In this thesis we consider isomorphisms amongst certain classes of cyclically presented groups. We give isomorphism theorems for two families of cyclically presented groups, the groups G_n(h, k, p, q,r,s, l), and the groups G^ε_n(m, k, h), which were introduced by Cavicchioli, Repovs and Spaggiari. These families contain many subfamilies of cyclically presented groups, we have results for two of them, the groups G_n(m, k), which were introduced by Johnson and Mawdesley, and the groups Γ_n(k, l), which were introduced by Cavicchioli, Repovs and Spaggiari. The abelianization of the Fibonacci groups F(2, n) was proved by Lyndon to be finite and its order can be expressed in terms of the Lucas numbers. Bardakov and Vesnin have asked if there is a formula for the order of the abelianization of G_n(m, k) groups that can be expressed in terms of Fibonacci numbers. We produce formulas that compute the order of G_{pm}(x_0x_mx^{±1}_k)^{ab} , G_{pk}(x_0x_mx^{±1}_k)^ab for certain values of p where m, k are coprime, and for the groups Γ_n(1,n/2− 1)^{ab} (this formula is given in terms of Lucas numbers). The values of the number of non-isomorphic G_n(m, k) groups was conjectured by Cavicchioli, O’Brien and Spaggiari for n = p^l, where p is prime and l is a positive integer, we show that these values provide an upper bound for the number of non-isomorphic G_n(m, k) groups. We also give lower bounds and upper bounds for the number of non- isomorphic G_n(m, k) and Γ_n(k, l) groups for certain values of n. Similar to the investigation of the type of isomorphisms of G_n(m, k) groups for n ≤ 27 that was carried by Cavicchioli, O’Brien and Spaggiari, we perform a similar investigation for Γn(k, l) groups for n ≤ 29.

This dissertion explores the relation between two formalisms and one algebraic framework for concurrency. Session Types and Concurrent Separation Logic are formalisms that support independent reasoning about concurrent processes, and our motivating question is whether their modularity springs from the same source despite the distance between their models. We first translate a small language we call Baby Session Types (BST), into a ‘basic’ version of Concurrent Separation Logic (BCSL), and we show that the translation is sound. We then describe a model for Separation Logic (SL) based on Actions, which exhibits some of the structure of a Concurrent Kleene Algebra, an algebra where operators for parallel and sequential composition are linked by a version of the exchange law from category theory. The model connects the algebraic notions to locality concepts that underlie Separation Logic. We then move on to provide a more general construction of an algebra model of BCSL, which can be built from (Baby) Session Types. Thus, we end up with a model that brings together concepts from all of Session Types, Separation Logic, and Concurrent Kleene Algebra. Thus, the model links diverse models of concurrency. In addition to this it suggests alterations of the algebraic axioms as well as the foundational models underlying Separation Logic. It is hoped that, apart from these specific results, this dissertation can in some modest way contribute to unification in concurrency theory, a theory (or theories) based presently on diverse models.

This dissertation concerns the computational interpretation of analysis via proof interpretations, and examines the variants of bar recursion that have been used to interpret the axiom of choice. It consists of an applied and a theoretical component. The applied part contains a series of case studies which address the issue of understanding the meaning and behaviour of bar recursive programs extracted from proofs in analysis. Taking as a starting point recent work of Escardo and Oliva on the product of selection functions, solutions to Godel's functional interpretation of several well known theorems of mathematics are given, and the semantics of the extracted programs described. In particular, new game-theoretic computational interpretations are found for weak Konig's lemma for 01 -trees and for the minimal-bad-sequence argument. On the theoretical side several new definability results which relate various modes of bar recursion are established. First, a hierarchy of fragments of system T based on finite bar recursion are defined, and it is shown that these fragments are in one-to-one correspondence with the usual fragments based on primitive recursion. Secondly, it is shown that the so called `special' variant of Spector's bar recursion actually defines the general one. Finally, it is proved that modified bar recursion (in the form of the implicitly controlled product of selection functions), open recursion, update recursion and the Berardi-Bezem- Coquand realizer for countable choice are all primitive recursively equivalent in the model of continuous functionals.

Bertrand Russell's works on logic, despite his reputation as a founder of mathematical logic, appear unnecessarily metaphysical and even naïve to contemporary logicians and philosophers. He offered several accounts of logic whilst pursuing the goal of logicism, the view of mathematics as reducible to logic. In their attempts to explain why those accounts look naïve nowadays, many commentators have sought one or another simple philosophical doctrine which can characterise his conception of logic. Instead of thus assuming a coherent theme underlying his works on logic, I propose to understand them as a shift from a conception of logic towards another. By looking into books, papers and manuscripts which he wrote during the period from 1898 to 1918, I argue that he inherited an antique, metaphysical conception of logic from his idealist predecessors and, through his attempts to replace some idealistic features of the conception with his realist alternatives, he became more sympathetic to—though never fully convinced of—a linguistic conception of logic, which was proposed by some of his contemporary logicians and has been widely accepted since then.

Fuzzy logic is a frequently used solution to control problems, especially when there are elements of human knowledge that may be incorporated into the system. Fuzzy logic comes in several varieties with the most common being based on either type-1 or type-2 fuzzy logic. Modifications to these standard varieties, termed Non-Stationary (NS) and Dual Surface (DS) are also investigated. Each variety allows a certain amount of flexibility in its expression. However, with this increased flexibility (and potentially performance) comes additional resource requirements: either during run time with higher processing and memory requirements; or at design time, with additional parameters requiring selection and optimisation. There have been several comparisons into the performance obtained from type-1 and type-2 investigating such factors as their internal configuration (such as membership functions as defined by their Footprint of Uncertainty), task difficulty and the environment in which the experiments are performed. However, no studies have been performed incorporating each of these factors with the goal of determining how they impact upon performance. The end goal of this work is the development of a methodology to understand which combination of conditions will cause type-2 control to consistently outperform type-1 based systems. This would enable the rationalisation of moving from a type-1 to a type-2 system, which is currently done without understanding if and how performance will increase with such a move. This thesis introduces a novel scheme by which several methods of comparing performance are employed to observe how the output and resulting performance levels change as factors including: controller configuration, task difficulty and environmental variability are varied. These methods are performed over three applications which gradually increase in complexity: a simple tipping example, a more developed simulation based on an autonomous sailing robots application and subsequent real-world experiments, which also involve the autonomous sailing problem. The first method of comparison studies how the rules which fire for a given input set change as the configuration of the fuzzy logic controller is increased. The second comparative technique investigates the control surfaces produced by a selection of fuzzy logic controllers to observe how they change as the internal configuration is changed. Observations such as the smoothing of the transitions between surfaces suggest that controllers with a larger FOU may give a better response. The third method for comparison is developed in which outputs from a controller operating in a simulated environment are compared to an ideal value, giving a single numeric output with which comparisons can be made. It was found that there are situations in which type-2 based fuzzy control outperforms type-1. However, these are found to be less common than expected. It is determined that this may be due to the simplicity of some of our case studies environments (especially the tipping example), where there may not be enough scope for large improvements to become apparent. These findings lay ground for future work in which (i) more developed and complex applications and (ii) a more tuned fuzzy system should be investigated to find if this will result in more obvious differences between configurations.

There exist well-known varieties of implication, such as strict, intuitionist, three-valued and rigorous, which are non-classical in the sense of being more restrictive than material implication. But there exists also a type of implication, intuitively plausible, which is nonclassical not only in being more restrictive, but in satisfying certain theses which are classically false. These theses are exceedingly venerable, dating back to Aristotle and Boethius, but, despite their plausibility, have been generally rejected by logicians since. It has not been noticed, however, that in Sextus Empiricus reference is made to a species of Stoic implication which fits them perfectly. In this work formal recognition is given to this species of implication, known as connexive implication. It is shown that none of the well-known systems of prepositional logic is connexive, and a new system is accordingly constructed. A proof of consistency is given, and a number of problems posed for further investigation.

In this thesis, we study Parisian excursions, which are defined as excursions of Brownian motion above or below a pre-determined barrier, exceeding a certain time length. Employing a new method, a recursion formula for the densities of single barrier and double barrier Parisian stopping times are computed. This new approach allows us to obtain a semi-closed form solution for the density of the one-sided stopping times, and does not require any numerical inversions of Laplace transforms. Further, it is backed by an intuitive argument which is premised on the recursive nature of the excursions and the strong Markov property of the Brownian motion. The same method is also employed in our computation of the two-sided and the double barrier Parisian stopping times. In turn, the resultant densities are used to price Parisian options. In particular, we provide numerical expressions for down-and-in Parisian calls. Additionally, we study the tail of the distribution of the two-sided Parisian stopping time. Based on the asymptotic properties of its distribution, we propose an approximation for the option prices, alleviating the heavy computational load arising from the recursions. Finally, we use the infinitesimal generator to obtain several results on other variations of Parisian excursions. Specifically, apart from the length, we are interested in the number of excursions and the maximum height achieved during an excursion. Using the same generator, we derive the joint Laplace transform of the occupation times of the Brownian motion above and below zero, but only starting the clock each time after a certain length.

On a regular semigroup S natural order relations have been defined by Nambooripad and by Lallement. Different characterisations and relationships between the Nambooripad order J, Lallement's order λ and a certain relation k are considered in Chapter I. It is shown that on a regular semigroup S the partial order J is left compatible if and only if S is locally R-unipotent. This condition in the case where S is orthodox is equivalent to saying that E(S) is a left seminormal band. It is also proved that λ is the least compatible partial order contained in J and that k = λ if and only if k is compatible and k if and only if J is compatible. A description of λ and J in the semigroups T(X) and PT(X) is presented. In Chapter II, it is proved that in an orthodox semigroup S the band of idempotents E(S) is left quasinormal if and only if there exists a local isomorphism from S onto an R-unipotent semigroup. It is shown that there exists a least R-unipotent congruence on any orthodox semigroup, generated by a certain left compatible equivalence R. This equivalence is a congruence if and only if E(S) is a right semiregular band. The last Chapter is particularly concerned with the description of R-unipotent congruences on a regular semigroup S by means of their kernels and traces. The lattice RC(S) of all R-unipotent congruences on a regular semigroup S is studied. A congruence≡ on the lattice RC(S) is considered and the greatest and the least element of each ≡-class are described.

My thesis is an analysis of classical problems in perceptual cognition as they appear in Descartes' mechanical philosophy. My primary focus will be on animals, as well as on the models and metaphors that Descartes used to explain how sense perception, information processing, self-regulation, and self-determination occur in natural automata. His models and metaphors typically include man-made devices of his age and a variety of natural processes taken from the inanimate part of nature, which will also be an integral part of my discussion. Throughout the analysis, I will approach these issues from the vantage point of the notion of physiological autonomy, a concept I develop to show how the inner mechanisms of organic bodies contribute to their autonomous functioning in the physical world in Descartes' conception. This is an important task because it allows us to have a better understanding of the mechanical approach to the living in the early modern period, but also because the approach I adopt here highlights the shortcomings of existing literature on the bête-machine theory which most often fail to appreciate Descartes' efforts to imagine a working cognitive system inside non-human living creatures. Even those commentators who direct their attention to Descartes' views about animals emphasise the limitations of natural automata resulting from what they are not, i.e. they are not mind-body unions as humans, whereas I shall maintain that if we understand correctly what the machinery of the body is capable of, we will understand better what Descartes has to say about human cognition as well, in particular, what he believes the body contributes to the cognitive economy of embodied minds.

Cylindrical algebraic decomposition is a powerful algorithmic technique in semi-algebraic geometry. Nevertheless, there is a disparity between what algorithms output and what the abstract definition of a cylindrical algebraic decomposition allows. Some work has been done in trying to understand what the ideal class of cylindrical algebraic decom- positions should be — especially from a topological point of view. We prove a special case of a conjecture proposed by Lazard in [22]; the conjecture relates a special class of cylindrical algebraic decompositions to regular cell complexes. Moreover, we study the properties that define this special class of cell decompositions, as well as their implications for the actual topology of the cells that make up the cell decompositions.