When studying the behaviour of dynamical systems, one particular goal is to find and isolate the periodic solutions and the equilibria. They are a subset of the chain-recurrent set of the dynamical system. In recent work, many improvements have been achieved in computing an approximation of a complete Lyapunov function of a given dynamical system and thus to identify the chain-recurrent set. A weak point in this approach, however, has been an over-estimation of the chain-recurrent set. In this work, we introduce a heuristic algorithm that reduces the overestimation in a simple and efficient way. Furthermore, a new and improved grid to evaluate the complete Lyapunov function is introduced to avoid unevaluated regions in the domain of the function.

Many phenomena in disciplines such as engineering, physics and biology can be represented as dynamical systems given by ordinary differential equations (ODEs). For their analysis as well as for modelling purposes it is desirable to obtain a complete description of a dynamical system. Complete Lyapunov functions, or quasi-potentials, describe the dynamical behaviour without solving the ODE for many initial conditions. In this paper, we use mesh-free numerical approximation to compute a complete Lyapunov function and to determine the chain-recurrent set, containing the attractors and repellers of the system. We use a homogeneous evaluation grid for the iterative construction, and thus improve a previous method. Finally, we apply our methodology to several examples, including one to compute an epigenetic landscape, modelling a bistable network of two genes. This illustrates the capability of our method to solve interdisciplinary problems.

In this thesis, I provide a statistical analysis of high-resolution contact pattern data within primary and secondary schools as collected by the SocioPatterns collaboration. Students are graphically represented as nodes in a temporally evolving network, in which links represent proximity or interaction between students. I focus on link- and node-level statistics, such as the on- and off-durations of links as well as the activity potential of nodes and links. Parametric models are fitted to the onand off-durations of links, interevent times and node activity potentials and, based on these, I propose a number of theoretical models that are able to reproduce the collected data within varying levels of accuracy. By doing so, I aim to identify the minimal network-level properties that are needed to closely match the real-world data, with the aim of combining this contact pattern model with epidemic models in future work.
I also provide Bayesian methods for parameter estimation using exact Bayesian and Markov Chain Monte Carlo methods, applying these in the case of Mittag-Leffler distributed data to artificially generated data and real-world examples. Additionally, I present probabilistic methods for model selection - namely the Akaike and Bayesian Information Criteria and apply them to the data and examples in the previous section.

Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a ‘kinetic under–cooling’ regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse–interface approximation of the model. Finite–element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions.

The blast fungus Magnaporthe oryzae gains entry to its host plant by means of a specialized pressure-generating infection cell called an appressorium, which physically ruptures the leaf cuticle. Turgor is applied as an enormous invasive force by septin-mediated reorganization of the cytoskeleton and actin-dependent protrusion of a rigid penetration hypha. However, the molecular mechanisms that regulate the generation of turgor pressure during appressorium-mediated infection of plants remain poorly understood. Here we show that a turgor-sensing histidine–aspartate kinase, Sln1, enables the appressorium to sense when a critical turgor threshold has been reached and thereby facilitates host penetration. We found that the Sln1 sensor localizes to the appressorium pore in a pressure-dependent manner, which is consistent with the predictions of a mathematical model for plant infection. A Δsln1 mutant generates excess intracellular appressorium turgor, produces hyper-melanized non-functional appressoria and does not organize the septins and polarity determinants that are required for leaf infection. Sln1 acts in parallel with the protein kinase C cell-integrity pathway as a regulator of cAMP-dependent signalling by protein kinase A. Pkc1 phosphorylates the NADPH oxidase regulator NoxR and, collectively, these signalling pathways modulate appressorium turgor and trigger the generation of invasive force to cause blast disease.

The chosen title for my PhD thesis is "Numerical and optimal control methods for partial differential equations arising in computational finance". The body of my research is divided into two parts. The first part is devoted to the application of an alternating direction implicit numerical method for solving stochastic volatility option pricing models. The second part focuses on a partial-integro differential equation constrained optimal control approach to parameter estimation for the forward jump-diffusion option pricing model. The body of the thesis is preceded by an extensive introduction, which seeks to contextualize my work with respect to the field of computational finance, this is followed by a brief conclusion. Finally, the thesis is completed by a list of refer ences. The first project proposes a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. The approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence. The second project proposes to solve a parameter calibration problem for the forward jump-diffusion option pricing model proposed by Andersen and Andreasen. A distributed optimal control approach is employed, with a partial-integro differential equation as our state equation. By approaching the problem from a functional analysis perspective, I investigate the necessary regularity conditions for our parameters of interest. Following this, the existence of optimal solutions is proven under certain analytical conditions. Furthermore, the first-order necessary conditions for optimality are also established. Finally, a projected-gradient optimization method is applied numerically to empirical market data and results are given.

The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from 123/122 ([9]) to 117/116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems.

Motivated by the formation of fingerprint patterns we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. For a given spatially homogeneous tensor field, we show that there exists a preferred direction of straight lines, i.e. straight vertical lines can be stable for sufficiently many particles, while many other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles and relate these results to the Kuecken-Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.

To respond rapidly and accurately to network and service outages, network operators must deal with a large number of events resulting from the interaction of various services operating on complex, heterogeneous and evolving networks. In this paper, we introduce the concept of functional connectivity as an alternative approach to monitoring those events. Commonly used in the study of brain dynamics, functional connectivity is defined in terms of the presence of statistical dependencies between nodes. Although a number of techniques exist to infer functional connectivity in brain networks, their straightforward application to commercial network deployments is severely challenged by: (a) non-stationarity of the functional connectivity, (b) sparsity of the time-series of events, and (c) absence of an explicit model describing how events propagate through the network or indeed whether they propagate. Thus, in this paper, we present a novel inference approach whereby two nodes are defined as forming a functional edge if they emit substantially more coincident or short-lagged events than would be expected if they were statistically independent. The output of the method is an undirected weighted graph, where the weight of an edge between two nodes denotes the strength of the statistical dependence between them. We develop a model of time-varying functional connectivity whose parameters are determined by maximising the model's predictive power from one time window to the next. We assess the accuracy, efficiency and scalability of our method on two real datasets of network events spanning multiple months and on synthetic data for which ground truth is available. We compare our method against both a general-purpose time-varying network inference method and network management specific causal inference technique and discuss its merits in terms of sensitivity, accuracy and, importantly, scalability.

For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks.

My PhD thesis contains a couple of results I obtained under the supervision of my advisor Filippo Cagnetti, during the past three years of my studies. In particular, I present two results about rigidity of perimeter inequality under symmetrization techniques. The first result, presented in Chapter 3, provides the characterization of rigidity for equality cases for the perimeter inequality under spherical symmetrization; whereas in Chapter 4 I will study the rigidity of equality cases for the Steiner's inequality for the anisotropic perimeter.

This thesis presents a mathematical model for cell migration that couples a system of reaction-advection-diffusion equations describing the interations between F-actin and myosin II toa force balance equation describing the velocity vector of the actin-myosin network.

Cell migration plays a crucial role in many biological processes. In eukaryotic cells, this migration is largely powered by a system of actin and myosin II. At the leading edge of the cell, cross-inked actin filaments polymerise by adding actin monomers to their ends while at the back of the cell, myosin II binds to a bundle of actin filaments. These processes create protrusive and contractile forces generated by the action of actin polymerisation and myosin II contraction.

Based on the idea that cell migration is powered by the actin-myosin network of the cell, we formulate model equations for migrating cell which comprises reaction-advection-diffusion equations that are coupled to a force balance describing the velocity vector of the network. This is a viscous model with active stresses coming from the actin-myosin system. These equations describing the migrating cells are highly nonlinear partial differential equations with no closed form solutions and we therefore result to numerical methods in order to compute the approximate solution.

F-actin and myosin II solution are the solution for the reaction-advection-diffusion equations while the speeds of the cell come from the solution of the force balance equation. We begin simulations on a unit disk at zero initial velocity with different data for the initial conditions of F-actin and myosin II concentrations. We also vary some parameters at a time while keeping all the other parameters constant: for example (i) total amount of actin ρtotα and (ii) contraction coefficient for myosin II η0m.

Actin polymerisation causes protrusive stress at the cell periphery which results in expansion of the cell. We observe that the initial conditions play an important role in the spatiotemporal dynamics of F-actin as well as the evolution of the cell shape. Actin changes from the active state (F-actin) to inactive state (G-actin) and vice-versa through polymerisation and depolymerisation processes and hence the total amount of actin is conserved at any time. We note that in our model, myosin II only diffuses inside the cell and exerts contractile stress in the cell. Its total concentration in the entire cell is conserved.

The aim of this thesis focuses on addressing several open questions in cell biology by using different mathematical approaches and numerical analysis methods to study the evolution of distinct protein families in various cellular phenomena, such as cell polarisation and cytoskeleton remodelling. Our approaches are based on conservation laws and compartmentalisation of proteins within appropriate geometrical subdomains representing different cellular structures, such as the cell membrane and cytosol.

The Rho GTPase are proteins responsible of coordinating the cell polarisation response, which is a biological process involving a huge number of different proteins and intricate networks of biochemical reactions. Rho GTPases localise their activity in specific cell regions where they interact with the cell cytoskeleton. Reducing the biological assumptions to a minimal level of complexity, we will present a simple qualitative model for cell polarisation in which proteins cycle between cell membrane and cytosol in an active and inactive form. This is described through a bulk-surface system of two reaction-diffusion equations coupled by the boundary condition. The model supports pattern formation and we will confirm this claim by using both mathematical analysis and simulations. The bulk-surface finite element method is presented and used to solve the model on different geometries.

Secondly, we will present a mathematical model for keratin intermediate filament dynamics in resting cells. This model, characterised by a quantitative approach, is a datadriven extension of a pre-existing model, initially introduced by Portet et al. (PlosONE, 2015). We will discuss the new assumptions and modelling ideas, and compare the solution of our model to the experimental data. Part of the biological impact of our model relies in its ability to estimate the amount of assembled and disassembled keratin material as a function of space and time, consistent with the biological model proposed by Windoffer et al. (Journal of Cell Biology, 2011).

In the last part we will introduce a second mathematical model for keratin spatiotemporal dynamics in non-resting cells. In this case, the model is derived on two- and three-dimensional geometries and accounts for a more detailed description of the processes involved in the keratin cytoskeleton remodelling process. The evolution of three different forms of keratin is modelled by a system composed of one reaction-diffusion equation and two reaction-advection-diffusion equations. Keratin kinetics are also described by the boundary conditions, which are posed both at the cell membrane and at the nuclear envelope. In solving the model, we will use the Streamline Upwind Petrov Galerkin method, as described in the text. In conclusion, in view of a future estimation of biologically relevant parameters, a simulation is presented, showing consistency of our mathematical model with the biological model proposed by Windoffer et al. (Journal of Cell Biology, 2011).

In summary, this thesis presents methods and techniques for data-driven modelling supported by rigorous mathematical analysis and novel numerical methods and simulations. Our approach involving the use of quantitative methods serves as a blue-print for how to study the synergy interplay between mathematics and its applications to experimental sciences.

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. We consider a contraction metric, i.e. a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable. In this paper we propose a construction method using meshfree collocation to approximately solve a matrix-valued PDE problem. We derive error estimates and show that the approximation is itself a matrix-valued PDE problem. We derive error estimates and show that the approximation is itself a contraction metric if the collocation points are sufficiently dense. We apply the method to several examples.

This thesis begins with the description of a tractable model for tumour growth. The unique feature of this model is that we pass through the thin rim limit. We derive the sharp interface weak form and finite element scheme. We discuss the mesh smoothing techniques used in the implementation of the sharp interface finite element scheme. We then introduce an unfitted finite element scheme, and a sharp interface finite element scheme in R3. We also write the model in the diffuse interface paradigm, along with the associated weak form. We prove the existence and uniqueness of the solution to the di_use interface version of the model, and prove convergence of the diffuse interface finite element method. We conclude this thesis with a number of simulations in R2 and R3. Here, we present rates of convergence, and also investigate the effect of parameter spaces on the morphology of the tumour. A biologically motivated investigation is made, and a brief comparison with in vivo tumours is presented.

We derive analytic bounds on the shape of the primordial power spectrum in the context of single-field inflation. In particular, the steepest possible growth has a spectral index of ns − 1 = 4 once transients have died down. Its primary implication is that any constraint on the power spectrum at a particular scale can be extrapolated to an upper bound over an extended range of scales. This is important for models which generate relics due to an enhanced amplitude of the primordial scalar perturbations, such as primordial black holes. In order to generate them, the power spectrum needs to grow many orders of magnitude larger than its observed value on CMB scales—typically achieved through a phase of ultra slow-roll inflation—and is thus subject to additional constraints at small scales. We plot all relevant constraints including CMB spectral distortions and gravitational waves sourced by scalar perturbations at second order. We show how this limits the allowed mass of PBHs, especially for the large masses of interest following recent detections by LIGO and prospects for constraining them further with future observations. We show that any transition from approximately constant epsilon slow-roll inflation to a phase where the power spectrum rapidly rises necessarily implies an intervening dip in power. We also show how to reconstruct a potential that can reproduce an arbitrary time-varying epsilon, offering a complementary perspective on how ultra slow-roll can be achieved.

We present a space- and phenotype-structured model of selection dynamics between cancer cells within a solid tumour. In the framework of this model, we combine formal analyses with numerical simulations to investigate in silico the role played by the spatial distribution of oxygen and therapeutic agents in mediating phenotypic selection of cancer cells. Numerical simulations are performed on the 3D geometry of an in vivo human hepatic tumour, which was imaged using computerised tomography. Our modelling extends our previous work in the area through the inclusion of multiple therapeutic agents, one that is cytostatic, whilst the other is cytotoxic. In agreement with our previous work, the results show that spatial inhomogeneities in oxygen and therapeutic agent concentrations, which emerge spontaneously in solid tumours, can promote the creation of distinct local niches and lead to the selection of different phenotypic variants within the same tumour. A novel conclusion we infer from the simulations and analysis is that, for the same total dose, therapeutic protocols based on a combination of cytotoxic and cytostatic agents can be more effective than therapeutic protocols relying solely on cytotoxic agents in reducing the number of viable cancer cells.

In the field of multiscale modelling of materials, a class of significant problems involves the atomistic-to-continuum coupling in crystals. Continuum models frequently fail to produce accurate predictions near singularities and defects and hence coupled atomistic/continuum methods have become popular. The ad-hoc coupling of atomistic and continuum energies results in numerical artifacts on the interface between the continuum and atomistic regions, known as ghost forces. The design and analysis of atomistic/continuum coupling methods that are ghost-force free is important in computational and mathematical modelling of materials and one of the very few well defined problems in multi-scale algorithm design for nonlinear phenomena.

In this thesis we developed a discontinuous ghost-force free bond volume based method in one dimensional and two dimensional crystal lattices. The design of the method was motivated by appropriately analysing the error both at the atomistic and the continuum region. Its design is consistent and transferable. Next, we were concerned about the energy consistency and the variational consistency of the coupled methods. Consistency is a quantity that measures the extent to which an exact smooth solution does satisfy the numerical scheme. We proved that in one dimension the local contributions of the energy were of second order in the lattice spacing ε, O(ε²). The total energy error in one and two dimensions was second order. We analysed the error for rst variations both in one and two dimensions. Our analysis confirmed that the proposed methods were indeed ghost-force free and their variational consistency error was bounded by (ε² + ε²-1/p ) in the discrete W-1,p norm. We implemented the static atomistic problem and compared it to the static coupled method in one dimension. We considered energies from multi-body potentials. By using the symmetry properties of the potentials we derived energy consistency error bounds of order O(ε²).

In this thesis, we branch into both harmonic and geometric analysis. Within harmonic analysis, we look at differential actions Lp on hypergeometric series and Jacobi polynomials, where the latter are known to represent the zonal spherical functions on compact rank-one symmetric spaces. Within geometric analysis, we examine spherical twists as solutions to the Euler-Lagrange system associated with the Dirichlet energy and certain of its nonlinear extensions for sphere-valued mappings in suitable Sobolev spaces.

Among various environmental factors associated with triggering or exacerbating autoimmune response, an important role is played by infections. A breakdown of immune tolerance as a byproduct of immune response against these infections is one of the major causes of autoimmune disease. In this paper we analyse the dynamics of immune response with particular emphasis on the role of time delays characterising the infection and the immune response, as well as on interactions between different types of T cells and cytokines that mediate their behaviour. Stability analysis of the model provides insights into how different model parameters affect the dynamics. Numerical stability analysis and simulations are performed to identify basins of attraction of different dynamical states, and to illustrate the behaviour of the model in different regimes

A fractional magnetic Sobolev inequality with two variables and critical exponents is considered in this paper, and the best constant in the inequality is determined. As an application of the inequality, we establish an existence result for the ground state solutions to a fractional magnetic critical system.

In this thesis, we present a Bayesian framework to solve inverse problems in the context of quantitative biology. We present a novel combination of the Bayesian approach to inverse problems, suitable for infinite-dimensional problems, with a parallel, scalable Markov Chain Monte Carlo algorithm to approximate the posterior distribution. Both the Bayesian framework and the parallelised MCMC were already known but they were not used in this context in the past. Our approach puts together existing results in order to provide a tool to easily solve inverse problems. We focus on models given by partial differential equations. Our methodology differs from previous results in its approach: it aims to be as transparent and independent of the model as possible, in order to make it flexible and applicable to a wide range of problems emerging from experimental and physical sciences. We illustrate our methodology with three of such applications in the areas of theoretical biology and cell biology.

The first application deals with parameter and function identification within a Turing pattern formation model. To the best of our knowledge, our results are the first attempt to use Bayesian techniques to study the inverse problem for Turing patterns. In this example, we show how our implementation can deal with both finite- and infinite-dimensional parameters in the context of inverse problems for partial differential equations.

The second example studies the spatio-temporal dynamics in cell biology. The study provides an example that seeks to best-fit a mathematical model to experimental data finding in the process optimal parameters and credible regimes and regions. We present a new derivation of the model, that corrects the short-comings of previous approaches. We provide all the details from techniques for data acquisition to the parameter identification, and we show in particular how the mathematical model can be used as a proxy to estimate parameters that are difficult to measure in the experiments, providing an novel alternative to more indirect estimates that also require more complex experiments.

Finally, our third example illustrates the flexibility of our implementation of the methodology by using it to study traction force microscopy (TFM) data with a solver implemented independent of the Bayesian approach for parameter identification. We limit ourselves to the classical TFM setting, that we model as a two-dimensional linear elasticity problem. The results and methods generalise to more complex settings where quantitative modelling driven by biological observations is a requirement.

We study the transformation of the vertices of a certain simple simplicial complex in n-dimensional Euclidian space and prove that the resulting set of simplices is a simplicial complex with an approximate rotational symmetry. Such simplicial complexes have applications in computing Lyapunov function for nonlinear dynamical systems using linear optimization and are also of interest for other applications.

Let PG(2; q) be the projective plane over the field Fq. Singer [19] notes that PG(2; q) has a cyclic group of order q2 + q + 1 that permutes the points of PG(2; q) in a single cycle. A k-arc set of k points no three of which are collinear. A k-arc is called complete if it is not contained in a (k + 1)-arc of PG(2; q).

By taking the orbits of points under a proper subgroup of a single cycle, one can decompose the projective plane PG(2; qk) into disjoint copies of subplanes isomorphic to PG(2; q) if and only if k is not divisible by three. Moreover, by taking the orbits of points under a proper subgroup, one can decompose the projective plane PG(2; q2) into disjoint copies of complete (q2 - q + 1)-arcs. In this thesis, our main problem is to classify (up to isomorphism) the different types of decompositions of PG(2 ;q2) for q = 3; 4; 5; 7, namely subplanes and arcs. We further illustrate some of the connections between these subgeometry decompositions and other areas of combinatorial interest; in particular, we explain the relationship between coding theory and projective spaces and describe the links with Hermitian unital. Furthermore, projective codes are obtained by taking the disjoint union of such subgeometries.

With the growing global demands on sustainable food production, one of the biggest challenges to agriculture is associated with crop losses due to parasitic nematodes. While chemical pesticides have been quite successful in crop protection and mitigation of damage from parasites, their potential harm to humans and environment, as well as the emergence of nematode resistance, have necessitated the development of viable alternatives to chemical pesticides. One of the most promising and targeted approaches to biocontrol of parasitic nematodes in crops is that of RNA interference (RNAi). In this study we explore the possibility of using biostimulants obtained from metabolites of soil streptomycetes to protect wheat (Triticum aestivum L.) against the cereal cyst nematode Heterodera avenae by means of inducing RNAi in wheat plants. Theoretical models of uptake of organic compounds by plants, and within-plant RNAi dynamics, have provided us with useful insights regarding the choice of routes for delivery of RNAi-inducing biostimulants into plants. We then conducted in planta experiments with several streptomycete-derived biostimulants, which have demonstrated the efficiency
of these biostimulants at improving plant growth and development, as well as in providing resistance against the cereal cyst nematode. Using dot blot hybridization we demonstrate that biostimulants trigger a significant increase of the production in plant cells of si/miRNA complementary with plant and nematode mRNA. Wheat germ cell-free experiments show that these si/miRNAs are indeed very effective at silencing the translation of nematode mRNA having complementary sequences, thus reducing the level of nematode infestation and improving plant resistance to nematodes. Thus, we conclude that natural biostimulants produced frommetabolites of soil streptomycetes provide an effective tool for biocontrol of wheat nematode.

In this thesis, a full repertoire of model formulation, model analysis, numerical analysis, sensitivity analysis and Bayesian method for parameter identification are presented, that seek to describe faithfully the temporal dynamics of GEF–Rho– Myosin signalling pathway as observed experimentally. The thesis is based on rigorous mathematical and numerical analysis to provide robust models and numerical results that exhibit the temporal dynamics as observed in experiments. We also explore the effect of spatial inhomogeneity on two of the models formulated. The modelling is based on experimental observations, and therefore three different mathematical models are formulated from first principles depending on the constitutive laws for the interaction between chemical species, entailing that new mathematical models are obtained. Detailed mathematical analysis of the stability of uniform steady states using nullcline theory, linear stability theory and sign pattern analysis is carried out, to characterise mathematically the key temporal dynamics of stability, oscillations, excitability and bistability as observed in experiments. Numerical bifurcation analysis using Matcont and numerical simulations carried using MATLAB illustrate theoretical analytical results through parameter variations for the key temporal dynamics. Rigorous sensitivity analysis provides a powerful tool for investigating the effects of parameter variations through local and global sensitivity. In particular, we use local sensitivity theory to characterise the limit cycle behaviour of an oscillatory dynamical system in terms of parameter variations and therefore, the thesis provides premises to characterise or study amplitude and period sensitivity to parameter variations. A full Bayesian approach is applied to the model for the identification of parameters that best-fits the model to experimental results. Therefore, the thesis provides a new framework for incorporating prior knowledge about parameters, which results in obtaining full probability distribution for parameters. Finally, the thesis explores and studies the spatially extended version on the ODE models. We analyse the existence of Turing instability for some parameter values. This proof-of-concept set premises to extend the temporal models to include spatial variations in the form of coupled bulk-surface reaction-diffusion systems through compartmentalisation of the spatial domain.

One of the major outstanding challenges in immunology is the development of a comprehensive, quantitative and accurate approach to understanding the causes and dynamics of immune responses. The immune system normally protects the body against infections, but at the same time it is possible that it can fail to distinguish the host’s own cells from the cells affected by the infection, which can lead to autoimmune disease. The question of what releases the auto-pathogenic potential of T lymphocytes is at the heart of understanding autoimmune disease. Among various possible causes of autoimmune disease, an important role is played by infections that can result in a breakdown of immune tolerance, primarily through the mechanism of molecular mimicry, where the introduction of pathogenic peptides that structurally resemble self-peptides, derived from infection, may induce T lymphocytes to proliferate and leave them with the ability to respond to self, as well as foreign antigens. Deterministic and stochastic models have been extensively used in the past to study the dynamics of immune responses and analyse a possible onset of autoimmunity. The main focus of this thesis is the development and analysis of mathematical models of immune response to infection, as well as the onset and progress of autoimmunity. Particular emphasis is made on developing new mathematical approaches for elucidating the roles played by various cytokines in the immune dynamics.
In the first part of the thesis I develop a mathematical model for dynamics of immune response to hepatitis B. This model explicitly includes contributions from innate and adaptive immune responses, as well as from cytokines. Analysis of the model identifies parameter regimes where the model exhibits clearance of infection, maintenance of a chronic infection, or periodic oscillations. Effects of nucleoside analogues and interferon treatments are analysed, and the critical drug efficiency is determined.
The second part of the thesis investigates the dynamics of immune response to a general viral infection and a possible onset of autoimmunity, which account for regulatory T cells, T cells with different activation thresholds, and cytokines. Feasibility and stability analyses of different steady states yield boundaries of stability and bifurcations in terms of system parameters. This model exhibits bi-stability and shows different regimes of normal clearance of viral infection, chronic infection, or autoimmune behaviour. Therefore, it can provide significant new insights into autoimmune dynamics.
To investigate the role of stochasticity in immune dynamics, I developed a stochastic version of the model, and the major result is that adding stochasticity can lead to the emergence of sustained oscillations around deterministically stable steady states, thus providing a possible explanation for experimentally observed variations in the progression of autoimmune disease. I also have investigated stochastic dynamics in the regime of bi-stability and computed the magnitude of these fluctuations.
I have also analysed the effects of different time delays, as well as the inhibiting effect of regulatory T cells on secretion of interleukin-2 on autoimmune dynamics. To this end, I have performed a systematic analysis of stability of all steady states of the corresponding model both analytically, and numerically. The identification of basins of attraction of different steady states and periodic solutions indicates that time delays can change the shape of these basins of attraction, and the new results show better qualitative agreement with the experimental observations.
My thesis culminates with the last part, where I explore stochastic effects in a time-delayed model for autoimmunity. The major achievement in this part of the thesis is the development of a new methodology for deriving an Itô stochastic delay differential equation (SDDE) from delay discrete stochastic models, as well as showing the equivalency of previously proposed methods. Using this equivalence, I derived a simpler SDDE model to perform numerical simulations. I have used a linear noise approximation (LNA) to determine the magnitude of stochastic fluctuations around deterministic steady states, and to obtain insights into how the coherence of stochastic oscillations around deterministically stable steady states depends on system parameters.

Directional cell migration in dense three-dimensional (3D) environments critically depends upon shape adaptation and is impeded depending on the size and rigidity of the nucleus. Accordingly, the nucleus is primarily understood as a physical obstacle, however, its pro-migratory functions by step-wise deformation and reshaping remain unclear. Using atomic force spectroscopy, timelapse fluorescence microscopy and shape change analysis tools, we determined nuclear size, deformability, morphology and shape change of HT1080 fibrosarcoma cells expressing the Fucci cell cycle indicator or being pre-treated with chromatin-decondensating agent TSA. We show oscillating peak accelerations during migration through 3D collagen matrices and microdevices that occur during shape reversion of deformed nuclei (recoil), and increase with confinement. During G1 cell cycle phase, nucleus stiffness was increased and yielded further increased speed fluctuations together with sustained cell migration rates in confinement as compared to interphase populations, or to periods of intrinsic nuclear softening in the S/G2 cell cycle phase. Likewise, nuclear softening by pharmacological chromatin decondensation or after lamin A/C depletion reduced peak oscillations in confinement. In conclusion, deformation and recoil of the stiff nucleus contributes to saltatory locomotion in dense tissues.

We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope. We show that along characteristic directions the order of the variance of the last passage time is of order N^{2/3} in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N) in the boundary model and to O(1) for the non-boundary model. This behaviour is the result of the two flat edges of the shape function.

We derive a mechanical model of human motion where an elderly person decides to step over an obstacle rather than avoiding it. Such a decision may be deliberate or
forced due to a sudden appearing obstacle in his/her way. The model is represented by a nonautonomous system of ordinary differential equations with discontinuous right hand side. We provide a notion of lateral stability. It is shown that increasing the angle between legs increases stability linearly. This implies that an individual reduces the risk of falling due to stepping over an obstacle by increasing the angle between legs.

There is a large literature of numerical methods for phase field models from materials science. The prototype models are the AllenCahn and Cahn-Hilliard equations. We present four benchmark problems for these equations, with numerical results validated using several computational methods with different spatial and temporal discretizations. Our goal is to provide the scientific community with a reliable reference point for assessing the accuracy and reliability of future software for this important class of problem.

This work presents the development, analysis and numerical simulations of a model for cell deformation and movement, which couples biochemical reactions and biomechanical forces. The way that cells move is key to the creation and development of most organisms on earth. Consequently a deeper understanding of cell motility is likely to have significant applications to medicine. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled with a force balancing equation for the displacement. The pressure and contractile forces are influenced by actin and myosin and we model these with a system of reaction-diffusion equations. The model consists of highly non-linear partial differential equations.
To analyse the model, we carry out linear stability analysis to determine key bifurcation parameters and find analytical solutions close to bifurcation points. We then approximate the equations and produce numerical solutions in multi-dimensions, using an evolving finite element method. The solutions predicted from linear stability theory are replicated in the early stages of cell movement. Subsequently, both simple and complex deformations, such as expansions, protrusions, contractions and translations of the cell are observed.
This theoretical and computational framework allows the study of more complex and experimentally driven reaction kinetics involving, actin, myosin and other molecular species that play an important role in cell movement and deformation.

The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.

In this paper we address questions on the existence and multiplicity of a class of geometrically motivated mappings with certain symmetries that serve as solutions to the nonlinear system in variation:
[equation not shown]
Here Ω ⊂ R n is a bounded domain, F = F(r, s, ξ) is a sufficiently smooth Lagrangian, Fs = Fs(|x|, |u| 2 , |∇u| 2 ) and Fξ = Fξ(|x|, |u| 2 , |∇u| 2 ) with Fs and Fξ denoting the derivatives of F with respect to the second and third variables respectively while P is an a priori unknown hydrostatic pressure resulting from the incompressibility constraint det ∇u = 1. Among other things, by considering twist mappings u with an SO(n)-valued twist path, we prove the existence of multiple and topologically distinct solutions to ELS for n ≥ 2 even versus the only (non) twisting solution u ≡ x for n ≥ 3 odd. An extremality analysis for twist paths and those of Lie exponential types and a suitable formulation of a differential operator action on twists relating to ELS are the key ingredients in the proof.

Many of the technological, social and biological systems we observe and partake in in our everyday lives can be described as networks of interacting elements. Network- based research can then be performed to improve our understanding about the structural features of such complex networks, the behaviour of processes occurring within such complex networks, and the interaction between the two. During my PhD I have considered neuronal and epidemiological dynamics occurring on complex networks, with the main aim of improving model realism by incorporating spatial or local structure whilst maintaining model tractability. In total I have considered three network-based research projects which are included in this thesis in chronological order.
This thesis begins with an introduction to the study of complex networks and processes occurring on complex networks. Comparisons are drawn between the approaches of neuroscience and epidemiology-based network studies, including consideration of the difficulties regarding modelling local spatial structure. Chapter 2 considers an existing model describing the activity-dependent growth and development of a network of excitatory and inhibitory neurons embedded in space. A systematic investigation of the effects of various spatial arrangements of neurons on the resultant electrical dynamics finds that increased spatial proximity between inhibitory neurons leads to oscillatory dynamics. Chapter 3 utilises the edge-based compartmental modelling approach. Existing research is extended to derive and validate equations describing the evolution of a susceptible-infected-recovered (SIR) epidemic process occurring on a dual-layer multiplex network incorporating heterogeneity in the structure, type and duration of connections between individuals. Chapter 4 considers pairwise models describing the SIR epidemic process and derives and validates analytic expressions for the epidemic threshold, an improvement on existing results. This thesis concludes with a discussion of the research contained in Chapters 2-4, including suggestions for improvements and future research ideas.

Regenerated silk (RS) is a natural polymer that results from the aggregation of liquid silk fibroin proteins. In this work, we observed that RS dispersed in aqueous solution undergoes a reversible solid/liquid transition by programmed heating/cooling cycles. Fourier transform infrared, atomic force microscopy imaging and Raman measurements of the RS reveal that the transition fromrandomcoil to b-sheet structures is involved in this liquid–solid transition. The reversible solid-liquid transition of silk fibroin was then found to be helpful to prepare polymer-like carbon nanotube (CNT) dispersions. We demonstrate that the gelation of RS makes the CNTs with the consistency of a dough with polymeric behavior. Such RS can disperse carbon nanotubes at high concentrations of tens of weight percent. Finally, such carbon nanotube dough has been used for the realization of rubber composites. With this method, we pave the way for handling nanopowders (e.g. CNTs or graphene related materials) with safety and reducing the filler volatility that is critical in polymer-processing.

This study empirically re-examines fat tails in stock return distributions by applying statistical methods to an extensive dataset taken from the Korean stock market. The tails of the return distributions are shown to be much fatter in recent periods than in past periods and much fatter for small-capitalization stocks than for large-capitalization stocks. After controlling for the 1997 Korean foreign currency crisis and using the GARCH filter models to control for volatility clustering in the returns, the fat tails in the distribution of residuals are found to persist. We show that market crashes and volatility clustering may not sufficiently account for the existence of fat tails in return distributions. These findings are robust regardless of period or type of stock group.

We show well-posedness for the parabolic Anderson model on 2-dimensional closed Riemannian manifolds. To this end we extend the notion of regularity structures to curved space, and explicitly construct the minimal structure required for this equation. A central ingredient is the appropriate re-interpretation of the polynomial model, which we build up to any order.

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable and we obtain an upper bound on the rate of exponential attraction. In this paper we study the converse question and show that, given an exponentially stable periodic orbit, a contraction metric exists on its basin of attraction and we can recover the upper bound on the rate of exponential attraction.

We use optimal control via a distributed exterior field to steer the dynamics of an ensemble of N interacting ferromagnetic particles which are immersed into a heat bath by minimizing a quadratic functional. Using the dynamic programming principle, we show the existence of a unique strong solution of the optimal control problem. By the Hopf–Cole transformation, the associated Hamilton–Jacobi–Bellman equation of the dynamic programming principle may be re-cast into a linear PDE on the manifold M=(S^2)^N, whose classical solution may be represented via Feynman–Kac formula. We use this probabilistic representation for Monte-Carlo simulations to illustrate optimal switching dynamics.

We extend the scheme developed in B. Düring, A. We extend the scheme developed in B. Düring, A. Pitkin, "High-order compact finite difference scheme for option pricing in stochastic volatility jump models", 2019, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is assessed through a number of numerical experiments, using comparisons against a standard second-order central difference scheme. We observe that the new high-order compact scheme achieves fourth order convergence and discuss the effects on efficiency and computation time.

We present the new spectrometer for the neutron electric dipole moment (nEDM) search at the Paul Scherrer Institute (PSI), called n2EDM. The setup is at room temperature in vacuum using ultracold neutrons. n2EDM features a large UCN double storage chamber design with neutron transport adapted to the PSI UCN source. The design builds on experience gained from the previous apparatus operated at PSI until 2017. An order of magnitude increase in sensitivity is calculated for the new baseline setup based on scalable results from the previous apparatus, and the UCN source performance achieved in 2016.

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

The thesis provides the discussion of three last passage percolation models. In particular, we focus on three aspects of probability theory: the law of large numbers, the order of the variance and large deviation estimates.

In Chapter 1, we give a brief introduction to the percolation models in general and we present some important results for this topic which are heavily used in the following proofs.

In Chapter 2, we prove a strong law of large numbers for directed last passage times in an independent but inhomogeneous exponential environment. Rates for the exponential random variables are obtained from a discretisation of a speed function that may be discontinuous on a locally finite set of discontinuity curves. The limiting shape is cast as a variational formula that maximises a certain functional over a set of weakly increasing curves.

Using this result, we present two examples that allow for partial analytical tractability and show that the shape function may not be strictly concave, and it may exhibit points of non-differentiability, at segments, and non-uniqueness of the optimisers of the variational formula. Finally, in a specific example, we analyse further the macroscopic optimisers and uncover a phase transition for their behaviour.

In Chapter 3, we discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope.

We show that along characteristic directions the order of the variance of the last passage time is of order N2=3 in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N) in the boundary model and to O(1) for the non-boundary model. This behavior is the result of the two at edges of the shape function.

In Chapter 4, we prove a large deviation principle and give an expression for the rate function, for the last passage time in a Bernoulli environment. The model is exactly solvable and its invariant version satisfies a Burke-type property. Finally, we compute explicit limiting logarithmic moment generating functions for both the classical and the invariant models. The shape function of this model exhibits a flat edge in certain directions, and we also discuss the rate function and limiting log-moment generating functions in those directions.

The standard additive coalescent starting with n particles is a Markov process which owns several combinatorial representations, one by Pitman as a process of coalescent forests, and one by Chassaing and Louchard as the block sizes in a parking scheme. In the coalescent forest representation, edges are added successively between a random node and a random root. In this paper, we investigate an alternative construction by, instead, adding edges between roots. This construction induces exactly the same process in terms of cluster sizes, meanwhile, it allows us to make numerous new connections with other combinatorial and probabilistic models: size biased percolation, parking scheme in a tree, increasing trees, random cuts of trees. The variety of the combinatorial objects involved justifies our interest in this construction.

We review some classical results in symmetrization theory, some recent progress in understanding rigidity, and indicate some open problems.

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its performance to standard finite difference and finite element methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial -factorisation of a sparse matrix to perform the option price valuation. Compared to finite element approaches, it is very parsimonious in terms of memory requirements and computational effort, since it achieves high-order convergence without requiring additional unknowns, unlike finite element methods with higher polynomial order basis functions. The new high-order compact scheme can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

The goal of these lecture notes is to present in a unified way various models for the dynamics of aligning self-propelled rigid bodies at different scales and the links between them. The models and methods are inspired from [12,13], but, in addition, we introduce a new model and apply on it the same methods. While the new model has its own interest, our aim is also to emphasize the methods by demonstrating their adaptability and by presenting them in a unified and simplified way. Furthermore, from the various microscopic models we derive the same macroscopic model, which is a good indicator of its universality.

We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek-Stokes system. The Vicsek model describes selfpropelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics (SOH)-Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek-Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.

We study tick-by-tick financial returns for the FTSE MIB index of the Italian Stock Exchange (Borsa Italiana). We confirm previously detected non-stationarities. Scaling properties reported before for other high-frequency financial data are only approximately valid. As a consequence of our empirical analyses, we propose a simple model for non-stationary returns, based on a non-homogeneous normal compound Poisson process. It turns out that our model can approximately reproduce several stylized facts of high-frequency financial time series. Moreover, using Monte Carlo simulations, we analyze order selection for this class of models using three information criteria: Akaike’s information criterion (AIC), the Bayesian information criterion (BIC) and the Hannan-Quinn information criterion (HQ). For comparison, we perform a similar Monte Carlo experiment for the ACD (autoregressive conditional duration) model. Our results show that the information criteria work best for small parameter numbers for the compound Poisson type models, whereas for the ACD model the model selection procedure does not work well in certain cases.

In this paper we present a novel approach for inferring functional connectivity within a large-scale network from time series of emitted node events. We do so under the following constraints: (a) non-stationarity of the underlying connectivity, (b) sparsity of the time-series of events, and (c) absence of an explicit model describing how events propagate through the network. We develop an inference method whose output is an undirected weighted network, where the weight of an edge between two nodes denotes the probability of these nodes being functionally connected. Two nodes are assumed to be functionally connected if they show significantly more coincident or short-lagged events than randomly picked pairs of nodes with similar levels of activity. We develop a model of time-varying connectivity whose parameters are determined by maximising the model’s predictive power from one time window to the next. We assess the accuracy, efficiency and scalability of our method on a real dataset of network events spanning multiple months.

Evidence suggests that both the interaction of so-called Merkel cells and the epidermal stress distribution play an important role in the formation of fingerprint patterns during pregnancy. To model the formation of fingerprint patterns in a biologically meaningful way these patterns have to become stationary. For the creation of synthetic fingerprints it is also very desirable that rescaling the model parameters leads to rescaled distances between the stationary fingerprint ridges. Based on these observations, as well as the model introduced by K¨ucken and Champod we propose a new model for the formation of fingerprint patterns during pregnancy. In this anisotropic interaction model the interaction forces not only depend on the distance vector between the cells and the model parameters, but additionally on an underlying tensor field, representing a stress field. This dependence on the tensor field leads to complex, anisotropic patterns. We study the resulting stationary patterns both

We consider the well-posedness and numerical approximation of a Hamilton-Jacobi equation on an evolving hypersurface in R3. Definitions of viscosity sub- and supersolutions are extended in a natural way to evolving hypersurfaces and provide uniqueness by comparison. An explicit in time monotone numerical approximation is derived on evolving interpolating triangulated surfaces. The scheme relies on a finite volume discretisation which does not require acute triangles. The scheme is shown to be stable and consistent leading to an existence proof via the proof of convergence. Finally an error bound is proved of the same order as in the at stationary case.

We study the stability of invariant sets such as equilibria or periodic orbits of a Dynamical System given by a general autonomous nonlinear ordinary differential equation (ODE). A classical tool to analyse the stability are Lyapunov functions, i.e. scalar-valued functions, which decrease along solutions of the ODE. An alternative to Lyapunov functions is contraction analysis. Here, stability (or incremental stability) is a consequence of the contraction property between two adjacent solutions, formulated as the local property of a Finsler-Lyapunov function. This has the advantage that the invariant set plays no special role and does not need to be known a priori. In this paper, we propose a method to numerically construct a Finsler-Lyapunov function by solving a first-order partial differential equation using meshless collocation. Depending on the expected attractor, the contraction only takes place in certain directions, which can easily be implemented within the method. In the basin of attraction of an exponentially stable equilibrium or periodic orbit, we show that the PDE problem has a solution, which provides error estimates for the numerical method. Furthermore, we show how the method can also be applied outside the basin of attraction and can detect the stability as well as the stable/unstable directions of equilibria. The method is illustrated with several examples.

In this paper we propose and study a new kinetic rating model for a large number of players, which is motivated by the well-known Elo rating system. Each player is characterised by an intrinsic strength and a rating, which are both updated after each game. We state and analyse the respective Boltzmann type equation and derive the corresponding nonlinear, nonlocal Fokker-Planck equation. We investigate the existence of solutions to the Fokker-Planck equation and discuss their behaviour in the long time limit. Furthermore, we illustrate the dynamics of the Boltzmann and Fokker-Planck equation with various numerical experiments.

The Dark matter Experiment using Argon Pulse-shape discrimination (DEAP) has been designed for a direct detection search for particle dark matter using a single-phase liquid argon target. The projected cross section sensitivity for DEAP-3600 to the spin-independent scattering of Weakly Interacting Massive Particles (WIMPs) on nucleons is 10−46cm2 for a 100 GeV/c2 WIMP mass with a fiducial exposure of 3 tonne-years. This paper describes the physical properties and construction of the DEAP-3600 detector.

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.

We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\Tr X_n^{-s}$ as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden \textit{et al.}~[F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order $n\to\infty$ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials.

Realistic human contact networks capable of spreading infectious disease, for example studied in social contact surveys, exhibit both significant degree heterogeneity and clustering, both of which greatly affect epidemic dynamics. To understand the joint effects of these two network properties on epidemic dynamics, the effective degree model of Lindquist et al. [28] is reformulated with a new moment closure to apply to highly clustered networks. A simulation study comparing alternative ODE models and stochastic simulations is performed for SIR (Susceptible–Infected–Removed) epidemic dynamics, including a test for the conjectured error behaviour in [40], providing evidence that this novel model can be a more accurate approximation to epidemic dynamics on complex networks than existing approaches.

The y-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic y-basin of attraction of the equilibrium.

In this study, we apply the Bayesian paradigm for parameter identification to a well-studied semi-linear reaction-difusion system with activatordepleted reaction kinetics, posed on stationary as well as evolving domains. We provide a mathematically rigorous framework to study the inverse problem of finding the parameters of a reaction-diffusion system given a final spatial pattern. On the stationary domain the parameters are finite dimensional, but on the evolving domain we consider the problem of identifying the evolution of the domain, i.e. a time dependent function. While others have considered these inverse problems using optimisation techniques, the Bayesian approach provides a rigorous mathematical framework for incorporating the prior knowledge on uncertainty in the observation and in the parameters themselves, resulting in an approximation of the full probability distribution for the parameters, given the data. Furthermore, using previously established results, we can prove wellposedness results for the inverse problem, using the well-posedness of the forward problem. Although the numerical approximation of the full probability is computationally expensive, parallelised algorithms make the problem solvable using high-performance computing.

This paper investigates the dynamics of immune response and autoimmunity with particular emphasis on the role of regulatory T cells (Tregs), T cells with different activation thresholds, and cytokines in mediating T cell activity. Analysis of the steady states yields parameter regions corresponding to regimes of normal clearance of viral infection, chronic infection, or autoimmune behavior, and the boundaries of stability and bifurcations of relevant steady states are found in terms of system parameters. Numerical simulations are performed to illustrate different dynamical scenarios, and to identify basins of attraction of different steady states and periodic solutions, highlighting the important role played by the initial conditions in determining the outcome of immune interactions.

Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshfree collocation with radial basis functions (RBF). In this paper, we propose two verification estimates combined with this RBF construction method to ensure that the constructed function is a Lyapunov function. We show that this combination of the RBF construction method and the verification estimates always succeeds in constructing and verifying a Lyapunov function for nonlinear ODEs in Rd with an exponentially stable equilibrium.

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium. The contraction metric is described by a matrix-valued function M(x) such that M(x) is positive definite and F(M)(x) is negative definite, where F denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation F(M)(x) = −C(x). We then use a construction method based on meshless collocation, developed in the companion paper [12], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.

A differential identity on the hypergeometric function ${}_2F_1(a,b;c;z)$ unifying and extending certain spectral results on the scale of Gegenbauer and Jacobi polynomials and leading to a new class of hypergeometric related scalars $\mathsf{c}_j^m(a,b,c)$ and polynomials $\mathscr{R}_m=\mathscr{R}_m(X)$ is established. The Laplace-Beltrami operator on a compact rank one symmetric space is considered next and for operators of the Laplace transform type by invoking an operator trace relation, the Maclaurin spectral coefficients of their Schwartz kernel are fully described. Other representations as well as extensions of the differential identity to the generalised hypergeometric function ${}_pF_q({\bf a}; {\bf b}; z)$ are formulated and proved.

We study the Boltzmann equation on the d-dimensional torus in a perturbative setting around a global equilibrium under the Navier-Stokes lineari- sation. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a C0-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently on the Knudsen number. Finally we show a Cauchy theory and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal and furthermore, this result only requires derivatives in the space variable and allows to connect to solutions to the incompressible Navier-Stokes equations in these spaces.

An a posteriori bound for the error measured in the discontinuous energy norm for a discontinuous Galerkin (dG) discretization of a linear one-dimensional stationary convection- diffusion-reaction problem with essential boundary conditions is presented. The proof is based on a conforming recovery operator inspired from a posteriori error bounds for the dG method for first order hyperbolic problems. As such, the bound remains valid in the singular limit of vanishing diffusion. Detailed numerical experiments demonstrate the independence of the quality of the a posteriori bound with respect to the Péclet number in the standard dG-energy norm, as well as with respect to the viscosity parameter.

A weak-strong uniqueness result is proved for measure-valued solutions to the system of conservation laws arising in elastodynamics. The main novelty brought forward by the present work is that the underlying stored-energy function of the material is assumed strongly quasiconvex. The proof employs tools from the calculus of variations to establish general convexity-type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics.

Consider a random walk Si = ξ1 + . . . + ξi , i ∈ N, whose increments ξ1, ξ2, . . . are independent identically distributed random vectors in R d such that ξ1 has the same law as −ξ1 and P[ξ1 ∈ H] = 0 for every affine hyperplane H ⊂ R d . Our main result is the distribution-free formula [see published version for formula] where the B(k, j)’s are defined by their generating function (t + 1)(t + 3). . .(t + 2k − 1) = Pk j=0 B(k, j)t j . The expected number of k-tuples above admits the following geometric interpretation: it is the expected number of k-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type Bn that are not intersected by a generic linear subspace L ⊂ R n of codimension d. The case d = 1 turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required.

We propose and analyse a lumped surface finite element method for the numerical approximation of reaction–diffusion systems on stationary compact surfaces in R3. The proposed method preserves the invariant regions of the continuous problem under discretization and, in the special case of scalar equations, it preserves the maximum principle. On the application of a fully discrete scheme using the implicit–explicit Euler method in time, we prove that invariant regions of the continuous problem are preserved (i) at the spatially discrete level with no restriction on the meshsize and (ii) at the fully discrete level under a timestep restriction. We further prove optimal error bounds for the semidiscrete and fully discrete methods, that is, the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. We provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up.