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Problem 1: Detecting imbedded thin strips
Detecting imbedded thin strips: metal or oil detection, conductivity determinations
Presenter:
Professor Neville Fowkes, University of Western Australia
Problem statement:
A thin strip of material is imbedded in a material of known but very different conductivity, either much larger or much smaller. Is it possible to determine the location, thickness and conductivity of the strip by observing changes in the temperature response at the surface brought about by the presence of the strip. How should this be done if the conductivity is known but the location is not, or if the location and strip thickness are known but the conductivity is not known. If all three strip parameters are unknown what can be determined.
The problem is analogous to, but simpler than, the problem of detecting oil under the ocean where electromagnetic waves are used for detection. Also the problem arises in the semiconductor conductivity measurements when oscillatory laser heat input is used.
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Problem 2. Designing an online shop
Presenter: Professor Jeff Sanders, African Institute for Mathematical Sciences and University of Stellenbosch
Problem statement:
Can we design an on-line shop which benefits from being on-line and digital? With the widespread availability of the web, e-commerce has become hugely popular. Apart from convenience to the buyer, on line sales are able to offer features not readily available traditionally: marketing data can be collected digitally and techniques of data analytics used to improve sales.
This project begins with a brief survey of types of on-line shops (amazon?), in order that we appreciate the range of potential behaviours and difficulties. It then considers the features and functionality desired of an on-line shop, including those not possible in a standard shop. Important is avoidance of undesirable features, where possible. (What about privacy? Accountability? Intrusion?)
A system is designed which incorporates just the desired features.
An on-line shop will be thought of as a ‘data structure’ i.e. as consisting of state on which certain operations are defined like Browse, Order, Replenish Stock, etc. We must decide what ‘state’ the system has in order to be able to express those operations mathematically. Indeed an important step is to understand what exactly is a ‘data structure’ (states supporting operations) and how to describe it using discrete mathematics.
A typical industrial problem requires a solution which optimises some quantity. When an optimal solution is unrealistic an acceptably efficient one is sought. It might be: what standard length of aluminium minimises wastage over various uses; or what treatment of the residue of sugar cane solution is typically described by differential equations and then constraints imposed to optimise the solution.
This project is an example of an important family of contemporary problems which fit that paradigm, but whose solution is, instead of a solution to a differential equation, a design (of optimal or acceptable efficiency). Ingenuity is required to invent the design and mathematics is needed to show that it functions as desired. The design might enable several robots to coordinate in exploring a mine; it might be for a secure electronic voting system; or it might be for on-line sales featuring specials that benefit from market data. The mathematics is discrete and straightforward though perhaps unfamiliar.
An implementation (i.e. code) is not sought. Indeed our time will be spent on expressing features and deciding how they would be captured in a design.
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Problem 3. Phenology model for the Jacaranda blossoms
Presenter: Dr Jennifer Fitchett, School of Geography, Archaeology and Environmental Studies, University of the Witwatersrand
Problem statement
Phenology refers to the timing of annually recurrent biological events and their biotic and abiotic drivers. These phonological events are triggered by the change in seasons, as temperatures change and rainfall begins or terminates. Under climate change, the timing of these events is shifting. In many instances, events such as blossoming that marked the beginning of spring are now occurring in late winter. Through historical records we can track the changes in the timing of phonological events for a given species in a specific location. This is valuable in determining that change is occurring, but provides insufficient information for long-term carbon-budget modelling, or understanding of species-interaction implications. For this reason mathematical phenology models are important. In this particular problem, you will be building a phenology model for the Jacaranda blossoms, using a suite of climate variables. These will be compared to historical records of Jacaranda flowering dates compiled from archival material to test for accuracy.
Supporting Materials
Blumel & Chmielewski 2012 (Modelling Camp Problem 3)
Chuine et al. 1999 (Modelling Camp Problem 3 2020)
Jacaranda Phenology SSAG (Modelling Camp Problem 3 2020)
Matzneller et al. 2014 (Modelling Camp Problem 3 2020)
Rea & Eccel 2006 (Modelling Camp Problem 3 2020)
Presentation
202008 Modelling Camp Fitchett
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Problem 4. Green roofs for managing stormwater runoff
Presenter: Professor Anne Fitchett, Assistant Dean, Faculty of Engineering and the Built Environment, University of the Witwatersrand
Problem statement
As cities become increasingly densified, open spaces with vegetation become more and more depleted. This has a negative effect on the stormwater performance of our cities, as conventional infrastructure cannot cope with the volume of rainfall. In extreme cases, this results in flooding, a phenomenon that will become increasingly common with the intensity of storm events that are predicted with climate change in our region.
Green (or vegetated) roofs can be one strategy for reducing these negative effects of urbanism and climate change. We have carried out some simulations using MUSIC that uses fairly simplistic hydrological methods which are functional for water balance analyses over long durations, but are not mathematically rigorous. The main interest we have with green roofs were:
- how much they reduce the overall hydraulic load (runoff) from the roof area of a building?
- what additional area of runoff can be accommodated by the same area of green roof? (i.e. assuming part of the roof area of a building is pitched and not suitable for green roof establishment, can that area drain to the green roof portion and still achieve reasonable reduction in roof runoff?)
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Problem 5. Mathematical modelling of tornadoes
Presenter: Dr Thama Duba, Durban, KwaZulu-Natal
Problem statement
In the week of 10-16 November 2019, Kwazulu-Natal Province in South Africa experienced severe thunderstorms that generated tornadoes. A tornado ripped through the semi-rural area of Hanover on Tuesday 12 November 2019, and two days later another one hit the Bergville community in the Drakensburg area.
The South African Weather Services (SAWS) was quick to say that such weather was not uncommon as November is usually characterized by frequent storms, hail and rain, which means they occur often in spring and summer. According to reports 17 tornadoes ripped across South Africa in the last decade.
What is a tornado? A tornado is a rapidly rotating column of air that is in contact with both the surface of the earth and a cumulonimbus cloud. The windstorm is often referred to as a twister, whirlwind or cyclone.
The tornado that occurred on the 12th November was classified as the strongest, an F5 tornado on the Fijuta Scale. The strength of a tornado is measured by the damage it caused – an F0 category being the weakest and damaging mainly the trees but not substantial structures. An F5 category rips buildings off their foundations and can deform large skyscrapers.
Occasionally, a single storm can produce more than one tornado either simultaneously or in succession.
Tornado forecasts and warnings are usually issued on a short-term basis since they involve rapidly developing weather systems. It is also not clear where a wind swirl will hit the ground and how far a tornado will travel before it dissipates away. Furthermore, it will be interesting to understand its propagation properties.
Tornadoes can be detected before or as they occur through the use of Pube-Doppler radar by recognizing patterns in velocity and reflectivity data as well as through the efforts of storm spotters.
Mathematical modelling has become essential in understanding the dynamics of storm formation and to answer some of the questions raised above.
The graduate modelling camp is required to answer the following questions:
- What is the mechanism that generates existence of a tornado?
- What mathematical model, assumptions and laws govern a tornado?
- What are the dynamics involved in the trajectory of a tornado?
References
https://doi.org/10.1016/j.gsf.2011.03.007
https://doi.org/10/1016/s0735-1933(00)00107-X
Gill A. Atmosphere-Ocean Dynamics, Academic Press, 1982.
Presentation
202008 Mathematical Modelling of Tornadoes
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Problem 6. Thermal plumes in Lake Kivu
Presenter: Professor David Mason, University of the Witwatersrand, Johannesburg
Problem statement
Lake Kivu is a deep African lake on the border between Rwanda and the Republic of the Congo. Groundwater carries volcanic carbon dioxide and methane into the lake which are dissolved in the lower regions due to the large hydrostatic pressure. Methane is extracted to generate electricity. There have been gas eruptions in two neighbouring lakes, Lake Monoun and Lake Nyos which have lead to loss of life. The carbon dioxide is heavier than air and lies on the surface of the lake and flows down the valleys. People and animals are suffocated. The minimum requirement for gas to be released is that the sum of the partial pressures of the dissolved carbon dioxide and methane exceeds the hydrostatic pressure at that depth. The water can be raised to achieve saturation by a rising thermal plume. The plumes can either be two-dimensional due to a line source of heat or axisymmetrical due to a point source of heat. There can also be wall plumes due to a heat source on a wall of the lake. We will consider a two-dimensional thermal plume and investigate exact analytical solutions.
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