MA469 Project Maths-in-Action Theme Guide 2011-12 Read This First! 1. Register your choices for MA469 Project by Sunday, 30 October 2011 at the latest. 2. This guide contains details of the 6 Maths-in-Action themes offered this year. For each theme you’ll find: • A Synopsis (a short introduction to help you decide) • The Brief (a formal definition of your terms of reference) • Further Information • Sources (a list of books and Web pages) 3. Each theme will be restricted to at most SIX takers (first come, first served). A collaborating pair counts as one taker. 1 Table of Contents Theme 1. Moving Data Across Networks......................................................................3 Theme 2. Optimal Vaccination...................................................................................... 6 Theme 3. The Mathematics of Voting........................................................................... 9 Theme 4. Data Compression........................................................................................ 12 Theme 5. Encryption on the Internet............................................................................14 Theme 6: The Human Cell .......................................................................................... 18 2 Theme 1. Moving Data Across Networks Theme 1. Moving Data Across Networks Synopsis Each day vast amounts of information are moved electronically around the world: across the Internet, through cell-phone networks, via satellites and along traditional telephone wires. To get from A to B data must be directed along available routes through a given network. How this can be done reliably and economically is a hot topic for engineers and mathematicians alike. The creative interplay of theory and practice continually brings improvements in communications (with faster downloads and cheaper phone calls, for instance). Read more about it at http://plus.maths.org/issue2/dar/index.html, a short informative article by Richard Gibbens and Stephen Turner at the Webby- Award-winning Plus Maths magazine (by the way, issue 15 of the on-line magazine Plus has a tribute to the famous architect of Communications Theory, Claude Shannon, who died in February 2001, aged 84). Your Brief You must address a coherent selection of the topics in following list, but are free to choose the emphasis you give them. Your work must highlight the connections between theory and practice, between mathematics and the real world. Your brief is: • To find out how messages are routed (i) through mobile `phone networks and (ii) across the Internet. • To discover how speed of connection and volume of traffic can be optimised. • To describe mathematical models of the routing process and measures of the network efficiency. • To describe the mathematical foundations that support these models and measures. • To conjecture how better mathematics and improved technology might interact and lead to greater network efficiency. Important: This brief will be strictly adhered to when assessing your work. If you wish to deviate from it, you must apply to the Course Organiser (either in person or by sending an email to [email protected]). Further Information A good place to start is the general survey article Modelling Communication Networks, Present and Future (see reference [1] below) by Frank Kelley, an active researcher in the field. The article, based on a memorial lecture he gave to the Royal Society in 1995, begins as follows: Modern communication networks are able to respond to randomly fluctuating demands and failures by allowing buffers to fill, by rerouting traffic and by reallocating resources. They are able to do this so well that, in many respects, large- 3 Theme 1. Moving Data Across Networks scale networks appear as coherent, almost intelligent, organisms. The design and control of such networks present challenges of a mathematical, engineering and economic nature. In this lecture I describe some of the models that have proved useful in the analysis of stability, statistical sharing and pricing, in systems ranging from the telephone networks of today to the information superhighways of tomorrow. I recommend two books: 1. Martha Steenstrup [2] is the editor of a collection of 11 chapters by experts in various aspects of routing. The chapters are divided into four parts according to the type of network under consideration: ◦ Circuit-switched networks, like the old physically-switched telephone networks. ◦ Packet-switched networks, where the data packets contend with each other for routes and resources ◦ Fast, broad-band networks, which can accommodate different types of traffic simultaneously ◦ Mobile networks, where A and B are on the move and their connections are passed from one hub to another to maintain contact. 2. The book by Bertsekas and Gallagher [3] covers all aspects of data networks and gives a good idea of the trade-offs between theory and practice. This will help you to understand the problems of routing in the broader context and its 130-page Chapter V is devoted specifically to routing. Interesting URLs are not hard to find. A short readable account of mobile phone networks can be found at http://www.sitefinder.ofcom.org.uk/mobilework.htm The Wikipedia offering at http://en.wikipedia.org/wiki/Cellular_network ends with a good list of internal links. A good introductory paper (with the mathematics in) by Frank Kelly can be downloaded from http://www.statslab.cam.ac.uk/~frank/mmi.html It is an article in Mathematics Unlimited - 2001 and Beyond, edited byB. Engquist and W. Schmid. Springer-Verlag, Berlin, 2001. pages 685-702,( ISBN: 3540669132). Sources 1. Modelling Communication Networks, Present and Future by Frank Kelley. Clifford Paterson Lecture in Philosophical Transactions of the Royal Society of London. Vol 354, No 1707. 437-463 (1996) A postscript version of this lecture can be found at http://www.statslab.cam.ac.uk/~frank/CP/. 2. Routing in Communications Networks by Martha Steenstrup (Editor) 600 pages, Prentice Hall (1995); ISBN 0130107522. 3. Data Networks, 2nd Edition by Dimitri Bertsekas and Robert Gallager, (both MIT) 592 pages, Prentice-Hall (1992) ISBN 0132009161. 4 Theme 1. Moving Data Across Networks 4. Data and Computer Communications (International Edition) by William Stallings 350 pages Prentice Hall (2003); ISBN 0131833111 5. Data Communications and Networks by David Miller, 512 pages, McGraw- Hill/Irwin (2005) ISBN 0072964049 Items 2,3, and 4 on this list are available from the Central Campus Library. 5 Theme 2. Optimal Vaccination Theme 2. Optimal Vaccination Synopsis Epidemiology is the branch of medicine that deals with the incidence and transmission of disease in populations, especially with the aim of controlling it. Vaccination is one method of control that has proved effective against diseases caused by viruses and bacteria, but it is expensive and can carry unwelcome risks. It is therefore important for politicians, who take the decisions on whether to implement a vaccination policy, to have an accurate assessment of the costs and benefits. Perhaps the most impressive example of a successful global vaccination policy is the eradication of smallpox in 1980 − see http://choo.fis.utoronto.ca/fis/courses/lis2102/KO.WHO.case.html Interestingly, smallpox has resurfaced in the contemporary context of bio-terrorism: should a new vaccination programme be started, and what level of the population should be vaccinated to prevent a resurgence of the disease? The British foot-and-mouth outbreak in 2001 inspired a lively debate about the pros and cons of vaccination, and the SARS panic in 2003 also raised some interesting epidemiological questions. Currently, the threat of avian ‘flu will continue to exercise official minds over the coming winter. Mathematics and statistics have played a central role in all these questions. Because epidemiology is such a large area, the brief of this project will be focused on vaccination models that predict the level of required to stem an outbreak of a given disease and to protect a population against further occurrences. Your Brief 1. Your first task is analyse the factors that influence the spread of a given disease: for instance, its virulence (what proportion survive the attack); how it is transmitted (airborne, carried by insects, through physical contact); its infection rate (is one sneeze in a lecture theatre enough to infect everyone?), the protection afforded by previous exposure and by the vaccination; and so on. 2. Your second task is to derive and describe one or more mathematical models that take these factors into account in predicting the level of vaccination required, It is also to assess how accurate and effective these models are, and where appropriate to compare their strengths and weaknesses and to speculate how they might be improved on. 3. Thirdly, you should illustrate your work with plenty of real-life examples, past and present; you should also discuss current issues (political, financial, medical, sociological, demographic) surrounding vaccination. Finally, these three threads should be woven into a coherent and absorbing story that will enthral our open day visitors and engross your fellow MMath students when they come to do a peer assessment. 6 Theme 2. Optimal Vaccination Important: This brief will be strictly adhered to when assessing your work. If you wish to deviate from it, you must apply to the MiA Course Organiser (either in person or by sending an email to [email protected]). Further Information There are