https://www.um.edu.mt/library/oar/handle/123456789/404 2026-04-23T16:53:16Z https://www.um.edu.mt/library/oar/handle/123456789/145690 Title: Non-commutative probability : from classical to quantum probability Abstract: In classical probability theory, it is implicitly assumed that random variables commute. However, this assumption does not necessarily hold in all mathemat ical frameworks, such as those involving matrices. In this thesis, we will explore quantum probability, a non-commutative extension of classical probability. The main aim of this thesis shall be to examine how key concepts from classical probability can be generalized in the quantum setting. Description: M.Sc.(Melit.) 2025-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/144070 Title: Intrinsic topologies on ordered structures : applications Abstract: This thesis is divided into two main parts. The first three chapters focus on the study of order and unbounded order convergence. We examine various well-established definitions of order convergence, that have emerged over time, each associated with a corresponding topology. These notions are compared in detail, with particular attention to the conditions under which they coincide or differ. Furthermore, in the context of a semi-finite measure space, we investigate the relationship between the topologies on L∞ arising from the duality (L∞, L1), and we compare these to the order topology. Notably, we establish a condition under which the Mackey topology is strictly weaker than the order topology. Compared to order convergence, unbounded order convergence is relatively new and is generally studied on Riesz spaces. In this thesis, we explore it in the broader context of lattices. Our results show that, similar to the order topology, the unbounded order topology is independent of the definition of order convergence. In addition, we extend key properties known to hold in Riesz spaces to lattices. We prove that order continuity of unbounded order convergence is equivalent to the lattice being infinitely distributive. Moreover, we show that the O-closure and uO-closure of a sublattice coincide and form a sublattice. Furthermore, we show that the uO-adherence of an ideal is an O-closed ideal. We also examine the MacNeille completion of a sublattice Y relative to that of a lattice L, identifying two conditions under which the completion of Y embeds regularly in that of L. The last chapter is dedicated to the study of lattice uniformities. It is known that for a locally solid Riesz space (X, τ ) there exists a locally solid linear topology uτ on X such that unbounded τ -convergence coincides with uτ -convergence. This topology is the weakest locally solid linear topology that agrees with τ on all order bounded subsets. Thus, for a uniform lattice (L, U), we introduce the weakest lattice uniformity U∗ on L that coincides with U on each order bounded subset of L. We see that if U is the uniformity induced by the topology of a locally solid Riesz space (X, τ ), then the U∗ -topology coincides with uτ . We also provide answers to several questions posed in [44, 37]. Description: Ph.D.(Melit.) 2026-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/143413 Title: Saved by the rook : a case of matchings and Hamiltonian cycles Authors: Abreu, Marién; Gauci, John Baptist; Zerafa, Jean Paul Abstract: The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set M containing pairs of cells such that each cell of the m1×m2 chessboard is in exactly one pair, we determine the values of the positive integers m1 and m2 for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in M and some edges of the rook graph. This is an alternative formulation of a graph-theoretical problem presented in [1] involving the Cartesian product G of two complete graphs Km1 and Km2 , which is, in fact, isomorphic to the m1×m2 rook graph. The problem revolves around determining the values of the parameters m1 and m2 that would allow any perfect matching of the complete graph on the same vertex set of G to be extended to a Hamiltonian cycle by using only edges in G. 2025-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/142588 Title: Analytical solutions to the Laplace equation on a hemispherical domain Authors: Sebu, Cristiana; Amaira, Andrei; Pidcock, Michael Abstract: In this paper, we derive analytical solutions to the Laplace equation in a hemispherical domain subject to two different idealized Neumann boundary conditions. The solutions are given as infinite series, and their convergence is analysed. The theoretical results have been validated by comparing them with numerical results obtained using EIDORS. 2025-01-01T00:00:00Z