A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, August 2015. / We construct, using rst principles, a number of non-trivial conservation laws of some partial di erence equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation. Symmetries and the more recent ideas and notions of characteristics (multipliers) for di erence equations are also discussed. We then determine the symmetry generators of some ordinary di erence equations and proceed to nd the rst integral and reduce the order of the di erence equations. We show that, in some cases, the symmetry generator and rst integral are associated via the `invariance condition'. That is, the rst integral may be invariant under the symmetry of the original di erence equation. We proceed to carry out double reduction of the di erence equation in these cases. We then consider discrete versions of the Painlev e equations. We assume that the characteristics depend on n and un only and we obtain a number of symmetries. These symmetries are used to construct exact solutions in some cases. Finally, we discuss symmetries of linear iterative equations and their transformation properties. We characterize coe cients of linear iterative equations for order less than or equal to ten, although our approach of characterization is valid for any order. Furthermore, a list of coe cients of linear iterative equations of order up to 10, in normal reduced form is given.

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in Pic(X)⊗C. We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

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Thesis (M.S.)--Georgia Southern University, 2007. / "A thesis submitted to the Graduate Faculty of Georgia Southern University in partial fulfillment of the requirements for the degree Master of Science." In Mathematics, under the direction of Billûr Kaymakc̦alan. ETD. Electronic version approved: July 2007. Includes bibliographical references (p. 39-43)

Thesis (M.S.E.C.E.)--Baylor University, 2007. / Includes bibliographical references (p. 99).

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 73-77).

Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains viii, 140 p.; also includes graphics. Includes bibliographical references (p. 139-140).

Thesis (M.S.)--University of Wisconsin--Madison, 1968. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2009. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed August 31, 2009) Includes bibliographical references (p. 146-149).