### Browsing by Subject "applied mathematics"

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- ItemOpen AccessA quantitative study of the relationship between mindset and academic performance in firstyear mathematics courses at the University of Cape Town(2021) Mokhithi, Mashudu; Campbell, Anita L; Shock, Jonathan PDespite attempts to decrease university drop-out rates, the graduation rate remains low both internationally and locally. Internationally, up to 40% of students who enter higher education do not graduate; in South Africa, the number is higher at 55%. Several studies have found that growth mindset interventions help improve performance in mathematics and language courses. However, most of these studies are carried out outside of South Africa and on children and adolescents. Very little is known about whether and how the growth mindset theory can help improve performance in first-year university courses in South Africa. In this study, the correlation between mindset and performance in first-year mathematics courses is investigated. First-year science, commerce, and engineering students (N=745) enrolled in four different introductory calculus courses participated in this study. Their mindsets were assessed using a survey questionnaire known as the Mindset Assessment Profile (MAP) tool. The reliability of the Mindset Assessment Profile was assessed using Cronbach's alpha coefficient. This was followed by a comparison of mindset scores of students enrolled in different degree programs. Moreover, the participants' average mindset scores in the current study were compared with international mindset scores. The participants' mathematics grades were collected for different assessments during the academic year. The changes in mathematics grades were compared with the mindset scores to examine the relationship between the two variables. The mathematics grade changes were used instead of the grades themselves; this is because the aim was to measure the improvement in mathematics grades rather than the final grade. In the face of failure, students with a growth mindset are predicted to put more effort and seek feedback to improve their grades in subsequent assessments. On average, the participants of this study were growth mindset oriented according to the Mindset Assessment Profile tool. The MAP was moderately reliable, with Cronbach's alpha coefficients ranging between 0.501 and 0.642. Item-by-item analysis showed that reliability could not be improved by the removal of any item in the Mindset Assessment Profile. There was a significant difference between the mindset scores of commerce students and the mindset scores of science and engineering students. Students enrolled in commerce degree programs scored significantly lower than students enrolled in science and engineering degree programs on the MAP. The University of Cape Town students scored higher than Hong Kong university students on the mindset scale but lower than the students in the US. There was no statistically significant correlation between mindset scores and academic performance in any of the degree programs. The correlations were assessed for (a) all the students, (b) students who failed their first mathematics test, and (c) students who scored 75% and above for their first mathematics test. The findings of this study provide a baseline of mindset scores for a South African university population. The tool for measuring mindset may need to be adapted to be better suited for the population outside of the United States. Furthermore, future research should investigate the effects of a growth mindset intervention on academic performance in mathematics grades at the University of Cape Town.
- ItemOpen AccessBiologically motivated reinforcement learning in spiking neural networks(2022) Rance, Dean; Shock, JonathanI consider the problem of Reinforcement Learning (RL) in a biologically feasible neural network model, as a proxy for investigating RL in the brain itself. Recent research has demonstrated that synaptic plasticity in the higher regions of the brain (such as the cortex and striatum) depends on neuromodulatory signals which encode, amongst other things, a response to reward from the environment. I consider which forms of synaptic plasticity rules might arise under the guidance of an Evolutionary Algorithm (EA), when an agent is tasked with making decisions in response to noisy stimuli (perceptual decision making). By proposing a general framework which captures many proposed biologically feasible phenomenological synaptic plasticity rules, including classical SpikeTime-Dependent Plasticity (STDP) rules and the triplet rules, and rate-based rules such as Oja's Rule and BCM rules, as well as their reward-modulated extensions (such as Reward-Modulated Spike-Time-Dependent Plasticity (R-STDP)), I allow a general biologically feasible neural network the ability to evolve the rules best suited for learning to solve perceptual decision-making tasks.
- ItemOpen AccessCyclic universes & direct detection of cosmic expansion by holonomy in the McVittie spacetime(2019) Campbell, Mariam; Dunsby, Peter KlausThis dissertation consists of two parts. They are separate ideas, but both fall into the context of General Relativity using dynamical systems. Part one is titled Cyclic Universes. It is shown that a Friedmann model with positive spatial sections and a decaying dark energy term admits cyclic solutions which is shown graphically by the use of phase planes. Coupling the modified Friedmann model to a scalar field model with cross-sectional terms in order to model the reheating phase in the early universe, it is found that there is a violation of the energy condition, i.e. when the universe is in the contracting phase and re-collapses again. We suspect that the cause for this violation is due to the asymmetry of the solution of w together with the cross-sectional terms at the bounce preceding slow-roll inflation. Part two is titled Thought Experiment to Directly Detect Cosmic Expansion by Holonomy. Two thought experiments are proposed to directly measure the expansion of the universe by the parallel transfer of a vector around a closed loop in a curved spacetime. Generally, expansion would cause a measurable deficit angle between the vector’s initial and final positions. Using the McVittie spacetime (which describes a spherically symmetric object in an expanding universe) as a backdrop to perform these experiments it is shown that the expansion of the universe can be directly detected by measuring changes in the components of a gyroscopic spin axis. We find these changes to be small but large enough (∆S ∼ 10−7 ) to be measured if the McVittie spacetime were a representation of our universe.
- ItemOpen AccessFrom statistical mechanics to machine learning: effective models for neural activity(2022) Schonfeldt , Abram; Rohwer, Christian; Shock, JonathanIn the retina, the activity of ganglion cells, which feed information through the optic nerve to the rest of the brain, is all that our brain will ever know about the visual world. The interactions between many neurons are essential to processing visual information and a growing body of evidence suggests that the activity of populations of retinal ganglion cells cannot be understood from knowledge of the individual cells alone. Modelling the probability of which cells in a population will fire or remain silent at any moment in time is a difficult problem because of the exponentially many possible states that can arise, many of which we will never even observe in finite recordings of retinal activity. To model this activity, maximum entropy models have been proposed which provide probabilistic descriptions over all possible states but can be fitted using relatively few well-sampled statistics. Maximum entropy models have the appealing property of being the least biased explanation of the available information, in the sense that they maximise the information theoretic entropy. We investigate this use of maximum entropy models and examine the population sizes and constraints that they require in order to learn nontrivial insights from finite data. Going beyond maximum entropy models, we investigate autoencoders, which provide computationally efficient means of simplifying the activity of retinal ganglion cells.
- ItemOpen AccessGradings of Lie Algebras(2022) Meyer, Thomas Leenen; Sanchez-Ortega, JuanaThe 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.