A remarkable inequality among integer numbers is given. Easily deduced from Fermat's Last Theorem, it would be nevertheless very difficult to establish through other means.

This thesis is the synthesis of theoretical and experimental works performed in the area of quantum foundations, particularly on quantum correlations and experimental tests of multiplicative Bell inequalities. First we begin with a comprehensive theoretical work performed on the foundations of quantum mechanics, focusing on the puzzling concepts of quantum entanglement, and hidden variable theories. Specifically, we present a broad overview of different classes of hidden variable theories such as local, crypto-nonlocal, contextual and non-local theories, along with several Bell like inequalities for these theories, providing theoretical proofs based on quantum mechanics for the falsification of some of these theories. Second we present a body of experimental, and theoretical works performed on a new class of Bell inequalities, i.e., the multiplicative Bell inequalities. We experimentally report the observation of the Bell parameters close to the Tsirelson (quantum) limit, upto a large number of measurement devices $(n)$, and compare the results with a particular deterministic strategy. We also obtain classical bounds for some $n$, and report the experimental violation of these classical limits. We theoretically derive new richer bounds on the CHSH inequality (named after John Clauser, Michael Horne, Abnor Shimony and Richard Holt) and the multiplicative Bell parameter for $n=2$, based on the principle of ``relativistic independence'', and experimentally observe the distribution of Bell parameters as predicted by these bounds.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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Heart failure is a common and serious chronic condition, which can be as ‘malignant’ as most forms of cancer (Stewart, MacIntyre, Hole, Capewell, & McMurray, 2001). Recent estimates are that around 40,000 men and 45,000 women are living with heart failure in Scotland (Stewart, MacIntyre, Capewell, & McMurray, 2003). Heart failure is significantly influenced by socioeconomic factors, with people on lower incomes being more likely to develop, and die faster from, heart failure (McAlister et al, 2004). There is a growing body of research on the experience of living with heart failure, however, none provides serious consideration of the role of socioeconomic factors in impacting the experience of heart failure, and some qualitative research may actually obscure such factors. There were two main aims in this thesis. One was to explore how qualitative research methods can better consider the relationship between experience and broader context, such as the influence of socio-economic factors on health. The other aim was to examine the experiences of people as they live with and die from heart failure in ways that situate their accounts in the broader context of their lives. An initial research study, on which I was the main researcher, focused on the experiences of 30 people living with advanced heart failure. These people, their carers and key health professionals were interviewed, where possible, three times over a six months period. This thesis re-examines that study, focusing on 20 of those participants, for which a total of 122 interviews were conducted. I used a dialogical approach to see whether the socioeconomic context of heart failure for these respondents, could be captured through exploring experiences, performance, relationships, discourses and institutional practices, the social processes that mediate the relationship between socioeconomic disadvantage and chronic diseases were explored. This offers important learning in relation to the experience of living with heart failure, along with the experience of providing care. The findings highlight the need to broaden our view of chronic illness beyond biomedical approaches, and grow our methodological approaches along with that, in order to develop knowledge and practice that has relevance for people who live with and die from heart failure.

In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalised by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of the Littlewood Offord result, a sharp anti-concentration inequality for products of independent random variables. (C) 2016 Elsevier Inc. All rights reserved.

Degree completion is as important as participation in higher education. To date, most of the research on this subject has investigated the association between socio-economic background and academic preparation to explain lower rates or non-completion. This thesis explores the role that individual, institutional and contextual factors have on degree completion chances. The first empirical chapter uses an individual-level longitudinal dataset to study the role that attainment on entry and socio-economic background have on students’ chances of completion. The chapter finds that attainment on entry significantly affects students’ chances of degree completion. It also reveals, however, that attainment on entry does not completely explain the socio-economic differences in degree completion chances in the system. The second empirical chapter examines the role that institutions have on students’ chances of degree completion. Like the first chapter, this one draws on the individual-level longitudinal dataset. Although the analyses discover institutional effects, they show them to have a small effect on the system. The third empirical chapter examines the role that teaching has on the probability that students complete their degrees in higher education. The findings show that teaching has a small but significant effect on students’ chances of degree completion. Teaching effects take place only at the most selective institutions of the system. The final empirical chapter examines the role that unemployment has on students’ decisions to complete a university degree. Using data from the Labour Force Survey, the research finds that higher unemployment rates have a positive effect on students’ chances of degree completion. This thesis contributes to the field by showing that stratification of higher education also translates to degree completion. The results confirm that selectivity and institutional effects are strongly associated. In addition, they show that where the institutional framework is concerned, there is a gradient for class effect in educational outcomes.

The first part of this thesis establishes a series of geometric ineqalities related to fractional integration in some geometric settings, including bilinear and multilinear forms. In the second part of this thesis, we study some kinds of rearrangement inequalities. In particular, some applications of rearrangement inequalities will be given, for instance, the determination of the extremals of some geometric problems. By competing symmetries and rearrangement inequalities, we prove the sharp versions of geometric inequalities introduced in the first part in Euclidean spaces. Meanwhile, there are the corresponding conformally equivalent formulations in unit sphere and in hyperbolic space. The last part is about collaborative work on the regularity of the Hardy-Littlewood maximal functions. We give a simple proof to improve Tanaka's result of the paper entitled "A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function". Our proof is based on the behaviour of the local maximum of the non-centered Hardy-Littlewood maximal function.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Vita. / Bibliography: leaves 135-137. / by Kenneth Sidney Alexander. / Ph.D.

We prove that the blowup term of a blowup solution of Jang's equation on an initial data set (M,g,k) near an arbitrary strictly stable MOTS Σ is exactly −1/√λlog τ, where τ is the distance from Σ and λ is the principal eigenvalue of the MOTS stability operator of Σ. We also prove that the gradient of the solution is of order τ^(-1). Moreover, we apply these results to get a Penrose-like inequality under additional assumptions.