scholarly journals On Total Positivity of Catalan-Stieltjes Matrices

10.37236/6270 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Qiongqiong Pan ◽  
Jiang Zeng
Keyword(s):  
Sufficient Conditions ◽  
Hankel Matrix ◽  
Total Positivity ◽  
Combinatorial Interpretation

Recently Chen-Liang-Wang (Linear Algerbra Appl. 471 (2015) 383—393) proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix $A$ we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of $A$ is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Catalan-Stieltjes matrix $A$.

scholarly journals A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1154
Author(s):  
Yunyi Fu ◽  
Yuanpeng Zhu
Keyword(s):  
Sufficient Conditions ◽  
Total Positivity ◽  
Basis Functions ◽  
Shape Functions ◽  
Bernstein Basis ◽  
B Spline ◽  
Special Cases ◽  
Chebyshev Space ◽  
Spline Basis ◽  
Bernstein Basis Functions

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions α(t) and β(t) are constructed in a generalized quasi cubic trigonometric space span{1,sin2t,(1−sint)2α(t),(1−cost)2β(t)}, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions αi(t) and βi(t) is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and C2 continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly.

scholarly journals Discrete Tracy–Widom operators

2009 ◽  
Vol 52 (3) ◽  
pp. 545-559
Author(s):  
Gordon Blower ◽  
Andrew McCafferty
Keyword(s):  
Matrix Theory ◽  
Sufficient Conditions ◽  
Hankel Matrix ◽  
Mathieu Equation ◽  
Integrable Operator ◽  
Asymptotic Eigenvalue ◽  
Mathieu’S Equation ◽  
Asymptotic Eigenvalue Distribution ◽  
Bessel Kernel ◽  
The Fourier Transform

AbstractIntegrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

scholarly journals Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Tomack Gilmore
Keyword(s):  
Total Positivity ◽  
Combinatorial Interpretation ◽  
Totally Positive ◽  
Labelled Trees ◽  
Planar Networks

We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and rooted labelled trees on $n+1$ vertices where $k$ children of the root are lower-numbered than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in eight indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.

scholarly journals Applications of Hankel and Regular Matrices in Fourier Series

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Hankel Matrix ◽  
Necessary And Sufficient Conditions ◽  
Regular Matrix ◽  
Conjugate Fourier Series ◽  
Necessary And Sufficient

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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