The Rigidity of Spherical Frameworks: Swapping Blocks and Holes

A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work noted a combinatorial analogy in which blocks and holes played equivalent roles. In this paper, we connect stresses in such a structure (P, p) to first-order motions in a swapped structure (P', p), where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use a projective geometric presentation of the statics and the motions, to make the key underlying correspondences transparent.Comment: 36 pages, 9 figure

Fire Containment in Planar Graphs

In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $\rho_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs \cG we are interested in the minimum value $k$ such that $\rho_k(G)\ge\epsilon$ for some constant $\epsilon>0$ and all G\in\cG i.e., such that linearly many vertices are expected to be saved in every graph from \cG). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.Comment: 15 pages, one reference adde

3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.Comment: 8 page

Multiplicity in the documentation of performance-based artworks: Displaying multi-media documentation in Rebecca Horn’s Body Sculptures at Tate

This is the author accepted manuscript. The final version is available from Taylor & Francis via the DOI in this recordThis paper addresses some of the key issues around authenticity within the ephemeral performance and durable document dichotomy. Engaging with these two artistic practices within the frame of the museum and in the context of displays and exhibitions, this paper considers some of the ways in which the role of performance documentation has been reassessed over the past 20 years. It will focus on the access the document provides to a now-absent ‘performance moment’, the benefit of acquiring and displaying multiple types of documentation, and the experience of the museum visitor within this.Arts and Humanities Research Council (AHRC

The Value of Performance Documentation in the Contemporary Art Museum: A Case Study of Tate

Performance and documentation have a complex historical relationship, based around perceived binaries of ephemerality and endurance, liveness and fixedness, originality and representation. This thesis explores this relationship and the ontological perspectives which underpin it, but moves beyond this by building on those contemporary theories which consider the potential of the performance documentation in relation to the performance moment, and the expanded, continuing performance artwork. Using the example of Tate as a contemporary art museum which has a history of creating and collecting performance documentation, this research engages the lens of value as an analytic tool through which to understand the positions and purposes of performance documentation in the contemporary art museum. Rather than attempting to measure the amount of value a performance document is perceived to have in economic terms, the intention here is to understand the nuanced types of value those within the museum apply to the performance document, based on an understanding of valuations as subjective, context-dependent, pluralistic and changeable. This thesis will explore both the museum’s creation of performance documents, tracing the variety of practices across Tate’s numerous departments, and how those within the museum approach acquiring, conserving, and displaying existing performance documents. Six case studies will be used to explore how different models of temporality, materiality, and authorship impact on the actions individuals and departments within Tate have taken around the creation, collection, and use of performance documents, and will explore what these indicate about the multiple, changeable types of value a performance document is perceived to have. The thesis will end by proposing how these findings around value and valuation can feed back into strategies and practices which are being developed at Tate to provide centralised, reflexive, mobile and easily accessible documentation of those live art works in the museum collection.AHR

Space and value in the contemporary art museum: The journey of a performance document at Tate

This is the author accepted manuscript. The final version is available from Intellect via the DOI in this recordThe space of the museum, rather than being monolithic and heterogeneous, is complex, fluid and fractured. As an institution, its multiple spaces relate to a variety of activities, motivations and attitudes towards the objects it collects, conserves and displays. By using Michel Foucault’s 1967 notion of the ‘heterotopia’ to read the museum as a space of spaces, and focusing on the complex object of the performance document, this article traces the link between the placement of objects in a specific space, and how this can be read as a perspective on their value. In tracing the journey of the Joseph Beuys performance document Four Blackboards (1972) through various spaces at Tate Gallery (now Tate Britain) and Tate Modern, this article will demonstrate those acts of valuation being undertaken over a 50-year period in the institution, and explore how changing value perspectives result in a changing space, both physically and conceptually, for the performance document.Arts and Humanities Research Council (AHRC

The dynamic modelling and development of a controller for a general purpose remotely operated underwater vehicle

A preliminary mathematical model for the UCT SEAHOG Remotely operated underwater vehicle (ROV) is developed, including estimation of the rigid body, hydrodynamic and hydrostatic properties of the robot. A single state thruster model is developed and verified according to real life test data. A closed-loop speed controller is developed for the thruster module using a standard PI scheme and is implemented on an MSP430 microcontroller using software fixed-point algorithms. The complete ROV system is simulated in Simulink® in an open-loop configuration to gain insight into the expected motion from the vehicle. Controllers for depth and heading holding are designed using standard PID linearized control methods with gain scheduling and are then assessed within the complete system in a simulation environment. In addition, upgrades and maintenance are performed on the Power Pod, light and camera modules. Redesign, manufacture and testing of the SEAHOG junction box is performed, including a design solution to connect the tether power and fibre-optic lines at the surface and on the ROV. An extensive overhaul of the SEAHOG GUI is performed, utilising multicore processing architecture in LabVIEW and resulting in a user-orientated interface capable of controlling and monitoring all existing system data from the robot

A note on bounds for the cop number using tree decompositions

In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth

γ\gamma-Graphs of Trees

For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in $G(\gamma)$ if they differ by a single vertex and the two different vertices are adjacent in $G$. In this paper, we consider $\gamma$-graphs of trees. We develop an algorithm for determining the $\gamma$-graph of a tree, characterize which trees are $\gamma$-graphs of trees, and further comment on the structure of $\gamma$-graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.Comment: 22 pages, 3 figure

On Packing Colorings of Distance Graphs

The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=\{1,t\}$ for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for $t\geq 447$: $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 40$ if $t$ is odd and $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 81$ if $t$ is even