/library/oar/handle/123456789/97952 2025-11-05T03:09:04Z /library/oar/handle/123456789/101432 Title: An introduction to the theory of distributions Abstract: When working with differential equations, it can be seen that a solution could not always be found. As centuries passed, the notion of the δ "function" started taking shape, as it could help solve these differential equations. Notable mathematicians such as Oliver Heaviside and George Green making contributions towards it. It was not until the early 20th century when P.A.M Dirac used it to further our understanding of Quantum Mechanics. This prompted a need for an entire theory centred around these type of functionals, named "distributions". In this dissertation we will discuss the theory behind distributions as well as arrive to one of its most important theorems, The Ehrenpreis-Malgrange Theorem. Description: B.Sc. (Hons)(Melit.) 2022-01-01T00:00:00Z /library/oar/handle/123456789/101426 Title: Hamiltonicity in Cayley graphs and digraphs Abstract: The existence of Hamiltonian paths and cycles has always been of interest, not necessarily just within graph theory. For example, the problem bears a strong relation with group theory: given a finite set of generators of a group, can one construct a finite sequence s1,s2, . . .sr from these generators such that every word s1s2 . . .si corresponds to a unique element of the group? Such a sequence of words would imply a Hamiltonian path in the Cayley graph of the group with the given generators forming the connecting set. We begin by introducing fundamental elements of graph and group theory, and the notion of a Cayley graph. We then introduce a conjecture of Lovász, on the existence of Hamiltonian cycles in finite connected Cayley graphs. In this manner we reconcile the Hamiltonian problem in graph and group theory. We then introduce a number of techniques, through which a number of classes of groups have been shown to have Hamiltonian Cayley graphs. We use these techniques to prove a result of Marušiˇc (1983), that every Cayley graph for every finite Abelian group has a Hamiltonian cycle. We will also consider the pioneering work of Rankin (1948, 1966) on groups with a generating set of size 2. We also consider Cayley digraphs and provide examples of infinite classes of such graphs which do not have a Hamiltonian cycle. Consequently, a variant of Lovász’s conjecture for Cayley digraphs cannot be stated. However, we prove a result of Holszty ´nski and Strube (1978) that every Cayley digraph of an Abelian group has a Hamiltonian path, hence providing a holistic overview of the problem in the case of Abelian groups. We conclude by proving a modern result, due to Pak and Radoici´c ˘ (2009), showing that every group has a small generating set such that the corresponding Cayley graph has a Hamiltonian cycle. This is followed by a short survey of further results and open problems. In this manner, we hope to present the reader with a foundation for carrying out research in this area, along with evidence suggesting that Lovász’s conjecture for Cayley graphs holds in the positive. Description: B.Sc. (Hons)(Melit.) 2022-01-01T00:00:00Z /library/oar/handle/123456789/101417 Title: Optimal control theory with applications in portfolio and consumption optimization Abstract: The main goal of a financial portfolio manager is to construct a high return portfolio. Additionally, different consumers have different risk exposure, which the portfolio manager has to identify and construct a portfolio well suited for their clients. Moreover, retirees have to spend their savings in the most efficient way. This can be done by maximizing their utility of consumption. This thesis treats the maximization the utility of consumption in two different methods. The first uses Calculus of variations and the other is using the maximum principle. On the other hand, financial portfolios were created using discrete and continuous maximum principle, and convex optimization. Finally, an example using stochastic optimal control constructs a portfolio using a risky asset and a risk free asset. One cannot compare the different methods used to construct the portfolios, because they all account for the risk differently. It is up to the portfolio manager to decide which methods to use. Description: B.Sc. (Hons)(Melit.) 2022-01-01T00:00:00Z /library/oar/handle/123456789/101404 Title: Application of optimal control theory to firm financing and investment Abstract: In modern society, one can notice the tendency to optimize every action that can be measured with a valuable unit. For instance, in economics, the unlimited demand has to be satisfied with limited supply, while in finance an unlimited number of investment opportunities arise for a finite amount of funds. One shouldn’t base certain allocation decisions purely on intuition or made-up signals as it would turn every economy and firm into a large gambling game. This dissertation aims to determine how allocation processes, specifically sequential investments, firm financing, price forecasting, and algorithmic trading, can be optimized based on certain mathematical techniques found in Optimal Control theory. The results obtained show that optimization can indeed happen, with the establishment of a switching point for a company to change its pay-out policy, various simulations for the prediction of crude oil prices, a systematic process on how to identify the optimal allocation in sequential investments, and the use of optimal filtering for algorithmic trading signals. To visualize the above results, the dissertation was structured so that the reader can comprehend the logic of optimal control theory prior to any application. Description: B.Sc. (Hons)(Melit.) 2022-01-01T00:00:00Z