Physical Sciences and Mathematics Commons^™

Fredholm-Regularity Of Holomorphic Discs In Plane Bundles Over Compact Surfaces, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has C2,α-close sections such that any (non-multiply covered) holomorphic disc with boundary in these sections are Fredholm regular. Fredholm regularity is also established when the complex surface is neutral K¨ahler, the action is both holomorphic and symplectic, and the section is Lagrangian with a single complex point.

Mean Curvature Flow Of Compact Spacelike Submanifolds In Higher Codimension, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We prove long-time existence for mean curvature flow of a smooth n-dimensional spacelike submanifold of an n + m dimensional manifold whose metric satisfies the timelike curvature condition.

A Global Version Of A Classical Result Of Joachimstha, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove the following global analogue of this result. Suppose that two closed convex surfaces intersect with constant angle along a curve that is not umbilic in either surface. We prove that the principal foliations of the two surfaces along the curve are either both orientable, or both nonorientable. We prove this by characterizing the constant angle …

A Decision Support Tool For Building Integrated Renewable Energy Microgrids Connected To A Smart Grid, Damilola A. Asaleye, Michael D. Murphy, Michael Breen

Publications

The objective of this study was to create a tool that will enable renewable energy microgrid (REμG) facility users to make informed decisions on the utilization of electrical power output from a building integrated REμG connected to a smart grid. A decision support tool for renewable energy microgrids (DSTREM) capable of predicting photovoltaic array and wind turbine power outputs was developed. The tool simulated users’ daily electricity consumption costs, avoided CO₂ emissions and incurred monetary income relative to the usage of the building integrated REμG connected to the national electricity smart grid. DSTREM forecasted climate variables, which were used …

Parabolic Classical Curvature Flows, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Hopf Hypersurfaces In Spaces Of Oriented Geodesics., Nikos Georgiou, Brendan Guilfoyle

Publications

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the space of oriented geodesics admits a canonical Kaehler–Einstein and para-Kaehler–Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in …

A Converging Lagrangian Flow In The Space Of Oriented Lines, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is the Steiner point of the initial surface, which remains constant under the flow.
To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition.Moreover, this flow converges to a holomorphic Lagrangian section, which forms …

Totally Null Surfaces In Neutral K¨Ahler 4-Manifolds, Nikos Georgiou, Brendan Guilfoyle, Wilhelm Klingenberg

Publications

We study the totally null surfaces of the neutral K¨ahler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the α-planes are integrable and α-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The β-surfaces are less known and our interest is mainly in their …

The Definitions Of Three-Dimensional Landmarks On The Human Face: An Interdisciplinary View, Stanislav Katina, Kathryn Mcneil, Ashraf Ayoub, Brendan Guilfoyle, Balvinder Khambay, Paul Siebert, Federico Sukno, Mario Rojas, Liberty Vittert, John Waddington, Paul F. Whelan, Adrian W. Bowman

Publications

The analysis of shape is a key part of anatomical research and in the large majority of cases landmarks provide a standard starting point. However, while the technology of image capture has developed rapidly and in particular three-dimensional imaging is widely available, the definitions of anatomical landmarks remain rooted in their two-dimensional origins. In the important case of the human face, standard definitions often require careful orientation of the subject. This paper considers the definitions of facial landmarks from an interdisciplinary perspective, including biological and clinical motivations, issues associated with imaging and subsequent analysis, and the mathematical definition of surface …

A Modified Surface On Titanium Deposited By A Blasting Process, Caroline O' Sullivan, Peter O'Hare, Greg Byrne, Liam O'Neill, Katie B. Ryan, Abina M. Crean

Publications

Abstract

: Hydroxyapatite (HA) coating of hard tissue implants is widely employed for its biocompatible and osteoconductive properties as well as its improved mechanical properties. Plasma technology is the principal deposition process for coating HA on bioactive metals for this application. However, thermal decomposition of HA can occur during the plasma deposition process, resulting in coating variability in terms of purity, uniformity and crystallinity, which can lead to implant failure caused by aseptic loosening. In this study, CoBlast™, a novel blasting process has been used to successfully modify a titanium (V) substrate with a HA treatment using a dopant/abrasive regime. …

On The Three-Dimensional Blaschke-Lebesgue Problem, Henri Anciaux, Brendan Guilfoyle

Publications

The width of a closed convex subset of n-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n ≥ 3. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n = 3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant and therefore are either spherical caps or pieces of tubes (canal surfaces).