[Abstract] This thesis is divided into two parts, both containing algorithms for dealing with matrices and matrix groups. Part I is concerned with individual matrices over an arbitrary field. Our algorithms make use of a sequence called the rank profile which is related to the linear dependence relations between the columns of a matrix. First we look at LSP decompositions of matrices as defined by Ibarra et al. in 1982. This decomposition is related to, and a little more general than, the LUP decomposition. The algorithm given by Ibarra et al. to compute an LSP decomposition was only defined for m?n matrices where m ≤ n and is claimed to have the same asymptotic cost as matrix multiplication. We prove that their cost analysis overlooked some aspects of the computation and present a new version of the algorithm which finds both an LSP decomposition and the rank profile of any matrix. The cost of our algorithm is the same as that claimed by Ibarra et al. when m ≤ n and has a similar cost when m > n. One of the steps in the Ibarra et al. algorithm is not completely explicit, so that any one of several choices can be made. Our algorithm is designed so that the particular choice made at this point allows for the simultaneous calculation of the rank profile. Next we study algorithms to find the characteristic polynomial of a square matrix. The current fastest algorithm to find the characteristic polynomial of a square matrix was developed by Keller-Gehrig in 1985. We present a new, simpler version of this algorithm with the same cost which makes the algorithm?s reliance on the rank profile explicit. In Part II we present generalised sifting, a scheme for creating Monte Carlo black box constructive group recognition algorithms. Generalised sifting is designed to facilitate computation in a known group, specifically re-writing arbitrary elements as words or straight-line programs in a standard generating set. It can also be used to create membership tests in black-box groups. Generalised sifting was inspired by the subgroup sifting techniques originally introduced by Sims in 1970 but uses a chain of subsets rather than subgroups. We break the problem down into a sequence of separately analysed and proven steps which sift down into each subset in turn ... All of the algorithms in Parts I and II are given with a theoretical proof and (where appropriate) complexity analysis. The LSP decomposition, characteristic polynomial and generalised sifting algorithms have all been implemented and tested in the computer algebra package GAP.

[Truncated abstract] Online games are an exciting new trend in the consumption of entertainment and provide the opportunity to examine selected antecedents of online game-playing based on studying the cognitive, experiential and habitual factors. This study was divided into two parts. The first part analysed the structural relations among research variables that might explain online game-playing using the Structural Equation Modeling (SEM) techniques. These analyses were conducted on a final sample of 218 online gamers. Specific issues examined were: If the variables of Perceived Game Performance, Satisfaction, Hedonic Responses, Flow and Habit Strength influence the Intention to Replay an online game. The importance of factors such as Hedonic Responses and Flow on Satisfaction in online game play. In addition to the SEM, analyses of the participants? reported past playing behaviour were conducted to test whether past game play was simply a matter of random frequency of past behaviour, or followed the specific pattern of the Negative Binomial Distribution (NBD). … The playing-time distribution was not significantly different to the Gamma distribution, in which the largest number of gamers plays for a short time (light gamers) and only a few gamers account for a large proportion of playing time (heavy gamers). Therefore, the reported time play followed a simple and predictable NBD pattern (Chisquare=. 390; p>.05). This study contributes to knowledge in the immediate field of online games and to the wider body of literature on consumer research. The findings demonstrate that gamers tend to act habitually in their playing behaviour. These findings support the argument that past behaviour (habit) is a better explanation of future behaviour than possible cognitive and affective explanations, especially for the apparent routinesed behaviour pattern on online games. The pattern of online game-playing is consistent with the finding of the NBD pattern in television viewing, in which the generalisability of the NBD model has been found in stable environments of repetitive behaviour. This supports the application of the NBD to areas beyond those of patterns in gambling and the purchase of consumer items. The findings have implications both for managerial and public policy decision-making.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.