It is well-known and easy to see that the zeros of both the associated polynomial and the derivative of an orthogonal polynomial pn(x) interlace with the zeros of pn(x) itself. The natural question of how these zeros interlace is under discussion. We give a sufficient condition for the mutual location of k-th, 1 ≤ k ≤ n − 1, zeros of the associated polynomial and the derivative of an orthogonal...

Newton-Cotes quadrature rules are based on polynomial interpolation in a set of equidistant points. They are very useful in applications where sampled function values are only available on a regular grid. Yet, these rules rapidly become unstable for high orders. In this paper we review two techniques to construct stable high-order quadrature rules using equidistant quadrature points. The stabil...

A decrease in arterial oxygen saturation (Sao2) has been observed to follow intravenous aminophylline administration in patients with chronic pulmonary disease (Halmagyi and Cotes, 1959). The decrease in Sao2 occurred 40 minutes or more after the injection in a group of patients who had shown an increase in maximum breathing capacity (M.B.C.) in response to the drug. The onset of symptomatic re...

This paper presents a framework for the calculation of functional requirements at several stages of the product life cycle. It uses the TTRS/MGRE concept for the determination of dimension chains. The geometry of the assembly is represented using a parametric model constituted of points and vectors. Finally, it also integrates the calculation of part deformation due to changing operating condit...

In many applications of ageand size-structured population models, there is an interest in obtaining good approximations of total population numbers rather than of their densities. Therefore, it is reasonable in such cases to solve numerically not the PDE model equations themselves, but rather their integral equivalents. For this purpose quadrature formulae are used in place of the integrals. Si...

Fractional directional integrals are the extensions of the Riemann-Liouville fractional integrals from oneto multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton-Cotes and Gauss-Legendre rules. It ...

Several deenitions of universality of an n-point quadrature formula Q n are discussed. Universality means that Q n is able to compete with the respective optimal formula in many classes of functions. It is proved in a certain sense that the Gaus-sian quadrature formula satisses such a universality criterion. The underlying classes of functions are In each of these classes, we loose at most the ...

This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on ...

In a previous paper it was shown how a probability measure (Eberlein measure) on the closed unit ball of the sequence space, h, can be used to find the variance a2 of the error functional for a quadrature formula for the fc-dimensional cube, regarded as a random variable. Here we give values of a for some specific formulae.