/library/oar/handle/123456789/83109 2025-11-14T21:17:38Z 2025-11-14T21:17:38Z /library/oar/handle/123456789/91974 2022-03-22T15:27:46Z 2021-01-01T00:00:00Z

Title: Differential equations and dynamical systems with applications in demography, epidemiology and economics Abstract: A theoretical overview of differential equations and dynamical systems is presented in the first four chapters of the dissertation. The theory is developed systematically, starting with linear and non-linear differential equations in one and two dimensions. Techniques such as proving the existence of closed orbits, bifurcation theory, chaos and iterated maps are dealt with in this section. A brief survey of population models is given in Chapter 5, followed by an in-depth analysis of the Leslie matrix model which is frequently used to describe the dynamics of an age-structured population. A two-sex model is used to project the population of the Maltese islands for the year 2024, taking into consideration the huge impact of the COVID-19 pandemic on migration. Two different scenarios are presented where the model predicts a total population of 529,846 people when considering ‘low’ migration and 580,955 people when considering ‘high’ migration. An overview of classical compartmental models is given in Chapter 6. This idea is extended to multi-site compartments with travelling patterns. An aggregated plot of infected individuals is both modelled and predicted given that the sites are connected via random trees. COVID-19 datasets are used to estimate model parameters applying data fitting. The extended Kalman filter is also implemented to estimate both states and parameter values. The last chapter is dedicated to two economic models which will be examined from a mathematical perspective: The Goodwin model and a model for the relationship between unemployment and inflation. Description: M.Sc.(Melit.)

2021-01-01T00:00:00Z /library/oar/handle/123456789/83336 2021-11-03T14:02:50Z 2021-01-01T00:00:00Z

Title: The labelling of graphs through colouring Abstract: Graph theory is a well-known area in the research field of mathematics. Its evolution started off with the quest of solving questions or games involving numbers. This led to the colouring of graphs; a way of labelling graph vertices or edges while having some constraints. In this dissertation, four chapters are presented. The first three, which constitute the major part of the thesis, are related to vertex graph colouring. The final one is a very short section on edge colouring. By researching vertex colouring, a number of key results were identified. A particular and important one is Brooks’ Theorem. This result can be proved to be true by making use of different techniques. In this study six of these methods are taken into account and presented in the second chapter as follows: a) A Greedy Colouring of G, b) A Partitioning Approach, c) Kempe Chains, d) Reducing to the cubic case, e) Kernel Perfection, and f) The Degree Choosable Graph. A well known subsection of vertex colouring and which has been recently regarded as an area of interest is list colouring and choosability. The famous result by Thomassen about the choosability of planar graphs is discussed, whilst mentioning other key results related to list colouring and choosability. The final chapter of this research is about Vizing’s Theorem, Vizing’s Theorem for multigraphs and Vizing’s Theorem for a precolouring extension of G. Description: M.Sc.(Melit.)

2021-01-01T00:00:00Z