The following list does not aim to be exhaustive, but a list of
things that seem like they might be interesting to investigate.
Feel free to suggest your own ideas.
- Applied non-linear difference equations
- Second-Order Difference Equations (MATH235 only)
In MATH111 we investigated first-order difference equations.
In this project we investigate second-order difference equations.
Topics covered could include:
- Analytical solutions
- Steady-state solutions and stability
- Modelling
- Chaos
- 0-1 test for chaos
- harvesting in predator-prey models
- piecewise-smooth dynamical systems
The assessment will contain `tutorial-type questions' and
Maple projects.
This project is will be jointly supervised with Dr Annette Worthy.
- Analysis (applications in image science)
- This project is for someone who is interested in analysis.
The detection of discontinuities are important in many fields.
For instance, in image processing it is often important
to detect the boundaries of the items in the image. An important
application of edge detection is in the
medical sciences. Another area of application
is in the movie industry: edge detection is important
in the automatic restoration of old film prints and the automatic
colouring of black and white film. In this project you will learn how
to use Fourier series to detect "edges". Mathematically the problem
is this: given the Fourier coefficients for a function how can we
determine the location and size of the jump discontinuities of the
function?
This project will be co-supervised by Associate Professor Rod Nillsen
and will be based around the following article.
Engelberg, S. (2008). Edge Detection Using Fourier Coefficients.
The American Mathematical Monthly, 115(6),
499-513.
- Business and politics
-
- Modelling alliances.
- Collective influence and social change. How does public opinion
effect issues such as voting choice? What's the relationship
between crime and punishment? Marriage as an economic institution.
- Electoral Systems.
Is it possible to design a `fair' electoral system? What do we mean
by a fair electoral system? How can mathematics be used to analyse
and improve electoral systems? By the end of this project you will
know!
- Calculators: How do they calculate?
- How do calculators calculate? What is
sin(1)? You might think they calculators use Taylor
polynomials to approximate trigonometric functions. This is a good guess,
but it is not correct! In this project we will investigate the
mathematics used by calculators to determine the values of
trigonometric functions
and other types of functions. We might also look into how
methods used by mathematicians over 1000 years ago to approximate
the values of trigonometric functions. For instance, how did
the Indian mathematician Aryabhata (476--550) compute a table of
sines for singles from 3o to 45o?
This project is suitable for second-year students only.
- Chemistry
- Wine Fermentation.
We investigate a physical and mathematical model
for the kinetics of wine fermentation.
The model predicts sugar utilisation curves based on
experimental data from wine fermentations with various initial nitrogen
and sugar concentrations in the juice.
- Combustion
- Spontaneous Combustion of Coal.
When coal reacts with oxygen it releases energy, which is why
it is used as a fuel. However, when too much coal is stockpiled
it can spontaneously ignite. In this project we will investigate the
spontaneous combustion of coal.
- see cube.dir for transient problem of estimating parameter values.
based upon malow:2004. Cylindrical problem.
- Drying of particles
- Drying of particles.
The drying of small particles is an important industrial process
with applications in many industries. A well-known example is the
spray drying of milk to produce powdered milk. In this project we will
examine some simple models for this process.
- Economics
-
- Can we predict the rhythms of the financial market? Do the
assumptions of classical economics ever hold?
- How does the interaction of individual traders lead to flucturations
in markets? Do traders act rationally?
- Can we predict market crashes?
- Modelling the growth of firms.
- Globalisation. How does cultural dissemination work? Is diversity
eliminated by globalisation?
- Game theory
- How do people/organisations learn how to cooperate?
- Is reciprocity good for us?
- Epidemiology & virus dynamics
- We will investigate the spread of infectious
diseases through a population and through an individual.
We will be interested in questions such as:
- Will a disease spread through a population?
- If so, how many individuals will be affected?
- If the disease is endemic (i.e. habitually present), what is
the prevalence of the infection?
- Suppose an infectious disease starts in one city. How quickly
will it spread to other cities? What can we do to stop a disease
spreading throughout the whole country?
H.W. Hethcote. 2000. The Mathematics of Infectious Diseases.
SIMA Review, Volume 42(4), 599-653.
For virus dynamics we will consider the modelling of HIV.
M.A. Nowak and R.M. May. 2000. Virus Dynamics: Mathematical
Principles of Immunology and Virology. Oxford University Press.
- Food Engineering
- Extraction of nutrients in the intestines
One of the big challenges in the field of biotechnology is the
study of food metabolism upon its consumption. The chemical
components making up a food product undergo a chain of reactions
in the gastro intestinal tract, which can be modelled as a chemical
reactor. An understanding of the processes that occur within the
intestines will have a major impact in the areas of nutrition science
and food engineering and will lead to new advances in targeted drug
delivery.
This is a huge problem, and would be a good topic for a PhD. However,
in this project we will examine some highly simplified models for
the extraction of nutrients in chemical reactors.
- Industrial Mathematics
- Cleaning Industrial Effluent.
Many industrial processes produce effluents that must be cleaned
before they can be discharged into rivers. In this project we will
analyse simple mathematical models for the degradation of
contaminated wastewater in order to maximise the efficiency of the
process.
Examples of processes where such models have been used in the
past include: the aerobic degradation of waste water originating in
the industrial treatment of black olives, the anaerobic treatment
of dairy manure, the anaerobic digestion of ice-cream wastewater,
the anaerobic treatment of textile wastewater and the aerobic
biodegradation of solid municipal organic waste.
- Laplace Transforms: Solving Linear PDEs
(MATH345)
- In MATH202 Laplace transforms were introduced as a technique to
solve ordinary differential equations. In this project we extend the
method to solve partial differential equations. This project will
involve
- extending the theory of Laplace transforms learnt in MATH202
- applying your new technique to solve PDEs arising in
mathematical modelling.
This project is will be jointly supervised with Dr Annette Worthy.
- Mathematical Medicine.
- Mathematical modelling of Hirschsprungs disease
The pre-requiste for this project is that you have passed MATH305.
This project has two components:
- Learning how to solve 1-d reaction-diffusion equations
numerically using the method of lines.
- Learning how to analyse 1-d reaction-diffusion equations
using phase-plane methods.
This project is based upon research carried out by
Professor Kerry Landman. For more information see her web page
http://www.ms.unimelb.edu.au/~kal.
- Tumor Dynamics.
Why do some tumors grow during treatment and others shrink after
the treatment stops?
To investigate this we consider a model with three types of cells
(immune cells, tumor cells, and normal cells) that takes into
account the competition for resources between tumor and normal cells.
This model has multiple equilibrium points, including a region in which
there is one unstable and two stable equilibrium points: this represents
a time during treatment at which the patient's tumor could be driven
to either to steadily shrink or to grow.
This model can be extended to include treatment, in the form of
chemotherapy.
SIAM News 42(8), October 2009.
- Non-linear oscillators
- Duffings Equation (3rd year students only really).
Duffing's equation is an example of a nonlinear oscillator.
It can be used to model a variety of mechanical problems including the
dynamics of a buckled beam or plate, the forced vibrations of a
cantilever beam in a nonuniform field of two permanent magnets
The plan would be to analyse the steady states and stability of the
damped Duffing eqn with no forcing (zero or positive damping), then the
Duffing equation with negative stiffness - which acts as a model for the
magneto-elastic. Apply increasing amounts of forcing to the system
to investigate how the basins of attraction change. See how
computing power has changed - write a matlab program to repeat on
your own PC the calculated that Moon and Li (1985) did on a
supercomputer!
Some references
T.H. Fay and S.V. Joubert. (2007). Nonlinear resonance and
Duffing's spring equation II.
International Journal of Mathematical Education in Science
and Technology, 38(4), 517-528.
Guckenheimer and Holmes.
Moon F.C. and Li G.-X. (1985). Fractal Basin Boundaries and Homoclinic
Orbits for Periodic Motion in a Two-Welled Potential.
Physical Review Letters, Volume 55, 14.
Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. Perseus
Books Publishing , LLC.
Wiggins S. (1990). Introduction to Applied Nonlinear Dynamical
Systems and Chaos, Springer-Verlag New York Inc.
- Forced van der Pol equation
- T.H. Fay. The forced can der Pol equation.
International Journal of Mathematical Education in Science
and Technology, 40(5), 669-677, 2009.
- Physics.
- Problems in classical physics can be solved by applying Newton's laws.
Problems in quantum physics can be solved working within the axios of
quantum mechanics. In both cases physical problems lead to well-defined
mathematical problems. Problems in the social sciences do not lead to
well defined mathematical problems - for how can the forces that act
between
individuals be cast into mathematical form?
HOWEVER, it is indeed possible to model the ways in which humans behave
and organise themselves, by applying ideas from statistical physics.
Possible problems that may be investigated in this project include:
- Dynamics of Growth and Movement.
- how do crowds move? What's the best strategy to escape from a
smoke-filled room when you don't know where the exits are?
What's the best way to disperse a large crowd?
- How do cities grow?
- The dynamics of traffic. How does traffic move and clog? Can we
predict when and where congestion will occur?
- Social Networks
- The Kevin Bacon Game, six degrres of freedom, social networks and
small worlds.
- What is the shape of the internet?
- Population Models.
- We investigate population models for single and
interacting species. We are interested in question such as
- Under what conditions will a population become extinct?
- What are the conditions for sustainable harvesting of a
renewable resource?
- What happens when several species compete for scarce resources?
- Ecosystem modelling, what is the effect of habitat destruction?
- the mathematics of Search
engines.
- How do internet search engines work? They work because the designers
understand the basics of linear algebra and matrices. Goggle's
success derives in large part from the way in which it's search engine
tanks the importance of web pages.
project you use basic ideas from MATH203 to understand the principles
behind popular internet search engines. Why does linear algebra
matter? Do this project to find out!
- Synchronisation.
- Interactions between individuals - be they fireflies, pendula or
people - can often lead to the emergence of coherent actions.
Mathematically, such models can be represented by a coupled system
of phase oscillators. Simply models can lead to surprising behaviour.
See, for example.
Y. Zhou, W. Gall and K. Nabb. (2006). Synchronizing Fireflies.
The College Mathematics Journal, 37(2),
187-193.
Notices of the American Mathematical Society,
Volume 51, Number 3,
pages 312-319, March 2004.
- the mathematics of Tennis and similar games
-
Tennis anyone?
In this project we will investigate very simple mathematical models
for games such as badminton, squash and tennis.
Simple thoughts. What are your chances of winning a game if you know
your chances of winning any given point? Did you know that if your
chances of winning a point are 50-50, then a 1% increase in your
probability of winning each point yields a 2.5% probability of
winning the game: so average players have the most to gain by
improving a bit.
The following list does not aim to be exhaustive, but a list of
things that I am particularly interested in investigating NOW.
Another good place to look for NEW projects that I might be willing
to supervisor at a non-honours level is my list of
potential honours projects.