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Many boundary value problems of mathematical physics are modelled by elliptic differential operators L in a given domain Ω . An effective method for treating such problems is the method of layer potentials, whose essence resides in reducing matters to solving a boundary integral equation. This, in turn, requires inverting a singular integral operator, naturally associated with L and Ω, on appropriate function spaces on ƌΩ. When the operator L is of second order and the domain Ω is Lipschitz (i.e., Ω is locally the upper-graph of a Lipschitz function) the fundamental work of B. Dahlberg, C. Kenig, D. Jerison, E. Fabes, N. Rivière, G. Verchota, R. Brown, and many others, has opened the door for the development of a far-reaching theory in this setting, even though several very difficult questions still remain unanswered. In this dissertation, the goal is to solve a number of open questions regarding spectral properties of singular integral operators associated with second and higher-order elliptic boundary value problems in non-smooth domains. Among other spectral results, we establish symmetry properties of harmonic classical double layer potentials associated with the Laplacian in the class of Lipschitz domains in R2. An array of useful tools and techniques from Harmonic Analysis, Partial Differential Equations play a key role in our approach, and these are discussed as preliminary material in the thesis: --Mellin Transforms and Fourier Analysis; --Calderón-Zygmund Theory in Uniformly Rectifiable Domains; -- Boundary Integral Methods. Chapter four deals with proving invertibility properties of singular integral operators naturally associated with the mixed (Zaremba) problem for the Laplacian and the Lamé system in infinite sectors in two dimensions, when considering their action on the Lebesgue scale of p integrable functions, for 1 < p < ∞. Concretely, we consider the case in which a Dirichlet boundary condition is imposed on one ray of the sector, and a Neumann boundary condition is imposed on the other ray. In this geometric context, using Mellin transform techniques, we identify the set of critical integrability indexes p for which the invertibility of these operators fails. Furthermore, for the case of the Laplacian we establish an explicit characterization of the Lp spectrum of these operators for each p є (1,∞), as well as well-posedness results for the mixed problem. In chapter five, we study spectral properties of layer potentials associated with the biharmonic equation in infinite quadrants in two dimensions. A number of difficulties have to be dealt with, the most significant being the more complex nature of the singular integrals arising in this 4-th order setting (manifesting itself on the Mellin side by integral kernels exhibiting Mellin symbols involving hyper-geometric functions). Finally, chapter six, deals with spectral issues in Lipschitz domains in two dimensions. Here we are able to prove the symmetry of the spectra of the double layer potentials associated with the Laplacian. This is in essence a two-dimensional phenomenon, as known examples show the failure of symmetry in higher dimensions. / Mathematics

Who uses they, and who can they be (or not be) used for? Singular they has been proscribed in formal grammars since the mid-18th century, yet it dates to at least the 14th century (Balhorn 2004; Curzan 2003), persevering in both writing and speech (e.g., Baranowski 2002; Balhorn 2009; Lagunoff 1997; Matossian 1997; Newman 1992; Strahan 2008). This thesis investigates the envelope of variation (e.g., LaScotte 2016; Maryna 1978; Meyers 1990) in which speakers make choices of third person singular pronouns based on a multiplicity of both linguistic (e.g., gender stereotypicality, antecedent type) and social (e.g., gender, age, LGBTQ2S+ identity) factors. The analysis is based on data from 620 participants from across Canada and the US between the ages 13 and 79. An online survey sought responses related to three occupations: LaScotte’s (2016) open ended ideal student question was replicated, and Martyna’s (1978) fill in-the-blank style was modelled for mechanic and secretary—nouns with observed and unambiguous gender stereotypes (masculine and feminine respectively; Deaux & Lewis 1986; Haines, Deaux, & Lofaro 2016). Participants self-identified their gender and were categorized into a ternary grouping: men (e.g., cis, trans, transmasculine), women (e.g., fem, cis, trans, female ish), and non-binary (e.g., genderqueer, genderfluid). LGBTQ2S+ identity was also collected, as well as personal pronouns. Use of third person pronouns in the survey responses is quantified by consistency (i.e., maintaining use of the same pronoun throughout a participant’s response) and by proportional frequency of use—the latter explored in depth. The most important quantitative finding is that singular they is the most consistently and frequently used third person pronoun overall. But, its patterns of use are not parallel across test occupations or participant social groups. The results indicate that student is gender-neutral, whereas mechanic and secretary remain gendered (he:they; she:they), results that are reflected by perceptual ratings: student remains neutral (they), mechanic skews masculine (he), and secretary skews feminine (she). The impact of social characteristics adds layers of complexity about the groups leading sociolinguistic change at societal levels and/or within their own communities and networks: Non-binary, LGBTQ2S+, users of gender neutral personal pronouns, and/or younger. Collectively, these findings suggest that gender stereotypical roles are not unilaterally weighted and biases can manifest through pronominal choice. There are multiple dimensions of influence, such as the referent, one’s identity, and the communities to which individuals are connected. Thus, this thesis both uncovers persistent gender biases and creates a dynamic display of pronominal variation across speakers. / Graduate

In this study, we consider numerical schemes for a class of singular integrodifferential equations with a nonatomic difference operator. Our interest in this particular class has been motivated by an application in aeroelasticity. By applying nonconforming finite element methods, we are able to establish convergence for a semi-discrete scheme. We use an ordinary differential equation solver for the semi-discrete scheme and then improve the result by using a fully discretized scheme. We report our numerical findings and comment on the rates of convergence. / Ph. D.

Yes / Voltage instability problems in power system is an important issue that should be taken into consideration during the planning and operation stages of modern power system networks. The system operators always need to know when and where the voltage stability problem can occur in order to apply suitable action to avoid unexpected results. In this paper, a study has been conducted to identify the weakest bus in the power system based on multi-variable control, modal analysis, and Singular Value Decomposition (SVD) techniques for both static and dynamic voltage stability analysis. A typical IEEE 3-machine, 9-bus test power system is used to validate these techniques, for which the test results are presented and discussed.