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Research article Special Issues

Assessment of the BTEX concentrations and health risk in urban nursery schools in Gliwice, Poland

  • Received: 29 August 2016 Accepted: 04 December 2016 Published: 07 December 2016
  • Indoor air quality (IAQ) in nursery school is believed to be different from elementary school. Moreover, younger children are more vulnerable to air pollution than higher grade children because they spend more time indoors, and their immune systems and bodies are less mature. The purpose of this study was to compare the concentrations of the monoaromatic volatile benzene, toluene, ethylbenzene m,p-xylene and o-xylene (BTEX) in urban nursery schools located in Gliwice, Poland. The nursery schools were chosen to include areas with different urbanization and traffic density characteristics in order to gather a more diverse picture of exposure risks in the various regions of the city. BTEX were sampled during winter and spring seasons in older and younger children classrooms. The samples were thermally desorbed (TD) and then analyzed with use of gas chromatography (GC). In addition, outdoor measurements were carried out in the playground at each nursery school. BTEX quantification, indoor/outdoor concentration, and correlation coefficients were used to identify pollutant sources. Elevated levels of o-xylene and ethylbenzene were found in all monitored classrooms during the winter season. Outdoor concentrations were lower than indoors for each classroom. Indicators based on health risk assessment for chronic health effects associated with carcinogenic benzene or non-carcinogenic BTEX were proposed to rank sites according to their hazard level.

    Citation: Anna Mainka, Barbara Kozielska. Assessment of the BTEX concentrations and health risk in urban nursery schools in Gliwice, Poland[J]. AIMS Environmental Science, 2016, 3(4): 858-870. doi: 10.3934/environsci.2016.4.858

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  • Indoor air quality (IAQ) in nursery school is believed to be different from elementary school. Moreover, younger children are more vulnerable to air pollution than higher grade children because they spend more time indoors, and their immune systems and bodies are less mature. The purpose of this study was to compare the concentrations of the monoaromatic volatile benzene, toluene, ethylbenzene m,p-xylene and o-xylene (BTEX) in urban nursery schools located in Gliwice, Poland. The nursery schools were chosen to include areas with different urbanization and traffic density characteristics in order to gather a more diverse picture of exposure risks in the various regions of the city. BTEX were sampled during winter and spring seasons in older and younger children classrooms. The samples were thermally desorbed (TD) and then analyzed with use of gas chromatography (GC). In addition, outdoor measurements were carried out in the playground at each nursery school. BTEX quantification, indoor/outdoor concentration, and correlation coefficients were used to identify pollutant sources. Elevated levels of o-xylene and ethylbenzene were found in all monitored classrooms during the winter season. Outdoor concentrations were lower than indoors for each classroom. Indicators based on health risk assessment for chronic health effects associated with carcinogenic benzene or non-carcinogenic BTEX were proposed to rank sites according to their hazard level.


    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 IR is said to be totally positive (TP) if, for any sequence of parameters T:=(t1,,tN+1) in I with t1<<tN+1 and Nn, the collocation matrix

    MT:=(uj1(ti))1iN+1;1jn+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)1i,jn+1 defined as

    gi,j:=ui1,uj1,1i,jn+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 AR(n+1)×(n+1) can be expressed as

    A=FnFn1F1DG1G2Gn, (1.3)

    where FiR(n+1)×(n+1) (respectively, GiR(n+1)×(n+1)) for i=1,,n are TP lower (respectively, upper) triangular bidiagonal matrices of the form:

    Fi=[11mi+1,11mn+1,n+1i1],GTi=[11˜mi+1,11˜mn+1,n+1i1],

    and D=diag(pi,i)1in+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,jpi1,j,

    with pi,1:=ai,1, 1in+1, and

    pi,j:=detA[ij+1,,i1,,j]detA[ij+1,,i11,,j1], (1.4)

    for 1<jin+1, where the submatrix of A formed by rows i1,,ir and columns j1,,js is denoted by A[i1,,irj1,,js] and A[i1,,ir] denotes the matrix A[i1,,iri1,,ir].

    Similarly, the entries ˜mi,j are given by

    ˜mi,j=˜pi,j˜pi1,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[ij+1,,i|1,,j],1jin+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 tJ. 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=limNbaNUTNKNUN, (2.2)

    where

    UN:=(uj1(ti))i=1,,N+1,j=1,,n+1,KN:=diag(κ(ti))i=1,,N+1, (2.3)

    and ti:=a+(i1)(ba)/N, i=1,,N+1, for NN.

    Proof. Using the definition of the Riemann integral, it is straightforward to verify that the matrix

    GN:=baNUTNKNUN (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 JI.

    Proof. Let us consider the compact interval J=[a,b]. If (u0,,un) is a TP basis on I, it remains TP on JI. Consequently, the matrices UN and KN defined in (2.3) are TP for all NN. Therefore, the matrix GN in (2.4) is TP for all NN, 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 1i1<<irn+1 and columns 1j1<<jrn+1. Since GN=(GNi,j)1in+1;1jn+1 is TP, we have

    0detGN[i1,,ir|j1,,jr]=σSrsgn(σ)GNi1jσ(1)GNirjσ(r),NN, (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 limNGNi,j=Gi,j and so,

    GNi,j=Gi,j+ϵNi,j,limNϵNi,j=0, (2.6)

    for 1i,jn+1. Using (2.5) and (2.6), we derive

    0detGN[i1,,ir|j1,,jr]=σSrsgn(σ)r=1(Gi,jσ()+ϵNi,jσ())=detG[i1,,ir|j1,,jr]+rk=1σSrsgn(σ)ϵNik,jσ(k)kGNi,jσ()detG[i1,,ir|j1,,jr]+rk=1σSr|ϵNik,jσ(k)|k|GNi,jσ()|,NN.

    By defining

    ϵN:=max{|ϵNi,j|i=i1,,ir,j=j1,,jr},ψN:=max{|GNi,j|i=i1,,ir,j=j1,,jr},

    we have

    0detG[i1,,ir|j1,,jr]+rk=1σSrϵNψr1N=detG[i1,,ir|j1,,jr]+rr!ϵNψr1N,NN.

    The value ψN is clearly bounded and ϵN0 as N. So,

    0=limNrr!ϵNψr1NdetG[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

    badx1bx1dx2bxj1dxjg(x1,,xj)=1j![a,b]jjl=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 ax1xjb. On the other hand, the hypercube is fully covered when considering all the points obtained by permuting the variables, that is,

    [a,b]j=σSj{xσ=(xσ(1),,xσ(j))|ax1xjb}.

    Two points xσ and xσ, with σσ, 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 σSj. 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[ij+1,,i|1,,j]=1j![a,b]jjl=1dxlκ(xl)detMTX[ij+1,,i|1,,j]detMX[1,,j], (3.2)

    where

    MX:=(uj1(xi))1i,jn+1

    is the square collocation matrix of (u0,,un) at the sequence of parameters X=(x1,,xn+1).

    Proof. Given NN, we define an equally spaced partition of [a,b] with ti:=a+(i1)(ba)/N, i=1,,N+1. Using basic properties of determinants, we can derive the following identities for the matrix GN in (2.4):

    detGN[ij+1,,i|1,,j]=(baN)j|N+1l=1uij+1(tl)u0(tl)κ(tl)N+1l=1uij+1(tl)uj1(tl)κ(tl)N+1l=1ui(tl)u0(tl)κ(tl)N+1l=1ui(tl)uj1(tl)κ(tl)|=(baN)jN+1k1,,kj=1jl=1ul1(tkl)κ(tkl)|uij+1(tk1)uij+1(tkj)ui(tk1)ui(tkj)|=(baN)jk1<<kjσSjjl=1ul1(tkσ(l))κ(tkσ(l))|uij+1(tkσ(1))uij+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[ij+1,,i|1,,j]=(baN)jk1<<kjσSjsgn(σ)jl=1ul1(tkσ(l))κ(tkl)|uij+1(tk1)uij+1(tkj)ui(tk1)ui(tkj)|=(baN)jk1<<kjjl=1κ(tkl)|uij+1(tk1)uij+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

    limNk1<<kj(baN)jg(tk1,,tkj)=badx1bx1dx2bxj1dxjg(x1,,xj).

    Thus,

    detG[ij+1,,i|1,,j]=limNdetGN[ij+1,,i|1,,j]=badx1bx1dx2bxj1dxjjl=1κ(xl)detMTX[ij+1,,i|1,,j]detMX[1,,j]=1j![a,b]jjl=1dxlκ(xl)detMTX[ij+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 λ1λ2λp>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+n2tλ1+nn+1tλ2+n11tλ2+n12tλ2+n1n+1tλn+11tλn+12tλn+1n+1]/det[111t1t2tn+1tn1tn2tnn+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 IR, and let (p0,,pn) be a basis of Pn(I) such that

    pi1(t)=n+1j=1ai,jtj1,tI,i=1,,n+1. (4.2)

    We denote by A=(ai,j)1i,jn+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:=(pj1(ti))1i,jn+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:=(tj1i)1i,jn+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[ij+1,,i|1,,j]=detVtij+1,,til1>>ljdetA[1,,j|lj,,l1]s(l1j,,lj1)(tij+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+rj1 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[ij+1,,i|lj,,l1]=detA[ij+1,,i|λj+1,,λ1+j]. (4.6)

    Since the indices satisfy l1>>lj and lkn+1, for 1kn+1, the corresponding partitions will have j parts, each with a maximum length of n+1j. In other words, the sum in (4.5) spans all Young diagrams that fit within a j×(n+1j) rectangular box, which can be expressed as:

    l(λ)j,λ1n+1j.

    With this notation, we have

    lj<<l1detA[1,,j|lj,,l1]s(l1j,,lj1)(tij+1,,ti)=l(λ)jλ1n+1jA[j,λ]sλ(tij+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[ij+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[ij+1,,i|1,,j]=l(λ)jλ1n+1jl(μ)jμ1n+1j|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]jjl=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+1j), 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[ij+1,,i|1,,j]=l(μ)jμ1n+1jA[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 cρμ=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(1t)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]jjl=1[dxlxαl(1xl)β](detVx1,,xj)2sρ(x1,,xj)=j!1i<kj(ρiρk+ki)ji=1Γ(α+ρi+ji+1)Γ(β+ji+1)Γ(α+β+2ji+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]jjl=1[dxlxαl](detVx1,,xj)2sλ(x1,,xj)sμ(x1,,xj)=j!1i<kj(ki+μiμk)(ki+λiλk)ji,k=11α+2jik+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[ij+1,,i|1,,j]=1j!l(λ)jλ1n+1jl(μ)jμ1n+1jA[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[ij+1,,i|1,,j], we present two notable examples that highlight its efficiency and utility.

    Bernstein polynomials, defined as

    Bni(t):=(ni)ti(1t)ni,i=0,,n,

    are square-integrable functions with respect to the inner product

    f,gα,β:=10tα(1t)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)1i,jn+1, where

    gα,βi,j=(ni1)(nj1)Γ(i+j+α1)Γ(2nij+β+3)Γ(2n+α+β+2),

    for 1i,jn+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=(121022001).

    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[ij+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+1j=1bjtj1,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=(b1000b1b200b1b2b30b1b2b3bn+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[ij+1,,i|m,ij+2,ij+3,,i]=bmjk=2bij+k,m=1,,ij+1.

    This way, the only nonzero contributions to (4.11) come from the partitions μ=(μ1,,μj) with

    μr=ij,r=1,,j1,μj=0,,ij.

    Applying (4.11) with (5.1), we obtain

    detG[ij+1,,i|1,,j]=b1jk=2bij+kbkj1l=1(jl)!2(il+1)jijm=01(m+1)jj1r=1(imr)bm+1,

    where (x)n:=x(x+1)(x+n1) 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 + 2j - 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))

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