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Let G be a group, and let b G be a group isomorphic to G. The constructive recognition problem for G with respect to b G is to find an isomorphism Á from G to b G such that the images under Á and Á¡1 of arbitrary elements may be computed efficiently. If the representation of b G is well-understood, then the representation of G becomes likewise by means of the action of Á. The problem is of foundational importance to the computational matrix group project in its ambitious desire to find an algorithmto construct a composition series for an arbitrarymatrix group over a finite field. This requires algorithms for the constructive recognition of all finite simple groups, which exist in the literature in varying degrees of practicality. Those for the exceptional groups of Lie type admit of improvement, and it is with these that we concern ourselves. Kantor and Magaard in [31] give Monte Carlo algorithms for the constructive recognition of black-box (i.e. opaque-representation) exceptional groups other than 2F4(22nÅ1). These run in time exponential in the length of the input at several stages. We specialise to the case of F4(q) for odd q, and in so doing develop a polynomial-time alternative to the preprocessing stage of the Kantor–Magaard algorithm; we then modify the procedure for the computation of images under the recognising isomorphisms to reduce this to polynomial running time also. We provide a prototype of an implementation of the resulting algorithm in MAGMA [10]. Fundamental to our method is the construction of involution centralisers using Bray’s algorithm [11]. Our work is complementary to that of Liebeck and O’Brien [40] which also uses involution centralisers; we make a comparison of the two approaches.

This thesis is divided into two parts. In the first of these we consider Ackermann-type set theories and many of our results concern natural models. We prove a number of results about the existence of natural models of Ackermann's set theory, A, and applications of this work are shown to answer several questions raised by Reinhardt in [56]. A⁺ (introduced in [56]) is another Ackermann-type set theory and we show that its set theoretic part is precisely ZF. Then we introduce the notion of natural models of A⁺ and show how our results on natural models of A extend to these models. There are a number of results about other Ackermann-type set theories and some of the work which was already known for ZF is extended to A. This includes permutation models, which are shown to answer another of Reinhardt's questions. In the second part we consider the different approaches to set theory; dealing mainly with the more philosophical aspects. We reconsider Cantor's work, suggest that it has frequently been misunderstood and indicate how quasi-constructive set theories seem to use a definite part of Cantor's earlier ideas. Other approaches to set theory are also considered and criticised. The section on NF includes some more technical observations on ordered pairs. There is also an appendix, in which we outline some results on extended ordinal arithmetic.

This thesis is concerned with models m of ZF that admit automorphisms of order greater than 1. We obtain such models using Boolean-valued models. Starting with a fixed o-non-standard countable m, and considering the algebra B epsilon M whose universe is B = RO (X^I) (X,I epsilon M), we construct a normal filter ^Gamma of subgroups of a group of automorphisms of Aut(B ), the ^Gamma-stable subalgebra B^Gamma of B, an automorphism of the replica B^Gamma and B^Gamma and, an ultrafilter U that in a natural sense is generic in B^Gamma, so that pi induces an automorphism of m^Gamma/U. Part of the construction is quite general and applies to any B = RO(X^I). (Chapters I-IV.) In Chapter I, by simulating the construction of B = RO(X^I) outside the model, we obtain a Boolean-algebra that is isomorphic to B. In Chapter II we list some known connections between generic ultrafilters and models of ZF which hold when m is non-standard and B is replaced by B. We introduce the concept of m-standardness. In Chapter III the concepts of 'extendability', of 'almost- genericity' and of 'locally-expressible' permutations and automorphisms are introduced. A generalised version of the "x&circ;'s" : x&circ;_b = {<&ycirc;_b,b>: y epsilon x} is given (x epsilon M, b epsilon B). Some of their properties are examined. It is shown that the condition pi[U] = U (*) is necessary and sufficient in order to induce automorphisms in m^Gamma/U, and that extendability constitutes a sufficient condition in order to obtain pi, U satisfying (*). Such pi,U are constructed simultaneously. In Chapter IV we construct automorphisms of two symmetric Boolean-valued submodels of m^B via locally expressible permutations pi (epsilon M) of the extension of I. If pi is locally-expressible, formulae of the form &phis; (pix,...,piX_n), (X₁,...,X_n epsilon M, piM, pi epsilon M, can be considered as formulae of the language of M. In chapter V, we consider the m^Gamma's introduced previously with B=RO^{(2oxox(kappa+1)}) kappa an o-non-standard number in m. Results from earlier chapters lead in each case to automorphisms pi of m^Gamma and generic ultrafilters U, so that pi induces an automorphism of m^Gamma/U.

The topics of this thesis are properties that distinguish between the 2²^X^o isomorphism-classes (called types) of non-principal ultrafilters on o. In particular we investigate various orders on ultrafilters. The Rudin-Frolik order is a topologically invariant order on types; it had been shown that there are types with 2^X^o predecessors in this order, and that, assuming the C.H., for every there are types with n predecessors. We shew that, assuming the C.H., there is a type with X_o predecessors. The next two main results can be phrased in terms of the minimal elements of these orders. Both assume the C.H. We find an ultrafilter that is a p-point (minimal in M.E.Rudin's "essentially greater than" order) that is not above any Ramsey ultrafilter (minimal in the Rudin-Keisler order). We also find an ultrafilter minimal in Blass' "initial segment" order that is not a p-point. These ultrafilters generate ultrapowers with interesting model-theoretic properties. We then investigate the classification of ultrafilters when the C.H. is no longer assumed. We find various properties of ultrafilters, sometimes by assuming some substitute for the C.H. such as Martin's Axiom, and sometimes without assuming any additional axiom of set-theory at all. Finally we relate the structure of ultrapowers to the existence of special sorts of ultrafilters.

The concept of ranking fuzzy numbers has received significant attention from the research community due to its successful applications for decision making. It complements the decision maker exercise their subjective judgments under situations that are vague, imprecise, ambiguous and uncertain in nature. The literature on ranking fuzzy numbers show that numerous ranking methods for fuzzy numbers are established where all of them aim to correctly rank all sets of fuzzy numbers that mimic real decision situations such that the ranking results are consistent with human intuition. Nevertheless, fuzzy numbers are not easy to rank as they are represented by possibility distribution, which indicates that they possibly overlap with each other, having different shapes and being distinctive in nature. Most established ranking methods are capable to rank fuzzy numbers with correct ranking order such that the results are consistent with human intuition but there are certain circumstances where the ranking methods are particularly limited in ranking non – normal fuzzy numbers, non – overlapping fuzzy numbers and fuzzy numbers of different spreads. As overcoming these limitations is important, this study develops an intuition based decision methodology for ranking fuzzy numbers using centroid point and spread approaches. The methodology consists of ranking method for type – I fuzzy numbers, type – II fuzzy numbers and Z – numbers where all of them are theoretically and empirically validated. Theoretical validation highlights the capability of the ranking methodology to satisfy all established theoretical properties of ranking fuzzy quantities. On contrary, the empirical validation examines consistency and efficiency of the ranking methodology on ranking fuzzy numbers correctly such that the results are consistent with human intuition and can rank more than two fuzzy numbers simultaneously. Results obtained in this study justify that the ranking methodology not only fulfills all established theoretical properties but also ranks consistently and efficiently the fuzzy numbers. The ranking methodology is implemented to three related established case studies found in the literature of fuzzy sets where the methodology produces consistent and efficient results on all case studies examined. Therefore, based on evidence illustrated in this study, the ranking methodology serves as a generic decision making procedure, especially when fuzzy numbers are involved in the decision process.

We have mentioned that each chapter in this thesis is conceived of as an independent paper, except for Chapter 3, which is a collection of results on non- continuous functions. Consequently each chapter contains a clearly marked introductory section, in which its back- ground and content are explained. In this abstract we shall summarize the remarks in these introductory sections. In chapter 1 we present an n-arc theorem for Peano spaces which is an extension of the theorem in §2 of [32], which Menger called the second n-arc theorem in [17]. Whereas in the second n-arc theorem n disjoint arcs are constructed joining two disjoint closed sets A and B, in chapter 1 we split the closed set A into n dis- joint closed subsets A1 A 2, ••• , An and give necessary and sufficient conditions for there to be n disjoint arcs joining A and B, one meeting each A1. At the end of chapter 1 we present a conjecture, which we have been able to verify in special cases. In [35] Whyburn proved a theorem concerning the weak connected separation of two non-degenerate connected closed sets A and B by a quasi-closed set L in a locally cohesive space X. In chapter 2 we show that A and B can in fact be taken as arbitrary closed sets in this theorem; that is, ,we remove the restriction of non-degeneracy and connectedness on A and B. In chapter 3 we study the circumstances under which a connectivity function is peripherally continuous. The study of the abstract relations between non- continuous functions was initiated by Stallings in [23]. In this paper he introduced the 1pc polyhedron and showed that a connectivity function was peripherally continuous on an 1pc polyhedron. Whyburn took up the study of non- continuous functions in [33]. [34] and [35]. He introduced the locally cohesive space, which is more general than the 1pc polyhedron, and proved that a connectivity function was peripherally continuous on a locally cohesive Peano space. For technical reasons, the locally cohesive space is not permitted to have local cut points. It is obvious, however, that on many Peano spaces having local, cut points a connectivity function remains peripherally continuous, In §2,3 of chapter 3 we formulate a sequence of properties Pn(X), which permit the space X to have local cut points, and we prove in each case that a connectivity function f : X →Y is peripherally continuous when X has property Pn(X). Each of these properties is an improvement on the last, and the final one, the U-space, satisfactorily incorporates the class of Peano spaces with local cut points on which we are able to prove that a connectivity function is peripherally continuous. An interesting feature of §3 of chapter 3 is provided by two "weak separation theorems," and more will be found about these in the introduction to chapter 3. In §4 of chapter 3 we show that a connectivity function is peripherally continuous on a locally compact ANR. This affirmatively answers a question that Stallings raised in [23]. The U-space that we have introduced in §3 of chapter 3 imposes a "unicoherence condition" in the space X (as do all the properties Pn(X) considered in §3, chapter 3). In §5 of chapter 3 we generalize the U-space to the S-space. This imposes a "multicoherence condition" on the space X, and we prove that a connectivity function is peripherally continuous on a cyclic S-space. We close chapter 3 by considering the question of placing weaker conditions than connectivity on the function f : X → Y which will still ensure that f is peripherally continuous. It is well known that if X is a unicoherent Peano continuum and A1, A 2, … is a sequence of disjoint closed subsets of X no one of which separates X, then Un=1 An does not separate X. In [28] van Est proved this theorem for the case where X is a Euclidean space of n dimensions. In chapter 4 we give an example which shows that this theorem does not hold if X is an arbitrary Peano space • In chapter 5 we provide a new angle to Lebesgue's covering lemma. We show that if the Lebesgue number ᵹ of an open covering U1, U2, •••• Un of a compact metric space X. ρ is finite. then it can be defined by the formula ᵹ = min ρ (E, F), where E and F are any compartments contained in no common U1 In chapter 6 we show that an involution on a cyclic Peano space leaves some simple closed curve setwise invariant. Whyburn has given a proof of R. L. Moore’s decomposition theorem for the 2-sphere in [31] (a refinement of this proof is presented in [36]). His proof is accomplished by showing that the decomposition space satisfies Zippin’s characterization theorem for the 2-sphere. In chapter 6 we present an alternative way of showing that the decomposition space satisfies Zippin's characterization theorem. Our proof closely follows Alexander's proof of the Jordan curve theorem as given by Newman in[21], and so consists of arguments that are well-known in another context. In [30] Whyburn gave a proof of the cyclic connectivity theorem. and in all subsequent appearances of this theorem in the literature Whyburn's proof has been used. Whyburn divided the proof of the theorem into three parts lemma 1, lemma 2, and the deduction of the theorem from lemmas 1 and2. In chapter 8 we give an alternative proof of lemma 1. Our proof is based on the fact that a cyclic Peano space has a base of regions whose closures do not separate the space, and it proceeds by an induction on a simple chain of these regions.

In the first part of this research we find an improvement to Huxley and Konyagin's current lower bound for the number of circles passing through five integer points. The improved lower bound is the conjectured asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem. In the second part of the research we consider questions linked to the distribution of different configurations of integer points of the circle passing through the unit square. We show that different configurations of points are distributed uniformly throughout the unit square for circles of fixed radius. Results are obtained by looking at the distribution of the crossing points of circles, where the circles form the boundaries of domains. The domain of a configuration is the set of possible positions of the centre of the circle within the configuration. We choose a rectangle within the unit square and then count the number of regions of the rectangle which are formed by domain boundaries.

With the advent of microarray technology, it is possible to monitor gene expression of tens of thousands of genes in parallel. In order to gain useful biological knowledge, it is necessary to study the data and identify the underlying patterns, which challenges the conventional mathematical models. Clustering has been extensively used for gene expression data analysis to detect groups of related genes. The assumption in clustering gene expression data is that co-expression indicates co-regulation, thus clustering should identify genes that share similar functions. Microarray data contains plenty of uncertain and imprecise information. Fuzzy c-means (FCM) is an efficient model to deal with this type of data. However, it treats samples equally and cannot differentiate noise and meaningful data. In this thesis, motivated by the preservation of local structure, a local weighted FCM is proposed which concentrate on the samples in neighborhood. Experiments show that the proposed method is not only robust to the noise, but also identifies clusters with biological significance. Due to FCM is sensitive to the initialization and the choice of parameters, clustering result lacks stability and biological interpretability. In this thesis, a new clustering approach is proposed, which computes genes similarity in kernel space. It not only finds nonlinear relationship between gene expression profiles, but also identifies arbitrary shape of clusters. In addition, an initialization scheme is presented based on Parzen density estimation. The objective function is modified by adding a new weighted parameter, which accentuates the samples in high density areas. Furthermore, a parameters selection algorithm is incorporated with the proposed approach which can automatically find the optimal values for the parameters in the clustering process. Experiments on synthetic data and real gene expression data show that the proposed method substantially outperforms conventional models in term of stability and biological significance. Time series gene expression is a special kind of microarray data. FCM rarely consider the characteristics of the time series. In this work, a fuzzy clustering approach (FCMS) is proposed by using splines to smooth time-series expression profiles to minimize the noise and random variation, by which the general trend of expression can be identified. In addition, FCMS introduces a new geometry term of radius of curvature to capture the trend information between splines. Results demonstrate that the new method has substantial advantages over FCM for time-series expression data.

We study the dynamics of consensus formation in a finite two-state voter model in which each agent has a confidence in their current opinion, k. The evolution of the distribution of k-values and the opinion change rules are coupled together to allow the opinion dynamics to dynamically develop heterogeneity of agent states in a simple way. We introduce two models. In both models pairs of agents with different opinions interact, which means k-values are compared and, with probability p, the agent with the lower value adopts the spin of the one with the higher value. In our nonconserved confidence model the agent with the higher k-value increments their confidence by one. In our conserved confidence model, additionally, the agent changing opinion reduces their k-value by one, so total confidence is conserved. The only parameter in both models is the probability p. We study the nonconserved model on the complete graph and compare the consensus time with the case p = 1/2 in which the opinion dynamics are decoupled from the k-values and are equivalent to standard voter model dynamics. When 1/2 < p < 1, agents with higher k{values are more persuasive and the consensus time is increased relative to the standard voter model although it still scales linearly with the number of agents, N. When p = 1, the consensus time scales as Nα with α = 1.4. When 0 < p < 1/2, agents with higher k-values are less persuasive and the consensus time is greatly decreased relative to the standard voter model and appears to be logarithmic in N. We provide some partial explanations for these observations using a mean-field model of the dynamics and a low-dimensional heuristic model which tracks only the sizes and mean k-values of each group. We also study the conserved model on the complete graph. When 1/2 < p < 1 this model also has consensus time that scales linearly with N and when p = 1 it scales as Nα with α = 1.4. However when 0 < p < 1/2 this conserved confidence model does not behave in the same way as the nonconserved model and the consensus time scales linearly with N. We compare the mean-field model dynamics with those of the nonconserved confidence model to partially explain model behaviour differences. We find consensus times for the nonconserved confidence model on low dimension regular lattices and compare with the complete graph. When 1/2 < p < 1, the consensus for the model on a 2d lattice is slower than the fully connected model, but in 3d results suggest that it is comparable. When p = 1, the population is prevented from reaching consensus by stable locally coordinated confidence arrangements. When 0 < p < 1/2, the consensus time is slow compared to the fully connected model and we notice spatial structures in the simulations. Also we implement a modification to the Axelrod model to introduce heterogeneity in the opinion space. We implement a bias in which opinions (features) are updated and separately a bias in the weight of influence of opinions on whether interaction occurs. Despite affecting the dynamics of the opinions and the time to absorbing state, we find the state, consensus or coexistence, the model typically reaches is robust under the effect of these two types of heterogeneity.