the aim of this paper is to prove some inequalities for p-valent meromorphic functions in thepunctured unit disk δ* and find important corollaries.

For any graph G, the k-improper chromatic number χ(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate the ratio of the kimproper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of McDiarmid and Reed (1999); McDiarmid (2003) (where they considered...

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [5], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general s...

Let S ⊂ R be a set of point sites. The unit disk graph UD(S) of S has vertex set S and an edge between two sites s, t if and only if |st| ≤ 1. We present a data structure that maintains the connected components of UD(S) when S changes dynamically. It takes O(log n) time to insert or delete a site in S and O(log n/ log log n) time to determine if two sites are in the same connected component. He...

We suggest a geometrical framework in which to discuss periodic layered structures in the unit disk. Bandgaps appear when the point representing the system approaches the unit circle. We show that the trace of the matrix describing the basic period allows for a classification in three families of orbits with quite different properties. The laws of convergence of the iterates to the unit circle ...

The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques. Given n points in the plane, the corresponding unit disk graph (UDG) has these points as vertices, and edges connecting points at distance at most 1. MCP in unit disk graphs is known to be NP-hard and several constant factor approximations are known, including a r...

Let V be a complete discrete valuation ring of mixed characteristic. We classify arithmetic D-modules on Spf(V[[t]]) up to certain kind of ‘analytic isomorphism’. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

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Motivated by a satellite communications problem, we consider a generalised colouringproblem on unit disk graphs. A colouring is k-improper if no vertex receives the same colouras k+1 of its neighbours. The k-improper chromatic number χ(G) the least number of coloursneeded in a k-improper colouring of a graph G. The main subject of this work is analysingthe complexity of computin...