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This thesis focusses on the notion of analysis (&vàXuatç) in Stoic logic, that is to say on the procedure which the Stoic logicians followed in order to reduce all valid arguments to five basic patterns. By reconsidering the uses of its Aristotelian homonym and by examining the evidence on the classification of Stoic arguments, I distinguish two methods of Stoic analysis and I discuss their rules: (i) the analysis of non-simple indemonstrables, which constitutes a process of breaking up an argument by means of general logical principles (8 prlpcxta) ; and (ii) the analysis of <syllogistic> arguments, which replaces demonstration (&toösttç) and is effected by employing standard well-determined rules (Oata). The ancient sources provide us with concrete examples illustrating the first type of analysis; however, there is no single text that reports the exact procedure of analysing <syllogistic> arguments. Modern scholars have reconstructed in different ways this type of Stoic analysis; I deal with all of them separately and show that the proposed reconstructions are insightful but historically implausible. Based on the textual materiel concerning the notion of analysis not only in its Stoic context but also in some other of its uses, and especially in mathematical practice, I suggest an alternative reconstruction of the Stoic method of reducing valid arguments to the basic indemonstrables

This thesis consists of two parts. The first part deals with the orthogonal projections of pairs of smooth surfaces and of triples of smooth surfaces onto planes. We take as a model of pairwise smooth surfaces the variety X= {(x, 0, z): x> 0} U {(0, y, z): y> 0} and classify germs of maps R3,3,0 -º R2,2,0 up to origin preserving diffeomorphisms in the source which preserve the variety X and any origin preserving diffeomorphisms in the target. This yields an action of a subgroup xA of the Mather group A on C3 2, the set of map-germs R3,0 -º R2,0. We list the orbits of low codimensions of such an action, and give a detailed description of the geometry of each orbit. We extend these results to triples of surfaces. In the second part of the thesis we analyse the shape of smooth embedded closed curves in the plane. A way of picking out the local reflexional symmetry of a given curve -y is to consider the centres of bitangent circles to the curve. ° The closure of the locus of these centres is called the Symmetry Set of y. We present an equivalent way of tracing the local reflexional symmetry of -r by considering the lines with respect to which a point on y and its tangent line are reflected to another point on the curve and to its tangent line. The locus of all these lines form the dual curve of the symmetry set of -y. We study the singularities occurring on duals of symmetry sets and their generic transitions in 1-parameter families of curves 7. A first attempt to define an analogous theory to study the local rotational symmetry in the plane is given. The Rotational Symmetry Set of a curve y is defined to be the locus of centres of rotations taking a point -y(ti) together with its tangent line and its centre of curvature, to y(t2) together with its tangent line and its centre of curvature. We study the properties of the rotational symmetry set and list the generic transitions of its singularities in 1-parameter families of curves ry. In the final chapter we investigate the local structure of the midpoint locus of generic smooth surfaces

In this thesis we study the relationship between secret sharing schemes and various discrete structures. Chapter 1 contains the mathematical background necessary for the rest of the work. In Chapter 2 t-affine designs are studied. Such designs with block class size two and the maximum number of parallel classes are classified and it is shown that a (3,1, q, q + 1) affine design with q even can always be extended to a (3,1, q, q + 2) affine design. Chapter 3 introduces secret sharing schemes and a new model for secret sharing is proposed in Chapter 4. This model is designed to facilitate the comparison with existing secret sharing models in the literature. We introduce the notion of equivalence of secret sharing schemes and propose a new definition of the information rate of a scheme. In Chapter 5 the problem of constructing schemes for general monotone access structures is considered by using manipulation of logical expressions. We use this method to construct geometrical secret sharing schemes for any monotone access structure. Chapter 6 considers ideal secret sharing schemes. It is shown that any ideal monotone access structure can be associated with a unique matroid. We enumerate the number of distinct rows of an ideal scheme and classify ideal threshold schemes as transversal TD1(t,k,n) designs. In Chapter 7 we look at several new constructions of new secret sharing schemes from existing ones. We use these constructions to show that the distinct rows of an ideal scheme occur with equal frequency and to find a lower bound for the maximum information rate of a monotone access structure. 2 Chapter 8 presents a solution to the problem of how to cope with a participant in a secret sharing scheme becoming untrustworthy after the scheme has been initiated. Finally in Chapter 9 we present a model for a different type of secret sharing scheme and construct such schemes for a family of two-level access structures.

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We start by giving a description of some generalised braids. For the cases of singular and virtual braids we give the algebraic definition and the geometric interpretation. Then we use combinatorial methods to give a proof that the singular braid group is torsion free. Motivated by the formation of labels for the edges of diagrams of virtual knots we give the abstract definiton for biracks and biquandles. Examples of these new algebraic objects are given and some properties are proved as well. We show how biracks provide representations of virtual braids and biquandles provide invariants of virtual knots. Finally, we give some applications of this theory using elements from the Alexander birack as labels for diagrams of virtual knots.

This thesis is organised and presented in nine chapters. The first chapter, the introduction, is in two broad sections and begins by discussing the origins of matched data and the reasons for matching. The general problems of censored data are mentioned and brief descriptions of the past attempts to analyse matched censored data are given, together with their shortcomings. The second section de-fines the notation used and presents the background to the failure time distributions and the types of censoring considered. Chapter 2 is concerned with the analysis of data from the proportional hazards model. The two existing methods are reviewed then a new solution, the integrated method, is proposed and the theory developed. These methods are com-pared in the following chapter, Chapter 3. Chapter 4 concentrates on data arising from the normal theory accelerated failure model. The previous solution is discussed and the results are derived for a new solution based upon the EM algorithm. This is extended to allow for right and interval censored data. The existing solution and the new solution are compared in Chapter 5. Chapter 6 provides analyses of some data sets to compare the results arising from the new methods and the existing solutions, in a practical framework. Chapter 7 discusses the relative merits of the new methods as compared with the previous solutions in the analysis of matched censored data and concludes with an outline of other areas in this field which require further research and the way in which the problems might be tackled. Chapter 8 comprises four appendices whilst Chapter 9 lists the references cited in the text.

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