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Thesis (Ph.D.)--George Mason University, 2009. / Vita: p. 158. Thesis director: Flavia Colonna. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics. Title from PDF t.p. (viewed Oct. 11, 2009). Includes bibliographical references (p. 150-157). Also issued in print.

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The important classes of normally solvable, ϴ₊ (ϴ₋) and strictly singular (strictly cosingular) operators have long been studied in the setting of bounded or closed operators between Banach spaces. Results by Kato, Lacey, et al (see Goldberg [16; III.1.9, III.2.1 and III.2.3] ) led to the definition of certain norm related functions of operators (Γ, Δ and Γ₀) which provided a powerful new way to study the classes of ϴ₊ and strictly singular operators (see for example Gramsch[19], Lebow and Schechter[28] and Schechter[36]). Results by Brace and R.-Kneece[4] among others led to the definition of analogous functions (Γ' and Δ') which were used to study ϴ₋ and strictly cosingular operators (see for example Weis, [37] and [38]). Again this problem was considered mainly for the case of bounded operators between Banach spaces. This thesis represents a contribution to knowledge in the sense that by considering the functions Γ', Δ' and Γ'₀, as well as the minimum modulus function in the more general setting of unbounded linear operators between normed linear spaces, we obtain the classes of F₋ and Range Open operators which turn out to be closely related to the classes of ϴ₋ and normally solvable operators respectively. We also define unbounded strictly cosingular operators and find that many of the classical results on ϴ₋, normally solvable and bounded strictly cosingular operators go through for F₋, range open and unbounded strictly cosingular operators respectively. This ties up with work done by R. W. Cross and provides a workable framework within which to study ϴ₋ and ϴ₊ type operators in the much more. general setting of unbounded linear operators between normed linear spaces.

A dissertation submitted to the University of the Witwatersrand in ful lment of the requirements for candidacy for the degree of Master of Science. Johannesburg, 2016 / Restricted Schur polynomials can be used to describe large N, non-planar limits of N = 4 super Yang-Mills theory. The R-symmetry generators commute with the dilatation operator. For small deformations of 1 2-BPS operators, the matrix elements of these generators have been computed and a set of recursion relations for the matrix elements of the dilatation operator are obtained from this commutation relation. Together with the knowledge that the smallest eigenvalues of the dilatation operator (corresponding to BPS operators) vanish, these recursion relations can be used to determine the matrix elements of the dilatation operator. Studies up to now have computed the matrix elements of the su(2) generators in the displaced corners approximation. Our first novel result is the computation of the exact su(2) generators. We obtain the matrix elements for the su(3) generators in the displaced corners approximation and exactly, for the first time. This is the first step to computing exact matrix elements of the dilatation operator. / TG2016

As with groups, one can study the left regular representation of a semigroup. If one considers such representations, then it is natural to ask similar questions to the group case. We start by formulating several questions in the semigroup case and then work towards understanding the structure of the representations given. We present results describing what the elements of the image under the representation map can look like (the semigroup problem), whether or not two semigroups will give isomorphic representations (the isomorphism problem), and whether or not the representation of a semigroup is reflexive (the reflexivity problem). This research has been funded in part by a scholarship from the Natural Sciences and Engineering Research Council of Canada.

As with groups, one can study the left regular representation of a semigroup. If one considers such representations, then it is natural to ask similar questions to the group case. We start by formulating several questions in the semigroup case and then work towards understanding the structure of the representations given. We present results describing what the elements of the image under the representation map can look like (the semigroup problem), whether or not two semigroups will give isomorphic representations (the isomorphism problem), and whether or not the representation of a semigroup is reflexive (the reflexivity problem). This research has been funded in part by a scholarship from the Natural Sciences and Engineering Research Council of Canada.

published_or_final_version / Physics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy