We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism. 1 Introduction. In this paper we recall from U2] the...

The parametric or geometric continuity of a rational polynomial curve has often been obtained by requiring the homogeneous polynomial curve associated with the rational curve to possess parametric or geometric continuity, respectively. Recently this approach has been shown overly restrictive. We make use of the necessary and su cient conditions of rational parametric continuity for de ning basi...

It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turr an, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. 0. Introduction The idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received...

Rational arithmetic functions are arithmetic functions of the form g1 ∗···∗ gr ∗ h−1 1 ∗ ···∗h−1 s , where gi, hj are completely multiplicative functions and ∗ denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type ide...

Acknowledgement I am very grateful to Professor Ferenc Schipp for initiating me on the field of the approximation theory, for his advices and remarks without them this thesis would have been impossible. I am also indebted to the members of the Department of Numerical Analysis of the Eötvös Loránd University for their support during the years. I am especially grateful to Professor József Bokor i...

Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degreeO(poly log(1/ )) which is -close, and similarly for any rational function there exists a ReLU network of size O(poly log(1/ )) which is -close. By contrast, polynomials need degree Ω(poly(1/ )) to approximate even a sing...

In this paper we introduce the D-resultant of two rational functions f(t), g(t) ∈ K(t) and show how it can be used to decide if K(f(t), g(t)) = K(t) or if K[t] ⊂ K[f(t), g(t)] and to find the singularities of the parametric algebraic curve define by X = f(t), Y = g(t). In the course of our work we extend a result about implicitization of polynomial parametric curves to the rational case, which ...

We find all polynomials f, g, h over a field K such that g and h are linear and f(g(x)) = h(f(x)). We also solve the same problem for rational functions f, g, h, in case the field K is algebraically closed.

Abstract. In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and power sum symmetric polynomials. We investigate properties of these families and focus on constructing their rational roots under a product induce...