The purpose of this paper is to develop an invariant integral for a compact topological group and, then to use that integral to prove the fundamental Peter-Weyl Theorem.

A dissertation submitted in fulfillment of the requirements for the degree of Master of Science in the school of Mathematics , University of the Witwatersrand, Johannesburg, 2018 / MT 2018

Purpose: To investigate reasons for patients not proceeding to resective epilepsy surgery after subdural grid evaluation (SDE). To correlate noninvasive investigation results with invasive EEG observations in a set of patients with nonlesional brain MRIs. Methods: Retrospective study of adult epilepsy patients undergoing SDE during an 8-year period at Cleveland Clinic. Construction of semiquantitative "scores" and Bayesian predictors summarizing the localizing value and concordance between noninvasive parameters in a subset with nonlesional MRIs. Results: One hundred forty patients underwent SDE, 25 of whom were subsequently denied resective surgery. In 10 of 25, this was caused by a nonlocalizing subdural ictal EEG onset. Eight of 10 such patients were nonlesional on MRI. Among all nonlesional patients (n = 34 of 140), n 1 = 10 of 34 patients had nonlocalizing and n2 = 24 of 34 had localizing, subdural ictal onsets. As groups, n1 and n 2 were statistically disjoint relative to their noninvasive scores. Bayesian measures predictive of focal invasive ictal EEG were highest for complete concordance of noninvasive parameters, decreasing with lesser degrees of concordance. A localizing scalp interictal EEG was a particularly good Bayesian prognosticator. Conclusions: A small but significant proportion of SDE patients are denied subsequent therapeutic resective surgery. This is due to several reasons, including a nonlocalizing intracranial ictal EEG. The majority of such patients have nonlesional MRIs. The noninvasive data may be summarized by a semiquantitative score, as well as Bayesian likelihood ratios, which correlate with subsequent invasive outcome. This approach may find use in the selection and counseling of potential surgical candidates offered SDE.

The aim of this work is to introduce differential forms on Euclidean space. The theory of differential forms provides a way of abstracting integration by formalising differentials over which an integral can be taken. The work builds towards Stokes’ Theorem for which a proof is given. Finally, using Stokes’ Theorem, three famous integral theorems from vector analysis are derived.

MacMahon's Master Theorem is an important result in the theory of algebraic combinatorics. It gives a precise connection between coefficients of certain power series defined by linear relations. We give a complete proof of MacMahon's Master Theorem based on MacMahon's original 1960 proof. We also study a specific infinite dimensional matrix inverse due to C. Krattenthaler.

<p> In the soft photon limit the bremsstrahlung cross-section is related to the corresponding elastic scattering cross-section in a model-independent manner. The cross-section for threshold photoproduction of pions becomes independent of the detailed pion-nucleon interaction if the ratio of pion to nucleon mass is made sufficiently small. These two results are a consequence of the gauge invariance of the scattering amplitude for a process involving one photon. We review the derivation of Low's bremsstrahlung theorem for spin-zero and spin-half particles, and outline an extension to all orders of photon energy. The theorem for photopion production is derived by the same method. A non-relativistic analogue to the Bremsstrahlung theorem is also considered.</p> / Thesis / Master of Science (MSc)

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