We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials p minimizing Dirichlet-type norms ‖pf − 1‖α for a given function f . For α ∈ [0, 1] (which includes the Hardy and Dirichlet spaces of the disk) and general f , we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative α, the weighted Bergman space case, the ext...

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain explicit upper envelope of it admits a superharmonic majorant. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant. 1. Definitions and statements Let D stand for the open unit disk in the complex plane, and H(D) ...

A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Since the minimum dominating set problem for unit disk graphs is NP-hard, several approximation algorithms with different merits have been proposed in the literature. On one extreme, there is a linear time 5-a...

The performance of striped disk arrays is governed by two parameters: the stripe unit size and the degree of striping. In this paper, we describe techniques for determining the stripe unit size and degree of striping for disk arrays storing variable bit rate continuous media data. We present an analytical model that uses the server configuration and the workload characteristics to predict the l...

This thesis studies the effects of latent sector faults on reliability, performance, and combined performability of disk arrays. This is done by developing two novel reliability models that include two fault types: disk unit faults and sector faults. The new reliability models study both hot swap and hot spare approaches in repairing disk unit faults. A traditional analytical method and also a ...

We look at the problem of coloring locally specially constructed spanners of unit disk graphs. First we present a local approximation algorithm for the vertex coloring problem in Unit Disk Graphs (UDGs) which uses at most four times as many colors as an optimal solution requires. Next we look at the colorability of spanners of UDGs. In particular we present a local algorithm for constructing a ...

Bidimensionality theory was introduced by Demaine et al. [JACM 2005 ] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of th...

Let P ⊂ R2 be a set of n point sites. The unit disk graph UD(P) on P has vertex set P and an edge between two sites p,q ∈ P if and only if p and q have Euclidean distance |pq| 6 1. If we interpret P as centers of disks with diameter 1, then UD(P) is the intersection graph of these disks, i.e., two sites p and q form an edge if and only if their corresponding unit disks intersect. Two natural ge...