# Topics for Honours Projects 2021 (random order)

### 2021

(Projects in Blue brought forward from 2020 - please confirm with Supervisors)

1. Weak solutions of parabolic and hyperbolic partial differential equations (Supervisor: Prof J H van der Walt) Parabolic and hyperbolic partial differential equations (PDEs) are used as mathematical models
for a number of physical processes, included heat conduction and the propagation of waves. In
fact, the heat equation and the wave equation are, respectively, the model parabolic and hyperbolic

2. The Laplace Transform (Supervisor: Prof J H van der Walt) The Laplace transform has a number of applications, in particular to differential equations. read more

3. The Riesz Representation Theorem (Supervisor: Prof J H van der Walt) In your third year analysis course, you encountered the Riemann-Stieltjes integral. Recall the definition: Definition 1 (Riemann-Stieltjes integral) If f;g: .... read more

4. Prof Baire's Marvelous Category Theorem (Supervisor: Prof J H van der Walt) A Baire Category Theorem is a topological theorem that guarantees that a particular space X is `large', in the sense that it cannot expresses as the union of countably many `small' pieces. Despite its apparently simple formulation, this is a deep and powerful theorem in Mathematics, with many applications in Functional Analysis, Topology, Measure Theory and Set Theory. What do we mean by `large' and `small'? read more

5. Relationship between topology and order (Supervisor: Dr Mokhwetha Mabula) Let X be non-empty set and < be a relation on X. We study the topologies induced by a relation < on X and also a relation induced by a topology t on X. read more

6. High dimensional geometry (Supervisor: Dr Daniel Fresen) This project deals with Rn when n is large, say n = 1010 or n → infinity. Such high dimensional Euclidean space has a very interesting geometry that sometimes contradicts the intuition that we have based on our 'experience' of R2 and R3read more

8. Alternative proofs of the compactness theorem in first-order logic (Supervisor: Dr R Kellerman) The Compactness Theorem, which is one of the fundamental results in first-order model theory, states that a set of first-order sentences Σ has a model if and only if each finite subset of Σ has a model....read more

9. A Zero-One Law For Graphs (Supervisor: Dr R Kellerman) Given a statement S about graphs (e.g. S could be the statement "there exists an Euler circuit"), let Pn (S) denote the probability that a randomly chosen graph with n vertices satisfies S. A property S is called... read more

10. The Polya Enumeration Theorem (Supervisor: Dr R Kellerman) The Polya Enumeration Theorem is a powerful theorem in combinatorics that can be used to count arrangements of objects in instances where certain arrangements are to be viewed as equivalent due to e.g. symmetry. read more

11. The Class Equation in Group Theory and its Applications (Supervisors: Prof James Raftery) In a group G, we say that b is a conjugate of a if b = cac-1 for some c. This is an equivalence relation on G, so G is partitioned by its conjugacy classes. The conjugacy class {cac-1: cЄG} containing a is denoted by (a). It consists of nothing but a if a commutes with all elements of G... read more

12. Energy Options (Supervisor: Dr R Kufakunesu) Pricing energy options differs in many respects to equity derivatives, say, but have many similarities with interest rate derivatives. For instance, the Heath-Jarrow-Merton (HJM) framework was originally introduced to model products in fixed income markets. How can this framework be applied to energy derivatives? In this project a student should explore this...read more

13. When does Lp (μ;X) (resp. C(K;X)) contain a copy of c0; l1; l? What about complemented copies?  (Supervisor: Dr Charles Maepa) Let X and Y be Banach spaces. We say that X has a (complemented) copy of Y if X has a (complemented) subspace which is isomorphic to Y. We'll denote these by XY and X(c) Y. Let X be a Banach space and let (Ω,Σ,µ) be a positive measure space... read more

14.The nearest point problem in metric spaces (Supervisor: Dr Charles Maepa) Notations and Conventions Assumed. Let (X; d) be a non-empty metric space, S a subset of X and x 2 X. Define the distance function from x to S to be dist(x; S) = inffd(x; s)js 2 Sg. This distance depends, of course, on the metric, and if it is necessary to specify the metric in question, we'll write distd(x; S) for the distance function from x to S ... read more

15. Comparison of Beam Models (Supervisor: Prof Nic van Rensburg)  Beams form part of many engineering structures, e.g. bridges, buildings and machinery. The Euler Bernoulli theory for a beam is well over 200 years old but is still used. In 1921 Timoshenko proposed a new model to improve the theory. There is some disagreement to this day about the relative merits of the two...read more

16. Modeling a Piano (Supervisor: Prof Nic van Rensburg)  In a piano vibrating strings are used to generate sound. However, to explain the quality of the sound is not that simple. The string’s oscillation activates a complex vibrating system that includes the sound board and sound radiation in a three dimensional domain. Since the string activates the system, it is important that...read more

17. Classification of low-dimensional real Lie algebras (Supervisor: Dr Rory Biggs) A real Lie (pronounced "Lee") algebra consists of a real vector space equipped with a special kind of bilinear map (called the Lie bracket). For instance, R3 with the cross product forms a Lie algebra. read more

18. Hamilton-Poisson systems on two-dimensional linear Poisson spaces (Supervisor: Dr Rory Biggs) A Poisson structure allows one to associate a Hamiltonian vector eld H to any smooth ("energy") function H; such structures can be viewed as generalizing classical Hamiltonian mechanics and account for a wide range of dynamical systems. read more

19. The RSA modulus (Supervisor: Dr G Maluleke) One of the most popular public key cryptosystem is RSA, whose name is derived from the algorithm’s developers Rivest, Shamir and Adleman. The RSA algorithm was the first public key cipher to be developed and published for commercial use. It is used for both encryption and digital signature ...read more

20. Wiener’s Attack on the RSA (Supervisor: Dr G Maluleke) One of the most popular public key cryptosystem is RSA, whose name is derived from the algorithm’s developers Rivest, Shamir and Adleman. The RSA algorithm was the first public key cipher to be developed and published for commercial use. It is used for both encryption and digital signature. The cipher is commonly used in securing e-commerce e-mail, implementing virtual private networks and providing authenticity of electronic documents. Security of the RSA cipher relies on the difficulty of factoring its modulus ...read more

21. Spatio-temporal modelling of Sterile Insect Technology control of wild insect populations (supervisor: Prof R Anguelov) The Sterile Insect Technology (SIT) is a nonpolluting method of control of the invading insects that transmit disease. The method relies on the release of sterile or treated males in order to reduce the wild insect population (fruit flies, mosquito). A model of SIT control was proposed in ...read more (reference; R Anguelov, Y Dumont, J M-S Lubuma, Mathematical modeling of sterile insect technology for control of anopheles mosquito, Computers and Mathematics with Applications 64 (2012) 374–389)

22. Solving the brachistochrone problem with calculus of variations (Supervisors: Dr M Messerschmidt and Dr E Kikianty) The brachistochrone problem can be roughly stated as follows: A massive particle slides under gravity along some track from some point A to some point B. What shape should the track be so that the time it takes the particle to slide from point A to point B is a minimum? read more

23. Riesz representation theorem in semi-inner product spaces (Supervisors: Dr M Messerschmidt and Dr E Kikianty) The Riesz representation theorem provides a connection between a Hilbert space and its dual space. read more

24. Representation theory (Supervisors: Dr Tung Lê and Dr SY Madanha) Cenarios: Let G be a finite group. An ordinary reprensentation of degree n of G is a homomorphism ø : G → GL(n;C) where GL(n;C) is the general linear group of degree n over the complex field C. read more

25. Does the Collatz-Algorithm show chaotic behaviour? (Co-Supervisor (CS): Prof. Stefan Gruner, Co-Supervisor (Math): Dr Ruaan Kellerman The Collatz-Function in the Natural Numbers is based on the following very simple conditioned iteration: "If some input N>1 is even, let N be N/2 (and repeat), otherwise let N be 1+3N (and repeat), HALT when N=1". read more

26. Orthogonal Polynomials and Computer-Algebra (Supervisor: Dr Alta Jooste) A sequence of real polynomials {Pn} N n=0, N ∈ N ∪ {∞}, where Pn is of exact degree n, is orthogonal on the interval (a, b), with respect to the weight function w(x) > 0, if, for m, n = 0, 1, . . . N, read more

27. Orthogonal Polynomials and Finances (Supervisor: Dr Alta Jooste) A sequence of real polynomials {Pn} N n=0, N ∈ N ∪ {∞}, where Pn is of exact degree n, is orthogonal on the interval (a, b), with respect to the weight function w(x) > 0, if, for m, n = 0, 1, . . . N, read more

28. Maximum principle for weak solutions of reaction-diffusion equations (Supervisor: Prof Roumen Anguelov) The maximum principle basically says that under certain assumptions the maximum/minimum value of the  solution of a partial differential equation is attained on the boundary of its domain. read more

29. Rate-induced tipping in dynamical systems (Supervisor: Dr Manjunath Gandhi) The classical theory of bifurcations concerns noting the qualitative asymptotic dynamics of a parametric dynamical system at all desired values of the parameter. read more

30. Image Processing and Synthesis using Alternating Projection onto Convex Sets (Supervisor: Dr Manjunath Gandhi) This project concerns reconstructing or re-synthesis of images by projections onto
convex sets (POCS) by employing a POCS-algorithm.

31. Discovering equations from data (Supervisor: Dr Manjunath Gandhi) This project concerns discovering ordinary di erential equations from data measurements on trajectories. read more

32. Mathematical modeling of oncolytic virotherapy (Supervisor: Prof Rachid Ouifki) The purpose of this project is to use ordinary-differential-equations-based models to study the effects of oncolytic viruses in treating cancer. Some models consisting of cancer cell population, oncolytic viruses and immune cells will be presented and analyzed. In the mathematical analysis, we investigate the models’ equilibria and stability behavior in terms of the virus basic reproduction number. This will be followed by some discussion on the conditions relating to tumor elimination. Then, we conduct a preliminary sensitivity analysis to identify the virus’ key parameters with the highest oncolytic effect on the tumor. Finally, numerical analyses will be performed to confirm the mathematical findings and simulate different scenarios of the tumor-virus-immune system interactions.

33. Quaternion linear algebra (Supervisor: Dr Marten Wortel) The quaternions form a four-dimensional algebra over the reals with basis read more

34. The Homeomorphic Measures Theorem (Supervisors: Prof Jan Harm van der Walt and Dr Marten Wortel) In this project we will prove the following theorem read more

35. Riesz spaces and equilibrium theory (Supervisors: Dr Mokhwetha Mabula and Ms Magadi Mabe) An ordered normed space (X, ||:||;<) is called a normed Riesz space if X is a Riesz space (that is, for x; y E X, sup{x,} and inf{x; y} exist in X) read more

36. The Path Partition Conjecture in graph theory (Supervisor: Dr Johan De Wet) The Path Partition Conjecture (PPC) is an unsettled conjecture in graph theory that was first put forward in 1983. Let G be a graph with a longest path that... read more

37. On the maximum number of subgroups of nite groups (Supervisor: Dr Chimere Stanley Anabanti) Let n be a positive integer. We write Gn for the set consisting of all groups of order n, and define M(n) to be the maximum number among all the number of subgroups of groups in Gn; i.e., read more

38. Similarity solution for a pre-existing fluid-driven fracture in a permeable medium (Supervisor: Mr Mathibele Nchabeleng) This research project is concerned with the analysis of a two-dimensional Newtonian hydraulic fracture in a permeable rock. Hydraulic fracturing is a key process in petroleum and mining engineering. In this process, fluid is pumped into a rock fracture at ultra-high pressure to extend it. Hydraulic fracturing has many applications in engineering, some of which include reservoir stimulation, drilling cuttings, underground caving operations, and contaminated land remediation.

39. Numerical techniques and solutions for Cauchy-type singular integral equations of the first kind(Supervisor: Mr Mathibele Nchabeleng) In this research, we will investigate numerical methods for singular integral equations. In particular, we will explore different techniques that can be used to deal with the singularity and evaluate their performance.