We introduce the Partition Principle PP, an axiom introduced by Russell in the context of its similarities and differences with the Axiom of Choice AC. We start by proving some properties of PP, and AC, and show that AC, entails PP. To address the problem of whether the converse holds, we develop the Zermelo-Fraenkel ZF set theory and examine its consistency and build a model in which AC, fails. We follow this with a discussion of forcing, a technique introduced by Paul Cohen to build new models of set theory from existing ones, which have differing properties from the starting model. We conclude by examining candidate models called permutation models where AC, fails, which may be useful as candidate models for forcing a model in which PP, holds but AC, does not. We conjecture that such a model exists, and that PP, does not entail AC. / Thesis / Master of Science (MSc)

<div>We further the research in o-minimal topology by studying in full generality definable topological spaces in o-minimal structures. These are topological spaces $(X, \tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ has a basis that is (uniformly) definable. Examples include the canonical o-minimal "euclidean" topology, “definable spaces” in the sense of van den Dries [17], definable metric spaces [49], as well as generalizations of classical non-metrizable topological spaces such as the Split Interval and the Alexandrov Double Circle.</div><div><br></div><div>We develop a usable topological framework in our setting by introducing definable analogues of classical topological properties such as separability, compactness and metrizability. We characterize these notions, showing in particular that, whenever the underlying o-minimal structure expands $(\mathbb{R},<)$, definable separability and compactness are equivalent to their classical counterparts, and a similar weaker result for definable metrizability. We prove the equivalence of definable compactness and various other properties in terms of definable curves and types. We show that definable topological spaces in o-minimal expansions of ordered groups and fields have properties akin to first countability. Along the way we study o-minimal definable directed sets and types. We prove a density result for o-minimal types, and provide an elementary proof within o-minimality of a statement related to the known connection between dividing and definable types in o-minimal theories.</div><div><br></div><div>We prove classification and universality results for one-dimensional definable topological spaces, showing that these can be largely described in terms of a few canonical examples. We derive in particular that the three element basis conjecture of Gruenhage [25] holds for all infinite Hausdorff definable topological spaces in o-minimal structures expanding $(\mathbb{R},<)$, i.e. any such space has a definable copy of an interval with the euclidean, discrete or lower limit topology.</div><div><br></div><div>A definable topological space is affine if it is definably homeomorphic to a euclidean space. We prove affineness results in o-minimal expansions of ordered fields. This includes a result for Hausdorff one-dimensional definable topological spaces. We give two new proofs of an affineness theorem of Walsberg [49] for definable metric spaces. We also prove an affineness result for definable topological spaces of any dimension that are Tychonoff in a definable</div><div>sense, and derive that a large class of locally affine definable topological spaces are affine.</div>

Prior work in the field of set theory has looked at the properties of union-free families. This thesis investigates families based on a different set operation, the symmetricc difference. It provides a complete characterization of maximal symmetric differencefree families of subsets of {1, . . . n}

<p>This work strengthens and extends an algorithm for computing Betti numbers of symmetric semialgebraic sets developed by Basu and Riener in, <em>Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-Algebraic Sets</em>. We first adapt a construction of Gabrielov and Vorobjov in, <em>Approximation of Definable Sets by Compact Families, and Upper Bounds on Homotopy and Homology,</em> for replacing arbitrary definable sets by compact ones to the symmetric case. The original construction provided maps from the homotopy and homology groups of the replacement set to those of the original; we show that for sets symmetric relative to the action of some finite reflection group <em>G</em>, we may construct these maps to be equivariant. This modification to the construction for compact replacement allows us to extend Basu and Riener's theorem on which submodules appear in the isotypic decomposition of each cohomology space to sets not necessarily closed and bounded. Furthermore, by utilizing this equivariant compact approximation, we may obtain a precise description of the aforementioned decomposition of each cohomology space, and not merely the final dimension of the space, from Basu and Riener's algorithm.</p> <p><br></p> <p>    Though our equivariant compact replacement holds for <em>G</em> any finite reflection group, Basu and Riener's results only consider the case of the action the of symmetric group, sometimes termed type <em>A</em>. As a first step towards generalizing Basu and Riener's work, we examine the next major class of symmetry: the action of the group of signed permutations (known as type <em>B</em>). We focus our attention on Vandermonde varieties, a key object in Basu and Riener's proofs. We show that the intersection of a type <em>B</em> Vandermonde variety with a fundamental region of type <em>B</em> symmetry is topologically regular. We also prove a result about the intersection of a type <em>B</em> Vandermonde variety with the walls of this fundamental region, leading to the elimination of factors in a different decomposition of the homology spaces.</p>

As soon as Bennett and Gill first demonstrated that, relative to a randomly chosen oracle, P is not equal to NP with probability 1, the random oracle hypothesis began piquing the interest of mathematicians and computer scientists. This was quickly disproven in several ways, most famously in 1992 with the result that IP equals PSPACE, in spite of the classes being shown unequal with probability 1. Here, we propose what could be considered strengthening of the random oracle hypothesis, using a stricter notion of what it means for a set to be 'large'. In particular, we suggest using largeness with respect to the Ramsey forcing notion. In this new context, we demonstrate that the set of oracles separating NP and coNP is 'not small', and obtain similar results for the separation of PSPACE from PH along with the separation of NP from BQP. In a related set of results, we demonstrate that these classes are all of the same descriptive complexity. Finally we demonstrate that this strengthening of the hypothesis turns it into a sufficient condition for unrelativized relationships, at least in the three cases considered here.

Ease of use differentiates products in a highly competitive market place. It also brings an added value that culminates in a higher degree of customer satisfaction, repeated business, increased sales, and higher revenue. User-centered design is a strategic asset that companies can use to improve their customer relationships by learning more about their customers, and increase their sales. In today's economy, the measurement of intangible assets such as user experience has become a major need for industries because of the relationship between user-centered design and organizational benefits such as customer loyalty. As companies realize that the inclusion of user-centered design concepts in product or system design are a key component of attracting and maintaining customers, as well as increasing revenue, the need for quantitative methods to describe these benefits has become more urgent. The goal of this research is to develop a methodology to characterize user-centered design features, customer benefits and organizational benefits resulting from developing products using user-centered design principles through the use of an integrated framework of critical factors. Therefore, this research focuses on the identification of the most significant variables required to assess and measure the degree of user-centered design (UCD) characteristics included in the various aspects of product development such as physical design features, cognitive design attributes, industrial design aspects and user experience design considerations. Also this research focuses on the development of assessment tools for developers to use when evaluating the incorporation of user-centered design features in the creation of products and systems. In addition, a mathematical model to quantify the inclusion of UCD factors considered in the design of a product and systems is presented in this research. The results obtained using the assessment tools and the mathematical model can be employed to assess the customer benefits and organizational benefits resulting from including user-centered design features in the creation of products and systems. Overall, organizational benefits such as customer loyalty, company image, and profitability are expected to be impacted by the company's capability to meet or exceed stated design claims and performance consistency while maintaining aesthetic appeal, long product life, and product usefulness. The successful completion of this research has produced many beneficial research findings. For example, it has helped characterize and develop descriptors for estimating critical quantitative and qualitative components, sub-components, and factors influencing user-centered design that are related to customer and organizational benefits through the use of fuzzy set modeling. In addition, the development of specific tools, methods, and techniques for evaluating and quantifying UCD components resulted from this study.

The objective of the research was to develop and validate a multifaceted model such as a fuzzy Analytical Hierarchy Process (AHP) model that considers both qualitative and quantitative elements with relative significance in assessing the likelihood of falls and aid in the design of NASA Ground Support Operations in aerospace environments. The model represented linguistic variables that quantified significant risk factor levels. Multiple risk factors that contribute to falls in NASA Ground Support Operations are task related, human/personal, environmental, and organizational. Six subject matter experts were asked to participate in a voting system involving a survey where they judge risk factors using the fundamental pairwise comparison scale. The results were analyzed and synthesize using Expert Choice Software, which produced the relative weights for the risk factors. The following are relative weights for these risk factors: Task Related (0.314), Human/Personal (0.307), Environmental (0.248), and Organizational (0.130). The overall inconsistency ratio for all risk factors was 0.07, which indicates the model results were acceptable. The results show that task related risk factors are the highest cause for falls and the organizational risk are the lowest cause for falls in NASA Ground Support Operations. The multiple risk factors weights were validated by having two teams of subject matter experts create priority vectors separately and confirm the weights are valid. The fuzzy AHP model usability was utilizing fifteen subjects in a repeated measures analysis. The subjects were asked to evaluate three scenarios in NASA KSC Ground Support Operations regarding various case studies and historical data. The three scenarios were Shuttle Landing Facility (SLF), Launch Complex Payloads (LCP), and Vehicle Assembly Building (VAB). The Kendall Coefficient of Concordance for assessment agreement between and within the subjects was 1.00. Therefore, the appraisers are applying essentially the same standard when evaluating the scenarios. In addition, a NASA subject matter expert was requested to evaluate the three scenarios also. The predicted value was compared to accepted value. The results from the subject matter expert for the model usability confirmed that the predicted value and accepted value for the likelihood rating were similar. The percentage error for the three scenarios was 0%, 33%, 0% respectively. Multiple descriptive statistics for a 95% confidence interval and t-test are the following: coefficient of variation (21.36), variance (0.251), mean (2.34), and standard deviation (0.501). Model validation was the guarantee of agreement with the NASA standard. Model validation process was partitioned into three components: reliability, objectivity, and consistency. The model was validated by comparing the fuzzy AHP model to NASA accepted model. The results indicate there was minimal variability with fuzzy AHP modeling. As a result, the fuzzy AHP model is confirmed valid. Future research includes developing fall protection guidelines.

Yes / Risk assessment methods have been widely used in various industries, and they play a significant role in improving the safety performance of systems. However, the outcomes of risk assessment approaches are subject to uncertainty and ambiguity due to the complexity and variability of system behaviour, scarcity of quantitative data about different system parameters, and human involvement in the analysis, operation, and decision-making processes. The implications for improving system safety are slowly being recognised; however, research on uncertainty handling during both qualitative and quantitative risk assessment procedures is a growing field. This paper presents a review of the state of the art in this field, focusing on uncertainty handling in fault tree analysis (FTA) based risk assessment. Theoretical contributions, aleatory uncertainty, epistemic uncertainty, and integration of both epistemic and aleatory uncertainty handling in the scientific and technical literature are carefully reviewed. The emphasis is on highlighting how assessors can handle uncertainty based on the available evidence as an input to FTA.

Yes / Risk assessment methods have been widely used in various industries, and they play a significant role in improving the safety performance of systems. However, the outcomes of risk assessment approaches are subject to uncertainty and ambiguity due to the complexity and variability of system behaviour, scarcity of quantitative data about different system parameters, and human involvement in the analysis, operation, and decision-making processes. The implications for improving system safety are slowly being recognised; however, research on uncertainty handling during both qualitative and quantitative risk assessment procedures is a growing field. This paper presents a review of the state of the art in this field, focusing on uncertainty handling in fault tree analysis (FTA) based risk assessment. Theoretical contributions, aleatory uncertainty, epistemic uncertainty, and integration of both epistemic and aleatory uncertainty handling in the scientific and technical literature are carefully reviewed. The emphasis is on highlighting how assessors can handle uncertainty based on the available evidence as an input to FTA.

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