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The main focus of this dissertation is to present an introduction to gradings of Lie algebras. The aim is twofold: to lay the necessary foundations to become (in the near future) an algebraist working in this area of research, and to tackle the problem of finding and classifying the Lie algebras arising as graded contractions of a specific (Z_2)^3-grading of the Lie algebra g_2. As a result, this dissertation consists of five chapters and six appendices which might appear very different, at first glance, but they are indeed connected since they are all very important tools for anyone interested in gradings of Lie algebras. The first chapter is devoted to introducing Lie algebras and the study of semisimple Lie algebras. This chapter will place into context much of the work of the second chapter. We begin by describing the basic notions of Lie algebras and their representations, and continue with the study of the Killing form of a Lie algebra, as well as, the root space decomposition. In the second chapter we study root systems and their bases. This leads to an investigation of the Weyl group associated to a root system. This work allows us to describe how one can uniquely extend isomorphisms between root systems to isomorphisms between Lie algebras to which those root systems correspond. We are then able to describe the special properties of Chevalley bases. Gradings make their first appearance in Chapter three. We quickly shift our focus to group gradings. We describe a process to obtain a universal grading group amongst equivalent gradings. We spend some time preparing and presenting an example of this process. The chapter ends with some results relating to the automorphisms of a grading. We present the construction of the exceptional Lie algebra g_2 in the fifth chapter. This chapter uses some definitions and results which are presented in Appendix B. We start by looking at useful results relating to alternative algebras. Then we introduce upper bounds to the dimension of g_2. Finally we show that g_2 is 14- dimensional and we construct an important (Z_2)^3-grading of g_2. In the fifth and final chapter we study graded contractions. This work continues into Appendix A, however this is the newest work and as such is still under revision. It is worth mentioning here that the bulk of Chapter 5 and Appendix A is original work under construction. It is the result of an ongoing collaboration with Dr Cristina Draper and Dr Juana Sánchez- Ortega. Although some of the proofs may be shortened in the future, we decided to include them as we are excited about the findings. After introducing the notions for general Lie algebras and gradings we look specifically at the grading on g2 which we constructed in the previous chapter. We are now in a position to attack the problem of finding and classifying the graded contractions relating to the Z3 2-grading of g2 presented in Chapter 4. The definitions in the first section of this chapter come from. The rest of Chapter 6 and Appendix A consist of original work completed for this dissertation. Tensor products of modules over a commutative ring R are the sole focus of Appendix B. We explicitly construct the tensor product of two R-modules, and see how all multi-linear maps filter through tensor products. This is followed by a collection of results chosen to help build intuition for the structure and workings of the tensor product. Lastly, we examine how tensor products interact with direct sums and how linear maps, between modules, may induce maps between the tensor products of those modules. Appendix C is centred around affine group schemes. We introduce the topic as familiarity with this area presents opportunities for future research problems and investigations. Our main aim in this chapter is to describe Hopf algebras. Appendix D is focussed on presenting a proof of Weyl's Theorem, used in the third chapter. Such a proof requires results about the Jordan canonical form of a matrix and the Casimir operator of a Lie algebra representation. The main goal of Appendix E is to describe the differential of a Lie group homomorphism. We make use of this work in Chapter 3. Before we can study the differential of a Lie group homomorphism we need to study matrix Lie groups and the exponential map. The last appendix, Appendix F, is a brief summary of important definitions and results related to the octonions. We need this work to accomplish our goals in Chapter 5.

Genomes exhibit a striking amount of complexity across a broad range of scales. This includes variation in the spatial distribution of features such as genes and transposable elements (TEs), which is observed both between species and among individuals in natural and artificial populations. Additionally, all eukaryotes studied to date have had gene duplications occur in their evolutionary history. In this dissertation, we develop a statistical method for analyzing relative changes in the expression of duplicated genes. We show that this method performs better than could otherwise be achieved using traditional methods of differential gene expression analysis. We apply this method to the analysis of subgenome expression dominance in two polyploid plant species. In both cases, it is shown that dominant subgenomes have a lower abundance of transposable elements. We then revisit the classical theory of the population genetics of transposable elements and show that the population variance in TE copy number predicted by this theory conflicts with empirical results from two naturally occurring and distinct populations. Finally, we develop both an analytic and simulation based approach to address this discrepancy, and discuss how these models can be connected with actual data as a step towards developing a more complete understanding of the evolution of the genome-wide distribution of genes and transposons.

We show how to compute the discrete symmetries for a given Black-Scholes (B-S) partial differential equation (PDE) with the aid of the full automorphism group of the Lie algebra associated to the standard B-S PDE. The paper determines the discrete symmetries using two methods. The first is by G. Silberberg which determines the full automorphism group by constructing the symmetry generators' centralizer and Lie algebra's radical. The other is by P. Hydon which is based on the observation that the adjoint action of any point symmetry of a partial differential equation is an automorphism of the PDE's Lie point symmetry algebra [27]. Automorphisms are essential for constructing discrete symmetries of a given partial differential equation. How does one _t in this mathematical concept in the application of finance? The concept of arbitrage which in certain circumstances allows us to establish the precise relationship between prices and thence how to determine prices, underlies the theory of financial derivatives pricing and hedging [40]. We use arbitrage together with the Black-Scholes model for asset price movements when trading derivative securities. 1Arbitrage is used to creating a portfolio and the discrete symmetries show how to create a portfolio. Gazizov and Ibragimov [10], computed the Lie point symmetries of the Black-Scholes PDE and found an infinite dimensional Lie algebra of infinitesimal symmetries generated by the operators. Discrete symmetries are more effective on PDEs since they are not held back by boundary conditions and are used in1. equivalent bifurcation theory; 2. construction of invariant solutions; 3. simplification of numerical schemes. 4. used in put-call parity relationship (see application in finance); 5. used in put-call symmetry relationship (see application in finance)

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[NOT OCR'D] Grazing animals in the semi-arid tropics are subjected to short or long periods of moderate to severe undernutrition. Many simulation models were developed to represent the mechanisms of ruminant adaptation. These mechanisms include, among others, the differential mobilization of tissues, the recycling of nitrogen to the rumen via the saliva and across the rumen wall, and the relation between intake and animal size. However, most simulation models have attempted to represent the mechanisms for above-maintenance nutritional restrictions. In this study, a simulation model, a linked rumen and intermediary metabolism model (RUMET), is developed to simulate rumen function and nutrient utilization during continuous growth, undernutrition (submaintenance) and realimentation for growing cattle.

Information Operations (IO) are campaigns waged by covert, powerful entities to distort public discourse in a direction that is advantageous for them. It is the behaviors of the underlying networks that signal these campaigns in action, not the specific content they are posting. In this dissertation we introduce a social media analysis system that uncovers these behaviors by analyzing the specific post timings of underlying accounts and networks. The presented method first clusters tweets based on content using Natural Language Processing (NLP). Each of these clusters - referred to as topics - are plotted in time using the attached metadata for each tweet. These topic signals are then analyzed using the Dynamic Wavelet Fingerprint (DWFP), which creates binary images of each topic that describe localized behaviors in the topic's propagation through Twitter. The features extracted from the DWFP and the underlying tweet metadata can be applied to various analyses. In this dissertation we present four applications of the presented method. First, we break down seven culturally significant tweet storms to identify characteristic, localized behavior that are common among and unique to each tweet storm. Next, we use the DWFP signal processing to identify bot accounts. Then this method is applied to a large dataset of tweets from the early weeks of the Covid-19 pandemic to identify densely connected communities, many of which display potential IO behaviors. Finally, this method is applied to a live-stream of Turkish tweets to identify coordinated networks working to push various agendas through a volatile time in Turkish politics.

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