Hàm dạng I với nhiều đối số ma trận và tích chập suy rộng của phép biến đổi I : Luận án TS. Toán học: 62 46 01 01

Đại học Quốc gia Hà Nội Trờng Đại học khoa häc tù nhiªn Trịnh Tuân Hàm dạng I với nhiều đối số ma trận tích chập suy rộng phép biến đổi I luận án Tiến sĩ Toán học Hà Nội - 2009 Đại học Quốc gia Hà Nội Trờng Đại học khoa học tự nhiên Trịnh Tuân Hàm dạng I với nhiều đối số ma trËn vµ tÝch chËp suy réng cđa phÐp biÕn đổi I Chuyên ngành: Toán Giải tích M số: 62.46.01.01 TËp thĨ h−íng dÉn khoa häc: PGS TS Ngun Xuân Thảo GS TSKH Nguyễn Văn Mậu luận án Tiến sĩ Toán học Hà Nội - 2009 ụ ụ ❉❛♥❤ ♠ơ❝ ❝➳❝ ❦Ý ❤✐Ư✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉❛♥❤ ♠ô❝ ❝➳❝ ❤➭♠ ✈➭ ❤➭♠ ➤➷❝ ❜✐Öt ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ I ❈❤➢➡♥❣ ✶ ✿ ❍➭♠ ❞➵♥❣ ✶✳✶ ❍➭♠ ❞➵♥❣ I ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✈í✐ ♥❤✐Ị✉ ➤è✐ sè ♠❛ tr❐♥ ✷✺ ✈í✐ ♥❤✐Ò✉ ➤è✐ sè ♠❛ tr❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✷ ❚Ý❝❤ ❝❤❐♣ ➤è✐ ✈í✐ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✶ M ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ❈❤➢➡♥❣ ✷ ✿ P❤Ð♣ ❜✐Õ♥ ➤æ✐ I ✈➭ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ➤è✐ ✈í✐ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ I ✹✻ ✷✳✶ P❤Ð♣ ❜✐Õ♥ ➤æ✐ I ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✷ ❚Ý❝❤ ❝❤❐♣ s✉② ré♥❣ ➤è✐ ✈í✐ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✷ I ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ❈❤➢➡♥❣ ✸ ✿ ❈➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ✈í✐ ❤➭♠ trä♥❣ ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ❑♦♥t♦r♦✈✐❝❤ ✲ ▲❡❜❡❞❡✈ ♥❣➢ỵ❝ ✈➭ ❝♦s✐♥❡ (K −1 )✱ ❋♦✉r✐❡r s✐♥❡ (Fs ) (Fc ) ✻✾ ✸✳✶ ❚Ý❝❤ ❝❤❐♣ s✉② ré♥❣ ✈í✐ ❤➭♠ trä♥❣ ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ K −1 ✱ Fs ✱ Fc ✈➭ ø♥❣ ❞ơ♥❣ ❣✐➯✐ ♠ét ❧í♣ ❤Ư ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✷ ❚Ý❝❤ ❝❤❐♣ s✉② ré♥❣ ✈í✐ ❤➭♠ trä♥❣ ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ Fc ✱ K −1 ✈➭ ø♥❣ ❞ơ♥❣ ❣✐➯✐ ♠ét sè ❤Ư ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑Õt ❧✉❐♥ ❝❤✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉❛♥❤ ♠ơ❝ ❝➠♥❣ tr×♥❤ ❝➠♥❣ ❜è ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✶ ❉❛♥❤ ♠ô❝ ❝➳❝ ❦Ý ❤✐Ư✉ • N = {1, 2, } ❧➭ t❐♣ sè tù ♥❤✐➟♥❀ Z ❧➭ t❐♣ sè ♥❣✉②➟♥ • R+ = {x|x > 0}✿ t❐♣ ❝➳❝ sè t❤ù❝ ❞➢➡♥❣✳ √ • ω = −i✱ i2 = −1 • (f ∗ g)(x)✿ tÝ❝❤ ❝❤❐♣ ❝đ❛ ❤❛✐ ❤➭♠ f γ • (f ∗ g)(x)✿ tÝ❝❤ ❝❤❐♣ ❝đ❛ ❤❛✐ ❤➭♠ f k • (f ∗ g)(x)✿ tÝ❝❤ ❝❤❐♣ ❝ñ❛ ❤❛✐ ❤➭♠ f ✈➭ g ✈➭ g ✈í✐ ❤➭♠ trä♥❣ ✈➭ g t❤❡♦ ❝❤Ø sè k✳ • F ✿ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ❋♦✉r✐❡r +∞ (F f )(y) = √ 2π f (x)e−iyx dx, y ∈ R −∞ • Fc ✿ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ ❋♦✉r✐❡r ❝♦s✐♥❡ +∞ (Fc f )(y) = π f (x) cos(xy)dx, y > 0 • Fs ✿ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ ❋♦✉r✐❡r s✐♥❡ +∞ (Fs f )(y) = π f (x) sin(xy)dx, y > 0 • M ✿ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ▼❡❧❧✐♥ +∞ f ∗ (x) = (M f )(y) = f (x)xy−1 dx • M −1 ✿ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ▼❡❧❧✐♥ ♥❣➢ỵ❝ c+i∞ (M −1 g)(x) = 2πi g(s)x−s ds c−i∞ ✷ γ(x) ✸ • L✿ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ ▲❛♣❧❛❝❡ +∞ e−xy f (x)dx (Lf )(y) = ã L1 é ế ổ ợ c+i∞ (L−1 f )(x) = 2πi exy f (y)dy c−i∞ −α • (xα Λ−1 + x )(.)✿ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ ▲❛♣❧❛❝❡ ❜✐Õ♥ ❞➵♥❣ 2πi −α (xα Λ−1 + x )f (x) = f ∗ (s)x−s ds Γ(s + α) σ tr♦♥❣ ➤ã Re(α) > − ✳ • H ✿ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❍✐❧❜❡rt +∞ (Hf )(y) = π f (x) dx x−y −∞ • H −1 ✿ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❍✐❧❜❡rt ♥❣➢ỵ❝ +∞ (H −1 f )(x) = π f (y) dy x−y −∞ • S ✿ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❙t✐❡❧t❥❡s +∞ f (x) dx x+y (Sf )(y) = • K ✿ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ ❑♦♥t♦r♦✈✐❝❤ ✲ ▲❡❜❡❞❡✈ +∞ (Kf )(y) = Kix (y)f (x) dx, y > 0 ✹ • K −1 ❧➭ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❑♦♥t♦r♦✈✐❝❤ ✲ ▲❡❜❡❞❡✈ ♥❣➢ỵ❝ +∞ (K −1 f )(x) = x sh(πx) π Kix (y)y −1 f (y)dy, x > 0 tr♦♥❣ ➤ã • Kix ❧➭ ❤➭♠ ▼❛❝❞♦♥❛❧❞✳ P❤Ð♣ ❜✐Õ♥ ➤ỉ✐ M✿ (det Z)ρ− Mρ (f ) = m+1 f (Z)dZ, Z>0 tr♦♥❣ ➤ã • Re ρ > P❤Ð♣ ❜✐Õ♥ ➤æ✐ m−1 G (α)1,p (β)1,p n (Gf )(x) =Gm pq = 2πi f (t) (x) ψ(s)f ∗ (s)x−s ds σ n tr♦♥❣ ➤ã Gm p q ❧➭ ❤➭♠ G− ▼❡✐❥❡r✱ σ = s ∈ C : Re(s) = ,0 n j=1 p ✈➭ j=1 q Γ(1 − βj − s) Γ(αj + s) j=n+1 • q; Γ(1 − αj − s) Γ(βj + s) f∗ m n m ψ(s) = p; j=m+1 ❧➭ ❜✐Õ♥ ➤æ✐ ▼❡❧❧✐♥ ❝ñ❛ f ✈➭  Re(β ) + > 0, j = 1, m; j Re(αj ) + > 0, j = n + 1, p; 2 − Re(αj ) > 0, − Re(βj ) > 0, j = 1, n m + 1, q ❚Ý❝❤ ♣❤➞♥ ▼❡❧❧✐♥ ✲ ❇❛r♥❡s ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿ γ+i∞ f (z) = 2πi Γ(a1 + A1 s) Γ(an + An s) Γ(c1 + C1 s) Γ(cp + Cp s) γ−i∞ Γ(b1 − B1 s) Γ(bn − Bn s) s z ds Γ(d1 − D1 s) Γ(dq − Dq s) tr♦♥❣ ➤ã γ ❧➭ t❤ù❝✱ Aj , Bj , Cj , Dj ❞➢➡♥❣✳ × ✺ +∞ |f (x)|dx < +∞} • L(R+ ) = {f (x) : +∞ |δ(x)||f (x)|dx < +∞}✿ ❦❤➠♥❣ ❣✐❛♥ ✈í✐ trä♥❣ • L(R+ , δ(x)) = {f (x) : δ • M−1 c,γ (L) = f (x) : f (x) = 2πi f ∗ (s)x−s ds σ ✈í✐ σ = s s❛♦ ❝❤♦ Re s = f (x) ✈➭ |s|γ f ∗ (s)eπc|s| ∈ L(σ)✳ ✈➭ f ∗ (s) ❧➭ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ▼❡❧❧✐♥ ❝đ❛ ❉❛♥❤ ♠ơ❝ ❤➭♠ ✈➭ ❤➭♠ ➤➷❝ ❜✐Ưt ex + e−x ch x = ❀ ex − e−x ❀ • sh x =     1, • sign(x) = • x>0 0,    −1, x=0 x (z) = ã q ị ♥❤➢ s❛✉ Γq (x) ≡ tr♦♥❣ ➤ã (x, q)∞ ❧➭ (q; q)∞ (1 − q)1−x , x (q ; q)∞ q ỗ ị + (1 aq k ) (a; q)∞ = k=0 • ❍➭♠ s✐➟✉ ❜é✐ +∞ r Fs = r Fs (a1 , ar ; b1 , bs ; z) = k=0 • (a1 )k (ar )k z k (b1 )k (bs )k k! ❍➭♠ ❇❡ss❡❧ ❧♦➵✐ ♠ét +∞ (−1)m ( z2 )ν+2m Jν (x) = m!Γ(ν + m + 1) m=0 • ❍➭♠ ❇❡ss❡❧ ❜✐Õ♥ ❞➵♥❣ ❧♦➵✐ ❜❛ ✭❤➭♠ ▼❛❝❞♦♥❛❧❞✮ Kν (z) = π I−ν (x) − Iν (z) sin(νπ) tr♦♥❣ ➤ã +∞ ( z2 )ν+2m Iν (z) = m!Γ(ν + m + 1) m=0 ✻ ✼ • ❍➭♠ ▼❛❝❞♦♥❛❧❞ ❞➵♥❣ tÝ❝❤ ♣❤➞♥ +∞ e−x ch u cos τ u du, x > 0, τ Kiτ (x) = 0 n • Gm pq ❧➭ ❤➭♠ G− ▼❡✐❥❡r✱ a1 , , ap b , , bq Gm.n x p,q m = 2πi n Γ(bj − s) j=1 p L xs ds, Γ(1 − bj + s) Γ(aj − s) j=m+1 j=n+1 • Γ(1 − aj + s) j=1 q ψ2 = ψ2 (a; c1 , , cn ; −Z1 , , −Zn ) ❧➭ ❤➭♠ s❛♦ ❝❤♦ ➯♥❤ q✉❛ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ M ủ ó ợ ị s M (2 ) =Mρ1 , ,ρn [ψ2 (a; c1 , , cn ; −Z1 , , −Zn )] n n { = Γp (cj )} Γp (a − ρ1 − · · · − ρn ){ j=1 Γp (ρj )} j=1 n Γp (a) Γp (cj − ρj )} { j=1 ✈í✐ Re(bj − ρ, cj − ρj , ρj , a − ρ1 − · · · − ρn ) > p−1 , j = 1, , n✱ ë ➤➞② tự ể t ĩ tự ủ ỗ t ♣❤➬♥ ➤Ị✉ ❧í♥ ❤➡♥ • p−1 ❍➭♠ φ2 = φ2 (b1 , , bn ; c; −Z1 , , −Zn ) ❧➭ ❤➭♠ s❛♦ ❝❤♦ M − ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ❝đ❛ ♥ã ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿ n Γp (c) M (φ2 ) = n { Γp (bj )} j=1 { Γp (bj − ρj )Γp (ρj )} j=1 Γp (c − ρ1 − · · · − ρn ) ✽ • ❈➳❝ ❤➭♠ ▲❛✉r✐❝❡❧❧❛✿ ❍➭♠ fA = fA (a, b1 , , bn ; c1 , , cn ; −Z1 , , −Zn ) t❤á❛ ♠➲♥ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ s❛✉ ❣ä✐ ❧➭ ❤➭♠ ▲❛✉r✐❝❡❧❧❛ |T1 |b1 − ··· M (fA ) = T1 >0 p+1 |Tn |bn − p+1 Tn >0 ×fA (a, b1 , , bn ; c1 , , cn ; −T1 , , −Tn )dT1 dTn n j=1 Γp (cj ) = Γp (a) n j=1 Γp (bj n j=1 Γp (bj ) n j=1 Γp (ρj ) Γp (a − ρ1 − · · · − ρn ) × − ρj ) n j=1 Γp (cj − ρj ) • q− ♠ị Eq (x) ✭①❡♠ ❬✶✵❪✮ ❧➭ ❤➭♠ ①➳❝ ➤Þ♥❤ ❜ë✐ +∞ Eq (z) = q n(n−1) zn n n=0 (1 − qi) i=0 • q− tÝ❝❤ ♣❤➞♥ ❧➭ f (x)dq x ①➳❝ ➤Þ♥❤ ❜ë✐ +∞ f (q i )q i , f (x)dq x = (1 − q) i=0 • ❍➭♠ trô ♣❛r❛❜♦❧✐❝ ✭❍➭♠ ❲❡❜❡r✮ ✈✐ ♣❤➞♥ ❲❡❜❡r y (z) + ν + Dν (z)✱ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ 1 − z y(z) = 0, ợ ị D (z) = 2/2+1/4 z −1/2 Wν/2+1/4,−1/4 z ✾✶ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ❤Ư ✭✸✳✷✳✶✾✱ ✸✳✷✳✷✵✮ ❝ã ♥❣❤✐Ư♠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ L(R+ )✳ ❑❤✐ ➤ã sư ❞ơ♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ✭✸✳✷✳✶✮ t❛ ✈✐Õt ❤Ư (3.2.19, 3.2.20) ❞➢í✐ ❞➵♥❣ s❛✉✳ γ f (x) + λ1 (ϕ ∗ g)(x) = h(x), λ2 (ψ ∗ f )(x) + g(x) = k(x), Fc x > ❙ư ❞ơ♥❣ ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤♦➳ ❝đ❛ ❝➳❝ tÝ❝❤ ❝❤❐♣ ✭✸✳✷✳✶✮ ✈➭ tÝ❝❤ ❝❤❐♣ ✭✸✳✶✳✶✵✮ t❛ ➤➢ỵ❝ (Fc f )(y) + λ1 γ(y)(K −1 ϕ)(y)(Fc g)(y) = (Fc h)(y), λ2 (Fc ψ)(y)(Fc f )(y) + (Fc g)(y) = (Fc k)(y), ∀y > ▲➵✐ ❝ã λ1 γ(y)(K −1 ϕ)(y) = ∆= λ2 (Fc ψ)(y) γ = − λ1 λ2 Fc (ϕ ∗ ψ)(y) = 0, ∀y > 0, (Fc h)(y) λ1 γ(y)(K −1 ϕ)(y) ∆1 = (Fc k)(y) γ = (Fc h)(y) − λ1 Fc (ϕ ∗ k)(y), ∆2 = (Fc h)(y) λ2 (Fc ψ)(y) (Fc k)(y) = (Fc k)(y) − λ2 Fc (h ∗ ψ)(y) Fc ❉♦ ➤ã γ [(Fc h)(y) − λ1 Fc (ϕ ∗ k)(y)] ∆   γ λ1 λ2 Fc (ϕ ∗ ψ)(y) γ   = 1+ [(F h)(y) − λ F (ϕ ∗ k)(y)] c c γ − λ1 λ2 Fc (ϕ ∗ ψ)(y) (Fc f )(y) = ✾✷ ❚❤❡♦ ➤Þ♥❤ ❧ý ❲✐❡♥❡r✲▲Ð✈② ✭①❡♠ ❬✷❪✮✱ tå♥ t➵✐ ❞✉② ♥❤✃t ❤➭♠ s❛♦ ❝❤♦ l(x) ∈ L(R+ ) γ λ1 λ2 Fc ((ϕ ∗ ψ)(y) γ − λ1 λ2 Fc ((ϕ ∗ ψ)(y) = (Fc l)(y) ❱❐② γ (Fc f )(y) = [1 + (Fc l)(y)][(Fc h)(y) − λ1 Fc (ϕ ∗ k)(y)] γ γ = (Fc h)(y) − λ1 Fc (ϕ ∗ k)(y) + Fc (l ∗ h)(y) − λ1 Fc (l ∗ (ϕ ∗ k))(y), Fc y > Fc ❙✉② r❛ γ γ f (x) = h(x) − λ1 (ϕ ∗ k)(x) + (l ∗ h)(x) − λ1 (l ∗ (ϕ ∗ k))(x) ◆❤ê sù tå♥ t➵✐ ❝ñ❛ tÝ❝❤ ❝❤❐♣ ✸✳✷✳✶ ❞➱♥ ➤Õ♥ Fc (· ∗ ·) Fc Fc tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ L(R+ ) ✈➭ ➤Þ♥❤ ❧ý f (x) ∈ L(R+ )✳ ❚➢➡♥❣ tù (Fc g)(y) =[1 + (Fc l)(y)][(Fc k)(y) − λ2 Fc (h ∗ ψ)(y)] Fc =(Fc k)(y) − λ2 Fc (h ∗ ψ)(y) + Fc (l ∗ k)(y) Fc − λ2 Fc (l ∗ (h ∗ ψ))(y), Fc Fc Fc y > 0, ✈➭ g(x) = k(x) + (l ∗ k)(x) − λ2 (h ∗ ψ)(x) − λ2 (l ∗ (h ∗ ψ))(x) Fc ◆❤ê sù tå♥ t➵✐ ❝ñ❛ tÝ❝❤ ❝❤❐♣ Fc (· ∗ ·) Fc Fc tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ Fc L(R+ ) ❞➱♥ ➤Õ♥ g(x) ∈ L(R+ )✳ ❙ư ❞ơ♥❣ ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤♦➳ (3.2.2) ❝đ❛ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♠í✐ ➤➢ỵ❝ ①➞② ❞ù♥❣ ✈➭ ❝➳❝ ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤♦➳ ❝đ❛ ❝➳❝ tÝ❝❤ ❝❤❐♣ ❦❤➳❝ ➤➲ ❜✐Õt t❛ ể tr ợ f g ị tr t ệ trì ệ ề ợ ứ ♠✐♥❤ ①♦♥❣✳ ✾✸ ❜✮ ❳Ðt ❤Ư ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ +∞ θ1 (x, u)g(u)du = h(x), x > 0, ✭✸✳✷✳✷✹✮ θ2 (x, v)f (v)dv + g(x) = k(x), x > 0, ✭✸✳✷✳✷✺✮ f (x) + λ1 +∞ λ2 tr♦♥❣ ➤ã λ1 , λ2 ❧➭ √ ❝➳❝ sè ♣❤ø❝❀ + x3 ✱ k, ψ, h ∈ L(R+ )❀ ϕ, ξ ∈ L R+ , √ x3 f ✈➭ g ❧➭ ❝➳❝ ➮♥ ❤➭♠ ❝➬♥ t×♠✱ ✈➭ +∞ θ1 (x, u) = 4π −u ch(x+v) e + e−u ch(x−v) ϕ(v)dv, u +∞ +∞ θ2 (x, v) = π2 √ ψ(z)ξ(u) sign(z − x) sh(|z − x| + v)e−u ch(|z−x|+v) 2π 0 + sign(z − x) sh(|z − x| − v)e−u ch(|z−x|−v) + sh(z + x + v)e−u ch(z+x+v) + sh(z + x − v)e−u ch(z+x−v) dudz ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ❦❤➻♥❣ ➤Þ♥❤ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❤Ư ✭✸✳✷✳✷✹✱ ✸✳✷✳✷✺✮ ✈➭ ❝❤♦ ❝✃✉ tró❝ ♥❣❤✐Ư♠ ❞➢í✐ ❞➵♥❣ ➤ã♥❣✳ ▼Ư♥❤ ➤Ị ✸✳✷✳✺✳ ●✐➯ sư r➺♥❣ γ γ1 − λ1 λ2 Fc (ϕ ∗ (ξ ∗ ψ) (y) = 0, ∀y > ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ tr♦♥❣ ✭✸✳✷✳✷✻✮ L(R+ ) ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ (3.2.24, 3.2.25) ❝❤♦ ❜ë✐ γ γ f (x) = h(x) − λ1 (ϕ ∗ k)(x) + (l ∗ h)(x) − λ1 l ∗ (ϕ ∗ k) (x), Fc Fc ✭✸✳✷✳✷✼✮ γ1 γ1 g(x) = k(x) − λ2 h ∗ (ξ ∗ ψ) (x) + (l ∗ k)(x) − λ2 l ∗ h ∗ (ξ ∗ ψ) (x), Fc Fc Fc Fc ✭✸✳✷✳✷✽✮ ✾✹ tr♦♥❣ ➤ã l ∈ L(R+ ) ✈➭ ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ γ γ1 λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) (Fc l)(y) = ✈➭ tÝ❝❤ ❝❤❐♣ γ γ1 , − λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) γ1 (· Ã) ợ ị ứ ●✐➯ sư ❤Ư ♣❤➢➡♥❣ tr×♥❤ (3.2.24, 3.2.25) ❝ã ♥❣❤✐Ư♠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ L(R+ )✳ ❑❤✐ ➤ã ❤Ư ❝ã t❤Ĩ ➤➢ỵ❝ ✈✐Õt ❧➵✐ ♥❤➢ s❛✉ γ f (x) + λ1 (ϕ ∗ g)(x) = h(x), λ2 ψ ∗ Fs,c γ1 ξ∗f x > 0, (x) + g(x) = k(x), x > ❙ư ❞ơ♥❣ ❝➳❝ ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤♦➳ ➤è✐ ✈í✐ ❝➳❝ tÝ❝❤ ❝❤❐♣ ✭✸✳✷✳✶✮✱ ✭✸✳✶✳✶✹✮✱ ✭✸✳✶✳✶✮✱ ✭✸✳✶✳✶✶✮ ✈➭ ✭✸✳✶✳✷✮ t❛ ❝ã (Fc f )(y) + λ1 γ(y)(K −1 ϕ)(y)(Fc g)(y) = (Fc h)(y), λ2 γ1 (y)(Fs ψ)(y)(K −1 ξ)(y) (Fc f )(y) + (Fc g)(y) = (Fc k)(y), ❍Ư tr➟♥ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ (Fc f )(y) + λ1 γ(y)(K −1 ϕ)(y)(Fc g)(y) = (Fc h)(y), γ1 λ2 Fc (ξ ∗ ψ)(y)(Fc f )(y) + (Fc g)(y) = (Fc k)(y), y > ❚❛ ❝ã ∆= λ1 γ(y)(K −1 ϕ)(y) γ1 λ2 Fc (ξ ∗ ψ)(y) γ1 = − λ1 λ2 γ(y)(K −1 ϕ)(y)Fc (ξ ∗ ψ)(y) γ γ1 =1 − λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) = 0, ∀y > ❚õ ➤ã s✉② r❛ γ γ1 λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) =1+ γ γ1 ∆ − λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) y > ✾✺ ❚❤❡♦ ➤Þ♥❤ ❧ý ❲✐❡♥❡r✲▲Ð✈② ✭①❡♠ ❬✷❪✮✱ tå♥ t➵✐ ❞✉② ♥❤✃t ♠ét ❤➭♠ l ∈ L(R+ ) s❛♦ ❝❤♦ γ γ1 λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) (Fc l)(y) = γ γ1 , − λ1 λ2 Fc ϕ ∗ (ξ ∗ ψ) (y) (Fc h)(y) λ1 γ(y)(K −1 ϕ)(y) ∆1 = (Fc k)(y) γ = (Fc h)(y) − λ1 Fc (ϕ ∗ k)(y) ❙✉② r❛ γ (Fc f )(y) = + (Fc l)(y) (Fc h)(y) − λ1 Fc (ϕ ∗ k)(y) γ γ = (Fc h)(y) − λ1 Fc (ϕ ∗ k)(y) + Fc (l ∗ h)(y) − λ1 Fc (l ∗ (ϕ ∗ k) (y) ❱❐② Fc Fc γ γ f (x) = h(x) − λ1 (ϕ ∗ k)(x) + (l ∗ h) − λ1 (l ∗ (ϕ ∗ k))(x) ◆❤ê sù tå♥ t➵✐ ❝ñ❛ tÝ❝❤ ❝❤❐♣ ✸✳✷✳✶ ❞➱♥ ➤Õ♥ Fc (· ∗ ·) Fc Fc tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ L(R+ ) ✈➭ ➤Þ♥❤ ❧ý f (x) ∈ L(R+ )✳ ❚➢➡♥❣ tù✱ ∆2 = (Fc h)(y) γ1 λ2 Fc (ξ ∗ ψ)(y) (Fc k)(y) γ1 = (Fc k)(y) − λ2 Fc (h ∗ (ξ ∗ ψ)(y) Fc ❙✉② r❛ γ1 (Fc g)(y) = [1 + (Fc l)(y)] (Fc k)(y) − λ2 Fc h ∗ (ξ ∗ ψ) (y) Fc γ1 = (Fc k)(y) − λ2 Fc (h ∗ (ξ ∗ ψ))(x) + Fc (l ∗ k)(x) Fc γ1 Fc − λ2 Fc l ∗ (h ∗ (ξ ∗ ψ)))(y) Fc Fc ❱❐② γ1 γ1 g(x) = k(x) − λ2 (h ∗ (ξ ∗ ψ))(x) + (l ∗ k)(x) − λ2 l ∗ (h ∗ (ξ ∗ ψ)))(x) Fc Fc Fc Fc ✾✻ ◆❤ê sù tå♥ t➵✐ ❝ñ❛ tÝ❝❤ ❝❤❐♣ ➤Õ♥ γ1 (· ∗ ·) ✈➭ (· ∗ ·) tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ L(R+ ) ❞➱♥ Fc g(x) ∈ L(R+ )✳ ❉Ô ❞➭♥❣ ❦✐Ĩ♠ tr❛ f ✱ g ①➳❝ ➤Þ♥❤ ♥❤➢ tr➟♥ ➤ó♥❣ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✷✳✷✹✮ ✲ ✭✸✳✷✳✷✺✮✳ ▼Ư♥❤ ➤Ị ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ①♦♥❣✳ ◆❤❐♥ ①Ðt ✸✳✷✳✶✳ ❚r♦♥❣ ♠ét sè ❦Õt q✉➯ ➤➲ ❜✐Õt ë ❬✶✹✱ ✸✺✱ ✹✼✱ ✺✵✱ ✺✶✱ ✺✸✱ ✺✹✱ ✺✺✱ ✺✻❪✱ ❝➳❝ tÝ❝❤ ❝❤❐♣ ➤➢ỵ❝ ①➞② ❞ù♥❣ tr➢í❝ ➤➞② ➤Ị✉ ❝ã ❝ï♥❣ ♠ét ➤➷❝ ➤✐Ĩ♠ ❧➭ tr♦♥❣ ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤ã❛ ❝đ❛ ♥ã ❝❤Ø ❝ã ❞✉② ♥❤✃t ♠ét ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ t❤❛♠ ❣✐❛ ❤♦➷❝ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ t❤❡♦ ❝❤Ø sè t❤✉é❝ ❝ï♥❣ ♠ét ❤ä t❤❛♠ ❣✐❛✳ ❈➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ✈í✐ ❤➭♠ trä♥❣ ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ Fc K −1 ✱ Fs ✱ ➤➢ỵ❝ ①➞② ❞ù♥❣ ë ➤➞② ❧➭ ❦❤➳❝ ❤♦➭♥ t♦➭♥ ✈í✐ ❝➳❝ tí ết trớ ó ỗ tr ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤ã❛ ❝đ❛ ❝❤ó♥❣ ❝ã ♥❤✐Ị✉ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ ❦❤➳❝ ♥❤❛✉ t❤❛♠ ❣✐❛✳ ❈➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♥➭② ❦❤➠♥❣ ❣✐❛♦ ❤♦➳♥ ✈➭ ❦❤➠♥❣ ❦Õt ợ ữ ệ trì tí ợ ❞ù♥❣ ë ➤➞② ♥❤➺♠ ♠ô❝ ➤Ý❝❤ ♠✐♥❤ ❤ä❛ ❝❤♦ ø♥❣ ❞ơ♥❣ ❝đ❛ ❝➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♠í✐ ♥❤❐♥ ➤➢ỵ❝✳ ệ ề ợ t ể s ỗ ệ trì tí ị tồ t ♥❤✃t ♥❣❤✐Ư♠ ❝đ❛ ❤Ư tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤➭♠ L(R+ ) ✈➭ ❝❤♦ ❝✃✉ tró❝ ♥❣❤✐Ư♠ ❞➢í✐ ❞➵♥❣ ➤ã♥❣✱ ❜✐Ĩ✉ ❞✐Ơ♥ t❤➠♥❣ q✉❛ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♠í✐ ✈➭ ❝➳❝ tÝ❝❤ ❝❤❐♣✱ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ➤➲ ❜✐Õt tr➢í❝ ➤ã✳ ❈➬♥ r ữ ệ trì tí ♥➭② ❝❤Ø ❝ã t❤Ĩ ❣✐➯✐ ➤➢ỵ❝ ❜➺♥❣ ❝➠♥❣ ❝ơ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♠í✐ ♥❤❐♥ ➤➢ỵ❝✳ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✸ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ❧➭✿ • ❳➞② ❞ù♥❣ ➤➢ỵ❝ ❜❛ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♠í✐ ✈í✐ ❤➭♠ trä♥❣ ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ (Fs , K −1 , Fc )❀ (Fc , K −1 , Fs )❀ (Fc , K −1 ) ❝ï♥❣ ✈í✐ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝ị♥❣ ♥❤➢ ♠ét sè ♠è✐ ❧✐➟♥ ❤Ư ✈í✐ ❝➳❝ tÝ❝❤ ❝❤❐♣ ➤➲ ❜✐Õt✳ • ø♥❣ ❞ơ♥❣ ❝➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♠í✐ ♥➭② ✈➭♦ ❣✐➯✐ ♠ét sè ❤Ư ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ❦✐Ĩ✉ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣✱ ♥❣❤✐Ư♠ ♥❤❐♥ ➤➢ỵ❝ ❞➢í✐ ❞➵♥❣ ➤ã♥❣✳ ✾✼ ❑Õt ❧✉❐♥ ❝❤✉♥❣ ◆❤÷♥❣ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ➳♥ ❧➭ ✶✳ ❳➞② ❞ù♥❣ ➤➢ỵ❝ ❤➭♠ ❞➵♥❣ I ✈í✐ ♥❤✐Ị✉ ➤è✐ sè ♠❛ tr❐♥✱ tõ ➤ã ➤✐ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ❳➞② ❞ù♥❣ ➤➢ỵ❝ tÝ❝❤ ❝❤❐♣ ➤è✐ ✈í✐ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ M ❝đ❛ ❤❛✐ ❤➭♠ f, g ✈í✐ ♥❤✐Ị✉ ➤è✐ sè ♠❛ tr❐♥✱ ✈➭ ø♥❣ ❞ơ♥❣ ➤Ĩ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ❦✐Ĩ✉ tÝ❝❤ ❝❤❐♣✳ ✷✳ ❳➞② ❞ù♥❣ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ I✱ ❝❤Ø r❛ ❦❤➠♥❣ ❣✐❛♥ ❤➭♠ ❝❤♦ sù tå♥ t➵✐ ❝đ❛ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ♥➭②✱ ♥❤❐♥ ➤➢ỵ❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ♥❣➢ỵ❝ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ➳♥❤ ①➵ ➤å♥❣ ♣❤➠✐ ❣✐÷❛ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ➤ã✳ ❚õ ➤ã ➤✐ ①➞② ❞ù♥❣ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ➤è✐ ✈í✐ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ I ✈➭ I −1 ✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➷t ❜✐Ưt ❝ñ❛ ❝➳❝ t❤❛♠ sè rk = ri = rj = 1✳ ◆❤❐♥ ➤➢ỵ❝ ❝➳❝ ➤➻♥❣ t❤ø❝ ♥❤➞♥ tư ❤ã❛ ✈í✐ sù t❤❛♠ ❣✐❛ ❝đ❛ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ H ✈➭ I ✳ ❳➞② ❞ù♥❣ t❤Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ sù tå♥ t➵✐ ❝ñ❛ ❝➳❝ t✐❝❤ ❝❤❐♣ s✉② ré♥❣ ❝ñ❛ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ I ✈➭ I −1 tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤➭♠ ❝❤Ø r❛ ✱ ♥❤ê ♠ét sè ❤➭♠ ➤➷❝ ❜✐Öt ❦❤➳❝ ♠➭ ú t ợ ệ ữ tí tr ữ trờ ợ ỉ số k = 3, 4, 5, 6✳ ✸✳ ❈❤ä♥ ❜❛ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ Fs ✱ K −1 ✱ Fc ➤Ĩ ①➞② ❞ù♥❣ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ✈í✐ ❤➭♠ trä♥❣ ➤è✐ ✈í✐ ❜❛ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ♥➭② ♥❤➢ ❧➭ ♠ét tr➢ê♥❣ ❤ỵ♣ ➳♣ ❞ơ♥❣ ❝❤♦ ❧Ý t❤✉②Õt tỉ♥❣ q✉➳t ë ❝❤➢➡♥❣ ✶ ✈➭ ❝❤➢➡♥❣ ✷✳ ◆❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ❝ñ❛ ❝➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ♥➭②✳ ➜➷❝ ❜✐Ưt ø♥❣ ❞ơ♥❣ ❝➳❝ tÝ❝❤ ❝❤❐♣ ♠í✐ ♥➭② ✈➭♦ ❣✐➯✐ ♠ét sè ❧í♣ ❤Ư ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ❦✐Ĩ✉ tÝ❝❤ ❝❤❐♣ ▲✉❐♥ ➳♥ ♠ë r❛ ♠ét sè ✈✃♥ ➤Ò ❝ã t❤Ĩ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉✿ ✶✳ ◆❣❤✐➟♥ ❝ø✉ ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ❝đ❛ ❤➭♠ ❞➵♥❣ I ✈í✐ ♥❤✐Ị✉ ➤è✐ sè ♠❛ tr❐♥✱ t❤❐♠ ❝❤Ý ♥❣❛② ❝➯ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ ❧➭ ❤➭♠ H ✈í✐ ♥❤✐Ị✉ ➤è✐ sè ♠❛ tr❐♥✳ ✷✳ ❳➞② ❞ù♥❣ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ❝❤♦ ❧í♣ ❤➭♠ ❤➭♠ ❞➵♥❣ I I ♥❤✐Ò✉ ❜✐Õ♥ sè ✈➭ ♥❤✐Ò✉ ❜✐Õ♥ sè✱ tõ ➤ã ♥❣❤✐➟♥ ❝ø✉ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ t❤❡♦ ✾✽ ✾✾ ❝❤Ø sè ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ♥➭② ❝ị♥❣ ♥❤➢ ➤❛ ❝❤❐♣ ❝đ❛ ♥ã✳ (f ∗ g) ❝đ❛ ❜é ❜❛ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ Fs ✱ K −1 ✱ Fc t❛ ❝ã t❤Ĩ ❝è ➤Þ♥❤ ♠ét tr♦♥❣ ❤❛✐ ❤➭♠ f ❤♦➷❝ g ➤Ó ♥❣❤✐➟♥ ✸✳ ỗ tí s rộ ứ é ế ổ tÝ❝❤ ♣❤➞♥ ❦✐Ĩ✉ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ❝ị♥❣ ♥❤➢ ➤❛ ❝❤❐♣ ❝đ❛ ❜❛ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ tÝ❝❤ ♣❤➞♥ ♥➭②✳ ➜➷❝ ❜✐Ưt ø♥❣ ❞ơ♥❣ ❝➳❝ tÝ❝❤ ❝❤❐♣ s✉② ré♥❣ ❝❤♦ ♥❤ã♠ ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ tÝ❝❤ ♣❤➞♥ Fs ✱ K −1 ✱ Fc ➤Ĩ ❣✐➯✐ ♠ét sè ❧í♣ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ✈í✐ ♥❤➞♥ ❚♦❡♣❧✐t③ ✰ ❍❛♥❦❡❧ ❞➢í✐ ❞➵♥❣ ♥❣❤✐Ư♠ ➤ã♥❣✳ ❉❛♥❤ ♠ơ❝ ❝➳❝ ❝➠♥❣ tr×♥❤ ❝đ❛ t➳❝ ❣✐➯ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❧✉❐♥ ➳♥ ➤➲ ➤➢ỵ❝ ❝➠♥❣ ❜è ✶✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ❛♥❞ ❚r✐♥❤ ❚✉❛♥ ✭✷✵✵✸✮✱ ❖♥ t❤❡ ●❡♥❡r❛❧✐③❡❞ ❈♦♥✲ ✈♦❧✉t✐♦♥ ❢♦r I ✲tr❛♥s❢♦r♠✱ ❆❝t❛ ▼❛t❤❡♠❛t✐❝❛ ❱✐❡t♥❛♠✐❝❛✳ ❱♦❧✳✷✽✱ ◆♦✳ ✷✱ ✶✺✾✲✶✼✹✳ ✷✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ❛♥❞ ❚r✐♥❤ ❚✉❛♥ ✭✷✵✵✹✮✱ ❇❛s✐❝ ❆♥❛❧♦❣✉❡ ♦❢ I− ❋✉♥❝t✐♦♥ ♦❢ ❙❡✈❡r❛❧ ▼❛tr✐① ❆r❣✉♠❡♥ts✳ ❱✐❡t♥❛♠ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✲ ✐❝s ✳ ❱♦❧✳ ✸✷✱ ◆♦✳ ✹✱ ✹✶✾ ✲ ✹✸✶✳ ✸✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ❛♥❞ ❚r✐♥❤ ❚✉❛♥ ✭✷✵✵✺✮✱ ❖♥ t❤❡ ●❡♥❡r❛❧✐③❡❞ ❈♦♥✈♦❧✉t✐♦♥s ♦❢ t❤❡ ■♥t❡❣r❛❧ ❑♦♥t♦r♦✈✐❝❤ ✲ ▲❡❜❡❞❡✈✱ ❋♦✉r✐❡r s✐♥❡ ❛♥❞ ❝♦s✐♥❡ ❚r❛♥s❢♦r♠s✱ ❆♥♥❛❧❡s ❯♥✐✈✳ ❙❝✐✳ ❇✉❞❛♣❡st✱ ❙❡❝t✳ ❈♦♠♣✳ ❱♦❧✳ ✷✺✱ ✸✼✲✺✶✳ ✹✳ ❚r✐♥❤ ❚✉❛♥ ✭✷✵✵✼✮✱ ❖♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✈♦❧✉t✐♦♥ ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❋♦✉r✐❡r ❈♦s✐♥❡ ❛♥❞ t❤❡ ■♥✈❡rs❡ ❑♦♥t♦r♦✈✐❝❤ ✲ ▲❡❜❡✲ ❞❡✈ ✐♥t❡❣r❛❧ tr❛♥❢♦r♠❛t✐♦♥s✳ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐✲ ❝❛t✐♦♥s✳ ❑♦r❡❛✳ ❱♦❧✳✶✷✱ ◆♦✳✷✱ ✸✷✺ ✲ ✸✹✶✳ ✶✵✵ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▼✳ ❆❜r❛♠♦✇✐t③ ❛♥❞ ■✳ ❆✳ ❙t❡❣✉♥ ✭✶✾✻✹✮✱ ❍❛♥❞❜♦♦❦ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋✉♥❝✲ t✐♦♥s ✇✐t❤ ❋♦r♠✉❧❛s✱ ●r❛♣❤s ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧ ❚❛❜❧❡s✱ ◆❛t✳ ❇✉r✳ ❙t❛♥ ❛♣♣❧✳ ▼❛t❤✳ ❙❡r✳ ✺✺✳ ❬✷❪ ◆✳ ■✳ ❆❝❤✐❡③❡r ✭✶✾✻✺✮✱ ▲❡❝t✉r❡s ♦♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❚❤❡♦r②✱ ❙❝✐❡♥❝❡ ♣✉❜❧✐s❤✲ ✐♥❣ ❍♦✉s❡✱ ▼♦s❝♦✇✳ ❬✸❪ ❋✳ ❆❧✲▼✉s❛❧❧❛♠ ❛♥❞ ❱✉ ❑✐♠ ❚✉❛♥ ✭✷✵✵✶✮✱ ❡t❡rs✿ ❊①✐st❡♥❝❡✳ ❬✹❪ H ✲❢✉♥❝t✐♦♥ ✇✐t❤ ❝♦♠♣❧❡① ♣❛r❛♠✲ ■♥t❡r♥❛t✳ ❏✳ ▼❛t❤✳ ▼❛t❤✳ ❙❝✐✳ ❍✳ ❇❛t❡♠❛♥ ❛♥❞ ❆✳ ❊r❞❡❧②✐ ✭✶✾✺✸✮✱ ✷✺ ✭✷✵✵✶✮✱ ♥♦✳ ✾✱ ✺✼✶✲✺✽✻✳ ❍✐❣❤❡r ❚r❛♥s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✱ ▼❝ ●r❛✇✲❍✐❧❧✱ ◆❡✇ ❨♦r❦✱ ❱✳✶✳ ❬✺❪ ❍✳ ❇❛t❡♠❛♥ ❛♥❞ ❆✳ ❊r❞❡❧②✐ ✭✶✾✺✹✮✱ ❚❛❜❧❡s ♦❢ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s✱ ▼❝✳●r❛✇✲ ❍✐❧❧✱ ◆❡✇ ❨♦r❦✱ ❱✳✶✱ ✷✳ ❬✻❪ ❘✳ ●✳ ❇✉s❝❤♠❛♥ ✭✶✾✼✽✮✱ H ✲❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s ■✱ ■♥❞✐❛♥✳ ▼❛t❤✳ ❏✳ ✷✵✱ ✶✸✾✲✶✺✸✳ ❬✼❪ ❘✳ ●✳ ❇✉s❝❤♠❛♥ ✭✶✾✼✾✮✱ ❏✳ ❬✽❪ H ✲❢✉♥❝t✐♦♥ ♦❢ N ✲ ✈❛r✐❛❜❧❡s ✱ ✶✵✱ ✽✶✲✽✽✳ ❨✉✳ ❆✳ ❇r②❝❤❦♦✈✱ ❍✳ ❏✳ ●❧❛❡s❦❡ ❛♥❞ ❖✳ ■✳ ▼❛r✐❝❤❡✈ ✭✶✾✽✸✮✱ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢ ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❝♦♥✈♦❧✉t✐♦♥ t②♣❡✱ ❆♥❛❧✳ ❱■◆■❚■✳ ❬✾❪ ❈✳ ❋♦① ✭✶✾✻✶✮✱ ❚❤❡ ●✳ ●❛s♣❡r✱ ■t♦❣✐ ◆❛✉❦✐ ✐ ❚❡❝❤♥✐❦✐✳ ▼❛t❤✳ ❱✳✷✶✱ ✸✲✹✶✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ❬✶✵❪ ❘❛♥❝❤✐✳ ❯♥✐✈✳ ▼❛t❤✳ G ❛♥❞ H ✲❢✉♥❝t✐♦♥s ❛s s②♠♠❡tr✐❝❛❧ ❋♦✉r✐❡r ❦❡r♥❡❧s✱ ❚r❛♥✳ ✾✽✱ ✸✾✺✲✲✹✷✾✳ ▼✳ ❘❛❤♠❛♥ ✭✶✾✾✵✮✱ ❇❛s✐❝ ❍②♣❡r❣❡♦♠❡tr✐❝ ❙❡r✐❡s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✶✶❪ ❲✳ ❍❛❤♥ ✭✶✾✹✾✮✱ ❇❡✐tr❛❣❡ ③✉r t❤❡♦r✐❡ ❞❡r ❍❡✐♥❡s❝❤❡♥ ❘❡✐❤❡♥✱ t❡❣r❛❧❡ ❞❡r ❤②♣❡r❣❡♦♠❡tr✐s❝❤❡♥ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠❛t✐♦♥✱ q ✲❉✐❢❢❡r❡♥③❡♥❧❡✐❝❤✉♥❣✱ ❉❛s ▼❛t❤✳ ◆❛❝❤r✳ ✶✵✶ ✷✱ ✸✹✵✲✲✸✼✾✳ q ❉✐❡ ✷✹ ✐♥✲ ✲❛♥❛❧♦❣❡♥ ❞❡r ✶✵✷ ❬✶✷❪ ◆❣✉②❡♥ ❚❤❛♥❤ ❍❛✐✱ ❖✳ ■✳ ▼❛r✐❝❤❡✈ ❛♥❞ ❘✳●✳ ❇✉s❝❤♠❛♥ ✭✶✾✾✷✮✳ ❚❤❡♦r② ♦❢ t❤❡ ❣❡♥❡r❛❧ H ✲ ❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐❛❜❧❡s✳ ❘♦❝❦② ▼♦✉♥t❛✐♥ ❏✳ ♦❢ ▼❛t❤✳ ❱♦❧✳ ✷✷✳ ◆✳ ✹✳ ✶✸✶✼ ✲ ✶✸✷✼✳ ❬✶✸❪ ■✳ ■✳ ❍✐r❝❤♠❛♥ ❛♥❞ ❖✳ ❱✳ ❲✐❞❞❡r ✭✶✾✺✺✮✱ ❚❤❡ ❈♦♥✈♦❧✉t✐♦♥ ❚r❛♥s❢♦r♠s✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡②✳ ❬✶✹❪ ❱✳ ❆✳ ❑❛❦✐❝❤❡✈ ✭✶✾✻✼✮✱ ❖♥ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ❢♦r ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠s✱ ❇❙❙❘✱ ❙❡r✳ ❋✐③✳ ▼❛t✳ ❬✶✺❪ ■③✈✳ ❆◆ ◆✳✷✱ ✹✽✲✺✼✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❱✳ ❆✳ ❑❛❦✐❝❤❡✈✱ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ❛♥❞ ◆❣✉②❡♥ ❚❤❛♥❤ ❍❛✐ ✭✶✾✾✻✮✱ ❈♦♠♣♦✲ s✐t✐♦♥ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t✐♥❣ ❝♦♥❝♦❧✉✐♦♥s ❢♦r t❤❡ ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠✳ ❚r❛♥s✳ ❙♣❡❝✐❛❧ ❋✉♥❝✳ ❬✶✻❪ ❱✳ ✹✳ ✷✸✺ ✲ ✷✹✷✳ ❱✳ ❆✳ ❑❛❦✐❝❤❡✈ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✶✾✾✹✮✱ ❖♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✈♦❧✉✲ t✐♦♥ ❢♦r ❬✶✼❪ H ✲tr❛♥s❢♦r♠s✱ ■③✈✳ ❱✉③♦✈ ▼❛t✳ ❱✳ ■③✈✳ ❱✉③♦✈✳ ▼❛t✱ ◆♦✳✶✱ ✸✶✲✹✵✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❆✳ ❑❛❦✐❝❤❡✈ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✷✵✵✵✮✱ ❆ ❜❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ❛♥❞ t✇♦ ✈❛r✐❛❜❧❡s✱ ❬✶✾❪ ◆♦✳ ✽✱ ✷✶✲✷✽✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❱✳ ❆✳ ❑❛❦✐❝❤❡✈ ❛♥❞ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✶✾✾✽✮✱ ❖♥ t❤❡ ❞❡s✐❣♥ ♠❡t❤♦❞ ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ ✐♥t❡❣r❛❧ ❝♦♥✈♦❧✉t✐♦♥✱ ❬✶✽❪ ■♥t❡❣r✳ ❨✳ ▲✳ ▲✉❦❡ ✭✶✾✻✾✮✱ ■③✈✳ ❱✉③♦✈ ▼❛t✳ H ✲ ✷✽✲✸✹✳ ❚❤❡ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❆♣r♦①✐♠❛t✐♦♥s✳ ❱♦❧✳ ■ ❛♥❞ ■■✳ ❆❝❛❞❡♠✐❝ Pr❡ss✳ ◆❡✇ ❨♦r❦✳ ❬✷✵❪ ❖✳ ■✳ ▼❛r✐❝❤❡✈ ✭✶✾✽✸✮✱ ❍❛♥❞❜♦♦❦ ♦❢ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ♦❢ ❍✐❣❤❡r ❚r❛♥✲ s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✳ ❚❤❡♦r② ❛♥❞ ❆❧❣♦r✐t❤♠✐❝ ❚❛❜❧❡s✳ ❬✷✶❪ ❖✳ ■✳ ▼❛r✐❝❤❡✈ ❛♥❞ ❱✉ ❑✐♠ ❚✉❛♥ ✭✶✾✽✺✮✱ ❚❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ ✐♥ t✇♦ s♣❛❝❡s ♦❢ ❢✉♥❝t✐♦♥s✳ ❬✷✷❪ ◆❡✇ ❨♦r❦✳ ❈♦♠♣❧❡① ❆♥❛❧✳ ❛♥❞ ❆♣♣❧✳ ❱❛r♥❛✳ ❆✳ ▼✳ ▼❛t❤❛✐ ❛♥❞ ❘✳ ❑✳ ❙❛①❡♥❛✱ ✭✶✾✼✸✮ G ✲tr❛♥s❢♦r♠ ✹✶✽ ✲ ✹✸✸✳ ●❡♥❡r❛❧✐③❡❞ ❍②♣❡r❣❡♦♠❡tr✐❝ ❋✉♥❝✲ t✐♦♥s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❙t❛t✐st✐❝s ❛♥❞ P❤②s✐❝❛❧ ❙❝✐❡♥❝❡s✱ ▲❡❝t✉r❡ ◆♦t❡s ◆♦✳ ✸✹✽✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❍❡✐❞❡❧❜❡r❣ ❛♥❞ ◆❡✇ ❨♦r❦✳ ❬✷✸❪ ❆✳ ▼✳ ▼❛t❤❛✐ ✭✶✾✾✸✮✱ ▲❛✉r✐❝❡❧❧❛ ❢✉♥❝t✐♦♥ ♦❢ r❡❛❧ s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ ❞❡❢✐♥✐t❡ ♠❛tr✐❝❡s✳ ■♥❞✐❛♥✳ ❏✳ P✉r❡✳ ❆♣♣❧✳ ▼❛t❤✳ ✷✹✱ ✺✶✸✲✺✸✶✳ ✶✵✸ ❬✷✹❪ ❆✳ ▼✳ ▼❛t❤❛✐ ✭✶✾✾✺✮✱ ❙♣❡❝✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ ♠❛tr✐① ❛r❣✉♠❡♥ts ✲■■■✳ ❆❝❛❞✳ ❙❝✐✳ ■♥❞✐❛✳ ❆✳ ❬✷✺❪ ❆✳ Pr♦❝✳ ◆❛t✳ ✻✺✱ ✸✻✼✲✸✾✸✳ ▼✳ ▼❛t❤❛✐ ❛♥❞ ❘✳ ❑✳ ❙❛①❡♥❛ ✭✶✾✼✽✮✱ ✐♥ ❙t❛t✐st✐❝s ❛♥❞ ❖t❤❡r ❉✐s❝✐♣❧✐♥❡✱ ❚❤❡ H ✲❋✉♥❝t✐♦♥ ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ❲✐❧❡② ❋❛st❡r♥ ▲✐♠✐t❡❞✱ ◆❡✇ ❉❡❧❧✐ ❇❛♥✲ ❣❛❧♦r❡ ❇♦♠❜❛②✳ ❬✷✻❪ ◆✐❡❧s❡♥ ◆✐❡❧s ❍❛♥❞❜✉❝❤ ❞❡r ❚❤❡♦r✐❡ ❞❡r ●❛♠♠❛✲❢✉♥❦t✐♦♥✳ ✭✶✾✵✻✮✱ ❇✳ ❈✳ ❚❡✉❜♥❡r✱ ▲❡✐♣③✐❣✳ ❬✷✼❪ ❆✳ P✳ Pr✉❞♥✐❦♦✈✱ ❨✉✳ ❆✳ ❇r✉❝❤❦♦✈ ❛♥❞ ❖✳ ■✳ ▼❛r✐❝❤❡✈ ✭✶✾✽✻✮✱ ❙❡r✐❡s✱ ❱✳✸✿ ▼♦r❡ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s✱ ■♥t❡❣r❛❧ ❛♥❞ ●♦r❞♦♥ ❛♥❞ ❇r❡❛❝❤ ❙❝✐❡♥❝❡ P✉❜❧✐s❤✲ ❡rs✱ ◆❡✇ ❨♦r❦✳ ❬✷✽❪ ▼✳ ❙❛✐❣♦ ❛♥❞ ❙✳ ❇✳ ❨❛❦✉❜♦✈✐❝❤ ✭✶✾✾✶✮✱ ❖♥ t❤❡ t❤❡♦r② ♦❢ ❝♦♥✈♦❧✉t✐♦♥ ✐♥t❡❣r❛❧s G ❢♦r ❬✷✾❪ ✲tr❛♥s❢♦r♠s✱ ❋✉❦✉♦❦❛✿ ❯♥✐✈✳ ❙❝✐✳ ❘❡♣♦rt✳ ✷✶✱ ✶✽✶✲✶✾✸✳ ❙✳ ●✳ ❙❛♠❦♦✱ ❆✳ ❆✳ ❑✐❧❜❛s ❛♥❞ ❖✳ ■✳ ▼❛r✐❝❤❡✈ ✭✶✾✽✼✮✱ ■♥t❡❣r❛❧ ❛♥❞ ❉❡r✐✈❛t✐✈❡ ♦❢ ❋r❛❝t✐♦♥❛❧ ❖r❞❡r ❛♥❞ ❙❡✈❡r❛❧ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥✱ ❬✸✵❪ ▼✐♥s❦✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❘✳ ❑✳ ❙❛①❡♥❛ ✭✶✾✽✷✮✱ ❋♦r♠❛❧ s♦❧✉t✐♦♥ ♦❢ ❝❡rt❛✐♥ ♥❡✇ ♣❛✐r ♦❢ ❞✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ H− ❢✉♥❝t✐♦♥s✳ Pr♦❝✳ ◆❛t✳ ❆❝❛❞✳ ❙❝✐✳ ■♥❞✐❛ ❙❡❝t✳ ❆✱ ◆♦✳ ✺✷✱ ✸✻✻ ✲ ✸✼✺✳ ❬✸✶❪ ❘✳ ❑✳ ❙❛①❡♥❛✱ ❢✉♥❝t✐♦♥✱ ❬✸✷❪ ❘✳ ❑✉♠❛r ▼❛t❡♠❛t✐❝❤❡ ✭✶✾✾✺✮✱ ❆ ❜❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ ❣❡♥❡r❛❧✐③❡❞ H ✲ ✺✵✱ ✷✻✸✲✷✼✶✳ ❘✳ ❑✳ ❙❛①❡♥❛✱ ❘✳ ❆✳ ❑✉♠❛r✱ ✭✶✾✾✵✮✱ ❈❡rt❛✐♥ ❢✐♥✐t❡ ❡①♣❛♥s✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❜❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ ❬✸✸❪ ❆✳ ❘✳ ❑✳ ❙❛①❡♥❛✱ ●✳ G ❘❡✈✳ ❚❡❝✳ ■♥❣✳ ❯♥✐✈✳ ❩✉❧✐❛✳ ❈✳ ▼♦❞✐✱ ❛♥❞ ❙✳ H ✲❢✉♥❝t✐♦♥✱ ❬✸✹❪ ■✳ ◆✳ ❙♥❡❞❞♦♥ ✭✶✾✺✶✮✱ ❬✸✺❪ ❍✳ t❤❡ ✲ ❢✉♥❝t✐♦♥✳ ❱✳ ✶✸✱ ✶✶✶ ✲ ✶✶✻✳ ▲✳ ❑❛❧❧❛ ✭✶✾✽✸✮✱ ❆ ❜❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ ❋♦①✬s ❘❡✈✳ ❚❡❝✳ ■♥❣✳ ❯♥✐✈✳ ❩✉❧✐❛✳ ✻✱ ✶✸✾✲✶✹✸✳ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠✱ ▼❝●r❛② ❍✐❧❧✱ ◆❡✇ ❨♦r❦✳ ▼✳ ❙r✐✈❛st❛✈❛ ❛♥❞ ❱✉ ❑✐♠ ❚✉❛♥ ✭✶✾✾✺✮✱ ❆ ♥❡✇ ❝♦♥✈♦❧✉t✐♦♥ t❤❡♦r❡♠ ❢♦r ❙t✐❡❧t❥❡s ❡q✉❛t✐♦♥✱ tr❛♥s❢♦r♠ ❆r❝❤✳ ▼❛t❤✳ ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ ✻✹✱ ✶✹✹✲✶✹✾✳ t♦ ❛ ❝❧❛ss ♦❢ s✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ✶✵✹ ❬✸✻❪ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✶✾✾✾✮✱ ❇❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ ❆r❣✉♠❡♥ts✳ H ✲ ❢✉♥❝t✐♦♥ ♦❢ ❙❡✈❡r❛❧ ▼❛tr✐① ❉♦♣♦✈✳ ◆❛ts ❆❦❛❞✳ ◆❛✉❦✳ ❯❦r✳ ▼❛t✳ Pr✐r♦❞♦③♥✳ ❚❡❦❤✳ ◆❛✉❦✐✳ ◆✳ ✶✷✳ ✶✷✲✶✼✳ ✭❘✉ss✐❛♥✮✳ ❬✸✼❪ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦✱ ❱✉ ❑✐♠ ❚✉❛♥ ❛♥❞ ◆❣✉②❡♥ ▼✐♥❤ ❑❤♦❛ ✭✷✵✵✹✮✱ ❖♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✈♦❧✉t✐♦♥ ✇✐t❤ t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❋♦✉r✐❡r ❝♦s✐♥❡ ❛♥❞ s✐♥❡ tr❛♥s❢♦r♠s✳ ❬✸✽❪ ✇✐t❤ tr❛♥s❢♦r♠s✳ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✶✾✾✽✮✱ H ❢♦r ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✷✵✵✶✮✱ t❤❡ ❋♦✉r✐❡r✱ ❋♦✉r✐❡r ❝♦s✐♥❡ ❛♥❞ s✐♥❡ ✸✸✳ ❱♦❧✳✹✱ ✹✷✶ ✲ ✹✸✻✳ ✲❢✉♥❝t✐♦♥ ♦❢ ♠❛tr✐① ❛r❣✉♠❡♥t✱ ●❯✳ ❙❡r✳✿ ❊st❡st✈✳ ❛♥❞ ❚❡❤♥✳ ◆❛✉❦✐ ❬✹✵❪ ❱♦❧✳ ✼✱ ◆♦✳✸✱ ✸✷✸ ✲ ✸✸✼✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ❛♥❞ ◆❣✉②❡♥ ▼✐♥❤ ❑❤♦❛ ✭✷✵✵✺✮✱ ❖♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✲ ✈♦❧✉t✐♦♥ ❬✸✾❪ ❋r❛❝t✳ ❈❛❧✳ ❛♥❞ ❆♣♣❧✳ ❆♥❛❧✳ ❖♥ ❱❡st♥✐❦✱ ◆♦✈✳ ✶✵✱ ✶✵✷✲✶✵✻ ✭✐♥ ❘✉ss✐❛♥✮✳ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❍✐❧❜❡rt✱ ❋♦✉r✐❡r ❝♦s✐♥❡ ❛♥❞ s✐♥❡ tr❛♥s❢♦r♠s✱ ❝♦♥✈♦❧✉t✐♦♥ ❯❦r✳ ▼❛t✳ ❏✳ ❢♦r ❙t✐❡❧t❥❡s✱ ✺✸✱ ✺✻✵✲✺✻✼✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❬✹✶❪ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦✱ ❱✳ ❆✳ ❑❛❦✐❝❤❡✈ ❛♥❞ ❱✉ ❑✐♠ ❚✉❛♥ ✭✶✾✾✽✮✱ ❖♥ t❤❡ ❣❡♥✲ ❡r❛❧✐③❡❞ ❝♦♥✈♦❧✉t✐♦♥ ❢♦r ❋♦✉r✐❡r ❝♦s✐♥❡ ❛♥❞ s✐♥❡ tr❛♥s❢♦r♠s✱ ▼❛t❤✳ ❬✹✷❪ ✶✱ ✽✺✲✾✵✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ❛♥❞ ◆❣✉②❡♥ ❚❤❛♥❤ ❍❛✐ ✭✶✾✾✼✮✱ ❈♦♥✈♦❧✉t✐♦♥ ❢♦r ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠ ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥✱ ❬✹✸❪ ❱♦❧✐♥✳ ▼❛t✳ ❱✐s♥✳ ▼❛t✳ ❩❛♠❡t✳ I ✲❢✉♥❝t✐♦♥ ♦❢ t✇♦ ✈❛r✐✲ ✶✵✷✲✶✵✻✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✷✵✵✵✮✱ ❆ ❜❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✱ ❬✹✺❪ ❘✉ss✐❛♥ ❆❝❛❞❡♠②✱ ▼♦s❝♦✇✳ ◆❣✉②❡♥ ❳✉❛♥ ❚❤❛♦ ✭✶✾✾✾✮✱ ❆ ❜❛s✐❝ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❛❜❧❡s✱ ❬✹✹❪ ❊❛st✲❲❡st ❏✳ H ✲❢✉♥❝t✐♦♥ ♦❢ s❡✈❡r❛❧ ✻✼✱ ✼✸✽✲✼✹✹✳ ❊✳ ❈✳ ❚✐t❝❤♠❛r❝❤ ✭✶✾✸✼✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❚❤❡♦r② ♦❢ ❋♦✉r✐❡r ■♥t❡❣r❛❧s✱ ❖①❢♦r❞ ❯♥✐✈✳ Pr❡ss✱ ❖①❢♦r❞✳ ❬✹✻❪ ❱✉ ❑✐♠ ❚✉❛♥ ✭✶✾✽✼✮✱ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❛♥❞ ❚❤❡✐r ❈♦♠♣♦s✐t✐♦♥ ❙tr✉❝t✉r❡✳ ❉r✳ ❙❝✐✳ ❉✐ss❡rt❛t✐♦♥✳ ▼✐♥s❦✱ ✷✼✺ ♣✳ ✶✵✺ ❬✹✼❪ ❱✉ ❑✐♠ ❚✉❛♥ ❛♥❞ ▼✳ ❙❛✐❣♦ ✭✶✾✾✺✮✱ ❈♦♥✈♦❧✉t✐♦♥ ♦❢ ❍❛♥❦❡❧ tr❛♥s❢♦r♠ ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s t♦ ❛♥ ✐♥t❡❣r❛❧ ✐♥✈♦❧✈✐♥❣ ❇❡ss❡❧ ❢✉♥❝t✐♦♥ ♦❢ ❢✐rst ❦✐♥❞✱ ▼❛t❤✳ ❛♥❞ ▼❛t❤✳ ❙❝✐✳ ❬✹✽❪ ■♥t✳ ❏✳ ❱✳ ✶✽✱ ◆✳ ✷✱ ✺✹✺✲✺✺✵✳ ❱✉ ❑✐♠ ❚✉❛♥ ✭✶✾✾✷✮✱ ❖♥ ❝r✐t❡r✐❛ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ❞♦✉❜❧❡ ▼❡❧❧✐♥ ✲ ❇❛r♥❡s ✐♥t❡❣r❛❧✳ ❱❡s❝✐✳ ❆◆✳ ❇❡❧♦r✉ss✐❛♥✱ ❋✐③ ✲ ▼❛t✳ ✷✺✲✸✶✳ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s✱ ■♥t❡r✳ P✉❜❧✳ ◆❡✇ ❨♦r❦ ❛♥❞ ▲♦♥❞♦♥✳ ❬✹✾❪ ❋✳ ●✳ ❚r✐❝♦♠✐ ✭✶✾✺✼✮✱ ❬✺✵❪ ❋✳ ●✳ ❚r✐❝♦♠✐ ✭✶✾✺✶✮✱ ❖♥ t❤❡ ❢✐♥✐t❡ ❍✐❧❜❡rt tr❛♥s❢♦r♠✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ✷ ✱ ✶✾✾✲✷✶✶✳ ❬✺✶❪ ◆✳ ❨❛✳ ❱✐❧❡♥❦✐♥ ✭✶✾✺✽✮✱ ❚❤❡ ♠❛tr✐① ❡❧❡♠❡♥ts ♦❢ ✐rr❡❞✉❝✐❜❧❡ ✉♥✐t❛r② r❡♣r❡✲ s❡♥t❛t✐♦♥s ♦❢ ❛ ❣r♦✉♣ ♦❢ ▲♦❜❛❝❡✈s❦✐✐✐ s♣❛❝❡ ♠♦t✐♦♥s ❛♥❞ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❋♦❦✲ ▼❡❤❧❡r tr❛♥s❢♦r♠❛t✐♦♥✱ ❬✺✷❪ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ❙❙❙❘✱ ✶✶✽✱ ✷✶✾➊✷✷✷✳ ✭■♥ ❘✉ss✐❛♥✮✳ ❋✳ ◆✐❦✐❢♦r♦✈✱ ❇✳ ❯✈❛r♦✈ ✭✶✾✽✽✮✱ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✿ ❆ ❯♥✐❢✐❡❞ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✳ ❬✺✸❪ ❙✳ ❇✳ ❨❛❦✉❜♦✈✐❝❤ ✭✶✾✾✵✮✱ ✐♥t❡❣r❛❧ ❝♦♥✈♦❧✉t✐♦♥s✱ ❬✺✹❪ ❙✳ ❇✳ ❨❛❦✉❜♦✈✐❝❤ ❖♥ t❤❡ ❉❆◆ ❇❙❙❘ ❛♥❞ ❆✳ ■✳ ❇✐r❦❤❛✉s❡r ❱❡r❧❛❣ ❇❛s❡❧✳ ❝♦♥str✉❝t✐♦♥ ♠❡t❤♦❞ ❢♦r ❝♦♥str✉❝t✐♦♥ ♦❢ ✸✹✱ ✺✽✽✲✺✾✶✳ ▼♦s✐♥s❦✐ ✭✶✾✾✸✮✱ ■♥t❡❣r❛❧ ✈♦❧✉t✐♦♥s ❢♦r tr❛♥s❢♦r♠s ♦❢ ❑♦♥t♦r♦✈✐❝❤✲▲❡❜❡❞❡✈ t②♣❡✱ ❡q✉❛t✐♦♥s ❛♥❞ ❝♦♥✲ ❉✐❢❢✳ ❯r❛✈♥❡♥✐❛ ✷✾✱ ✶✷✼✷✲✶✷✽✹✳ ❬✺✺❪ ❙✳ ❇✳ ❨❛❦✉❜♦✈✐❝❤ ❛♥❞ ◆❣✉②❡♥ ❚❤❛♥❤ ❍❛✐ ✭✶✾✾✶✮✱ ■♥t❡❣r❛❧ ❝♦♥✈♦❧✉t✐♦♥s ❢♦r H ❬✺✻❪ ✲tr❛♥s❢♦r♠s✱ ■③✈✳ ❱✉③♦✈✳ ▼❛t✳ ◆♦✳✽✱ ✼✷✲✼✾✳ ❙✳ ❇✳ ❨❛❦✉❜♦✈✐❝❤ ✭✶✾✽✼✮✱ ❖♥ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ❢♦r ❑♦♥t♦r♦✈✐❝❤✲▲❡❜❡❞❡✈ ✐♥t❡✲ ❣r❛❧ tr❛♥s❢♦r♠ ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠✱ ❇❙❙❘ ❬✺✼❪ ❙✳ ❇✳ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦✳ ✸✶✱ ✶✵✶✲✶✵✸ ✭✐♥ ❘✉ss✐❛♥✮✳ ❨❛❦✉❜♦✈✐❝❤ ✭✶✾✾✻✮✱ ■♥❞❡① ❚r❛♥s❢♦r♠s✱ ❲♦r❧❞ ❙❝✐❡♥t✐❢✐❝ P✉❜❧✐s❤✐♥❣ ❈♦♠♣❛♥②✱ ❙✐♥❣❛♣♦r❡✱ ◆❡✇ ❏❡rs❡②✱ ▲♦♥❞♦♥ ❛♥❞ ❍♦♥❣ ❑♦♥❣✱ ✷✹✽ ♣✳