We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of P. Turán): namely, for n < N , one can freely prescribe the n-th polynomial and N − n zeros of the N...

Let A(p) denote the class of functions which are analytic in the open unit disk U. By making use of certain integral operator,we obtain some interesting properties of multivalent analytic functions.

We model a problem about networks built from wireless devices using identifying and locating-dominating codes in unit disk graphs. It is known that minimising the size of an identifying code is NP-complete even for bipartite graphs. First, we improve this result by showing that the problem remains NP-complete for bipartite planar unit disk graphs. Then, we address the question of the existence ...

By a Riemann mapping we mean an analytic function which is defined and univalent on the unit disk. Such a function is said to be normalized if it has value zero and positive derivative at the origin. We are concerned with normalized Riemann mappings of the unit disk into itself. If B(z) is such a function, the operator f(z) into f(B(z)) maps the Dirichlet space contractively into itself. Elemen...

A quadriculated disk is a juxtaposition of finitely many squares along sides forming a closed topological disk such that vertices of squares which lie in the interior of the disk belong to precisely four squares. In this paper, by a square we mean a topological disk with four privileged boundary points, the vertices. A simple example of a quadriculated disk is the n ×m rectangle divided into un...

Choose random pointsX1, X2, X3, . . . independently from a uniform distribution in a unit ball in <. Call Xn a dominator iff distance(Xn, Xi) ≤ 1 for all i < n, i.e. the first n points are all contained in the unit ball that is centered at the n’th point Xn. We prove that, with probability one, only finitely many of the points are dominators. For the special casem = 2, we consider the unit disk...

We prove that the max-cut and max-bisection problems are NP-hard on unit disk graphs. We also show that λ-precision graphs are planar for λ > 1/ √ 2 and give a dichotomy theorem for max-cut computational complexity on λ-precision unit disk graphs.

We propose a polynomial-time algorithm which takes as input a finite set of points of R3 and compute, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii. Almost three decades ago, an elegant...

We obtain separation and growth results for meromorphic functions f in the unit disk such that for some positive integer k, f 6= 0 and f (k) 6= 1 in the disk, or such that f ′fk 6= 1 in the disk. These results are only slightly weaker than those for functions f such that |f | > δ in the disk.

We derive some criteria for univalence of certain integral operators for analytic functions in the open unit disk. 1. Introduction. Let Ꮽ be the class of the functions f (z) which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and f (0) = f (0) − 1 = 0.