https://www.um.edu.mt/library/oar/handle/123456789/24337 2026-04-15T13:05:14Z https://www.um.edu.mt/library/oar/handle/123456789/40487 Title: Independent sets of graphs Abstract: A set I of vertices of a graph G is called an independent set of G if no edge of G is a subset of I. The independence number of G is the size of a largest independent set of G and is denoted by α(G). The study of independent sets is a fundamental and widely-studied area of graph theory. We record several central results in the literature, and we include proofs of some of them with the aim of highlighting key ideas. The main focus is on bounds for α(G) and relations it has with other parameters of G. The performance of the bounds is investigated and some of their theoretical implications in different areas of study are discussed. Additionally, some approaches to solving the problem of finding a largest independent set of a graph are studied. Description: M.SC.MATHS 2018-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24356 Title: Cardinal functions for compact and metrizable topological spaces Abstract: Cardinal functions are mappings from the class of topological spaces into the class of infinite cardinal numbers. A systematic study of cardinal functions began in the mid-1960s but most of the fundamental results were obtained long before. Notable researchers such as Alexandroff, Urysohn, Cech, Jones, Hajnal, Juhàsz, Arhangel'skii, and others contributed to the study of cardinal functions. Cardinal functions are very useful to obtain bounds on the cardinality of a topological space X. The importance and study of cardinal functions can also be justified by the fact that such functions where fundamental in solving long standing important problems in topology. We will introduce the basic global and local cardinal functions on a topological space X. Then we will outline some basic inequalities between these cardinal functions. The Pol-Sapirovskii technique will then be applied to obtain some interesting bounds on ƖXƖ. Finally we consider two of the most important classes of topological spaces, namely compact and metrizable spaces. We will see that cardinal functions on compact spaces are very interesting and useful, one only has to mention that weight and net weight coincide for such spaces. Due to the nice qualities of metrizable spaces, inequalities among cardinal functions tend to be dull. However, it is possible to obtain precise information about the cardinality of such spaces. Description: M.SC.MATHS 2017-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24354 Title: Cardinal functions for linearly ordered topological spaces Abstract: Cardinal functions extend such important properties as countable base, separable, and first countable to higher cardinality. Cardinal functions then allow one to formulate, generalise, and prove related results in a systematic and elegant manner. In addition, cardinal functions allow one to make precise quantitative comparisons between certain topological properties. Experience indicates that the idea of a cardinal function is one of the most useful and important unifying concepts in all of set-theoretic topology. A cardinal function is a function ϕ from the class of all topological spaces (or some precisely defined subclass) into the class of all infinite cardinals such that ϕ(X) = ϕ (Y) whenever X and Y are homeomorphic. An obvious example of a cardinal function is cardinality, denoted ƖXƖ and equal to the number of points in X plus ω. Perhaps the most useful cardinal function is weight, defined by ω (X) = min{ƖB Ɩ : B a base for X} + ω . In the thesis, we first define carefully the most important cardinal functions in general topological spaces and study the relationships between them. Important results related to bounds on jXj in terms of other cardinal functions are then investigated. In particular, difficult inequalities like jXj _ 2L(X)__(X) for X 2 T2, jXj ≤ 2c(X)__(X) for X 2 T2, jXj _ 2s(X)_ (X) for X 2 T1, and jXj ≤ 22s(X) for X 2 T2, are studied. Finally, and most importantly, the behaviour of the above mentioned cardinal functions for the specific class of linearly ordered topological spaces is investigated. Description: M.SC.MATHS 2017-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24345 Title: Thermography imaging for medical applications Abstract: The Pennes equation is a heat equation for biological tissue. In the first part of the dissertation, dealing with the direct problem, we investigate the temperature distribution inside a domain consisting of two regions with different thermic properties subject to mixed boundary conditions, using five different numerical methods: the finite element method (FEM), boundary element method, method of fundamental solutions, finite difference method, and spectral method with FEM found to be the best. In the second part dealing with the inverse problem, FEM is used to estimate thermal and geometrical parameters of the system from the temperature distribution on the surface where measurements are available (i.e. skin surface). This scenario pertains to the medical application of infrared thermography. Description: M.SC.MATHS 2017-01-01T00:00:00Z