Doctoral Theses (Mathematical & Computer Sciences)
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Item Nonlocal macroscopic limits of kinetic equations in biological and robotic systems(Heriot-Watt University, 2019-07) Estrada-Rodriguez, Gissel; Gimperlein, Doctor HeikoThis thesis addresses the derivation, analysis and numerical analysis of nonlocal partial differential equations from individual movement. More specifically, my main results concern how fractional diffusion and swarming behaviour arise as a limit of microscopic kinetic models for interacting particles. Applications of these results to biology, swarm robotic systems and networks are discussed. In the following I summarize the content of this thesis, which is motivated by using the analysis of PDEs to shed light on macroscopic behaviour in complex systems. Nonlocal diffusion from microscopic movement We assumed the motion of the individuals, in the case of chemotaxis, follows a velocity jump process, characterized by long runs according to an approximate Lévy distribution, interrupted by instantaneous reorientations. From the resulting kinetic equation obtained from this microscopic movement, and using a perturbation argument, we derived nonlocal Patlak-Keller-Segel equations in the appropriate limit. The resulting system involves fractional Laplacians that describe the nonlocal movement of the individuals. Subsequent work studied more complex search strategies. Motivated by recent experimental results from T. Harris et al. [105] in the case of T cells migrating through chronically-infected brain tissues, we considered that the runs are further interrupted by long pauses according to a Lévy distribution. No directional motion is present in this case. We obtained two coupled kinetic equations for the moving and resting populations. Solving the equation for the resting population introduces a nonlocal delay in time, consistent with those observed in experiments. The simple structure of this equation allows analytic insights not directly visible from the microscopic model. In particular, we found an explicit fundamental solution in R n , allowing us to study hitting times, showing the effect of the delay and the advantages of a Lévy search strategy over Brownian motion. Interacting particles with Lévy strategies and alignment We considerd Lévy robotic systems which combine superdiffusive random movement with emergent collective behaviour from local communication and alignment. We de rived a fractional PDE from the movement strategies of the individual robots, intro ducing long range interactions and alignment into the analysis. The resulting kinetic model is studied at short and long time scales. Applications we study include targeting efficiency and optimal search strategies. We showed that the system allows efficient parameter studies for a search problem, ad dressing basic questions such as the optimal number of robots needed to cover an area. Validation against concrete robotic simulations with e-puck robots are also included in collaboration with computer scientists from the Edinburgh Centre for Robotics. Superdiffusion in complex systems: metaplex networks In complex, non convex geometries, the fractional Laplacian, based on the Euclidean distance, is not the physically relevant operator to describe superdiffusive behaviour. Based on an approach used in complex systems to study superdiffusion in networks, where particles are allowed to hop to non-nearest neighbours, we propose to study diffusion using a network of subdomains, corresponding to the nodes of a graph. We introduced a general framework for diffusion in networks with internal structure in the nodes: metaplex networks. We illustrate its use in a toy model and in real-world networks. The results shed light on the rich and substantially different nature of the dynamics of metaplexes and the interplay of their internal and external structure.Item Mathematical analysis of a gas and plate model for a micro-electro-mechanical system (MEMS) capacitor(Heriot-Watt University, 2018-12) Simmons, Timothy James; Lacey, Andrew Alfred; Schmuck, MarcusAbstract unavailable. Please refer to PDF.Item Dispersal and periodic travelling waves in ecology(Heriot-Watt University, 2018-09) Bennett, Jamie J. R.; Sherratt, Professor Jonathan A.The research in this thesis is the culmination of four separate studies on periodic travelling wave generation in three ecological systems: cyclic predator-prey interactions, intertidal mussel beds and semi-arid vegetation. Patterns in space and time generated in mathematical models are analysed with the aim of identifying the underlying mechanisms in real-world ecosystems, and determining the impact of ecological change and variation. In particular, pattern formation theory is extended to include more realistic and justifiable model assumptions about population dispersal. In Chapter 1, we discuss the general ideas behind pattern formation theory; providing ecological examples and an overview of important mathematical techniques. In Chapter 2, we derive an equation for the amplitude of periodic travelling waves generated by an ecological invasion in cyclic predator-prey systems when populations disperse at different rates. In Chapter 3, we demonstrate how both stripe and spotted patterns can arise in intertidal mussel beds as a result of algal dispersal via tidal flow by performing an analysis in two space dimensions. In Chapter 4, we discover that the non-local seed dispersal of banded vegetation in semi-arid regions can increase ecosystem resilience to climate change via oscillating peaks of vegetation and stationary patterns. In Chapter 5, we return to invasive predator-prey systems and locate the transition from periodic travelling waves to spatio-temporal irregularity when populations disperse at different rates. In Chapter 6, we provide a summary of our conclusions.Item A proof-theoretic approach to coinduction in Horn clause logic(Heriot-Watt University, 2019-09) Li, Yue; Komendantskaya, Dr Ekaterina; Lawson, Professor Mark V.The thesis is on coinduction in Horn clauses. Specifically, it considers productive corecursion, and presents a framework called Coinductive Uniform Proof as a principled approach to coinduction in first-order Horn clause logic. It addresses the challenges of 1) discovering sufficient conditions for logic programs to be productive, 2) providing an explanation of why unification (without occur-check) between goals in a SLD derivation can be exploited to capture productive corecursion, and 3) identifying the principle that unifies the diverse approaches to Horn clause coinduction which are scattered across the literature. The thesis advances the state of the art by 1) providing a sufficient condition for productive corecursion which requires that a logic program does not admit perpetual term-rewriting steps nor existential variables, 2) showing that the goal-unification technique can be used to capture productive corecursion if a goal is no less general than some previous goal to which it unifies, and 3) defining a coinductively sound proof construction method for Horn clauses where a Horn clause to be proved is first asserted as an assumption and then used for its own proof.Item Automated strategic visualisations and user confidence(Heriot-Watt University, 2019-03) Le Bras, Pierre; Padilla, Doctor Stefano; Chantler, Professor MikeData visualisations aim at providing accessible and interpretable information for people. At a strategic level, such representations can be used to stimulate decision making. We have found that users are however hesitant to exploit unfamiliar visualisations, and require more material to be confident about their description of an unbiased representation of data. In this thesis we aim at exploring which characteristics affect users’ confidence in their ability to interpret and explain Topic Maps. These visualisations display the multi-dimensional thematic abstraction of large document collections, and as such require an automated generation process. In three qualitative studies, we challenge participants’ confidence with stimuli and scenarios, and analyse their responses. The studies focus on: Explanation Systems, Topic Map layouts, and mapping processes. In our studies, we demonstrate that the use of data-driven and interactive Explanation Systems gives users a sense of control, allowing for an enhanced interpretability and confidence. We then found that structure and narrative are both equally important characteristics of layouts for a confident presentation of Topic Maps. We finally explore mapping processes in detail, and establish that constructive mapping methods are more fit to improve user confidence than reductive ones. This thesis, in summary, defines a comprehensive understanding of user confidence in automatically generated visualisations.Item Route discovery schemes in Mobile Ad hoc Networks with variable-range transmission power(Heriot-Watt University, 2019-02) Alghamdi, Atif A.; King, Doctor Peter; Taylor, Doctor Hamish; Abdelshafy, Doctor MohamedBroadcasting in MANETs is important for route discovery but consumes significant amounts of power that is difficult to renew for devices that rely heavily on batteries. Most existing routing protocols make use of a broadcast scheme known as simple flooding. In such an on-demand routing protocol (e.g. AODV) the source node originates a Route Request (RREQ) packet that is blindly rebroadcast via neighbouring nodes to all nodes in the network. Simple flooding leads to serious redundancy, together with contention, and collisions, which is often called the broadcast storm problem. This thesis proposes two improvement strategies: topology control (adjusting transmission power) and reduced retransmissions (reducing redundant rebroadcasts) to reduce energy consumption. For energy efficient route discovery the main idea is to reduce the energy consumed per broadcast during route discovery. An Energy Efficient Adaptive Forwarding Algorithm (called EEAFA) is proposed to reduce the impact of RREQ packet flooding in on-demand routing protocols. The algorithm operates in two phases: 1) Topology construction phase, which establishes a more scalable and energy efficient network structure where nodes can adjust their transmission power range dynamically, based on their local density. 2) A Forwarding Node Determination phase, that utilises network information provided by the constructed topology, where nodes independently decide to forward a RREQ packet or not without relying on GPS or any distance calculations. A further Enhanced EEAFA (called E-EEAFA) algorithm is also proposed, which combines two techniques: graph colouring and sectoring techniques. Graph colouring increases awareness at network nodes to improve the determination of a forwarding node, while the sectoring technique divides neighbours into different forwarding sectors. This helps to reduce overlap between forwarding nodes and select suitable nodes in each sector to forward RREQ packets. These techniques are employed in a distributed manner and collaborate to reduce the number of forwarding nodes, which thus reduces the volume of RREQ packets populating the network. These algorithms have been validated as effective by NS2 simulation studies that are detailed in the thesis.Item The geometry of integrable vortices(Heriot-Watt University, 2019-05) Ross, Calum Duncan Hugh; Schroers, Professor BerndThis thesis is concerned with geometric interpretations of vortices. We demonstrate that all five of the integrable Abelian vortex equations can be encoded in terms of the flatness and holomorphic trivialisation of a non-Abelian connection. There is a natural lift of this story to three dimensional group manifolds where the flat connection is related to the Maurer-Cartan one-form. In particular we present a detailed study of vortices on the two-sphere and on two dimensional hyperbolic space. For these cases the lifted vortices give rise to solutions of coupled equations including a massless gauged Dirac equation on flat three dimensional space. Squaring the Dirac equation arising from vortices on the 2-sphere gives rise to a Schrodinger like equation for a spinor wave function in the background of a magnetic field with non-trivial linking. Both the wave function and the magnetic field pick up non-trivial topology related to the vortex. In the final Section a potential realisation scheme for magnetic fields with non-trivial linking as synthetic fields in a Bose-Einstein condensate is discussed.Item Stochastic differential equations with multiple invariant measures and related problems(Heriot-Watt University, 2019-08) Dobson, Paul; Ottobre, Associate Professor MichelaWe study problems related to SDEs which admit multiple invariant measures. The main problem we address is determining the long time behaviour of a large class of diffusion processes on R N , generated by second order differential operators of (possibly) degenerate type. The operators that we consider need not satisfy the Hormander condition and need not admit a unique invariant measure. Instead, we consider the so-called UFG condition, introduced by Hermann, Lobry and Sussmann in the context of geometric control theory and later by Kusuoka and Stroock, this time with probabilistic motivations. In this thesis we will demonstrate the importance of UFG diffusions in several respects: We show that UFG processes constitute a family of SDEs which exhibit multiple invariant measures and for which one is able to describe a systematic procedure to determine the basin of attraction of each invariant measure (equilibrium state). We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs. We prove that there exists a change of coordinates such that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with ODE, where the ODE evolves independently of the SDE part of the dynamics. As a result, UFG diffusions are inherently “less smooth” than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. We introduce a novel pathwise approach to obtain (long-time) derivative estimates for Markov semigroups. The content of this thesis has resulted in two long papers [1] and [2], both submitted for publication.Item Multiscale modelling analysis and computations of complex heterogeneous multiphase systems(Heriot-Watt University, 2019-05) Ververis, Antonios; Goddard, Doctor Ben; Duncan, Professor DugaldIn this thesis, we analytically and computationally investigate various aspects related to the multiphase-multicomponent interfacial processes and reactive transport in homogeneous domains and heterogeneous periodic perforated media. More precisely, we perform formal homogenization arguments to the microscopic Cahn-Hilliard type equations governed the dynamics in binary and ternary mixtures, in the presence of two or more phases. We additionally consider the coupling of the Cahn-Hilliard type species diffusion to fluid flow, a coupling which gives rise to more complex systems since a Navier-Stokes momentum balance is involved. Each particular model can be formally derived by an Energetic Variational Approach, that combines the classical idea of gradient flows for free energy minimization as a direct consequence of the second law of thermodynamics, together with the Least Action and Maximum Dissipation Principles. Moreover, as an extension of the already established two-scale convergence approach, we investigate further a reiterated homogenization procedure over three separated scales of periodic oscillations. Finally, we examine the General Equations for Non-Equilibrium Reversible-Irreversible Coupling commonly known by the abbreviation GENERIC, an extended two-generator variational framework, which was initially developed in order to model the rheological properties of complex fluids, far from thermodynamic equilibrium.Item Aspects of mathematical biology : from self-organisation of the cytoskeleton to transport of migratory species(Heriot-Watt University, 2019-09) Płochocka, Aleksandra Zofia; Chumakova, Doctor Lyuba; Painter, Professor KevinThis thesis spans scales of mathematical biology, from single molecules to groups of organisms. We explore questions regarding the self-organisation of the cytoskeleton and the long distance migration of animals. Though disparate at first glance, both topics revolve around transport and self-organisation of biological particles. We first model the microtubule cytoskeleton: a self-organising dynamic scaffolding along which cellular components, e.g. proteins, are transported. Its organisation is crucial for correct cellular functions; for example, maintaining the correct distribution of E-cadherin (the epithelial cell adhesion protein) along the cell boundary to ensure tissue integrity. Using stochastic simulations, genetic manipulations of the Drosophila epithelial cells and a probabilistic model we show that microtubule cytoskeleton selforganisation principally depends on cell geometry and microtubule seed density and is robust at the tissue scale. We then extend this work. Specifically, we build and explore an analytical model and perform stochastic simulations to explain microtubule self-organisation in crowded cytoplasm, i.e. containing various highly anisotropic barriers. We consider Drosophila follicular epithelium cells, which contain actin cables throughout. We find that anisotropy in the cell interior leads to a significant increase in the number of microtubules pointing in the direction of the anisotropy. This allows us to deduce the type of interaction between microtubules and actin cables. We introduce a new measure of self-organisation of microtubules, the bundling factor, and use it to explore the persistent direction of transport created by microtubule bundles. A second research topic is subsequently discussed. Many animals navigate long distances for purposes including foraging or nesting. While often mysterious, various lines of research support the idea that navigation is aided by a combination of cues whose magnitudes change with distance from the target. Motivated by agent-based simulations from a study of green sea turtle migration, we construct an abstract model for taxis-based animal navigation. We investigate the key properties of various navigating cues and their impact on animal migration, and discuss how the starting location can affect the mean first passage time of a migratory journey.