https://www.um.edu.mt/library/oar/handle/123456789/119307 Sat, 11 Apr 2026 03:00:11 GMT 2026-04-11T03:00:11Z https://www.um.edu.mt/library/oar/handle/123456789/132829 Title: On graph crossing number Abstract: This thesis explores the theory of crossing numbers, a significant area in graph theory focused on minimizing the number of edge crossings in graph drawings. This field, originating from Turán’s Brick Factory problem, has expanded significantly over the past five decades. We present original results on the crossing numbers of several graph families, namely, generalized Petersen graphs, circulant graphs, and the Cartesian product of graphs. In the first chapter we give an introduction to crossing numbers. The theory of crossing numbers has evolved since Turán’s Brick Factory problem, which sought the crossing number of a complete bipartite graph. This chapter outlines the current state of research. We highlight that while exact crossing numbers have been determined for some graph families, many others remain unsolved, often with only conjectured bounds. In Chapter 2, we present our initial results which focus on generalized Petersen graphs. We prove two conjectures by Zhou and Wang (2008) regarding the crossing numbers of two subfamilies of generalized Petersen graphs. Additionally, we develop a method using Watkins’ results to efficiently determine if two generalized Petersen graphs with different parameters are isomorphic. In Chapter 3, we examine the planar crossing number of circulant graphs. We confirm a previously proposed conjecture by Wang and Huang (2008). Additionally, we determine the crossing number for a range of other circulant graphs and initiate the study of a subfamily of circulant graphs that has not been previously investigated. Chapter 4 extends our investigation to the projective plane. The majority of the work that has been done involves the drawing of graphs on the sphere (or equivalently the plane), but few results are known for the crossing number of graphs drawn on other surfaces. Building on the work of Ho (2012) and Cheon (2020), we determine the projective plane crossing numbers for a further subfamily of quartic circulant graphs, and propose a conjecture. In Chapter 5, we obtain the crossing number of the Cartesian product S3□Pn of the star S3 and the path Pn on an infinite number of surfaces. To our knowledge, there is only one other infinite family of graphs for which the crossing number has been determined on every orientable and nonorientable surface, namely the complete bipartite graph K3,n. Our result thus means that we now have the second infinite family of graphs for which the crossing number has been completely determined for any orientable surface and for nonorientable surfaces of sufficiently large Euler genus. This thesis makes significant contributions to the understanding of the crossing number of various graph families on different surfaces. We provide both theoretical advancements and conjectures, offering a foundation for future research in this domain. Description: Ph.D.(Melit.) Mon, 01 Jan 2024 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/132829 2024-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/132112 Title: The Yamabe problem Abstract: Differential geometry is a field of mathematics which deals with the study of smooth manifolds, which generalize the notion of smooth surfaces to n dimensions. The importance of differential geometry, and its subfield of Riemannian geometry, have only increased over time as they have become the language of one of the foundational theories of modern physics, general relativity [27]. A problem of great prominence in the field of Riemannian geometry is called the Yamabe problem, which was formulated as both as an extension to the uniformisation theorem, proved independently by H. Poincaré and P. Koebe in 1907. as well as a step towards solving the Poincaré conjecture. The statement is as follows [29]: The Yamabe Problem. Given a compact Riemannian Fmanifold (M, g) of dimension n > 3, find a metric conformal to g with constant scalar curvature. A solution was presented by Hidehiko Yamabe in 1960 [45], but an error was discovered in the proof and the problem remained as a conjecture until it was proved for all cases in 1984 by R. Schoen, following the work of Yamabe, N. Trudinger and T. Aubin. In this dissertation, we start by presenting a solution to the uniformisation theorem with an emphasis on its historical context, as well as any necessary analytic preliminaries. Next, we develop the language of differential geometry and more specifically geometric analysis, which rose to prominence in no small part due to the Yamabe problem and the Poincaré conjecture. We then present the solution to the Yamabe problem as it developed over time, highlighting the various geometric, analytical and physical concepts which were required for its total solution. Finally, we present some of the mathematical developments which followed the solution of the Yamabe problem, while also exploring some of the connections between the Yamabe problem and the Poincaré conjecture. Description: M.Sc.(Melit.) Mon, 01 Jan 2024 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/132112 2024-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/125423 Title: Solar system dynamics Abstract: This dissertation investigates the complex dynamics of planetary motion, with a specific emphasis on explaining the mechanisms governing Milankovitch cycles. The primary objective is to identify the three fundamental orbital parameters responsible for Earth’s cycle variations and to investigate their impact on fluctuations in solar insolation. To accomplish this, the Milankovitch terms — eccentricity, precession and obliquity — are introduced and their periods are calculated. The computation for the precession period relies on the foundational concept of gyroscope precession. In contrast, numerical apsidal precession is calculated based on the definition of torque and gravitational theory, incorporating gravitational forces from both outer and inner planets. Additionally, the derivation of the Milankovitch equations, highlighting variations in eccentricity and angular momentum, is achieved based on Hamiltonian theory and Poisson brackets. Moreover, this dissertation strives to establish a relationship between solar insolation and the Milankovitch cycles. A relationship between the solar constant at perihelion and aphelion is expressed as a function dependent on eccentricity. A corresponding graph illustrating the variation in solar insolation ratio between perihelion and aphelion is presented. Additionally, the solar insolation values across different latitudes are discussed by modifying one or all of the three Milankovitch terms. To present these outcomes, a brief review of conic properties, including the equation of an ellipse, and Kepler’s three laws, is provided. Additionally, the methodology to calculate planetary positions relative to the Sun and to other planets at specific times is outlined. In investigating orbital dynamics, the role of the disturbing function, in understanding how the gravitational pull from massive bodies influences the motion of other bodies, is analyzed alongside secular perturbations. Finally, the dissertation concludes by exploring the possibility that the Milankovitch parameters, accountable for Earth’s orbit variations around the Sun, could explain Earth’s current warming trend. Description: B.Sc. (Hons)(Melit.) Mon, 01 Jan 2024 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/125423 2024-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/125328 Title: Proximities on Tychonoff spaces : Smirnov theorem Abstract: In the following text we define the axioms of a proximity relation δ as defined by Efremovic and explore some properties derived from these basic rules. We briefly discuss two alternative approaches to defining a proximity relation on a set X, first by the dual binary relation ≪ called a proximity neighbourhood relation, and later by the use of a collection of clusters σ ∈ P (P (X)). We recall the topological separation axioms up to T4, and define a relation between proximities and topologies by way of “compatibility”. We further explore proximal mappings and equimorphisms as parallel structures to continuous mappings and homeomorphisms, and their associated properties. The properties of clusters are explored with consideration for how they mirror ultrafilters, and we expand upon this relation to prove many of their properties. Next, we introduce Hausdorff compactifications, i.e. T2 compact extensions, of a Tychonoff topological space and prove some very important results in this area. From there we follow in the footsteps of Smirnov to establish a one-to-one correspondence between the set of all compatible proximities and the set of all T2 compactifications on a Tychonoff space X. Description: B.Sc. (Hons)(Melit.) Mon, 01 Jan 2024 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/125328 2024-01-01T00:00:00Z