Typescript. Includes vita and abstract. Includes bibliographical references (leaves 53-54) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 240-243).

An important trend in fuzzy group theory in recent years has been the notion of classification of fuzzy subgroups using a suitable equivalence relation. In this dissertation, we have successfully used the natural equivalence relation defined by Murali and Makamba in [81] and a natural fuzzy isomorphism to classify fuzzy subgroups of some finite abelian p-groups of rank three of the form Zpn + Zp + Zp for any fixed prime integer p and any positive integer n. This was achieved through the usage of a suitable technique of enumerating distinct fuzzy subgroups and non-isomorphic fuzzy subgroups of G. We commence by giving a brief discussion on the theory of fuzzy sets and fuzzy subgroups from the perspective of group theory through to the theory of sets, leading us to establish a linkage among these theories. We have also shown in this dissertation that the converse of theorem 3.1 proposed by Das in [24] is incorrect by giving a counter example and restate the theorem. We have then reviewed and enriched the study conducted by Ngcibi in [94] by characterising the non-isomorphic fuzzy subgroups in that study. We have also developed a formula to compute the crisp subgroups of the under-studied group and provide its proof. Furthermore, we have compared the equivalence relation under which the classification problem is based with various versions of equivalence studied in the literature. We managed to use this counting technique to obtain explicit formulae for the number of maximal chains, distinct fuzzy subgroups, non-isomorphic maximal chains and non-isomorphic fuzzy subgroups of these groups and their proofs are provided.

The thesis shows that, under suitable conditions, a pseudo-differential operator, defined on some "nice" set of functions on Rn x Zm, with continuous negative definite symbol q(x,xi,o) extends to a generator of a Feller semigroup. Sections 1-5 are the preliminary sections, these sections discuss some harmonic analysis concerning locally compact Abelian groups. The essence of this thesis are Sections 6-13, which deals with obtaining the estimates required for the fulfilment of the conditions of the Hille-Yosida-Ray theorem.

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Thesis (Ph.D.)--York University, [2004]. Graduate Programme in [Mathematics]. / Typescript. Includes bibliographical references (leaves 163-166). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ99158

We study the Fourier restriction phenomenon in settings where there is no underlying proper smooth subvariety. We prove an (Lp, L2) restriction theorem in general locally compact abelian groups and apply it in groups such as (Z/pLZ)n, R and locally compact ultrametric fields K. The problem of existence of Salem sets in a locally compact ultrametric field (K, | · |) is also considered. We prove that for every 0 < α < 1 and ǫ > 0 there exist a set E ⊂ K and a measure μ supported on E such that the Hausdorff dimension of E equals α and |bμ(x)| ≤ C|x|−α 2 +ǫ. We also establish the optimal extension of the Hausdorff-Young inequality in the compact ring of integers R of a locally compact ultrametric field K. We shall prove the following: For every 1 ≤ p ≤ 2 there is a Banach function space Fp(R) with σ-order continuous norm such that (i) Lp(R) ( Fp(R) ( L1(R) for every 1 < p < 2. (ii) The Fourier transform F maps Fp(R) to ℓp′ continuously. (iii) Lp(R) is continuously included in Fp(R) and Fp(R) is continuously included in L1(R). (iv) If Z is a Banach function space with the same properties as Fp(R) above, then Z is continuously included in Fp(R). (v) F1(R) = L1(R) and F2(R) = L2(R).

This paper is an investigation of several basic properties of ordered Abelian groups, valuations, the relationship between valuation rings, valuations, and their value groups and valuation rings. The proofs to all theorems stated without proof can be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1858. In Chapter I several basic theorems which are used in later proofs are stated without proof, and we prove several theorems on the structure of ordered Abelian groups, and the basic relationships between these groups, valuations, and their valuation rings in a field. In Chapter II we deal with valuation rings, and relate the structure of valuation rings to the structure of their value groups.

A research project submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment for the degree of Master of Science in Physics. May 25, 2017. / This project addresses the study of the memory effect. We review the effect in electromagnetism, which is an abelian gauge theory. We prove that we can shift the phase factor by performing a gauge transformation. The gauge group is U(1). We extend the study to the nonabelian gauge theory by computing the memory in SU(2) which vanishes up to the first order Taylor expansion. Keywords: Memory Effect, Aharonov-Bohm effect, Nonabelian Gauge Theory, Supersymmetry / GR2018

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