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▲❛✇ ♦❢ t❤❡ ❏✉♥❣❧❡ ✐♥ ❛ ▲✐♥❡❛r ◆❡t✇♦r❦

▼❛t❤✐❡✉ ❋❡✉✐❧❧❡t

❘❆P✱ ■◆❘■❆ ❘♦❝q✉❡♥❝♦✉rt✱ ❋r❛♥❝❡

◆♦✈❡♠❜❡r ✷✵✱ ✷✵✵✾

❏♦✐♥t ✇♦r❦ ✇✐t❤ ❚❤♦♠❛s ❇♦♥❛❧❞ ✭❖r❛♥❣❡✮ ❛♥❞ ❆❧❡①❛♥❞r❡ Pr♦✉t✐èr❡ ✭▼❙❘✮

■♥tr♦❞✉❝t✐♦♥

◮ ❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

◮ ❆✈♦✐❞❛♥❝❡ ♦❢ ♣❛❝❦❡t ❧♦ss ◮ ❆❞❛♣t❛t✐♦♥ ♦❢ t❤❡ t❤r♦✉❣❤♣✉t ❛❝❝♦r❞✐♥❣ t♦ ♣❛❝❦❡t ❧♦ss

◆♦ ❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

◆♦ ❛✈♦✐❞❛♥❝❡ ♦❢ ♣❛❝❦❡t ❧♦ss ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡✿ t❤❡ ❛❝❝❡ss r❛t❡ ❘❡❝♦✈❡r② ♠❡❝❤❛♥✐s♠ ❢r♦♠ ♣❛❝❦❡t ❧♦ss ✭s♦✉r❝❡ ❝♦❞✐♥❣✮

■s t❤❛t s✉st❛✐♥❛❜❧❡❄ ■s t❤❡r❡ ❛♥② ❝♦♥❣❡st✐♦♥ ❝♦❧❧❛♣s❡❄

■♥tr♦❞✉❝t✐♦♥

◮ ❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

◮ ❆✈♦✐❞❛♥❝❡ ♦❢ ♣❛❝❦❡t ❧♦ss ◮ ❆❞❛♣t❛t✐♦♥ ♦❢ t❤❡ t❤r♦✉❣❤♣✉t ❛❝❝♦r❞✐♥❣ t♦ ♣❛❝❦❡t ❧♦ss

◮ ◆♦ ❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

◮ ◆♦ ❛✈♦✐❞❛♥❝❡ ♦❢ ♣❛❝❦❡t ❧♦ss ◮ ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡✿ t❤❡ ❛❝❝❡ss r❛t❡ ◮ ❘❡❝♦✈❡r② ♠❡❝❤❛♥✐s♠ ❢r♦♠ ♣❛❝❦❡t ❧♦ss ✭s♦✉r❝❡ ❝♦❞✐♥❣✮

■s t❤❛t s✉st❛✐♥❛❜❧❡❄ ■s t❤❡r❡ ❛♥② ❝♦♥❣❡st✐♦♥ ❝♦❧❧❛♣s❡❄

▲❛✇ ♦❢ t❤❡ ❏✉♥❣❧❡ ❙t❛❜✐❧✐t② ♦❢ ❛ ❧✐♥❡❛r ♥❡t✇♦r❦ ❚❤❡♦r❡♠ ♦❢ ❛♥❛r❝❤②

❋❧♦✇ ▼♦❞❡❧

❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧ ♠♦❞❡❧ ✐♥tr♦❞✉❝❡❞ ❜② ▼❛ss♦✉❧✐é ❛♥❞ ❘♦❜❡rts ✐♥ ✷✵✵✵✳

◮ ◆❡t✇♦r❦ ♠♦❞❡❧❡❞ ❛t ❛ ✢♦✇ ❧❡✈❡❧✱ ♥♦t ❛ ♣❛❝❦❡t ❧❡✈❡❧ ◮ ❋❧♦✇s ❛r❡ ❣♦✐♥❣ ✐♥ t❤❡ ♥❡t✇♦r❦ ❧✐❦❡ ❛ ✢✉✐❞ ◮ ❯s❡rs ❞✐✈✐❞❡❞ ✐♥ ❝❧❛ss❡s t♦ ♠♦❞❡❧ ❤❡t❡r♦❣❡♥❡✐t② ♦❢ t❤❡

tr❛✣❝

◮ ❉②♥❛♠✐❝ tr❛✣❝ ◮ ❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ❞❡t❡r♠✐♥❡❞ ❜② ❝♦♥❣❡st✐♦♥ ♣♦❧✐❝②

▲✐♥❡❛r ◆❡t✇♦r❦

✶ ✵ ✷

◮ ❚✇♦ ❧✐♥❦s ♦❢ ❝❛♣❛❝✐t② ✶ ◮ ✸ ❝❧❛ss❡s ♦❢ ✢♦✇s

◮ ❈❧❛ss ✵ ❣♦✐♥❣ t❤r♦✉❣❤ ❜♦t❤ ❧✐♥❦s ✇✐t❤ ❛❝❝❡ss r❛t❡ ✶ ◮ ❈❧❛ss ✶ ❣♦✐♥❣ t❤r♦✉❣❤ ❧✐♥❦ ✶ ✇✐t❤ ❛❝❝❡ss r❛t❡ ✶ ◮ ❈❧❛ss ✷ ❣♦✐♥❣ t❤r♦✉❣❤ ❧✐♥❦ ✷ ✇✐t❤ ❛❝❝❡ss r❛t❡ ❛

◮ ❋❧♦✇ ❣❡♥❡r❛t✐♦♥ r❛t❡s✿ λ✐ ✭✢♦✇s✴s❡❝✮ ◮ ❋❧♦✇ s✐③❡ r❛t❡s✿ µ✐ ✭❜✐ts−✶✮ ◮ ❚r❛✣❝ ✐♥t❡♥s✐t✐❡s✿ ρ✐ = λ✐/µ✐ ✭❜✐ts✴s❡❝✮

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

◮ ❇❛♥❞✇✐❞t❤ ❛❧❧♦❝❛t✐♦♥ ❞❡t❡r♠✐♥❡❞ ❜② ❈♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

❛❧❣♦r✐t❤♠

◮ ■♥♣✉t r❛t❡ ✐s ❡q✉❛❧ t♦ ♦✉t♣✉t r❛t❡ ◮ ❲❡❧❧ st✉❞✐❡❞ ❡①❛♠♣❧❡s✿ α✲❢❛✐r ❛❧❧♦❝❛t✐♦♥s ✭♣r♦♣♦rt✐♦♥❛❧

❢❛✐r♥❡ss✱ ♠❛①✲♠✐♥ ❢❛✐r♥❡ss✮ ✭▼♦ ❛♥❞ ❲❛❧r❛♥❞ ✷✵✵✵✮

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

◆♦ ❝♦♥❣❡st✐♦♥ ❝♦♥tr♦❧

◮ ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡ ✐♥ t❤❡ ♥❡t✇♦r❦ ◮ ❇❛♥❞✇✐❞t❤ ❛❧❧♦❝❛t✐♦♥ ❞❡t❡r♠✐♥❡❞ ❜② ❜✉✛❡r ♣♦❧✐❝② ✐♥ t❤❡

r♦✉t❡rs

◮ ❖✉t♣✉t r❛t❡ ✐s ❞✐✛❡r❡♥t ♦❢ ✐♥♣✉t r❛t❡ ◮ ❋❛✐r ♣♦❧✐❝②✿ ❋❛✐r ❉r♦♣♣✐♥❣ ✭❙❤❡♥❦❡r ❡t ❛❧ ✾✽✮ ◮ ❙✐♠♣❧❡st ♣♦❧✐❝②✿ ❚❛✐❧ ❉r♦♣♣✐♥❣ ✭▲❛✇ ♦❢ t❤❡ ❏✉♥❣❧❡✮

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

❚❛✐❧ ❞r♦♣♣✐♥❣✿

◮ ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡ ◮ ❆t ❡❛❝❤ ❧✐♥❦✱ ♦✉t♣✉t r❛t❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐♥♣✉t r❛t❡s

♥✶ ♥✵ ♥✷

✶ ♥✶ ♥✵ ♥✶ ♥✵ ♥✵ ♥✶ ✵

♠✐♥

♥✷❛ ✷

♠✐♥ ♥✷❛

♥✷❛ ♥✷❛

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

❚❛✐❧ ❞r♦♣♣✐♥❣✿

◮ ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡ ◮ ❆t ❡❛❝❤ ❧✐♥❦✱ ♦✉t♣✉t r❛t❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐♥♣✉t r❛t❡s

♥✶ ♥✵ ♥✷

✶ ♥✶ ♥✵ ♥✶ ♥✵ ♥✵ ♥✶ ✵

♠✐♥

♥✷❛ ✷

♠✐♥ ♥✷❛

♥✷❛ ♥✷❛

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

❚❛✐❧ ❞r♦♣♣✐♥❣✿

◮ ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡ ◮ ❆t ❡❛❝❤ ❧✐♥❦✱ ♦✉t♣✉t r❛t❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐♥♣✉t r❛t❡s

♥✶ ♥✵ ♥✷ φ✶ =

♥✶ ♥✵+♥✶

α =

♥✵ ♥✵+♥✶ ✵

♠✐♥

♥✷❛ ✷

♠✐♥ ♥✷❛

♥✷❛ ♥✷❛

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

❚❛✐❧ ❞r♦♣♣✐♥❣✿

◮ ❙♦✉r❝❡s s❡♥❞ ❛t t❤❡✐r ♠❛①✐♠✉♠ r❛t❡ ◮ ❆t ❡❛❝❤ ❧✐♥❦✱ ♦✉t♣✉t r❛t❡s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐♥♣✉t r❛t❡s

♥✶ ♥✵ ♥✷ φ✶ =

♥✶ ♥✵+♥✶

α =

♥✵ ♥✵+♥✶

φ✵ = ♠✐♥(α,

α ♥✷❛+α)

φ✷ = ♠✐♥(♥✷❛,

♥✷❛ α+♥✷❛)

▼❛r❦♦✈ Pr♦❝❡ss ✭❙✉♠♠❛r②✮

✶ ✵ ✷ ❲❡ st✉❞② t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss ❞❡s❝r✐❜✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✢♦✇s ✐♥ ❡❛❝❤ ❝❧❛ss✿ (◆✵(t), ◆✶(t), ◆✷(t)) ✇✐t❤ tr❛♥s✐t✐♦♥ r❛t❡s ✿ ♥ → ♥ + ❡✐ : λ✐ ♥ → ♥ − ❡✐ : µ✐φ✐(♥)

P❡r❢♦r♠❛♥❝❡ ♠❡tr✐❝s

◮ ▼❡❛♥ ✢♦✇ r❡s♣♦♥s❡ t✐♠❡

◮ ❊r❣♦❞✐❝✐t② ❝♦♥❞✐t✐♦♥

◮ ❖♣t✐♠❛❧ ♣❡r❢♦r♠❛♥❝❡

◮ ❚❤❡ ♥❡t✇♦r❦ ❝❛♥ ❜❡ st❛❜✐❧✐③❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢

∀❧

• ✐:❧∈r✐

ρ✐ < ❈❧

◮ α✲❢❛✐r ♣♦❧✐❝✐❡s ❛r❡ ♦♣t✐♠❛❧ ✭❇♦♥❛❧❞ ❛♥❞ ▼❛ss♦✉❧✐é

✷✵✵✶✱ ❞❡ ❱❡❝✐❛♥❛ ❡t ❛❧ ✷✵✵✶✮

◮ ❋❛✐r ❞r♦♣♣✐♥❣ ✐s ♦♣t✐♠❛❧ ✭❇♦♥❛❧❞ ❡t ❛❧ ✷✵✵✾✮ ◮ ❚❛✐❧ ❉r♦♣♣✐♥❣ ✐s ♥♦t ♦♣t✐♠❛❧

▲❛✇ ♦❢ t❤❡ ❏✉♥❣❧❡ ❙t❛❜✐❧✐t② ♦❢ ❛ ❧✐♥❡❛r ♥❡t✇♦r❦ ❚❤❡♦r❡♠ ♦❢ ❛♥❛r❝❤②

❙t❛❜✐❧✐t② ♦❢ ❝❧❛ss ✷

♥✷ φ✵ = ♠✐♥(α,

α ♥✷❛+α)

φ✷ = ♠✐♥(♥✷❛,

♥✷❛ α+♥✷❛)

φ✶ =

♥✶ ♥✵+♥✶

α =

♥✵ ♥✵+♥✶

♥✶ ♥✵ ♥✶ ♥✵

✶ ♥✶ ♥✵ ♥✶

❲❡ ❤❛✈❡ ✵ ≤ α ≤ ✶✳ ❚❤✉s✱ ❝❧❛ss ✷ ✐s st❛❜❧❡ ✐❢ ρ✷ < ✶ ❛♥❞ tr❛♥s✐❡♥t ✐❢ ρ✷ > ✶ ✇❤❛t❡✈❡r t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ ✵ ❛♥❞ ✶✳ ◆♦✇✱ ✇❡ s✉♣♣♦s❡ t❤❛t ρ✷ < ✶

❙t❛❜✐❧✐t② ♦❢ ❝❧❛ss ✷

♥✷ φ✵ = ♠✐♥(α,

α ♥✷❛+α)

φ✷ = ♠✐♥(♥✷❛,

♥✷❛ α+♥✷❛) ✶ ♥✶ ♥✵ ♥✶ ♥✵ ♥✵ ♥✶

♥✶ ♥✵ ♥✶ ♥✵ φ✶ =

♥✶ ♥✵+♥✶

α ❲❡ ❢r❡❡③❡ t❤❡ ♥✉♠❜❡r ♦❢ ✢♦✇s ✐♥ ❝❧❛ss❡s ✵ ❛♥❞ ✶ ❛♥❞ t❤❡♥ α✳ ❚❤❡r❡ ✐s ❛ st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ πα ❢♦r ❝❧❛ss ✷✳ ❲❡ t❤❡♥ ❞❡✜♥❡ t❤❡ ❛✈❡r❛❣❡❞ t❤r♦✉❣❤♣✉t ♦❢ ❝❧❛ss ✵✿ ¯ φ✵(α) =

• ♥✷∈N

πα(♥✷) ♠✐♥

• α,

α ♥✷❛ + α

❋❧✉✐❞ ❧✐♠✐t ♦❢ t❤❡ s②st❡♠

❈❧❛ss ✷ ✐s st❛❜❧❡✱ s♦ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ s❝❛❧❡ ✐t✦ ❲❡ ♣❡r❢♦r♠ t❤❡ s❝❛❧✐♥❣ ♦♥❧② ♦♥ ❝❧❛ss❡s ✵ ❛♥❞ ✶✿ (◆✵(✵), ◆✶(✵)) = ♥❦, ❧✐♠

❦→∞ ♥❦ = ∞,

❧✐♠

❦→∞

✶ ♥❦(◆✵(♥❦t), ◆✶(♥❦t))

❞

= (③✵(t), ③✶(t)) ❛♥❞ (③✵(t), ③✶(t)) s❛t✐s✜❡s✿ ˙ ③✵(t) = λ✵ − µ✵ ¯ φ✵

• ③✵(t)

③✵(t) + ③✶(t)

• ˙

③✶(t) = λ✶ − µ✶ ③✶(t) ③✵(t) + ③✶(t) ❚❤❡ ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ♦❢ ❍✉♥t ❛♥❞ ❑✉rt③ ✐♥ ✶✾✾✹ ❛❜♦✉t ▲♦ss ◆❡t✇♦r❦s✳

❋❧✉✐❞ ▲✐♠✐t ♦❢ t❤❡ s②st❡♠

˙ ③✵(t) = λ✵ − µ✵ ¯ φ✵

• ③✵(t)

③✵(t) + ③✶(t)

• ˙

③✶(t) = λ✶ − µ✶ ③✶(t) ③✵(t) + ③✶(t) ❚❤❡r❡ ✐s ❛ s❡♣❛r❛t✐♦♥ ♦❢ t✐♠❡ s❝❛❧❡ ❜❡t✇❡❡♥ t❤❡ ✢✉✐❞ ❧✐♠✐t (③✵, ③✶) ❛♥❞ ❝❧❛ss ✷✿ ❝❧❛ss ✷ ✐s ❛❧✇❛②s ❛t ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ t❤❡r❡ ✐s ❛♥ ❛✈❡r❛❣✐♥❣ ♦♥ ❝❧❛ss ✷ ❢♦r ✢✉✐❞ ❧✐♠✐t ③✵✳

❋❧✉✐❞ ▲✐♠✐t ❞②♥❛♠✐❝s

③✶ ③✵

③✵ ③✵+③✶ = ¯

φ−✶

✵ (ρ✵) ③✶ ③✵ ③✶ ✶

③✵(t) ③✵(t) ③✶ t ③✶ t ③✵ t ③✶ t ③✵ t ③✶ t ③✵ t ③✶ t

③✵ ③✵ ③✶ ✶ ✵ ✵ ③✶ ③✵ ③✶ ✶

③✵ t ③✶ t ˙ ③✵ = λ✵ − µ✵ ¯ φ✵

• ③✵

③✵ + ③✶

❋❧✉✐❞ ▲✐♠✐t ❞②♥❛♠✐❝s

③✶ ③✵

③✵ ③✵ ③✶ ✶ ✵ ✵ ③✶ ③✵+③✶ = ρ✶

③✵ t ③✵ t ③✶(t) ③✶(t) ③✵ t ③✶ t ③✵ t ③✶ t ③✵ t ③✶ t

③✵ ③✵ ③✶ ✶ ✵ ✵ ③✶ ③✵ ③✶ ✶

③✵ t ③✶ t ˙ ③✶ = λ✶ − µ✶ ③✶ ③✵ + ③✶

❋❧✉✐❞ ▲✐♠✐t ❞②♥❛♠✐❝s

③✶ ③✵

③✵ ③✵+③✶ = ¯

φ−✶

✵ (ρ✵) ③✶ ③✵+③✶ = ρ✶

③✵ t ③✵ t ③✶ t ③✶ t (③✵(t), ③✶(t)) (③✵(t), ③✶(t)) (③✵(t), ③✶(t))

③✵ ③✵ ③✶ ✶ ✵ ✵ ③✶ ③✵ ③✶ ✶

③✵ t ③✶ t ρ✵ < ¯ φ✵(✶ − ρ✶)

❋❧✉✐❞ ▲✐♠✐t ❞②♥❛♠✐❝s

③✶ ③✵

③✵ ③✵ ③✶ ✶ ✵ ✵ ③✶ ③✵ ③✶ ✶

③✵ t ③✵ t ③✶ t ③✶ t (③✵(t), ③✶(t)) (③✵(t), ③✶(t)) ③✵ t ③✶ t

③✵ ③✵+③✶ = ¯

φ−✶

✵ (ρ✵) ③✶ ③✵+③✶ = ρ✶

(③✵(t), ③✶(t)) ρ✵ > ¯ φ✵(✶ − ρ✶)

❙t❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥s

❚❤❡ ❡①❛❝t ❝♦♥❞✐t✐♦♥s ❢♦r st❛❜✐❧✐t② ✉♥❞❡r t❤❡ ▲❛✇ ♦❢ t❤❡ ❏✉♥❣❧❡ ❛r❡✿ ρ✶ < ✶, ρ✷ < ✶, ρ✵ < ¯ φ✵(✶ − ρ✶) ❚❤❡ ♦♣t✐♠❛❧ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ❛r❡✿ ρ✶ < ✶, ρ✷ < ✶, ρ✵ < ♠✐♥(✶ − ρ✶, ✶ − ρ✷) ❇✉t✿ ¯ φ✵(✶ − ρ✶) < ♠✐♥(✶ − ρ✷, ✶ − ρ✶) ❚❤❡ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ❛r❡ ♥♦t ♦♣t✐♠❛❧✦

▲❛✇ ♦❢ t❤❡ ❏✉♥❣❧❡ ❙t❛❜✐❧✐t② ♦❢ ❛ ❧✐♥❡❛r ♥❡t✇♦r❦ ❚❤❡♦r❡♠ ♦❢ ❛♥❛r❝❤②

❲❤❛t ✐s t❤❡ ❛♥❛r❝❤②❄

❚❤❡ ♣r✐❝❡ ♦❢ ❛♥❛r❝❤② q✉❛♥t✐✜❡s ❤♦✇ ❢❛r t❤❡ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ❛r❡ ❢r♦♠ ♦♣t✐♠❛❧ ♦♥❡s✳ ■♥ ♦✉r ❝❛s❡✱ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s s✐♠♣❧②✿ P(❛) = ♠❛①

ρ✶,ρ✷

• ♠✐♥(✶ − ρ✶, ✶ − ρ✷) − ¯

φ✵(✶ − ρ✶)

• ❚❤❡ ♣r✐❝❡ ♦❢ ❛♥❛r❝❤② ❞❡♣❡♥❞s ♦♥ ❛✱ t❤❡ ❛❝❝❡ss r❛t❡ ♦❢ ❝❧❛ss ✷✳

❚❤❡♦r❡♠ ♦❢ ❛♥❛r❝❤②

ρ✶ ¯ φ✵(✶ − ρ✶)

✶ ✶ ✶ − ρ✷ ❛ = ✶ ❛ = ✵.✶ ❛ = ✵.✵✶ ❛ → ✵ : ♠✐♥(✶ − ρ✶, ✶ − ρ✷)

❚❤❡♦r❡♠✿ ❧✐♠

❛→✵ P(❛) = ✵

❋♦r ❛♥② (ρ✵, ρ✶, ρ✷) s❛t✐s❢②✐♥❣ t❤❡ ♦♣t✐♠❛❧ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s✱ t❤❡r❡ ❡①✐sts ❛ s♠❛❧❧ ❡♥♦✉❣❤ s✉❝❤ t❤❛t t❤❡ ♥❡t✇♦r❦ ✐s st❛❜❧❡✳

❈♦♥❝❧✉s✐♦♥

◮ ❉♦ ✇❡ ♥❡❡❞ ❝♦♥❣❡st✐♦♥ ❝♦♥tr♦❧ ✐♥ t❤❡ ■♥t❡r♥❡t❄ ◮ ❊✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ✐♠♣❛❝t ♦❢ ❜✐❣ ❝❧✐❡♥ts ♦♥ t❤❡ ♥❡t✇♦r❦

✭♦♣t✐❝❛❧ ♥❡t✇♦r❦s✮✳

◮ ❆ ✢✉✐❞ ❧✐♠✐t ✇✐t❤ ❛♥ ✐♥t❡r❡st✐♥❣ ❛✈❡r❛❣✐♥❣ ♣❤❡♥♦♠❡♥♦♥ ◮ ❈❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❧✐♥❡❛r ♥❡t✇♦r❦s ✇✐t❤ ♠♦r❡ t❤❛♥ t✇♦

❧✐♥❦s

◮ ■♥ ♦t❤❡r ❝♦♥t❡①ts❄ ◮ ❈♦♥❥❡❝t✉r❡✿ ❚❤❡ t❤❡♦r❡♠ ♦❢ ❛♥❛r❝❤② ✐s tr✉❡ ✐♥ ❛❝②❝❧✐❝

♥❡t✇♦r❦s✳