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Literature Review Background: Mathematical and reading abilities are predictive of academic achievement. To date, limited research has examined the relationship between processing speed and academic achievement in typically developing children. Greater insight into this relationship could help to identify the impact that reduced processing speed may have on long-term academic achievement. This review aimed to explore the relationship between these variables in typically developing children. Method: Studies conducted in the past twenty years measuring mathematics and/or reading abilities and processing speed in typically developing children using a standardised assessment measure were included in the review. In total 1278 studies were screened, which led to the identification of eight eligible studies that were included in the review. Results: No relationship was found between processing speed and reading ability. The findings on mathematics abilities were conflicting, with some studies identifying a relationship and other finding no significant association between these variables. Age appeared to be a moderating factor in studies that reported a significant relationship between mathematics and processing speed. Conclusions: The findings suggest that the relationship between processing speed and academic achievement is complex and the following review was unable to ascertain the direct relationship between these variables. It is recommended that future research examines the relationship between age and academic achievement in further detail. Empirical Paper Processing speed interventions have been found to be acceptable in children; however, there is limited evidence that they are effective in this population. This study investigated whether a processing speed intervention was effective in improving processing speed (PS) in children with suspected white matter disorders. The study hypothesised that children would demonstrate improvement on a daily outcome measure and between pre-baseline and post-intervention measures of PS. A single case experimental design utilising a multiple baseline approach was used to observe the effect of the intervention within and across participants. Three participants were recruited, each completing a choice reaction time (CRT) task three times a week that acted as the outcome measure. The processing speed intervention involved playing single player, multiplayer and iPad/android games. Overall there was no significant change in CRT between phases; however two participants demonstrated a medium effect size. There was no significant change in pre- or post-PS measures but there was evidence of reliable change in overall and cognitive fatigue. These findings suggest that the processing speed intervention was not effective in improving PS abilities. This paper highlights a number of challenges in implementing a processing speed intervention and explores the clinical implications of these findings.

There is increasing understanding that many complex systems of interest - everything from the global economy, to social group dynamics, to biochemical processes in the brain - require holistic modelling, rather than the consideration of units of the population in isolation. Network science techniques, which commence by viewing the system as a set of vertices or nodes (the units of the population, e.g. individual people) and edges (the relationships between them, e.g. friendship), are one popular approach to do so. Naturally, such complex systems express a wide array of important properties that we ought to account for when modelling them, beyond simply the presence or absence of a particular relationship. Most pertinently for this work, they evolve over time - i.e. they are 'dynamic' - and the units of the population may have distinct properties, or attributes, which further differentiate them from each other. We define any such extra information we might possess outside of the simple node/edge paradigm to be 'metadata'. Despite the potential utility of such metadata, it is only quite recently that methods have begun to jointly model both network and metadata together. In this thesis, we provide a new class of models that do so - specifically, with the purpose of finding groups in networks that change over time. We describe distinct versions of this class of models that allow the networks to be weighted and directed, as well as avoid the potential issue of placing nodes with similar degrees in the same group. In addition to elaborating such models, we derive novel requirements for the efficient detectability of groups given the presence of metadata - and in the process explain why a recent paper which claims to do the same for a similar static model is flawed. The inference method we leverage to investigate detectability is also highly scalable, and we further accelerate the process by proposing both a 'greedy' scheme, and a recursive procedure that effectively provides a top-down hierarchy of the network groups. We conclude by using our models as one component of a larger method, that provides an entirely novel means of estimating the influence of an author. We use a causal framing of the problem that to our knowledge has not previously been explored in this context, and depends upon recent ideas from the causal inference literature.

This thesis concerns the principled use of mathematical models for informing control interventions in the context of an infectious disease outbreak context. We address two main related questions. First, how can interventions be evaluated and compared in a principled manner using a dynamic (time-varying), mathematical model? The second question pertains to structural identifiability, a prerequisite for successful parameter inference in dynamic models: how does the type of model output influence the structural identifiability of dynamic, mathematical models for infectious disease transmission? We address the first question using two outbreaks of infectious disease - hepatitis A and African swine fever (ASF) - as illustrative examples, and the second question by conducting structural identifiability analysis of various ordinary differential equation (ODE) model versions. Links to all code and files needed to reproduce the results in this thesis have been provided. Our contributions are detailed in three papers: a published article and two manuscripts, now summarized. Chapter 3: Dankwa et al. (2021) developed a dynamic, deterministic model to explain transmission in the 2017-2019 hepatitis A virus (HAV) outbreak in Louisville, Kentucky, US, among persons experiencing homelessness or who use drugs, known to be at a high risk of HAV infection. With the model, alternative vaccination scenarios were examined for effectiveness and cost, and an estimate for the critical vaccination threshold required for herd immunity in this population was derived. Prior to this study, no such estimate had been obtained for this population anywhere in the US. Chapter 4: Dankwa et al. (2022b) developed a dynamic, stochastic, spatial model to explain ASF virus (ASFV) transmission among wild boar and domestic pig herds, and to evaluate alternative outbreak management measures. The model was developed such that it could be refined to account for more outbreak data and hence it is suitable for real- time outbreak analysis. This study is relevant, given the ongoing ASF epidemic in Europe, and the scarcity of ASFV transmission models which account for transmission at the wildlife-livestock interface, despite the evidence of such transmission in Europe. Chapter 5: Dankwa et al. (2022a) conducted structural identifiability analysis of unknown parameters, including initial conditions, of 26 ODE model versions to demonstrate the influence of the type of model output(s) on models' structural identifiability. Data types such as incidence and prevalence, typically encountered in disease surveillance, were studied as model outputs. This analysis emphasizes the importance of a careful consideration of model outputs prior to performing inference with transmission models.

The bounded cohomology of a group G with coefficients in a normed G-module V was first systematically studied by Gromov in 1982 in his seminal paper [Gro82] in connection to the minimal volume of manifolds. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for the most basic groups: There is no finitely generated group G for which the full bounded cohomology with real coefficients is known except where it is known to vanish in all degrees (see [Mon06]). In this thesis we discuss several new constructions for classses in bounded cohomology. There is a well-known interpretation of ordinary group cohomology in degrees 2 and 3 in terms of group extensions. We establish an analogous interpretation in the context of bounded cohomology. This involves certain maps between arbitrary groups called quasihomomorphisms, which were defined and studied by Fujiwara and Kapovich in [FK16]. A key open problem is to compute the full bounded cohomology of a non-abelian free group F with trivial real coefficients. It is known that the bounded cohomology in dimension n is trivial for n = 1 and infinite dimensional for n = 2, 3, but essentially nothing is known about for n ≥ 4. For n = 4, one may construct classes by taking the cup product between two 2-classes, but it is possible that all such cup-products are trivial. We show that all such cup-products do indeed vanish if both classes are induced by the quasimorphisms of Brooks or Rolli.

The aim of this research was to investigate how I engaged in constructivist teaching (CT) when helping a group of low-performing Grade 7 students to develop new meanings of notation as they started to learn formal algebra. Data was collected over a period of one scholastic year, in which I explored the teacher-student dynamics during my mathematics lessons, where students learnt new representations and interpretations of notation with the help of the computer software Grid Algebra. Analysing video recordings of my lessons, I observed myself continuously changing my teaching purpose as I negotiated between the mathematics I intended to teach and the mathematics being constructed by my students. These shifts of focus and purpose were used to develop a conceptual framework called Mathematics-Negotiation-Learner (M-N-L). Besides serving as a CT model, the M-N-L framework was found useful to determine the extent to which I managed to engage in CT during the lessons and also to identify moments where I lost my sensitivity to students constructions of knowledge. The effectiveness of my CT was investigated by focusing on students learning, for which reason I developed the analytical framework called CAPS (Concept-Action-Picture-Symbol). The CAPS framework helped me to analyse how students developed notions about properties of operational notation, the structure and order of operations in numerical and algebraic expressions, and the relational property of the equals sign. Grid Algebra was found to be a useful tool in helping students to enrich their repertoire of representations and to develop new interpretations of notation through what I defined as informal- and formal-algebraic activities. All students managed to transfer these representations and interpretations of notation to pen-and-paper problems, where they successfully worked out traditionally set substitution-and-evaluation tasks.

This thesis is primarily focused on problems in extremal combinatorics, although we will also consider some questions of analytic and algorithmic nature. The d-dimensional hypercube is the graph with vertex set {0,1}d where two vertices are adjacent if they differ in exactly one coordinate. In Chapter 2 we obtain an upper bound on the 'saturation number' of Qm in Qd. Specifically, we show that for m ≥ 2 fixed and d large there exists a subgraph G of Qd of bounded average degree such that G does not contain a copy of Qm but, for every G' such that G ⊊ G' ⊆ Qd, the graph G' contains a copy of Qm. This result answers a question of Johnson and Pinto and is best possible up to a factor of O(m). In Chapter 3, we show that there exists ε > 0 such that for all k and for n sufficiently large there is a collection of at most 2(1-ε)k subsets of [n] which does not contain a chain of length k+1 under inclusion and is maximal subject to this property. This disproves a conjecture of Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós. We also prove that there exists a constant c ∈ (0,1) such that the smallest such collection is of cardinality 2(1+o(1))ck for all k. In Chapter 4, we obtain an exact expression for the 'weak saturation number' of Qm in Qd. That is, we determine the minimum number of edges in a spanning subgraph G of Qd such that the edges of E(Qd)\E(G) can be added to G, one edge at a time, such that each new edge completes a copy of Qm. This answers another question of Johnson and Pinto. We also obtain a more general result for the weak saturation of 'axis aligned' copies of a multidimensional grid in a larger grid. In the r-neighbour bootstrap process, one begins with a set A0 of 'infected' vertices in a graph G and, at each step, a 'healthy' vertex becomes infected if it has at least r infected neighbours. If every vertex of G is eventually infected, then we say that A0 percolates. In Chapter 5, we apply ideas from weak saturation to prove that, for fixed r ≥ 2, every percolating set in Qd has cardinality at least (1+o(1))(d choose r-1)/r. This confirms a conjecture of Balogh and Bollobás and is asymptotically best possible. In addition, we determine the minimum cardinality exactly in the case r=3 (the minimum cardinality in the case r=2 was already known). In Chapter 6, we provide a framework for proving lower bounds on the number of comparable pairs in a subset S of a partially ordered set (poset) of prescribed size. We apply this framework to obtain an explicit bound of this type for the poset 𝒱(q,n) consisting of all subspaces of 𝔽qnordered by inclusion which is best possible when S is not too large. In Chapter 7, we apply the result from Chapter 6 along with the recently developed 'container method,' to obtain an upper bound on the number of antichains in 𝒱(q,n) and a bound on the size of the largest antichain in a p-random subset of 𝒱(q,n) which holds with high probability for p in a certain range. In Chapter 8, we construct a 'finitely forcible graphon' W for which there exists a sequence (εi)∞i=1 tending to zero such that, for all i ≥ 1, every weak εi-regular partition of W has at least exp(εi-2/25log∗εi-2) parts. This result shows that the structure of a finitely forcible graphon can be much more complex than was anticipated in a paper of Lovász and Szegedy. For positive integers p,q with p/q ❘≥ 2, a circular (p,q)-colouring of a graph G is a mapping V(G) → ℤp such that any two adjacent vertices are mapped to elements of ℤp at distance at least q from one another. The reconfiguration problem for circular colourings asks, given two (p,q)-colourings f and g of G, is it possible to transform f into g by recolouring one vertex at a time so that every intermediate mapping is a p,q-colouring? In Chapter 9, we show that this question can be answered in polynomial time for 2 ≤ p/q < 4 and is PSPACE-complete for p/q ≥ 4.

Literature review abstract Background: Domain-general processes, such as working memory (WM), short-term memory (STM), and attention, have been found to be related to mathematical performance in children. The relationship between these abilities, however, is not well understood. Objective: This systematic review aimed to evaluate the literature investigating the relationship between mathematical performance and WM, STM, and attention in typically developing primary school aged children. Methods: Three databases were searched for studies published between January 1974 and February 2015 reporting associations between mathematics performance and at least one measure of WM, STM, and attention. Study selection was undertaken by applying inclusion and exclusion criteria and 43 studies were selected for inclusion. The methodological quality of the included studies was assessed using a validated checklist. Results: WM, STM, and attention were all significantly related to mathematics performance. Visuospatial STM and WM were strongly related to mathematics performance in younger children, while verbal STM and WM were more strongly related in older children; although some studies found the opposite pattern. The relationship between attention and mathematics performance increased in strength with age. Conclusions: There are many factors relevant to the relationship between mathematical performance and WM, STM, and attention which can affect the strength of the association, including the types of tasks used to measure the constructs, the confounding variables considered, and the age of the participants. Future research needs to focus on the construction of an integrated model of mathematical development.   Empirical paper abstract Background: Moderate to late preterm children (MLPT; born between 32 weeks and 36 weeks and 6 days) are at increased risk of developing cognitive difficulties compared to term children (born between 37 weeks and 41 weeks and 6 days). Mathematical attainment is an important area of academic development. Domain-general cognitive abilities, which constrain all learning, and domain-specific mathematical precursors are both important for mathematical development. Objectives: The current study had two aims: 1) to investigate the relationship between gestational age (GA), mathematical attainment, working memory (WM), short-term memory (STM), and attentional control; and 2) to investigate WM, STM, and attentional control as domain-general predictors of mathematical attainment. It was hypothesised that WM would predict additional variance in mathematical attainment after attentional control, STM, and demographic variables (intellectual ability (IQ) and socioeconomic status) were accounted for. Methods: A cross-sectional and correlational design was used to investigate the study aims. Participants were 34 MLPT children and 25 term children who were between 72 and 107 months at the time of the study. Children who weighed less than 1500 grams at birth, had cerebral palsy, epilepsy, severe hearing or vision loss, or had a diagnosed learning disability were excluded. Each participant completed a cognitive assessment which measured their mathematical attainment and components of WM, STM, and attentional control. Results: GA was only significantly correlated with IQ. In the model of mathematical attainment, GA also significantly moderated the relationship between attentional switching and mathematical attainment. The hypothesis regarding the role of WM in predicting mathematical attainment was partially supported as only verbal WM predicted significant additional variance in mathematical attainment. Attention behaviour and IQ also predicted significant additional variance in mathematical attainment. Conclusion: These findings suggest that birth weight greater than 1500g, higher socioeconomic status, and lower levels of co-morbid medical conditions may serve as protective factors against the potential negative consequences of MLPT birth. Findings regarding the domain-general predictors of mathematical attainment supported some previous findings and highlighted the need for a variety of tasks to be used to measure each domain-general ability. Longitudinal studies in MLPT children would be helpful for further understanding the role of GA and domain-general abilities in the development of mathematical attainment.

This thesis contains four related papers which study different aspects of double constructions for braided Hopf algebras. The main result is a categorical action of a braided version of the Drinfeld center on a Heisenberg analogue, called the Hopf center. Moreover, an application of this action to the representation theory of rational Cherednik algebras is considered. Chapter 1 : In this chapter, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result of Lu (1994) that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras. Chapter 2 : In this chapter, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (2004) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2. Chapter 3 : The universal enveloping algebra U(trn) of a Lie algebra associated to the classical Yang-Baxter equation was introduced in 2006 by Bartholdi-Enriquez-Etingof-Rains where it was shown to be Koszul. This algebra appears as the An-1 case in a general class of braided Hopf algebras in work of Bazlov-Berenstein (2009) for any complex reection group. In this chapter, we show that the algebras corresponding to the series Bn and Dn, which are again universal enveloping algebras, are Koszul. This is done by constructing a PBW-basis for the quadratic dual. We further show how results of Bazlov-Berenstein can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra representations of different rank and type for the classical series of Coxeter groups. Chapter 4 : Quantum groups can be understood as braided Drinfeld doubles over the group algebra of a lattice. The main objects of this chapter are certain braided Drinfeld doubles over the Drinfeld double of an irreducible complex reflection group. We argue that these algebras are analogues of the Drinfeld-Jimbo quantum enveloping algebras in a setting relevant for rational Cherednik algebra. This analogy manifests itself in terms of categorical actions, related to the general Drinfeld-Heisenberg double picture developed in Chapter 2, using embeddings of Bazlov and Berenstein (2009). In particular, this work provides a class of quasitriangular Hopf algebras associated to any complex reflection group which are in some cases finite-dimensional.

This study applies statistical models to mood in patients with bipolar disorder. Three analyses of telemonitored mood data are reported, each corresponding to a journal paper by the author. The first analysis reveals that patients whose sleep varies in quality tend to return mood ratings more sporadically than those with less variable sleep quality. The second analysis finds that forecasting depression with weekly data is not feasible using weekly mood ratings. A third analysis shows that depression time series cannot be distinguished from their linear surrogates, and that nonlinear forecasting methods are no more accurate than linear methods in forecasting mood. An additional contribution is the development of a new k-nearest neighbour forecasting algorithm which is evaluated on the mood data and other time series. Further work is proposed on more frequently sampled data and on system identification. Finally, it is suggested that observational data should be combined with models of brain function, and that more work is needed on theoretical explanations for mental illnesses.

In 1965 J. Ginibre introduced an ensemble of random matrices with no symmetry conditions imposed as the mathematical counterpart to hermitian random matrix theory. In his original paper he treats the case of matrices with i.i.d. normally distributed real, complex or quaternion entries. Since then, mainly due to interest from applications, the development of non-hermitian random matrix theory has further evolved, though the eigenvalue statistics of non-hermitian random matrices are far from being as thoroughly understood as their hermitian counterpart. A characteristic of non-hermitian random matrices are eigenvalue distributions in the complex plane. Real asymmetric random matrices have the additional caveat of having real and complex eigenvalues and thus are technically more challenging. In the following work a new three-fold family of non-hermitian random matrices is introduced via a quadratization procedure. As a consequence the entries of these matrices are highly dependent. For all three ensembles the joint eigenvalue probability density functions and eigenvalue correlations are derived for β = 1, 2. In the limit of large matrix dimensions a classification of eigenvalue correlation functions for different asymptotic regimes is undertaken. In tune with the title of this work for all three ensembles there exists an asymptotic regime, in which the eigenvalues are supported on an annulus around the origin. Thus the induced family of non-hermitian random matrix ensembles serves as an example, for ensembles of the Feinberg-Zee type with logarithmic potential.

Noncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention. The theory of quantum groups provides a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been investigated as noncommutative complex spaces. In a series of papers, Heckenberger and Kolb showed that for each of these spaces, there exists a q-deformed Dolbeault double complex. In this thesis a comprehensive framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully at quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. It is shown that when applied to the quantum projective spaces, this theory reproduces the q-Dolbeault double complexes of Heckenberger and Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{ Weil theory, and to produce the beginnings of a theory of noncommutative Kahler geometry.

My supervisors Ian Chiswell and Thomas M¨uller have found a new class of groups of functions defined on intervals of the real line, with multiplication defined by analogy with multiplication in free groups. I have extended this idea to functions defined on a densely ordered abelian group. This doesn’t give rise to a class of groups straight away, but using the idea of exponentiation from a paper by Myasnikov, Remeslennikov and Serbin, I have formed another class of groups, in which each group contains a subgroup isomorphic to one of Chiswell and M¨uller’s groups. After the introduction, the second chapter defines the set that contains the group and describes the multiplication for elements within the set. In chapter three I define exponentiation, which leads on to chapter four, in which I describe how it is used to find my groups. Then in chapter five I describe the structure of the centralisers of certain elements within the groups.

In this thesis, we obtain new results for convolution operators on homogeneous spaces and give applications to the Laplacian on a homogeneous graph. Some of these results have been published in joint papers [13, 14] with my supervisor. Let be a homogeneous space of a locally compact group G and let T : Lp( ) ! Lp( ) be a convolution operator induced by a measure on G, where 1 p < 1. When is symmetric and absolutely continuous, we describe the L2- spectrum of T in terms of the Fourier transform of . An operator T is said to be hypercyclic if there is a vector x 2 Lp( ) such that the orbit fx; Tx; : : : ; Tnx; : : :g is dense in Lp( ). Given a positive weight w on , we consider the weighted convolution operator T ;w(f) = wT (f) on Lp( ) and study hypercyclic properties of T ;w. For a unit point mass , we show that T ;w is hypercyclic under some condition on the weight w. This condition is also necessary in the discrete case, and is equivalent to hereditary hypercyclicity of the operator. The condition can be strengthened to characterise topologically mixing weighted translation operators on discrete spaces. A weighted homogeneous graph is a homogeneous space of a discrete group G and the Laplacian L on can be viewed as a convolution operator. We can therefore apply the above result on L2-spectrum to describe the spectrum of L in terms of irreducible representations of G. We compare the eigenvalues of L with eigenvalues of the Laplacian on a regular tree, and obtain a Dirichlet eigenvalue comparison theorem. We also prove a version of the Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.

There have been many contributions of work in different fields of convexity giving various metrical properties of convex sets. In this thesis we shall consider some further ideas which seem interesting to study. A standard way of tackling certain types of problems is to prove the existence of an 'extremal' convex set with respect to the property in consideration and by a series of arguments determine its construction. Generally speaking the extremal set turns out to be regular in some sense with a correspondingly easy geometry. In Chapters 1 and 2 we shall concern ourselves entirely with polytopes and we shall give some results on the metric properties of their faces. Following these results, we shall in Chapter 3 consider some continuity properties of the more general class of cell-complexes. In Chapters 4, 5 and 6, we shall confine ourselves to the plane. In Chapter 4, we shall consider sets which incertain senses correspond to the sets of constant width. This leads us in Chapter 5 to give some results concerning the minimal widths of triangles circumscribing convex sets. Finally, in Chapter 6 we consider the areas of certain subsets of a convex set which are determined by partitions of that set by three concurrent lines. Papers which are relevant to the field of study in a particular chapter are mentioned briefly in an introduction to that chapter.

The object of this thesis is to solve, in integers X and Y, various equations of the form [equation] where d and N are given square free integers. The work stems from two papers by J.H.E. Cohn in which the equations [equation] are solved for certain values of d. It is well known that the solutions of X2-dY2=4, and those of X2-dY2 = -4 where such solutions exist, may be expressed in terms of the least positive solutions of these equations. Solutions of the equations [equations] may now be sought among those of x2-dY2 = +1,+4. Extensive work in this direction has been done by W. Ljunggren working in the quadratic field R(d2) and other allied algebraic fields. His methods are powerful but deep and complicated. It is possible to show that the solutions of the equations X2 for sequences which satisfy a three-term recurrence relation. By applying the elementary theory of quadratic residues to these equations Cohn has solved the equations and for those d for which either of the equations has solutions (X,Y) for which X and Y are both odd. This thesis extends Cohn's work, Using similar methods, to solve the equations and for the same values of d as above and any given integer N. A few limited results are given for other values of d. L.J. Mordell has given simple conditions under which the equation can have no solutions. A theorem of a similar type concerning the equations x4-dY2 = 1,4 is proved. Finally the results proved in this thesis are compared with those of Cohn, Ljunggren and Mordell.

The problem we consider here arises quite naturally from Crum's Problem which asks: What is the maximum number of non-overlapping polyhedra such that any pair of them have a common boundary of positive area? In (1) Besicovitch, by constructing a sequence of polyhedra satisfying the required conditions, showed that the answer to Crum's Problem is infinity. In this thesis we ask: What is the answer when the polyhedra of Crum's Problem are restricted to be tetrahedra? As stated in the Abstract, we prove that the answer is either 8 or 9 and the evidence tends to point to the answer in fact being 8. As the difference in answers would suggest, the methods we use to establish these results are completely different from those used by Besicovitch in his paper. In Chapter One we show that an n-con (for definition see the Abstract) can be represented by an n-Towed matrix whose minors satisfy certain conditional we then develop arguments from which we deduce that n is less than 18. Chapter Two shows that the bound may be reduced to n less than 14. The subsequent five chapters are mainly concerned with the conditions under which a 9-con can exist and we eventually show that if an n-con for n > 9 exists then no plane contains six faces of the tetrahedra of the n-con.5.Our analysis of the 9-con continues in Chapter Eight where we show that what one may describe as the most symmetrical case for a 9-con cannot exist and also that the faces of the tetrahedra of the 9-con must be so arranged that they are contained in either nine or ten planes. To demonstrate that the existence or not of a 9-con is critical, we show in Chapter Nine that a 10-con cannot exist and in Chapter Ten that a 8-con does exist. Chapter Eleven discusses the results obtained.

In Chapter I we begin by considering a theorem of J. Dieudonne on the minimum radius of starlikeness of a class of analytic functions. We give a simple new proof of this theorem. By this new proof also we find the minimum radius of univalence of this class and we determine all the cases which give the minimum radius of univalence and the minimum radius of starlikeness. We then use a method similar to that in this new proof to obtain the minimum radius of univalence and the minimum radius of starlikeness of some other classes of analytic functions. For each class we determine all the cases giving the minimum radius of univalence and the minimum radius of starlikeness. Then we give some similar results for the minimum radius of convexity. In Chapter II first we deal with Heawood's Lemma which was established and used by P.J. Heawood to prove the theorem known as the Grace-Heawood Theorem. The same lemma was used by S. Kakeya in the proof of another theorem. We show that Heawood's Lemma is false and we give new proofs of these results. Then for some special cases we improve the value of the radius of univalencegiven by Kakeya's Theorem. In this connection we firstgive L.N. Cakalov's result and then we obtain some improvements of his result. In Appendix I we give some examples related with Chapter I and Chapter II. In Appendix II we give an example which shows that there is an error in a paper by M. Robertson.