James Cruickshank

• Doctor of Philosophy
• Lecturer at Ollscoil na Gaillimhe – University of Galway

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex polyhedron $P$ is rigid i.e. every continuous motion of the vertices of $P$ in $\mathbb R^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex poytopes in $\mathbb R^d$ for all $d\geq 3$ by Whiteley, and to...

We show that if $\Gamma $ is a point group of $\mathbb {R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal {S}$ is a k-pseudomanifold which has a free automorphism of order two, then either $\mathcal {S}$ has a $\Gamma $-symmetric infinitesimally rigid realisation in ${\mathbb R}^{k+1}$ or $k=2$ and $\Gamma $ is a half-turn rotation group. Thi...

We show that minimally 3-rigid block-and-hole graphs with one block are characterised as those that are constructible from K 3 by vertex splitting, and also as those having associated looped face graphs that are (3, 0)-tight. This latter property can be verified in polynomial time by a form of pebble game algorithm. We also indicate an application...

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for t...

Identifiability of data is one of the fundamental problems in data science. Mathematically it is often formulated as the identifiability of points satisfying a given set of algebraic relations. A key question then is to identify sufficient conditions for observations to guarantee the identifiability of the points. This paper proposes a new general...

We show that, if $\Gamma$ is a point group of $\mathbb{R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal S$ is a $k$-pseudomanifold which has a free automorphism of order two, then either $\mathcal S$ has a $\Gamma$-symmetric infinitesimally rigid realisation in $\mathbb{R}^{k+1}$ or $k=2$ and $\Gamma$ is a half-turn rotation group.This verif...

We give a relatively short graph theoretic proof of a result of Jordán and Tanigawa that a 4-connected graph which has a spanning plane triangulation as a proper subgraph is generically globally rigid in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepacka...

We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for (2,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{...

We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in $\mathbb R^d$. In our setting we allow multiple vertices to be constrained to the same line. Under a mild assumption on the given set of lines we give a complete combinatorial characterisation of graphs that are generical...

We prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case $d=3$ resolves a conjecture of Connelly. Our results actually apply to a much larger class of simplicial comp...

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer $b$ there is such an inductive construction of triangulations with $b$ braces, having finitely many base graphs. In particular we establish a bound f...

We present representations of tight surface graphs by contacts of circular arcs where the vertices are interior disjoint circular arcs in the flat surface and each edge is realised by an endpoint of one arc touching the interior of another. We try to find necessary and/or sufficient conditions for a surface graph to be the contact graph of a collec...

We characterise the quotient surface graphs arising from symmetric contact systems of line segments in the plane and also from symmetric pointed pseudotriangulations in the case where the group of symmetries is generated by a translation or a rotation of finite order. These results generalise well known results of Thomassen, in the case of line seg...

Let B be a ring, not necessarily commutative, having an involution ⁎ and let U2m(B) be the unitary group of rank 2m associated to a hermitian or skew hermitian form relative to ⁎. When B is finite, we construct a Weil representation of U2m(B) via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive...

We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for \( (2,2) \)-tight surface graphs in the case of the sphere, the plane, the twice punctured sphere and the torus. In the case of the torus we identify all 116 irreducible base graphs and provide a geometric application to con...

Let $B$ be a ring, not necessarily commutative, having an involution $*$ and let ${\mathrm U}_{2m}(B)$ be the unitary group of rank $2m$ associated to a hermitian or skew hermitian form relative to $*$. When $B$ is finite, we construct a Weil representation of ${\mathrm U}_{2m}(B)$ via Heisenberg groups and find its explicit matrix form on the Bruh...

A simple graph is 3‐rigid if its generic embeddings in R3 are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal 3‐rigidity of a simple graph obtained from a triangulated torus by the deletion of edges interior to an embedded triangulated disc.

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite.

We consider the problem of characterising the generic rigidity of bar-joint frameworks in Rd in which each vertex is constrained to lie in a given affine subspace. The special case when d=2 was previously solved by I. Streinu and L. Theran in 2010. We will extend their characterisation to the case when d≥3 and each vertex is constrained to lie in a...

The influence of our peers is a powerful reinforcement for our social behaviour, evidenced in voter behaviour and trend adoption. Bootstrap percolation is a simple method for modelling this process. In this work we look at bootstrap percolation on hyperbolic random geometric graphs, which have been used to model the Internet graph, and introduce a...

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Theran in 2010. We will extend their characterisation to the case when $d\geq 3$ and each vertex is con...

Let $(S,*)$ be an involutive local ring and let $U(2m,S)$ be the unitary group associated to a nondegenerate skew hermitian form defined on a free $S$-module of rank $2m$. A presentation of $U(2m,S)$ is given in terms of Bruhat generators and their relations. This presentation is used to construct an explicit Weil representation of the symplectic g...

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.

We study the classification problem of possibly degenerate hermitian and skew hermitian bilinear forms over local rings where 2 is a unit.

We study the classification problem of possibly degenerate hermitian and skew hermitian bilinear forms over local rings where 2 is a unit.

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite.

A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal $3$-rigidity of a simple graph which is obtained from the $1$-skeleton of a triangulated torus by the deletion of edges interior to a triangulated disc.

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal $3$-rigidity are obtaine...

We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and, as a consequence, are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid gr...

Given a weighted graph (G,l)(G,l) and its associated moduli space M(G,l)M(G,l), then a sufficient condition is provided which ensures that M(G,l)M(G,l) is a smooth manifold whenever G is a series-parallel graph.

We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and as a consequence are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid grap...

Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified extension of finite local rings.

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unita...

We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected. Comment: 18 pages, 11 figures

Blood lactate markers are used as summary measures of the underlying model of an athlete's blood lactate response to increasing work rate. Exercise physiologists use these endurance markers, typically corresponding to a work rate in the region of high curvature in the lactate curve, to predict and compare endurance ability. A short theoretical back...

We formulate and prove a rearrangement type inequality and use it to give a description of the face lattice of a certain polytope that is naturally associated to the alternating group An.

We develop methods for computing the equivariant homotopy set [M,SV]G, where M is a manifold on which the group G acts freely, and V is a real linear representation of G. Our approach is based on the idea that an equivariant invariant of M should correspond to a twisted invariant of the orbit space M/G. We use this method to make certain explicit c...