In this thesis, we present a number of results in set theory, particularly in the areas of forcing, large cardinals, and combinatorial set theory. Chapter 2 concerns covering matrices, combinatorial structures introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of this proof and subsequent work with Sharon, Viale isolated two reflection principles, CP and S, which can hold of covering matrices. We investigate covering matrices for which CP and S fail and prove some results about the connections between such covering matrices and various square principles. In Chapter 3, motivated by the results of Chapter 2, we introduce a number of square principles intermediate between the classical and (+). We provide a detailed picture of the implications and independence results which exist between these principles when is regular. In Chapter 4, we address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik-Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik-Sharon model and various other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales. In Chapter 5, we prove that, assuming large cardinals, it is consistent that there are many singular cardinals such that every stationary subset of + reflects but there are stationary subsets of + that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth and is joint work with James Cummings. In Chapter 6, we extend a result of Gitik, Kanovei, and Koepke regarding intermediate models of Prikry-generic forcing extensions to Radin generic forcing extensions. Specifically, we characterize intermediate models of forcing extensions by Radin forcing at a large cardinal using measure sequences of length less than. In the final brief chapter, we prove some results about iterations of w1-Cohen forcing with w1-support, answering a question of Justin Moore.

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θK, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The mth moment of the rth longest cycle of a random permutation is Θ(nm), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or negatively correlated, depending on θ. The mth moments of the rth shortest cycle of a random permutation is Θ(nm−θ/(ln n)r−1) when θ < m, Θ((ln n)r) when θ = m, and Θ(1) when θ > m. The exponent of cycle lengths at the 100qth percentile goes to q with zero variance. The exponent of the expected cycle lengths at the 100qth percentile is at least q due to the Jensen’s inequality, and the exact value is derived.

The final Chapter concerns a problem of partitioning graphs subject to certain restrictions. We prove that several subproblems are NP-complete.

The advantage of one-bead-one-compound combinatorial libraries is that hundreds of thousands to millions of compounds can be rapidly synthesized and screened simultaneously. The beads supporting the compounds of interest are then isolated and analyzed to decipher the structure of the desired compound. Many methods are currently used to allow deconvolution of the compound on the individual beads. Herein is described a novel method to encode TentaGel beads using absorption of different ratios of lanthanide ions. The encoding process is completed in parallel with the synthesis of the library of compounds. Once the desired beads are identified, the lanthanide ions can be released from the bead and analyzed using Inductively Coupled Plasma-Mass Spectrometry (ICP-MS).

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

Bibliography: leaves 125-131. / xi, 131 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis examines the performace of optimisation networks. The main objectives are to determine if there exist any factors which limit the solution quality that may be achieved with optimisation networks, to determine the reasons for any such limitations and to suggest remedies for them. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 1996

Bibliography: leaves 231-239. / x, 290 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Demonstrates the significance and usefulness of representation in geometry by generating various floor design layouts for a typical Australian house, a single fronted cottage and the building enclosures. / Thesis (Ph.D.)--University of Adelaide, School of Architecture, Landscape Architecture and Urban Design, 2000?

Includes bibliography. / 281 p ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--Dept. of Pure Mathematics, University of Adelaide, 1978

Thesis (Ph. D.)--University of Washington, 2007. / Includes bibliographical references (p. 134-142).

Thesis (Ph.D.)--University of Tulsa, 1995. / Includes bibliographical references (leaves 118-121).