2024

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 equations. read more

2. Similarity solution for a pre-existing fluid-driven fracture in a permeable medium (Supervisor: Dr Mathibele Nchabeleng) 

3. Numerical techniques and solutions for Cauchy-type singular integral equations of the first kind (Supervisor: Dr Mathibele Nchabeleng)

4. 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

5. 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

6. 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

7. 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 R3. read 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. 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

12. Applications of generating functons (Supervisor: Dr HR (Maya) Thackeray) In combinatorics, a generating function is an infinite series of the form read more

13. Investigating the influence of the number of Sylow subgroups (Supervisor: Dr Chimere Stanley Anabanti)  Let G be a finite group. We write np(G) for the number of  read more

14. Comparison of Beam Models (Supervisor: Dr Belinda Stapelberg)  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

15. Modeling a Piano (Supervisor: Dr Madelein Labuschagne)  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  

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. Data-driven Molecular Dynamics Modeling (Supervisor: Dr Manjunath Gandhi) This project falls at the interface of mathematics, machine learning, and molecular dynamics. Molecular dynamics helps study properties of materials without actually synthesizing them, but by simulating the positions of atoms and molecules as a function of time using Newton’s equations. read more

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

26. 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

27. 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

28. 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

29. 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.

30. 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.

31. Group partitions yielding lower bounds for Ramsey numbers (Supervisor: Dr CS Anabanti)  The Ramsey number Rn(3) is the smallest positive integer such that colouring the edges of a complete graph on Rn(3) vertices in n colours forces the appearance of a monochromatic triangle. It is known that R1(3)=3, R2(3)=6 and R3(3)=17. On the other hand, only bounds are known for n>3. An upper bound that Rn+1(3)<(n+1)(Rn(3)-1) + 3 for n>1 was given by Greenwood and Gleason in 1955. A lower bound on Rn(3) is obtainable by partitioning the non-identity elements of a finite group into disjoint union of n symmetric product-free sets. The goal of this project is to improve existing lower bounds on R_n(3) by determining finite groups whose non-identity elements form a disjoint union of n symmetric product-free sets.

32. On Baer’s theorem on products of finite groups (Supervisor: Dr SY Madanha) Let A and B be subgroups of G. The group G = AB is a normal product of subgroups A and B if A and B is normal. An interesting problem in group theory is to determine how A and B affect the structure of G. read more

33. Widely supersouble groups (Supervisor: Dr SY Madanha) A group G is supersouble if there exists normal subgroups of G; 1 = G0, G1, G2, . . . , Gn = G for some positive integer n such that Gi/Gi−1 is a cyclic subgroup for all i = 1, . . . , n. read more

34. The Riesz-Kantorovich Formulas (Supervisor: Dr. CM Schwanke) An ordered vector space that is also a lattice (i.e. every set of two elements has a supremum and infimum) is called a vector lattice or a Riesz space. Analogous to norms in normed vector spaces,vector lattices possess an absolute value,which is used to make sense of natural notions of convergence. As such, one can do analysis in vector lattices. read more

35. Extensions of Positive Operators (Supervisor: Dr. CM Schwanke) An elementx ofan ordered vector space E iscalled positive ifx ≥ 0,where 0 denotes the zero element of E. Interestingly, the collection of all linear operators T : E → F between two ordered vector spaces E and F is itselfan ordered vector space. Here such an operator is positive if T(x)≥ 0 holds in F whenever x ≥ 0 holds in E. read more

36. Study of Chaotic Dynamics (Supervisor: Dr Manjunath Gandhi) It is a well-known fact that when a function performs the geometric stretching and folding it frequently leads to chaotic dynamics on some part of the phase space. A classical example is the Smale Horseshoe map (see Figure 1). read more

37. Real Analysis in Reverse (Supervisors: Dr Eder Kikianty, Dr Miek Messerschmidt) The Dedekind completeness property states: "Every non-empty set that is bounded above has a least upper bound". This property is commonly set as an axiom for the real numbers, known as the completeness axiom, in most real analysis courses (and actually characterizes the reals as the unique Dedekind complete ordered field). Using the ordered field axioms and the completeness axiom, one proves a number of standard results about the reals in real analysis courses. Some of these results, when taken as axioms along with the ordered field axioms, again imply the Dedekind completeness property, while others do not. We will study these different axiom systems, either, by proving their equivalence to the usual axioms for the reals, or, by exhibiting their non-equivalence by constructing exotic ordered fields that satisfy our different axiom system while failing to be Dedekind complete.

38. Exploring the Brownian motion (Supervisor: Dr Lesedi Mabitsela) One of the most widely used stochastic processes in finance is Brownian motion. This project will research Brownian motion. The student will give a brief overview of the construction of Brownian motion. In addition, the student will study the martingale and Markov properties of the Brownian motion. Applications in finance will be discussed.

39. Mathematical modelling and analysis of inhibition of signaling pathways in cancer cells (Supervisor: Prof Roumen Anguelov) Cancer cells have an enhanced capacity for proliferation (cancer growth) and spreading (metastasis). The receptors of activators of signaling pathways for the synthesis of proteins promote adhesion, proliferation and migration. Understanding the signaling pathways presents an opportunity to design appropriate therapeutic interventions. The aim of this project is to construct differential equations model of activation and inhibition of the mentioned signaling pathways. The project is part of interdisciplinary research with the departments of Physiology and Anatomy. In their current research, the biology part of the team conducts tests on the inhibiting properties of various substances on signaling pathways of melanoma cells. The model developed in this project will assist in deriving the inhibitory properties of the investigated substance(s) in a comprehensive and reliable way. Honours students selecting this project are expected to have good knowledge of differential equations and dynamical systems as well as some programming skills. Having some chemistry and/or physics in the undergraduate transcript will be helpful, but is not a prerequisite. Through this project, the student will develop: (i) skills to relate mathematics to application; (ii) knowledge and skills for analysis of dynamical systems; (iii) ability to work in interdisciplinary team.

40. L-Series and primes in Arithmetic progressions (Supervisor: Dr HR (Maya) Thackeray) It is well known that there are infinitely many prime numbers. In number theory, Dirichlet’s theorem on primes in arithmetic progressions says that read more

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

42. Nordhaus-Gaddum type inequalities​ (Supervisor: Dr Valisoa Rakotonarivo) Let G be a graph, the complement read more

43. The Wiener index and its variants for the class of trees​ (Supervisor: Dr Valisoa Rakotonarivo) The Wiener index of a graph is the sum of distances between all pairs of vertices in the graph. It has been thoroughly studied, especially for the class of trees (acyclic connected graph), where several results have been obtained. In this project, we want to explore several variants of the Wiener index. read more

44. All groups of size less than 60 are soluble (Supervisor: Dr Chimere Stanley Anabanti) A non–identity group G is said to be soluble if there exists a sequence of subgroup read more

45. Disease in mutualistic systems (Supervisor: Dr Mihaja Ramanantoanina) Mutualism is one of the three interaction types between species in ecology. Broadly defined, mutualism refers to. read more

46. Stage structure in mutualistic interactions (Supervisor: Dr Mihaja Ramanantoanina) Mutualism is one of the three interaction types between species in ecology. It is broadly defined an association that reciprocally benefits the interacting species. Understanding the consequences of mutualism is crucial for ecosystem management. However, mutualism is still poorly understood compared with interactions that negatively affect species, such as predation. In this project, the candidate will investigate the effect of stage structure in mutualistic interactions. read more   -- This Project numbered 46 has been taken by a 2024 student.

47. The polynmial method in additive number theory (Supervisor: Dr Taboka P. Chalebgwa) This project falls within the read more 

48. The Hermite polynomials and their applications (Supervisor: Dr AS Jooste) A sequence read more

49. The use of orthogonal polynomials in Approximation theory (Supervisor: Dr AS Jooste) A sequence read more

50. Stochastic processes and their connection with orthogonal polynomials (Supervisor: Dr AS Jooste) A sequence read more

51. Zeilberger’s Algorithm (Supervisor: Dr AS Jooste) A sequence read more

52. Projective Geometry and Relevance Logic (Supervisor: Dr JJ Wannenburg) Models of Projective Geometry one can define a natural ternary ‘colinearity’ relation a∗b∗c which means that points a, b, c lie on the same line. The relational models of the ‘Routley-Meyer’ semantics for Relevance logic also feature a more opaque ternary relation meant to capture read more

53. Monotone dynamical systems and applications (Supervisor: Prof Jacek Banasiak) There is a class of systems of differential equations which whose solutions preserve partial order in Rⁿ. This property has significant implications for the existence and stability of equilibria of such systems. The aim of the project is to present a survey of results on monotone systems and a selection of examples of applications in natural sciences.