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Let R ̂ = (d̂(t);ĥ1(t),ĥ2(t)) be the compression of a double Riordan array. We show that if d̂(t),ĥ1(t) and ĥ2(t) are all Pólya frequency formal power series, then R ̂ is totally positive. [ABSTRACT FROM AUTHOR]

Variation diminution (VD) is a fundamental property in total positivity theory, first studied in 1912 by Fekete–Pólya for one-sided Pólya frequency sequences, followed by Schoenberg, and by Motzkin who characterized sign regular (SR) matrices using VD and some rank hypotheses. A classical theorem by Gantmacher–Krein characterized the strictly sign regular (SSR) m \times n matrices for m>n using this property. In this article we strengthen these results by characterizing all m \times n SSR matrices using VD. We further characterize strict sign regularity of a given sign pattern in terms of VD together with a natural condition motivated by total positivity. We then refine Motzkin's characterization of SR matrices by omitting the rank condition and specifying the sign pattern. This concludes a line of investigation on VD started by Fekete–Pólya [Rend. Circ. Mat. Palermo 34 (1912), pp. 89–120] and continued by Schoenberg [Math. Z. 32 (1930), pp. 321–328], Motzkin [Beiträge zur Theorie der linearen Ungleichungen, PhD Dissertation, Jerusalem, 1936], Gantmacher–Krein [ Oscillyacionye matricy i yadra i malye kolebaniya mehaničeskih sistem , Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950], Brown–Johnstone–MacGibbon [J. Amer. Statist. Assoc. 76 (1981), pp. 824–832], and Choudhury [Bull. Lond. Math. Soc. 54 (2022), pp. 791–811; Bull. Sci. Math. 186 (2023), p. 21]. In fact we show stronger characterizations, by employing single test vectors with alternating sign coordinates – i.e., lying in the alternating bi-orthant. We also show that test vectors chosen from any other orthant will not work. [ABSTRACT FROM AUTHOR]

This paper presents an accurate method to obtain the bidiagonal decomposition of some generalized Pascal matrices, including Pascal k-eliminated functional matrices and Pascal symmetric functional matrices. Sufficient conditions to assure that these matrices are either totally positive or inverse of totally positive matrices are provided. In these cases, the presented method can be used to compute their eigenvalues, singular values and inverses with high relative accuracy. Numerical examples illustrate the high accuracy of our approach. [ABSTRACT FROM AUTHOR]

The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the n -th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the n -th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every n -th production matrix can be factorized into a product of n matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the n -th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the n -th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases. [ABSTRACT FROM AUTHOR]

ABSTRACT Little q$$ q $$‐Jacobi polynomials belong to the field of quantum calculus. This article obtains the bidiagonal decomposition of the collocation matrices of these polynomials, showing that, in many cases, it can be constructed to high relative accuracy (HRA). Then, it can be used to compute with HRA the inverses, eigenvalues, and singular values of these matrices. Numerical experiments are provided and illustrate the excellent results obtained when applying the presented methods. [ABSTRACT FROM AUTHOR]

A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula. [ABSTRACT FROM AUTHOR]

We consider generalized Green matrices that, in contrast to Green matrices, are not necessarily symmetric. In spite of the loss of symmetry, we show that they can preserve some nice properties of Green matrices. In particular, they admit a bidiagonal decomposition. Moreover, for convenient parameters, the bidiagonal decomposition can be obtained efficiently and with high relative accuracy and it can also be used to compute all eigenvalues, all singular values, the inverse, and the solution of some linear system of equations with high relative accuracy. Numerical examples illustrate the high accuracy of the performed computations using the bidiagonal decompositions. Finally, nonsingular and totally positive generalized Green matrices are characterized. [ABSTRACT FROM AUTHOR]

Summary: The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included. [ABSTRACT FROM AUTHOR]

Using the Blossom method in the quasi extended Chebyshev space, we first construct a class of generalized cubic Bernstein functions in the generalized cubic polynomial space { 1 , 3 t 2 - 2 t 3 , (1 - t) 2 λ (t) , t 2 μ (t) } . Our work includes many previous works as special cases by selecting specific shape functions λ (t) and μ (t) . The resulting generalized cubic Bézier curves and their properties are discussed. The corner cutting method is also proposed to compute curves efficiently and stably. A generalized Bernstein operator is given and spectral analysis on a specific case is carried out. The results show that the obtained generalized Bézier curves can better fit the control polygon compared to the Bézier curves. We further deduce the closed-form solution of generalized cubic B-splines that possess local shape functions. Some important properties are presented, including local support property, nonnegativity, partition of unity, total positivity property, C 2 continuity, linear independence and so on. Some numerical examples shows that the resulting generalized cubic B-spline curves can approximate the control polygon more flexibly. [ABSTRACT FROM AUTHOR]

Generalized order statistics (GOSs) are often adopted as a tool for providing a unified approach to several stochastic models dealing with ordered random variables. In this contribution, we first recall various useful results based on the notion of total positivity. Then, some stochastic comparisons between spacings of GOSs from one sample, as well as two samples, are developed under the more general assumptions on the parameters of the model. Specifically, the given results deal with the likelihood ratio order, the hazard rate order and the mean residual life order. Finally, an application is demonstrated for sequential systems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of monotonicity according to a direction for a set of random variables. This concept extends well-known multivariate dependence notions, such as corner set monotonicity, and can be used to detect dependence in multivariate distributions not detected by other known concepts of dependence. Additionally, we establish relationships with other known multivariate dependence concepts, outline some of their salient properties, and provide several examples. [ABSTRACT FROM AUTHOR]

In this work, block checkerboard sign pattern matrices are introduced and analyzed. They satisfy the generalized Perron–Frobenius theorem. We study the case related to total positive matrices in order to guarantee bidiagonal decompositions and some linear algebra computations with high relative accuracy. A result on intervals of checkerboard matrices is included. Some numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory.

Consider bivariate observations (X 1 , Y 1) , … , (X n , Y n) ∈ R × R with unknown conditional distributions Q x of Y, given that X = x . The goal is to estimate these distributions under the sole assumption that Q x is isotonic in x with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution L (X , Y) under the sole assumption that it is totally positive of order two. An algorithm is developed which estimates the unknown family of distributions (Q x) x via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data. [ABSTRACT FROM AUTHOR]

In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory. [ABSTRACT FROM AUTHOR]

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U 4 (I α) = span { 1 , cos t , sin t , cos 2 t , sin 2 t } defined on compact intervals I α = [ 0 , α ] , where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U 4 (I α) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α → 0 . As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C (3 Hermite interpolation problems. [ABSTRACT FROM AUTHOR]

Given a sequence α = (a k) k ≥ 0 of nonnegative numbers, define a new sequence L (α) = (b k) k ≥ 0 by b k = a k 2 - a k - 1 a k + 1 . The sequence α is called r-log-concave if L i (α) = L (L i - 1 (α)) is a nonnegative sequence for all 1 ≤ i ≤ r . In this paper, we study the r-log-concavity and its q-analogue for r = 2 , 3 using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann ξ -function. We give some criteria for r-q-log-concavity for r = 2 , 3 . As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. [ABSTRACT FROM AUTHOR]

Let $V_{(r,n,\tilde {m}_n,k)}^{(p)}$ and $W_{(r,n,\tilde {m}_n,k)}^{(p)}$ be the $p$ -spacings of generalized order statistics based on absolutely continuous distribution functions $F$ and $G$ , respectively. Imposing some conditions on $F$ and $G$ and assuming that $m_1=\cdots =m_{n-1}$ , Hu and Zhuang (2006. Stochastic orderings between p -spacings of generalized order statistics from two samples. Probability in the Engineering and Informational Sciences 20: 475) established $V_{(r,n,\tilde {m}_n,k)}^{(p)} \leq _{{\rm hr}} W_{(r,n,\tilde {m}_n,k)}^{(p)}$ for $p=1$ and left the case $p\geq 2$ as an open problem. In this article, we not only resolve it but also give the result for unequal $m_i$ 's. It is worth mentioning that this problem has not been proved even for ordinary order statistics so far. [ABSTRACT FROM AUTHOR]

Green matrices are interpreted as discrete version of Green functions and are used when working with inhomogeneous linear system of differential equations. This paper discusses accurate algebraic computations using a recent procedure to achieve an important factorization of these matrices with high relative accuracy and using alternative accurate methods. An algorithm to compute any minor of a Green matrix with high relative accuracy is also presented. The bidiagonal decomposition of the Hadamard product of Green matrices is obtained. Illustrative numerical examples are included. [ABSTRACT FROM AUTHOR]

We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties. [ABSTRACT FROM AUTHOR]

This paper studies total positivity and relative convexity properties in option pricing models. We introduce these properties in the Black-Scholes setting by showing the following: out-of-the-money calls are totally positive in strike and volatility; out-of-the-money puts have a reverse sign rule property; calls and puts are convex with respect to at-the-money prices; and relative convexity of option prices implies a convexity-in-time property of the underlying. We then extend these properties to other models, including scalar diffusions, mixture models, and certain Lévy processes. We show that relative convexity typically holds in time-homogeneous local volatility models through the Dupire equation. We develop implications of these ideas for empirical option prices, including constraints on the at-the-money skew. We illustrate connections with models studied by Peter Carr, including the variance-gamma, CGMY, Dagum, and logistic density models. [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]

Due to the importance of generalized order statistics (GOS) in many branches of Statistics, a wide interest has been shown in investigating stochastic comparisons of GOS. In this article, we study the likelihood ratio ordering of $p$ -spacings of GOS, establishing some flexible and applicable results. We also settle certain unresolved related problems by providing some useful lemmas. Since we do not impose restrictions on the model parameters (as previous studies did), our findings yield new results for comparison of various useful models of ordered random variables including order statistics, sequential order statistics, $k$ -record values, Pfeifer's record values, and progressive Type-II censored order statistics with arbitrary censoring plans. Some results on preservation of logconvexity properties among spacings are provided as well. [ABSTRACT FROM AUTHOR]

Given a root system R and the corresponding finite reflection group W let \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2) be the group of homomorphisms from W into \widehat {\mathbb Z}_2, where \widehat {\mathbb Z}_2=\{1,-1\} with multiplication. We propose a procedure of constructing subroot systems of R by using homomorphisms \eta \in \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2). This construction is next used for establishing a relation between concepts of total positivity and \eta-total positivity. [ABSTRACT FROM AUTHOR]

In this paper we extend the theory of total positivity for reductive groups to the case of symmetric spaces. [ABSTRACT FROM AUTHOR]

In this article, we first study the likelihood ratio ordering of generalized order statistics (GOS) in both one-sample and two-sample problems. Then, we establish the transmission of the increasing hazard rate and decreasing reversed hazard rate aging properties of GOS. To do this, we extend Karlin's basic composition theorem for the functions of three variables. Then, we settle certain open problems in this regard by providing some counterexamples. We further investigate similar transmission cases which have not been addressed in the literature so far. [ABSTRACT FROM AUTHOR]

There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of k-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most k - 1. For k = 1, this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of k-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for k = 1, this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of k-positivity by analyzing internally Hankel k-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are k-positive. We show that the existing notions of external Hankel and internal k-positivity are subsumed under internal Hankel k-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first k compound systems. As such, this class provides new means to verify external Hankel k-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel k-positive realizations. [ABSTRACT FROM AUTHOR]

We show that the totally nonnegative part of a partial flag variety G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov. [ABSTRACT FROM AUTHOR]

We construct an embedding of the space of electrical networks to the totally nonnegative Lagrangian Grassmannian in a generic situation with the help of the technique of vertex integrable models of statistical mechanics. [ABSTRACT FROM AUTHOR]

Based on the Stirling triangle of the second kind, the Whitney triangle of the second kind and one triangle of Riordan, we study a Stirling–Whitney–Riordan triangle [ T n , k ] n , k satisfying the recurrence relation: T n , k = (b 1 k + b 2) T n - 1 , k - 1 + [ (2 λ b 1 + a 1) k + a 2 + λ (b 1 + b 2) ] T n - 1 , k + λ (a 1 + λ b 1) (k + 1) T n - 1 , k + 1 , where initial conditions T n , k = 0 unless 0 ≤ k ≤ n and T 0 , 0 = 1 . We prove that the Stirling–Whitney–Riordan triangle [ T n , k ] n , k is x -totally positive with x = (a 1 , a 2 , b 1 , b 2 , λ) . We show that the row-generating function T n (q) has only real zeros and the Turán-type polynomial T n + 1 (q) T n - 1 (q) - T n 2 (q) is stable. We also present explicit formulae for T n , k and the exponential generating function of T n (q) and give a Jacobi continued fraction expansion for the ordinary generating function of T n (q) . Furthermore, we get the x -Stieltjes moment property and 3- x -log-convexity of T n (q) and show that the triangular convolution z n = ∑ i = 0 n T n , i x i y n - i preserves Stieltjes moment property of sequences. Finally, for the first column (T n , 0) n ≥ 0 , we derive some properties similar to those of (T n (q)) n ≥ 0. [ABSTRACT FROM AUTHOR]

This article shows that the bidiagonal decomposition of many important matrices of q‐integers can be constructed to high relative accuracy (HRA). This fact can be used to compute with HRA the eigenvalues, singular values, and inverses of these matrices. These results can be applied to collocation matrices of q‐Laguerre polynomials, q‐Pascal matrices, and matrices formed by q‐Stirling numbers. Numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A, 74 (1996), pp. 57--70]. Schirmacher [J. Combin. Theory Ser. A, 85 (1999), pp. 127--134] extended them to q-log-concavity and almost q-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula, and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle, and some triangles related to binomial coefficients, rook polynomials, and Laguerre polynomials. We not only get total positivity of these lower-triangles, and q-Stieltjes moment properties and 3-q-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve the Stieltjes moment property. In particular, we solve a conjecture of Sokal on the q-Stieltjes moment property of rook polynomials. [ABSTRACT FROM AUTHOR]

We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log‐supermodular (MTP2) distributions and log‐L♮‐concave (LLC) distributions. In both cases we also assume log‐concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0,1}d or in ℝ2 under MTP2, and for samples in ℚd under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate. [ABSTRACT FROM AUTHOR]

Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The aim of this paper is to study the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of generalized m -Jacobi-Rogers triangles and their applications. Using the theory of production matrices, we present the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of the output matrices of certain production matrices. In particular, we gain a criterion for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the generalized m -Jacobi-Rogers triangle. This immediately implies that the corresponding generalized m -Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and log-convex of higher order in all the indeterminates. In consequence, for m = 1 , we immediately obtain some results on Hankel-total positivity for the Catalan-Stieltjes matrices. In particular, we in a unified manner apply our results to some combinatorial triangles or polynomials including the generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a refined Stirling cycle polynomial and a refined Eulerian polynomial. For the general m , combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of exponential Rirodan arrays. In addition, we also derive some results for coefficientwise Hankel-total positivity in terms of compositional functions and m -branched Stieltjes-type continued fractions. Finally, we apply our criteria to: (1) rook polynomials and signless Laguerre polynomials (confirming a conjecture of Sokal on coefficientwise Hankel-total positivity of rook polynomials), (2) labeled trees and forests (proving some conjectures of Sokal on total positivity and Hankel-total positivity), (3) r th-order Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total positivity of r th-order Eulerian polynomials, which in particular implies the conjecture of Sokal on the coefficientwise Hankel-total positivity of reversed 2th-order Eulerian polynomials), (4) multivariate Ward polynomials, labeled series-parallel networks and nondegenerate fanout-free functions, (5) an array from the Lambert function and a generalization of Lah numbers and associated triangles, and so on. [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 provide a full classification of all attainable term structure shapes in the two-factor Vasicek model of interest rates. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes, up to four additional shapes can be produced. Our results apply to both forward and yield curves and show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The key mathematical tool is the theory of total positivity, pioneered by Samuel Karlin and others in the 1950s. [ABSTRACT FROM AUTHOR]

First, we introduce the two-Motzkin-like number as the weight of vertically constrained Motzkin-like path with no leading vertical steps from (0, 0) to (n, 0) consisting of up steps, down steps, horizontal steps, vertical steps in the down direction and vertical steps in the up direction. Secondly, we provide sufficient conditions under which the two-Motzkin-like numbers (resp. the q-analogue of the two-Motzkin-like numbers) are Stieltjes moment sequences (resp. are q-Stieltjes moment sequences) and therefore infinitely log-convex sequences. As applications, on the one hand, we show that many well-known counting coefficients, including the central trinomial 2 n 2 n 2 and pentanomial 2 n 4 n 4 numbers of even indices respectively are Stieltjes moment sequences and, therefore, infinitely log-convex sequences in a unified approach. On the other hand, we prove that the sequence of polynomials of square trinomials ∑ k = 0 2 n n k 2 2 q k are q-Stieltjes moment sequence of polynomials. Finally, we provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences in more generalized triangular array. [ABSTRACT FROM AUTHOR]

The quantum analogue of Bernstein operators B m , q reproduce the linear polynomials which are therefore eigenfunctions corresponding to the eigenvalue 1 , ∀ q > 0 . In this article the rest of eigenstructure of q-Bernstein operators and the distinct behaviour of zeros of eigenfunctions for cases (i) 1 > q > 0 , and (ii) q > 1 are discussed. Graphical analysis for some eigenfunctions and their roots are presented with the help of MATLAB. Also, matrix representation for diagonalisation of q-Bernstein operators is discussed. [ABSTRACT FROM AUTHOR]

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms. [ABSTRACT FROM AUTHOR]

We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary Jacobi matrices. [ABSTRACT FROM AUTHOR]

We use an argument of Madan and Yor to construct associated submartingales to a class of two-parameter processes that are ordered by increasing convex dominance. This class includes processes whose integrated survival functions are multivariate totally positive of order 2 ( MTP 2 ). We prove that the integrated survival function of an integrable two-parameter process is MTP 2 if and only if it is totally positive of order 2 ( TP 2 ) in each pair of arguments when the remaining argument is fixed. This result cannot be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. Since the MTP 2 property is closed under several transformations, it allows us to exhibit many other processes having the same total positivity property. [ABSTRACT FROM AUTHOR]

A method for solving least squares problems (A ⊗ Bi)x = b whose coefficient matrices have generalized Kronecker product structure is presented. It is based on the exploitation of the block structure of the Moore-Penrose inverse and the reflexive minimum norm g-inverse of the coefficient matrix, and on the QR method for solving least squares problems. Firstly, the general case where A is a rectangular matrix is considered, and then the special case where A is square is analyzed. This special case is applied to the problem of bivariate polynomial regression, in which the involved matrices are structured matrices (Vandermonde or Bernstein-Vandermonde matrices). In this context, the advantage of using the Bernstein basis instead of the monomial basis is shown. Numerical experiments illustrating the good behavior of the proposed algorithm are included. [ABSTRACT FROM AUTHOR]

We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian's inequality also holds true for BLM distributions. [ABSTRACT FROM AUTHOR]

We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, it proves the total positivity of the generalized Pascal triangle (classical and q-analogue versions). Second, it studies of the log-convexity (resp. strong q-log-convexity) of the first column of certain infinite generalized triangular array { A n , k } 0 ≤ k ≤ 2 n of nonnegative numbers (resp. of polynomials in q with nonnegative coefficients), of the form A n , k = e k A n - 1 , k - 2 + f k A n - 1 , k - 1 + g k A n - 1 , k + h k A n - 1 , k + 1 + i k A n - 1 , k + 2 . This allows a unified treatment of the log-convexity of the peers sequence of central trinomials 2 n 2 n 2 n ≥ 0 , as well as that of the strong q-log-convexity of the polynomial of square trinomial coefficients, i.e. A n (q) = ∑ k = 0 2 n n k 2 2 q k . [ABSTRACT FROM AUTHOR]