The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes.

We study the Brauer classes rising from the obstruction to the existence of tautological line bundles on the Picard scheme of curves. We establish various properties of the Brauer classes for families of smooth curves. We compute the period and index of the Brauer class associated with the universal smooth curve for a fixed genus. We also show such Brauer classes are trivialized when we specialize to certain generalized theta divisors. If we consider the universal totally degenerate curve with a fixed dual graph, using symmetries of the graph, we give bounds on the period and index of the Brauer classes. As a result, we provide some division algebras of prime degree, serving as candidates for the cyclicity problem. As a byproduct, we re-calculate the period and index of the Brauer class for universal smooth genus g curve in an elementary way. We study certain conic associated with the universal totally degenerate curve with a fixed dual graph. We show the associated conic is non-split in some cases. We also study some other related geometric properties of Brauer groups.

In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. Next we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. Finally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.

Not available / text

The key research question which was addressed by the study was whether gaps existed between the sal ient attributes employed for hotel selection by managers and customers and the inspection criteria used by the UK hotel classification and grading schemes. If so, to identify whether such unassessed attributes were appraisable by the hotel inspectorate. A review of the literature indicated that no such published study had previously been attempted. A literature review examined the criteria identified to assess service quality, and in particular its provision within the hotel industry. The historical development and operational characteristics of the major grading schemes were presented. A unique numerical analysis of the schemes provided the incidence of classified and graded hotels by country. This formed the basis for the establishment of a representative stratified random sample. The determination of the hotel selection attributes was achieved by literature review and in-depth and focus group interviews. An extensive questionnaire asking recipients to rate the importance of the selection attributes was distributed to 500 hotel managers, producing a 62.4~ response. Equivalent customer contacts were provided by the managers, and 500 customers were surveyed, producing a 57.8~ response. Attribute analysis defined important, interjacent and unimportant groups. Comparisons were made between leisure and business, gender, grading categories and forms of business ownersh ip for both data sets. The closeness of association between the total manager and customer data sets allowed a merging into a consolidated attribute set. An analysis of the schemes' grading criteria was compared with the important attributes to indicate those which were not specifically assessed by the schemes. A survey of hotel inspectors asked them to indicate whether such attributes were specifically, generally or not assessab 1e during a routine inspect ion, and if they were specifically assessable, to provide suggested methodologies for such assessment. The aim was achieved. Sixty five attributes were identified as important but not assessed by the schemes. Of these, 45 were capable of being specifically assessed. It was recommended that the scheme operators should take account of these findings when reviewing their hotel grading methodologies.

No description available.

Certain numerical methods have been well developed for solving one-dimensional two-phase flow (e.g. gas and liquid) problems in the literatures during the last two decades. Based on the existing methods, the present work compares the computational efficiency, accuracy, and robustness of various numerical schemes by predicting the numerical solutions of fluid properties for a specific case to find the proper numerical method. One of the numerical schemes introduced in this work is a practical, semi-implicit upwind method used for fluid flow simulations in different flow patterns,stratified flow and slug flow. This method implements the iterative and non-iterative schemes using a two-fluid model that consists of sets of non-hyperbolic equations. A numerical error term is applied in the pressure equation to maintain the volume balance of the two-phase flow model. If the temperature varies, the discretised energy equations use similar error terms as in the pressure equation. In some cases, the small values of the numerical errors are negligible and do not influence the numerical results. These errors are, however, important factors to consider when maintaining the stability and robustness of the above numerical schemes for strong non-linear cases. The computational efficiency ofthe non-iterative scheme, where the inner iterations are deactivated, is better than the iterative scheme. Different grid arrangements are compared with respect to computational accuracy and efficiency. A staggered structured grid implements the same semi-implicit upwind method as in the non-iterative scheme; the non-staggered grid arrangement uses an existing flux-splitting scheme (Evje and Flåtten, 2003) as a reference. All the above schemes produce numerical solutions with a single precision that normally satisfy the requirements of computational accuracy of industrial two-phase pipe flows. However, if one pursues a higher-order accuracy scheme, e.g. a Roe-averaged algorithm, the governing equations should be strictly a hyperbolic system of partial differential equations, which is achieved by introducing the nonviscous force terms in the two-fluid model (LeVeque, 2002).By properly incorporating the non-conservative terms in the formulation of the numerical fluxes, the capability of the Roe-averaged algorithm is demonstrated by capturing shock waves. Results from the present research include the following. A one-dimensional scheme that solves a system of discretised equations with the staggered semi-implicit upwind method is presented and validated for its computational efficiencyand robustness. This scheme can be widely used in the industry with sufficient accuracy. The other first-order semi-implicit numerical schemes producestable numerical results, especially in the dynamic cases of two-phase flow, except when the gas phase nearly disappears or appears in pipes. The Roe-averaged algorithm is recommended due to the high-resolution numerical results obtained, but at the costs of computational time and effort.

In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. Next we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. Finally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.

This thesis explains the theoretical background of stochastic differential equations in one dimension. We also show how to solve such differential equations using strong It o-Taylor expansion schemes over large time grids. We also attempt to solve a problem regarding a specific approximation of a stochastic integral for which there is no explicit solution. This approximation, which utilizes the distribution of this particular stochastic integral, gives the wrong order of convergence when performing a grid convergence study. We use numerical integration of the stochastic integral as an alternative approximation, which is correct with regards to convergence.

In his influential essay “On the Very Idea of a Conceptual Scheme”, Donald Davidson argues that we cannot make sense of conceptual relativism, the doctrine that there could be incommensurably different systems of concepts applicable to a single world. According to Davidson, there is no criterion of identity for language that does not imply or presuppose the possibility that we interpret that language by means of our own language. Given some plausible assumptions, this implies that there is at most one conceptual scheme, one way of interpreting or representing the world. But then the very idea of a conceptual scheme is empty. The dissertation is an examination of Davidson’s reasoning, and a defence of a different position regarding conceptual relativism. I reject much of Davidson’s argumentation, and his radical (subordinate) conclusion that we would be able, at least in principle, to make sense of any language. Languages that we would be unable to translate or interpret, even in principle, are at least logically possible, in my view. However, this possibility should not be thought to imply or encourage conceptual relativism. In this respect, I think that Davidson and many of his critics have conflated the notion of a difference in conceptual scheme, which requires incommensurability between languages or systems of concepts, and mere conceptual difference. I argue that a genuinely alternative conceptual scheme would be associated with language unintelligible to us because of its relation to our language. For what is at issue, supposedly, is a conceptual relation: a relation between languages, not a relation between speakers, or their capacities, on the one hand, and languages, on the other. I try to show how some of Davidson’s arguments, suitably modified, can be deployed against the possibility of an alternative scheme, so understood, and provide some additional arguments of my own. My position is thus significantly weaker than Davidson’s: there could not be languages that we would be unable to interpret because they are incommensurable with our own.