Let Ω be a disk of radius R in the plane. A set F of closed unit disks contained in Ω forms a maximal packing if the unit disks are pairwise disjoint and the set is maximal: i.e., it is not possible to add another disk to F while maintaining the packing property. A point p is hidden within the “forest” defined by F if any ray with apex p intersects some disk of F : that is, a person standing at...

Fill an n × n matrix with independent complex Gaussians of variance 1/n. As n → ∞, the eigenvalues {zk} converge to a sum of an H1-noise on the unit disk and an independent H1/2-noise on the unit circle. More precisely, for C1 functions of suitable growth, the distribution of ∑n k=1(f(zk) − Ef(zk)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the di...

We show that finding a minimal number of landmark nodes for a unique virtual addressing by hop-distances in wireless ad-hoc sensor networks is NPcomplete even if the networks are unit disk graphs that contain only Gabriel edges. This problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The Gabriel edges of a unit disc graph induce a planar O( √ n) distance and an optimal ener...

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szegö measure on the unit circumference is perturbed...

The radio frequency assignment problem is to minimize the number of frequencies used by transmitters with no interference in radio communication networks; it can be modeled as the minimum vertex coloring problem on unit disk graphs. In this paper, we consider the on-line first-fit algorithm for the problem and show that the competitive ratio of the algorithm for the unit disk graph G with χ(G)=...

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact su...

We give a new local test, called a Half-Space Proximal or HSP test, for extracting a sparse directed or undirected subgraph of a given unit disk graph. The HSP neighbors of each vertex are unique, given a fixed underlying unit disk graph. The HSP test is a fully distributed, computationally simple algorithm that is applied independently to each vertex of a unit disk graph. The directed spanner ...

We present a distributed, localized and integrated approach for establishing both low-level (i.e. exploration of 1-hop neighbors, interference avoidance) and high-level (a subgraph of the unit-disk graph) infrastructure in wireless sensor networks. More concretely, our proposed scheme constructs a subgraph of the unit-disk graph which is connected, planar and has power stretch factor of 1 (the ...

If a function f has finite Henstock integral on the boundary of the unit disk of R 2 then its Poisson integral exists for |z| < 1 and is o((1 − |z|) −1) as |z| → 1 −. It is shown that this is the best possible uniform pointwise estimate. For an L 1 measure the best estimate is O((1 − |z|) −1). In this paper we consider estimates of Poisson integrals on the unit circle with respect to Alexiewicz...

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szegö measure on the unit circumference is perturbed...