<p>Polya’s theorem can be used to enumerate objects under permutation groups. Using grouptheory, combinatorics and some examples, Polya’s theorem and Burnside’s lemma arederived. The examples used are a square, pentagon, hexagon and heptagon under theirrespective dihedral groups. Generalization using more permutations and applications tograph theory.Using Polya’s Enumeration theorem, Harary and Palmer [5] give a function whichgives the number of unlabeled graphs n vertices and m edges. We present their work andthe necessary background knowledge.</p>

Polya’s theorem can be used to enumerate objects under permutation groups. Using grouptheory, combinatorics and some examples, Polya’s theorem and Burnside’s lemma arederived. The examples used are a square, pentagon, hexagon and heptagon under theirrespective dihedral groups. Generalization using more permutations and applications tograph theory.Using Polya’s Enumeration theorem, Harary and Palmer [5] give a function whichgives the number of unlabeled graphs n vertices and m edges. We present their work andthe necessary background knowledge.

The goal of this paper is to develop extensions of Polya enumeration methods which count orbits of functions. De Bruijn, Harary, and Palmer all worked on this problem and created generalizations which involve permuting the codomain and domain of functions simultaneously. We cover their results and specifically extend them to the case where the group of permutations need not be a direct product of groups. In this situation, we develop a way of breaking the orbits into subclasses based on a characteristic of the functions involved. Additionally, we develop a formula for the number of orbits made up of bijective functions. As a final extension, we also expand the set we are acting on to be the set of all relations between finite sets. Then we show how to count the orbits of relations.

This dissertation proposes stress testing of a bank’s corporate credit portfolio in a Basel Internal Ratings Based (IRB) framework, using publicly available macroeconomic variables. Corporate insolvencies are used to derive a credit cycle index, which is linked to macroeconomic variables through a multiple regression model. Probability of default (PD) and loss given default (LGD) that are conditional on the worst state of the credit cycle are derived from through-the-cycle PDs and LGDs. These are then used as stressed inputs into the Basel regulatory and Economic capital calculation for credit risk. Contrary to the usual expert judgement stress testing approaches, where management apply their subjective view to stress the portfolio, this approach allows macroeconomic variables to guide the severity of selected stress testing scenarios. The result is a robust stress testing framework using Rösch and Scheule (2008) conditional LGD that is correlated to the stressed PD. The downturn LGD used here is an alternative to the widely used Federal Reserve downturn LGD which assumes no correlation between PDs and LGDs. / Dissertation (MSc)--University of Pretoria, 2014. / gm2014 / Mathematics and Applied Mathematics / Unrestricted