Klaus Schiefermayr

Description of Inverse Polynomial Images which Consist of Two Jordan Arcs with the Help of Jacobi’s Elliptic Functions

First we discuss the description of inverse polynomial images of [−1,1], which consists of two Jordan arcs, by the endpoints of the arcs only. The polynomial which generates the two Jordan arcs is given explicitly in terms of Jacobi’s theta functions. Then we concentrate on the case where the two arcs are symmetric with respect to the real line. In particular it is shown that the endpoints vary monotonically with respect to the modulus k of the associated elliptic functions.

Estimates for the asymptotic convergence factor of two intervals

Abstract Let E be the union of two real intervals not containing zero. Then L n r ( E ) denotes the supremum norm of that polynomial P n of degree less than or equal to n , which is minimal with respect to the supremum norm provided that P n ( 0 ) = 1 . It is well known that the limit κ ( E ) ≔ lim n → ∞ L n r ( E ) n exists, where κ ( E ) is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor κ ( E ) can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for κ ( E ) in terms of elementary functions of the endpoints of E .

Inverse polynomial images which consists of two Jordan arcs---An algebraic solution

Inverse polynomial images of [-1,1], which consists of two Jordan arcs, are characterised by an explicit polynomial equation for the four endpoints of the arcs.

The Optimal Control Of A General Tandem Queue

We consider a scheduling problem with two interconnected queues and two flexible servers. It is assumed that all jobs are present at the beginning and that there are no further arrivals to the system at any time. For each job, there are waiting costs per unit of time until the job leaves the system. A job of queue 1, after being served, joins queue 2 with probability p and leaves the system with probability 1 − p. The objective is how to allocate the two servers to the queues such that the expected total holding costs until the system is empty are minimized. We give a sufficient condition such that for any number of jobs in queue 1 and queue 2, it is optimal to allocate both servers to queue 1 (resp. queue 2).

Geometric Properties of Inverse Polynomial Images

Given a polynomial \({\mathcal{T}}_{n}\) of degree n, consider the inverse image of \(\mathbb{R}\) and [−1,1], denoted by \({\mathcal{T}}_{n}^{-1}(\mathbb{R})\) and \({\mathcal{T}}_{n}^{-1}([-1,1])\), respectively. It is well known that \({\mathcal{T}}_{n}^{-1}(\mathbb{R})\) consists of n analytic Jordan arcs moving from ∞ to ∞. In this paper, we give a necessary and sufficient condition such that (1)\({\mathcal{T}}_{n}^{-1}([-1,1])\) consists of ν analytic Jordan arcs and (2)\({\mathcal{T}}_{n}^{-1}([-1,1])\) is connected, respectively.

An upper bound for the logarithmic capacity of two intervals

The logarithmic capacity (also called Chebyshev constant or transfinite diameter) of two real intervals [−1, α]  ∪ [β, 1] has been given explicitly with the help of Jacobis elliptic and theta functions already by Achieser in 1930. By proving several inequalities for these elliptic and theta functions, an upper bound for the logarithmic capacity in terms of elementary functions of  α and  β is derived.

Random walks with similar transition probabilities

We consider random walks on the nonnegative integers with a possible absorbing state at -1. A random walk is called α-similar to a random walk X if there exist constants Cij such that for the corresponding n-step transition probabilities Pij(n)= α-nCijPij(n), i,j ≥ 0, hold. We give necessary and sufficient conditions for the α-similarity of two random walks both in terms of the parameters and in terms of the corresponding spectral measures which appear in the spectral representation of the n-step transition probabilities developed by Karlin and McGregor.

Inverse Polynomial Images Consisting of an Interval and an Arc

In this paper, some geometric properties of inverse polynomial images which consist of a real interval and an arc symmetric with respect to the real line are obtained. The proofs are based on properties of Jacobi’s elliptic and theta functions.

A density result concerning inverse polynomial images

In this paper, we consider polynomials of degree n, for which the inverse image of [−1, 1] consists of two Jordan arcs. We prove that the four endpoints of these arcs form an O(1/n)-net in the complex plane.

Zolotarev's conformal mapping and Chebotarev's problem

Consider the Chebotarev problem of finding a continuum S in the complex plane including some given points such that the logarithmic capacity of S is minimal. In this paper, we give a complete solution of this problem for the case of three given points with the help of Zolotarevs conformal mapping using Jacobian elliptic and theta functions. Moreover, for four given points, some special cases can be treated.