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The Horadam polynomials unify many well-known polynomials, such as the Fibonacci polynomials, the Lucas polynomials, the Pell polynomials, the Jacobsthal polynomials and the Chebyshev polynomials. We investigate some properties of various combinatorial triangles related to Horadam polynomials, including their properties as almost-Riordan arrays and Riordan arrays, their total positivity, the real-rootedness of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems). [ABSTRACT FROM AUTHOR]

Catalan–Stieltjes matrices and their corresponding Catalan-like numbers are very important in combinatorics. In this paper, we give a sufficient condition for the Catalan–Stieltjes matrix to be triangular strictly totally positive (Δ STP), that is, all its minors are positive except those which are zero due to the triangular structure. On this basis, we show that it is possible to compute the bidiagonal factorization of a class of Catalan–Stieltjes matrices with high relative accuracy (HRA). We also demonstrate that this class of matrices includes many well-known combinatorial triangles, such as the Catalan triangle of Aigner and the Bell triangle. Moreover, we construct an accurate algorithm to compute the bidiagonal factorization of some Catalan–Stieltjes matrices. The algorithm guarantees HRA for several linear algebraic computations associated with these matrices, like computing their singular values, their inverses, as well as the solutions of some linear systems. We show the accuracy and effectiveness of the proposed algorithm through numerical experiments. [ABSTRACT FROM AUTHOR]

Let S be a punctured surface of negative Euler characteristic. We show that given a generic representation ρ : π 1 (S) → PSL n (C) , there exists a positive representation ρ 0 : π 1 (S) → PSL n (R) that dominates ρ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space X n = PSL n (C) / PSU (n) . Moreover, the ρ 0 -lengths of peripheral curves remain unchanged. The dominating representation ρ 0 is explicitly described via Fock–Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks. [ABSTRACT FROM AUTHOR]

The article "Hyperbolic Stable Polynomials and Total Positivity" published in Mathematical Notes explores the localization of roots of polynomials using the technique of parametrizing spaces of totally positive matrices. The main result of the paper establishes a theorem linking the negativity of roots of a real polynomial with the oscillatory nature of a related matrix. The research delves into the applications of total positivity matrices in various mathematical fields, such as integrable systems, cluster algebras, and statistical mechanics. The study was funded by the Ministry of Science and Higher Education of the Russian Federation and the Theoretical Physics and Mathematics Advancement Foundation "BASIS." [Extracted from the article]

We present two complementary techniques called catalysis and inhibition which allow one to determine if a given pattern is TP completable or TP non-completable, respectively. Empirically, these techniques require considering only one unspecified entry at a time in a vast majority of cases, which makes these techniques ripe for automation and a powerful framework for future work in the TP completion problem. With small modifications, these techniques are also applicable to the TN completion problem. We provide two major applications. First, we characterize all 4-by-4 patterns by completability. There are a total of 78 new obstructions of this size, six times as many as the 3-by- n case for all n combined. Second, we provide a characterization of the so-called 1-variable obstructions in the TN case, which includes as a corollary a characterization of patterns with a single unspecified entry. This also provides a novel partial result towards proving the conjecture that all TN-completable patterns are TP-completable. [ABSTRACT FROM AUTHOR]

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included. [ABSTRACT FROM AUTHOR]

Lagrange–Vandermonde matrices are the collocation matrices corresponding to Lagrange‐type bases, obtained by removing the denominators from each element of a Lagrange basis. It is proved that, provided the nodes required to create the Lagrange‐type basis and the corresponding collocation matrix are properly ordered, such matrices are strictly totally positive. A fast algorithm to compute the bidiagonal decomposition of these matrices to high relative accuracy is presented. As an application, the problems of eigenvalue computation, linear system solving and inverse computation are solved in an efficient and accurate way for this type of matrices. Moreover, the proposed algorithms allow to solve fastly and to high relative accuracy some of the cited problems when the involved matrices are collocation matrices corresponding to the standard Lagrange basis, although such collocation matrices are not totally positive. Numerical experiments illustrating the good performance of our approach are also included. [ABSTRACT FROM AUTHOR]

We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight m ! ϕ m for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays. [ABSTRACT FROM AUTHOR]

A fast and accurate algorithm to compute the bidiagonal decomposition of collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition are also included. Starting from this bidiagonal decomposition, the accurate and efficient solution of several linear algebra problems involving these matrices is addressed: linear system solving, eigenvalue and singular value computation, and computation of the inverse and the Moore-Penrose inverse. The numerical experiments carried out show the good behaviour of the algorithm. [ABSTRACT FROM AUTHOR]

The 3-by-n TP-completable patterns are characterized by identifying the minimal obstructions up to natural symmetries. They are finite in number.

Let R = R (d (t) , h (t)) be a Riordan array, where d (t) = ∑ n ≥ 0 d n t n and h (t) = ∑ n ≥ 0 h n t n. We show that if the matrix [ d 0 h 0 0 0 ⋯ d 1 h 1 h 0 0 d 2 h 2 h 1 h 0 ⋮ ⋮ ⋱ ] is totally positive, then so is the Riordan array R. [ABSTRACT FROM AUTHOR]

It is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.

We give a unified proof of the interlacing properties of eigenvalues of principle submatrices of totally positive matrices. [ABSTRACT FROM AUTHOR]

We study three combinatorial models for the lower-triangular matrix with entries t n , k = ( n k ) n n − k : two involving rooted trees on the vertex set [ n + 1 ] , and one involving partial functional digraphs on the vertex set [ n ]. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials t n , k (y , z) that count improper and proper edges, and further to polynomials t n , k (y , ϕ) in infinitely many indeterminates that give a weight y to each improper edge and a weight m ! ϕ m for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays. [ABSTRACT FROM AUTHOR]

We correct an error in the proof of Theorem 3.2 in our paper arXiv:1612.02210 [math.AC] , published in Fallat et al. (2017) [1]. [ABSTRACT FROM AUTHOR]

A fast and accurate algorithm to compute the bidiagonal decomposition of rectangular totally positive h-Bernstein-Vandermonde matrices is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices are addressed. The computation of this bidiagonal decomposition is used as the first step for the accurate and efficient computation of the singular values of rectangular totally positive h-Bernstein-Vandermonde matrices and for solving least squares problems whose coefficient matrices are such matrices. [ABSTRACT FROM AUTHOR]

We consider a recent result which claims that if the series d (t) and h (t) are Pólya frequency formal power series, then the Riordan array (d (t) , h (t)) is totally positive. Namely, we pose and answer the question if the various types of the converse of this claim are also true. [ABSTRACT FROM AUTHOR]

The normalized totally positive bases are widely used in many fields. Based on the generalized Vandermonde determinant, the normalized total positivity of a kind of generalized toric-Bernstein basis is proved, which is defined on a set of real points. By this result, then the progressive iterative approximation property of the generalized toric-Bézier curve is obtained and some examples are provided to illustrate this property. [ABSTRACT FROM AUTHOR]

An algorithm for the computation of the bidiagonal decomposition of strictly totally positive h-Bernstein-Vandermonde matrices is presented. These matrices are collocation matrices of h-Bernstein bases (a generalization of the Bernstein basis) of the space of polynomials of degree less than or equal to n. The algorithm is fast (its computational complexity is O (n 2)) and has high relative accuracy. The computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices: linear system solving, eigenvalue computation and inverse computation. Numerical experiments are also included. [ABSTRACT FROM AUTHOR]

Let X = (x 1 , ... , x n) be a vector of distinct positive integers. The n × n matrix S = S (X) : = (gcd (x i , x j)) i , j = 1 n , where gcd (x i , x j) denotes the greatest common divisor of x i and x j , is called the greatest common divisor (GCD) matrix on X. By a surprising result of Beslin and Ligh (1989) [7] , all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when n ≥ 3 , and is TN if and only if it is T N 2 , i.e., all its 2 × 2 minors are nonnegative. We next demonstrate that a GCD matrix is TN 2 if and only if the exponents of each prime divisor in the prime factorization of the x i 's form a monotonic sequence. Reformulated in the language of kernels, our results characterize the subsets of integers over which the kernel K (x , y) = gcd (x , y) is totally nonnegative. The proofs of our characterizations depend on Gantmacher and Krein's notion of a Green's matrix. We conclude by showing that a GCD matrix is TN if and only if it is a Green's matrix. As a consequence, we obtain explicit formulas for all the minors and for the inverse of totally nonnegative GCD matrices. [ABSTRACT FROM AUTHOR]

In this paper we establish a connection between full spark frames and totally nonsingular matrices. Then we provide a method for constructing infinite totally positive matrices which make up a subclass of the class of totally nonsingular matrices. Using this method we then construct a family of infinite totally positive matrices parameterized by non-negative numbers which contains, as the simplest case, the infinite Pascal matrix. The paper ends with some examples and comments on full spark frames. [ABSTRACT FROM AUTHOR]

Abstract A matrix is called strictly sign-regular of order k (denoted by S S R k) if all its k × k minors are non-zero and have the same sign. For example, totally positive matrices, i.e., matrices with all minors positive, are S S R k for all k. Another important subclass are those that are S S R k for all odd k. Such matrices have interesting sign variation diminishing properties, and it has been recently shown that they play an important role in the analysis of certain nonlinear cooperative dynamical systems. In this paper, the spectral properties of nonsingular matrices that are S S R k for a specific value k are studied. One of the results is that the product of the first k eigenvalues is real and of the same sign as the k × k minors, and that linear combinations of certain eigenvectors have specific sign patterns. It is then shown how known spectral properties for matrices that are S S R k for several values of k can be derived from these results. Using these theoretical results, the notion of a totally positive discrete-time system (TPDTS) is introduced. This may be regarded as the discrete-time analogue of the important notion of a totally positive differential system. It is shown that TPDTSs can be applied to prove that certain time-varying nonlinear dynamical systems entrain to periodic excitations. [ABSTRACT FROM AUTHOR]

Abstract The computation of the Moore–Penrose inverse of structured strictly totally positive matrices is addressed. Since these matrices are usually very ill-conditioned, standard algorithms fail to provide accurate results. An algorithm based on the Q R factorization and which takes advantage of the special structure and the totally positive character of these matrices is presented. The first stage of the algorithm consists of the accurate computation of the bidiagonal decomposition of the matrix. Numerical experiments illustrating the good behavior of our approach are included. [ABSTRACT FROM AUTHOR]

When we speak about parametric programming, sensitivity analysis or related topics, we usually mean the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied. In this paper, we turn to another question. Suppose that A is a matrix having a specific property . What are the maximal allowable variations of the data such that the property still remains valid for the matrix? We study two basic forms of perturbations. The first is a perturbation in a given direction, which is closely related to parametric programming. The second type consists of all possible data variations in a neighbourhood specified by a certain matrix norm; this is related to the tolerance approach to sensitivity analysis or to stability. The matrix properties discussed in this paper are positive definiteness; M-matrix, H-matrix and P-matrix property; total positivity; inverse M-matrix property and inverse nonnegativity. [ABSTRACT FROM AUTHOR]

The problem of polynomial interpolation with the Lagrange-type data when using the Bernstein basis instead of the monomial basis is addressed. The extension to the bivariate case, which leads to the use of a generalized Kronecker product, is also developed. In addition to the matricial description of the solution and the proof of unisolvence, algorithms for the computation of the coefficients of the interpolating polynomial are presented. Numerical experiments illustrating the advantage of computing with Bernstein-Vandermonde matrices instead of with Vandermonde matrices are included. [ABSTRACT FROM AUTHOR]

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t ≥ r − 2 . We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices. [ABSTRACT FROM AUTHOR]

Cauchy–Vandermonde matrices play a fundamental role in rational interpolation theory and in other fields. When all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings, these matrices are totally positive. In this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy. These algorithms can be used to solve with high relative accuracy other algebraic problems, such as the computation of all singular values, all eigenvalues or the solution of certain linear systems. The error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed. [ABSTRACT FROM AUTHOR]

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given by d A d t = [ A u − A l , A ] , A ( 0 ) = A 0 , where A is a real n × n matrix (not necessarily symmetric), [ A , B ] = A B − B A is the matrix commutator (also known as the Lie bracket), A u is the strictly upper triangular part of A and A l is the strictly lower triangular part of A . We prove that if the initial matrix A 0 is staircase, so is A ( t ) . Moreover, we prove that this flow preserves the certain positivity properties of A 0 . Also we prove that if the initial matrix A 0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A ( t ) converges to a diagonal matrix while preserving the spectrum of A 0 . Some simulations are provided to confirm the convergence properties. [ABSTRACT FROM AUTHOR]

We investigate coordination numbers of the cubic lattices with emphases on their analytic behaviors, including the total positivity of the coordination matrices, the distribution of zeros of the coordination polynomials, the asymptotic normality of the coefficients of the coordination polynomials, the log-concavity and the log-convexity of the coordination numbers. [ABSTRACT FROM AUTHOR]

Let Γ be a subset of { 0 , 1 , 2 ,... }. We show that if Γ has 'gaps' then the completeness and frame properties of the system { t k e 2 π i n t : n ∈ Z , k ∈ Γ } differ from those of the classical exponential systems. This phenomenon is closely connected with the existence of certain uniqueness sets for lacunary polynomials. [ABSTRACT FROM AUTHOR]

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q -Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers. [ABSTRACT FROM AUTHOR]

If T is a labelled tree, a matrix A is totally positive relative to T , principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T . This conclusion has been incorrectly conjectured under weaker hypotheses. [ABSTRACT FROM AUTHOR]

A close relationship between the class of totally positive matrices and anti-Monge matrices is used for suggesting a new direction for investigating totally positive matrices. Some questions are posed and a partial answer in the case of Vandermonde-like matrices is given. [ABSTRACT FROM AUTHOR]

An algorithm for computing the bidiagonal decomposition of Said–Ball–Vandermonde matrices in the rectangular case is presented, which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is fast and possesses high relative accuracy. The error analysis of the algorithm is also addressed, along with the perturbation theory for the bidiagonal factorization of totally positive Said–Ball–Vandermonde matrices. Some numerical experiments showing the good behavior of the algorithm are included. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schröder numbers, are Stieltjes moment sequences in a unified approach. [ABSTRACT FROM AUTHOR]

In this paper totally nonnegative (positive) matrices are considered which are matrices having all their minors nonnegative (positive); the almost totally positive matrices form a class between the totally nonnegative matrices and the totally positive ones. An efficient determinantal test based on the Cauchon algorithm for checking a given matrix for falling in one of these three classes of matrices is applied to matrices which are related to roots of polynomials and poles of rational functions, specifically the Hankel matrix associated with the Laurent series at infinity of a rational function and matrices of Hurwitz type associated with polynomials. In both cases it is concluded from properties of one or two finite sections of the infinite matrix that the infinite matrix itself has these or related properties. Then the results are applied to derive a sufficient condition for the Hurwitz stability of an interval family of polynomials. Finally, interval problems for a subclass of the rational functions, viz. R -functions, are investigated. These problems include invariance of exclusively positive poles and exclusively negative roots in the presence of variation of the coefficients of the polynomials within given intervals. [ABSTRACT FROM AUTHOR]

Here, we define and consider (linear) TP-directions and TP-paths for a totally nonnegative matrix, in an effort to more deeply understand perturbation of a TN matrix to a TP matrix. We give circumstances in which a TP-direction exists and an example to show that they do not always exist. A strategy to give (nonlinear) TP-paths is given (and applied to this example). A long term goal is to understand the sparsest TP-perturbation for application to completion problems. [ABSTRACT FROM AUTHOR]

Let A = [ a n , k ] n , k ≥ 0 be an infinite lower triangular matrix defined by the recurrence a 0 , 0 = 1 , a n + 1 , k = r k a n , k − 1 + s k a n , k + t k + 1 a n , k + 1 , where a n , k = 0 unless n ≥ k ≥ 0 and r k , s k , t k are all nonnegative. Many well-known combinatorial triangles are such matrices, including the Pascal triangle, the Stirling triangle (of the second kind), the Bell triangle, the Catalan triangles of Aigner and Shapiro. We present some sufficient conditions such that the recursive matrix A is totally positive. As applications we give the total positivity of the above mentioned combinatorial triangles in a unified approach. [ABSTRACT FROM AUTHOR]

Let n and b be positive integers. Define the amazing matrix P n , b = [ P (i , j) ] i , j = 0 n − 1 to be an n × n matrix with entries P (i , j) = 1 b n ∑ r ≥ 0 (− 1) r ( n + 1 r ) ( n − 1 − i + (j + 1 − r) b n ). Diaconis and Fulman conjectured that the amazing matrix is totally positive. We give an affirmative answer to this conjecture. [ABSTRACT FROM AUTHOR]

For any pattern of the specified entries, an explicit finite list of polynomial inequalities in the specified entries is given that characterizes the completability of a partial positive matrix to a TPmatrix. Each polynomial happens to be a monomial, and an algorithm, whose input is the pattern, is given for finding the polynomials. The method uses some new partial orders on matrices and the logarithmic method to reduce the question to determining the generators of a certain finitely generated, pointed cone. [ABSTRACT FROM PUBLISHER]

The aforementioned paper contains some mistakes corrected here. [ABSTRACT FROM AUTHOR]

We study monic polynomials Q( x) generated by a high-order three-term recursion xQ( x)= Q( x)+ a Q( x) with arbitrary p≥1 and a>0 for all n. The recursion is encoded by a two-diagonal Hessenberg operator H. One of our main results is that, for periodic coefficients a and under certain conditions, the Q are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev-Kalyagin-Saff, where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_{n}-a|<\infty$ for some a>0. An important tool in this paper is the study of 'Riemann-Hilbert minors', or equivalently, the 'generalized eigenvalues' of the Hessenberg matrix H. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients a, we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann-Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Abstract: Delsarte and Seidel found that the dimension of a certain linear space can serve as a Fisher type lower bound for the size of a weighted regular t-wise balanced design. They put it as a problem to explicitly determine this dimension. This note calculates the rank of a special set inclusion matrix and then solves the above problem. [Copyright &y& Elsevier]

Abstract: The notions of total positivity and of TP k are generalized to “shapes” (a generalization of matrices). In particular, the relationship between positivity of “contiguous” minors and all minors is characterized for general shapes and for certain special types of shapes. This and other ideas are used to address the TP k -completion problem and TP k -completable patterns. In case , a near characterization of TP2-completable patterns is given. [Copyright &y& Elsevier]

Abstract: The combined matrix of a nonsingular matrix A is the matrix , where means the Hadamard (entrywise) product. It has simple properties, its row- as well as column-sums are always one, and it is not changed if A is multiplied from either side by a nonsingular diagonal matrix. Although it is usually difficult to compute, its further properties deserve attention. In the paper, we concentrate on the sequence of the diagonal entries of combined matrices in various classes of matrices. [Copyright &y& Elsevier]