https://www.um.edu.mt/library/oar/handle/123456789/10990 2026-04-11T14:24:23Z 2026-04-11T14:24:23Z https://www.um.edu.mt/library/oar/handle/123456789/91635 2022-03-17T10:54:17Z 2016-01-01T00:00:00Z

Title: An integrated system for telescope control using FPGA(s) Abstract: Astronomical radio interferometers typically consist of an array of antennas, widely separated and connected using a transmission line. Signals recorded from each antenna are sent to a back-end where they are digitised, channelised and superposed, giving rise to constructive / destructive interference depending on their phase difference. This technique of interferometry is called aperture synthesis and it results in a combined telescope with a resolution equivalent to that of a single antenna with a diameter equal to the spacing of the antennas furthest apart in the array. When we consider a radio signal from a celestial source, incident on two antennas separated by a distance, the antennas receive the radio signal with a time difference, causing the signals to be out of phase. In order to achieve power amplification, the recorded signals must be phased using a technique called beamforming. A beamformer, has the added capability of steering the beam of a radio telescope using constructive / destructive interference. The aim of this project was to design and implement an integrated control system (ICS) for the digital steering of beams in a radio telescope. The system design was based on the PAPER F-engine and implemented on the ROACH2 FPGA board. A beamformer module was designed using the CASPER XPS tool-flow to be capable of continually steering beams in the sky, whilst monitoring the behaviour of a phased array system to ensure that the beams are continually tracking a source. Description: B.SC.(HONS)MATHS&PHYSICS

2016-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/13461 2018-06-27T06:26:05Z 2016-01-01T00:00:00Z

Title: Σ-hypergraphs : colouring, independence, matchings and hamiltonicity Abstract: A hypergraph H is a nite set V (H) of vertices together with a family E(H) of subsets of V (H) called hyperedges (or simply edges). An r-uniform hypergraph is one in which all the edges are of the same size r. So a standard graph can be defined as a 2-uniform hypergraph. Hypergraphs arose naturally as an extension to graphs, with the main intent of extending important and interesting results from Graph Theory to this generalised setting. The results were sometimes simplified, and in other instances unexpected. In this thesis we study various properties of a special family of hypergraphs, the -hypergraphs, a generalisation of -hypergraphs, first defined by Caro and Lauri and studied in the context of Voloshin colourings, focusing mainly on non-monochromatic-non-rainbow (NMNR) colourings. The most interesting aspect of these types of colourings is arguably the appearance of gaps in the chromatic spectrum of the hypergraph. Results in this area of study often required ad-hoc and complex hypergraph structures such as designs. Both - and -hypergraphs proved to be a simple unifying construction in this respect giving interesting results and chromatic spectra by controlling the parameters which define the structure. Furthermore, they also proved to have interesting properties with respect to other hypergraph theoretic properties such as independence, matchings and Hamiltonicity, as we shall see in this thesis. Description: PH.D.MATHS

2016-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/13460 2018-06-27T06:35:45Z 2016-01-01T00:00:00Z

Title: On the spectra and walks of molecular and controllable graphs Abstract: Walks in a graph G relate to its configuration and to the eigenspaces of its adjacency matrix G. In this thesis, we extend the notion of walks. It impacts on the inverse of G, settling a conjecture on the conductivity properties of molecules modelled by chemical graphs, whose vertices represent carbon atoms having degree at most three. Walks also bring out structural properties of a controllable system represented by G. The product of the weights of the edges in a walk of length on a weighted looped graph G is called the walk weight. We deduce closed form expressions for the generating function of the sum of walk weights of G between vertices among two subsets of the vertices of G. We produce three expressions for this generating function: one in terms of the eigenvalues and eigenspaces of G, one in terms of the inverse of the matrix, where I is the identity matrix, and one in terms of the characteristic polynomials of G and those of related graphs, called overgraphs, obtained from G by adding a vertex. In the latter case, characteristic polynomials with a reversal of their coefficients, called reverse polynomials, are employed. We introduce the notion of a walk of negative length on the graph G as being a walk on the weighted looped graph whose adjacency matrix is the inverse of that of G. Similar to the case for the usual, positively{long walks, we provide generating functions producing sums of walk weights of negative length, and relate these walk weight sums with those of positive length. Remarkably, if we use characteristic polynomials to produce generating functions of the sum of the weights of walks of negative lengths, the usual characteristic polynomials are used, rather than their reversal as in the case for positively long walks. In the literature, the walk matrix of a graph G is a square matrix whose ijth entry is the number of walks of length between vertex j and any other vertex of G. We extend this notion to walk matrices whose entries represent walks between each vertex and a subset of the vertex set of G. Hankel matrices whose entries enumerate walk weights of a graph are also discussed. They turn out to be Gram matrices of the columns of the walk matrix of the graph. Another matrix in the form of a Toeplitz matrix, combining walks of both positive and negative length, is also introduced. A new walk matrix that mimics the properties of the commonly used walk matrix, but containing the sum of walk weights of closed walks in G instead of the number of walks, is also proposed. Our study of properties revealed in walk matrices contributes to new discoveries in molecular chemistry and in control theory. Description: PH.D.MATHS

2016-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/10995 2018-06-27T06:55:37Z 2016-01-01T00:00:00Z

Title: Two-fold isomorphisms Abstract: In this dissertation, we present a natural generalization of the notion of isomorphism of a graph or digraph G, namely two-fold isomorphisms from a graph or digraph G to a graph or digraph H. A mixed graph is a pair G = (V (G),A(G)) where V (G) is a finite set and A(G) is a set of ordered pairs of elements of V (G). The elements of V (G) are called vertices and the elements of A(G) are called arcs. When referring to an arc (u, v), we say that u is adjacent to v and v is adjacent from u. The vertex u is the tail and v is the head of a given arc (u, v). An arc of the form (u, u) is a loop. A mixed graph cannot contain multiple arcs, that is, it cannot contain the arc (u, v) more than once. A set S of arcs is self-paired if, whenever (u, v) 2 S, (v, u) is also in S. If S = {(u, v), then(v, u)}, then we identify S with the unordered pair {u, v}; this unordered pair is called an edge. It will be useful to consider two special cases of mixed graphs. An undirected graph or, in short graph, is a mixed graph without loops whose arc-set is self-paired. The edge set of a graph is denoted by E(G). A digraph is a mixed graph with no loops in which no arc is self-paired. A two-fold isomorphism (TF-isomorphism) is a pair (α, β) of permutations of the vertex set V (G) which acts on ordered pairs of vertices of G in the natural way and such that (u, v) is an arc in G if and only if (α(u), β(v)) is also an arc in H. When α = β the TF-isomorphism is just a usual isomorphism. The following is a brief outline of this dissertation. Chapter 1: In this chapter, we define canonical double covers, incidence double covers and alternating double covers. We also define alternating trails or A-trails. Furthermore, we introduce the notion of stability of graphs. In later chapters, seemingly unrelated concepts and ideas are brought together through the concept of two-fold isomorphism. Chapter 2: As far as we know, TF-isomorphisms of graphs were first introduced by Zelinka [35, 36] who was motivated by the study of isotopy of groupoids, referring to what we call ‘TF-automorphisms’ as ‘autotopies’. Zelinka’s work had been overlooked for decades. Here we review the early work by Zelinka [34, 35, 36, 37, 38], Porcu [26], Pacco and Scapellato [24], and Maruˇsiˇc, Scapellato and Zagaglia-Salvi [21, 22] which, in hindsight, involved TFisomorphisms. Chapter 3: In this chapter we start developing material to tackle our problems. We start by addressing questions concerning invariance under TF-isomorphisms. Then we use alternating double covers of graphs to be able to extend certain results previously known to hold for graphs to mixed graphs. We conclude this chapter by defining an equivalence relation on the arc-set of any mixed graph G and bringing together the concept of A-trails and the construction of graph covers to ultimately explain how TF-isomorphisms arise. Chapter 4: In this chapter, we use and develop new theory with the aim of constructing graphs with specific properties. In particular, we are interested in asymmetric graphs which have a non-trivial TF-automorphism group. We construct a smallest unstable graph using two different constructions, one of which will be generalized in Chapter 5. We also show that a graph and a digraph can be TF-isomorphic. Chapter 5: In this chapter, we show how TF-isomorphisms provide a fresh outlook when considering the concept of unstable graphs. We first show that there is a link between TF-automorphisms and graph stability. We then present simple arguments concerning triangles to ultimately present a method for constructing unstable graphs of arbitrarily large diameter. We also generalize one of the methods used to construct a smallest asymmetric unstable graph and asymmetric unstable graphs of arbitrarily large instability index. Chapter 6: In this chapter, we look at TF-orbitals in some more detail. We define the TF-rank of a group of TF-permutations and look at the situation where the TF-rank equals 1 or 2. We then apply this to unstable rank 3 strongly regular graphs. We consider further properties on a TF-permutation group which force the TF-orbitals to satisfy structure constants similar to those of a coherent configuration. Chapter 7: In the final chapter, we review in a little more detail the connection between TF-isomorphisms and problems such as the Neighbourhood Reconstruction Problem. We also discuss a method suggested by Klin to extend the notion of stability to digraphs and finally we reveal new avenues for research which we plan to explore in the near future. Chapter 1 and Chapter 3 include results found in [16, 17, 18]. Chapter 4 includes constructions found in [16]. Most of the contents of Chapter 5 appear in [18], whereas the contents of Chapter 6 are included in a paper which at the time of writing has been accepted for publication in the Journal of Combinatorial Mathematics and Combinatorial Computing. At the time of writing, a paper which describes the construction of a smallest unstable graph and of a family of graphs with an arbitrarily large index of stability is being prepared. These constructions are found in Chapter 4 and Chapter 5 respectively. Description: PH.D.MATHS

2016-01-01T00:00:00Z