Honours thesis topics

The branch of dynamical systems theory known as symbolic dynamics interacts in many ways with the theory of formal languages, a theory of fundamental importance in many areas of computer science. This project would explore some of the many connections between the two theories. Some examples of specific areas that might be investigated include the connection between sofic shifts and regular languages, the application of the theory of context free languages to symbolic dynamics, the application of symbolic dynamics to coding problems such as the coding of data for hard disk storage.

Just as some dynamical systems have conservation laws, some have monotonicity laws such as dissipation of energy or increase of entropy. In this project, entropy laws will be found systematically for dynamical systems that are governed by practical nonlinear partial differential equations. These will be used to prove stability of states of maximum entropy and to prove stability of solutions that are invariant under some symmetry. A connecdpalmertion will be made between smoothing properties or positivity properties and decrease of Shannon information. Applications might include heat and mass transfer, metal surface evolution and quantum mechanics.

Since the late 1990s, it has been observed that the universe is not only expanding but accelerating, and that most of the energy is in the form of “dark energy” not associated with known types of waves or particles. If we allow a scalar field to be minimally coupled in an invariant way to the simplest consistent expanding accelerating universe, then the field has energy eigenstates that become unstable at discrete times. These non-oscillatory states have a classical analogue in an oscillating spring on an accelerating platform – eventually the spring ceases to oscillate. The quantized non-oscillatory states have a continuous spectrum (like that of an ionized atom), unlike the quantum particle states that  have a well defined particle number and discrete spectrum (like the bound states of a neutral atom).  This project will predict the partitioning of the scalar field energy into the particle states and the continuous “jelly” states, and will estimate how this non-particulate energy increases in time.

Recently, a formal series solution in terms of Kummer functions, has been constructed for an integrable nonlinear convection-diffusion equation. Up to 180 series coefficients have been calculated explicitly. The series converges in practical terms but its radius of convergence  has not been estimated in any way. This solution leads to an exact infiltration series, a power series in square root time, for the total depth of surface water having infiltrated a soil. The coefficients of this series have been in dispute for some time. These exact results may help to explain pre-existing partial results that seem to be in mutual contradiction. At a practical level, the integrable equation seems to well model water infiltration in soil but the goodness of fit with experimental data needs to be quantified and compared with that of more popular phenomenological models.

The Harry Dym equation is a well known integrable nonlinear wave equation. There is another less well known nonlinear  water wave equation that appears to be very similar but it can be solved more simply because it becomes linear after a change of variables. This will allow an investigation of how special is the soliton behaviour of the Harry-Dym equation. The Harry Dym equation has a multi-soliton solution for which the finite number of peaks remains constant. What does the other similar integrable equation predict in similar circumstances? Is there a trace of soliton behaviour? If not, how is the soliton energy dissipated?

I have a range of possible thesis topics in Combinatorial Game Theory and Number Theory. Some are classical, some are open problems. For further information, come and see me to discuss topics that might interest you.

Mort Brown discovered an unusual periodic homeomorphism of the plane that has sparked interest in several fields. This project investigates this and related examples.

The 'classical' kinds of algebras include groups, rings and vector spaces. I am mainly interested in more 'modern' kinds of algebras, like lattices and Boolean algebras, which arise in logic and theoretical computer science. Possible topics in general algebra include:

1. Primal and quasi-primal algebras. Primal and quasi-primal algebras are important generalisations of the two-element Boolean algebra, which is familiar from discrete mathematics, and arise in many parts of general algebra. The aim of this project will be to find simple and effective tests for quasi-primality based on some old and some recent tests for primality.
2. Graph algebras. Every graph (in the sense of discrete mathematics) can be converted in a natural way into an algebra with a single binary operation. This project will review the many applications that graph algebras have found and will include some recent papers illustrating their use on the fascinating interface between complexity theory and algebra.
3. NU algebras. Algebras with a near-unanimity term are an important generalisation of lattices. This project will review the various properties that these algebras possess and study some very recent papers, including the proof that it is decidable whether or not a finite algebra actually has a near-unanimity term.

There are a number of logical and algebraic approaches to analysing the correctness of computer programs. This project would involve an exposition of some of these approaches and their interrelationships, and lead to an investigation into the relationship between these and algebras of functions and relations. This project is best suited to students with some extra background in logic, or with a strong background in algebra with a computer science interest. No knowledge of programming is necessary, since the focus is on algorithms, rather than actual programming languages.

This topic concerns the problem of classifying various algebras of functions and binary relations.  For example: the Cayley representation for groups shows that groups are precisely the algebras arising as systems of permutations of a set under the operations of composition and inverse. What about algebras of binary relations under the operation of composition and intersection? There are a plethora of such questions, many motivated from the kind of issue considered in the previous topic "Algebras of programs".

How hard is it to decide if two arbitrary finite graphs are isomorphic? What about two finite groups? This project would involve a study of computational questions such as these: the focus would be on general algebraic structures, and connections between computational problems and logical properties of algebras. This project could be directed to computational problems on finite structures (a popular current trend) or for infinite structures (more classical, but still active and important).

Finite state automata (as encountered in, say MAT1DM) and algebraic structures known as "semigroups" have a close relationship: finite semigroups all arise as the algebra of input transitions of a FSA under composition, and every finite semigroup gives rise to a finite automata for which it is the algebra of transitions.  Simon's Theorem relates languages recognised by machines that read a fixed multiple number of bits at a time to an elementary structural property of finite semigroups. It has a number of different proofs.  This topic is suitable for a student with an algebra background.

In this project, starting with the definition and general properties of Lie groups and Lie algebras, we will study the geometric structure of metric Lie groups. In the first part of the project, we will classify low-dimensional Lie algebras and compute their curvatures, geodesics and curves of constant curvature. No advanced techniques will be used, but a good understanding of metric and curvature (MAT3AC ) and symmetric and skew-symmetric matrices, eigenvalues and eigenspaces (MAT2LAL) is desirable.

The project will also involve some calculations using computer algebra packages (Maple or Mathematica) and visualisation of some distinguished curves and surfaces (geodesic balls, etc).

This project deals with an exciting and challenging topic in modern Riemannian geometry. It is oriented on a motivated student who considers a future career in mathematics. The project combines a relative accessibility with a potential of resolving a long-standing conjecture of David Blair and producing a publishable result. No heavy machinery will be used, but a good understanding of the material of MAT3AC (metric and curvature) is required. The project will also involve extensive calculations using computer algebra packages (Maple or Mathematica).

The aim of this project is to compute the curvature and the Weyl conformal curvature of so-called symmetric spaces of dimension 4 and 5. Such spaces may produce a counterexample to generalising one recent result from conformal geometry of symmetric spaces to low-dimensional cases. The project will require a good understanding of the material of MAT3AC (metric and curvature) and may involve some calculations using computer algebra packages (Maple or Mathematica).

This project is about graph drawings. A pseudothrackle is a graph drawing on the plane such that any two edges have exactly one point in common (a vertex or a crossing or a touching point). A recent conjecture of J.Pach, motivated by the famous (and still unresolved) Conway’s Thrackle Conjecture, states that there must be a linear bound for the number of edges of a pseudothrackle in terms of the number of vertices. The thesis will focus on studying small graphs and also the graphs of ‘popular’ classes (complete, bipartite, etc). No prior knowledge is necessary, but a general understanding of main concepts of Graph Theory is desirable.

Orthogonal polynomial systems are sequences of polynomials that are orthonormal with respect to some linear form. The study of non-standard orthogonal polynomials have many applications to classically integrable systems such as the Schrödinger wave equation.

The integrable systems that arise are a ubiquitous part of various quantum mechanical models. This project may be directed towards any number of integrable aspects of orthogonal polynomial systems.

When are the solutions of a system of second order ordinary differential equations (ode’s)

the solution of an Euler-Lagrange equation

This is a famous question in the calculus of variations, important in differential geometry, mechanics and relativity. This project involves an exploration of this inverse problem for some important classes of differential equations using differential geometry and computer algebra.

Numerical integration of differential equations uses approximation by straight line segments to construct solutions. This project explores what seems to be a completely new idea: approximation by circular arcs. It involves development and implementation of analogues of Euler's method and Runge-Kutta methods and comparison with the traditional algorithms.

Amongst all differential/difference equations, or mappings/correspondences, the integrable ones are the nice ones. They always carry special structures. Examples are a Lax-pair, infinitely many symmetries, sufficiently many integrals of motion, and polynomial growth (as opposed to exponential growth) of degrees of iterates and multivaluedness. Often one is able to calculate the first few terms of an infinite series from which one can guess the general form. At this point you want to write down a proof! What does the identity, which makes it happen, look like?

Cox regression is the most established statistical method for analysing time-to-event data, where the objective is to ascertain how identified factors are associated with an event of interest (eg, time to hospitalisation, recovery from an illness, or death). It takes into account the presence of incomplete records (say premature withdrawal, late recruitment to the study), which hinder the observation of the event of interest.  This topic attempts to analyse time-to-event data with an alternative strategy known as competing risk. In general, a competing risk situation arises when an individual can experience more than one type of event and the occurrence of one of such events hinders the occurrence of other types of events. For example, a group of patients with a specific condition are followed up in order to observe an event of interest, say hospitalization. If by the end of the study, each patient is either hospitalized, alive or remain free from disease, then the conventional statistical methods may apply. But if some patients might have died from other causes (e.g., accidents), then a competing risk situation has arisen because death has hindered the occurrence of hospitalization.  Hence, the patients in question should not be considered as censored because their data are not incomplete.  The R and Stata packages are recommended for this topic.

Note: Please contact Dr Siew Pang Chan if you would like to propose other applied topics, such as data mining, medical decision analysis, longitudinal data analysis, structural-equation model and reliability engineering.

Suppose that X1;:::;Xm and Y1;:::;Yn are independent random variables. Also suppose that X1;:::;Xm are independent and identically N(11;¾12) distributed and Y1;:::;Yn are independent and identically N(11;¾12) distributed. We do not require the population variances ¾12 and ¾22 to be equal. A classic problem in statistics, called the Behrens-Fisher problem, is how one should carry out inference about '1¡ '2. The purpose of the project is to compare several confidence intervals for '1¡ '2 that have coverage probability that never falls below a specified value '1¡ ®. Included in this comparison is the method described in section 3 of Kabaila (2005).

Reference

• Kabaila, P. (2005)  Assessment of a preliminary F-test solution to the Behrens_Fisher problem. Communications in Statistics – Theory and Methods, 34, 291 – 302.

Suppose that the distribution of the data is determined by the parameter vector (µ;¸), where µ is the scalar parameter of interest and the vector parameter ¸ is of no direct interest to us. The presence of the parameter ¸ (called a nuisance parameter) causes difficulties for statistical inference about  µ.  One method of carrying out this inference is to use a weighted integration, with respect to ¸ , of the likelihood function. This is called an integrated likelihood method. An introduction to such methods is provided by Berger et al (1999). The purpose of the project is to analyse the properties of such methods by considering some novel integrated likelihoods.

Reference

• Berger, J.O., Liseo, B. & Wolpert, R.L. (1999) Integrated likelihood methods for eliminating nuisance parameters. Statistical Science, 14, 1 – 28.

Suppose that the random vector X has a multivariate normal distribution with mean ? and, for simplicity, covariance the identity matrix. Also suppose that X has dimension 3 or greater and that the parameter of interest is ?. The usual estimator of  ? is X. However, in 1961, James and Stein found an estimator that is, according to a particular plausible criterion, better than X. This is a very surprising result that is sometimes referred to as Stein's Paradox (see section 10.7 of Casella & Berger, 1990).

Furthermore, one can construct a confidence set centred at this estimator that has both smaller expected volume and larger coverage probability than the usual confidence set, which is centred at X (see e.g. Casella & Hwang, 1983). Casella & Hwang (1987) state that these confidence sets provide improved analysis of one-way ANOVA data. The purpose of this project is two-fold:

1. After reading over see section 10.7 of Casella & Berger (1990), to provide some of the details of the derivations of Casella & Hwang (1983). 
2. To examine the extent to which the confidence sets of Casella & Hwang (1983) can be used to provide an improvement of the analysis of one-way ANOVA data.

References

• Casella, G. & Berger, R.L. (1990) Statistical Inference. Wadsworth. 
• Casella, G. & Hwang, J.T. (1983) Empirical Bayes confidence sets for the mean of a multivariate normal distribution. Journal of the American Statistical Association, 78, 688-698. 
• Casella, G. & Hwang, J.T. (1987) Employing vague prior information in the construction of confidence sets. Journal of Multivariate Analysis, 21, 79-104.

Each of N experts assesses an object using some finite numerical scale. The coefficient of agreement is a measure of similarity of experts’ preferences. We will study new entropy-based coefficients. The aims of the project are to:

1. study properties of the coefficients using simulations and real data
2. compare entropy-based coefficients with other coefficients of agreement.

Biomedical engineering often needs to assess the morphological similarity among temporal waveforms. Most current similarity indices overlook any amplitude (or is it "amplitude offset") difference among the waveforms, thereby mistakenly concluding curves of similar shapes as very different.

We will generalize ASCI coefficient and proposes new metrics for multi-class classification of a set of waveforms. We will discuss some limitations of ASCI for which usage of new metrics can gives substantial improvement.

Random fields are multi-dimensional analogues of one-dimensional stochastic processes. They are used to model spatial data in environmental, atmospheric, and geological sciences. The aim of the project is to study currently implemented methods for the simulation of stationary and isotropic random fields, and propose new methods for random fields which exhibit long-range dependence.

Geometric characteristics of random surfaces play a crucial role in areas such as geosciences, environmetrics, astrophysics and medical imaging, to mention a few examples. The aim of the project is to investigate various properties of long-range dependent random fields with spectral densities which are unbounded at some frequencies.

In many areas of research it is useful to measure individuals at multiple time points to analyse the effect of time on a variable of interest.  For example, suppose that a new treatment is available that is specifically designed to assist in lowering blood pressure for hypertensive patients.  To assess the effectiveness of this new treatment, scientists recruit a number of patients and place them on either the new treatment or an existing treatment.  Patients are then monitored for 24 hours and have their blood pressure recorded every hour.  The data is then used to compare the effectiveness of the treatments.  The analysis of data of this type is not so simple and it is common in many areas of research to focus on ‘area under curve’ (AUC) summary measures.  The data for each patient can be displayed as a line plot with time on the horizontal axis and the variable of interest on the vertical axis.  The AUC for each patient is then simply the total area under this line and can be used as a single summary measure in place of the multiple measures recorded for each patient.  For the example described above, the average AUC for each group can then be used to compare the effectiveness of the treatments.  Whilst such an approach is quite simple and easy to understand, important information specifically related to the variable time is lost.  This project will discuss the limitations of such an approach by looking at both real and simulated data sets.

Early R functionality that allowed the fitting of linear mixed effect (LME) models was designed for nested random effects.  However, crossed random effects arise commonly in practice and newer R functionality allows for the fitting of such models.  A drawback of using this functionality is that p-values associated with the testing of model parameters are not provided in the summary output.  This is not simply an oversight and there are good reasons as to why they are not given.  Douglas Bates, the author of the R functionality that allows for the fitting of models with crossed random effects, provides a detailed discussion as to why this is the case and recommends using a Markov Chain Monte Carlo approach to assess the model parameters.  This project will discuss the fitting of LME’s in R with crossed random effects and consider the effectiveness of this Monte Carlo approach via simulation.

For details please speak with Professor Robert Staudte.