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A long standing unanswered question in number theory concerns the existence (or not) of odd perfect numbers. Over time many properties of an odd perfect number have been established and refined. The initial goal of this research was to improve the lower bound on the number of distinct prime factors of an odd perfect number, if one exists, to at least 9. Previous approaches to this problem involved the analysis of a carefully chosen set of special cases with each then being eliminated by a contradiction. This thesis applies an algorithmic, factor chain approach to the problem. The implementation of such an approach as a computer program allows the speed, accuracy and flexibility of modern computer technology to not only assist but even direct the discovery process. In addition to considering odd perfect numbers, several related problems were investigated, concerned with (i) harmonic, (ii) even multiperfect and (iii) odd triperfect numbers. The aim in these cases was to demonstrate the correctness and versatility of the computer code and to fine tune its efficiency while seeking improved properties of these types of numbers. As a result of this work, significant improvements have been made to the understanding of harmonic numbers. The introduction of harmonic seeds, coupled with a straightforward procedure for generating most harmonic numbers below a chosen bound, expands the opportunities for further investigations of harmonic numbers and in particular allowed the determination of all harmonic numbers below 10 to the power 12 and a proof that there are no odd harmonic numbers below 10 to the power 15. When considering even multiperfect numbers, a search procedure was implemented to find the first 10-perfect number as well as several other new ones. As a fresh alternative to the factor chain search, a 0-1 linear programming model was constructed and used to show that all multiperfect numbers divisible by 2 to the power of a, for a being less than or equal to 65, subject to a modest constraint, are known in the literature. Odd triperfect numbers (if they exist) have properties which are similar to, but simpler than, those for odd perfect numbers. An extended test on the possible prime factors of such a number was developed that, with minor differences, applies to both odd triperfect and odd perfect numbers. When applicable, this test allows an earlier determination of a contradiction within a factor chain and so reduces the effort required. It was also shown that an odd triperfect number must be greater than 10 to the power 128. While the goal of proving that an odd perfect number must have at least 9 distinct prime factors was not achieved, due to mainly practical limitations, the algorithmic approach was able to show that for an odd perfect number with 8 distinct prime factors, (i) if it is exactly divisible by 3 to the power of 2a then a = 1, 2, 3, 5, 6 or a is greater than or equal to 31 (ii) if the special component is pi to the power of alpha, pi less than 10 to the 6 and pi to the (alpha+1) less than 10 to the 40, then alpha = 1.

Our aim is to investigate perfect isometry groups, which are invariants for blocks of finite groups. There are two subgoals. First is to study some properties of perfect isometry groups in general. We found that every perfect isometry has essentially a unique sign. This allowed us to show that, in many cases, a perfect isometry group contains a direct factor generated by -id. The second subgoal is to calculate perfect isometry groups for various blocks. Notable results include the perfect isometry groups for blocks with defect 1, abelian p-groups, extra special p-groups, and the principal 2-block of the Suzuki group Sz(q). In the case of blocks with defect 1, we also showed that every perfect isometry can be induced by a derived equivalence. With the help of a computer, we also calculated perfect isometry groups for some blocks of sporadic simple groups.Apart from perfect isometries, we also investigated self-isotypies in the special case where C_G(x) is a p-group whenever x is a p-element. We applied our result to calculate isotypies in cyclic p-groups and the principal 2-blocks of the Suzuki group Sz(q).

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This thesis examines informal and corruptive activities agents may pursue within organisations. Chapter 1 is a brief introduction to the general theme and the related literature. Chapter 2 develops a simple theory of non-monetary collusion, where agents collude by exchanging favours. It examines the optimal use of supervisory information in a simple hierarchy under potential collusion. It is shown that when only the supervisor's information about the agent is used, collusion does not arise, since favours can not be exchanged. Secondly, it is analysed whether the agent's information about his superior should be used. In this case collusion is possible, and there is an interesting trade-off between the benefits of using additional information and the costs of collusion. It is then shown that sometimes the principal may be better off when using less than all available information. Chapter 3 considers task assignment and whistle-blowing as measures a principal may use to break collusion. The principal's response to potential collusion is to allocate less time to monitoring, and he breaks collusion with money. It is shown that the principal may also break collusion by hiring a third worker, and the decision how to break collusion optimally is endogenously determined. Breaking collusion by task assignment is costly, and therefore we consider whistle-blowing as a collusion breaking device. It provides the principal strictly higher welfare than the collusion-proof solution. It is also shown that under reasonable conditions, the collusion-free outcome will be achieved with no further cost. Chapter 4 develops a model of blackmail, where a piece of information an agent prefers to keep private may facilitate blackmail when another agent, namely a blackmailer, threatens to reveal that information. The crucial feature of the blackmail game is the commitment problem from the blackmailer's side. The blackmailer can not commit not to come back in future to demand more despite the payments received in the past. The chapter outlines conditions under which successful extortion may arise, and shows that there is a unique Markov Perfect Equilibrium, which gives a precise prediction how much money the blackmailer is able to extort from the victim. It is also shown that the blackmailer receives a blackmail premium that compensates the blackmailer for not taking money from the victim and revealing information anyway.

The functional properties of a cognate morpheme often vary between languages or dialects. The prefix me- in the Swahili dialects, derived from Proto-Sabaki *-mala ‘finish’ (Proto-Bantu *-màd-), is one such case. In this article, I discuss the functional differences and similarities of this prefix between Kimakunduchi, the dialect spoken in the southern part of Unguja, and Kiunguja, the prestigious dialect of Swahili originally spoken in Zanzibar Town and environs. Me- has been labelled both ‘perfect’ and ‘perfective’. I maintain, however, that the former should be assigned to Kimakunduchi me-, the latter to Kiunguja me-, based on the contexts in which they are used. This contrast not only sheds light on the difference in the TA (tense-aspect) system for anterior reference between the dialects, but also suggests that me- in the Swahili dialects has taken a diachronic path from perfect to perfective, a cross-linguistically observable tendency. The process of diachronic functional change can be shown more clearly with the me-cognate ma- in the Kitumbatu dialect spoken on Tumbatu island off the northwest tip of Unguja. Kitumbatu ma- occurs almost in the same contexts as Kimakunduchi me-, but differs in co-occurrence with hodiernal temporal adverbials, which supports the claim that the restriction on temporal closeness relaxes in parallel with the change from perfect to perfective.

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