66 research outputs found
The Gilbert Arborescence Problem
We investigate the problem of designing a minimum cost flow network
interconnecting n sources and a single sink, each with known locations in a
normed space and with associated flow demands. The network may contain any
finite number of additional unprescribed nodes from the space; these are known
as the Steiner points. For concave increasing cost functions, a minimum cost
network of this sort has a tree topology, and hence can be called a Minimum
Gilbert Arborescence (MGA). We characterise the local topological structure of
Steiner points in MGAs, showing, in particular, that for a wide range of
metrics, and for some typical real-world cost-functions, the degree of each
Steiner point is 3.Comment: 19 pages, 7 figures. arXiv admin note: text overlap with
arXiv:0903.212
Algorithms for the power-p Steiner tree problem in the Euclidean plane
We study the problem of constructing minimum power- Euclidean -Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of (where ), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio of the beaded-MST heuristic satisfies . We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the case
The algebra of entanglement and the geometry of composition
String diagrams turn algebraic equations into topological moves that have
recurring shapes, involving the sliding of one diagram past another. We
individuate, at the root of this fact, the dual nature of polygraphs as
presentations of higher algebraic theories, and as combinatorial descriptions
of "directed spaces". Operations of polygraphs modelled on operations of
topological spaces are used as the foundation of a compositional universal
algebra, where sliding moves arise from tensor products of polygraphs. We
reconstruct several higher algebraic theories in this framework.
In this regard, the standard formalism of polygraphs has some technical
problems. We propose a notion of regular polygraph, barring cell boundaries
that are not homeomorphic to a disk of the appropriate dimension. We define a
category of non-degenerate shapes, and show how to calculate their tensor
products. Then, we introduce a notion of weak unit to recover weakly degenerate
boundaries in low dimensions, and prove that the existence of weak units is
equivalent to a representability property.
We then turn to applications of diagrammatic algebra to quantum theory. We
re-evaluate the category of Hilbert spaces from the perspective of categorical
universal algebra, which leads to a bicategorical refinement. Then, we focus on
the axiomatics of fragments of quantum theory, and present the ZW calculus, the
first complete diagrammatic axiomatisation of the theory of qubits.
The ZW calculus has several advantages over ZX calculi, including a
computationally meaningful normal form, and a fragment whose diagrams can be
read as setups of fermionic oscillators. Moreover, its generators reflect an
operational classification of entangled states of 3 qubits. We conclude with
generalisations of the ZW calculus to higher-dimensional systems, including the
definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures;
University of Oxford doctoral thesi
Recommended from our members
Accelerating Materials Discovery with Machine Learning
As we enter the data age, ever-increasing amounts of human knowledge are being recorded in machine-readable formats.
This has opened up new opportunities to leverage data to accelerate scientific discovery.
This thesis focuses on how we can use historical and computational data to aid the discovery and development of new materials.
We begin by looking at a traditional materials informatics task -- elucidating the structure-function relationships of high-temperature cuprate superconductors.
One of the most significant challenges for materials informatics is the limited availability of relevant data.
We propose a simple calibration-based approach to estimate the apical and in-plane copper-oxygen distances from more readily available lattice parameter data to address this challenge for cuprate superconductors.
Our investigation uncovers a large, unexplored region of materials space that may yield cuprates with higher critical temperatures.
We propose two experimental avenues that may enable this region to be accessed.
Computational materials exploration is bottle-necked by our ability to provide input structures to feed our workflows.
Whilst \textit{ab-intio} structure identification is possible, it is computationally burdensome and we lack design rules for deciding where to target searches in high-throughput setups.
To address this, there is a need to develop tools that suggest promising candidates, enabling automated deployment and increased efficiency.
Machine learning models are well suited to this task, however, current approaches typically use hand-engineered inputs.
This means that their performance is circumscribed by the intuitions reflected in the chosen inputs.
We propose a novel way to formulate the machine learning task as a set regression problem over the elements in a material.
We show that our approach leads to higher sample efficiency than other well-established composition-based approaches.
Having demonstrated the ability of machine learning to aid in the selection of promising compound compositions, we next explore how useful machine learning might be for identifying fabrication routes.
Using a recently released data-mined data set of solid-state synthesis reactions, we design a two-stage model to predict the products of inorganic reactions.
We critically explore the performance of this model, showing that whilst the predictions fall short of the accuracy required to be chemically discriminative, the model provides valuable insights into understanding inorganic reactions.
Through careful investigation of the model's failure modes, we explore the challenges that remain in the construction of forward inorganic reaction prediction models and suggest some pathways to tackle the identified issues.
One of the principal ways that material scientists understand and categorise materials is in terms of their symmetries.
Crystal structure prototypes are assigned based on the presence of symmetrically equivalent sites known as Wyckoff positions.
We show that a powerful coarse-grained representation of materials structures can be constructed from the Wyckoff positions by discarding information about their coordinates within crystal structures.
One of the strengths of this representation is that it maintains the ability of structure-based methods to distinguish polymorphs whilst also allowing combinatorial enumeration akin to composition-based approaches.
We construct an end-to-end differentiable model that takes our proposed Wyckoff representation as input.
The performance of this approach is examined on a suite of materials discovery experiments showing that it leads to strong levels of enrichment in materials discovery tasks.
The research presented in this thesis highlights the promise of applying data-driven workflows and machine learning in materials discovery and development.
This thesis concludes by speculating about promising research directions for applying machine learning within materials discovery
Eisenstein series and automorphic representations
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of the Langlands constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker function associated to unramified automorphic
representations of G(Q_p). In addition, we explain how the classical theory of
Hecke operators fits into the modern theory of automorphic representations of
adelic groups, thereby providing a connection with some key elements in the
Langlands program, such as the Langlands dual group LG and automorphic
L-functions. Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
also introduce some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In particular, we
provide a detailed treatment of supersymmetry constraints on string amplitudes
which enforce differential equations of the same type that are satisfied by
automorphic forms. Our treatise concludes with a detailed list of interesting
open questions and pointers to additional topics which go beyond the scope of
this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with
highlighted applications to string theory. v2: 375 pages. Substantially
extended and small correction
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