# Graduate Student Workshop Problems

### Problem 1: E-vota

**Presenter: **Professor Jeff Sanders, University of Stellenbosch and African Institute of Mathematical Sciences.

**Problem Statement:**

A typical MISG problem requires a solution which optimises some quantity. When an optimal solution is unrealistic an acceptably efficient one is sought. For example the `target' might be a standard length of aluminium that minimises wastage when multiples of that length are cut by a glassmaker for use in its various designs of doors and windows.

There is an important family of contemporary problems which fit that paradigm, but whose solution is instead a design (of optimal or acceptable efficiency) rather than an instantiation of parameters. Mathematics that before was required to show that the solution was valid and efficient is now needed to show that the design functions as it should. The design might, for example, be a distributed algorithm. So the mathematics is likely to be discrete and perhaps unfamiliar.

We nowadays rely on the internet to perform tasks that used to be performed in person (like e-commerce, e-governance and e-banking). In this project we will consider election, and produce a design to run an election in a distributed setting (like the internet). Let's call it e-vota.

We will think of e-vota as a `system' on which certain operations are defined. We must decide what operations we want: like casting a vote; validating a voter; verifying that a vote has been counted; publishing the election result. Then we must decide what state the system should have in order to make those operations possible. Indeed the first step is to understand what exactly is a `system' (states supporting operations) and how to describe it using discrete mathematics.

Next we must decide what properties e-vota should have. This is exciting because an information system can offer benefits not possible in the standard setting (like verification that a specific vote has indeed been counted). But it can also offer new possibilities for corruption and insecurity (like impersonation, and leaking of information). Our design for e-vota must conform to the desired features and avoid the undesirable ones. For that purpose we must formalise the desired and undesired features.

Finally the result of this project will be a design, expressed in discrete mathematics. An implementation (i.e. code) is not required; indeed given our work the coding of an implementation would be entirely routine. Of more interest would be proofs that e-vota exhibits the features required of it.

#### Presentations

E-vota - Alice, Melusi and Jeff

### Problem 2: Safety barriers for dump trucks

**Presenter: **Professor Neville Fowkes, University of Western Australia, Perth, Western Australia.

**Problem Statement:**

Enormous dump trucks 5m high are used to move soil and rocks from vast open cut mines. An example is the “Super Pit” in Kalgoorlie, Western Australia. The loads carried are typically 500 tons and typical speeds are 15 to 20 km/hr. The roads circle around the deep open pit. Safescape has invented a concrete filled safety barrier about 2m high with soil backing, referred to as the bund, that may cushion the impact of an out of control dump truck and so prevent major accidents. Will the procedure work? If so how should the barrier be designed: its height width and bund height.

#### Presentations

Safety barriers for dump trucks - Prof. N Fowkes

### Problem 3: The design of a state change instrument

**Presenter: **Professor Neville Fowkes, University of Western Australia, Perth, Western Australia

**Problem Statement**The aim is to design an instrument for determining the state change characteristics of a range of organic substances (think fats and oils). The apparatus is to consist of a closed copper cylinder (approximate length 1m, radius 10cm) with atemperature probe as illustrated in Figure 1. The cylinder is filled with organic material and the temperature raised above the melting point of the substance (think boiling water). The cylinder is then placed in a cold bath (think ice) so that a solidication front propagates through the substance from the cylindrical wall to its axis. The temperature probe measures the temperature variation in time along the axis of the cylinder. The conductivity of the solid and liquid substance are known and the aim is to determine the latent heat and solidification temperature of the substance using the probe results.

#### Presentations

The design of a state change intrument - Prof. N Fowkes

### Problem 4: Legalisation of rhino horn trade

**Presenter**: Ashleigh Hutchinson, University of the Witwatersrand

**Problem Statement:**Legalisation of rhino horn trade is a topic of significant interest. The importance of preserving our rhino population cannot be overstated. In this problem, we will study the consequences of legalisation. The legal market obtains its rhino horns from existing stockpiles from public parks. In private parks, rhino horns are harvested in an attempt to deter poachers. Currently, harvested rhino horns and horns from the existing stockpiles cannot be legally sold to overseas buyers which consist of the majority of this consumer market. The purpose of this problem is to determine whether legalising rhino horn trade to overseas buyers can result in a decrease of the horns sold illegally. We will develop an optimal strategy that ensures the legal market will out-compete the illegal market without depleting the stock of horns available to be sold legally.

#### Presentations

Legalisation of rhino horn trade - Ashleigh Hutchinson

### Problem 5: Linear Least Square Problem and the Bias-Variance Trade-off in Machine Learning

**Presenter:** Professor Montaz Ali, University of the Witwatersrand

**Problem Statement:**

Linear Least Square

Given an *m* by* n* matrix A and an* m* by 1 vector b, the linear least squares problem is to find an *n* by 1 vector *x* minimizing ||A*x- b|| _{2}*

Read the full problem statement

#### Presentations

Linear Least square Problem - Prof. Montaz Ali

### Problem 6: Blood Platelet Production and Inventory Problem

**Presenter: **Dr Micheal Olusanya and Professor Aderemi Adewumi, University of KwaZulu-Natal

**Problem Statement:**Blood platelets are a vital and scarce live-saving perishable product. The inventory management of blood platelets is of high importance in healthcare practice. Platelets are critical component of today’s therapies and their very short shelf life makes their production and inventory management a challenging task. Production and Operational costs are high but a shortage can result into higher costs of loss of lives. Large outdate rates are also thought to be a threat to the stability of the platelet supply chain because donor participation rates are typically small. In this problem, given the initial inventory state at the beginning of the planning horizon, the statistical information of demand over the period, and fixed and variable costs data, the goal is to come up with an optimal production rule that prevents large outdate and shortage rates as well as minimize costs for the blood platelet producer.

### Problem 7: Interaction between birds and wind turbines

**Presenter: **Dr Craig Symes, University of the Witwatersrand

**Problem Statement:**Wind turbines are an important source of alternative clean energy which is sustainable in the face of rapid global change. There is strong growth in the number of wind farms in South Africa. Wind turbines do, however, have an impact on the environment. The population of the bearded vulture could decline and reach extinction if wind farms are constructed in their environment.

The aim of this problem is to investigate the effect of wind turbines on birds in flight, both resident and transient. The group is asked to develop a collision risk model to determine how the interaction of birds with wind turbines affects bird population. Do rotating blades create lift and attract birds? can alyertnative turbine designs be suggested to reduce collision risk? Can the group suggest other ideas to protect birds in flight in wind farms?

#### Presentations

Windfarms - student Maths workshop - Jan 2016

Wind Turbine and Bird Flight Interaction