Parallel Chip Firing Game associated with n-cube orientations

We study the cycles generated by the chip firing game associated with n-cube orientations. We show the existence of the cycles generated by parallel evolutions of even lengths from 2 to $2^n$ on $H_n$ (n >= 1), and of odd lengths different from 3 and ranging from 1 to $2^{n-1}-1$ on $H_n$ (n >= 4)

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics

Problems and results on linear hypergraphs

In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint. We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an $n\times n$ Latin square $A$ one can define a binary operation $\circ:[n]\times[n]\to [n]$, and $\circ$ is associative if and only if $A$ is a group multiplication table. Hrushovski asked whether, if $\circ$ is only associative a positive proportion of the time, $A$ must still in some sense be close to a group multiplication table. This problem manifests a well-studied combinatorial theme, in which a local structural constraint is relaxed (first to a `99$\%$' version and then to a `1$\%$' version) and the global consequences of the relaxed constraints are analysed. We show that the partial associativity condition is sufficient to deduce powerful global information, allowing us to find within $A$ a large subset with group-like structure. Since Latin squares can be regarded as 3-uniform, linear hypergraphs, and the partial associativity condition can be formulated in terms of the count of a particular subhypergraph, we are able to apply purely combinatorial methods to a problem that touches algebra, model theory and geometric group theory. We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square $A$, proves that $A$ is in fact the multiplication table of an abelian group. It is then natural to ask whether a relaxed version of this result is also attainable, and by extending our methods we are able to prove a result of this flavour. Since the combinatorial obstructions to commutativity of $\circ$ are far more complex than those for associativity, topological complications arise that are not present in the earlier work. We also study a problem of Loh concerning sequences of triples of integers from $[n]$ satisfying a certain `increasing' property. Loh studied the maximum length of such a sequence, improving a trivial upper bound of $n^2$ to $n^2/\exp(\log^*n)$ using the triangle removal lemma and conjecturing that a natural construction of length $n^{3/2}$ is best possible. We provide the first power-type improvement to the upper bound, showing that there exists $\epsilon>0$ such that the length is bounded by $n^{2-\epsilon}$. By viewing the triples as edges in a 3-uniform hypergraph, the increasing property shows that the hypergraph is linear and provides further restrictions in terms of forbidden subhypergraphs. By considering this formulation, we provide links to various important open problems including the Brown--Erd\H os--S\'os conjecture. Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S\'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili\'cevi\'c and Tan concerning partitions of boxes, consider a problem on projective cubes in $\mathbb{Z}_{2^n}$, and resolve a conjecture concerning a diffusion process on graphs

SpiNNaker - A Spiking Neural Network Architecture

20 years in conception and 15 in construction, the SpiNNaker project has delivered the world’s largest neuromorphic computing platform incorporating over a million ARM mobile phone processors and capable of modelling spiking neural networks of the scale of a mouse brain in biological real time. This machine, hosted at the University of Manchester in the UK, is freely available under the auspices of the EU Flagship Human Brain Project. This book tells the story of the origins of the machine, its development and its deployment, and the immense software development effort that has gone into making it openly available and accessible to researchers and students the world over. It also presents exemplar applications from ‘Talk’, a SpiNNaker-controlled robotic exhibit at the Manchester Art Gallery as part of ‘The Imitation Game’, a set of works commissioned in 2016 in honour of Alan Turing, through to a way to solve hard computing problems using stochastic neural networks. The book concludes with a look to the future, and the SpiNNaker-2 machine which is yet to come

SpiNNaker - A Spiking Neural Network Architecture

20 years in conception and 15 in construction, the SpiNNaker project has delivered the world’s largest neuromorphic computing platform incorporating over a million ARM mobile phone processors and capable of modelling spiking neural networks of the scale of a mouse brain in biological real time. This machine, hosted at the University of Manchester in the UK, is freely available under the auspices of the EU Flagship Human Brain Project. This book tells the story of the origins of the machine, its development and its deployment, and the immense software development effort that has gone into making it openly available and accessible to researchers and students the world over. It also presents exemplar applications from ‘Talk’, a SpiNNaker-controlled robotic exhibit at the Manchester Art Gallery as part of ‘The Imitation Game’, a set of works commissioned in 2016 in honour of Alan Turing, through to a way to solve hard computing problems using stochastic neural networks. The book concludes with a look to the future, and the SpiNNaker-2 machine which is yet to come

A family of matrix-tree multijections

For a natural class of $r \times n$ integer matrices, we construct a non-convex polytope which periodically tiles $\mathbb R^n$. From this tiling, we provide a family of geometrically meaningful maps from a generalized sandpile group to a set of generalized spanning trees which give multijective proofs for several higher-dimensional matrix-tree theorems. In particular, these multijections can be applied to graphs, regular matroids, cell complexes with a torsion-free spanning forest, and representable arithmetic matroids with a multiplicity one basis. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.Comment: Several edits from the previous version including a new title (the previous title was "A Combinatorial Mapping for the Higher-Dimensional Matrix-Tree Theorem"). There are also many added references to the author's dissertatio

Virtual camera selection using a semiring constraint satisfaction approach

Players and viewers of three-dimensional computer generated games and worlds view renderings from the viewpoint of a virtual camera. As such, determining a good view of the scene is important to present a good game or three-dimensional world. Previous research has developed technologies to nd good positions for the virtual camera, but little work has been done to automatically select between multiple virtual cameras, similar to a human director at a sporting event. This thesis describes a software tool to select among camera feeds from multiple virtual cameras in a virtual environment using semiring-based constraint satisfaction techniques (SCSP), a soft constraint approach. The system encodes a designer's preferences, and selects the best camera feed even in over-constrained or under-constrained environments. The system functions in real time for dynamic scenes using only current information (i.e. no prediction). To reduce the camera selection time the SCSP evaluation can be cached and converted to native code. This SCSP approach is implemented in two virtual environments: a virtual hockey game using a spectator viewpoint, and a virtual 3D maze game using a third person perspective. Comparisons against hard constraints are made using constraint satisfaction problems