t srs - - PowerPoint PPT Presentation

❈♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣s ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥ ♠♦♥♦✐❞s ✇✐t❤ ❝❛♥❝❡❧❧❛t✐♦♥

▼❛r❣❛r✐❞❛ ❘❛♣♦s♦

❯♥✐✈❡rs✐t② ♦❢ t❤❡ ❆③♦r❡s ❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❙t❛t✐st✐❝s ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆✳ P❛✉❧❛ ●❛rrã♦✱ ◆✳ ▼❛rt✐♥s✲❋❡rr❡✐r❛ ❛♥❞ ▼✳ ❙♦❜r❛❧✮

❈❚ ✷✵✶✽

❈❛t❡❣♦r② ❚❤❡♦r② ✷✵✶✽ ❏✉❧② ✵✽✲✶✹✱ P♦♥t❛ ❉❡❧❣❛❞❛✱ ❆③♦r❡s

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶ ✴ ✷✹

❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s

❆ ❝♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣ ✐s ❛ s❡♠✐❣r♦✉♣ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

✭✶✮ ✭✷✮ ✭✸✮

❊①❛♠♣❧❡s ❆♥② ❣r♦✉♣ ✇✐t❤

✶✳

❆♥② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ✇✐t❤ ✳ ✵ ✶ ✇✐t❤ q✉❛t❡r♥✐♦♥ ♣r♦❞✉❝t ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷ ✴ ✷✹

❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s

❆ ❝♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣ (S, +, ()) ✐s ❛ s❡♠✐❣r♦✉♣ (S, +) ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ () : S → S s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

✭✶✮

x + x = x + x

✭✷✮

x + y + y = y + y + x

✭✸✮

(x + y) = y + x ❊①❛♠♣❧❡s ❆♥② ❣r♦✉♣ ✇✐t❤

✶✳

❆♥② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ✇✐t❤ ✳ ✵ ✶ ✇✐t❤ q✉❛t❡r♥✐♦♥ ♣r♦❞✉❝t ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷ ✴ ✷✹

❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s

❆ ❝♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣ (S, +, ()) ✐s ❛ s❡♠✐❣r♦✉♣ (S, +) ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ () : S → S s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

✭✶✮

x + x = x + x

✭✷✮

x + y + y = y + y + x

✭✸✮

(x + y) = y + x ❊①❛♠♣❧❡s ❆♥② ❣r♦✉♣ ✇✐t❤ x = x−✶✳ ❆♥② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ✇✐t❤ x = e✳ ✵ ✶ ✇✐t❤ q✉❛t❡r♥✐♦♥ ♣r♦❞✉❝t ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷ ✴ ✷✹

❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s

❆ ❝♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣ (S, +, ()) ✐s ❛ s❡♠✐❣r♦✉♣ (S, +) ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ () : S → S s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

✭✶✮

x + x = x + x

✭✷✮

x + y + y = y + y + x

✭✸✮

(x + y) = y + x ❊①❛♠♣❧❡s ❆♥② ❣r♦✉♣ ✇✐t❤ x = x−✶✳ ❆♥② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ ✇✐t❤ x = e✳ S = {q ∈ H|✵ < q < ✶} ✇✐t❤ q✉❛t❡r♥✐♦♥ ♣r♦❞✉❝t ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷ ✴ ✷✹

❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s

❆ ❝♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣ (S, +, ()) ✐s ❛ s❡♠✐❣r♦✉♣ (S, +) ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ () : S → S s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s✿

✭✶✮

x + x = x + x

✭✷✮

x + y + y = y + y + x

✭✸✮

(x + y) = y + x ❚❤❡ q✉❛s✐✈❛r✐❡t② S ♦❢ ❝♦♥❥✉❣❛t✐♦♥ s❡♠✐❣r♦✉♣s ✇✐t❤ ❝❛♥❝❡❧❧❛t✐♦♥ ✐s ❛ ✇❡❛❦❧② ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✸ ✴ ✷✹

❲❡❛❦❧② ▼❛❧✬ts❡✈ ❈❛t❡❣♦r②

❆ ✜♥✐t❡❧② ❝♦♠♣❧❡t❡ ❝❛t❡❣♦r② ✐s ✇❡❛❦❧② ▼❛❧✬ts❡✈ ✐❢ ❢♦r ❛❧❧ ♣✉❧❧❜❛❝❦s ♦❢ s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s ❛❧♦♥❣ s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s

✶ ✷ ✷ ✶

t❤❡ ♣❛✐r

✶ ✷ ✱ ✇✐t❤ ✶

✶ ❛♥❞

✷

✶ ✱ ✐s ❥♦✐♥t❧② ❡♣✐♠♦r♣❤✐❝✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✹ ✴ ✷✹

❲❡❛❦❧② ▼❛❧✬ts❡✈ ❈❛t❡❣♦r②

❆ ✜♥✐t❡❧② ❝♦♠♣❧❡t❡ ❝❛t❡❣♦r② ✐s ✇❡❛❦❧② ▼❛❧✬ts❡✈ ✐❢ ❢♦r ❛❧❧ ♣✉❧❧❜❛❝❦s ♦❢ s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s ❛❧♦♥❣ s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s A ×B C

π✶

• π✷

C

g

• e✷
• A

e✶

• f

B

s

• r
• t❤❡ ♣❛✐r (e✶, e✷)✱ ✇✐t❤ e✶ =< ✶A, sf > ❛♥❞ e✷ =< rg, ✶C >✱ ✐s ❥♦✐♥t❧②

❡♣✐♠♦r♣❤✐❝✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✹ ✴ ✷✹

❲❡❛❦❧② ▼❛❧✬ts❡✈ ❈❛t❡❣♦r②

❊①❛♠♣❧❡s ♦❢ ✇❡❛❦❧② ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s ❛r❡ ✲ ❉▲❛t✱ ♣r♦♣❡rt② ❝❤❛r❛❝t❡r✐③✐♥❣ ✐t ❛♠♦♥❣st t❤❡ ✈❛r✐❡t✐❡s ♦❢ ❧❛tt✐❝❡s ✲ q✉❛s✐✈❛r✐❡t✐❡s ♦❢ ❛❧❣❡❜r❛s ✇✐t❤ ❛ t❡r♥❛r② ♦♣❡r❛t✐♦♥ p(x, y, z) s❛t✐s❢②✐♥❣ p(x, y, y) = p(y, y, x) ❛♥❞ p(x, y, y) = p(x′, y, y) ⇒ x = x′✳ ■♥ ✇❡ ❤❛✈❡

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✺ ✴ ✷✹

❲❡❛❦❧② ▼❛❧✬ts❡✈ ❈❛t❡❣♦r②

❊①❛♠♣❧❡s ♦❢ ✇❡❛❦❧② ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s ❛r❡ ✲ ❉▲❛t✱ ♣r♦♣❡rt② ❝❤❛r❛❝t❡r✐③✐♥❣ ✐t ❛♠♦♥❣st t❤❡ ✈❛r✐❡t✐❡s ♦❢ ❧❛tt✐❝❡s ✲ q✉❛s✐✈❛r✐❡t✐❡s ♦❢ ❛❧❣❡❜r❛s ✇✐t❤ ❛ t❡r♥❛r② ♦♣❡r❛t✐♦♥ p(x, y, z) s❛t✐s❢②✐♥❣ p(x, y, y) = p(y, y, x) ❛♥❞ p(x, y, y) = p(x′, y, y) ⇒ x = x′✳ ■♥ S ✇❡ ❤❛✈❡ p(x, y, z) = x + y + z

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✺ ✴ ✷✹

❆❞♠✐ss✐❜✐❧✐t② ❞✐❛❣r❛♠s

❆♥ ❛❞♠✐ss✐❜✐❧✐t② ❞✐❛❣r❛♠ A

f

• α
• B

r

• s
• β
• C

g

• γ
• D

✭✶✮

fr = ✶B = gs✱ αr = β = γs

❣✐✈❡s r✐s❡ t♦ C

e✷

• g
• γ
• A ×B C

π✷

• π✶
• B

r

• s
• β

D

A

f

• e✶
• α
• ❚❤❡ tr✐♣❧❡ (α, β, γ) ✐s ❛❞♠✐ss✐❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ (f , r, g, s) ✐❢ t❤❡r❡ ❡①✐sts ❛

✉♥✐q✉❡ ♠♦r♣❤✐s♠ ϕ: A ×B C → D s✉❝❤ t❤❛t ϕe✶ = α ❛♥❞ ϕe✷ = γ✳ ❚❤❡♥ ✇❡ s❛② t❤❛t t❤❡ ❞✐❛❣r❛♠ (✶) ✐s ❛❞♠✐ss✐❜❧❡✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✻ ✴ ✷✹

❆❞♠✐ss✐❜✐❧✐t② ✐♥ S

❚❤❡♦r❡♠✿

❆ ❞✐❛❣r❛♠ ✐♥ S

A

f

• α
• B

r

• s
• β
• C

g

• γ
• D

fr = ✶B = gs✱ αr = β = γs, ✐s ❛❞♠✐ss✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (Ad✶) t❤❡ ❡q✉❛t✐♦♥ x + β(b) + β(b) = α(a) + β(b) + γ(c) ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❢♦r ❛❧❧ a ∈ A ❛♥❞ c ∈ C s✉❝❤ t❤❛t f (a) = g(c) = b ∈ B✳ (Ad✷) t❤❡ ❡q✉❛t✐♦♥ α(a✶+a✷)+β(b✶ + b✷)+γ(c✶+c✷) = α(a✶)+β(b✶)+γ(c✶)+α(a✷)+β(b✷)+γ(c✷) ✐s s❛t✐s✜❡❞ ❢♦r a✶, a✷ ∈ A ❛♥❞ c✶, c✷ ∈ C s✉❝❤ t❤❛t f (a✶) = g(c✶) = b✶ ∈ B ❛♥❞ f (a✷) = g(c✷) = b✷ ∈ B✳

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

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✼ ✴ ✷✹

❆❞♠✐ss✐❜✐❧✐t② ✐♥ S

❚❤❡♦r❡♠✿

❆ ❞✐❛❣r❛♠ ✐♥ S

A

f

• α
• B

r

• s
• β
• C

g

• γ
• D

fr = ✶B = gs✱ αr = β = γs, ✐s ❛❞♠✐ss✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (Ad✶) t❤❡ ❡q✉❛t✐♦♥ x + β(b) + β(b) = α(a) + β(b) + γ(c) ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❢♦r ❛❧❧ a ∈ A ❛♥❞ c ∈ C s✉❝❤ t❤❛t f (a) = g(c) = b ∈ B✳ (Ad✷) t❤❡ ❡q✉❛t✐♦♥ α(a✶+a✷)+β(b✶ + b✷)+γ(c✶+c✷) = α(a✶)+β(b✶)+γ(c✶)+α(a✷)+β(b✷)+γ(c✷) ✐s s❛t✐s✜❡❞ ❢♦r a✶, a✷ ∈ A ❛♥❞ c✶, c✷ ∈ C s✉❝❤ t❤❛t f (a✶) = g(c✶) = b✶ ∈ B ❛♥❞ f (a✷) = g(c✷) = b✷ ∈ B✳

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

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✼ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

❊①✐st❡♥❝❡ ♦❢ ❛ ♠❛♣ ϕ : A ×B C → D ✇✐t❤ ϕe✶ = α ❛♥❞ ϕe✷ = γ ✐♠♣❧✐❡s t❤❛t✱ ❢♦r f (a) = g(c) = b✱ α(a) = ϕ(a, s(b))✱ γ(c) = ϕ(r(b), c)✱ β(b) = ϕ(r(b), s(b))✳ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛♥❞

✷ ✐s ❢✉❧✜❧❧❡❞✳

■❢

✶ ❛♥❞ ✷ ❤♦❧❞✱ t❛❦✐♥❣

t❤❡ s♦❧✉t✐♦♥ ♦❢

✶ t❤❡♥ ✶

❛♥❞

✷

❛♥❞

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✽ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

❊①✐st❡♥❝❡ ♦❢ ❛ ♠❛♣ ϕ : A ×B C → D ✇✐t❤ ϕe✶ = α ❛♥❞ ϕe✷ = γ ✐♠♣❧✐❡s t❤❛t✱ ❢♦r f (a) = g(c) = b✱ α(a) = ϕ(a, s(b))✱ γ(c) = ϕ(r(b), c)✱ β(b) = ϕ(r(b), s(b))✳ ϕ ∈ S ⇒ ϕ(a, c) ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ x + β(b) + β(b) = α(a) + β(b) + γ(c) ❛♥❞ (Ad✷) ✐s ❢✉❧✜❧❧❡❞✳ ■❢

✶ ❛♥❞ ✷ ❤♦❧❞✱ t❛❦✐♥❣

t❤❡ s♦❧✉t✐♦♥ ♦❢

✶ t❤❡♥ ✶

❛♥❞

✷

❛♥❞

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✽ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

❊①✐st❡♥❝❡ ♦❢ ❛ ♠❛♣ ϕ : A ×B C → D ✇✐t❤ ϕe✶ = α ❛♥❞ ϕe✷ = γ ✐♠♣❧✐❡s t❤❛t✱ ❢♦r f (a) = g(c) = b✱ α(a) = ϕ(a, s(b))✱ γ(c) = ϕ(r(b), c)✱ β(b) = ϕ(r(b), s(b))✳ ϕ ∈ S ⇒ ϕ(a, c) ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ x + β(b) + β(b) = α(a) + β(b) + γ(c) ❛♥❞ (Ad✷) ✐s ❢✉❧✜❧❧❡❞✳ ■❢ (Ad✶) ❛♥❞ (Ad✷) ❤♦❧❞✱ t❛❦✐♥❣ ϕ(a, c) t❤❡ s♦❧✉t✐♦♥ ♦❢ (Ad✶) t❤❡♥ ϕe✶ = α ❛♥❞ ϕe✷ = γ ❛♥❞ ϕ ∈ S.

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✽ ✴ ✷✹

❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s ♦❢ ♠♦♥♦✐❞s

■♥ ✇✐t❤ ✶ ❛♥❞ ✐s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ✐❢ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s❡t✲t❤❡♦r✐❝❛❧ ♠❛♣ ✱ ❝❛❧❧❡❞ t❤❡ ❙❝❤r❡✐❡r r❡tr❛❝t✐♦♥✱ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✳ ❚♦ ❝♦rr❡s♣♦♥❞s ❛♥ ❛❝t✐♦♥ ♦❢ ♦♥ ✱ ❈♦♥✈❡rs❡❧② t♦ ❡❛❝❤ ❛❝t✐♦♥ ✐t ❝♦rr❡s♣♦♥❞s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ✈✐❛ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✾ ✴ ✷✹

❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s ♦❢ ♠♦♥♦✐❞s

■♥ Mon X

k

A

f

B

r

• ✇✐t❤ fr = ✶B ❛♥❞ X = kerf

✐s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ✐❢ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s❡t✲t❤❡♦r✐❝❛❧ ♠❛♣ q : A → X✱ ❝❛❧❧❡❞ t❤❡ ❙❝❤r❡✐❡r r❡tr❛❝t✐♦♥✱ s✉❝❤ t❤❛t a = kq(a) + rf (a) ❢♦r ❛❧❧ a ∈ A✳ ❚♦ ❝♦rr❡s♣♦♥❞s ❛♥ ❛❝t✐♦♥ ♦❢ ♦♥ ✱ ❈♦♥✈❡rs❡❧② t♦ ❡❛❝❤ ❛❝t✐♦♥ ✐t ❝♦rr❡s♣♦♥❞s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ✈✐❛ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✾ ✴ ✷✹

❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s ♦❢ ♠♦♥♦✐❞s

■♥ Mon X

k

A

f

B

r

• ✇✐t❤ fr = ✶B ❛♥❞ X = kerf

✐s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ✐❢ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s❡t✲t❤❡♦r✐❝❛❧ ♠❛♣ q : A → X✱ ❝❛❧❧❡❞ t❤❡ ❙❝❤r❡✐❡r r❡tr❛❝t✐♦♥✱ s✉❝❤ t❤❛t a = kq(a) + rf (a) ❢♦r ❛❧❧ a ∈ A✳ ❚♦ X

k

A

q

• f

B

r

• ❝♦rr❡s♣♦♥❞s ❛♥ ❛❝t✐♦♥ ♦❢ B ♦♥ X✱ ϕ : B → End(X)

b · x := ϕ(b)(x) = q(r(b) + k(x)) ❈♦♥✈❡rs❡❧② t♦ ❡❛❝❤ ❛❝t✐♦♥ ϕ : B → End(X) ✐t ❝♦rr❡s♣♦♥❞s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ✈✐❛ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✾ ✴ ✷✹

❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ✐♥ M

• ✐✈❡♥ ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ✐♥

✇❡ ❤❛✈❡✿

✭❛✮

✶ ❀

✭❜✮

✵❀

✭❝✮

✵ ✵❀

✭❞✮

❀

✭❡✮

❀

✭❢✮ ▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✵ ✴ ✷✹

❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ✐♥ M

• ✐✈❡♥ ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ✐♥ M

X

k

A

q

• f

B

r

• ✇❡ ❤❛✈❡✿

✭❛✮

qk = ✶X❀

✭❜✮

qr = ✵❀

✭❝✮

q(✵) = ✵❀

✭❞✮

k(b · x) + r(b) = r(b) + k(x)❀

✭❡✮

q(a + a′) = q(a) + q(rf (a) + q(a′))❀

✭❢✮

q(a) = f (a) · q(a).

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✵ ✴ ✷✹

■♥❞✉❝✐♥❣ ✐♥t❡r♥❛❧ str✉❝t✉r❡s

• ✐✈❡♥ h : X → B ❛♥❞ ❛ ❙❝❤r❡✐❡r s♣❧✐ ❡♣✐♠♦r♣❤✐s♠ ✐♥ M

X

k

• h
• A

q

• f

B

r

• ✇❤❡♥ ❞♦❡s h ✐♥❞✉❝❡✿

❛ r❡✢❡①✐✈❡ ❣r❛♣❤✱ ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r②✱ ❛♥ ✐♥t❡r♥❛❧ ❣r♦✉♣♦✐❞❄

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✶ ✴ ✷✹

■♥❞✉❝✐♥❣ r❡✢❡①✐✈❡ ❣r❛♣❤s

Pr♦♣♦s✐t✐♦♥

• ✐✈❡♥ ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ❛♥❞ ❛ ♠♦r♣❤✐s♠ h ✐♥ M

X

k

• h
• A

q

• f

B

r

• ,

h ✐♥❞✉❝❡s ❛ r❡✢❡①✐✈❡ ❣r❛♣❤ A

˜ h

• f

B

r

• ✱

✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥ (C✶) h(b · x) + b = b + h(x)

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✷ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

■❢ t❤❡r❡ ❡①✐sts ❛ ♠❛♣ ˜ h✱ ♣r❡s❡r✈✐♥❣ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❝❤ t❤❛t ˜ hk = h ❛♥❞ ˜ hr = ✶B✱ t❤❡♥ ˜ h(a) = ˜ h(kq(a) + rf (a)) = hq(a) + f (a), ❢r♦♠ ✇❤✐❝❤ ✐t ❢♦❧❧♦✇s t❤❛t ˜ h(a) = f (a) + hq(a) ❛♥❞ s♦ ˜ h(a) = ˜ h(a). ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦

✶ ✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✸ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

■❢ t❤❡r❡ ❡①✐sts ❛ ♠❛♣ ˜ h✱ ♣r❡s❡r✈✐♥❣ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❝❤ t❤❛t ˜ hk = h ❛♥❞ ˜ hr = ✶B✱ t❤❡♥ ˜ h(a) = ˜ h(kq(a) + rf (a)) = hq(a) + f (a), ❢r♦♠ ✇❤✐❝❤ ✐t ❢♦❧❧♦✇s t❤❛t ˜ h(a) = f (a) + hq(a) ❛♥❞ s♦ ˜ h(a) = ˜ h(a). ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ˜ h ✐s ❡q✉✐✈❛❧❡♥t t♦ (C✶)✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✸ ✴ ✷✹

■♥❞✉❝✐♥❣ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r✐❡s

Pr♦♣♦s✐t✐♦♥

• ✐✈❡♥ ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ❛♥❞ ❛ ♠♦r♣❤✐s♠ h ✐♥ M

X

k

• h
• A

q

• f

B

r

• h ✐♥❞✉❝❡s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r②

A ×B A

m

A

˜ h

• f

B

r

• ✐❢ ❛♥❞ ♦♥❧② ✐❢

(C✶) h(b · x) + b = b + h(x), ∀x ∈ X, ∀b ∈ B (C✷) h(y) · x + y = y + x, ∀x, y ∈ X✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✹ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

❚❤❡ r❡✢❡①✐✈❡ ❣r❛♣❤ A

˜ h

• f

B

r

• ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡

❞✐❛❣r❛♠

A

f

B

r

• r
• r
• A

˜ h

• A

✐s ❛❞♠✐ss✐❜❧❡✳ ❚❤❡♥ ✐❢

✷ ❤♦❧❞s✱ s✉❝❤ ❛♥

❞❡✜♥✐♥❣ ❛ ❙❝❤r❡✐❡r ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ❡①✐sts✱ ❛♥❞ ✐s ❞❡✜♥❡❞ ❜② ❆♥❞

✷ ✐s ❛❧s♦ ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✺ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

❚❤❡ r❡✢❡①✐✈❡ ❣r❛♣❤ A

˜ h

• f

B

r

• ✐s ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡

❞✐❛❣r❛♠

A

f

B

r

• r
• r
• A

˜ h

• A

✐s ❛❞♠✐ss✐❜❧❡✳ ❚❤❡♥ ✐❢ (C✷) ❤♦❧❞s✱ s✉❝❤ ❛♥ m : A ×B A → A ❞❡✜♥✐♥❣ ❛ ❙❝❤r❡✐❡r ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② A ×B A

m

A

˜ h

• f

B

r

• ❡①✐sts✱ ❛♥❞ ✐s ❞❡✜♥❡❞ ❜②

m(a, a′) = kq(a) + a′ ❆♥❞ (C✷) ✐s ❛❧s♦ ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✺ ✴ ✷✹

■♥❞✉❝✐♥❣ ✐♥t❡r♥❛❧ ❣r♦✉♣♦✐❞s

Pr♦♣♦s✐t✐♦♥

• ✐✈❡♥ ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠ ❛♥❞ ❛ ♠♦r♣❤✐s♠ h ✐♥ M

X

k

• h
• A

q

• f

B

r

• ,

h ✐♥❞✉❝❡s ❛♥ ✐♥t❡r♥❛❧ ❣r♦✉♣♦✐❞

A ×B A

m A t

• f
• ˜

h

B

r

• ✐❢ ❛♥❞ ♦♥❧② ✐❢

(C✶) h(b · x) + b = b + h(x), ∀x ∈ X, ∀b ∈ B (C✷) h(y) · x + y = y + x, ∀x, y ∈ X✳ (C✸) X ✐s ❛ ❣r♦✉♣ ❛♥❞ −x = (−x)

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✻ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿

❚❤❡ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② A ×B A

m

A

˜ h

• f

B

r

• ✐s ❛♥ ✐♥t❡r♥❛❧ ❣r♦✉♣♦✐❞ ✇✐t❤ t❤❡ ✐♥✈❡rs❡s ❞❡✜♥❡❞ ♦♥ t❤❡ ✧♦❜❥❡❝t ♦❢

♠♦r♣❤✐s♠✧ A ❜② t(a) = −kq(a) + r˜ h(a) ❡①❛❝t❧② ✇❤❡♥ (C✸) ✐s s❛t✐s✜❡❞✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✼ ✴ ✷✹

❊①❛♠♣❧❡

B = {q ∈ H : q = ✶} X = {q ∈ H : ✵ < q ≤ ✶} b · x = bxb−✶ = bxb X

<✶,✵>

X ×ϕ B

π✶

• π✷

B

<✵,✶>

• ✇✐t❤ (x, b) = (b · x, b) ✐s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ ✐♥ M✳
• ✐✈❡♥ h : X → B✱ s✉❝❤ t❤❛t h(x) =

x x✱ h s❛t✐s✜❡s (C✶) ✭❛♥❞ s♦ ✐t ✐♥❞✉❝❡s

❛ r❡✢❡①✐✈❡ ❣r❛♣❤✮ ❜✉t ♥♦t (C✷) ✭❞♦❡s ♥♦t ✐♥❞✉❝❡ ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r②✱ ✐♥ ❣❡♥❡r❛❧✮✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✽ ✴ ✷✹

✧❙♠✐t❤ ✐s ❍✉q✧

❚❤❡♦r❡♠

■♥ t❤❡ ❝❛t❡❣♦r② M ♦❢ ❝♦♥❥✉❣❛t✐♦♥ ♠♦♥♦✐❞s ✇✐t❤ ❝❛♥❝❡❧❧❛t✐♦♥✱ t✇♦ ❙❝❤r❡✐❡r ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s R ❛♥❞ S ♦♥ t❤❡ s❛♠❡ ♦❜❥❡❝t X ❝♦♠♠✉t❡ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❙♠✐t❤✲P❡❞✐❝❝❤✐♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡✐r ♥♦r♠❛❧✐③❛t✐♦♥s ❝♦♠♠✉t❡ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❍✉q✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✶✾ ✴ ✷✹

✧❙♠✐t❤ ✐s ❍✉q✧

• ✐✈❡♥ t✇♦ ❙❝❤r❡✐❡r ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s (R, r✶, r✷) ❛♥❞ (S, s✶, s✷) ♦♥ X

Y

l

• R ×X S

p✶

• p✷

S

e✷

• s✶
• qg
• s✷
• X

k

R

e✶

• r✷
• qf
• r✶
• X

iR

• iS
• idX
• X,

r✶k, s✷l ❝♦♠♠✉t❡ ✐♥ ❍✉q s❡♥s❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ∃ϕ : R ×X S → X s✉❝❤ t❤❛t ϕe✶ = r✶ ❛♥❞ ϕe✷ = s✷✱ ❛♥❞ t❤✐s ♠❡❛♥s t❤❛t R ❛♥❞ S ❝♦♠♠✉t❡✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷✵ ✴ ✷✹

❋r♦♠ ❧♦❝❛❧ t♦ ❣❧♦❜❛❧

❚❤❡ ❞✐❛❣r❛♠ Y

l

• A ×B C

p✶

• p✷

C

e✷

• g
• qg
• γ
• X

k

A

e✶

• f
• qf
• α
• B

r

• s
• β
• D,

✐s ❛❞♠✐ss✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ αk ❛♥❞ γl ❍✉q✲❝♦♠♠✉t❡✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷✶ ✴ ✷✹

❋r♦♠ ❧♦❝❛❧ t♦ ❣❧♦❜❛❧

■❢ ❥✉st (f , r) ✐s ❛ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐ t❤❡♥ t❤❡ ❞✐❛❣r❛♠ A ×B C

p✶

• p✷

C

e✷

• g
• γ
• X

k

A

e✶

• f
• qf
• α
• B

r

• s
• β
• D,

✐s ❛❞♠✐ss✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ αk(qf (c) · x) + γ(c) = γ(c) + αk(x) ❢♦r ❛❧❧ x ∈ X ❛♥❞ c ∈ C✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷✷ ✴ ✷✹

❋r♦♠ ❧♦❝❛❧ t♦ ❣❧♦❜❛❧

■❢ C = A ❛♥❞ s = r✱ t❤❛t ✐s ✐❢ ✇❡ ❤❛✈❡ ❛ r❡✢❡①✐✈❡ ❣r❛♣❤ ✐♥❞✉❝❡❞ ❜② h = gk✱ t❤❡♥ t❤❡ ❞✐❛❣r❛♠ A ×B A

p✶

• p✷

A

e✷

• g
• γ
• X

k

A

e✶

• f
• qf
• α
• B

r

• r
• β
• D,

✐s ❛❞♠✐ss✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ αk(h(y) · x) + γk(y) = γk(y) + αk(x)✱ ❢♦r ❛❧❧ x, y ∈ X✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷✸ ✴ ✷✹

❘❡❢❡r❡♥❝❡s

❬✶❪ ❉✳❇♦✉r♥✱ ◆✳ ▼❛rt✐♥s✲❋❡rr❡✐r❛✱ ❆✳ ▼♦♥t♦❧✐ ❛♥❞ ▼✳ ❙♦❜r❛❧✱ ❙❝❤r❡✐❡r s♣❧✐t ❡♣✐♠♦r♣❤✐s♠s ✐♥ ♠♦♥♦✐❞s ❛♥❞ ✐♥ s❡♠✐r✐♥❣s ✱ ❚❡①t♦s ❞❡ ▼❛t❡♠át✐❝❛ ✭sér✐❡ ❇✮✱ ✈♦❧✳ ✹✺✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❈♦✐♠❜r❛ ✭✷✵✶✸✮✳ ❬✷❪ ◆✳ ▼❛rt✐♥s✲❋❡rr❡✐r❛✱ ❲❡❛❦❧② ▼❛❧✬❝❡✈ ❝❛t❡❣♦r✐❡s✱ ❚❤❡♦r② ❆♣♣❧✳ ❈❛t❡❣✳ ✷✶ ✭✷✵✵✽✮ ✾✶✕✶✶✼✳ ❬✸❪ ◆✳ ▼❛rt✐♥s✲❋❡rr❡✐r❛ ❛♥❞ ❆✳ ▼♦♥t♦❧✐✱ ❖♥ t❤❡ ✏❙♠✐t❤ ✐s ❍✉q✧ ❝♦♥❞✐t✐♦♥ ✐♥ ❙✲♣r♦t♦♠♦❞✉❧❛r ❝❛t❡❣♦r✐❡s✱ ❆♣♣❧✳ ❈❛t❡❣✳ ❙tr✉❝t✳ ✷✺✭✷✵✶✼✮ ✺✾✕✼✺✳ ❬✹❪ ◆✳ ▼❛rt✐♥s✲❋❡rr❡✐r❛ ❛♥❞ ❚✳ ❱❛♥ ❞❡r ▲✐♥❞❡♥✱ ❆ ♥♦t❡ ♦♥ t❤❡ ✧❙♠✐t❤ ✐s ❍✉q✧ ❝♦♥❞✐t✐♦♥✱ ❆♣♣❧✳ ❈❛t❡❣✳ ❙tr✉❝t✉r❡s ✷✵ ✭✷✵✶✷✮ ✶✼✺✕✶✽✼✳

▼✳❘❛♣♦s♦ ✭❋❈❚✲❯❆❝✮ ❈♦♥❥✉❣❛t✐♦♥ ❙❡♠✐❣r♦✉♣s ❏✉❧② ✵✽✲✶✹✱ ✷✵✶✽ ✷✹ ✴ ✷✹