arXiv subjects

Richard Cowan

Publications and source records attributed to Richard Cowan.

5 recordsLinked to original sources

arXiv2016

Tessellations derived from random geometric graphs

In this paper we consider a random partition of the plane into cells, the partition being based on the nodes and links of a {\it random planar geometric graph}. The resulting structure generalises the \emph{random \tes}\ hitherto studied in the literature. The cells of our partition process, possibly with holes and not necessarily closed, have a fairly general topology summarised by a functional which is similar to the Euler characteristic. The functional can also be extended to certain cell-unions which can arise in applications. Vertices of all valencies, $0, 1, 2, ...$ are allowed. Many of the formulae from the traditional theory of random tessellations with convex cells, are made more general to suit this new structure. Some motivating examples of the structure are given.

math.PR↗

arXiv2014

Column tessellations

A new class of random spatial tessellations is introduced -- the so-called column tessellations of three-dimensional space. The construction is based on a stationary planar tessellation. Each cell of the spatial tessellation is a prism whose base facet is congruent to a cell of the planar tessellation. Thus intensities, topological and metric mean values of the spatial tessellation can be calculated by suitably chosen parameters of the planar tessellation. A column tessellation is not facet-to-facet.

math.PR↗

arXiv2013

Constraints on the fundamental topological parameters of spatial tessellations

Tessellations of $R^3$ that use convex polyhedral cells to fill the space can be extremely complicated, especially if they are not facet-to-facet, that is, if the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. In a recent paper (Weiss and Cowan, Adv.Appl.Prob. 2011), we have developed a theory which covers these complicated cases, at least with respect to their combinatorial topology. The theory required seven parameters, three of which suffice for facet-to-facet cases; the remaining four parameters are needed for the awkward adjacency concepts that arise in the general case. This current paper establishes constraints that apply to these seven parameters and so defines a permissible region within their seven-dimensional space, a region which we discover is not bounded. Our constraints in the relatively simple facet-to-facet case are also new.

math.PR↗

arXiv2013

The Moment Generating function for ray lengths in the Half Gilbert Model with Rectangular Cells

In the full rectangular version of Gilbert's tessellation lines extend either horizontally (with east- and west--growing rays) or vertically (north- and south--growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east and south growing rays do not interact with west and north rays. Using techniques developed in our previous paper, we derive an exact expression for the moment generating function for the ray length distribution in the half rectangular model.

math.PR↗

arXiv2012

Full and Half Gilbert Tessellations with Rectangular Cells

We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation. In the full rectangular version, lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east and south growing rays do not interact with west and north rays. For the half rectangular tessellation we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst for the full rectangular version we develop an accurate simulation technique, based in part on the stopping-set theory of Zuyev, to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. Our paper explores this coincidence mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moment of ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model as much better approximations than those provided by Mackissack and Miles.

math.PR↗