Empty pentagons in point sets with collinearities

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.Comment: 15 pages, 11 figure

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]

Happy endings for flip graphs

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio

The design and analysis of mixture experiments with applications to glazes.

This thesis is concerned with mixture experiments. A mixture experiment is one in which certain properties of a product depend on the relative proportions of its ingredients. This thesis looks at existing and new methods for generating optimal mixture designs, as well as the form of the optimal design. A mixture experiment of industrial relevance is designed and analysed. The example used is the production of suspensions of ceramic glaze. Johnson Matthey Ceramics plc (JM) are interested in producing a model for predicting the viscosity of these suspensions. This viscosity will depend on the proportions of each ingredient and certain processing variables. In seeking optimal designs attention is restricted to designs over candidate sets consisting of points which are known to occur frequently in optimal designs. A variety of computer based methods for generating optimal designs were investigated by comparing the designs they produce for the regular simplex with known optima. It is shown that exchange algorithms using these candidate sets produced near efficient designs. Little is known about the form of the optimal design for constrained mixture problems. An algorithm is developed to generate continuous D optimal designs over a given candidate set. The form of these D optimal designs is investigated for the quadratic and cubic models for three components (q=3), and the quadratic model for q=4. The designs for the quadratic models included all vertices plus selections from the mid points of long edges and the centroid. Designs for the cubic model also included selections from thirds of edges, Mikaeili points and axial check points. The weights of points were found to be particularly dependent on those of their near neighbours. The form of the V optimal design for a quadratic model and q=3 was also investigated. The conclusions were similar to those for D optimality, although the centroid was often weighted considerably more highly. Practical situations often require repeated trials. Various algorithms were developed for generating optimal designs with r trials included twice. For q > 3, the best results were obtained using an exchange algorithm which only considered designs with r trials repeated. Following an extensive programme of experimental work, and detailed discussion with JM to identify important responses, models were produced to describe the relationship between viscosity and the explanatory variables. It was found that increasing the amount of clay and frit in the mixture increased the viscosity, while increasing water decreased it. As particle size was reduced and flocculant was added the effects which the mixture variables had on the top plateau and yield point was found to increase in magnitude

Combinatorial geometry of point sets with collinearities

© 2014 Dr. Michael S. PayneIn this thesis various combinatorial problems relating to the geometry of point sets in the Euclidean plane are studied. The unifying theme is that all the problems involve point sets that are not in general position, but have some collinearities. Topics addressed include; Dirac's conjecture related to the maximum degree of visibility graphs, Erdős' general position subset selection problem, connectivity properties of visibility graphs, visibility in bichromatic point sets, and empty pentagons in point sets with collinearities