Green’s matrices

Matrix methods for quadrature formulas on the unit circle. A survey

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel–Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.

Eigenstructure of order-one-quasiseparable matrices. Three-term and two-term recurrence relations

Stein’s method provides a way of finding approximations to the distribution of a random variable, which at the same time gives estimates of the approximation error involved. The strengths of the method are that it can be applied in many circumstances in which dependence plays a part. A key tool in Stein’s theory is the generator method developed by Barbour [10]. In this chapter we show how orthogonal polynomials appear in this context in a natural way. They are used in spectral representations of the transition probabilities of Barbour’s Markov processes. A good introduction to Stein’s approximation method can be found in [95].KeywordsOrthogonal PolynomialDeath ProcessDuality RelationStein EquationKrawtchouk PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

Distortion correction for the orthogonally-splitting-imaging pose sensor

The algorithm of ∇V-factorization, suggested earlier for decomposing one- and two-parameter polynomial matrices of full row rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix, whereas the regular spectrum is empty), is extended to the case of q-parameter (q ≥ 1) polynomial matrices. The algorithm of ∇V-q factorization is described, and its justification and properties for matrices with arbitrary number of parameters are presented. Applications of the algorithm to computing irreducible factorizations of q-parameter matrices, to determining a free basis of the null-space of polynomial solutions of a matrix, and to finding matrix divisors corresponding to divisors of its characteristic polynomial are considered. Bibliogrhaphy: 4 titles.

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one extra free parameter to them. In particular, we generalize generating functions for the continuous \(q\)-ultraspherical/Rogers, little \(q\)-Laguerre/Wall, and \(q\)-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and \(q\)-integrals.

Darboux transformations for CMV matrices

Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

In this paper, we will present a strong (or pathwise) approximation of\nstandard Brownian motion by a class of orthogonal polynomials. The coefficients\nthat are obtained from the expansion of Brownian motion in this polynomial\nbasis are independent Gaussian random variables. Therefore it is practical\n(requires $N$ independent Gaussian coefficients) to generate an approximate\nsample path of Brownian motion that respects integration of polynomials with\ndegree less than $N$. Moreover, since these orthogonal polynomials appear\nnaturally as eigenfunctions of an integral operator defined by the Brownian\nbridge covariance function, the proposed approximation is optimal in a certain\nweighted $L^{2}(\\mathbb{P})$ sense. In addition, discretizing Brownian paths as\npiecewise parabolas gives a locally higher order numerical method for\nstochastic differential equations (SDEs) when compared to the standard\npiecewise linear approach. We shall demonstrate these ideas by simulating\nInhomogeneous Geometric Brownian Motion (IGBM). This numerical example will\nalso illustrate the deficiencies of the piecewise parabola approximation when\ncompared to a new version of the asymptotically efficient log-ODE (or\nCastell-Gaines) method.\n

Transverse vibration of completely-free elliptic and circular plates using orthogonal polynomials in the Rayleigh-Ritz method

Applying the methods of BCOP and matrix exponential in the vibration analysis of multi-span composite plates under opposing moving inertia loads

Numerical computation of BCOPs in two variables for solving the vibration problem of a CF-elliptical plate