In any population of cells, individual cells grow for some period of time and then divide into two or more parts, called daughters. To describe this process mathematically, we need to specify functions describing the growth rate, size at division, and proportions into which each cell divides. In this thesis, it is assumed that the growth rate of a cell can be determined precisely from its size, but that both its size at division and the proportions into which it divides may be described stochastically, by probability density functions whose parameters are dependent on cell size and age (or birth-size). Special cases are also considered where all cells with the same birth-size divide at the same size, or where all cells divide exactly in half. We consider a population of cells growing and dividing steadily, such that the total cell population is increasing, but the proportion of cells in any size class remains constant. In Chapter 1, equations are derived which need to be solved in order to deduce the shape of the steady size distribution (or steady size/age or size/birth-size distributions) from any given growth rate and probability distributions describing the division rate and division proportions. In the general case, a Fredholm-type integral equation is obtained, but if the probability of cell division depends on cell size only (i.e. not age or birth-size), and all cells divide into equal-sized daughters, then we obtain a functional differential equation. In two special cases, the resulting equations simplify considerably, and it is these cases which are explored further in this thesis. The first is where the probability of a cell dividing in any instant of time is a constant, independent of cell age or size. In Chapter 2, the functional differential equation resulting when cells divide into equal-sized daughters is solved for the special case where the growth rate is constant, and in an appendix the case where the growth rate is described by a power law is dealt with. The second case which simplifies is where the time-independent part of the growth rate of a cell is proportional to cell size. This case is particularly important, as it is a good first-order approximation to the real cell growth rate in some structured tissues, and in some bacteria. The special case in which this leads to a functional differential equation is discussed in Chapter 3, and the integral equation arising in the general case is dealt with in Chapter 4. Finally, the conditions under which the integral operator in Chapter 4 will be both square-integrable and non-factorable are discussed in Chapter 5. It is shown that if these conditions are satisfied then a unique, stable, steady size distribution will exist.

Each aspect of this thesis is motivated by the recent paper of Beresnevich, Dickinson and Velani (BDV03]. Let 'ljJ be a real, positive, decreasing function i.e. an approximation function. Their paper considers a general lim sup set A( 'ljJ), within a compact metric measure space (0, d, m), consisting of points that sit in infinitely many balls each centred at an element ROt of a countable set and of radius 'I/J(130) where 130 is a 'weight' assigned to each ROt. The classical set of 'I/J-well approximable numbers is the basic example. For the set A('ljJ) , [BDV03] achieves m-measure and Hausdorff measure laws analogous to the classical theorems of Khintchine and Jarnik. Our first results obtain an application of these metric laws to the set of 'ljJ-well approximable numbers with restricted rationals, previously considered by Harman (Har88c]. Next, we consider a generalisation of the set of badly approximable numbers, Bad. For an approximation function p, a point x of a compact metric space is in a general set Bad(p) if, loosely speaking, x 'avoids' any ball centred at an element ROt of a countable set and of radius c p(I3Ot) for c = c(x) a constant. In view of Jarnik's 1928 result that dim Bad = 1, we aim to show the general set Bad(p) has maximal Hausdorff dimension. Finally, we extend the theory of (BDV03] by constructing a general lim sup set dependent on two approximation functions, A('ljJll'ljJ2)' We state a measure theorem for this set analogous to Khintchine's (1926a) theorem for the Lebesgue measure of the set of ('l/Jl, 1/12)-well approximable pairs in R2. We also remark on the set's Hausdorff dimension.

The pricing of options is one of the key problems in mathematical finance. In recent years, pricing models that are based on the continuous time random walk (CTRW), an anomalous diffusive random walk model widely used in physics, have been introduced. In this thesis, we investigate the pricing of European call options with CTRW and generalized CTRW models within the Black-Scholes framework. Here, the non-Markovian character of the underlying pricing model is manifest in Black-Scholes PDEs with fractional time derivatives containing memory terms. The inclusion of non-zero interest rates leads to a distinction between different types of \forward" and \backward" options, which are easily mapped onto each other in the standard Markovian framework, but exhibit significant dfferences in the non-Markovian case. The backward-type options require us in particular to include the multi-point statistics of the non-Markovian pricing model. Using a representation of the CTRW in terms of a subordination (time change) of a normal diffusive process with an inverse L evy-stable process, analytical results can be obtained. The extension of the formalism to arbitrary waiting time distributions and general payoff functions is discussed. The pricing of path-dependent Asian options leads to further distinctions between different variants of the subordination. We obtain analytical results that relate the option price to the solution of generalized Feynman-Kac equations containing non-local time derivatives such as the fractional substantial derivative. Results for L evy-stable and tempered L evy-stable subordinators, power options, arithmetic and geometric Asian options are presented.

Share price returns on different time scales can be well modeled by a superstatistical dynamics. We provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while chi-square-superstatistics works well on a time scale of days, on a much smaller time scale of minutes the price changes are better described by lognormal superstatistics. The system dynamics thus exhibits a transition from lognormal to chi-square-superstatistics as a function of time scale. We discuss a more general model interpolating between both statistics which fits the observed data very well. We also present results on correlation functions of the extracted superstatistical volatility parameter, which exhibits exponential decay for returns on large time scales, whereas for returns on small time scales there are long-range correlations and power-law decays. We also apply the symbolic dynamics technique from dynamical system theory to analyse the coarse-grained evolution of share price returns. A nontrivial spectrum of Renyi entropies is found. We study how the spectrum depends on the time scale of returns, the sector of stocks considered, as well as the number of symbols used for the symbolic description. Overall our analysis confirms that in the symbol space transition probabilities of observed share price returns depend on the entire history of previous symbols, thus emphasizing the need for a model of share price evolution based on non-Markovian stochastic processes. Our method allows for quantitative comparisons of entirely different complex systems, for example the statistics of coarse-grained share price returns using 4 symbols can be compared with that of other complex systems.

The dose regimen of a drug gives important information about the dose sizes, dose frequency and the duration of treatment. Optimisation of dose regimens is critical to ensure therapeutic success of the drug and to minimise its possible adverse effects. The central theme of this thesis is the Efficient Dosing (ED) algorithm - a computation algorithm developed by us for optimisation of dose regimens. In this thesis, we have attempted to develop a quantitative framework for measuring the efficiency of a dose regimen for specified criteria and computing the most efficient dose regimen using the ED algorithm. The criteria considered by us seek to prevent over- and under-exposure to the drug. For example, one of the criteria is to maintain the drug's concentration around a desired target level. Another criterion is to maintain the concentration within a therapeutic range or window. The ED algorithm and its various extensions are programmed in MATLAB R . Some distinguishing features of our methods are: mathematical explicitness in the optimisation process for a general objective function, creation of a theoretical base to draw comparisons among competing dose regimens, adaptability to any drug for which the PK model is known, and other computational features. We develop the algorithm further to compute the optimal ratio of two partner drugs in a fixed dose combination unit and the efficient dose regimens. In clinical trials, the parameters of the PK model followed by the drug are often unknown. We develop a methodology to apply our algorithm in an adaptive setting which enables estimation of the parameters while optimising the dose regimens for the typical subject in each cohort. A potential application of the ED algorithm for individualisation of dose regimens is discussed. We also discuss an application for computation of efficient dose regimens for obliteration of a pre-specified viral load.

The subject of this thesis is the asymptotic behaviour of generating functions of different combinatorial models of two-dimensional lattice walks and polygons, enumerated with respect to different parameters, such as perimeter, number of steps and area. These models occur in various applications in physics, computer science and biology. In particular, they can be seen as simple models of biological vesicles or polymers. Of particular interest is the singular behaviour of the generating functions around special, so-called multicritical points in their parameter space, which correspond physically to phase transitions. The singular behaviour around the multicritical point is described by a scaling function, alongside a small set of critical exponents. Apart from some non-rigorous heuristics, our asymptotic analysis mainly consists in applying the method of steepest descents to a suitable integral expression for the exact solution for the generating function of a given model. The similar mathematical structure of the exact solutions of the different models allows for a unified treatment. In the saddle point analysis, the multicritical points correspond to points in the parameter space at which several saddle points of the integral kernels coalesce. Generically, two saddle points coalesce, in which case the scaling function is expressible in terms of the Airy function. As we will see, this is the case for Dyck and Schröder paths, directed column-convex polygons and partially directed self-avoiding walks. The result for Dyck paths also allows for the scaling analysis of Bernoulli meanders (also known as ballot paths). We then construct the model of deformed Dyck paths, where three saddle points coalesce in the corresponding integral kernel, thereby leading to an asymptotic expression in terms of a bivariate, generalised Airy integral.

In this thesis I investigate the effects of positive and negative connections on social and organization networks, and the presence and role of homophily in networks of scientific collaborations and citations through the combination of methodologies borrowed from complexity science, statistics, and organizational sciences. In the first part of the thesis, I study the differences between patterns of positive and negative connections among individuals in two online signed social networks. Findings suggest that the sign of links in a social network shapes differently the network's topology: there is a positive correlation between the degrees of two nodes, when they share a positive connection, and a negative correlation when they share a negative connection. I then move my focus to the study of a dataset on start-ups from which I construct and analyse the competition and mobility networks among companies. Results show that the presence of competition has negative effects on the mobility of people among companies and on the success of the start-up ecosystem of a nation. Competitive behaviours may also emerge in science. Therefore, in the second part of this thesis, I focus on a database of all papers and authors who have published in the American Physical Society (APS) journals. Through the analysis of the citation network of the APS, I propose a method that aims to statistically validate the presence (or absence) of a citation between any two articles. Results show that homophily is an important mechanism behind the citation between articles: the more two articles share similar bibliographies, i.e., deal with similar arguments, the more likely there is a citation between them. In the last chapter, I investigate the presence of homophily in the APS data set, this time at the level of the collaboration network among sci- entists. Results show that homophily can be responsible in fostering collaboration, but above a given point the effect of similarity decreases the probability of a collaboration. Additionally, I propose a model that successfully reproduces the empirical findings.

This work addresses the development and application of gene set analysis methods to problems in microarray-based data sets. The work consists of three parts. In the first part a gene set analysis method (PCOT2) is developed. It utilizes inter-gene correlation to detect significant alteration in gene sets across experimental conditions. The second part is focused on the exploration of correlation-based gene sets in conjunction with the application of the PCOT2 testing method in the investigation of biological mechanisms underlying breast cancer recurrence. In the third part, statistical models for analyzing combined microarray-based expression and genomic copy number data are developed. In addition, an analysis which incorporates tumour subgroups is shown to provide more accurate prognosis assessment for breast cancer patients.

The effective integration of technology into the teaching and learning of mathematics remains one of the critical challenges facing tertiary mathematics, which has traditionally been slow to respond to technological innovation. This thesis reveals that the term integration is widely used in the literature with respect to technology and the curriculum, although its meaning can vary substantially, and furthermore, the term is seldom well defined. A review of the literature provides the basis for a survey of undergraduate mathematics educators, to determine their use of technology, their views of what an Integrated Technology Mathematics Curriculum (ITMC) may resemble, and how it may be achieved. Responses to this survey, and factors identified in the literature, are used to construct a taxonomy of integrated technology. The taxonomy identifies six defining characteristics of an ITMC, each with a number of associated elements. A visual model using radar diagrams is developed to compare courses against the taxonomy, and to identify aspects needing attention in individual courses. T Evidence from an observational study of initiatives to introduce Computer Algebra Systems into undergraduate mathematics courses at The University of Auckland, firstly using CAS-calculators and latterly computer software, is examined against the taxonomy. A number of critical issues influencing the integration of these technologies are identified. These include mandating technology use in official departmental policy, attention to congruency and fairness in assessment, re-evaluating the value of topics in the curriculum, re-establishing the goals of undergraduate courses, and developing the pedagogical technical knowledge of teaching staff. The thesis concludes that effective integration of technology in undergraduate mathematics requires a recognition of, and comprehensive attention to, the interdependence of the taxonomy components. An integrated, holistic approach, which aims for curricular congruency across all elements of the taxonomy, provides the basis for a more consistent, effective and sustainable ITMC.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.