ ✐♥ t❤❡ ❦❡r♥❡❧✳ ◮ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❍❡ss✐❛♥ ♦♥ t❤❡ ♦rt❤♦❣♦♥❛❧ ❝♦♠♣❧❡♠❡♥t✳ ◮ ❍❡r❡ ✐s ❛ ♥✉♠❡r✐❝ ♣❧♦t ♦❢ t❤❡ ❣r❛♣❤ ♦❢ ✐ts ❞❡t❡r♠✐♥❛♥t✿

0.50 0.55 0.60 200 150 100 50

0.50 0.55 0.60 200 150 100 50

◮ H ✐s ♥❡❣❛t✐✈❡ ❞❡✜♥✐t❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣✱ ❤❛s t✇♦ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s

✐♥ t❤❡ ✈❛❧❧❡②✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❞♦✉❜❧❡ r♦♦t❄ ♦r t✇♦ r♦♦ts ✈❡r② ❝❧♦s❡ t♦ ♦♥❡ ❛♥♦t❤❡r❄

❚❤❡♦r❡♠

❚❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ D5 ✐s ❛ s②♠♠❡tr② ❣r♦✉♣ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥✳ ❚❤✉s✱ ✐t ❛❝ts ♦♥ t❤❡ t❛♥❣❡♥ts s♣❛❝❡✱ ♦♥ ✇❤✐❝❤ t❤❡ ❍❡ss❡ ❢♦r♠ ✐s ❞❡✜♥❡❞✳ ❚❤❡ ❍❡ss❡ ❢♦r♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ D5✳ ❚❤✉s✱ t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs ♦❢ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❍❡ss❡ ♠❛♣✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ♦❜t❛✐♥ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❞✐❛❣♦♥❛❧✐③✐♥❣ t❤❡ ❍❡ss❡ ❢♦r♠✳ ❚❤❡ ❍❡ss❡ ♠❛♣ ❤❛s ❝♦♥s❛♥t ❡✐❣❡♥✈❛❧✉❡ ♦♥ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs✳

❊✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ❍❡ss❡ ♠❛♣

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦✦✦