Thesis (M.Sc. (Mathematics)) - Rhodes University, 2009. / A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Mathematics.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2007. / Includes bibliographical references (p. 65-66).

Thesis (Ph. D.)--University of Chicago. / Includes bibliographical references. Also available on the Internet.

Inaug.-Diss.--Bonn. / Bibliography: p. 46-47.

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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 character table of a finite group G is constructed by computing the eigenvectors of matrix equations determined by the centre of the group algebra. The numerical character values are expressed in algebraic form. A variant using a certain sub-algebra of the centre of the group algebra is used to ease problems associated with determining the conjugacy classes of elements of G. The simple group of order 50,232,960 and its subgroups PSL(2,17) and PSL(2,19) are constructed using general techniques. A combination of hand and machine calculation gives the character tables of the known simple groups of order < 106 excepting Sp(4,4) and PSL(2,q). The characters of the non- Abelian 2-groups of order < 2 6 are computed. Miscellaneous computations involving the symmetric group Sn are given.

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The mutual admittance between two waveguide-fed apertures radiating into a multilayered region is expressed as a double integral of a combination of the Fourier transforms of the respective aperture electric fields and the solutions to a set of transformed wave equations for the layered region. The result is an expression which is stationary about variations in the assumed aperture electric field distributions. The special case of two circular apertures whose electric field distributions are assumed to be the same as the TE_mn and TM_mn circular waveguide modes is analyzed in detail. In this case, the admittance expression can be reduced to a single integral which simplifies the numerical calculation. A computer program was written which can include up to four external layers, two of which may be inhomogeneous normal to the aperture plane. Numerical results for the mutual coupling between two circular apertures with TE₁₁ mode excitation agree very well with measurements for both free space and one homogeneous dielectric layer. Numerical results are compared with measurements of two circular apertures in a 12 inch by 24 inch flat plate for various combinations of frequency, polarization, and spacing in order to verify all aspects of the solution. It was noted that in some cases, the diffractions from the edges of the ground plane caused some appreciable variations in the measured data, although the measurements in all cases tended to scatter about the calculated values. A numerical study was performed in order to determine the effect of higher order TE_mn and TM_mn modes upon the TE₁₁ mode coupling of two circular apertures radiating into free space. The only noticeable effect was a change in the phase of the coupling coefficient when the TM₁₁ mode was included in the calculations. A study of the effects of array size upon the performance of a triangular grid arrangement of circular apertures indicated that the elements near the center of a large phased array have similar radiation characteristics to those of an infinite array, except near the scan angle at which the"blind spot" occurs. At this angle, the reflection coefficient of the center element of the finite array exhibited a sharp peak, but not total reflection as in the infinite array. Also major differences were observed between the reflection coefficients of the center element and the edge elements of the finite array as a function of beam scan. / Ph. D.