In this thesis, we seek to extend some results of group theory to a new structure in algebra, called partial groups. Initially, we will prove a number of basic results of partial groups, introducing the elementary concepts of partial groups as abelian, nilpotent, homomorphism partial groups and Coprime Action on partial groups and some other ideas. After that, we are going to prove some results of characteristic p members in partial groups. These results are two uniqueness theorems of characteristic p members and further uniqueness theorems in partial groups. The principle result of this work is an extension of the Solvable Signalizer Functor Theorem to partial groups.

We discuss two extremal problems in extremal graph theory. First we establish a precise characterisation of 4-uniform hypergraphs with minimum codegree close to n/2 which contain a Hamilton 2-cycle. As a corollary we determine the exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs, and we provide a polynomial-time algorithm which answers the corresponding decision problem for 4-graphs with minimum degree close to n/2. In contrast we also show that the corresponding decision problem for tight Hamilton cycles in dense k-graphs is NP-complete. Furthermore we study the following bootstrap percolation process: given a connected graph G, we infect an initial set A of vertices, and in each step a vertex v becomes infected if at least a p-proportion of its neighbours are infected. A set A which infects the whole graph is called a contagious set. Our main result states that for every pin (0,1] and for every connected graph G on n vertices the minimal size of a contagious set is less than 2pn or 1. This result is best-possible, but we provide a stronger bound in the case of graphs of girth at least five. Both proofs exploit the structure of a minimal counterexample.

In this pure mathematics thesis we study realizations of fusion systems on finite groups and determine minimal right characteristic bisets for infinite families of saturated fusion systems over 3-groups of maximal nilpotency class.

Brauer's induction theorem states that every irreducible character of a finite group G can be expressed as an integral linear combination of induced characters from elementary subgroups. The goal of this thesis is to develop our own induction theorem inspired by both Brauer's induction theorem and Global-Local conjectures. Specifically we replace the set of elementary subgroups of G by the set of subgroups of index divisible by the prime power divisors of the given character's degree. We aim to do this by using a reduction theorem to almost simple and quasisimple groups, using the Classification of Finite Simple Groups to deal with the remaining cases.

This thesis is an investigation into the modelling of compressible viscoelastic fluids. It can be divided into two parts: (i) the development of continuum models for compressible and nonisothermal viscoelastic fluids using the generalised bracket method and (ii) the numerical modelling of compressible viscoelastic flows using a stabilised finite element method. We introduce the generalised bracket method, a mathematical framework for deriving systems of transport equations for viscoelastic fluids based on an energy/entropy formulation. We then derive nonisothermal and compressible generalisations of the Oldroyd-B, Giesekus and FENE-P constitutive equations. The Mackay-Phillips (MP) class of dissipative models for Boger fluids is developed within the bracket framework, complimenting the class of phenomenological models that already exist in the literature. Advantages of the MP models are their generality and consistency with the laws of thermodynamics. A Taylor-Galerkin finite element scheme is used as a basis for numerical simulations of compressible and nonisothermal viscoelastic flow. Numerical predictions for four 2D benchmark problems: lid-driven cavity flow, natural convection, eccentric Taylor-Couette flow and axisymmetric flow past a sphere are presented. In each case numerical comparisons with both empirical and numerical data from the literature are presented and discussed. Numerical drag predictions for the FENE-P-MP model are presented, displaying good agreement with both numerical and experimental data for the drag behaviour of Boger fluids.

This thesis deals with two main research topics in the field of complex network science, including the identification of influential spreaders from the view of nodes and their community discovery from the view of groups of nodes. It aims at understanding the mechanism of a small number of initial nodes known as information sources leading to wide information propagation in social networks. The discovery of communities that these spreaders belong to is another key factor of identifying influential spreaders. Heterogeneous community detection is a vital step towards the understanding of network topological structures and propagation dynamics. These two questions are not isolated but have certain relationship in our study. In this thesis, we have made the following novel contributions. There are a small number of vital nodes which will lead to wide propagation if they are selected as spreaders at the very beginning. The previous method VoteRank is limited because some of the spreaders are clustered in one community. Thus in this thesis, a novel approach ComVote is proposed to uncover multiple influential spreaders. Further experimental study on real networks shows that in Susceptible-Infected-Recovered (SIR) propagation, ComVote outperforms existing methods including VoteRank, CI (Collective Influence), H-index, K-shell and etc. on the final affected scale in SIR. But it also has limitations in the case that the influential spreaders are not connected with each other. Furthermore, VoteRank is merely applied to unweighted networks. A new approach WVoteRank is proposed to find multiple spreaders on weighted networks. WVoteRank is generalised to deal with both unweighted and weighted networks by considering both degree and weight in the voting process. Experimental studies in the present research on synthetic and real networks show that in the context of Susceptible-Infected Recovered (SIR) propagation, WVoteRank outperforms existing methods such as weighted H-index, weighted K-shell, weighted degree centrality and weighted betweenness centrality on the final affected scale. The improvement of final affected scale is as much as 8.96%. In social networks, communities are the natural partition of groups of nodes of the underlying networks where nodes within the same group are closely connected while the edges between different groups are loosely connected. This thesis defines overlapping communities as belonging to such groups where one node can belong to more than one group. Chen et al. firstly proposed a community game (Game) to study this problem. In this thesis, we have investigated how similar vertices affect the formation of the community game. The Adamic-Adar Index (AA Index) has been employed to define the new utility function in chapter 4. This result implicates that 'friend circles of friends' of Facebook are valuable to understand overlapping community partition. Many real bipartite networks are naturally divided into two-mode communities including user-item bipartite communities in e-commerce, user-news bipartite communities in personalized news web sites and etc. In the present research, a two-mode community detection algorithm termed BiAttractor was formulated. It is based on distance dynamics model Attractor proposed by Shao et al. with extension from unipartite to bipartite networks. This new idea makes clear assumptions in linear time complexity O(|E|) in sparse networks, where |E| is the number of edges. Experiments in the present research on synthetic networks demonstrate it is capable to overcome resolution limit compared with other existing methods. Further study in the present research on real networks shows that BiAttractor has excellent accuracy compared with other existing methods and it is fast to deal with large-scale networks with millions of nodes and edges. Finally, the influential spreaders during the Malaysian General Election in 2013 (MGE2013) were investigated. It was found that representative of political party located at the center of the propagation network was the winner of the presidential election. It was also found that a number of non-politicians also significantly influenced the election because they located at the central area of the Twitter propagation sphere. This new point of view also supports the Jurgen Habermass ideal of the public sphere, especially in the General Election of Malaysia, a sphere that permits citizens to interact, and debate on the public issues without fear of political powers. In this thesis, influential spreaders are studied to uncover the mechanisms governing the information propagation on complex networks. Overlapping communities and two-mode communities on bipartite networks are investigated to suggest new methods to explain the formation of communities. Studies on influential spreaders and community partitions shed light on future study of patterns and dynamics of complex networks.

When students work productively together on a mathematical problem they are required to simultaneously manage their own efforts, while understanding what others are doing, and saying. Consequently, it is argued that monitoring and regulating the construction of meaning involves individual metacognitive processes, operating socially. This can take three forms: learners regulate their own cognitive processes; scaffold their partner's understandings by taking on the role of a tutor; or mutually regulate their joint understanding. Engaging in these ways when solving unstructured non-routine problems can substantially extend students' understanding and raise levels of achievement as measured by standard assessment criteria. Unfortunately, extant literature suggests that these processes of social metacognitive regulation (SMR) can remain elusive for many students. The aim of this study was to develop an intervention to support students in this endeavour. In so doing, local theory concerning the intervention was established. Theoretically guided classroom resources were created for teachers to use in UK secondary school mathematics classrooms. They were designed to provide opportunities for students to work with solutions to unstructured non-routine problems, authored by others. These solutions were either those constructed by peers or provided by the teacher - I designed the latter. These were worked-out solutions to problems, in the form of designed student responses (DSRs). The intention was that features such as their coherence, anonymity, and accuracy would engender a less demanding situation than when students jointly-solved problems. Rather than focusing on performance, students would attempt to comprehend and evaluate the DSRs. Integral to this undertaking was an opportunity for students to practise social metacognitive regulation (SMR). These developed practices could then be applied to the more challenging environment of constructing a joint solution to a problem. Working within a design research paradigm, three intervention studies were undertaken to establish the veracity of this claim and, simultaneously, develop theory. In each study, a structured sequence of activities was enacted in a mathematics classroom of 13 and 14 year old students. The scope of the sequence varied from study to study. Students tackled between one and seven unstructured problems and then worked with DSRs to each of the problems. Through theory-driven iterative cycles of design, implementation, and analysis, the intervention was refined. A rich range of instruments to measure SMR processes through a duel focus on what students did and said were also developed. Accordingly, the class's written work and the dialogue between one or two pairs of students as they worked together were analysed. In general, the results supported the efficacy of the intervention. They revealed that between the start and end of an intervention, students' problem solving improved. There was a modest shift from students producing concrete solutions towards better quality, more abstract ones. The shift, however, also coincided with an increased demand inherent within the problem. It was found that the way students formulated the problem and the strategy they employed was key to their success. The results from the studies also indicated that DSRs acted as a mediating tool to support students to practise SMR. Two important phenomena underpinned this finding. Firstly, compared to constructing a joint solution, when students worked together with DSRs, SMR manifested itself more frequently in their conversations. Students also generated a higher proportion of utterances that included reasoning. Secondly, DSRs that incorporated a visual strategy prompted better quality conversations than those that did not. When students worked with these DSRs, SMR episodes were characterised by extended interactions, in which contributions were acknowledged and built on rather than ignored. Furthermore, such DSRs provoked more student evaluations that considered specific mathematical criteria as opposed to non-specific, personal criteria. The research also considered the factors that determined when students produced a better joint solution than their initial, individual conceptions of a problem. The findings suggested that when there was little divergence in initial conceptions, students more frequently introduced a new, improved strategy into their joint solution. In these cases, students more effectively adopted the role of the tutor to support their partner's understanding. Students were less successful in the role when their initial conceptions were highly divergent. When students worked with the DSRs, however, they were consistently successful as tutor. Although the research on SMR is growing, there is a pronounced lacuna in relation to unstructured problems used in a classroom setting, and none with regard to DSRs. This study contributes to this research agenda. The findings support the notion that DSRS can act as a mediating tool for social metacognitive regulation (SMR). They also illuminate the circumstances under which students productively collaborate when solving a problem.

In the first part of the thesis, the problem of determining the optimal capacity expansion strategy for a firm operating within a random economic environment is studied. The underlying market uncertainty is modelled by means of a general one-dimensional positive diffusion with possible absorption at 0. The objective is to maximise a performance criterion that involves a general running payoff function and associates a cost with each capacity increase up to the first hitting time of 0, at which time the firm defaults. The resulting optimisation problem takes the form of a degenerate twodimensional singular stochastic control problem that is explicitly solved. The general results are further illustrated in the special cases in which market uncertainty is modelled by a Brownian motion with drift, a geometric Brownian motion or a square-root mean-reverting process such as the one in the CIR model. The second part of the thesis presents a study of mean-variance portfolio selection for asset prices modelled by Lévy processes under conic constraints on trading strategies. In this context, the combination of the price processes’ jumps and the trading constraints gives rise to a new qualitative behaviour of the optimal strategies. The existence and the behaviour of the optimal strategies are related to different no-arbitrage conditions that can be directly expressed in terms of the Lévy triplet. This allows for a fairly complete characterisation of mean-variance optimal portfolios under conic constraints.

The Baum-Connes conjecture gives a description of the K-theory of the reduced group C*-algebra of a locally compact second countable group. In the case of a connected Lie group G, Connes reformulated the conjecture in terms of a deformation of G provided by a certain continuous field of C*-algebras. The conjecture is known to be true for complex semisimple Lie groups, and in 2008 Higson provided a new proof of this result, using Connes reformulation and an observation due to Mackey about the representation theories of a complex semisimple Lie group and an associated group called the Cartan motion group. In this thesis, we present and prove an analogue of the conjecture for the quantum group quantum SL(2, C) in the spirit of Connes reformulation and Higson's proof. In particular, we define a quantum version of Connes' field, which provides a deformation from quantum SL(2, C) to a quantum analogue of the Cartan motion group. We show that Mackey's observation carries over to the quantum setting, and we then prove an analogue of the conjecture using Higson's method. We also show there is compatibility between the Baum-Connes conjecture for SL(2, C) and our quantum result, in that we can construct a continuous field which encodes Connes' field and our quantum field, as well as a deformation of SL(2, C) to quantum SL(2, C) and a deformation of the Cartan motion group to the quantum Cartan motion group.

This thesis explores the relationship between Khovanov homology and strongly invertible knots through the use of a geometric construction due to Sakuma. On the one hand, new homological and polynomial invariants of strongly invertible knots are extracted from Sakuma's construction, all of which are related to Khovanov homology. Conversely, these invariants are used to study the two-component links and annular knots obtained from Sakuma's construction, the latter of which are almost entirely disjoint from the class of braid closures. Applications include the problem of unknot detection in the strongly invertible setting, the efficiency of an invariant when compared with the η-polynomial of Kojima and Yamasaki, and the use of polynomial invariants to bound the size of the intrinsic symmetry group of a two-component Sakuma link. We also define a new quantity, κA, and conjecture that it is an invariant of strongly invertible knots.