Citation: Emily M. Barker, Ashli R. Toles, Kyle A. Guess, Janice Paige Buchanan. C60 and Sc3N@C80(TMB-PPO) derivatives as constituents of singlet oxygen generating, thiol-ene polymer nanocomposites[J]. AIMS Materials Science, 2016, 3(3): 965-988. doi: 10.3934/matersci.2016.3.965
[1] | Huyuan Chen, Laurent Véron . Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data. Mathematics in Engineering, 2019, 1(3): 391-418. doi: 10.3934/mine.2019.3.391 |
[2] | Filippo Gazzola, Gianmarco Sperone . Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations. Mathematics in Engineering, 2022, 4(5): 1-24. doi: 10.3934/mine.2022040 |
[3] | Antonio Vitolo . Singular elliptic equations with directional diffusion. Mathematics in Engineering, 2021, 3(3): 1-16. doi: 10.3934/mine.2021027 |
[4] | Mouhamed Moustapha Fall, Veronica Felli, Alberto Ferrero, Alassane Niang . Asymptotic expansions and unique continuation at Dirichlet-Neumann boundary junctions for planar elliptic equations. Mathematics in Engineering, 2019, 1(1): 84-117. doi: 10.3934/Mine.2018.1.84 |
[5] | David Arcoya, Lucio Boccardo, Luigi Orsina . Hardy potential versus lower order terms in Dirichlet problems: regularizing effects. Mathematics in Engineering, 2023, 5(1): 1-14. doi: 10.3934/mine.2023004 |
[6] | Elena Beretta, M. Cristina Cerutti, Luca Ratti . Lipschitz stable determination of small conductivity inclusions in a semilinear equation from boundary data. Mathematics in Engineering, 2021, 3(1): 1-10. doi: 10.3934/mine.2021003 |
[7] | Boumediene Abdellaoui, Ireneo Peral, Ana Primo . A note on the Fujita exponent in fractional heat equation involving the Hardy potential. Mathematics in Engineering, 2020, 2(4): 639-656. doi: 10.3934/mine.2020029 |
[8] | María Ángeles García-Ferrero, Angkana Rüland . Strong unique continuation for the higher order fractional Laplacian. Mathematics in Engineering, 2019, 1(4): 715-774. doi: 10.3934/mine.2019.4.715 |
[9] | La-Su Mai, Suriguga . Local well-posedness of 1D degenerate drift diffusion equation. Mathematics in Engineering, 2024, 6(1): 155-172. doi: 10.3934/mine.2024007 |
[10] | Patrizia Pucci, Letizia Temperini . On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces. Mathematics in Engineering, 2023, 5(1): 1-21. doi: 10.3934/mine.2023007 |
Traditionally, a matrix is defined as totally positive (TP) if all its minors are nonnegative, and strictly totally positive (STP) if all its minors are strictly positive (see [1,11,25]). It is worth noting that in some literature, TP and STP matrices are referred to as totally nonnegative and totally positive matrices, respectively [9]. A fundamental property of TP matrices is that their product also results in a TP matrix.
A basis (u0,…,un) of a given space U of functions defined on I⊆R is said to be totally positive (TP) if, for any sequence of parameters T:=(t1,…,tN+1) in I with t1<⋯<tN+1 and N≥n, the collocation matrix
MT:=(uj−1(ti))1≤i≤N+1;1≤j≤n+1 | (1.1) |
is TP.
Collocation matrices form a significant class of structured matrices, which has become a prominent research topic in numerical linear algebra, attracting increasing attention in recent years. Extensive studies have focused on algebraic computations for various types of collocation matrices associated with specific bases of polynomials [2,3,7,19,20,21,22,23], rational bases [4,26], or bases of functions of the form tkeλt [16], among others. Once recognized as TP matrices, collocation matrices have found diverse and interesting applications in numerical mathematics, computer-aided geometric design, and statistics.
If U is a Hilbert space of functions equipped with an inner product ⟨⋅,⋅⟩, and (u0,…,un) is a basis of U, the corresponding Gram (or Gramian) matrix is the symmetric matrix G=(gi,j)1≤i,j≤n+1 defined as
gi,j:=⟨ui−1,uj−1⟩,1≤i,j≤n+1. | (1.2) |
Gramian matrices have a wide range of applications. For example, they can be used to transform non-orthogonal bases of U into orthogonal ones, with the Gramian matrix defined in (1.2) serving as the transformation matrix. In least-squares approximation, where a function is represented as a linear combination of a basis in U, the solution involves solving a system of normal equations. The coefficient matrix for this system is precisely the Gramian matrix (1.2) corresponding to the chosen basis. Furthermore, Gramian matrices are also linked to inverse scattering problems [6], highlighting their significance in applied mathematics and physics.
For certain polynomial bases [18] and bases of the form tkeλt [16], these matrices have been efficiently represented through bidiagonal factorizations. These factorizations facilitate the design of highly accurate algorithms for addressing key problems in linear algebra.
As shown in [10,11], any nonsingular TP matrix A∈R(n+1)×(n+1) can be expressed as
A=FnFn−1⋯F1DG1G2⋯Gn, | (1.3) |
where Fi∈R(n+1)×(n+1) (respectively, Gi∈R(n+1)×(n+1)) for i=1,…,n are TP lower (respectively, upper) triangular bidiagonal matrices of the form:
Fi=[1⋱1mi+1,11⋱⋱mn+1,n+1−i1],GTi=[1⋱1˜mi+1,11⋱⋱˜mn+1,n+1−i1], |
and D=diag(pi,i)1≤i≤n+1 is a nonsingular diagonal matrix whose nonzero diagonal entries are called pivots. The off-diagonal entries mi,j, known as multipliers, satisfy
mi,j=pi,jpi−1,j, |
with pi,1:=ai,1, 1≤i≤n+1, and
pi,j:=detA[i−j+1,…,i∣1,…,j]detA[i−j+1,…,i−1∣1,…,j−1], | (1.4) |
for 1<j≤i≤n+1, where the submatrix of A formed by rows i1,…,ir and columns j1,…,js is denoted by A[i1,…,ir∣j1,…,js] and A[i1,…,ir] denotes the matrix A[i1,…,ir∣i1,…,ir].
Similarly, the entries ˜mi,j are given by
˜mi,j=˜pi,j˜pi−1,j, | (1.5) |
where ˜pi,1:=a1,i, and the terms ˜pi,j can be computed as in (1.4) for the transpose AT of the matrix A. For symmetric matrices, it holds that mi,j=˜mi,j, which implies Gi=FTi for i=1,…,n.
The factorization (1.3) offers an explicit expression for the determinant of TP matrices. Furthermore, when its computation avoids inaccurate cancellations, it provides a matrix representation suitable for developing algorithms with high relative accuracy (HRA) to address relevant algebraic problems (cf. [15,16]). Achieving HRA is crucial, as such algorithms ensure that relative errors are on the order of machine precision and remain unaffected by the matrix dimension or condition number. Outstanding results have been obtained for collocation matrices (see [2,3,4,5,19,20,21,22]) as well as for Gramian matrices of bases such as the Poisson and Bernstein bases on the interval [0,1]. Similarly, significant progress has been made with non-polynomial bases like {xkeλx} (see [17], and [16]).
A symmetric function is a function in several variables which remains unchanged for any permutation of its variables. In contrast, a totally antisymmetric function changes sign with any transposition of its variables. If MT is a collocation matrix defined as in (1.1), any minor detMT[i1,…,ir|j1,…,js] is a totally antisymmetric function of the parameters ti1,…,tir. The same applies to the minors of the transpose of MT. These statements may be concisely expressed through the following relations:
detMT[i1,…,ir|j1,…,jr]=g(ti1,…,tir)detVti1,…,tir,detMTT[i1,…,ir|j1,…,jr]=˜g(tj1,…,tjr)detVtj1,…,tjr, |
for suitable symmetric functions g(x1,…,xr), ˜g(x1,…,xr), and Vtj1,…,tjr, the Vandermonde matrix at nodes tj1,…,tjr.
The above observation, together with formula (1.4) for computing the pivots and multipliers of the factorization (1.3), reveals an intriguing connection between symmetric functions and TP bases, previously explored in [7] and [8]. For collocation matrices of polynomial bases, the diagonal pivots and multipliers involved in (1.3) can be expressed in terms of Schur functions, leading to novel insights into the total positivity properties (cf. [7]). Subsequently, [8] extended these results to the class of Wronskian matrices, also deriving their bidiagonal decomposition in terms of symmetric functions.
In this paper, we extend this line of research by considering Gramian matrices (1.2). Due to their inherent symmetry (see (1.4) and (1.5)), computing the pivots and multipliers in the factorization (1.3) reduces to determining minors with consecutive rows and initial consecutive columns, specifically:
detG[i−j+1,…,i|1,…,j],1≤j≤i≤n+1. | (1.6) |
These minors will be central objects of study in this work. Moreover, it will be shown that Gramian matrices can be represented as a specific limit of products of matrices involving collocation matrices of the given basis (see formula (2.2) in Section 2). This framework enables us to derive results analogous to those obtained in [7] for collocation matrices, or in [8] for Wronskian matrices now in the context of Gramian matrices. Ultimately, we establish a connection between the total positivity of Gramian matrices and integrals of symmetric functions.
In the following sections, we first demonstrate that any Gramian matrix for a given basis can be written as a limit of products involving the collocation matrices of the system. Consequently, we establish that Gramian matrices of TP bases are themselves TP. These findings are applied in Section 3, where we represent any minor (1.6) as an integral of products of minors of collocation matrices. In Section 4, we further refine this representation for polynomial bases, expressing the minors in terms of integrals of Schur polynomials. These integrals, known as Selberg-like integrals, have been explicitly computed in the literature and arise naturally in various contexts of Physics and Mathematics, such as the quantum Hall effect and random matrix theory. Relevant results on these integrals, essential for our purposes, are summarized in Section 5. Finally, we include an appendix providing the pseudocode of an algorithm designed for the computation of the determinants in (1.6), specifically tailored for polynomial bases.
Consider U to be a Hilbert space of functions defined on J=[a,b], equipped with the inner product
⟨u,v⟩:=∫Jκ(t)u(t)v(t)dt, | (2.1) |
for a weight function κ satisfying κ(t)≥0, for all t∈J. In this section, we focus on the Gramian matrix G, as defined in (1.2), corresponding to a basis (u0,…,un) of U with respect to the inner product (2.1).
The following result shows that G can be represented as the limit of products involving collocation matrices for (u0,…,un) evaluated at equally spaced sequences of parameters on [a,b] and diagonal matrices containing the values of the weight function κ at those parameters.
Lemma 2.1. Let G be the Gramian matrix (1.2) of a basis (u0,…,un) with respect to the inner product (2.1). Then,
G=limN→∞b−aNUTNKNUN, | (2.2) |
where
UN:=(uj−1(ti))i=1,…,N+1,j=1,…,n+1,KN:=diag(κ(ti))i=1,…,N+1, | (2.3) |
and ti:=a+(i−1)(b−a)/N, i=1,…,N+1, for N∈N.
Proof. Using the definition of the Riemann integral, it is straightforward to verify that the matrix
GN:=b−aNUTNKNUN | (2.4) |
converges component-wise to the Gramian matrix G as N→∞.
Using Lemma 2.1, we establish the total positivity property of Gramian matrices corresponding to TP bases under the inner product (2.1).
Theorem 2.1. Let (u0,…,un) be a TP basis of a space U of functions defined on the interval I. The Gramian matrix G, as defined in (1.2), is TP if J⊆I.
Proof. Let us consider the compact interval J=[a,b]. If (u0,…,un) is a TP basis on I, it remains TP on J⊆I. Consequently, the matrices UN and KN defined in (2.3) are TP for all N∈N. Therefore, the matrix GN in (2.4) is TP for all N∈N, as it is the product of TP matrices.
Let us analyze the sign of the r×r minor detG[i1,…,ir|j1,…,jr] corresponding to rows 1≤i1<⋯<ir≤n+1 and columns 1≤j1<⋯<jr≤n+1. Since GN=(GNi,j)1≤i≤n+1;1≤j≤n+1 is TP, we have
0≤detGN[i1,…,ir|j1,…,jr]=∑σ∈Srsgn(σ)GNi1jσ(1)⋯GNirjσ(r),N∈N, | (2.5) |
where Sr denotes the group of permutations of {1,…,r} and sgn(σ) is the signature of the permutation σ, taking the value 1 if σ is even and −1 if σ is odd. Recall that a permutation is even (or odd) if it can be expressed as the product of an even (or odd) number of transpositions.
From (2.2), we have limN→∞GNi,j=Gi,j and so,
GNi,j=Gi,j+ϵNi,j,limN→∞ϵNi,j=0, | (2.6) |
for 1≤i,j≤n+1. Using (2.5) and (2.6), we derive
0≤detGN[i1,…,ir|j1,…,jr]=∑σ∈Srsgn(σ)r∏ℓ=1(Giℓ,jσ(ℓ)+ϵNiℓ,jσ(ℓ))=detG[i1,…,ir|j1,…,jr]+r∑k=1∑σ∈Srsgn(σ)ϵNik,jσ(k)∏ℓ≠kGNiℓ,jσ(ℓ)≤detG[i1,…,ir|j1,…,jr]+r∑k=1∑σ∈Sr|ϵNik,jσ(k)|∏ℓ≠k|GNiℓ,jσ(ℓ)|,N∈N. |
By defining
ϵN:=max{|ϵNi,j|∣i=i1,…,ir,j=j1,…,jr},ψN:=max{|GNi,j|∣i=i1,…,ir,j=j1,…,jr}, |
we have
0≤detG[i1,…,ir|j1,…,jr]+r∑k=1∑σ∈SrϵNψr−1N=detG[i1,…,ir|j1,…,jr]+r⋅r!ϵNψr−1N,N∈N. |
The value ψN is clearly bounded and ϵN→0 as N→∞. So,
0=limN→∞−r⋅r!ϵNψr−1N≤detG[i1,…,ir|j1,…,jr]. |
Finally, since any r×r minor of G is nonnegative, we conclude that G is a TP matrix.
In this section, we use formula (2.2) to write the determinants in (1.6) as integrals involving the product of minors of specific collocation matrices associated with the considered basis.
Before presenting the main result of the section, we first prove the following auxiliary lemma on the integrals of general symmetric functions.
Proposition 3.1. Let g(x1,…,xj) be a symmetric function. Then
∫badx1∫bx1dx2⋯∫bxj−1dxjg(x1,…,xj)=1j!∫[a,b]jj∏l=1dxlg(x1,…,xj). | (3.1) |
Proof. The integration region of the LHS of (3.1) covers all the points (x1,…,xj) of the hypercube [a,b]j such that a≤x1≤⋯≤xj≤b. On the other hand, the hypercube is fully covered when considering all the points obtained by permuting the variables, that is,
[a,b]j=∪σ∈Sj{xσ=(xσ(1),…,xσ(j))|a≤x1≤⋯≤xj≤b}. |
Two points xσ and xσ′, with σ≠σ′, can be equal only if two or more variables take the same value, say xi=xj. So, only points lying on a face of a simplex are in the intersection set. But the collection of these points forms a set of null measure. Thus the integral over [a,b]j can split into j! integrals, each corresponding to a region labeled by a permutation σ∈Sj. Moreover, since g(x1,…,xj) is a symmetric function, the permutation of variables does not alter the value of the integral. So, all the j! integrals are identical and (3.1) follows.
Theorem 3.1. Let G be the Gramian matrix (1.2) of a basis (u0,…,un) with respect to the inner product (2.1). Then
detG[i−j+1,…,i|1,…,j]=1j!∫[a,b]jj∏l=1dxlκ(xl)detMTX[i−j+1,…,i|1,…,j]detMX[1,…,j], | (3.2) |
where
MX:=(uj−1(xi))1≤i,j≤n+1 |
is the square collocation matrix of (u0,…,un) at the sequence of parameters X=(x1,…,xn+1).
Proof. Given N∈N, we define an equally spaced partition of [a,b] with ti:=a+(i−1)(b−a)/N, i=1,…,N+1. Using basic properties of determinants, we can derive the following identities for the matrix GN in (2.4):
detGN[i−j+1,…,i|1,…,j]=(b−aN)j|∑N+1l=1ui−j+1(tl)u0(tl)κ(tl)…∑N+1l=1ui−j+1(tl)uj−1(tl)κ(tl)⋮⋱⋮∑N+1l=1ui(tl)u0(tl)κ(tl)…∑N+1l=1ui(tl)uj−1(tl)κ(tl)|=(b−aN)jN+1∑k1,…,kj=1j∏l=1ul−1(tkl)κ(tkl)|ui−j+1(tk1)…ui−j+1(tkj)⋮⋱⋮ui(tk1)…ui(tkj)|=(b−aN)j∑k1<⋯<kj∑σ∈Sjj∏l=1ul−1(tkσ(l))κ(tkσ(l))|ui−j+1(tkσ(1))…ui−j+1(tkσ(j))⋮⋱⋮ui(tkσ(1))…ui(tkσ(j))|. | (3.3) |
In the last line in (3.3), we have taken into account that the determinant cancels whenever two of the dummy variables (k1,…,kj) take the same value and so, the total sum can be reorganized into ascending sequences and permutations of the variables (k1,…,kj). Next, we can sum over the permutation group Sj, and obtain
detGN[i−j+1,…,i|1,…,j]=(b−aN)j∑k1<⋯<kj∑σ∈Sjsgn(σ)j∏l=1ul−1(tkσ(l))κ(tkl)|ui−j+1(tk1)…ui−j+1(tkj)⋮⋱⋮ui(tk1)…ui(tkj)|=(b−aN)j∑k1<⋯<kjj∏l=1κ(tkl)|ui−j+1(tk1)…ui−j+1(tkj)⋮⋱⋮ui(tk1)…ui(tkj)||u1(tk1)…uj(tk1)⋮⋱⋮u1(tkj)…uj(tkj)|. |
In general, for any integrable function g(x1,…,xj) defined on [a,b]j, we have
limN→∞∑k1<⋯<kj(b−aN)jg(tk1,…,tkj)=∫badx1∫bx1dx2⋯∫bxj−1dxjg(x1,…,xj). |
Thus,
detG[i−j+1,…,i|1,…,j]=limN→∞detGN[i−j+1,…,i|1,…,j]=∫badx1∫bx1dx2⋯∫bxj−1dxjj∏l=1κ(xl)detMTX[i−j+1,…,i|1,…,j]detMX[1,…,j]=1j!∫[a,b]jj∏l=1dxlκ(xl)detMTX[i−j+1,…,i|1,…,j]detMX[1,…,j], | (3.4) |
where, in the last step of (3.4), we were able to use Proposition 3.1 since the integrand is always a symmetric function in its variables (x1,…,xj).
Theorem 3.1 exhibits an explicit connection between Gramian matrices and collocation matrices. Namely, it shows the relation between the determinants (1.6) and the analogous minors of the collocation matrices that can be constructed within the range of integration. Two comments about Theorem 3.1 are in order. First, it serves as a consistency check of Theorem 2.1, since the integral of the product of positive minors and a positive definite function κ is always positive. Thus, the total positivity of a basis (u0,…,un) translates into the total positivity of G. Second, since the integrand of (3.4) is a symmetric function, the pivots and multipliers of the bidiagonal decomposition of the Gramian matrix associated to a TP basis can be expressed in terms of integrals of symmetric functions. In the following section, we will flesh out the last statement in the case of polynomial bases, for which the integrand is an explicit linear combination of Schur polynomials.
Given a partition λ:=(λ1,λ2,…,λp) of size |λ|:=λ1+λ2+⋯+λp and length l(λ):=p, where λ1≥λ2≥⋯≥λp>0, Jacobi's definition of the corresponding Schur polynomial in n+1 variables is expressed via Weyl's formula as:
sλ(t1,t2,…,tn+1):=det[tλ1+n1tλ1+n2…tλ1+nn+1tλ2+n−11tλ2+n−12…tλ2+n−1n+1⋮⋮⋱⋮tλn+11tλn+12…tλn+1n+1]/det[11…1t1t2…tn+1⋮⋮⋱⋮tn1tn2…tnn+1]. | (4.1) |
By convention, the Schur polynomial associated with the empty partition is defined as s∅(t1,…,tn+1):=1. This serves as the multiplicative identity in the algebra of symmetric functions. When considering all possible partitions, Schur polynomials form a basis for the space of symmetric functions, allowing any symmetric function to be uniquely expressed as a linear combination of Schur polynomials.
Let Pn(I) denote the space of polynomials of degree at most n defined on I⊆R, and let (p0,…,pn) be a basis of Pn(I) such that
pi−1(t)=n+1∑j=1ai,jtj−1,t∈I,i=1,…,n+1. | (4.2) |
We denote by A=(ai,j)1≤i,j≤n+1 the matrix representing the linear transformation from the basis (p0,…,pn) to the monomial polynomial basis of Pn(I). Specifically,
(p0,…,pn)T=A(m0,…,mn)T, | (4.3) |
where (m0,…,mn) denotes the monomial basis.
Let MT be the collocation matrix of (p0,…,pn) at T:=(t1,…,tn+1) on I with t1<⋯<tn+1, defined as
MT:=(pj−1(ti))1≤i,j≤n+1. | (4.4) |
The collocation matrix of the monomial polynomial basis (m0,…,mn) at T corresponds to the Vandermonde matrix VT at the chosen nodes:
VT:=(tj−1i)1≤i,j≤n+1. |
In [7], it was shown how to express the bidiagonal factorization (1.3) of M:=MT in terms of Schur polynomials and some minors of the change of basis matrix A satisfying (4.3). For this purpose, it was shown that
detM[i−j+1,…,i|1,…,j]=detVti−j+1,…,ti∑l1>⋯>ljdetA[1,…,j|lj,…,l1]s(l1−j,…,lj−1)(ti−j+1,…,ti). | (4.5) |
To effectively apply the product rules of Schur polynomials, we express the linear combination in (4.5) using partitions. Consider partitions λ=(λ1,…,λj), where λr=lr+r−j−1 for r=1,…,j. Given that l1>⋯>lj, λ is a well-defined partition. For the minors of matrix A, we use the following notation:
A[i,λ]:=detA[i−j+1,…,i|lj,…,l1]=detA[i−j+1,…,i|λj+1,…,λ1+j]. | (4.6) |
Since the indices satisfy l1>⋯>lj and lk≤n+1, for 1≤k≤n+1, the corresponding partitions will have j parts, each with a maximum length of n+1−j. In other words, the sum in (4.5) spans all Young diagrams that fit within a j×(n+1−j) rectangular box, which can be expressed as:
l(λ)≤j,λ1≤n+1−j. |
With this notation, we have
∑lj<⋯<l1detA[1,…,j|lj,…,l1]s(l1−j,…,lj−1)(ti−j+1,…,ti)=∑l(λ)≤jλ1≤n+1−jA[j,λ]sλ(ti−j+1,…,ti). | (4.7) |
The derived formula (4.5) for the minors of the collocation matrices MT of polynomial bases in terms of Schur polynomials, combined with known properties of these symmetric functions, facilitates a comprehensive characterization of total positivity on unbounded intervals for significant polynomial bases (p0,…,pn) (cf. [7]). Furthermore, considering Eqs (4.5) and (1.4), the bidiagonal factorization (1.3) of Mt1,…,tn+1 was obtained, enhancing high relative accuracy (HRA) computations in algebraic problems involving these matrices.
Equation (3.2) applies to a general basis of functions (u0,…,un). For polynomial bases, fully characterized by the transformation matrix A via (4.3), explicit formulas for the minors (1.6) can be derived in terms of Schur polynomials. In this context, it is important to recall the role of Littlewood-Richardson numbers, denoted cνλ,μ, which describe the coefficients in the expansion of the product of two Schur polynomials. Specifically, given Schur polynomials sλ and sμ associated with partitions λ and μ, their product can be expressed as:
sλ(x1,…,xj)⋅sμ(x1,…,xj)=∑ρcρλμsρ(x1,…,xj). | (4.8) |
The following result provides a compact formula for the minors detG[i−j+1,…,i|1,…,j], representing a significant contribution of this paper.
Theorem 4.1. Let G be the Gramian matrix of a basis (p0,…,pn) of Pn(J) with respect to an inner product (2.1). Let A be the matrix of the linear transformation satisfying (4.3). Then
detG[i−j+1,…,i|1,…,j]=∑l(λ)≤jλ1≤n+1−j∑l(μ)≤jμ1≤n+1−j∑|ρ|=|λ|+|μ|A[j,λ]A[i,μ]cρλμfρ,j,⟨⋅,⋅⟩, | (4.9) |
where the determinants A[j,λ] and A[i,μ] are defined in (4.6), cρλμ are the Littlewood-Richardson numbers,
fρ,j,⟨⋅,⋅⟩:=1j!∫[a,b]jj∏l=1dxlκ(xl)(detVx1,…,xj)2sρ(x1,…,xj), | (4.10) |
and Vx1,…,xj denotes the Vandermonde matrix corresponding to the variables x1,…,xj.
Proof. Consider the general formula for the initial minors of Gramian matrices provided in (3.2). By substituting the minors of the collocation matrices in terms of Schur polynomials, as shown in (4.7), we derive a compact expression for these minors. The derivation relies on the relation (4.8) for the product of Schur polynomials, and the fact that the numbers cρλμ are zero unless |ρ|=|λ|+|μ|.
The evaluation of minors of Gramian matrices using the expression (4.9) involves summing over all partitions λ and μ whose Young diagrams fit within a box of size j×(n+1−j), as well as over partitions ρ for which the Littlewood-Richardson numbers cρλμ are nonzero. For a general matrix A, the complexity of computing minors through (4.9) grows rapidly with n, primarily due to the need to calculate the Littlewood-Richardson numbers. Although combinatorial methods exist for their computation, these coefficients become increasingly costly to determine as n increases. Indeed, it has been conjectured that no algorithm can compute Littlewood-Richardson numbers in polynomial time [24].
However, this limitation is mitigated when considering lower triangular change of basis matrices A, where the computation of initial minors becomes significantly simpler, as we will now show. Notably, polynomial bases corresponding to lower triangular change of basis matrices constitute a broad and commonly used family.
Corollary 4.1. Let G be the Gramian matrix (1.2) of a basis (p0,…,pn) of Pn(J) with respect to an inner product (2.1). If the matrix A satisfying (4.3) is lower triangular, then
detG[i−j+1,…,i|1,…,j]=∑l(μ)≤jμ1≤n+1−jA[j,∅]A[i,μ]fμ,j,⟨⋅,⋅⟩, | (4.11) |
where the determinants A[j,∅] and A[i,μ] are defined in (4.6) and fμ,j,⟨⋅,⋅⟩ is defined in (4.10).
Proof. In (4.9), be aware that, for lower triangular matrices A, A[j,λ]≠0 only in the case where λ=∅. Then use the following property of Littlewood-Richardson numbers cρ∅μ=δρμ.
In this section, we address the computation of the values fμ,j,⟨⋅,⋅⟩. For a specific inner product, the integrals in (4.10) have been explicitly calculated and are commonly referred to as Selberg-like integrals. The case of integrals with the inner product
κ(t):=tα(1−t)β,J=[0,1], |
has been studied by Kadell, among others. In [13,14], it was found that for α,β>−1 and a partition ρ=(ρ1,…,ρj), we have
Ij(α,β;ρ)=∫[0,1]jj∏l=1[dxlxαl(1−xl)β](detVx1,…,xj)2sρ(x1,…,xj)=j!∏1≤i<k≤j(ρi−ρk+k−i)j∏i=1Γ(α+ρi+j−i+1)Γ(β+j−i+1)Γ(α+β+2j−i+1+ρi). | (5.1) |
Also interesting for us is the result for the integral involving the product of two Schur polynomials with the same arguments in the case that β=0. In [12], it was found that for α>−1 and partitions λ=(λ1,…,λj) and μ=(μ1,…,μj), we have
Ij(α;λ,μ)=∫[0,1]jj∏l=1[dxlxαl](detVx1,…,xj)2sλ(x1,…,xj)sμ(x1,…,xj)=j!∏1≤i<k≤j(k−i+μi−μk)(k−i+λi−λk)j∏i,k=11α+2j−i−k+1+λi+μk. | (5.2) |
Now, the integral (5.2) can be used by substituting
∑|ρ|=|λ|+|μ|cρλμfρ,j,⟨⋅,⋅⟩=1j!Ij(α;λ,μ) |
in (4.9). Thus, for the case κ(t)=tα and J=[0,1], we have
detG[i−j+1,…,i|1,…,j]=1j!∑l(λ)≤jλ1≤n+1−j∑l(μ)≤jμ1≤n+1−jA[j,λ]A[i,μ]Ij(α;λ,μ). | (5.3) |
So, for this particular case of inner product, the computation of (5.3) does not involve the Littlewood-Richardson numbers, which significantly reduces the computational complexity. This simplification arises from the remarkable properties of (5.2), which implicitly incorporates these numbers.
Algorithm 1 (see Appendix) provides an implementation of (5.3). To illustrate the application of this formula for computing the minors detG[i−j+1,…,i|1,…,j], we present two notable examples that highlight its efficiency and utility.
Bernstein polynomials, defined as
Bni(t):=(ni)ti(1−t)n−i,i=0,…,n, |
are square-integrable functions with respect to the inner product
⟨f,g⟩α,β:=∫10tα(1−t)βf(t)g(t)dt,α,β>−1. | (5.4) |
The Gramian matrix of the Bernstein basis (Bn0,…,Bnn) under the inner product (5.4) is denoted as Gα,β=(gα,βi,j)1≤i,j≤n+1, where
gα,βi,j=(ni−1)(nj−1)Γ(i+j+α−1)Γ(2n−i−j+β+3)Γ(2n+α+β+2), |
for 1≤i,j≤n+1, and Γ(x) is the Gamma function (see [17]). In the special case where α=β=0, the Gramian matrix M:=G(0,0) is referred to as the Bernstein mass matrix.
For n=2, G:=G0,0 is
G=(1/51/101/301/102/151/101/301/101/5). |
It can be easily checked that (B20,B21,B22)T=A(1,t,t2)T, with
A=(1−2102−2001). |
Be aware that the matrix of change of basis A is not lower triangular, so we cannot use (4.11) to compute the minors. Instead, we will use formula (5.2).
For detM[2,3|1,2], the partitions used for λ are {(1,1),(1,0),(0,0)} and for μ are {(1,1),(1,0),(0,0)}. From (4.6), we can obtain A[2,λ] and A[3,μ], respectively, as follows:
A[2,(1,1)]=detA[1,2|2,3]=2,A[2,(1,0)]=detA[1,2|1,3]=−1,A[2,(0,0)]=detA[1,2|1,2]=2,A[3,(1,1)]=detA[2,3|2,3]=2,A[3,(1,0)]=detA[2,3|1,3]=0,A[3,(0,0)]=detA[2,3|1,2]=0. |
Then, taking into account (5.2) and I(λ,μ):=I2(α;λ,μ) for j=2 and α=0, we can obtain
I((1,1),(1,1))=2/240,I((1,1),(1,0))=I((1,0),(1,1))=2/60,I((1,1),(0,0))=I((0,0),(1,1))=2/72,I((1,0),(1,0))=8/45,I((1,0),(0,0))=2/12,I((0,0),(0,0))=2/12. |
Now, by (5.3), we obtain
detM[2,3|1,2]=(1/2)(A[2,(1,1)](A[3,(1,1)]I((1,1),(1,1))+A[3,(1,0)]I((1,1),(1,0))+A[3,(0,0)]I((1,1),(0,0)))+(A[2,(1,0)](A[3,(1,1)]I((1,0),(1,1))+A[3,(1,0)]I((1,0),(1,0))+A[3,(0,0)]I((1,0),(0,0)))+(A[2,(0,0)](A[3,(1,1)]I((0,0),(1,1))+A[3,(1,0)]I((0,0),(1,0))+A[3,(0,0)]I((0,0),(0,0)))=1/180. |
Following the same reasoning, we can obtain the other determinants.
While the matrix A is not lower triangular, the computation of Littlewood-Richardson numbers can be avoided by using (5.2). Specifically, (5.3) can be efficiently applied to any polynomial basis, provided that the inner product is defined as in (5.4) with β=0.
As previously noted, the computation of Littlewood-Richardson numbers is also unnecessary for polynomial bases associated with lower triangular matrices A. In such cases, the formula (4.11) is applicable for any inner product, provided the corresponding Selberg-like integrals can be efficiently evaluated. This approach achieves a comparable level of computational efficiency and is demonstrated in the following example, which features a generic recursive basis.
The example of recursive bases illustrates that Eq (4.11) can be highly effective for computing the minors detG[i−j+1,…,i|1,…,j], especially when the structure of the basis change matrix A facilitates systematic computation of its minors involving consecutive rows.
For given values b1,…,bn+1 with bi>0 for i=1,…,n+1, the recursive basis (p0,…,pn) is defined by the polynomials:
pi=i+1∑j=1bjtj−1,i=0,…,n. |
The corresponding change of basis matrix B, which satisfies (p0,…,pn)T=B(m0,…,mn)T, where mi:=ti for i=0,…,n, is a nonsingular, lower triangular, and the TP matrix is structured as follows:
B=(b100⋯0b1b20⋯0b1b2b3⋯0⋮⋮⋮⋱⋮b1b2b3⋯bn+1). |
Thus, the basis (p0,…,pn) is TP for t∈[0,∞). As before, we consider the inner product defined as:
⟨f,g⟩=∫10f(t)g(t)dt, |
which corresponds to the special case of the inner product (5.4) with α=β=0.
Let us note that the only nonzero minors of B are
B[i−j+1,…,i|m,i−j+2,i−j+3,…,i]=bmj∏k=2bi−j+k,m=1,…,i−j+1. |
This way, the only nonzero contributions to (4.11) come from the partitions μ=(μ1,…,μj) with
μr=i−j,r=1,…,j−1,μj=0,…,i−j. |
Applying (4.11) with (5.1), we obtain
detG[i−j+1,…,i|1,…,j]=b1j∏k=2bi−j+kbkj−1∏l=1(j−l)!2(i−l+1)ji−j∑m=01(m+1)jj−1∏r=1(i−m−r)bm+1, |
where (x)n:=x(x+1)⋯(x+n−1) denotes the Pochhammer symbol for ascending factorials.
We have shown that Gramian matrices can be expressed as limits of products of collocation matrices associated with the corresponding bases. This formulation allows the total positivity property of the bases to be extended to their Gramian matrices, whose minors can be written in terms of Selberg-like integrals, in the polynomial case.
Several open lines of research will be dealt with in future works. A logical continuation of this work is to consider other internal products. We would like to point out that aside from the conceptual and theoretical interest that Eqs (4.9) and (4.11) may have, their computational operativeness crucially depends on the Selberg-like integrals to be solved. Different bases may be TP for different ranges, and therefore the use of suitable inner products will be necessary. The chosen inner product will enter explicitly the computation of the minors of G in Theorem 4.1 through the multivariable integrals fρ,j,⟨⋅,⋅⟩. For this reason, in the future, it will be desirable to solve other Selberg-like integrals.
Besides the above proposal, Gramian matrices of non-polynomial basis could be considered, and their bidiagonal decomposition studied using Theorem 3.1.
Pablo Díaz: Conceptualization, methodology, investigation, writing—original draft, writing—review and editing; Esmeralda Mainar: Conceptualization, methodology, investigation, writing—original draft, writing—review and editing; Beatriz Rubio: Conceptualization, methodology, investigation, writing—original draft, writing—review and editing. All authors have read and agreed to the published version of the manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research was partially supported by Spanish research grants PID2022-138569NB-I00 (MCI/AEI) and RED2022-134176-T (MCI/AEI) and by Gobierno de Aragón (E41_23R).
Esmeralda Mainar is the Guest Editor of special issue "Advances in Numerical Linear Algebra: Theory and Methods" for AIMS Mathematics. Esmeralda Mainar was not involved in the editorial review and the decision to publish this article.
The authors declare no conflicts of interest.
Implementation of the code of formula (5.3)
Algorithm 1: MATLAB code formula (5.3) |
Require: alpha, i, j, A |
Ensure: G [i-j+1, …, i—1, …, j] (see (5.3)) |
n = size(A, 1) |
parts = partitions(j, n) |
partsize = size(parts, 1) |
total = 0 |
for k1 = 1:partsize |
rho = parts(k1, :) |
sum = 0 |
for k2 = 1:partsize |
mu = parts(k2, :) |
sum = sum + Alambda(A, j, i, mu)∗ f(alpha, j, rho, mu) |
end |
G = G + Alambda(A, j, j, rho) ∗ sum |
end |
function s = s(j, vector) |
comb = transpose(nchoosek(1:j, 2)) |
s = 1 |
for c = comb |
i = c(1) |
k = c(2) |
s = s ∗ (vector(i)-vector(k)+k-i) |
end |
function part = partitionsR(from, level) |
part = [] |
for value = from:-1:0 |
if level > 1 |
res = partitionsR(min(from, value), level-1) |
part = [part; [value.∗ ones(size(res, 1), 1) res]] |
else |
part = [part; value] |
end |
end |
function partitions = partitions(j, n) |
partitions = partitionsR(n-j, j) |
function I = f(alpha, j, rho, mu) |
I = s(j, rho) ∗ s(j, mu) |
for i = 1:j |
for k = 1:j |
I = I ∗ 1 / (alpha + 2∗j - i - k + 1 + rho(i) + mu(k)) |
end |
end |
function Alambda = Alambda(A, j, i, lambda) |
rows = (i-j+1):i |
cols = flip(lambda) + (1:j) |
Alambda = det(A(rows, cols)) |
[1] |
Arbogast JW, Darmanyan AP, Foote CS, et al. (1991) Photophysical properties of sixty atom carbon molecule (C60). J Phys Chem 95: 11–12. doi: 10.1021/j100154a006
![]() |
[2] |
Ching WY, Huang MZ, Xu YN, et al. (1991) First-principles calculation of optical properties of the carbon sixty-atom molecule in the fcc. lattice. Phys Rev Lett 67: 2045–2048. doi: 10.1103/PhysRevLett.67.2045
![]() |
[3] |
Maser W, Roth S, Anders J, et al. (1992) P-Type doping of C60 fullerene films. Synth Met 51: 103–108. doi: 10.1016/0379-6779(92)90259-L
![]() |
[4] | Sun Y-P, Lawson GE, Riggs JE, et al. (1998) Photophysical and Nonlinear Optical Properties of [60]Fullerene Derivatives. J Phys Chem A 102: 5520–5528. |
[5] |
Accorsi G, Armaroli N (2010) Taking Advantage of the Electronic Excited States of [60]-Fullerenes. J Phys Chem C 114: 1385–1403. doi: 10.1021/jp9092699
![]() |
[6] | Allemand PM, Khemani KC, Koch A, et al. (1991) Organic molecular soft ferromagnetism in a fullerene C60. Science 253: 301–303. |
[7] | Stephens PW, Cox D, Lauher JW, et al. (1992) Lattice structure of the fullerene ferromagnet TDAE-C60. Nature 355: 331–332. |
[8] | Hebard AF, Rosseinsky MJ, Haddon RC, et al. (1991) Superconductivity at 18 K in potassium-doped fullerene (C60). Nature 350: 600–601. |
[9] | Dubois D, Moninot G, Kutner W, et al. (1992) Electroreduction of Buckminsterfullerene, C60, in aprotic solvents. Solvent, supporting electrolyte, and temperature effects. J Phys Chem 96: 7137–7145. |
[10] |
Schon TB, Di Carmine PM, Seferos DS (2014) Polyfullerene Electrodes for High Power Supercapacitors. Adv Energy Mater 4: 1301509–1301515. doi: 10.1002/aenm.201301509
![]() |
[11] |
Pupysheva OV, Farajian AA, Yakobson BI (2008) Fullerene Nanocage Capacity for Hydrogen Storage. Nano Lett 8: 767–774. doi: 10.1021/nl071436g
![]() |
[12] |
Nadtochenko VA, Vasil'ev IV, Denisov NN, et al. (1993) Photophysical properties of fullerene C60: picosecond study of intersystem crossing. J Photochem Photobiol, A 70: 153–156. doi: 10.1016/1010-6030(93)85035-7
![]() |
[13] | Foote CS (1994) Photophysical and photochemical properties of fullerenes. Electron Transfer I. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 347–363. |
[14] | Sun R, Jin C, Zhang X, et al. (1994) Photophysical properties of C60. Wuli 23: 83–87. |
[15] |
Qu B, Chen SM, Dai LM (2000) Simulation analysis of ESR spectrum of polymer alkyl-C60 radicals formed by photoinitiated reactions of low-density polyethylene. Appl Magn Reson 19: 59–67. doi: 10.1007/BF03162261
![]() |
[16] |
Guldi DM, Asmus K-D (1997) Photophysical Properties of Mono- and Multiply-Functionalized Fullerene Derivatives. J Phys Chem A 101: 1472–1481. doi: 10.1021/jp9633557
![]() |
[17] |
McEwen CN, McKay RG, Larsen BS (1992) C60 as a radical sponge. J Am Chem Soc 114: 4412–4414. doi: 10.1021/ja00037a064
![]() |
[18] | Tzirakis MD, Orfanopoulos M (2013) Radical Reactions of Fullerenes: From Synthetic Organic Chemistry to Materials Science and Biology. Chem Rev 113: 5262–5321. |
[19] | Krusic PJ, Wasserman E, Keizer PN, et al. (1991) Radical reactions of C60. Science 254: 1183–1185. |
[20] |
Krusic PJ, Wasserman E, Parkinson BA, et al. (1991) Electron spin resonance study of the radical reactivity of C60. J Am Chem Soc 113: 6274–6275. doi: 10.1021/ja00016a056
![]() |
[21] |
Wu S-H, Sun W-Q, Zhang D-W, et al. (1998) Reaction of [60]fullerene with trialkylphosphine oxide. Tetrahedron Lett 39: 9233–9236. doi: 10.1016/S0040-4039(98)02131-5
![]() |
[22] |
Cheng F, Yang X, Fan C, et al. (2001) Organophosphorus chemistry of fullerene: synthesis and biological effects of organophosphorus compounds of C60. Tetrahedron 57: 7331–7335. doi: 10.1016/S0040-4020(01)00670-6
![]() |
[23] |
Cheng F, Yang X, Zhu H, et al. (2000) Synthesis and optical properties of tetraethyl methano[60]fullerenediphosphonate. Tetrahedron Lett 41: 3947–3950. doi: 10.1016/S0040-4039(00)00491-3
![]() |
[24] |
Liu Z-B, Tian J-G, Zang W-P, et al. (2003) Large optical nonlinearities of new organophosphorus fullerene derivatives. Appl Opt 42: 7072–7076. doi: 10.1364/AO.42.007072
![]() |
[25] |
Ford WT, Nishioka T, Qiu F, et al. (1999) Structure Determination and Electrochemistry of Products from the Radical Reaction of C60 with Azo(bisisobutyronitrile). J Org Chem 64: 6257–6262. doi: 10.1021/jo990346w
![]() |
[26] |
Ford WT, Nishioka T, Qiu F, et al. (2000) Dimethyl Azo(bisisobutyrate) and C60 Produce 1,4- and 1,16-Di(2-carbomethoxy-2-propyl)-1,x-dihydro[60]fullerenes. J Org Chem 65: 5780–5784. doi: 10.1021/jo000686d
![]() |
[27] |
Shustova NB, Peryshkov DV, Kuvychko IV, et al. (2011) Poly(perfluoroalkylation) of Metallic Nitride Fullerenes Reveals Addition-Pattern Guidelines: Synthesis and Characterization of a Family of Sc3N@C80(CF3)n (n = 2-16) and Their Radical Anions. J Am Chem Soc 133: 2672–2690. doi: 10.1021/ja109462j
![]() |
[28] |
Shu C, Slebodnick C, Xu L, et al. (2008) Highly Regioselective Derivatization of Trimetallic Nitride Templated Endohedral Metallofullerenes via a Facile Photochemical Reaction. J Am Chem Soc 130: 17755–17760. doi: 10.1021/ja804909t
![]() |
[29] |
Shu C, Cai T, Xu L, et al. (2007) Manganese(III)-Catalyzed Free Radical Reactions on Trimetallic Nitride Endohedral Metallofullerenes. J Am Chem Soc 129: 15710–15717. doi: 10.1021/ja0768439
![]() |
[30] |
Shustova NB, Popov AA, Mackey MA, et al. (2007) Radical Trifluoromethylation of Sc3N@C80. J Am Chem Soc 129: 11676–11677. doi: 10.1021/ja074332g
![]() |
[31] |
Cardona CM, Kitaygorodskiy A, Echegoyen L (2005) Trimetallic nitride endohedral metallofullerenes: Reactivity dictated by the encapsulated metal cluster. J Am Chem Soc 127: 10448–10453. doi: 10.1021/ja052153y
![]() |
[32] | Yu G, Gao J, Hummelen JC, et al. (1995) Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions. Science 270: 1789–1791. |
[33] |
Troshin PA, Hoppe H, Renz J, et al. (2009) Material Solubility-Photovoltaic Performance Relationship in the Design of Novel Fullerene Derivatives for Bulk Heterojunction Solar Cells. Adv Funct Mater 19: 779–788. doi: 10.1002/adfm.200801189
![]() |
[34] |
Jiao F, Liu Y, Qu Y, et al. (2010) Studies on anti-tumor and antimetastatic activities of fullerenol in a mouse breast cancer model. Carbon 48: 2231–2243. doi: 10.1016/j.carbon.2010.02.032
![]() |
[35] |
Xu J-Y, Su Y-Y, Cheng J-S, et al. (2010) Protective effects of fullerenol on carbon tetrachloride-induced acute hepatotoxicity and nephrotoxicity in rats. Carbon 48: 1388–1396. doi: 10.1016/j.carbon.2009.12.029
![]() |
[36] |
Mikawa M, Kato H, Okumura M, et al. (2001) Paramagnetic Water-Soluble Metallofullerenes Having the Highest Relaxivity for MRI Contrast Agents. Bioconjugate Chem 12: 510–514. doi: 10.1021/bc000136m
![]() |
[37] |
Chen C, Xing G, Wang J, et al. (2005) Multihydroxylated [Gd@C82(OH)22]n Nanoparticles: Antineoplastic Activity of High Efficiency and Low Toxicity. Nano Lett 5: 2050–2057. doi: 10.1021/nl051624b
![]() |
[38] |
Aoshima H, Kokubo K, Shirakawa S, et al. (2009) Antimicrobial activity of fullerenes and their hydroxylated derivatives. Biocontrol Sci 14: 69–72. doi: 10.4265/bio.14.69
![]() |
[39] |
Guldi DM, Asmus K-D (1999) Activity of water-soluble fullerenes towards ·OH-radicals and molecular oxygen. Radiat Phys Chem 56: 449–456. doi: 10.1016/S0969-806X(99)00325-4
![]() |
[40] |
Lai HS, Chen WJ, Chiang LY (2000) Free radical scavenging activity of fullerenol on the ischemia-reperfusion intestine in dogs. World J Surg 24: 450–454. doi: 10.1007/s002689910071
![]() |
[41] |
Sun D, Zhu Y, Liu Z, et al. (1997) Active oxygen radical scavenging ability of water-soluble fullerenols. Chin Sci Bull 42: 748–752. doi: 10.1007/BF03186969
![]() |
[42] |
Dugan LL, Gabrielsen JK, Yu SP, et al. (1996) Buckminsterfullerenol free radical scavengers reduce excitotoxic and apoptotic death of cultured cortical neurons. Neurobiol Dis 3: 129–135. doi: 10.1006/nbdi.1996.0013
![]() |
[43] | Chiang LY, Lu F-J, Lin J-T (1995) Free radical scavenging activity of water-soluble fullerenols. J Chem Soc, Chem Commun: 1283–1284. |
[44] |
Xiao L, Takada H, Maeda K, et al. (2005) Antioxidant effects of water-soluble fullerene derivatives against ultraviolet ray or peroxylipid through their action of scavenging the reactive oxygen species in human skin keratinocytes. Biomed Pharmacother 59: 351–358. doi: 10.1016/j.biopha.2005.02.004
![]() |
[45] |
Oberdorster E (2004) Manufactured nanomaterials (fullerenes, C60) induce oxidative stress in the brain of juvenile largemouth bass. Environ Health Perspect 112: 1058–1062. doi: 10.1289/ehp.7021
![]() |
[46] | Hamano T, Okuda K, Mashino T, et al. (1997) Singlet oxygen production from fullerene derivatives: effect of sequential functionalization of the fullerene core. Chem Commun 21–22. |
[47] |
Guldi DM, Prato M (2000) Excited-State Properties of C60 Fullerene Derivatives. Acc Chem Res 33: 695–703. doi: 10.1021/ar990144m
![]() |
[48] |
Jensen AW, Daniels C (2003) Fullerene-Coated Beads as Reusable Catalysts. J Org Chem 68: 207–210. doi: 10.1021/jo025926z
![]() |
[49] |
Jensen AW, Maru BS, Zhang X, et al. (2005) Preparation of fullerene-shell dendrimer-core nanoconjugates. Nano Lett 5: 1171–1173. doi: 10.1021/nl0502975
![]() |
[50] |
Foote CS (1994) Photophysical and photochemical properties of fullerenes. Top Curr Chem 169: 347–363. doi: 10.1007/3-540-57565-0_80
![]() |
[51] |
McCluskey DM, Smith TN, Madasu PK, et al. (2009) Evidence for Singlet-Oxygen Generation and Biocidal Activity in Photoresponsive Metallic Nitride Fullerene-Polymer Adhesive Films. ACS Appl Mater Interfaces 1: 882–887. doi: 10.1021/am900008v
![]() |
[52] | Alberti MN, Orfanopoulos M (2010) Recent mechanistic insights in the singlet oxygen ene reaction. Synlett 999–1026. |
[53] | Foote CS, Wexler S, Ando W (1965) Singlet oxygen. III. Product selectivity. Tetrahedron Lett 4111–4118. |
[54] | Dallas P, Rogers G, Reid B, et al. (2016) Charge separated states and singlet oxygen generation of mono and bis adducts of C60 and C70. Chem Phys 465–466: 28–39. |
[55] | Yano S, Naemura M, Toshimitsu A, et al. (2015) Efficient singlet oxygen generation from sugar pendant C60 derivatives for photodynamic therapy [Erratum to document cited in CA163:618143]. Chem Commun 51: 17631–17632. |
[56] |
Prat F, Stackow R, Bernstein R, et al. (1999) Triplet-State Properties and Singlet Oxygen Generation in a Homologous Series of Functionalized Fullerene Derivatives. J Phys Chem A 103: 7230–7235. doi: 10.1021/jp991237o
![]() |
[57] |
Tegos GP, Demidova TN, Arcila-Lopez D, et al. (2005) Cationic Fullerenes Are Effective and Selective Antimicrobial Photosensitizers. Chem Biol 12: 1127–1135. doi: 10.1016/j.chembiol.2005.08.014
![]() |
[58] |
Schinazi RF, Sijbesma R, Srdanov G, et al. (1993) Synthesis and virucidal activity of a water-soluble, configurationally stable, derivatized C60 fullerene. Antimicrob Agents Chemother 37: 1707–1710. doi: 10.1128/AAC.37.8.1707
![]() |
[59] | Dai L (1999) Advanced syntheses and microfabrications of conjugated polymers, C60-containing polymers and carbon nanotubes for optoelectronic applications. Polym Adv Technol 10: 357–420. |
[60] |
Phillips JP, Deng X, Todd ML, et al. (2008) Singlet oxygen generation and adhesive loss in stimuli-responsive, fullerene-polymer blends, containing polystyrene-block-polybutadiene- block-polystyrene and polystyrene-block-polyisoprene-block-polystyrene rubber-based adhesives. J Appl Polym Sci 109: 2895–2904. doi: 10.1002/app.28337
![]() |
[61] |
Lundin JG, Giles SL, Cozzens RF, et al. (2014) Self-cleaning photocatalytic polyurethane coatings containing modified C60 fullerene additives. Coatings 4: 614–629. doi: 10.3390/coatings4030614
![]() |
[62] |
Phillips JP, Deng X, Stephen RR, et al. (2007) Nano- and bulk-tack adhesive properties of stimuli-responsive, fullerene-polymer blends, containing polystyrene-block-polybutadiene- block-polystyrene and polystyrene-block-polyisoprene-block-polystyrene rubber-based adhesives. Polymer 48: 6773–6781. doi: 10.1016/j.polymer.2007.08.050
![]() |
[63] |
Samulski ET, DeSimone JM, Hunt MO, Jr., et al. (1992) Flagellenes: nanophase-separated, polymer-substituted fullerenes. Chem Mater 4: 1153–1157. doi: 10.1021/cm00024a011
![]() |
[64] |
Chiang LY, Wang LY, Kuo C-S (1995) Polyhydroxylated C60 Cross-Linked Polyurethanes. Macromolecules 28: 7574–7576. doi: 10.1021/ma00126a042
![]() |
[65] | Ahmed HM, Hassan MK, Mauritz KA, et al. (2014) Dielectric properties of C60 and Sc3N@C80 fullerenol containing polyurethane nanocomposites. J Appl Polym Sci 131: 40577–40588. |
[66] | Kokubo K, Takahashi R, Kato M, et al. (2014) Thermal and thermo-oxidative stability of thermoplastic polymer nanocomposites with arylated [60]fullerene derivatives. Polym Compos: 1–9. |
[67] |
Shin J, Nazarenko S, Phillips JP, et al. (2009) Physical and chemical modifications of thiol-ene networks to control activation energy of enthalpy relaxation. Polymer 50: 6281–6286. doi: 10.1016/j.polymer.2009.10.053
![]() |
[68] |
Hoyle CE, Bowman CN (2010) Thiol-ene click chemistry. Angew Chem Int Ed 49: 1540–1573. doi: 10.1002/anie.200903924
![]() |
[69] |
Hoyle CE, Lee TY, Roper T (2004) Thiol–enes: Chemistry of the past with promise for the future. J Polym Sci A Polym Chem 42: 5301–5338. doi: 10.1002/pola.20366
![]() |
[70] |
Cramer NB, Scott JP, Bowman CN (2002) Photopolymerizations of Thiol-Ene Polymers without Photoinitiators. Macromolecules 35: 5361–5365. doi: 10.1021/ma0200672
![]() |
[71] |
Li Q, Zhou H, Hoyle CE (2009) The effect of thiol and ene structures on thiol–ene networks: Photopolymerization, physical, mechanical and optical properties. Polymer 50: 2237–2245. doi: 10.1016/j.polymer.2009.03.026
![]() |
[72] |
Northrop BH, Coffey RN (2012) Thiol-Ene Click Chemistry: Computational and Kinetic Analysis of the Influence of Alkene Functionality. J Am Chem Soc 134: 13804–13817. doi: 10.1021/ja305441d
![]() |
[73] |
Singh R, Goswami T (2011) Understanding of thermo-gravimetric analysis to calculate number of addends in multifunctional hemi-ortho ester derivatives of fullerenol. Thermochimica Acta 513: 60–67. doi: 10.1016/j.tca.2010.11.012
![]() |
[74] |
Barker EM, Buchanan JP (2016) Thiol-ene polymer microbeads prepared under high-shear and their successful utility as a heterogeneous photocatalyst via C60-capping. Polymer 92: 66–73. doi: 10.1016/j.polymer.2016.03.091
![]() |
[75] | Jockusch S, Turro NJ (1998) Phosphinoyl Radicals: Structure and Reactivity. A Laser Flash Photolysis and Time-Resolved ESR Investigation. J Am Chem Soc 120: 11773–11777. |
[76] |
Ruoff RS, Tse DS, Malhotra R, et al. (1993) Solubility of fullerene (C60) in a variety of solvents. J Phys Chem 97: 3379–3383. doi: 10.1021/j100115a049
![]() |
[77] | Ginzburg BM, Shibaev LA, Melenevskaja EY, et al. (2004) Thermal and Tribological Properties of Fullerene-Containing Composite Systems. Part 1. Thermal Stability of Fullerene-Polymer Systems. J Macromol Sci Phys 43: 1193–1230. |
[78] |
Leifer SD, Goodwin DG, Anderson MS, et al. (1995) Thermal decomposition of a fullerene mix. Phys Rev B Condens Matter 51: 9973–9981. doi: 10.1103/PhysRevB.51.9973
![]() |
[79] | Mackey MA (2011) Exploration in metallic nitride fullerenes and oxometallic fullerenes: A new class of metallofullerenes [Ph.D. Dissertation]. Hattiesburg, MS: The University of Southern Mississippi. |