The following thesis is divided in two main topics. The first part studies variations of optimal prediction problems introduced in Shiryaev, Zhou and Xu (2008) and Du Toit and Peskir (2009) to a randomized terminal-time set up and different families of utility measures. The work presents optimal stopping rules that apply under different criteria, introduces a numerical technique to build approximations of stopping boundaries for fixed terminal time problems and suggest previously reported stopping rules extend to certain generalizations of measures. The second part of the thesis is concerned with analysing optimal wealth allocation techniques within a defaultable financial market similar to Bielecki and Jang (2007). It studies a portfolio optimization problem combining a continuous time jump market and a defaultable security; and presents numerical solutions through the conversion into a Markov Decision Process and characterization of its value function as a unique fixed point to a contracting operator. This work analyses allocation strategies under several families of utilities functions, and highlights significant portfolio selection differences with previously reported results.

Portfolio optimization involves the optimal assignment of limited capital to different available financial assets to achieve a reasonable trade-off between profit and risk objectives. Markowitz’s mean variance (MV) model is widely regarded as the foundation of modern portfolio theory and provides a quantitative framework for portfolio optimization problems. In real market, investors commonly face real-world trading restrictions and it requires that the constructed portfolios have to meet trading constraints. When additional constraints are added to the basic MV model, the problem thus becomes more complex and the exact optimization approaches run into difficulties to deliver solutions within reasonable time for large problem size. By introducing the cardinality constraint alone already transformed the classic quadratic optimization model into a mixed-integer quadratic programming problem which is an NP-hard problem. Evolutionary algorithms, a class of metaheuristics, are one of the known alternatives for optimization problems that are too complex to be solved using deterministic techniques. This thesis focuses on single-period portfolio optimization problems with practical trading constraints and two different risk measures. Four hybrid evolutionary algorithms are presented to efficiently solve these problems with gradually more complex real world constraints. In the first part of the thesis, the mean variance portfolio model is investigated by taking into account real-world constraints. A hybrid evolutionary algorithm (PBILDE) for portfolio optimization with cardinality and quantity constraints is presented. The proposed PBILDE is able to achieve a strong synergetic effect through hybridization of PBIL and DE. A partially guided mutation and an elitist update strategy are proposed in order to promote the efficient convergence of PBILDE. Its effectiveness is evaluated and compared with other existing algorithms over a number of datasets. A multi-objective scatter search with archive (MOSSwA) algorithm for portfolio optimization with cardinality, quantity and pre-assignment constraints is then presented. New subset generations and solution combination methods are proposed to generate efficient and diverse portfolios. A learning-guided multi-objective evolutionary (MODEwAwL) algorithm for the portfolio optimization problems with cardinality, quantity, pre-assignment and round lot constraints is presented. A learning mechanism is introduced in order to extract important features from the set of elite solutions. Problem-specific selection heuristics are introduced in order to identify high-quality solutions with a reduced computational cost. An efficient and effective candidate generation scheme utilizing a learning mechanism, problem specific heuristics and effective direction-based search methods is proposed to guide the search towards the promising regions of the search space. In the second part of the thesis, an alternative risk measure, VaR, is considered. A non-parametric mean-VaR model with six practical trading constraints is investigated. A multi-objective evolutionary algorithm with guided learning (MODE-GL) is presented for the mean-VaR model. Two different variants of DE mutation schemes in the solution generation scheme are proposed in order to promote the exploration of the search towards the least crowded region of the solution space. Experimental results using historical daily financial market data from S &P 100 and S & P 500 indices are presented. When the cardinality constraints are considered, incorporating a learning mechanism significantly promotes the efficient convergence of the search.

Tissue engineering aims to regenerate, repair or replace organs or tissues which have become defective due to trauma, disease or age related degeneration. This engineering may take place within the patient's body or tissue can be regenerated in a bioreactor for later implantation into the patient. Regeneration of soft tissue is one of the most demanding applications of tissue engineering. Producing proper nutrient supply, uniform cell distribution and high cell density are the important challenges. Many experimental models exist for tissue growth in a bioreactor. It is important to put experiments into a theoretical framework. Mathematical modelling in terms of physical and biochemical mechanisms is the best tool to understand experimental results. In this work a mathematical model of convective and diffusive transport of nutrients and cell growth in a perfusion bioreactor is developed. A cell-seeded porous scaffold is placed in a perfusion bioreactor and fluid delivers the nutrients to the cells for their growth. The model describes the key features of the tissue engineering processes which includes the interaction between the cell growth,variation of material porosity, flow of fluid through the material and delivery of nutrients to the cells. The fluid flow through the porous scaffold is modelled by Darcy's law, and the delivery of nutrients to the cells is modelled by the advection-diffusion equation. A non-linear reaction diffusion system is used to model the cell growth. The cell diffusion depends on the cell density and growth of cells is modelled by logistic growth. The effect of shear stress on nutrient consumption and cell growth is also included in the model. COMSOL (a commercial finite element solver) is used to numerically solve the model. The results show that the distribution of cells and total cell number in the scaffold depends on the initial cell density and porosity. We suggest various seeding strategies and scaffold designs to improve the cell growth rate and total cell yield.

A promising method for growing functional liver tissue in vitro involves culturing hepatocytes as spheroidal cell aggregates. In this thesis, we develop mathematical models of cell aggregation, and use them to determine how hepatocytes' interactions with the extracellular matrix (ECM) on which they are seeded, and with stellate cells, affect the process. Chapters 2-4 focus on the effect that cell-ECM coupling has on the aggregation process. We use a novel formulation that couples a mechanical model for the ECM with a two-phase model for the cell-culture region. A combination of linear stability analysis and numerical simulations are used to identify parameter regimes in which aggregation occurs, and investigate the effect of changing key parameters. In Chapter 2, we assume a one-dimensional geometry, whereas in Chapters 3 and 4, the slender two-dimensional geometry is exploited to obtain two alternative one-dimensional models in which the mechanisms dominating aggregation are chemotaxis and surface tension. In Chapter 5, we focus on interactions between hepatocytes and stellates, neglecting the role of the ECM.We develop new non-local models to investigate the relative contributions of hepatocyte-heaptocyte and hepatocyte-stellate interactions in controlling spheroid formation. Comparison with experimental results suggests that the hepatocyte-stellate interaction is the stronger, in which case a 1:1 seeding ratio of hepatocytes to stellates is likely to be optimal for promoting swift aggregate formation.

Growing populations in combination with the effects of climate change make the task of ensuring global food security in the future challenging. Water and various nutrients contained in soils are essential for the growth and survival of crop plants. Processes governing the dynamics of these substances are often highly complex and occur at various spatial scales. Because of that, and also due to limited possibilities for direct studies of soil processes in general, the ability to model such processes across these scales will most likely be crucial to address the food security challenge. Therefore, in this thesis, we model water and nutrient uptake by plant roots at various spatial scales. As all our models will be simulated numerically, we first test whether the software used throughout this thesis (FEniCS) gives us reliable numerical results (Chapter 2). We then proceed with the central part of this thesis, where we study nutrient uptake by root hairs using the method of homogenisation (introduced in Section 1.4.4). In Chapter 3, we first rederive the homogenisation result from [80] using a framework of periodic arrays of uptaking cylinders (hairs). Noticing that this framework can also be used to model nutrient or water uptake by a field of crops, we further study how well the homogenisation result compares with full-geometry numerics using various continuity equations and boundary conditions. In Chapter 4, we study the case where the radius of the root hair is much smaller than the inter-hair distance, which eventually leads us to a distinguished limit. In Chapter 5, we first establish that the framework from Chapters 3 and 4 is a suitable geometry for modelling nutrient uptake by root hairs, if the hair length is much smaller than the root radius. However, this is rarely the case. Therefore, we then investigate the effects of root surface curvature and hair length on the homogenised equation, and obtain a better approximation for the case where the hair length is comparable to the root radius. In the final two chapters, we introduce different complex problems relating to uptake by plants, and show how even simple multiscale techniques can provide us with useful insights into these problems. In Chapter 6, we show how to upscale nitrate uptake kinetics from a single transporter level to a root segment level, and then propose a model for nitrate uptake via low and high affinity transporters (see Section 6.1.2). Model predictions for the time of depletion, and a threshold nitrate concentration at which uptake ceases, are both in accordance with empirical values (Section 6.3.6). Finally, we demonstrate that under certain conditions, three-dimensional descriptions of the root system architecture are not necessary to estimate overall water and nitrate uptake, and that simple one-dimensional models can be used instead (Chapter 7).

Age-related macular degeneration and diabetic retinopathy, in which the cells at the back of the eye degrade due to age and diabetes respectively, are prevalent causes of vision loss in adults. We formulate mathematical models of retinal metabolic regulation to investigate defects that may be responsible for pathology. Continuum PDE models are developed to test whether rod photoreceptors, light detecting cells in the eye, may regulate their energy demand by adapting their length under light and dark conditions. These models assume photoreceptor length depends on the availability of nutrients, such as oxygen, which diffuse and are consumed within the photoreceptor. Our results suggest that the length is limited by oxygen and phosphocreatine shuttle-derived ATP under dark and light conditions respectively. Parameter sensitivity analysis indicates that lowered mitochondrial efficiency due to ageing may be responsible for the damage to and death of photoreceptors that are characteristic of age-related macular degeneration. In the latter part of this thesis we shift our focus to the inner retina and examine how metabolite levels in the tissue surrounding the neurons (highly sensitive, excitable cells that transmit electrical signals) are regulated by glial cells. For instance, stimulated neurons activate their neighbours via the release of the neurotransmitter glutamate, while glial cells regulate neuronal activity via glutamate uptake. Diabetes produces large fluctuations in blood glucose levels, and eventually results in neuronal cell death, causing vision loss. We generate an ODE model for the exchange of key metabolites between neurons and surrounding cells. Using numerical and analytical techniques, we use the model to show that the fluctuations in blood glucose and metabolic changes associated with diabetes may result in abnormally high glutamate levels in the inner retina, which could lead to neuronal damage via excitotoxicity (unregulated neuronal stimulation).

The Jiang-Su algebra Z and the notion of Z-stability (i.e. tensorial absorption of the Jiang-Su algebra) are now widely acknowledged to be of particular importance in the classification and structure theory of separable nuclear C*-algebras. The key results in this thesis are early attempts to explore Z-stability outside the constraints of unital and of nuclear C*-algebras. Standard unitisations of a separable Z-stable C*-algebra are not Z-stable and we therefore explore possible unitisations that preserve Z-stability. We construct the minimal Z-stable unitisation of a separable Z-stable C*-algebra and show that it satisfies an appropriate universal property. An interesting area in which to exploit Z-stability outside of the context of nuclear C*-algebras is the so-called Kadison’s similarity problem. We show that the tensor product of two separable unital C*-algebras has Kadison’s similarity property if one of them is nuclear and admits a unital *-homomorphism from (the building blocks of) the Jiang-Su algebra. An immediate consequence of this is that any separable unital Z-stable C*-algebra also has this property.

Spatially structured bursts of propagating neural activity revealed in cortical slice experiments and in vivo tantalise many scientists on their possible functional mechanisms. Theoretical studies suggest waves with complex firing patterns afford a great capacity for the transmission of information across the brain. This thesis develops a framework for analysing the dynamics of such waves within spiking neuronal networks. We seek to investigate important questions concerning how the wave’s spatiotemporal voltage properties, propagation speed and spike time interval distributions depend on the underlying network structure and the intrinsic features of the neurons that make up the network. These are often difficult to extract with biophysically detailed network models. We therefore analyse simplified spiking networks of synaptically connected neurons, capable of supporting a rich repertoire of propagating activity, yet, amenable to mathematical analysis. Useful information is then obtained on the dynamics of waves found in this network in relation to the model’s parameters. These results can be compared to the findings obtained from more detailed computational studies and experimental observations. Numerical simulations in discrete networks of integrate-and-fire neurons reveal localised bumps that can wander diffusively across the network. These wandering bumps are seen to evolve into persistent synchronous coherent propagating structures, where neurons fire multiple times as the wave envelope passes over. We call these structures multiple-spike waves. An intrinsic feature of the neuron, describing how quickly neurons process synaptic current, is shown to be an important determinant in the emergent network activity. Waves with different number of spiking events co-exist across most parameter regimes, and with lateral-inhibition synaptic connectivity structure, can exhibit large variability in wave speed that has not been reported in studies of networks with purely excitatory connectivity. As a result, we investigate the interaction dynamics of multiple-spike waves on a large spatial domain. Here we find that multiple-spikes waves can merge to form a composite system, with greater complexity in the firing patterns, increasing the wave’s information content. Mathematical progress is made by studying a partial integro-differential equation that is equivalent to the discrete network as the number of neurons tends to infinity. We develop a method of solving the wave speed of the multiple spike waves and its set of spike-times, which then allows us to construct the network’s exact voltage and synaptic profiles and formulate a non-local eigenvalue problem to compute asymptotic stability. This is achieved by considering general perturbations around the wave’s firing times. An in-depth numerical study on the multiple-spike wave’s bifurcation structure is performed, uncovering various mechanisms behind propagation failure and how the wave’s dynamics depend on the network’s system parameters. The analysis of waves with a large number of spikes poses interesting questions regarding the existence of stationary bump solutions in the continuum limit. Uncertainty quantification is performed on waves, revealing how different types of uncertainty in system parameters influence the wave solutions statistical properties. This allows for predictions of the spatial regions of the waves profile most vulnerable to destabilisation. We finally analyse synaptically generated waves in a similar spiking network of Morris-Lecar neurons, where we find interesting transitions from single to double spike waves. Also, similar to what was seen in the integrate-and-fire network, the wave’s dynamics at the network level is strongly influenced by the neuron’s intrinsic features.

This thesis examines various aspects of time series and their applications. In the rst part, we study numerical and asymptotic properties of Box-Pierce family of portmanteau tests. We compare size and power properties of time series model diagnostic tests using their asymptotic c2 distribution and bootstrap distribution (dynamic and fixed design) against various linear and non-linear alternatives. In general, our results show that dynamic bootstrapping provides a better approximation of the distribution underlying these statistics. Moreover, we find that Box-Pierce type tests are powerful against linear alternatives while the CvM due to Escanciano (2006b) test performs better against non linear alternative models. The most challenging scenario for these portmanteau tests is when the process is close to the stationary boundary and value of m, the maximum lag considered in the portmanteau test, is very small. In these situations, the c2 distribution is a poor approximation of the null asymptotic distribution. Katayama (2008) suggested a bias correction term to improve the approximation in these situations. We numerically study Katayama's bias correction in Ljung and Box (1978) test. Our results show that Katayama's correction works well and conrms the results as shown in Katayama (2008). We also provide a number of algorithms for performing the necessary calculations efciently. We notice that the bootstrap automatically does bias correction in Ljung-Box statistic. It motivates us to look at theoretical properties of the dynamic bootstrap in this context. Moreover, noticing the good performance of Katayama's correction, we suggest a bias correction term for the Monti (1994) test on the lines of Katayama's correction. We show that our suggestion improves Monti's statistic in a similar way to what Katayama's suggestion does for Ljung-Box test. We also make a novel suggestion of using the pivotal portmanteau test. Our suggestion is to use two separate values of m, one a large value for the calculation of the information matrix and a smaller choice for diagnostic purposes. This results in a pivotal statistic which automatically corrects the bias correction in Ljung-Box test. Our suggested novel algorithm efciently computes this novel portmanteau test. In the second part, we implement lasso-type shrinkage methods to linear regression and time series models. We look through simulations in various examples to study the oracle properties of these methods via the adaptive lasso due to Zou (2006). We study consistent variable selection by the lasso and adaptive lasso and consider a result in the literature which states that the lasso cannot be consistent in variable selection if a necessary condition does not hold for the model. We notice that lasso methods have nice theoretical properties but it is not very easy to achieve them in practice. The choice of tuning parameter is crucial for these methods. So far there is not any fully explicit way of choosing the appropriate value of tuning parameter, so it is hard to achieve the oracle properties in practice. In our numerical study, we compare the performance of k-fold cross-validation with the BIC method of Wang et al. (2007) for selecting the appropriate value of the tuning parameter. We show that k-fold crossvalidation is not a reliable method for choosing the value of the tuning parameter for consistent variable selection. We also look at ways to implement lasso-type methods time series models. In our numerical results we show that the oracle properties of lasso-type methods can also be achieved for time series models. We derive the necessary condition for consistent variable selection by lasso-type methods in the time series context. We also prove the oracle properties of the adaptive lasso for stationary time series.

This thesis deals with a two component reaction-diffusion system (RDS) for competing and cooperating species. We have analyse in detail the stability and bifurcation structure of equilibrium solutions of this system, a natural extension of the Lotka-Volterra system. We find seven topologically different regions separated by bifurcation boundaries depending on the number and stability of equilibrium solutions, with four regions in which the solutions are similar to those in the Lotka-Volterra system. We study RDS in the small parameter of the range $0< \lambda \ll 1 $ (fast diffusion and slow reaction), and in a few cases we assume $\lambda=O(1)$. We consider three types of initial conditions, and we find three types of travelling wave solutions using numerical and asymptotic methods. However, neither numerical nor asymptotic methods were able to find a particular travelling wave solution which connects a coexistence state say, $(u_0,w_0)$ to an extinction state $(0,0)$ when $0< \lambda \ll 1 $. This type can be found when the reaction-diffusion system satisfy the symmetry property and $\lambda=1$.