Una introducción a la teoría de transformaciones espectrales
Transcripción
Una introducción a la teoría de transformaciones espectrales
Polinomios ortogonales: Una introducción a la teoría de transformaciones espectrales Luis E. Garza Universidad de Colima Encuentro Nacional de Jóvenes Investigadores en Matemáticas, IMATE, UNAM Diciembre 2, 2015. LEGG (UdeC) Diciembre 2, 2015. 1 / 42 Contents 1 Polinomios ortogonales en la recta y matrices de Jacobi 2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg 3 La representación CMV 4 Algunas generalizaciones LEGG (UdeC) Diciembre 2, 2015. 2 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Contents 1 Polinomios ortogonales en la recta y matrices de Jacobi 2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg 3 La representación CMV 4 Algunas generalizaciones LEGG (UdeC) Diciembre 2, 2015. 3 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Orthogonal polynomials in R Given a nontrivial probability measure µ supported on some infinite subset E of the real line, a (unique) sequence of orthonormal polynomials {pn }n>0 can be defined as Z pm (x)pn (x)dµ(x) = δm,n , n, m > 0, (1) E where pn (x) = γn xn + ζn xn−1 + lower degree terms, (2) with γn > 0, n > 0. Classical orthogonal polynomials: Jacobi dµ(x) = (1 − x)α (1 + x)β dx in [−1, 1]. (Tchebychev, Gegenbauer, Legendre) Laguerre dµ(x) = xα e−x dx in R+ . 2 Hermite dµ(x) = e−x dx in R. LEGG (UdeC) Diciembre 2, 2015. 4 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Some applications OP appear in a wide range of applications such as: Approximation theory Integrable systems Numerical integration Signal theory Image processing Etc, etc, etc. LEGG (UdeC) Diciembre 2, 2015. 5 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Three term recurrence relation Starting from p0 (x) = 1 and p−1 (x) = 0, {pn }n>0 satisfies xpn (x) = an+1 pn+1 (x) + bn pn (x) + an pn−1 (x), where an = Z xpn−1 (x)pn (x)dµ(x) = E and bn = Z xp2n (x)dµ(x) = E γn−1 > 0, γn ζn ζn+1 − , γn γn+1 n > 0, (3) n > 1, n > 0. Favard’s theorem: Given any sequences {an }n>1 , {bn }n>0 of real numbers, the polynomials constructed with (3) are orthogonal with respect to some measure dµ(x). LEGG (UdeC) Diciembre 2, 2015. 6 / 42 Polinomios ortogonales en la recta y matrices de Jacobi The monic Jacobi matrix On the other hand, the monic OP with respect to µ are given by Pn (x) = pn (x)/γn , n > 0. In such a case, (3) becomes Pn+1 (x) = (x − bn )Pn (x) − dn Pn−1 (x), n > 0, (4) with dn = a2n , and has the matrix representation xP(x) = JP(x), where J = b0 d1 1 b1 0 1 0 0 0 d2 b2 1 0 .. . 0 .. . d3 .. . b3 .. . ··· ··· .. . , . . . . . . is known as monic Jacobi matrix. LEGG (UdeC) Diciembre 2, 2015. 7 / 42 Polinomios ortogonales en la recta y matrices de Jacobi The LU factorization of J Notice that Pn (0) , 0, n > 1 ⇐⇒ J has a unique LU factorization, where L and U are bidiagonal matrices L = 1 l1 0 1 0 0 0 0 0 l2 1 0 0 .. . 0 .. . l3 .. . 1 .. . ··· ··· .. . , U = . . . .. . u1 0 1 u2 0 1 0 0 0 0 u3 1 0 .. . 0 .. . 0 .. . u4 .. . ··· ··· .. . , . . . . . . (5) where LEGG (UdeC) l1 = u1 = dn d1 , ln = , n > 2, b0 bn−1 − ln−1 b0 , un = bn−1 − ln−1 , n > 2. (6) (7) Diciembre 2, 2015. 8 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Darboux transformations Darboux transformation without parameter J = LU, J p := UL Darboux transformation (not unique) J = UL, Jd := LU Notice that J p and Jd are again tridiagonal matrices with ones as entries on the upper diagonal and, according to Favard’s theorem, they are monic Jacobi matrices associated with some nontrivial measure µ̃. LEGG (UdeC) Diciembre 2, 2015. 9 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Canonical spectral transformations on R Christoffel transformation (RC ) dµ̃ = (x − β)dµ, β < supp(µ). Uvarov transformation (UU ) dµ̃ = dµ + Mr δ(x − β), Mr ∈ R. Geronimus transformation (RG ) dµ̃ = dµ + Mr δ(x − β), x−β β < supp(µ), Mr ∈ R. Proposition LEGG (UdeC) RC ◦ RG = I Identity transformation RG ◦ RC = RU Diciembre 2, 2015. 10 / 42 Polinomios ortogonales en la recta y matrices de Jacobi LST and Stieltjes functions The Stieltjes function associated with µ is S (x) = ∞ Z E dµ(t) X µk = , x−t xk+1 k=0 where µk = E xk dµ(x) are the moments of µ. It has been shown that the previous transformations can be expressed as R A(x)S (x) + B(x) , Se(x) = D(x) (8) e(x) is the Stieltjes function associated with µ̃, and A(x), B(x), D(x) are where S polynomials in the variable x, which are known. Furthermore, Proposition (Zhedanov, 97) All transformations of the form (8) can be obtained as a composition of Christoffel and Geronimus transformations. LEGG (UdeC) Diciembre 2, 2015. 11 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Rational spectral transformations Associated polynomials From a OPS {Pn }n>0 , define the monic associated polynomials or order k, {P(k) n }n>0 , by the shifted recurrence relation (k) (k) P(k) n+1 (x) = (x − bn+k )Pn (x) − dn+k Pn−1 (x), n > 0, i.e. removing the first k rows and columns of J. Anti-associated polynomials If we "push" the first k rows and columns of J, and introduce new coefficients b−i (i = k, k − 1, ..., 1) and d−i (i = k − 1, k − 2, ..., 0), then the anti-associated polynomials of order k are defined by (−k) (−k) P(−k) n+1 (x) = (x − b̃n+k )Pn (x) − d̃n+k Pn−1 (x), where {b̃i }i>0 = {b−i }1i=k LEGG (UdeC) n > 0, S S {bi }i>0 and {d̃i }i>1 = {d−i }0i=k−1 {di }i>1 . Diciembre 2, 2015. 12 / 42 Polinomios ortogonales en la recta y matrices de Jacobi RST and Stieltjes functions It has been shown that the previous transformations can be expressed as A(x)S (x) + B(x) Se(x) = , C(x)S (x) + D(x) (9) e(x) is the transformed Stieltjes function, and A(x), B(x), C(x), D(x) are where S polynomials in the variable x, which are known. Furthermore, Proposition (Zhedanov, 97) All transformations of the form (9) can be obtained as a combination of Christoffel, Geronimus, associated and anti-associated transformations. LEGG (UdeC) Diciembre 2, 2015. 13 / 42 Polinomios ortogonales en la recta y matrices de Jacobi ST and Jacobi matrices Question Can we express RC , RU , and RG in terms of the corresponding monic Jacobi matrices? Proposition Let J be the monic Jacobi matrix associated with µ, and β ∈ R such that Pn (β) , 0, n > 1. Then, J − βI = LU, J̃ := UL + βI, then J̃ is the monic Jacobi matrix associated with dµ̃ = (x − β)dµ, i.e. the Christoffel transformation. LEGG (UdeC) Diciembre 2, 2015. 14 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Christoffel transformation Proposition Let µ and J be as before. Consider the following transformations C1 := J − β1 I = L1 U1 , C̃1 := U1 L1 + β1 I, C2 .. . := C̃1 − β2 I = L2 U2 , C̃2 := U2 L2 + β2 I, Cm := C̃m−1 − βm I = Lm Um , C̃m := Um Lm + βm I, with β1 , β2 , . . . , βm ∈ R. If {Pn,i } is the MOPS associated with C̃i , 1 6 i 6 m − 1, and assuming that Pn (β) , 0, Pn,i (βi+1 ) , 0, n > 1, 1 6 i 6 m − 1, then C̃m is the monic Jacobi matrix associated with the measure dµ̃ = (x − β1 )(x − β2 ) . . . (x − βm )dµ. LEGG (UdeC) Diciembre 2, 2015. 15 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Uvarov transformation Proposition Let J0 be the monic Jacobi matrix associated with µ. Consider J0 − βI = L1 U1 , J1 := U1 L1 , = U2 L2 , J2 := L2 U2 + βI. J1 Then J2 is the monic Jacobi matrix associated with the measure dµ̃ = dµ + Mr δ(x − β), i.e. the Uvarov transformation of µ, where Mr = with µ0 = of J1 . R E µ0 (b0 − β − s) , s dµ(x) and s is the free parameter associated with the UL factorization LEGG (UdeC) Diciembre 2, 2015. 16 / 42 Polinomios ortogonales en la recta y matrices de Jacobi Geronimus transformation Proposition Let J1 be the monic Jacobi matrix associated with µ̂. Suppose there exists µ s.t. dµ̂ = (x − β)dµ. If J1 − βI = U1 L 1 , J2 := L1 U1 + βI, then J2 is the monic Jacobi matrix associated with dµ̃ = dµ̂ + Mr δ(x − β), x−β R dµ̂ i.e. the Geronimus transformation of µ̂, where Mr = E s parameter associated with the UL factorization of J1 . LEGG (UdeC) and s is the free Diciembre 2, 2015. 17 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Contents 1 Polinomios ortogonales en la recta y matrices de Jacobi 2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg 3 La representación CMV 4 Algunas generalizaciones LEGG (UdeC) Diciembre 2, 2015. 18 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Measures on T and Toeplitz matrices If σ is a nontrivial positive Borel measure supported on the unit circle, then we can consider the inner product hp, qi = Z p(z)q(z)dσ(z), R The moments are defined by cn := h1, z i = T zn dσ(z), n ∈ Z. Notice that we have Z Z n n z dσ(z) = z−n dσ(z) = z−n , 1 = h1, z−n i = c̄−n , cn := h1, z i = T n T T and thus the Gram matrix in terms of the standard basis {1, z, z2 , . . .} is the Toeplitz matrix c0 c −1 . T = .. c −n . .. LEGG (UdeC) c1 c0 .. . ··· ··· .. . cn cn−1 .. . c−n+1 .. . ··· c0 .. . ··· ··· · · · . . . (10) Diciembre 2, 2015. 19 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Orthogonal polynomials on T We can apply G-S to get a sequence {ϕn }n>0 , where ϕ(z) has the form ϕ(z) = κn zn + lower order terms. We have Φn (z) = ϕn (z)/κn , satisfying Φn+1 (z) = Φn+1 (z) = zΦn (z) + Φn+1 (0)Φ∗n (z), 1 − |Φn+1 (0)|2 zΦn (z) + Φn+1 (0)Φ∗n+1 (z), (11) (12) Φ∗n (z) = zn Φ̄n (z−1 ) (reversed polynomial), {Φn (0)}n>1 (Verblunsky, Schur, reflection parameters). |Φn (0)| < 1, n > 1. Furthermore, if kn = kΦn k2 = κn−2 , then kn = (1 − |Φn (0)|2 )kn−1 LEGG (UdeC) Diciembre 2, 2015. 20 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Hessenberg matrices The multiplication operator with respect to {ϕn }n>0 is represented in a matrix form by zϕ(z) = Hϕ ϕ(z), (13) where ϕ(z) = ϕ0 (z), ϕ1 (z), . . . , ϕn (z), . . . t and Hϕ is a lower Hessenberg matrix whose entries are hn, j κn κn+1κ j = − κn Φn+1 (0)Φ j (0) 0 if if if j = n + 1, j 6 n, j > n + 1. (14) Notice that Hϕ is defined in terms of {Φn (0)}n>1 . LEGG (UdeC) Diciembre 2, 2015. 21 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Hessenberg matrices (cont.) Proposition Hϕ satisfies (i) Hϕ H∗ϕ = I, (ii) H∗ϕ Hϕ = I − λ∞ (0)ϕ(0)ϕ(0)∗ , where I is the semi-infinite identity matrix and λ∞ (0) = Q∞ n=0 (1 − |Φn+1 (0)|2 ). Remark Hϕ is unitary ⇐⇒ P∞ n=0 |Φn (0)|2 = +∞ ⇐⇒ log σ0 < L1 dθ 2π (σ < Szegő class). Remark In the monic case, HΦ has as entries hn, j LEGG (UdeC) 1 kn − k j Φn+1 (0)Φ j (0) = 0 if if if j = n + 1, j 6 n, j > n + 1. (15) Diciembre 2, 2015. 22 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Canonical spectral transformations on T Christoffel transformation (FC ) dσ̃ = |z − α|2 dσ, α ∈ C. Uvarov transformation (FU ) dσ̃ = dσ + Mc δ(z − α) + M c δ(z − ᾱ−1 ), α ∈ C {0}, Mc ∈ C. Geronimus transformation (FG ) dσ̃ = dσ + Mc δ(z − α) + M̄c δ(z − ᾱ−1 ), |z − α|2 α ∈ C {0}, Mc ∈ C. Proposition LEGG (UdeC) FC ◦ FG = I Identity transformation FG ◦ FC = FU Diciembre 2, 2015. 23 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg ST and Carathéodory functions Define F(z) = c0 + 2 ∞ X c−k zk , k=1 In the positive definite case, F(z) is analytic, Re[F(z)] > 0 in D, and F(z) = Z T w+z dσ(w). w−z The previous transformations can be expressed as e = A(z)F(z) + B(z) , F(z) D(z) (16) e is associated with σ̃ and A(z), B(z), D(z) are known polynomials in z. where F(z) LEGG (UdeC) Diciembre 2, 2015. 24 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Rational spectral transformations Associated polynomials (N) Denote by {Φn }n>0 the associated polynomials of order N , defined by (N) ∗ (N) Φ(N) n+1 (z) = zΦn (z) + Φn+N+1 (0)(Φn ) (z), n > 0, i.e. the first N coefficients are removed. Anti-associated polynomials Let ν1 , ν2 , . . . , νN ∈ C with |ν j | < 1, 1 6 j 6 N . Define S {Φ̂n (0)}n>1 = {ν j }Nj=1 {Φ j (0)}∞j=1 . Then, the polynomials (−N) Φn+1 (z) = zΦ(−N) (z) + Φ̂n+1 (0)(Φ(−N) )∗ (z), n n n > 0, are called anti-associated polynomials of order N . LEGG (UdeC) Diciembre 2, 2015. 25 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg RST and Carathéodory functions Aleksandrov transformation Define {Φλn (0)}n>1 , where Φλn (0) = λΦn (0), with λ ∈ C, |λ| = 1. Then, Φλn+1 (z) = zΦλn (z) + Φλn+1 (0)(Φλn )∗ (z), are called Aleksandrov polynomials. These transformations can be expressed as e = A(z)F(z) + B(z) , F(z) C(z)F(z) + D(z) (17) e is the transformed Carathéodory function and A(z), B(z), C(z), D(z) are where F(z) known polynomials in z. LEGG (UdeC) Diciembre 2, 2015. 26 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg ST and Hessenberg matrices Question Can we express FC , FU , and FG in terms of the corresponding Hessenberg matrices? LEGG (UdeC) Diciembre 2, 2015. 27 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Christoffel transformation Let dσC = |z − α|2 dσ, and {ψn }n>0 its OPS. The relation between both families of polynomials is s (z − α)ψn (z) = where Kn (z, y) = n X n X ϕn+1 (α)ϕ j (α) Kn (α, α) ϕn+1 (z) − ϕ j (z), √ Kn+1 (α, α) Kn+1 (α, α)Kn (α, α) j=0 (18) ϕk (z)ϕk (y). k=0 In matrix form (z − α)ψ(z) = MC ϕ(z), (19) where MC has entries ϕ (α)ϕ (α) − √K i+1(α,α)Kj (α,α) , i+1 i q Ki (α,α) mi, j = Ki+1 (α,α) , 0, LEGG (UdeC) if j 6 i, if j = i + 1, if j > i + 1. (20) Diciembre 2, 2015. 28 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Christoffel transformation Proposition MC satisfies (i) MC MC∗ = I. (ii) MC∗ MC = I − λ∞ (α)ϕ(α)ϕ(α)∗ , Proposition Let MC n be the n × n principal submatrix of MC . Then, (i) MC n MC ∗n = In − (ii) Kn−1 (α,α) ∗ t Kn (α,α) en en , where en = [0, . . . , 0, 1] 1 MC ∗n MC n = In − Kn (α,α) ϕ(n) (α)ϕ(n)∗ (α), where (n) ϕ (α) = [ϕ0 (α), ϕ1 (α), . . . , ϕn−1 (α)]t LEGG (UdeC) ∈ C(n,1) . Diciembre 2, 2015. 29 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Christoffel transformation (cont.) Furthermore, if Lϕψ is the lower triangular matrix such that ϕ(z) = Lϕψ ψ(z), then Proposition We have Hϕ − αI = Lϕψ MC , (21) Hψ − αI = MC Lϕψ . (22) An "almost" QR factorization appears, since (MC )n is a quasi-unitary matrix, i.e. its first n − 1 rows constitute an orthonormal set, and the last row is orthogonal with respect to this set, but is not normalized. LEGG (UdeC) Diciembre 2, 2015. 30 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Uvarov transformation Let σU be the Uvarov transformation of σ. If we assume {υn }n>0 is its associated OPS, and define by Hυ its corresponding Hessenberg matrix, then Proposition Hϕ − αI = Lϕψ MC , (23) Hυ − αI = LU MU , (24) where LU = Lυϕ Lϕψ , MU = MC L−1 υϕ , and L are the matrices of change of bases for the orthonormal polynomial families denoted by their subindices. LEGG (UdeC) Diciembre 2, 2015. 31 / 42 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg Geronimus transformation Let σG be the Geronimus transformation of σ. If {Gn }n>0 is its OPS and MG a Hessenberg matrix such that (z − α)Φ(z) = MG G(z). Then we get Proposition Let LG be such that G(z) = LG Φ(z) and denote by HG the Hessenberg matrix associated with {Gn }n>0 . Then, HΦ − αI = MG LG (25) HG − αI = LG MG . (26) and LEGG (UdeC) Diciembre 2, 2015. 32 / 42 La representación CMV Contents 1 Polinomios ortogonales en la recta y matrices de Jacobi 2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg 3 La representación CMV 4 Algunas generalizaciones LEGG (UdeC) Diciembre 2, 2015. 33 / 42 La representación CMV Laurent polynomials space Let Λ(k,l) be span{z j }lj=k , k 6 l, and P(k,l) the orthogonal projection over Λ(k,l) with respect to a bilinear functional L. Set Λ (n) n = 2k, Λ(−k,k) = Λ(−k,k+1) n = 2k + 1, and let P(n) be the orthogonal projection over Λ(n) . Furthermore, define χ(0) n z−k = zk+1 n = 2k, n = 2k + 1. Applying Gram-Schmidt, we obtain the CMV basis from χn = (1 − P(n−1) )χ(0) n . LEGG (UdeC) Diciembre 2, 2015. 34 / 42 La representación CMV The CMV basis {χn }n>0 can be expressed in terms of {Φn (z)}n>0 as follows χ2n (z) χ2n−1 (z) = z−n Φ∗2n (z), = z −n+1 n > 0, Φ2n−1 (z), n > 1, and satisfies the following recurrence relations zχ0 ! χ2n−1 z χ2n = −Φ1 (0)χ0 + ρ0 χ1 , ! ! χ2n−2 χ2n T b = Ξ2n−1 + Ξ2n , χ2n−1 χ2n+1 n > 1, with −ρn−1 Φn+1 (0) Ξn := −Φn (0)Φn+1 (0) ! ρn−1 ρn , Φn (0)ρn −ρ̂n−1 Φn+1 (0) b Ξn := −Φn (0)Φn+1 (0) ! ρ̂n−1 ρ̂n , Φn (0)ρ̂n where ρn = |1 − |Φn+1 (0)|2 |1/2 and ρ̂n = ςn ρn , with ςn = sign(1 − |Φn |2 ). LEGG (UdeC) Diciembre 2, 2015. 35 / 42 La representación CMV A five diagonal matrix Thus, the five diagonal matrix C of CMV representation is defined as D E Ci, j = χi , zχ j , L in such a way that −Φ1 (0) ρ0 0 C = 0 0 ... LEGG (UdeC) −Φ2 (0)ρ̂0 −Φ2 (0)Φ1 (0) −Φ3 (0)ρ1 ρ2 ρ1 0 ... ρ̂1 ρ̂0 Φ1 (0)ρ̂1 −Φ3 (0)Φ2 (0) Φ2 (0)ρ2 0 ... 0 0 −Φ4 (0)ρ̂2 −Φ4 (0)Φ3 (0) −Φ5 (0)ρ3 ... 0 ... 0 ... ρ̂3 ρ̂2 ... Φ3 (0)ρ̂3 ... −Φ5 (0)Φ4 (0) . . . ... ... Diciembre 2, 2015. . 36 / 42 La representación CMV CMV factorization Furthermore, C = WM, where 1 M = Θ0 W = Θ1 Θ2 with Θj = LEGG (UdeC) −Φ j+1 (0) ρ̂ j Θ3 Θ4 , . . . , . . . ρj Φ j+1 (0) ! . Diciembre 2, 2015. 37 / 42 La representación CMV ST and CMV matrices Open question Can we express FC , FU , and FG in terms of the corresponding CMV matrices? Partial answer: Yes (Cantero-Marcellán-Velázquez, 2015) LEGG (UdeC) Diciembre 2, 2015. 38 / 42 Algunas generalizaciones Contents 1 Polinomios ortogonales en la recta y matrices de Jacobi 2 Ortogonalidad en la circunferencia unidad y matrices de Hessenberg 3 La representación CMV 4 Algunas generalizaciones LEGG (UdeC) Diciembre 2, 2015. 39 / 42 Algunas generalizaciones Matrix orthogonal polynomials A matrix polynomial has the form P(x) = An zn + . . . A0 , where Ai are q × q matrices. A matrix inner product can be defined by Z P(x)dµ(x)QT (x), E where dµ(x) is a q × q symmetric matrix of measures with support in E ∈ R. Orthogonality is defined by Z Pn (x)dµ(x)PTm (x) = δn.mCn , E where Cn is a nonsingular matrix. LEGG (UdeC) Diciembre 2, 2015. 40 / 42 Algunas generalizaciones Spectral transformation for matrix polynomials Christoffel transformation (Marcellán, Mañas - 2015) Uvarov transformation (Marcellán, Piñar, 2000s) Geronimus transformation (Marcellán, LG - 2015) Other perturbations studied by Choque, Domínguez de la Iglesia, LG. LEGG (UdeC) Diciembre 2, 2015. 41 / 42 Algunas generalizaciones ¡Gracias por su atención! LEGG (UdeC) Diciembre 2, 2015. 42 / 42