C O N TE N TSma.rhul.ac.uk/files/undergradhandbook.pdf · 3.1 A c a d e m ic W e lf a re 3.2 S ta ff -S tu d e n t C o m m itte e 3.3 Q u e stio n n a ire s 3.4 S tu d y A b ro a

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CONTENTS

Page No.

1. Introduction 2

1.1 Teaching Objectives

1.2 Term Dates

1.3 What’s New in 200 6-7?

2. About the Department 3

2.1 Location

2.2 Methods of Communication

2.3 Department Staff

3. Student Welfare 8

3.1 Academic Welf are

3.2 Staff -Student Committee

3.3 Questionnaires

3.4 Study Abroad

3.5 Equal Opportunities

3.6 Support for Special Needs Students

3.7 English Language Support

3.8 Personal Development Plans

3.9 Careers Advice

3.10 Volunteering

4. The Academic Structure 11

4.1 The Course Unit System

4.2 Degree Programmes

4.3 Course Registration

5. Teaching Methods 12

5.1 Lectures

5.2 Tutorials and Workshops

5.3 Worksheets

5.4 Projects and Post -Examination Essays

5.4.1 First year essays

5.4.2 Second year group projec ts

5.4.3 Third and fourth year projects

5.5 Textbooks

5.6 Computer Facilities

5.6.1 Mathematica

5.6.2 MINITAB

5.6.3 Computer Centre training

5.7 Departmental Skills Matrix

6. The Monitoring Process 15

6.1 Attendance

6.2 Work Requirements

6.3 Monitoring

Page No.

7. Assessment 16

7.1 Course Unit Assessment

7.2 Marking Criteria for Essays etc.

7.3 Progression

7.4 Titles of Degrees

7.5 Degree Classification

7.6 Adverse Circums tances

7.7 Plagiarism

7.8 Calculators in Examinations

8. Prizes 19

9. Lecture List 21

10. Programmes of Study 22

10.1 Mathematics and Mathematics

Major

Degree Programmes

10.2 Joint Degrees involving Mathematics

10.3 Mathematics Minor Degree

Programmes

10.4 Mathematics MSci (4 year degree)

11. Course Unit Descriptions 28

12. The Greek Alphabet 83

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1. Introduction This booklet is prepared for undergraduate students registered for degree programmes involving mathematics as a single subject, in a joint degree with another department or as a major component in the Major/Minor scheme. The aim of the booklet is to give general information about the topics listed on the contents page. The contents are also on the Departmental web site at www.ma.rhul.ac.uk/current/bluebook/

1.1 Teaching Objectives

The teaching objectives of the Department are that all students in the above categories should: have gained a solid grounding in Mathematics, a working knowledge of its tool-kit of methods, and an understanding of its basic concepts; have developed their capacity to think logically and clearly; be able to communicate Mathematics effectively in writing; be able to use appropriate computer packages as an aid to their study and problem solving; have developed the ability to work efficiently and independently, and to organize their work to meet deadlines; have an understanding of a further range of mathematical topics; be equipped for employment, training or further study across a wide spectrum of mathematical and non-mathematical activities. The Department aims to comply with the College Regulations, Student Charter and Codes of Practice. The Codes of Practice cover Academic Welfare, Freedom of Speech, Student Union Affairs, Personal Harassment, and Health and Safety. No interpretation of the information presented here should conflict with these regulations or a Code of Practice. In the case of any apparent difference, the College regulations will prevail. Students should consult the College Undergraduate Regulations for general information, at www.rhul.ac.uk/Registry/academic_regulations/Undergraduate_Regulations.html .

1.2 Term dates

First term: Monday 25 September to Friday 15 December 2006 (lectures start Monday 2 October)

Second term: Monday 15 January to Friday 30 March 2007 Third term: Monday 30 April to Friday 15 June 2007 Graduation ceremonies: Monday 16 July to Friday 20 July 2007.

1.3 What’s new in 2006-2007?

The Periodical Departmental Review last year said that we should introduce some project work for all students, including oral presentations, as part of the core curriculum. After discussion with various people, including the Careers Service and the Student-Staff Committee, we have decided to introduce this in the first term of the second year, as part of the algebra course MT282 (which therefore becomes MT280 with a longer title). 20% of the credit will be for the project (which can be in any area of mathematics), and 80% for the examination on linear and abstract algebra (the content has been reduced in line with this). MT280 will be core for all Mathematics, Mathematics major and Mathematics joint students other than Economics and Mathematics (LG11) students. Economics and Mathematics and Mathematics minor students may take MT280 as an option.

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In 2005 there were insufficient students for us to give MT251 Computational Mathematics II. We plan therefore to replace it with a third year option, starting in 2007-8; it is clear that the material is useful for many employers and we do not wish to lose the opportunity for students to study it. Because MT465/5465 Network Algorithms is taken largely by MSc students who took their first degrees elsewhere and often have not covered the material in MT365 Algorithmic Graph Theory, we have altered MT5465 (the MSc version) so that the lectures will consist of those for MT365 together with a set of lectures specifically for the MSc students. The assessment is quite separate, however. This means that fourth year students who took MT365 in their third year cannot take MT465: however those who did not take MT365 may take MT5465 as a fourth-level course. (The same applies to MT361/5461, MT362/5462 and MT364/5464.) To balance the fourth year units between the terms, MT447 Advanced Financial Mathematics will be in term 2. The Computer Science and Mathematics Departments have decided that (from October 2005) students may not take both MT262 and CS349 in their programme, as there is substantial overlap in the material.

2. About the Department

2.1 Location

The Department – that is, the staff and secretarial offices and the Computer Room – is located on the second and third floors of the McCrea Building. Some staff have offices in Founders Building. The McCrea building is open from 8.00am to 5.30pm each working day; it is closed at weekends and on Public or College holidays. Maps of levels 2 and 3 of the McCrea Building are on pages 3 and 4. The Department Office (C243) is open to student enquiries from 9.00am to 5.00pm. Past examination papers may be purchased from C243 at the advertised times. Computing facilities room C103 is a shared lab with computer science; it will be open from 9.00am to 5.30pm and is available for use except when booked for course teaching purposes. The times when it is booked are shown on the door. A further 6 PCs will be available for general use during office hours in C356. Smoking is not allowed in the McCrea Building.

2.2 Methods of Communication

Members of staff communicate with students via announcements in lectures, the notice boards, by letters addressed to student rooms (for residents) or posted in the pigeonholes in the foyer of McCrea and by e-mail. Therefore, all students should check the main notice board, their pigeonhole and their college e-mail daily. Students communicate with staff by visiting them in their offices, by telephone, by e-mail or by letter (through the post or hand-delivered to the box outside the relevant office). A complete list of the notice boards is: Main notice board McCrea foyer College Official Notices McCrea foyer Student Union Notices McCrea foyer Staff-Student Committee by C223 Postgraduate Courses by C251 and C253

Statistics Notices by C349

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Student Society Notices in Lecture Rooms Job Opportunities by C224 Mathematical News by C223 You will also find the Departmental web site, http://www.ma.rhul.ac.uk, a useful source of information. It contains most of the contents of this book, as well as other details of the Department’s activities.

2.3 Department Staff

Rooms given by just a number are in the McCrea Building, and rooms prefixed by E or W are in Founders Building. When telephoning staff from outside College dial the Egham code (01784) if required and follow with 44 if the extension number starts with 3 and 41 if the extension number starts with 4, then the extension number. After 5 rings the call, if unanswered, is diverted to the appropriate secretary. Teaching Staff Room Phone e-mail address Mr John Austen 348 3974 [email protected] Dr Yiftach Barnea 228 4689 [email protected] Professor Simon Blackburn Head of Department

234 3422 [email protected]

Professor Ken Bowen 356B 3082 Dr Carlos Cid 224 4685 [email protected] Dr Chez Ciechanowicz 341 3112 [email protected] Dr Jason Crampton 344 3117 [email protected] Dr Mark Damerell 214 3090 [email protected] Dr Christine Davies Admissions Tutor


Dr Gar de Barra 352 3079 [email protected] Dr Alex Dent 223 4922 [email protected] Dr Christian Elsholtz 240 4021 [email protected] Professor John Essam 256B 3080 [email protected] Dr Christine Farmer Year 1 Coordinator


Mr Andreas Fuchsberger 220 3094 [email protected] Dr Steven Galbraith 346 4396 [email protected] Mrs Hilary Ganley 342 3084 [email protected] Mr Ed Godolphin E145 3435 [email protected] Professor Glyn Harman 244 4235 [email protected] Dr James McKee 231 3670 [email protected] Dr Kostas Markantonakis W158 4409 [email protected] Dr Keith Martin 349 3099 [email protected] Dr Keith Mayes W156 4408 [email protected] Professor Chris Mitchell 347 3423 [email protected] Dr Francisca Mota-Furtado 241 3096 [email protected] Professor Sean Murphy 354 3699 [email protected] Dr Siaw-Lynn Ng 350 4397 [email protected] Dr Chris Norman 352 3079 [email protected] Professor Pat O’Mahony 351 3088 p.o’[email protected]

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Dr Tobias Osborne 232 4983 [email protected] Professor Kenny Paterson 345 4393 [email protected] Professor Fred Piper 233 3098 [email protected] Dr Geraint Price 225 4160 [email protected] Professor Rüdiger Schack 245 3097 [email protected] Dr Scarlet Schwiderski-Grosche 340 3089 scarlet.schwiderski-

[email protected] Dr Eira Scourfield 356A 3671 [email protected] Dr Teo Sharia 236 4331 [email protected] Dr Andrew Sheer Academic Coordinator


Professor Peter Wild 227 3081 [email protected] Dr Stephen Wolthusen 353 3270 [email protected] Dr David Yates 356A 3078 [email protected]

Secretarial Staff Room Phone e-mail address Joy Fitzsimmons 243 3093 [email protected] Jenny Lee 243 3091 [email protected] Drs Anja Steenkiste Administrator, Mathematics


Mrs Pauline Stoner Administrator, ISG


Technical Staff Room Phone e-mail address Tristan Findley Systems Administrator, ISG


Jon Hart Systems Administrator, ISG


Liz Jenkins Computer Technician, Mathematics


Lisa Nixon 243 3106 [email protected]

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3. Student Welfare and Support

3.1 Academic Welfare

The academic welfare of undergraduates is conducted through the (personal) adviser, and in the first year also the Year Coordinator. Lists allocating students to advisers are posted on the Main Notice Board at the beginning of the Academic Year.

Advisers

Each student is assigned a member of staff to act as personal adviser. The role of the adviser is to guide the academic progress and provide pastoral care for the advisee throughout the duration of the programme of study. It is important that students discuss any academic, financial, medical or other problems with their adviser as soon as they arise. The adviser may then be able to recommend an appropriate source of help, and be able to act on the student’s behalf. Any personal information you impart will be treated in strict confidence and disclosed to other staff only with your consent. See section 6.6 below for what to do if your examination preparation is affected by adverse circumstances. Students should see their adviser at least at the beginning and end of each term. These meetings combine routine aspects of course administration with planning and reviewing the programme of study.

3.2 Student-Staff Committee

Liaison between staff and students occurs informally on a daily basis through personal contact and formally through the Student-Staff Committee. The Committee consists of two student representatives elected from each year group, with postgraduate and staff representatives, and has the wide-ranging brief to discuss any and all matters of interest or concern to students. It normally meets two or three times a term in the first two terms, and once in the final term. Photographs of the student representatives, the Agenda and Minutes of meetings are displayed on the notice board by C223. Items for the Agenda should be raised with your year representatives or with the Academic Coordinator. Note: Problems requiring urgent attention should be directed to your adviser or to the Academic Coordinator immediately they arise – a large problem is often a small problem which has been allowed to grow.

3.3 Questionnaires

Course questionnaires, issued at the end of each lecture course, and occasional general questionnaires are used as a means of assessing student views. These form an essential part of our procedures to monitor the quality of our provision and they are invaluable in course planning and improvement. For the system to work it is imperative that we have a high percentage of returns and that the questionnaires have been filled in thoughtfully. The completed anonymous questionnaires are scanned by an outside contractor, and the summaries are presented to the Student-Staff Committee and to all staff. The comments on the back are passed to the lecturer and to the Head of Department and the Academic Coordinator.

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3.4 Study Abroad

If you are interested in gaining experience of higher education in another country you may be able to take advantage of one of two schemes. The SOCRATES programme encourages exchanges mostly in European Union states, and the Mathematics Department has links with universities in Austria, Belgium, Germany, Greece, Italy and Spain. Royal Holloway also has exchange partner universities in Australia, Canada, Hong Kong, Japan, Russia, Singapore and the USA. In each case they allow you to spend an academic year abroad, which in some cases may count towards your degree and in some cases means an extra year. If the year abroad is to count, it must be the second year (the second or third year in the case of MSci students). The deadline for applications is in January each year. For further details see www.rhul.ac.uk/for-students/socrates.html for Europe and www.rhul.ac.uk/for-students/studyabroad.html for outside Europe.

3.5 Equal Opportunities

This is the College's statement on equal opportunities: The University of London was established to provide education on the basis of merit without regard to race, creed or political belief and was the first university in the United Kingdom to admit women to its degrees. Royal Holloway is proud to continue this tradition, and to commit itself to equality of opportunity. Specifically, the College’s policy in relation to the recruitment of staff and the admission of students is as follows: The only consideration in appointment, training, appraisal and promotion of employees is how the genuine requirements of the post are met or are likely to be met by the individual under consideration. The requirements (including retirement at the appropriate age) being met, no regard will be taken of any of the following factors; race, gender, age, marital status, details of dependants, nationality, disabilities, sexual orientation, religion, political beliefs or social origins. All persons of the requisite academic standard, whether resident in the United Kingdom or elsewhere, are eligible for consideration for admission as registered students of the College. Royal Holloway does not discriminate against any applicant on grounds of race, gender, age, marital status, details of dependants, the nature of any disability, nationality, sexual orientation, religion, political beliefs or social origins

3.6 Support for Special Needs Students

The Department is fully compliant with legislation concerning students with specific learning disabilities or other special needs. The Department Special Needs Coordinator, Dr Mota-Furtado, liaises with the central College support mechanisms provided by the Student Support Services. Any students diagnosed with special needs, or who think they may have special needs, are encouraged to contact either Dr Mota-Furtado, their personal adviser, or the Educational Support Office (Founders W151). Applications for extra time in examinations must be submitted during the first term of each academic session.

3.7 English Language Support

The College Language Centre offers a wide range of English Language programmes throughout the year to suit the diverse needs of international students. For details look at the Language Centre’s website on www.rhul.ac.uk/language-centre/ .

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3.8 Personal Development Plans

Personal Development Plans or Records have been used to help employees analyse their progress within private companies and certain branches of the public sector for a number of years, and Royal Holloway, like most universities in the UK, offers a version of Personal Development Planning. The steps entailed in generic PDP can be summarized as follows:

• Assessing your academic and non-academic strengths and areas for improvement.

• Monitoring your recent progress against a set of personal and interpersonal ‘competency benchmarks’.

• Reflecting on your values and priorities and engaging in active decision-making with respect to the future.

• Identifying and pursuing opportunities for skills development both on and off campus.

• Documenting and interpreting your learning highlights, achievements and career-related activities.

• Setting goals and constructing an action plan for personal and academic development.

Personal Development Planning is entirely optional and you are not obliged to undertake all or any of the activities outlined above. Should you decide to do so, the records you produce remain your private property. Staff in your department will not require you to share their contents or retain them. However, it is often helpful to discuss learning, achievement and employment goals, as well as completed Personal Development Records, with someone else: ideally your adviser, who should know your full academic background. For further details look at www.rhul.ac.uk/pdp/

3.9 Careers Advice

The Department works closely with the College Careers Service. The Department’s Careers Officer is Dr Farmer, and our liaison Adviser in the Careers Service is Hilary Moor. Between them they organise talks and events of particular relevance to Mathematics students. You should also look regularly at the notice boards by rooms C224 and C349, and the Careers website on www.rhul.ac.uk/Careers/ .

3.10 Volunteering

Student Community Action – get volunteering!

Whoever you are, whatever you’re into, we’re always looking for volunteers to support local projects (one-off and ongoing). Try out new things, meet new people, learn new skills, build your CV and give something back! You could help people with special needs, do conservation/environmental work, work for a charity shop, rescue animals, coach disabled sports, tutor in local schools, help people with mental health difficulties or work with asylum seekers/refugees. Or start you own project! Training and on-going support is provided and you can get your volunteering accredited under the QUANTA-scheme. Visit http://www.rhul.ac.uk/services/volunteering , contact Matt Rosenberg on (01784) 414078, e-mail [email protected] or pop into the Community Action Office (room FE115, Founder’s East 1st floor).

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4. The Academic Structure

4.1 The Course Unit System

The College operates a course unit system. In this system the study material is divided into packages called course units or course half units. The terms course unit and half-unit are used to describe a lecture package and also, through successful completion of the course assessment (loosely, by passing the examination!), to represent the appropriate amount of credit towards the qualification for a degree (see section 5.3). In Mathematics all undergraduate courses – except the Year 4 MSci project – are half unit courses taught within a term. Undergraduate courses are here (and within the Mathematics Department generally) referred to by MT + a three-digit number. The College administration refers to courses by a two-letter departmental code (MT) + a four-digit number . To convert the number given here to the College reference, add 0 (so MT182 becomes MT1820 and MT369 becomes MT3690). Fourth year courses which also form part of an MSc programme have two numbers: for example MT4540 and MT5454.

4.2 Degree Programmes

A degree programme is a combination of course units/half units taken over three years (BSc, or BA for some combined programmes) or four years (MSci) of full-time study, normally including courses to the value of 4 units per year. Each named degree programme contains a core of prescribed courses and a choice of options from the courses given by the Department (or Departments in the case of combined degree programmes). Each student is registered for a named degree programme; either that programme to which the student was admitted to College or a programme to which the student has formally transferred. In each year you must take the prescribed courses specified for your programme. Students wishing to change their degree programme must:

(i) consult their adviser to discuss the change; (ii) if it is their first term at the College, consult the Admissions Tutor, Dr

Christine Davies; (iii) obtain the consent of all departments involved in the existing and

proposed new degree programme – by consulting the Academic Co-ordinators in the departments;

(iv) collect a Change of Degree Programme Form from the Registry, room Founders W141; you must get this signed by the Academic Coordinators of all departments involved and return it to the Registry.

4.3 Course Registration

At the beginning of the Academic Year, students should see their adviser and complete the course registration form listing the courses the student will take during the year. Changes in registration for optional courses are permitted as follows: (i) In the first two weeks of each term students may sample the optional courses by attending all the lectures; (ii) At the end of the two weeks, a firm commitment to the choice of courses is required – no change will be allowed after the first two weeks of each term.

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For each change you should fill in a form (available from Drs Steenkiste or Dr Sheer) giving the details, which must be signed by you and by your adviser. It should then be given to Drs Steenkiste. Early in the third term students will be asked to pre-register for their courses in the succeeding year. This pre-registration is not binding but is required for forward planning to construct a Faculty timetable which allows every student to take the combination of courses they have chosen, and to allocate appropriately sized lecture rooms to each course.

5. Teaching Methods

5.1 Lectures

For most mathematics (half-unit) courses the main method of instruction is through three hours of formal lectures per week delivered throughout a term. The aim of the lectures is to develop the subject matter step-by-step and thereby guide the student to comprehension of the concise notation and precise use of technical language of written mathematics. Students are expected to construct their own set of lecture notes using the material displayed on the board or screen to augment any duplicated handouts. The resulting set of notes plus the weekly worksheets forms the definitive text for the course. Copies of the lecture timetable are available from the box outside C243. You must ensure that any mobile phones or similar devices are switched off in lectures, tutorials or other classes. Failure to comply with this requirement, or engaging in any other disruptive behaviour, may lead to a formal warning.

5.2 Tutorials and Workshops

Weekly tutorials, consisting of a small group of students (typically 4) meeting with a staff member or postgraduate in his or her office, are used to complement the lectures in the first term of the first year. The aim of a tutorial is to deepen the understanding of the course material, to develop good study skills and to increase motivation. You will gain more from tutorials if you identify subjects for discussion beforehand, possibly in conjunction with the other students in your group. Workshop sessions, consisting of a group of about twenty students meeting with a staff member, are used in most first and some second year courses. These aim to help you develop your understanding and skills by working on specified problems at your own pace, either singly or in a group. The staff member is there to give you instant feedback on your work, to help you in difficulties, and to suggest ideas to enrich your learning. We do expect you to attend tutorials and workshops regularly; if you are prevented from attending you should tell your tutor or workshop leader in advance. An Office Hours system is used for most second year and all third year and fourth year courses. Staff members will put on their doors times at which they may be consulted.

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5.3 Worksheets

In most courses the lecturer will give out weekly worksheets. You should hand in solutions on time each week for the following reasons: (i) It is a college regulation that all course work is completed and submitted for

assessment. Failure to comply may lead to a formal warning and the award of incomplete status on that course (see also 6.2).

(ii) The learning process in mathematics is concentrated on the solution of

problems. Those problems which form the weekly coursework are designed to reinforce the lecture material and progressively develop the ability to solve problems in mathematics. This forms an essential part of the learning process, and the corrections and comments on your attempts give feedback to you and to the lecturer on your progress. Model solutions are provided afterwards.

(iii) If due to illness, or other good cause, you fail to attend an examination or your

performance was affected, your record of worksheets will be taken into account. The examiners may then return an outcome of 'Allowed' for the course which might enable you to progress or to graduate.

5.4 Projects and Post-Examination Essays

When asked what qualities employers looked for in Mathematics graduates, a leading industrialist put ‘literacy’ and ‘the ability to communicate’ at the top of his list. Other valued skills are the ability to work with others on a problem, to research a topic, using a library, the web, and other resources, and to present information from a variety of sources in a coherent report. So we provide an opportunity to develop these skills at various stages.

5.4.1 First year essay

After the examinations, at the end of the first year, we encourage you to write an essay or project on a mathematical topic of your choice. Detailed instructions and guidelines will be sent to all first students during May, and the deadline for submission is July 31. The best essay is normally awarded a Driver Prize.

5.4.2 Second year group projects

These form part of MT280, and are core for all students studying Mathematics, Mathematics with anything and Mathematics and anything (except Economics and Mathematics students). They are optional for students in any other programme involving Mathematics. See the entry on MT280 for further details.

5.4.3 Third and fourth year projects

Third year students may if they wish prepare a mathematical project as a half-unit, MT300; all fourth year MSci students must take a full-unit project, MT400. See these entries for more details.

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5.5 Textbooks

Though (as explained above) your lecture notes and worksheets form the definitive version of the course, you will need to consult other sources. The descriptions in section 11 below list the main texts for each course; the lecturer will tell you which books are most useful, and possibly suggest others. You may also find your A-level textbooks useful for some first year courses. We welcome comments about books in the library. If you think that more copies of an important textbook are needed in the library, or you find a book that you think should be there, tell the lecturer concerned. Remember, though, that it may take a little while for a book to be bought and appear on the shelves. The Liaison Librarian for Mathematics, Graham Firth, is happy to help you with your queries about the Library.

5.6 Computer Facilities

As well as the PCs available in C103 (shared with computer science) and C356, you have access to PCs in the Computer Centre and elsewhere on campus.

5.6.1 Mathematica

All students attend two introductory sessions, on Mathematica, ‘a system for doing mathematics by computer,’ in their first week, and must produce some work done in Mathematica as part of MT171. A basic knowledge of Mathematica may be assumed in all subsequent courses. Mathematica is available on the PCs in C103, and some other PCs on campus, including Lab 1 in the Computer Centre and the Statistics Laboratory in Queens Annexe: the instructions for using it there are given on the inside of the door to C103. As a student you can buy your own copy of Mathematica for Students at a very reduced price (about £90) from some branches of Blackwell’s bookshops, including that on Charing Cross Road, London, or direct from Wolfram Research on www.wolfram.co.uk . You will need evidence that you are a bona fide student.

5.6.2 MINITAB

The statistical package Minitab is used in all the statistics half-units; it is introduced in MT130. Like Mathematica, it is on the PCs in C103, and on most other PCs around the campus (where you usually enter via Programs->Current Applications). A student version of Minitab is on sale, from www.olc.co.uk for £79.

5.6.3 Computer Centre training

If you are a first year student you will take a basic course as part of Induction Week. For all students we strongly recommend that you develop your IT skills, and take some of the Computer Centre’s courses. Look at www.rhul.ac.uk/Information-Services/Computer-Centre/training/isis/students .

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5.7 Departmental skills matrix

As part of the College’s Learning and Teaching Strategy, each department provides a matrix showing in which part of each degree programme each skill is learnt. The Mathematics Department’s matrix is shown overleaf.

6. The Monitoring Process

6.1 Attendance

We expect you to attend tutorials, workshops, lectures and practical sessions regularly. Attendance registers are kept for all tutorials and workshops, and a random sample of lectures and practical sessions. If you are unable to turn up you should contact the member of staff in charge, if necessary by e-mail or telephone. An absence of more than one week for medical reasons should be reported to the Department Office and a medical certificate supplied. Permission for absence on non-medical grounds must be obtained from the Academic Coordinator. Students may only leave early, before the end of any term, if they have completed the academic and administration requirements and have obtained the agreement of their adviser.

6.2 Work Requirements

Students normally follow four half units per term and should expect to study each half unit for about ten hours per week - this total includes lectures, workshops, tutorials and practicals. The coursework is an integral part of the learning process and you must allow time for this.

It is expected that students will:

• hand in on time (attempted) solutions to the weekly coursework; • submit project components by the specified deadlines;

The Academic Development Committee has decided that in the absence of acceptable extenuating cause, late submission of work should be penalised as follows:

• for work submitted up to 24 hours late, the mark will be reduced by ten percentage marks, subject to a minimum mark of a minimum Pass;

• for work submitted more than 24 hours late, the maximum mark will be zero. This was passed by the Academic Board of the College on 14 June 2005, and applies to all students in all subjects. Your attention is drawn to the College regulation which states that students are required to complete all coursework set by the Department, whether or not that coursework counts towards assessment. Failure to complete the coursework for a half-unit may result in the award of incomplete status for that course.

6.3 Monitoring

Student progress is monitored on a course basis by the course lecturers, by the Year Coordinator (in the first year), and the Academic Coordinator. Any student whose attendance or work falls below an acceptable standard will be so informed and appropriate remedial action or help will be suggested. Failure to act on this advice and remedy deficiencies will result in the operation of the formal warning procedure which

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could lead to the termination of the student’s registration for a course-unit or for a programme of study.

7. Assessment

7.1 Course Unit Assessment

Each course is assessed in the year in which it is taken. All course units in mathematics, except for the project based courses MT300, MT309, MT364 and MT400, are examined by a written paper in the third term. A number of half units contain both a written paper and project work, and in these a satisfactory standard in each of the separate components is required for a pass. Written papers are set by the course lecturer, checked by another member of the department and then scrutinized and approved by an external examiner. The scripts are marked, second marked and scrutinized by the same three people. Project work follows a similar pattern. The result of each examination is recorded as a score in the range 0 -100. Students who hand in a blank or nearly blank script will be awarded an incomplete status on that course (see also 7.2). Note that an incomplete result means that, if the course is to be attempted again, it must be as a repeat course. Subject to the agreement of the Sub-Board of Examiners in Mathematics, any half unit which has been failed may be attempted once again either by resitting the examination in the succeeding year (in which case the maximum mark available is 50%); or by repeating the course – that is attending the lectures as part of the course of study in the succeeding year. The highest mark thus obtained will be used in the scheme for classification for honours – the weighting of a retaken course will be that which would have been used if it had been passed at the first attempt. Notes: (i) The Year 1 Term 1 end-of-term examinations for MT141 and MT194 are progress-

check examinations and are not part of the assessment for those course-units. (ii) In some very limited cases a student who fails to progress at the end of the first

year may be able to resit failed courses in early September. The conditions are described in a handout from the College: see also 6.2 below.

(iii) To pass a course-unit you must have completed and presented for assessment all work specified for the course within specified deadlines.

7.2 Marking Criteria for Essays etc.

These criteria are used in courses MT300, MT309, MT364 and MT400, and the relevant sections of other courses. First class (85 – 100%): It shows deep understanding and near-comprehensive knowledge of the subject. There should be significant originality in interpretation or analysis; coherent structure, intensive, detailed and critical reading with independent reading including research papers. The essay or report should be excellently presented, with referencing and bibliography of publishable quality; the style should be incisive and fluent with no errors of spelling, punctuation or grammar. First class (70 – 84%): It shows deep understanding of the topic, and demonstrates excellent analytical and problem-solving skills. The arguments are clearly constructed, and the material is very well organized and presented. There are no errors or omissions.

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There is good use of outside reading and of originality of thought and analytical skill. There are no or very minor errors of spelling, punctuation and grammar. Upper second class (60 – 69%): It shows sound understanding of the topic, and demonstrates good analytical and problem-solving skills. The arguments are well constructed, and the material is well organized and presented. There are no significant errors or omissions. There is some use of outside reading and of critical evaluation of the evidence relevant to the topic. The essay or report has a coherent structure, with few errors of spelling, punctuation and grammar. Lower second class (50 – 59%): It shows understanding of the main issues and demonstrates sound knowledge of information relevant to the topic. There may be minor flaws in the arguments and in the organization of the material. There is some evidence of outside reading. The essay or report has a recognizable structure, with some errors of spelling, punctuation and grammar. Third class (40 – 49%): It shows partial understanding of the main issues and there may be major flaws in the arguments and in problem-solving. The essay or report may not be fully focussed on the topic. The essay or report is disorganized, with frequent errors of spelling, punctuation and grammar. Fail (up to 39%): It shows little or no understanding of the main issues and there may be serious errors or misunderstandings. Arguments are seriously deficient or absent. The essay or report is disorganized or fragmentary, with many errors of spelling, punctuation and grammar.

7.3 Progression

The College regulations for students to proceed to the next year of their course are given at http://www.rhul.ac.uk/Registry/academic_regulations , together with a lot of other material. If you have any query ask your adviser or the Academic Coordinator. The Mathematics Department does not require that any specific course-units must be passed before progression to the next year, but many other departments do so.

7.4 Titles of Degrees

Students who have passed 9 units have qualified for a BSc degree, with 14 units over four years required for the MSci degree. The class of the degree will be determined by application of the appropriate classification scheme (see the College regulations at the web site given in 6.2 above) and the detailed considerations of the internal and visiting examiners at the meeting of the Sub-Board of Examiners. The title of the degree will generally be that of the degree programme for which the student is registered. The rules governing the degree title are as follows. Students whose main subject is Mathematics will have their degree described as: Mathematics if the total number of courses passed outside Mathematics is less than a quarter of the total number of units passed; Mathematics with xxx if the proportion of the courses passed in the second subject is at least 1/4 and less than 1/3; Mathematics and xxx if the proportion of courses passed in the second subject is at least 1/3. However, in the case of certain joint degrees where the other subject requires that certain core courses be passed before allowing the name in the degree title, the outcome may be different. For this purpose the courses MT130, MT230, MT232, MT331, MT332, MT334 and MT336 are regarded as courses in Statistics.

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7.5 Degree classification

The College-wide system is described in the Undergraduate Regulations, section 7: Consideration for the Award. Broadly speaking the rules are that students starting the first year of an Undergraduate programme in September 2000 or later will be assessed equally across all marks in each year;

• the year weighting for all Bachelors degrees will be 0:1:2 ; • the year weighting for all MSci degrees will be 0:1:2:2.

The section below gives the specific criteria for considering borderline candidates by the Mathematics Sub-Board.

Discipline-specific criteria for considering borderline candidates

College Undergraduate Regulation 86 states that a candidate on the borderline between two classes of degree will be considered for raising into the next class if both the following are satisfied: a) the final average must fall 2.00% or less below one of the classification boundaries; b) at least five half-unit marks counting in the final year, and a further three half units

marks counting in the penultimate and/or final year(s), must be in or above the higher class.

The Mathematics Sub-Board has determined the following factors which may be identified in favour of raising a candidate into the next class: a) performance in assessed work across the subjects studied in the final year which in

the professional judgement of the examiners is worthy of the higher class, using the QAA MSOR Modal Benchmark (reproduced below) as the guiding principle for such judgement. The Examiners will look more positively on candidates who have produced complete answers to questions

b) extenuating circumstances.

For joint degrees, the meeting of the criteria for any one of the two subject areas will normally be identified by the Sub-board of Examiners as a factor in favour of raising a candidate to the higher class. For Major/Minor degrees, candidates will normally be expected to show performance in the higher class in the Major subject in at least five of the eight half unit marks over the penultimate and final year specified in Regulation 86(b) above. The discipline-specific criteria for the Major subject will then apply, although strong (but not weak) performance in the Minor subject may also be taken into account where this is considered appropriate in the academic judgement of the examiners.

QAA MSOR Modal Benchmark

5.3.1 It is intended that students should meet this standard in an overall sense, not necessarily in respect of each and every of the statements listed. 5.3.2 A graduate who has reached the modal level should be able to: • Demonstrate a reasonable understanding of the main body of knowledge for the

programme of study • Demonstrate a good level of skill in calculation and manipulation of the material

within this body of knowledge • Apply a range of concepts and principles in loosely-defined contexts, showing

effective judgement in the selection and application of tools and techniques

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• Develop and evaluate logical arguments • Demonstrate skill in abstracting the essentials of problems, formulating them

mathematically and obtaining solutions by appropriate methods • Present arguments and conclusions effectively and accurately • Demonstrate appropriate transferable skills and the ability to work with relatively

little guidance or support. ("Reasonable" is interpreted in the benchmark document to mean "according to the professional judgement of the examiners")

7.6 Adverse Circumstances

Students wishing to draw adverse circumstances – such as illness or other personal problems – to the attention of the Sub-Board of Examiners must discuss their problem with their Adviser and write to the Chair of the Sub-Board, Prof. G. Harman. You must support your request for adverse circumstances to be taken into consideration with written evidence. Where confidential information should be considered you must indicate that you give permission for this information to be used. Medical certificates are required when illness interferes with studies. Note that it is advisable to obtain a medical certificate at the time of the illness – this should then be presented to the Department Office and you may refer to this documentation if needed.

7.7 Plagiarism

All work submitted by students as part of the requirements for any examination or other assessment must be expressed in their own words and incorporate their own ideas and judgements. Any other material used should be referenced carefully. Failure to observe this rule will result in a fail or non-examined grade in the relevant course and may lead to an application of the college disciplinary procedure. Further details are given in the College Undergraduate Regulations, at the web site given above. Note that, notwithstanding this rule, students are encouraged to discuss the problems in the (non-assessed) weekly worksheets and even help each other towards obtaining a solution. However, the written solutions to these problems must be prepared and submitted on an individual basis.

7.8 Calculators in Examinations

For examinations in which the use of calculators is permitted, calculators will be supplied by the College. The calculators are of an easy-to-use standard type: shortly before the exams start sample calculators will be available in the Departmental Office and you can borrow one briefly so that you can familiarize yourself with it. You are not permitted to take your own calculator into the examination room.

8. Prizes The following prizes are available to Mathematics students. The names of members of the Department who won prizes in 2002, 2003, 2004 and 2005 are given below. College prizes (All these are awarded when funds allow) Martin-Holloway prize for the best finalist in each faculty, Lillian F Heather and Murgoci prizes for the best first year students in each Faculty. Harrison prize for the best combined honours finalist.

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2002 – Martin-Holloway Prize: Polly Woodcock; Harrison Prize: Sarah Vaughan; 2004 – Martin-Holloway Prize: James Ashton; 2005 – Harrison Prizes: Duncan Pallant and Sarah Malt; Lillian F Heather Prize: Helen Thornewell Mathematics Department prizes The Bailey Prize marks Professor Wilfred Bailey, Head of Mathematics at Bedford College 1944-1958. It is awarded to a student of Mathematics at the start of his or her third year of study. 2003 – James Ashton and Barbara Sandison (shared); 2004 – Matthew England; 2005 – Richard Biegler-König and Timothy Jones (shared); 2006 – Helen Thornewell The Coulter McDowell Prize marks Professor Coulter McDowell, member of staff at Royal Holloway 1957-1964 and 1969-1986, Head of Mathematics 1981-1986. It is awarded to a third or fourth year undergraduate in the Department of Mathematics in recognition of outstanding performance in the examinations in Operational Research. 2003 – William Byatt; 2004: not awarded; 2005 – Sarah Speirs; 2006 – Tom Pierpoint The Simpson Prize is in memory of Professor Harold Simpson, Head of Mathematics at Bedford College 1907-1944 (until 1939 his surname was Hilton). It is awarded to a first-year undergraduate in the Department of Mathematics in recognition of outstanding academic effort or achievement. 2003 – Alex Channing and Faiza Yousaf (shared); 2004 – Timothy Jones and Marc Jourdain de Muizon (shared); 2005 – Elizabeth Chen and Helen Thornewell (shared); 2006 – Li Liang The Thewlis/Wilks Prize is in memory of Madeline Thewlis and Ruth Wilks, former students at Bedford College. It is awarded to a student of the College entering for the final examinations in Mathematics to assist him or her in postgraduate work. 2003 – Irfan Sheikh; 2004 – James Ashton; 2005 – Matthew England; 2006 – Tim Jones The Harding Prize marks Professor Percy Harding, Head of Mathematics at Bedford College 1870-1907. It is awarded to a student in the Department of Mathematics who has completed at least five terms of study, usually for a post-examination essay. 2002: not awarded; 2003 – Caroline Doherty; 2004: not awarded; 2005 – Timothy Jones The Mary Bradburn Prizes were given in memory of Dr Mary Bradburn, student at Royal Holloway College 1935-1940 and staff 1945-1980. They are awarded to undergraduate students of the Mathematics Department in their third or fourth year for outstanding effort or achievement, and were awarded for the first time in 2004. 2004 – Paolo Giambrone, Abdul Kobir and Barbara Sandison; 2005 – Yuan Tian; 2006 – Simon Robinson and Rayhan Miah The Driver prizes, in all subjects taught at Royal Holloway at the time, were endowed by Miss Mary Ann Driver in 1887, just after the opening of the College. Miss Driver was the sister of Jane, wife of Thomas Holloway. Up to two are awarded in Mathematics, for post-examination essays. 2002 – James Ashton and Irfan Sheikh; 2003 – Claudia Yong; 2004 – Jonathan Cooley; 2005 – Abtin Pourgive and Helen Thornewell The two IMA Prizes are awarded by the Institute of Mathematics and its Applications to the two best finalists in Mathematics at the College. The prizes consist of free membership to the IMA. 2003 – Jonathan Andrews and Kirsty Ford; 2004 – James Ashton and Paolo Giambrone; 2005 – Matthew England and Yuan Tian; 2006 – Richard Biegler-König and Tim Jones. Apart from the IMA and Thewlis/Wilks prizes, the Mathematics prizes are to be spent on books. There are also various College prizes, for work unrelated to mathematics, occasionally advertised.

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9. Lecture List 2006 – 2007

9.1 First Year Courses

Term Lecturer MT141 From Bernoulli to Mandelbrot 1 TJO MT171 Calculus 1 CMF MT181 Number Systems 1 SRB MT194 Numbers and Functions 1 GH MT121 Introduction to Applied Mathematics 2 PO’M MT130 Principles of Statistics 2 AFS MT151 Computational Mathematics I 2 SDG MT172 Functions of Several Variables 2 CMD MT182 Matrix Algebra 2 JFM

9.2 Second Year Courses

Term Lecturer

MT222 Vector Analysis and Fluids 1 CMD MT230 Linear Statistical Methods 1 RS MT261 Discrete Mathematics 1 CE MT280 Lin and Abs Algebra and Project 1 RMD/GH MT294 Real Analysis 1 JFM MT232 Probability 2 RMD MT262 Mathematical Programming 2 YB MT272 Ordinary DEs and Fourier Analysis 2 TJO MT283 Primes and Factorisation 2 GH MT290 Complex Variable 2 CMF

9.3 Third Year Courses

Term Lecturer

MT300 Mathematics Project 1, 2 MT301 History and Development of Mathematics 1 DLY MT311 Number Theory 1 EJS MT320 Quantum Theory I 1 PO’M MT322 Dynamics of Real Fluids 1 CMD MT328 Non-Linear Dynamical Systems 1 FMF MT331 Experimental Design 1 EJG MT336 Applied Probability 1 EJG MT347 Mathematics of Financial Markets 1 CMF MT362 Cipher Systems 1 SPM MT364 Applications of OR Techniques 1 AFS/KCB MT373 Control Systems 1 GdeB MT422 Advanced EM and Special Relativity 1 TJO MT309 Mathematics in the Classroom 2 CMF MT332 Inference 2 TS MT334 Statistical Systems 2 EJG

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Term Lecturer

MT345 Quantum Information and Coding 2 PO’M MT361 Error Correcting Codes 2 SRB MT365 Algorithmic Graph Theory 2 JFM MT369 Game Theory 2 RMD MT421 Aerodynamics and Geophysical FD 2 CMD PH242 Electromagnetic Theory 2 Prof. B Cowan

SC3002 Science Communication 2 Dr A Lewis

9.4 Fourth Year Courses

Term Lecturer

MT400 MSci Project (one unit) 1, 2 MT422 Advanced EM and Special Relativity 1 TJO MT441 Channels 1 CE MT454 Combinatorics 1 YB MT485 Applications of Field Theory 1 RMD MT421 Aerodynamics and Geophysical FD 2 CMD MT447 Advanced Financial Mathematics 2 AFS MT466 Public Key Cryptography 2 SDG

10. Programmes of Study This section gives the prescribed or recommended mathematical component of the degree programmes offered by the department. An entry such as '2 units MT' means that you choose mathematics options to that value. In the case of joint programmes full details of courses in the other subject are available from the relevant department, and an entry here such as (for example) '2 units EC' may include core, compulsory or optional courses in Economics. Notes: (i) When choosing your second year programme you must ensure that you include those courses prerequisite for your intended third year programme (and fourth year, for MSci students). The flow diagram on page 21 should be useful for this purpose. (ii) You should not include more than two half-units chosen from a lower year. This does not apply to MT130 taken by Mathematics and Management students in Year 2, MT323, MT381 or MT394 taken by MSci students in Year 4, or to Mathematics Minor students. (iii) Students whose degree programme includes the word mathematics may not take courses with a mathematical content which are primarily intended for students of other disciplines, for example EC1102, GL1700, MN1025, PH1110 and PH1120. Exceptionally, students whose programme is xxx with mathematics may be able to do so: the Academic Coordinator should be consulted for details.

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10.1 Mathematics and Mathematics major degree programmes

Mathematics BSc (three year course) Year 1 Year 2 Year 3

Term 1 MT141, 171, 181, 194 MT280, 294 plus 1 unit MT

2 units MT

Term 2 MT172, 182, and two of MT121, 130, 151

MT290 plus 1½ units MT

2 units MT

In addition we recommend that students take MT261 and MT272 in Year 2. Mathematics MSci (four year course)

Year 1 Year 2 Years 3 and 4 Term 1 MT141, 171, 181, 194 MT280, 294 plus 1 unit

MT See below

Term 2 MT172, 182, and two of MT121, 130, 151


In addition we recommend that students take MT261 and MT272 in Year 2. Mathematics with Statistics

Year 1 Year 2 Year 3 Term 1 MT141, 171, 181, 194 MT230, 280 plus 1 unit

MT MT331 plus 1½ units MT

Term 2 MT130, 172, 182, and one of MT121, 151

MT232, 290 plus 1 unit MT


Mathematics with Economics

Year 1 Year 2 Year 3 Term 1 MT171, 181, 194 plus ½

unit EC MT230, 280 plus ½ unit MT, ½ unit EC

1½ units MT, ½ unit EC

Term 2 MT130, 172, 182 plus ½ unit EC

MT290 plus 1 unit MT, ½ unit EC

1½ units MT, ½ unit EC

Mathematics with Management


unit MN MT230, 280 plus ½ unit MT, ½ unit MN

1½ units MT, ½ unit MN

Term 2 MT130, 172, 182 plus ½ unit MN

MT262, 290 plus ½ unit MT, ½ unit MN

1½ units MT, ½ unit MN

Mathematics Major (other than with Economics or Management)


unit other subject MT280 plus 1 unit MT, ½ unit other sub

1½ units MT, ½ unit other subject

Term 2 MT172, 182, and one of MT121, 130 or 151 plus ½ unit other subject

MT290 plus 1 unit MT, ½ unit other subject

1½ units MT, ½ unit other subject

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10.2 Joint degree programmes including Mathematics

Computer Science and Mathematics Year 1 Year 2 Year 3 Term 1 MT171, 181 plus 1 unit

CS MT261, 280 plus 1 unit CS

1 unit MT, 1 unit CS

Term 2 MT172, 182 plus 1 unit CS

1 unit MT, 1 unit CS 1 unit MT, 1 unit CS

Economics and Mathematics

Year 1 Year 2 Year 3 Term 1 MT171, 181 plus 1 unit

EC MT230 plus ½ unit MT, 1 unit EC*

1 unit MT, 1 unit EC

Term 2 MT130, 172, 182 plus ½ unit EC

MT262 plus 1½ units EC*

1 unit MT, 1 unit EC

* Economics and Mathematics students take 2½ units of Mathematics in Year 1 and 1½ units in Year 2: they may take the extra Economics half-unit in either term. Mathematics and Management


MN MT280 plus ½ unit MT, 1 unit MN

1 unit MT, 1 unit MN

Term 2 MT172, 182 plus 1 unit MN

MT130 plus ½ unit MT, 1 unit MN

1 unit MT, 1 unit MN

Geology and Mathematics Mathematics and Physics


GL or PH MT280 plus ½ unit MT and 1 unit GL or PH

1 unit MT and 1 unit GL or PH

Term 2 MT172, 182 plus 1 unit GL or PH

MT290 plus ½ unit MT and 1 unit GL or PH

1 unit MT and 1 unit GL or PH

We recommend that the other two second year half units in mathematics are MT222 and MT272. Mathematics and Music Mathematics and Psychology


MU or PS MT280 plus ½ unit MT and 1 unit MU or PS*

1 unit MT and 1 unit MU or PS*

Term 2 MT172, 182 plus 1 unit MU or PS



* Mathematics and Psychology students who wish to obtain GBR (Graduate Basis for Registration) status from the British Psychological Society should take a total of 1½ units of Mathematics and 2½ units of Psychology in Years 2 and 3.

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10.3 Mathematics minor degree programmes

Year 1* Year 2* Year 3 or Year 4 Term 1 MT181 plus 1½ units in

the other subject MT171 plus 1½ units in the other subject

½ unit MT plus 1½ units in the other subject

Term 2 MT182 plus 1½ units in the other subject

MT172 plus 1½ units in the other subject

½ unit MT plus 1½ units in the other subject

*The Year 1 and Year 2 programmes may be interchanged to suit your timetable. Exceptionally, Economics with Mathematics students may take MT194 in place of MT171.

10.4 Mathematics MSci (four-year degree)

Students on this programme follow the same first and second year pathway as those taking the three-year Mathematics BSc degree programme, but in the third and fourth years their selection of course units is more tightly controlled. To progress in the MSci programme it is necessary to reach at least an average mark of 50% in each year of study; otherwise students will be transferred to the BSc programme. The structure in years 3 and 4 is as follows. Year 3: You must take at least four half-units chosen from MT311, MT320, MT322, MT332, MT336, MT362, MT364, MT365, MT381 and MT394. Year 4: You must take the full-unit project course MT400. Your other six half-units may include no more than two from the Year 3 list (except that MT323, MT381 and MT394, which are given in alternate years, count as Year 4 half-units for this purpose). When you choose your year 3 courses you should also consider your likely year 4 choices, and remember that some options are given in alternate years only. Your year 3 and 4 choices together should make a coherent set of course-units which is approved by your adviser. Notes: (i) Students with LEA awards must enter by the end of Term 1 of Year 2 at

the latest. (ii) Entry to the MSci must in any case be by the start of Term 1 of Year 3

and will only be permitted for students whose earlier programme fulfils the MSci structural requirements.

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11. Course Unit Descriptions The aim of each course half unit is to introduce and explain the subject material which forms the course syllabus, and to develop the student’s understanding and skills through problem-solving exercises. Notes: (i) All Mathematics courses (except for the MSci Project MT400) are half-unit courses.

(ii) Prerequisites normally show only those courses required in addition to MT171 and MT182, which should have been taken by all students in the first year.

(iii) A pass or near pass in a prerequisite course is normally expected. (iv) Exceptions must be approved by the Academic Coordinator.

(v) The books listed as indicative texts should be viewed as suitable course companions which may be consulted in the Library or (if in print) purchased. In each case the Library catalogue reference is given. More detailed alternative recommendations may be made at the beginning of each course.

(vi) MT280 is subject to validation.

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MT121 Introduction to Applied Mathematics (Term 2: Professor P O’Mahony)

Prerequisite: MT171 Teaching: 33hr lectures, 8hr workshop classes Assessment: 2hr written examination

Aims

The aim of this course is to provide an introduction to some key ideas and methods of classical mechanics, chaos theory and special relativity. The course covers Newton’s equations of motion for a single particle, shows how these equations can give rise to chaotic behaviour, and shows how they need to be modified for velocities close to the speed of light.

Learning objectives

On completion of the course, the student should be able to: • solve Newton’s equations of motion for a variety of problems, including the

damped, driven harmonic oscillator; • use the conservation laws for energy and momentum; • work with co-ordinate systems that accelerate or rotate; • explain how chaos arises for the forced pendulum; • state Einstein’s principle of relativity and explain how it leads to special relativity; • use the Lorentz transformation and draw Minkowski diagrams.

Content

Classical Dynamics: Dimensional analysis, units, forces. Newton's laws, One dimensional motion; Conservation of energy and momentum. Stable and unstable equilibrium points. Simple Harmonic motion, damped and harmonic forced motion. Three dimensional motion. Projectile in the presence of friction. Circular motion. Accelerating and rotating coordinate systems. Coriolis force. Chaos: The damped forced pendulum. Special Relativity: Galilean invariance. Inertial systems. Einstein's principle of relativity. The Lorentz transformation. Length contraction, time dilation, the twin paradox. Energy-mass equivalence E = mc2.

Indicative Text Mechanics − P. Smith & R. C. Smith (Wiley) 2nd edition. Library Ref. 531 SMI

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MT130 Principles of Statistics (Term 2: Dr A F Sheer)

Prerequisite: MT171 Teaching: 33hr lectures, 7hr practical work, 5hr tutorials Assessment: Two-hour written examination 80%, MINITAB project 20%.

Aims

This course introduces the notion of probability and the basic theory and methods of statistics, aiming to give an understanding of random variables and their distributions, data sets and their initial analysis, estimation and inference concerning means and variances. The overall aim of the course is to show students how to analyse a variety of different sorts of data sets in a scientific way.

Learning outcomes

At the end of the course the students should be able to: • calculate probabilities of events that arise from the standard distributions; • examine data critically, calculate summary statistics and display main features

graphically; • calculate estimates of means and variances, deriving the corresponding sampling

distributions; • derive confidence intervals for means and differences of means; • carry out t tests for means and differences of means; • analyse two-factor contingency tables using χ2; • specify null/alternative hypotheses and calculate the corresponding

acceptance/rejection regions. The student should be familiar with the notions of types of error, power and significance level. The student will have had a good experience of MINITAB, and should be proficient in its use for the applied parts of the course.

Content

Descriptive Statistics: Organizing data; histogram dotplot, boxplot and stem-and-leaf; descriptive measures; plots of bivariate data; empirical distribution function. Probability: Elementary notion of probability in terms of distribution of random variables as models for experiments. The Binomial, Poisson, Discrete and Geometric distributions; the

normal distribution, !2

and t distributions; the Exponential distribution. Expectation, variance and covariance. Moment generating function methods. Statistics: Simple random sampling, estimation (point and interval); maximum likelihood estimation; tests of hypotheses, null and alternative hypotheses, error types and power, sample size/power relation, large and small samples. One sample, two sample and

paired comparison t tests, !2

and contingency tests.

Indicative Text

A Basic Course in Statistics (4th edition) – G.M. Clarke and D. Cooke (Arnold 1998). Library Ref. 518.3 CLA

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MT141 From Bernoulli to Mandelbrot: Chance, Information and Chaos (Term 1: Dr T J Osborne)

Prerequisite: Mathematics A-level or equivalent Teaching: 33hr lectures, 5hr tutorials Assessment: 2hr written examination Aims: This course aims to show how mathematics can be used to give a precise description of concepts such as chance, uncertainty, information and chaos. Motivated by this, the course introduces the students to a range of aspects of the subject. Learning outcomes: On completion of the course, students should be able to • solve simple problems involving probability; • define and compute the entropy of a probability distribution; • explain and use the concept of a uniquely decipherable code; • understand simple techniques of data compression; • understand the concepts of self-similarity and fractal dimension; • use simple arguments to distinguish between countable and uncountable sets; • analyse the logistic map and similar iterated maps; • explain the period-doubling route to chaos. Content: Chance: Simple probability, conditional probability, the law of total probability. Information: Entropy, memoryless sources, instantaneous codes, the Kraft inequality, the noiseless coding theorem for memoryless sources, Huffman coding. Fractals: Self-similarity, fractal dimension, Koch snowflake, Cantor dust, Sierpinski gasket. Countability: Countability of rationals, uncountability of reals and of the Cantor set. Iteration: Iterative maps, cobwebbing, fixed points, limit cycles, stability, logistic equation, period doubling. Chaos. The Mandelbrot set. Bilinear transformations. Fibonacci numbers, the golden mean.

Indicative texts

Taking Chances – J. Haigh (Oxford UP, 1999) Library reference: 518.1 HAI Fractals, Chaos, Power Laws – M. Schroeder (Freeman 1991) Library Ref. 530.15 SCH

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MT151 Computational Mathematics I (Term 2: Dr S D Galbraith)

Prerequisite: MT171 and MT181 Teaching: 22hr lectures, 22hr practicals Assessment: 2hr written examination (75%), project (25%).

Aims

This course teaches how to program in C++ and how to solve a range of mathematical problems using a computer. Programming techniques and numerical algorithms are developed in parallel and applied to practical examples.

Learning outcomes

On completion of the course, the student should be able to • edit, compile, run and debug C++ programs; • handle input and output; • use standard constructs like loops, arrays and functions; • analyse numerical errors and the precision of results; • understand a range of simple mathematical algorithms; • implement those algorithms in C++.

Content

C++ Programming: Editing, compiling and running a program; input and output; common errors and debugging; types of variables; branching and loops; arrays and pointers; functions; structures. Theory and applications: Numerical error and precision. Applications from numerical analysis, such as numerical integration, numerical derivatives, the Horner scheme to evaluate a polynomial, evaluation of series, finding zeros of a function, finding maxima of a function. Applications from algebra, such as modular arithmetic, the Euclidean algorithm, random number generation. Applications from combinatorics, such as recurrence relations, using a computer to solve simple enumeration problems, counting integer points in a polygon. Searching and sorting.

Indicative Text

An Introduction to C++ and numerical methods – J M Ortega & A S Grimshaw (Oxford UP 1999) Library Ref. 001.6424 ORT

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MT171 Calculus (Term 1: Dr C M Farmer)

Prerequisite: A-level Mathematics Teaching: 33hr lectures, 11hr workshop classes Assessment: 2hr written examination (85%), one test in term 1 (0%, but you must attend it) and one test at the start of term 2 (10%) and a Mathematica notebook (5%).

Aims:

This course aims to develop the students’ confidence and skill in dealing with mathematical expressions, to extend their understanding of calculus, and to introduce some topics which they may not have met at A-level. It also aims to ease the transition to university work and to encourage the student to develop good study skills. Mathematica is to be used as a calculating and graphical aid.

Learning outcomes:

On completion of the course the student should be able to: • factorize polynomials and separate rational functions into partial fractions; • sketch the graphs of polynomials, rational functions and other elementary functions,

identifying turning points and asymptotes where appropriate; • differentiate commonly occurring functions, and find indefinite and definite

integrals of a wide variety of functions, using substitution or integration by parts; • recognize the standard forms of first-order differential equations, reduce other

equations to these forms, and solve them; • solve certain second and higher order differential equations; • demonstrate that he or she can use Mathematica as an aid in the solution of

problems or to illustrate the ideas met in the course.

Content:

Polynomials and rational functions: asymptotes, sketching, differentiation.

Transcendental functions: ex , ln x , trigonometric and hyperbolic functions (differentiation, zeros, turning points, sketching, symmetry, periodicity). Calculus: chain rule, integration by parts, substitution, use of trigonometric formulae, partial fractions. First-order differential equations: separable equations, linear equations. Second-order differential equations: constant coefficients, complementary function and particular integral. Use of the Mathematica package: including polynomials, integrals and derivatives, plots, and general applications to many of the above topics.

Indicative texts

Calculus (5th edition) – J Stewart (Brooks-Cole 2003). Library Ref. 515.34 STE Elementary Differential Equations and Boundary Value Problems − W E Boyce & R C di Prima (Wiley). Library Ref. 515.41 BOY

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MT172 Functions of Several Variables (Term 2: Dr C M Davies)

Prerequisite: MT171 Teaching: 33hr lectures, 11hr workshop classes Assessment: Two-hour written examination

Aims

This course aims to introduce students to the calculus of functions of more than one variable, and show how it may be used in such areas as geometry and optimization, and to demonstrate how simple functions may be represented as a power series under certain conditions.

Learning outcomes

On completion of the course, students should be able to • manipulate partial derivatives; • use partial derivatives to determine the nature of stationary points and to analyse

certain properties of surfaces; • construct and manipulate line integrals; • evaluate double integrals, including the use of change of order of integration and

change of coordinates;

• expand functions such as e x , trigonometrical and hyperbolic functions, ln( )1+ x ,

arctan x and simple variants as power series; • generate Taylor and Maclaurin series, including the remainder terms.

Content:

Partial differentiation: partial derivatives (using Mathematica to check), exact first order differential equations; chain rule for differentiation; stationary points; use of Mathematica for visualization; geometry: gradient, directional derivative, normals, tangents. Applications of calculus: intuitive notions of continuity and differentiability, intermediate value theorem, Rolle’s theorem and mean value theorem, all stated without proof but illustrated by examples; l’Hôpital’s rule. Series: idea of a power series, Taylor and Maclaurin series; binomial, geometric, exponential, sin and cos, ln( )1+ x , arctan x . Remainder terms (one type only).

Integration in more than one dimension: curves in three dimensions: parametric equations, distance along a curve; line integrals; line integral of a gradient; double integrals; use of Mathematica; change of order; change of variables, Jacobian; plane polar coordinates.

Indicative text

Calculus (5th edition) – J Stewart (Brooks-Cole 2003). Library Ref. 515.34 STE

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MT181 Number Systems (Term 1: Professor S R Blackburn)

Prerequisite: A-level Mathematics Teaching: 33hr lectures, 11hr workshop classes Assessment: 2hr written examination (90%), one in-term test in term 1 (0%, but you must attend it) and one test at the start of term 2 (10%)

Aims:

This course aims to introduce fundamental algebraic structures used in subsequent courses and the notion of formal proofs, and to illustrate these concepts with examples.

Learning outcomes:

On completion of the course, students should be able to: • apply Euclid’s algorithm to find the greatest common divisor of two integers; • use mathematical induction in a careful and logical way to prove simple results;

• perform arithmetic operations on complex numbers, using x iy+ and rei! forms,

locate points on the Argand diagram, and extract roots of complex numbers; • prove De Morgan’s laws and the distributive laws of set theory, and use the

principle of inclusion/exclusion in simple cases; • determine whether a given mapping is bijective and if so find its inverse; • establish whether a given relation on a set is an equivalence relation and find the

corresponding equivalence classes; • compile truth tables to determine whether two statements are logically equivalent; • define a ring, integral domain and field, establish some of their simple properties.

Content:

The integers: division with quotient and remainder, binary numbers, the Euclidean algorithm, greatest common divisors, gcd( , ) ,m n sm tn= + primes, statement of the

fundamental theorem of arithmetic, the principle of mathematical induction. Complex numbers: Cartesian addition and multiplication, the complex conjugate, rules of manipulation (the field axioms), inversion, the Argand diagram, modulus and argument, extraction of nth roots (quadratic equations and roots of unity (cyclic

groups)), e ii! ! != +cos sin , ez and log z .

Sets: intersection, union, complement, Venn diagrams, De Morgan’s laws. Mappings: composition, associative law, injections, surjections, bijections and inverses. Equivalence relations and partitions. Propositional logic, truth tables. Rings and fields: the ring Z

n of integers modulo n, the field Zp . The ring F x[ ] of

polynomials over a field, analogy with Z (division law, monic polynomials, gcds), zeros, remainder and factor theorems, a polynomial of degree n over F has at most n zeros. The ring of 2 2! matrices over a field.

Indicative Texts

A Concise Introduction to Pure Mathematics – M Liebeck (Chapman and Hall/CRC Mathematics 2000) Library Ref. 510 LIE Discrete Mathematics (2nd edition)− N L Biggs (Oxford UP 2002). Library Ref. 510 BIG

36

MT182 Matrix Algebra (Term 2: Dr J F McKee)

Prerequisite: MT181 Teaching: 33hr lectures, 11hr workshop classes Assessment: 2hr written examination

Aims:

This course aims to give students a working knowledge of basic linear algebra, with an emphasis on an approach via matrices and vectors. The course introduces some of the basic theoretical and computational techniques of matrix theory, and illustrates them by examples.

Learning outcomes:

On completion of the course, students should be able to: • appreciate the power of vector methods, use vector methods to describe three-

dimensional space, and apply scalar and vector products of two vectors and triple products appropriately;

• understand the notions of field, vector space and subspace; • calculate the determinant of an n n! matrix; calculate the inverse of a non-

singular matrix; • appreciate the significance of the characteristic polynomial of a matrix, compute

the eigenvalues and eigenspaces of a matrix, and diagonalize it when possible; • understand the notions of linear independence and dimension; • reduce a matrix to row-reduced echelon form.

Content:

Vectors in R3 : vectors as directed line segments; addition, scalar multiplication, parallelogram law. Dot product, length, distance, perpendicular vectors, angle

between vectors, u v u v. cos= ! . Lines and planes in R3 . Cross product, area u v!

and volume wvu ).( ! .

2 × 2 matrices over a field: determinants, inverses, rotation and reflection matrices. Eigenvalues and eigenvectors, the characteristic polynomial and trace.

Diagonalization P AP!1 with applications.

3 × 3 matrices: permutations, compositions, parity. Definition of the determinant of an n n! matrix, cofactors, row and column expansion, the adjugate matrix, inverse of a non-singular matrix. Examples of characteristic polynomial and diagonalization of 3 × 3 matrices A. Statement of the properties of determinants, including AB A B= . Vector spaces: axioms, linear independence, span, dimension. Subspaces. Row-reduction: elementary row operations, echelon and row-reduced echelon form, rank of a matrix. Applications: solution of systems of linear equations, deriving a basis from a spanning set, computing the inverse of a matrix.

Indicative Texts Linear Algebra (Schaum Series ) − S. Lipschutz (McGraw-Hill) Library Ref. 510.76 LIP Undergraduate Algebra − C. W. Norman (Oxford 1986) Library Ref. 512.11 NOR Linear Algebra: a Modern Introduction – D Poole (Brooks-Cole 2003). Library Ref. 512.3 POO

37

MT194 Numbers and Functions (Term 1: Professor G Harman)

Prerequisite: Mathematics A-level or its equivalent Teaching: 33hr lectures, 5hr tutorials Assessment: Two-hour written examination

Aims

• To kindle an interest in analysis, and to provide a taste of what the subject is about; • To give a user-friendly introduction to key ideas of analysis, illustrated with copious

examples; • To provide a structure that enables students to gain confidence in handling

concepts in analysis.

Learning outcomes

On completion of the course, the student should be able to: • appreciate the significance of the completeness property distinguishing R from Q; • find out whether a given (simply defined) function is continuous at a point, and

apply properties of continuous functions in examples; • determine whether a given (simple) sequence tends to a limit, using the

monotonicity when appropriate; • use standard tests to investigate the convergence of infinite series with positive

(uncomplicated) terms, including power series.

Content

The real number system: The natural numbers N, the binomial theorem using induction. The integers Z and the rational field Q. Order properties, manipulation of inequalities,

the triangular inequality. Irrationality of 2 . Decimal representation of real numbers. Null sequences. Subsets, maximum, upper bound, least upper bound. Every non-empty subset of the reals which is bounded above has a least upper bound. Continuity: Discussion of continuity, continuity of f at a point. A continuous function is

specified by its values at rational points. Discussion of f x x( ) = 2 . Continuity on an

interval. A continuous function on a closed interval is bounded and achieves its bounds. Discussion of the intermediate value theorem . Sequences and series: Sequences, limits of sequences, limits of sequences defined recursively. Monotonic sequence theorem. Connection with continuity. Infinite series. Tests for convergence: comparison test, ratio test. Examples of real power series, including the geometric series.

Indicative Text

Yet Another Introduction to Analysis – Victor Bryant (Cambridge 1990). Library Ref. 515 BRY

38

MT222 Vector Analysis and Fluids (Term 1: Dr C M Davies)

Prerequisites: MT171, MT172 Teaching: 33hr lectures Assessment: 2hr written examination

Aims

• to study the integration and differentiation of vectors and scalars defined at points in space, introducing the concepts of scalar and vector fields;

• to familiarize the student with the use of general orthogonal curvilinear coordinates and the evaluation of differential operators;

• to introduce integral theorems and demonstrate their usefulness; • to show how simple partial differential equations may be solved by the technique of

separation of variables; • to show how the acquired concepts can be applied in the field of dynamics of

inviscid fluids.

Learning outcomes

On completion of the course students should be able to: • identify scalar and vector fields; • calculate the gradient of a scalar field and the divergence and curl of a vector

field; • use general orthogonal curvilinear co-ordinates and, in particular, cylindrical and

spherical polar co-ordinates; • use the divergence theorem and Stokes’ theorem; • recognize when and how variables separate in a partial differential equation; • apply the equations of continuity and motion for an inviscid fluid and use Bernoulli’s

equation; • use velocity potential and apply it to examples of irrotational flow.

Content

Vector analysis: scalar and vector fields. Field lines for a vector field. Gradient of a scalar field, divergence and curl of a vector field. The del-operator. Cylindrical and spherical polar coordinates. General orthogonal curvilinear coordinates. Surface and volume integrals. The divergence theorem and Stokes’ theorem. Green’s theorem. Partial differential equations: Laplace’s equation, the diffusion equation and the wave equation in Cartesian coordinates. Separation of variables method, used in Cartesian, plane polar and spherical polar coordinates. Dynamics of inviscid fluids: equation of continuity. Velocity and acceleration. Equation of motion. Bernoulli’s equation. Irrotational flow and velocity potential. Examples of potential flow of incompressible fluids.

Indicative Text

Calculus (5th edition) – J Stewart (Brooks-Cole 2003). Library Ref. 515.34 STE Vector Analysis and Cartesian Tensors − (Third Edition) D.E. Bourne and P.C. Kendall (Chapman and Hall 1992). Library Ref. 515.34 BOU

39

MT230 Linear Statistical Methods (Term 1: Professor R Schack)

Prerequisite: MT130 Teaching: 25hr lectures, 8hr workshop classes Assessment: 2 hour written examination 80%, project 20%

Aims

To study important aspects of statistical modelling in an integrated way and develop some expertise both in the theory and applications of linear models.

Learning outcomes

On completion of the course, students should be able to • demonstrate familiarity with the main methods based on linear models; • apply these methods to analyse data and interpret the results from such analysis; • use MINITAB effectively in the analysis of relevant data.

Content

Principles of statistical modelling and terminology: Systematic and random components, types of variables. Simple and multiple linear regression: Matrix notation, fitting the model, inferences about individual regression parameters, prediction, assessing the regression. Some special cases: Polynomial models, models that incorporate factors. Model building: Testing significance of specified subsets of variables, examining all subsets, sequential methods. Model validation and comparison of regressions: Examination of residuals, influential observations, some possible problems and remedial actions, dummy variables. Qualitative explanatory variables - analysis of variance: One-way and two-way ANOVA, point estimation, linear contrasts, a general approach via multiple regression.

Indicative Texts Introduction to Statistical Modelling − W J Krzanowski (Arnold) Library Ref. 518.3 KRZ Applied Regression Analysis and Other Multivariable Methods − D G Kleinbaum, L L Kupper and K E Muller (Duxbury Press 1998) Library Ref. 518.3 KLE Data Analysis Using Regression Models – E W Frees (Prentice Hall 1995) Library Ref. 300.183 FRE A Second Course in Statistics: Regression Analysis – W Mendenhall and T Sincich (Prentice Hall 1996) Library Ref. 518.3 MEN

40

MT232 Probability (Term 2: Dr R M Damerell)

Prerequisite: MT181 and MT172 Teaching: 33hr lectures Assessment: 2hr written examination Aims: To introduce the formalism of the mathematical theory of probability and thereby to lay a firm foundation for applications of probability in virtually all areas of science, including statistics, economics, the mathematics of financial markets, operational research, information theory, number theory, quantum theory and statistical physics.

Learning outcomes

On completion of the course, the student should be able to • demonstrate an understanding of the basic principles of the mathematical theory

of probability; • use the fundamental laws of probability to solve a range of problems; • prove simple theorems involving both discrete and continuous random variables; • explain the weak law of large numbers and the central limit theorem.

Content

Elements of probability: Sets and events. Axioms of probability. Independent events. Conditional probability. Bayes’ theorem. Discrete random variables: Probability distribution function and cumulative distribution function. Joint distribution, marginal distribution, independence. Distribution of a function of a random variable. Expectation, variance, covariance. Binomial and Poisson distributions, with application to simple combinatorial problems. Chebychev’s inequality and the weak law of large numbers. Further topics, such as probability generating functions, the hypergeometric and negative binomial distributions. Continuous random variables: Definition as R!SX : such that

!=<<

b

a

dxxfbXa )()Pr( . Expectation and variance. Normal and exponential

distributions. Joint and marginal densities, independence. Transformations of random variables. Normal approximation to the binomial distribution. Statement of the central limit theorem. Further topics such as moment generating functions and the gamma distribution. Further topics: Possible further topics include applications in the fields of geometric probability and simple random walks, such as Buffon’s needle problem, simple problems related to the geometry of points chosen at random in the interior of squares and circles, the gambler's ruin problem, the reflection principle, the ballot theorem, first passage times.

Indicative text:

S M Ross – A First Course in Probability (Prentice Hall 1998). Library Ref. 518.1 ROS

41

MT261 Discrete Mathematics (Term 1: Dr C Elsholtz)

Prerequisite: MT181 Teaching: 33hr lectures Assessment: 2hr written examination

Aims • To introduce the idea of graphs and some of their important properties, such as

connectedness, Eulerian and Hamiltonian paths, planarity. • To show how the pigeonhole principle and its extensions, sieve methods, and

other simple ideas can be used to prove quite deep results on counting. • To define Latin squares, explaining their use for constructing magic squares, and

to introduce block designs.

Objectives

On completion of the course, students should be able to: • understand the fundamental concepts of graph theory; • to be able to apply them to the determination of the Eulerian and Hamiltonian

characters of a (reasonably straight forward) graph; • prove and apply various forms of the pigeonhole principle and sieve methods; • prove the basic theorem concerning the multinomial coefficients and apply it to

various combinatorial problems; • apply modular arithmetic and (simple) finite fields to construct orthogonal

systems of Latin squares, magic squares and simple block designs.

Content

Graphs: Definitions of graph, adjacency matrix, incidence matrix, graph isomorphism, valency lists. The handshaking lemma; complete graphs, bipartite graphs, simple relationships between n and m. Connected graphs, trees; Eulerian trails and Hamiltonian cycles, planar graphs, Euler’s theorem. Methods of counting: Functions on finite sets; the pigeonhole principle, permutations and combinations, binomial and multinomial coefficients, partitions, Stirling numbers. Finite mathematical structures: Modular arithmetic, definition and construction of Latin squares; an introduction to the theory of block designs, and relations between their parameters, difference sets and finite geometries.

Indicative Text A first course in discrete Mathematics − I Anderson (Springer 2001). Library Ref. 512.23 AND

42

MT262 Mathematical Programming (Term 2: Dr Y Barnea)

Prerequisite: MT182. Students may not take both MT262 and CS349. Teaching: 33hr lectures Assessment: Two hour written examination

Aims

• To show how a wide range of problems may be formulated as linear programming (LP) problems;

• To show how two-variable problems may be solved using graphical methods; • To show how the general LP problem can be solved using the simplex algorithm; • To show how duality in LP fits into a more general framework using Lagrange

multipliers; • To show how the transportation and allocation problems may be solved using

special algorithms; • To show how the integer constraint may be taken into account.

Learning outcomes

On completion of the course the students should be able to: • formulate problems as LP problems; • solve two-variable problems by graphical means; • use the simplex algorithm to solve small LP and transportation problems; • use the Hungarian algorithm to solve assignment problems; • solve integer programming problems using the branch and bound method.

Content

Linear programmes: Examples of formulation of problems as LPs. Examples with solutions by diagram. Unbounded and infeasible LPs. General LPs: feasible and basic feasible solutions. Simplex method: How you solve with the simplex method, with the emphasis on elementary row operations. The formula for the δj. Getting started: the two-phase method, infeasibility. Proof of termination, cycling. Brief mention of complexity. Illustration using a package. Duality: Its meaning; dual of the dual is the primal; the dual solution; the duality theorem; the simplex tableau displays the dual solution. Complementary slackness and testing for optimality. Dual simplex method. Duality in a wider context: graphical description of Lagrange multipliers in a constrained optimization; the dual variables as Lagrange multipliers. Sensitivity analysis: Examples of different types. Transportation problem: Loops and basic feasible solutions; setting out the algorithm; formulation of problems as transportation problems. Assignment problems: Why the transportation method is inefficient. Hungarian algorithm, proof of convergence. Integer programming: Formulation of problems using binary variables; capital budget problems, fixed charge problems. Solution methods: branch and bound, Gomory’s method.

Indicative Text

Elementary Linear Programming with Applications – B Kolman and R E Beck (Academic Press). Library Ref. 519.41 KOL

43

MT272 Ordinary Differential Equations and Fourier Analysis (Term 2: Dr T J Osborne)

Prerequisites: MT171, MT172 and MT182 Teaching: 33hr lectures Assessment: 2hr written examination

Aims

This course aims to introduce the concepts of eigenvalues and eigenfunctions in the familiar situation of the trigonometric differential equation and to show how these yield Fourier series expansions for a general function. These Fourier series can be generalized to (a) generate more general eigenfunction expansions for a given function and (b) develop the Fourier transform, which is used in a variety of applications; its properties are investigated. The final step is to introduce a technique for solving differential equations where the coefficients are no longer constant.

Learning outcomes

On completion of the course, students should be able to: • locate eigenvalues both analytically and graphically; • determine the Fourier series for a periodic function, including odd and even functions,

and recognize the function represented by a given Fourier series; • understand the role of eigenfunctions in building up a general function; • orthogonalize a set of polynomials over a specified interval; • manipulate the Dirac delta-function; • manipulate and apply the Fourier transform; • complete a solution-in series in straightforward cases.

Content

Introduction to Sturm-Liouville theory: eigenvalues and eigenfunctions; self-adjoint operators, orthogonal functions and their properties, orthogonalization, completeness of eigenfunctions. Laguerre polynomials, Legendre polynomials. Fourier series: Fourier-Euler formulae and statement of Fourier Theorem on ],[ !!" ,

Fourier sine and cosine formulae, extension to general analysis. The Fourier transform: Fourier transform of derivatives, statement of Inversion Theorem, Dirac delta-function, Convolution Theorem, Parseval Theorem. Ordinary differential equations: The Cauchy-Euler equation. Solution in series for a second-order linear differential equation, for two out of the four cases that can arise.

Indicative Text Elementary Differential Equations and Boundary Value Problems − W E Boyce and R C di Prima (Wiley). Library Ref. 515.41 BOY

44

MT280 Linear and Abstract Algebra and a Group Project (Term 1: Dr R M Damerell and Professor G Harman)

Prerequisite: MT182 Teaching: 28hr lectures, 10hr workshops Assessment: 2hr written examination 80%, written report 10%, oral presentation 10%

Aims

• To develop the matrix theory covered in MT182 • To introduce the abstract concepts of groups, rings and fields. • To learn how to put together different aspects of mathematics via a project. • To improve spoken and written communication.

Learning outcomes

On completion of the course, the student should be able to: • understand rank and nullity and the connection between them; • diagonalize real symmetric matrices; • prove Lagrange’s theorem on the subgroups of a finite group; • perform calculations involving polynomials and elements of finite fields; • work with others to prepare a mathematical report and a presentation.

Content

Linear mappings, rank plus nullity. Groups: axioms, group tables, subgroups, order of a group element, cosets, Lagrange’s theorem, groups of order 4, 6 and 8. Group of units of a ring. Fields: construction of finite fields, especially quadratic extensions of Zp. Characteristic. Euclidean spaces, orthogonal matrices, the spectral theorem. Preparation of a small project on a mathematical topic chosen from an approved list, together with three or four other students. Writing of an agreed report on the project, and a joint oral presentation.

Key points about the projects

The skills and confidence gained in these will be valuable when applying for jobs: it will also show that you are able to work as a team member. Each group will consist of five (occasionally four) people, and they will be allocated in the first teaching week by the course team. The group choose a topic from an approved list about fifty covering a wide range of mathematics. They then prepare a written report (about 2500 words) and give an oral presentation (about 15 minutes) involving all members of the group. We envisage that the Bailey Prize will be awarded for the best performance on these.

Indicative Texts Linear Algebra (3rd edition) − J B Fraleigh and R A Beauregard (Addison Wesley). Library Ref. 512.3 FRA Rings, Fields and Groups (2nd edition) − R Allenby (Edward Arnold). Library Ref. 512.61 ALL

45

MT283 Primes and Factorisation (Term 2: Professor G Harman)

Prerequisite: MT181 and MT282 Teaching: 33hr lectures Assessment: 2hr written examination

Aims:

Prime numbers and divisibility have been of interest to mathematicians for thousands of years. More recently, non-mathematicians have become interested too: many modern ciphers (examples are found in such places as web browsers and bank cash machines) are based on the surprising fact that it is easy to tell whether a number is composite, but it is difficult to factor a composite number into its prime factors. This course aims to explore some of the mathematics that concerns itself with divisibility (in the integers and in more general settings), and to present some primality testing and factorisation algorithms that arise from this theory.

Learning outcomes

On completion of the course, students should be able to • carry out calculations in factor rings; • show that certain rational polynomials are irreducible; • determine those rings

nZ with a cyclic group of units;

• state the first isomorphism theorem for groups and for rings, and use the theorems in simple applications;

• apply primality testing algorithms to show certain integers are composite.

Content

Group theory: Direct products of groups. Structure of the group of units of nZ .

Group homomorphisms, normal subgroups and the first isomorphism theorem. Ring theory: Ring homomorphisms, subrings, ideals and the first isomorphism theorem for rings. Factor rings, maximal and prime ideals. Principal ideals and principal ideal domains. Kronecker's theorem that every field can be extended to include a root of a

given irreducible polynomial. Characteristic of a field. A finite field has np elements

and has a cyclic multiplicative group Primality and factorisation: Unique factorisation domains. Euclidean rings. Gaussian integers and the 2-square theorem. Trial division. Irreducibility tests for polynomials. Pseudoprimes and Carmichael numbers. The Miller-Rabin primality test. Pollard rho factorisation algorithm.

Indicative texts:

Rings, Modules and Linear Algebra – B Hartley and T Hawkes (Chapman and Hall, 1978) Library reference 512.61 HAR An Introductory Course in Commutative Algebra – A N Chatters and C R Hajarnavis (Oxford 1998) Library reference 512.61 CHA Factorization and Primality Testing – D M Bressoud (Springer 1989) Library reference 512.91 BRE A Course in Number Theory and Cryptography – N Koblitz (Springer 1994) Library reference 512.91 KOB

46

MT290 Complex Variable (Term 2: Dr C M Farmer)

Prerequisites: MT171, MT172, MT181 Teaching: 33hr lectures, 11hr tutorials Assessment: 2hr written examination

Aims

This course is designed to provide an outline of the basic complex variable theory with some proofs. Applications are exhibited as used in other areas of mathematics. The object is to equip students to be able to use complex analysis to solve specific problems.

Learning outcomes

On completion of the course, the students should be able to: • use the definitions of continuity and differentiability of a complex valued function at a

point, establish the necessity of the Cauchy-Riemann equations and apply this result; • use a power series to define the complex exponential function and hence define the

trigonometric and hyperbolic functions and the complex logarithm, and establish their properties;

• use the parametric definition of a contour integral in specific simple examples; • state and use Cauchy’s Theorem, and apply Cauchy’s Integral Formulae to evaluate

integrals; • obtain Taylor series of rational and other functions of standard type; • determine zeros and poles of given functions, and the residue at a simple pole and at

higher order poles; • state Cauchy’s Residue Theorem and apply it to evaluate real integrals (using

Jordan's lemma when relevant) and to sum certain series, and state and use Rouché’s Theorem.

Content

Special functions: Power series and radius of convergence. Discussion of the exponential, trigonometric and hyperbolic functions for both real and complex

variable. Definition of zlog and az .

Topology: An open (pathwise) connected set of the plane. Functions of a complex variable: Continuity and differentiability of functions defined on an open set. The Cauchy-Riemann equations and Laplace's equation. Contour integrals along piecewise smooth curves C, defined by

! != .)('))(()( dttztzfdzzf

Cauchy's theorem and Cauchy's integral formulae. Taylor series with examples, removable singularities, zeros and poles. Residue theorem and applications: calculation of simple integrals, including use of Jordan's lemma, and summation of infinite series. Principle of the argument. Rouché's theorem and the location of zeros of polynomials.

Indicative Texts

Complex Analysis – J M Howie (Springer 2003). Library Ref. 515.24 HOW Theory and Problems of Complex Variables − M R Spiegel (Schaum 1997). Library Ref. 510.76 SPI Advanced Engineering Mathematics, 8th ed. – E Kreyszig (Wiley 1999). Library Ref. 510.245 KRE

47

MT294 Real Analysis (Term 1: Dr J F McKee)

Prerequisite: MT194 Teaching: 33hr lectures, 5hr tutorials Assessment: 2hr written examination

Aims:

• To explain the rigorous definition of limit of a function of a positive integer variable; • To discuss convergence of series, including power series; • To discuss the concepts of continuity and differentiability of functions of a real

variable x; • To show how the Riemann integral is constructed.

Learning outcomes

On completion of the course, students should be able to: • quote the Weierstrass definition of a limit and verify it in simple cases; • use standard tests to investigate the convergence of commonly occurring series; • specify the power series of standard functions; • understand the Intermediate Value Theorem and the Mean Value Theorems; • understand the constructive approach of the Riemann integral.

Content

Sequences and series: Sequences which tend to a limit (such as n n1

, ( )11

+n

n ).

Absolute convergence of series; use of comparison and ratio tests for absolute convergence; absolute convergence implies convergence. Conditional convergence,

alternating series test, rearrangement of ( )! !

"1 1n

n.

Differentiation: Formal definition of ‘ !")(xf as x a! ’ with connection to

continuity. The intermediate value theorem. Differentiability at a point – definition and geometric interpretation, with examples. Differentiability implies continuity. Derivative of a sum, product, quotient, and the chain rule (with application to inverse functions). Differentiability on an open interval; Rolle’s theorem, Mean Value Theorem, Cauchy’s Mean Value Theorem, with applications including l’Hôpital’s rule. Taylor’s theorem with (one) remainder. Power series: Existence of radius of convergence, and use of ratio test to find it. Power series can be differentiated term-by-term within the circle of convergence. Formal definition and properties of exp, sin, cos, etc., and (using the inverse function) of log, sin-

1 etc. Periodicity of sin and cos. Riemann integral: Upper and lower sums, leading to definition and properties of Riemann integral. Fundamental theorem of calculus. Integral test for convergence of series.

Indicative Text

Yet Another Introduction to Analysis – V Bryant (Cambridge 1990). Library Ref. 515 BRY

48

MT300 Mathematics Project (Either Term: see Dr D L Yates for queries)

Prerequisites: At least three Mathematics half units taken in the second year. Assessment: Small project at the end of the previous year with a presentation (20%

together), dissertation on a larger project (70%) and presentation on this (10%). There may also be an oral examination at the request of the examiners.

Aims

• To enable students to make a detailed investigation into one topic in mathematics; • To develop their skills in finding information from a variety of sources; • To develop their skills in writing and talking about mathematics.

Learning outcomes

On completion of this course the student should • have acquired detailed knowledge of their chosen topic from several sources; • show understanding of how the facets of this topic fit together; • have written a well-constructed dissertation on this topic; • be able to give a well-prepared presentation on this topic.

Arrangements

In the third term of the previous year (normally the second year) the student is expected to write an essay of about 2,000 words on a mathematical subject chosen by themselves after consultation with the course co-ordinator. After deciding on the subject the student should find a member of staff to supervise this, and to suggest references. The subject should be agreed by the end of May and the essay should be handed to the course co-ordinator by the end of July. The student will give a ten-minute presentation on this at the start of the next term. Some suggested subjects for the larger project, which must be of a mathematical nature, will be available at the start of the session, together with a guide to the other areas in which members of staff are willing to supervise projects. A project title, outline, and supervision plan should be agreed with the supervisor and given to the course co-ordinator by October 31st. The results of the project must be submitted, as a dissertation of 7000 to 8000 words, to the project supervisor by the first Thursday of the third term. Later in May the student must give a ten-minute presentation on the results of the project. The dissertation will be examined by the project supervisor and a second examiner, who will submit a report on the dissertation and presentation as well as a grade to the Mathematics Sub-Board of Examiners. The examiners also have the right to request a separate oral examination of the candidate if they wish. The Department will provide advice and facilities (if required) to enable the student to produce a neat final version of the dissertation and visual aids for the presentation.

49

MT301 History and Development of Mathematics (Term 1: Dr D L Yates)

Prerequisite: Available to any third year Mathematics student Teaching: 33hr lectures Assessment: 2hr written examination (70%), 3 pieces of coursework (30%)

Aims

To give an overview of the history and development of mathematics from ancient times to the present day. As well as covering all the major mathematicians and their work, the main branches of the subject are followed from their beginnings in antiquity to recent advances, showing (where relevant) their relationships with Astronomy, Physics and other sciences.

Learning outcomes

On completion of the course the student should: • have acquired an overall view of the history of mathematics; • be able to place individual mathematicians into their historical context; • have understood the development of the different strands of mathematics; • have written three short essays on given topics, combining their facts and comments

from several sources.

Content

A summary of the development of mathematics, starting from Ancient Egypt and Mesopotamia, via Greek, Indian, Chinese and Arabic mathematics, medieval and renaissance Europe, to the present day.

Indicative Text History of Mathematics − C B Boyer and U C Merzbach (Wiley, 2nd Edition). Library Ref. 510.9 BOY

50

MT309 Mathematics in the Classroom (Term 2: see Dr C M Farmer for queries) (The Undergraduate Ambassadors Scheme)

UAS is a national scheme, endorsed by the DTI and DfES joint project 'Science and Engineering Ambassadors Scheme' and was started by the author Simon Singh (who has links with the Department). It provides an opportunity for third year undergraduates to gain valuable transferable skills by giving them first-hand experience of science education. Each student spends half a day each week for the second term in a local school under the supervision of a teacher, and writes a report on the experience. This differs from other schemes in that it counts towards your degree. Prerequisite: At least five Mathematics half-units taken in the second year; it may also be taken by students registered for Mathematics and Physics or Mathematics and Psychology. Because a Criminal Records Bureau check is required only UK citizens may take this course. Time commitment: A one-day training session and a half-day each week in a local school, plus preparation time; weekly meetings as a group. Assessment: An end-of-course report (70%), and a presentation (30%). You must also write a journal of how your time was used, but this is not part of the assessment process.

Aims

• To develop a range of skills; • To act as a role model; • To gain confidence in communicating mathematics; • To learn how to develop projects and teaching methods suitable for pupils.

Learning outcomes

On completing the scheme, students should be able to • understand the needs of pupils and answer questions on mathematics; • assess and devise appropriate ways to communicate principles and concepts; • prepare lesson plans and teaching materials; • report on what they have learnt; • critically evaluate their experience.

Selection, training and support

The number of students allowed from this department will be about six. There will be an initial one day of training in the first term, including an introduction to working with children and conduct in the school environment. You will be asked to give a ten-minute mock lesson on a topic of your choice, and will be interviewed. Each successful candidate will be matched with a specific teacher in a school, who will act as trainer and mentor, and determine the tasks and responsibilities of the student. The cost of travel to the school will be reinbursed.

Registration

If you are interested in entering this scheme for credit, you should discuss it with your adviser. Since it is likely that not all will be chosen, you should ensure that you include an alternative second term half-unit on your form. You may not take more than one course out of MT300, MT309 and PH309.

51

MT311 Number Theory (Term 1: Dr E J Scourfield)

Prerequisite: MT181 Teaching: 33hr lectures Assessment: 2 hr written examination

Aims

To acquaint students with some of the elementary tools used to analyse the additive and multiplicative structures of the set of integers.

Learning outcomes

On completion of this course students should: • Be confident in handling congruences, including the use of the Chinese Remainder

theorem, and the Fermat-Euler theorem; • Be able to manipulate arithmetic functions such as )(),(),( nnn µ!" and )(n! and

derive some of their basic properties; • Be able to prove the existence of primitive roots modulo a prime and use them in

solving certain congruences; • Be able to test for quadratic residues, and use them to answer questions on primes

in arithmetic progressions, and representing numbers as sums of two squares; • Be able to find the continued fraction expansion of real numbers, in particular

quadratic irrationals, and apply continued fractions to the solution of Pell’s equation.

Syllabus

Introduction. Revision of material seen in previous years. Integers, primes, factorization, congruences including Chinese remainder theorem and the Fermat-Euler theorem. Arithmetic functions. Introduction to the functions )(),(),( nnn !µ" and );(n! product

and summation form, Mobius inversion. Primitive roots. The order of an integer modulo p, primitive roots modulo p, the proof of their existence, the index of an integer modulo p. Quadratic residues. Legendre’s symbol. Euler’s criterion for a quadratic residue. Gauss’s Lemma. The law of quadratic reciprocity. Applications to quadratic congruences and primes in arithmetic progressions. Continued fractions. Definition. Diophantine approximation. Quadratic irrationals and Pell’s equation. Arithmetic functions II. The function r(n) – representing numbers as sums of two squares.

Indicative Texts

Introduction to the Theory of Numbers – I Niven, H S Zuckerman and H L Montgomery. 5th edition (Wiley 1991). Library Ref. 512.91 NIV Elementary Number Theory in Nine Chapters – J J Tattersall. (Cambridge UP 1999) Library Ref. 512.91 TAT

52

MT320 Quantum Theory I (Term 1: Professor P F O’Mahony)

Prerequisite: MT272. Students may not take both MT320 and PH221. Teaching: 33hr lectures Assessment: 2hr written examination

Aims

To provide a complete introduction to the major methods and concepts of quantum theory at a level suitable for third year students. The course will stress applications and will cover many of the classic problems of quantum theory. The probabilistic theory of measurement is explained and its philosophical implications are touched upon.

Learning outcomes

On completion of the course students should be able to: • show whether a given operator is linear and hermitian • to understand the probabilistic interpretation of quantum theory • write down the Schrödinger equation for an arbitrary dynamical system • obtain the expectation value of a hermitian operator for a given wavefunction • to solve the Schrödinger equation and obtain the eigenenergies and energy

eigenfunctions for a constant potential, the harmonic oscillator and the hydrogen atom

• to write down the uncertainty relationship between two conjugate hermitian operators.

Content

Historical origins of quantum theory and formal background: Linear Hermitian operators; Dirac delta functions. Closure, orthogonality; postulates of quantum mechanics. Applications: Schrödinger equation: free particle, particle in an infinite well, particle in a box. potential barriers, quantum tunnelling. Particle in a finite well, quantum parity. Simple harmonic oscillator. Angular momentum. The hydrogen atom. The momentum representation. More basic principles: Heisenberg uncertainty principle. Connections with classical physics, Ehrenfest's theorem. Measurement theory.

Indicative Texts

Quantum Physics – S Gasiorowicz (John Wiley). Library Ref. 530.12 GAS An Introduction to Quantum Mechanics – B H Bransden and C J Joachain (Longmans). Library Ref. 530.12 BRA

Note

MT420, which follows on from MT320, is intended for both third and fourth year students, and is given in alternate years. It will be given in 2007-8.

53

MT322 Dynamics of Real Fluids (Term 1: Dr C M Davies)


Aims

This course aims to give an overview of how the theory of ideal fluids met in MT222 can be extended to a more realistic model. It will show how the equations can be solved in simple cases and how other methods such as conservation laws and dimensional analysis can be used in more complicated cases.

Learning outcomes

On completion of the course the student should be able to: • demonstrate an understanding of the essential features of compressible flow, sound

waves and shock waves; • tackle a variety of problems involving surface waves on a liquid; • solve simple problems in viscous flows; • apply appropriately and with confidence basic vector analysis techniques and the

additional general mathematical techniques introduced in this course.

Content

Compressible flow: Relation between pressure and density. Linear small amplitude waves in a gas and possible solutions of the wave equation. Plane and spherical waves. Waves in pipes: harmonics and normal modes. Shock waves. Surface waves on a liquid: Small amplitude waves on a fluid of arbitrary depth. Stationary waves on a moving stream. Waves on an interface. What happens if wave speed depends upon wavelength: dispersion and group velocity. Viscous fluids: Discussion of the effects of viscosity by means of a stress tensor leading to the extra terms that need to be included in the equation of motion. Problems for which exact solutions can be found. The Reynolds number as a measure of the importance of viscosity.

Indicative Text A First Course in Fluid Dynamics − A R Paterson (CUP 1983). Library Ref. 532.05 PAT Fluid Mechanics – P K Kundu and I M Cohen (Academic Press 2002) Library ref. 532 KUN

Note

The two second term courses which follow on from MT322, MT323 and MT421, are intended for both third and fourth year students. They are given in alternate years, and MT421 will be taught in this session.

54

MT323 Magnetohydrodynamics (given in 2007-8)


Aims

This course aims to introduce the study of the motion of conducting fluids in the presence of a magnetic field. Practical applications and a discussion of the structure of sunspots and the origin of the Earth’s magnetic field will be given.

Learning outcomes

On completion of the course the student should be able to: • demonstrate an understanding of the basic principles of MHD; • apply appropriate mathematical techniques to solve a wide variety of problems in

MHD.

Content

Foundations of Magnetohydrodynamics (MHD): Consideration of the electrodynamics of moving media and MHD approximations, leading to the induction equation - an equation central to MHD. Alfvén's theorem for a medium of infinite electrical conductivity - its proof and physical importance. The necessity for an additional term in the equation of motion - the electromagnetic body force. Alternative description in terms of electromagnetic stresses. MHD waves: Alfvén waves in a medium of infinite electrical conductivity, reflection and transmission at a discontinuity in density, effect of finite electrical conductivity and/or viscosity, waves in a compressible medium. MHD shock waves. Steady flow problems: including Hartmann flow. Magnetohydrostatics: Pressure balanced configurations. Force-free fields.

Indicative Texts An Introduction to Magneto-fluid Mechanics − V C A Ferraro & C Plumpton (2nd edition) (OUP 1966). Library Ref. 538.6 FER An Introduction to Magnetohydrodynamics – P A Davidson (CUP 2001). Library Ref. 538.6 DAV

55

MT324 Electromagnetic Theory (Term 2) Students should take PH242: details of PH242 below

Prerequisite: MT222. Teaching: 33hr lectures Assessment: 2hr written examination (90%) and the best four of five fortnightly worksheets (10%)

Aims

An understanding of the development from elementary ideas of electromagnetism up to Maxwell’s equations and the existence of electromagnetic waves.

Learning outcomes

On completion of the course students should be able to: • calculate electric fields and electric potentials from given fixed charge distributions; • calculate magnetic fields and vector potentials from given steady current

distributions; • understand, explain and perform calculations on electromagnetic induction and

displacement currents; • synthesise the above phenomena into the Maxwell equations; • derive properties of electromagnetic waves.

Content

Electrostatics: the electric field, Coulomb’s and Gauss’ laws, electric field energy,

equations of Poisson and Laplace, physical meaning of 2! .

Steady currents: continuity equation, Kirchoff’s laws, Laplace’s equation in conductors. Magnetic effects of currents: Biot-Savart law, magnetic field, Ampere’s law, energy of a magnetic field. Maxwell’s equations, electromagnetic waves in free space. Electric and magnetic dipoles and the electromagnetism of matter.

Indicative Texts

Introduction to Electrodynamics – D J Griffiths (Prentice-Hall International, 1989). Library Ref. 537.6 GRI Electricity and Magnetism – W J Duffin (McGraw-Hill 1980). Library Ref. 537 DUF Electromagnetic Fields and Waves – D Lorrain, D R Corson and F Lorrain (W H Freeman and Company, 1990). Library Ref. 530.141 LOR

56

MT328 Non-linear Dynamical Systems: Routes to Chaos (Term 1: Dr F Mota-Furtado)

Prerequisite: MT172 and MT182, or PH213 Teaching: 33hr lectures Assessment: 2hr examination

Aims

To introduce the fundamentals of the analysis of nonlinear dynamical systems and, in particular, to investigate whether the behaviour of a nonlinear system can be predicted from the corresponding linear system.

Learning outcomes

On completion of the course, students should be able to: • identify and classify the critical points for both discrete and continuous dynamical

systems; • understand when and why the direct and indirect Liapunov methods are

appropriate and use them both; • understand when a limit cycle can, and cannot, occur and prove the non-existence

as appropriate; • recognize the role of the linear system in predicting the long-term behaviour of the

non-linear system.

Content

Systems of first order linear differential equations: Similarity types for 22! matrices and their connection with linear systems. Classification of two-dimensional linear phase portraits. Extension to three dimensions. Nonlinear differential equations: Liapunov’s stability analysis, periodic solutions and limit cycles, Poincaré-Bendixson theorem. Applications to problems from physics, biology and economics. Non-linear difference equations: Poincaré surface of section, stability of critical points, routes to chaos.

Indicative Texts

Dynamical systems, differential equations, maps and chaotic behaviour- D K Arrowsmith and C M Place (Chapman & Hall). Library Ref. 515.41 ARR Differential Equations, dynamical systems and an introduction to chaos-M W Hirsch, S. Smale and R Devaney (Academic Press). Library Ref. 515.41 HIR Elementary differential equations and boundary value problems-W E Boyce and R C di Prima (Wiley). Library Ref. 515.41 BOY

57

MT331 Experimental Design (Term 1: Mr E J Godolphin)

Prerequisite: MT130; MT230 preferred. Teaching: 25hr lectures, 8hr practical work Assessment: 2 hour written examination 80%, project 20%

Aims

• To enlarge on the treatment of inference given in MT130. • To present more advanced topics in the field of experimental design than given in

MT230.

Learning outcomes

At the end of the course, the student should • have a good overall understanding of some of the theoretical aspects of inference,

and a particular understanding of the notions of power, efficiency, sufficiency, unbiasedness, minimum variance estimation and likelihood ratio tests;

• be able to design and analyse the results from two- and three-way classifications, taking account of possible interactions; Latin square, balance complete block and

Youden Square experiments; fractional replicates of 2n experiments; confounding with blocks;

• be able to use MINITAB in the analysis of some of the above experiments.

Content:

Estimation: Maximum likelihood, efficiency, sufficiency, minimum-variance unbiased estimation. Hypothesis testing: Neyman-Pearson lemma, likelihood ratio tests, types of error and power. Experimental designs: Two- and three-way classification, balanced incomplete block, Latin square, Graeco-Latin square and Youden Square.

Factorial designs: 2n factorial models and fractional replicates, confounding with blocks; generating principal block and cosets. Estimability and aliasing. Yates algorithm.

Indicative Text

Design and analysis of experiments – D C Montgomery (Wiley). Library Ref. 001.434 MON

58

MT332 Inference (Term 2: Dr T Sharia)

Prerequisite: MT232 Teaching: 33hr lectures Assessment: 2 hr written examination

Aims

To provide the theory underlying the main principles and methods of statistics, in particular, to provide an introduction to the theory of parametric estimation and hypotheses.

Learning outcomes

On completion of the course, students should be able to • demonstrate a familiarity with the theoretical background of the concepts and

results in the theory of estimation and hypothesis testing; • formulate statistical problems in mathematical terms.

Content

Estimation: Maximum likelihood, method of moments, Bayes estimators, sufficiency, unbiasedness, efficiency, asymptotic properties of maximum likelihood estimators. Hypothesis testing: Neyman-Pearson framework, uniformly most powerful tests, likelihood ratio tests. Introduction to decision theory: Formulation, Bayes and minimax rules.

Indicative texts

Statistical Inference – G Casella and R L Berger (Duxbury 1990) Library reference 518.1 CAS Mathematical Statistics and Data Analysis – J A Rice (Duxbury 1995) Library reference 518.3 RIC John E Freund’s Mathematical Statistics – I Miller and M Miller (Prentice Hall 1999) Library reference: 518.3 FRE Probability and Statistical Inference – R V Hogg and A T Tanis (Prentice Hall 2001) Library reference: 518.1 HOG

59

MT334 Statistical Systems (Term 2: Mr E J Godolphin)

Prerequisite: MT130; MT230 recommended Teaching: 33hr lectures Assessment: 2hr written examination

Aims

To introduce the concept of time-related statistical and dynamic systems. Methods for system modelling and state estimation in real time are described and Bayesian forecasting is introduced as a concept of measurement projection. The course is not formally project-based but several industrial and real-time data series are presented for analysis by MINITAB.

Learning outcomes

At the end of the course the students should be able to: • obtain weighted least-squares estimators from the generalized linear model and

study their properties; • generate the recursive formulation for estimators of observable systems and extend

to multivariate observations; • derive the unbiasedness and optimality relations for the Kalman filter in discrete

state-space models; • apply these results to simulated data series occurring in real time; • express the conditional posterior distribution for the state vector under conditions of

Gaussian noise; • analyse parametric and trend-projecting representations of time series in state

space form and derive their predictor distributions; • extend the state space equations to non-linear situations using a Taylor series

approximation and apply to the constant velocity tracking problem.

Content

General linear models: The linear regression model, unbiased estimation; Gauss-Markov theorem. The normal regression model; maximum likelihood estimation, likelihood ratio criteria, residual analysis. Block regression model: Multivariate observations; covariance weighting matrix; generalised least-squares. Recursive formulation of estimates; recursive covariance estimation; variable parameter vectors; discrete linear systems; deterministic system vectors. State space model: Stochastic system vectors and their estimation, unbiased stochastic estimation; Minimum mean-squared error. Kalman filter equations; Kalman gain matrix. Estimation under Gaussian assumptions. Conditional distributions. Bayesian estimation. Observability criteria. Asymptotic estimates. System extrapolation: The projection of measurement equations. Extrapolation lead times. Conditional estimates. Dual representations. Extrapolation cases: Projection of the steady state; smoothing coefficients. Dimension theorem. Polynomial-projecting models. Forecast systems. Parameter-restructured models. Extension to non-linear problems: Deterministic input vector. Constant velocity tracking model. State estimator extrapolation, state estimator updating. Linearising measurement and system equations. Extended Kalman filter, Bearings only. Observations in two dimensions.

Indicative Text Bayesian Forecasting and Dynamic Models (2nd edition) − M West and J Harrison (Springer-Verlag 1997). Library Ref. 518.3 WES

60

MT336 Applied Probability (Term 1: Mr E J Godolphin)


Aims

To introduce the student to a range of examples of probabilistic methods used to model systems that exhibit random behaviour.

Learning outcomes

On completion of the course the student should be able to: • understand the structure and concepts of discrete and continuous time Markov

chain with countable state space; • use the method of conditioning and the method of conditional expectation; • use the method of generating functions; • construct a probability model for a variety of problems.

Content

Preliminaries: Conditional expectation; generating functions; Distribution of random sums; Stochastic perocesses - basic Notions. Poisson Process: Interarrival and waiting times; Conditional distribution of the waiting times; Nonhomogeneous processes; Compound Poisson process. Renewal theory: Renewal processes; Some limit theorems; Alternating renewal processes; Delayed renewal processes; Cumulative renewal processes. Markov processes: Markov chains, classification of states, Some limit theorems; Stationary distributions; Absorption probabilities.

Indicative Texts Stochastic Processes − S M Ross (Wiley 1996). Library Ref. 519.2 Introduction to Stochastic Modeling – H M Taylor and S Karlin (Academic Press 1998). Library Ref. 518.2 TAY Introduction to Probability Models – S M Ross (Academic Press 2003). Library Ref. 518.1 ROS Probability and Random Processes – G R Grimmett and D R Stirzaker (Oxford UP 1992). Library Ref. 518.1 GRI

61

MT345 Quantum Information and Coding (Term 2: Professor P O’Mahony)


Aims

'Anybody who is not shocked by quantum theory has not understood it' (Niels Bohr). This course aims to provide a sufficient understanding of quantum theory in the spirit of the above quote. Many applications of the novel field of quantum information theory can be studied using undergraduate mathematics. The course relies almost exclusively on tools from linear algebra – prior knowledge of applied mathematics or quantum theory is neither required nor particularly useful.

Learning outcomes

On completion of the course the student should be able to: • demonstrate an understanding of the principles of quantum superposition and

quantum measurement; • use the basic linear algebra tools of quantum information theory confidently; • manipulate tensor-product states and use and explain the concept of

entanglement; • explain applications of entanglement such as quantum teleportation or quantum

secret key distribution; • describe the Einstein-Podolsky-Rosen paradox and derive a Bell inequality; • solve a range of simple problems involving one or two quantum bits; • explain Deutsch's algorithm and its implications for the power of a quantum

computer.

Content

Linear algebra: Complex vector space, inner product, Dirac notation, projection operators, unitary operators, Hermitian operators, Pauli matrices. One qubit: Pure states of a qubit, the Poincaré sphere, von Neumann measurements, quantum logic gates for a single qubit. Tensor products: 2 qubits, 3 qubits, quantum logic gates for 2 qubits, Deutsch's algorithm, the Schmidt decomposition. Mixed states: Partial trace, probability, entropy, von Neumann entropy. Entanglement: The Einstein-Podolsky-Rosen paradox, Bell inequalities, quantum teleportation, measures of entanglement, decoherence. Further applications, such as e.g. the quantum Fourier transform, Shor's factoring algorithm, the BB84 key distribution protocol, Grover's search algorithm, quantum channel capacity, the Holevo bound.

Indicative Text

M A Nielsen and I L Chuang – Quantum Computation and Quantum Information (Cambridge 2000). Library Ref. 001.64 NIE

62

MT347 Mathematics of Financial Markets (Term 1: Dr C M Farmer)

Prerequisite: MT130 and MT172, also MT262 preferred Teaching: 33hr lectures Assessment: 2hr written examination

Aims

This course aims to show how mathematics and statistics are used (and sometimes misused) by those who work in securities markets. Since many of our graduates find employment in this area, the topics in the course are chosen to demonstrate the most important applications. They are portfolio theory, two simple asset pricing models, the general behaviour of markets (how random, how chaotic are they?) and the theory of derivative securities.

Learning outcomes

On completion of the course the student should be able to: • understand the ideas of risk and return and how they can be measured; • formulate Markowitz portfolio theory as an optimization problem and use simple

algorithms to solve it; • understand the assumptions behind asset pricing models and the mathematical

arguments leading to them; • appreciate the consequences of a random walk model of price change and the

arguments for and against such a model; • understand the Black and Scholes formulation of option pricing and find simple

solutions of the equation.

Content

Portfolio analysis: Risk and return. Mean-variance portfolio theory, the efficient frontier. Lending and borrowing: finding the market portfolio. Utility theory. Correlation models: single-index and multi-index. Pricing models: Capital asset pricing model, arbitrage pricing model. Looking for opportunities. Market movements: The random walk model and its shortcomings. The efficient market hypothesis. Skewness and kurtosis. Futures and options: Introduction to derivatives. Pricing of futures. Options: payoff at expiry, use in hedging positions. Put-call parity and related inequalities. Pricing by binomial trees. Brief discussion of Wiener and Ito processes. Delta-hedging and the Black-Scholes equation. Reduction to a diffusion equation and solution for a European call. The American put problem.

Indicative Texts

Paul Wilmott Introduces Quantitative Finance – P Wilmott (Wiley 2001) Library reference 332.632 WIL Modern Portfolio Theory and Investment Analysis − E J Elton and M J Gruber (Wiley 1995). Library Ref. 332.6 ELT The Mathematics of Financial Derivatives − P Wilmott, S Howison and J Dewynne (Cambridge 1995). Library Ref. 332.632 WIL

63

MT361 Error Correcting Codes (Term 2: Professor S R Blackburn)


Aims

To provide an introduction to the theory of error correcting codes employing the methods of elementary enumeration, linear algebra and finite fields.

Learning outcomes

On completion of the course, students should be able to: • calculate the probability of error of the necessity of retransmission under various

assumptions for a binary symmetric channel with given cross-over probability; • prove and apply various bounds on the number of possible code words in a code

of given length and difference; • reduce a linear code to standard form, finding a parity check matrix, building

standard array and syndrome decoding tables, including for partial decoding; • use MOLSs to construct large linear codes of certain parameters; • know/prove/apply the theorem that a cyclic code of length n over a field consists

of the code words corresponding to all multiples of any factor of xn !1.

Content

Geometry in Vn (q): The Hamming distance in the space Vn(q) of n-tuples over an

alphabet of q symbols (the emphasis is on V qn ( ) .= Zn

2 ). The Hamming, Singleton,

Gilbert-Varshamov and Plotkin bounds. General theory of coding: Words, codes, errors, t-error detection and t-error correction. Probability calculations. Efficiency of codes. Perfect codes. Linear Codes: Generator and Parity check matrices, standard array and syndrome decoding, incomplete decoding. Special codes. Hamming codes, Hadamard codes and Levenstein's theorem, Reed-Muller codes. Cyclic Codes: Generator and parity check polynomials, construction of cyclic codes, encoding and decoding. Golay codes.

Indicative Texts A First Course in Coding Theory − R Hill (OUP). Library Ref. 001.539 HIL Coding Theory – a First Course − S Ling and C Xing (Cambridge UP 2004) Library Ref. 001.539 LIN

64

MT362 Cipher Systems (Term 1: Professor S P Murphy)

Prerequisite: MT182 and some probability Teaching: 33hr lectures Assessment: 2hr written examination

Aims

To introduce both symmetric key cipher systems and public key cryptography covering methods of obtaining the two objectives of privacy and authentication.

Learning outcomes

On completion of the course the student should be able to: • understand the concepts of secure communications and cipher systems; • understand and use statistical information and the concept of entropy in the

cryptanalysis of cipher systems; • understand the structure of stream ciphers and block ciphers; • know how to construct as well as have an appreciation of desirable properties of

key stream generators, understand and manipulate the concept of perfect secrecy;

• understand the modes of operation of block ciphers and their properties; • understand the concept of public key cryptography, including details of the RSA

and ElGamal cryptosystems both in the description of the schemes and in their cryptanalysis;

• understand the concepts of authentication, identification and signature, be familiar with techniques that provide these, including one way functions, hash functions and interactive protocols, including the Fiat-Shamir scheme;

• understand the problems of key management, be aware of key distribution techniques.

Content

Cipher systems: An introductory overview of the aims and types of ciphers. Methods and types of attack. Information theory. Statistical tests. Stream ciphers: The one time pad. Pseudo-random key streams - properties and generation. Block ciphers: Confusion and diffusion. Iterated ciphers - substitution/ permutation. The Feistal principle, DES, FEAL, Modes of operation. Public key ciphers: Discussion of key management. Diffie-Hellman key exchange. One-way functions and trap-doors. RSA; ElGamal cryptosystem. Authentication/Identification: Protocols. Challenge/response. MACs. Zero-knowledge protocols; Fiat-Shamir protocol. Digital signatures: Digital signature methods - arbiters. Hash functions. DSS. Certificates.

Indicative Text

Codes and Cryptography – D Welsh (Oxford UP 1988). Library Ref. 001.5436 WEL

65

MT364 Applications of Operational Research Techniques (Term 1: Dr A F Sheer and Professor K C Bowen)

Prerequisite: MT262 Teaching: 33hr lectures, discussions and case studies Assessment: Based on reports on three other case studies (30% each) and a short oral examination (10%).

Aims

This course has two main aims: to show how techniques met in other courses (mathematical programming, combinatorial optimization, statistics and others), have been used in real situations, and to develop a critical faculty, so that students can distinguish between good and bad approaches to a problem. Its secondary aims are to give some idea of how an OR analyst works, to give practice in obtaining information from journals, and to improve communication skills.

Learning outcomes

On completion of the course the student should be able to: • be able to make constructive criticism based on the published description of an OR

project; • decide whether a problem has been tackled by a suitable method, and whether or

not a satisfactory solution has been found; • collect information from journals, books, people and other sources and decide what

is appropriate; • write a coherent report on what they have found; • talk about their report.

Content

This course discusses the problems faced by the OR analyst; choice of methodology, communication with the client, social aspects and also how techniques work in practice, and is based on case studies.

Indicative Text

None: the course involves reading recent papers, mostly in the Journal of the Operational Research Society.

Deadlines

The first report is to be submitted by November 16, the second by December 14 and the third by the end of the first week of Term 2.

66

MT365 Algorithmic Graph Theory (Term 2: Dr J F McKee)


Aims

• to introduce the formal idea of an algorithm; • to develop feeling for when an algorithm is good; • to investigate techniques for constructing and verifying algorithms; • to explore connectivity and colourings of graphs, from an algorithmic perspective.

Learning outcomes

On completion of the course the students should be able to: • use particular algorithms which optimize various properties for graphs and networks; • understand elementary ideas of complexity exemplified in particular by the

Travelling Salesman problem; • understand when greediness works, exemplified by minimum spanning tree

algorithms; • apply Fleury’s and Tucker’s algorithms to find Eulerian trails; • find chromatic polynomials and illustrate Vizing's theorem on edge colourings.

Content

Trees: Algorithms for minimum spanning trees. Sorting and searching: Sorting methods including bubble sort and heap sort. Depth first search and breadth first search. Shortest paths. The Travelling Salesman Problem: Branch and bound method, upper and lower bounds, approximate methods. Flows in networks: The max-flow min-cut theorem. An algorithm for finding maximum flows. Matching problems: Hall's theorem. Maximum and complete matchings. Alternating paths and applications. Menger's theorems on edge and vertex connectivity. Eulerian trails: Algorithms for finding them: Fleury's algorithm; Tucker's algorithm. Hamiltonian paths: Ore’s and Dirac’s theorems on Hamiltonian cycles. Colouring graphs: Vertex and edge colourings; chromatic polynomials. Brook's, Vizing's and König's theorems. Colouring maps, the four-colour theorem.

Indicative Texts Algorithmic Graph Theory − A Gibbons. (Cambridge UP 1985). Library Ref. 512.23 GIB Discrete Mathematics − N L Biggs (Oxford UP 2002). Library Ref. 510 BIG

67

MT369 Theory of Games (Term 2: Dr R M Damerell)

Prerequisites: MT172, MT182, MT262 Teaching: 33hr lectures Assessment: 2hr written examination

Aims

• to analyse a way in which mathematics can be applied to real problems; • to apply game theory to a wide variety of situations.

Learning outcomes

On completion of the course the students should be able to: • model a simple game in a mathematical framework. • understand the formal setting of a model and its relation to the real world. • understand how mathematical tools can be used for modelling business and

economics problems; • solve a zero-sum matrix model and a simple continuous game.

Content

Zero-sum games: Non-cooperative games, extensive and normal form, pure and mixed strategies, use of dominance to reduce these,saddle points, the minimax theorem for matrix games. Ad hoc methods of solutions for particular games such as 2 × m games or symmetric games. Continuous games: Continuous games on the unit square, continuous games with no value. Differential games, kinematic equations, main equation, path equations. Non-zero sum games: Models of duopoly, Farber’s model of final offer arbitration. Bargaining models.

Indicative Texts Games, Theory and Applications − L. C. Thomas. (Ellis Horwood 2003). Library Ref. 519.42 THO Game Theory − A.J. Jones (Ellis Horwood 2000). Library Ref. 519.3 JON Games and Information − An Introduction to Game Theory − E. Rasmusen. (Blackwell 2001). Library Ref. 519.4 RAS

68

MT373 Control Systems (Term 1: Dr G de Barra)


Aims

• To show how physical control systems can lead to systems of differential equations. • To extend the methods of solution available from earlier courses. • To study the stability of these systems, especially using complex analysis.

Learning outcomes

At the end of this course the student should be able: • to show how physical systems can be described in terms of block diagrams. • to get the differential equations describing the systems in the standard form

y Cx Ax Bu= = +, x.

.

• to show how these equations can be solved (i) in the time-independent case using Laplace transforms; (ii) defining the matrix exp(At) using a matrix norm and showing its use in the solution in the time-independent case; (iii) expressing the solution in the general case in terms of the Fundamental Matrix.

• to define different types of stability including asymptotic and BIBO stability; to apply the methods to examples.

• to examine stability using poles of the transfer function, the Routh criterion and Nyquist plots.

• to establish criteria (in terms of rank) for the complete controllability and for the complete observability of linear systems, and to apply these to examples.

Content

State space: the underlying concepts of state, control and output variables, open and closed loop control. Solution of time-independent and time-dependent controlled equations, using matrix and Laplace transform techniques as appropriate. Realizations of a system. Controllability and observability: criteria for controllability and observability: duality. Minimal realizations. Stability: eigenvalue analysis. Routh analysis. Root locus method. Nyquist criterion. Optimal control: dynamic programming approach; time-independent and dependent systems. Pontryagin's Theorem (statement). Constrained control, using Lagrange multiplier approach. Bang-bang control.

Indicative Texts Introduction to Mathematical Control Theory − S Barnett and R G Cameron (Oxford). Library Ref. 519.3 BAR.. Schaum's Outline of Feedback and Control Systems – J J Distefano, A R Stubberud and I J Williams (McGraw-Hill). Library Ref. 629.3814 DIS Introduction to Control Theory − O L R Jacobs (Macmillan). Library Ref. 519.3 JAC

69

MT381 Algebra III (given in 2007-8)

Prerequisite: MT282 and MT283 Teaching: 33hr lectures Assessment: 2hr written examination

Aims

The aim of this course is to give the students a working knowledge of finitely-generated abelian groups and the analogous theory of modules.

Learning outcomes

On completion of the course, students should be able to: • demonstrate the existence of normal subgroups of certain finite groups, and to use

this to determine the isomorphism classes of groups of certain orders; • classify all finitely-generated abelian groups; • understand the concepts of module, submodule and lattice; • prove that the minimum polynomial of a matrix is invariant under similarity, but not

conversely; • prove that A is similar to the companion matrix of its characteristic polynomial if and

only if the corresponding module is cyclic; • obtain the rational canonical form, primary canonical form or Jordan form as

appropriate.

Content

Group Theory: Homomorphisms, normal subgroups, factor groups, the first

isomorphism theorem. The centre, centralizers, groups of order 2p .

Modules: The ][xF –module determined by a square matrix over F, the order of a

module element, cyclic submodules, companion matrices, minimum polynomials. Abelian groups as Z–modules. Module decomposition: Direct sum, primary decomposition. Free R-modules and their rank, invertible matrices over R. Equivalent matrices over R, reduction to invariant factor form (Smith normal form) over ][xF and Z. The rational, primary and Jordan

forms of a square matrix over F. The classification of finitely–generated abelian groups.

Indicative Text Rings, Modules and Linear Algebra − B Hartley & T Hawkes (Chapman & Hall). Library Ref. 512.61 HAR Topics in Algebra – I N Herstein (Wiley 1975) Library Ref. 512.61 HER

70

MT394 Metric Spaces (not given 2006-7)


Aims

• to extend the understanding of real analysis begun in MT194 and MT294; • to introduce the concept of a metric space, including the notion of completeness; • to illustrate the power of these techniques with the contraction mapping theorem

and some of its applications.

Learning outcomes

On completion of the course the student should be able to: • understand and use the concept of countability; • understand the concept of uniform convergence and establish the uniform

convergence, or not, of sequences and series; • understand and use upper and lower limits in analytical arguments; • understand and apply the basic concepts of metric space topology including

completeness; • know and be able to apply the contraction mapping.

Content

Countability; upper and lower limits; uniform convergence. Metric space topology; contraction mapping theorem; applications to algebraic and integral equations.

Indicative Texts

Metric Spaces: Iteration and Application – V Bryant (Cambridge 1985) Library reference 515.25 BRY Introduction to Metric and Topological Spaces – W A Sutherland (Oxford 1975). Library reference 514.3 SUT Principles of Mathematical Analysis – W Rudin (McGraw-Hill 1976) Library reference 515.23 RUD

71

SC3002 Science Communication (Term 2: Dr A Lewis) Prerequisite: none Suitable: year 3 Value: half course unit Teaching: Monday 2-5pm (for Mathematics students)

Media lab available all hours for course work only Assessment: 2hr written examination

Aims

Communicating science to a wide audience is a vital skill for practicing scientists, teachers at all levels or for those wanting to pursue a career in the media. This course will give you the skills needed to communicate your subject while also increasing the depth of your knowledge of your chosen study area. You will learn to convey a variety of messages and ideas through demonstrations, Powerpoint, TV, radio, print, the internet and other relevant media. It is a combination of lectures and highly practical sessions in the Queen’s Studio and out on location. This is a Science Faculty course, and available to Mathematics, Mathematics major (that is, Mathematics with something), and Mathematics and Computer Science/Geology/Physics/Psychology students, but not to Mathematics minor or Mathematics and Economics/Management/Music students. Due to the limited facilities, only four students from each department can enrol for this course, so if you include this course in your choices make sure that you include a reserve choice. If more than four students from the Mathematics Department apply, we are told to choose the four ‘on the basis of academic merit’. You may not take both SC3002 and MT309.

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MT400 The MSci Project (Both Terms) (1 unit value)

The project is compulsory for all fourth-year MSci students. Prerequisite: Successful completion of the third year of the MSci course. Teaching: Regular meetings with your supervisor. Assessment: By project (90%) and a presentation on it (10%).

Aims

• To enable students to make a detailed study of one topic in mathematics; • To develop their skills in finding and analysing information from a variety of sources; • To develop their skills in writing and talking about mathematics.

Learning outcomes

On completion of the project you should • be able to extract information from books and research papers; • be able to write a coherent report; • be able to give a clear presentation on a mathematical topic; • understand the relevance of the project material to the appropriate branch of

mathematics. Each student will have a supervisor, whose task is to advise, not to direct the project. You should discuss the project regularly with your supervisor; in particular you must show an outline plan to your supervisor early in Term 1, and a draft at an early stage. The choice of supervisor is by agreement with the student, the supervisor and the MSci Co-ordinator. A list of members of staff and the areas of mathematics in which they are willing to supervise projects is available from the MSci coordinator. You will be expected to investigate in depth some branch of mathematics and write a report upon the investigation. The subject studied may vary from giving an overview of a broad area to a detailed analysis of a very narrow field. The level is expected to be at a substantially higher level than the material in lecture courses, and there should be either a review of research level work or original work. You should be able to word process mathematical material (formulae etc.). This may be in TeX, LateX, Mathtype, Word Equation Editor or a similar package. The examiners will look for evidence that you have understood the material and are able to present it coherently. The report should be word processed (see above) and approximately 10,000 words (30 pages of A4) in length. Overlong reports will be penalised. Factors which will be taken into account include the following: (1) Appreciation of the relations of the results described to one another and to the rest of mathematics. (2) Elaboration and extension of your source material. (3) Selection of material where choices exist. Original research will be encouraged and rewarded, but is not expected.

Submission

Two copies of the project must be handed in to the Departmental Office by the first day of Term 3. The presentation will be later in Term 3.

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MT420 Quantum Theory II (given in 2007-8)


Aims

• To derive methods, such as the Rayleigh-Ritz variational principle and perturbation theory, in order to obtain approximate solutions of the Schrödinger equation.

• To introduce spin and the Pauli exclusion principle and hence explain the mathematical basis of the Periodic table of elements.

• To introduce the quantum theory of the interaction of electromagnetic radiation with matter using time dependent perturbation theory.

• To show how scattering theory is used to probe interactions between particles and hence to show how the probability or cross section for a scattering event to occur can be derived from quantum theory.

Learning outcomes

On completion of the course students should be able to: • use various methods to obtain approximate eigenvalues and eigenfunctions of any

given Schrödinger equation, • to understand the importance of spin in quantum theory, • to appreciate how the Periodic Table of elements follows from quantum theory, • to write down the Schrödinger equation for the interaction of electromagnetic

radiation with the hydrogen atom and to work out photoabsorption cross sections for hydrogen,

• to define the scattering cross section and to work it out for some simple systems.

Content

Variational principles in quantum mechanics: the Rayleigh-Ritz variational principle. Bounds on energy levels for quantum systems. Perturbation theory: Rayleigh-Schrödinger time-independent perturbation theory. Perturbations of energy levels due to external electromagnetic fields. The electron’s spin: the eigenfunctions and eigenvalues of the spin operator. The Pauli exclusion principle. The periodic table of elements. Spin precession in an external magnetic field. Radiative transitions: the absorption and emission of electromagnetic radiation by matter. Photoabsorption cross-sections for the hydrogen atom. Scattering theory: definition of the scattering cross-section and the scattering amplitude. Decomposition of the scattering amplitude into partial waves. Phase shifts and the S-matrix. Integral representations of the scattering amplitude. The Born approximation. Potential scattering.

Indicative texts

Quantum Physics – S Gasiorowicz (Wiley 1974) Library reference 530.12 GAS Quantum Mechanics – P C W Davies (Chapman and Hall 1984) Library reference 530.12 DAV

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MT421 Aerodynamics and geophysical fluid dynamics (Term 2: Dr C M Davies)


Aims

This course aims to show how the mathematical models of MT222 and MT322 are successful in describing how aircraft are able to fly, and how the motions of the atmosphere and the oceans are caused. It also gives insight into the effect that individual terms in the mathematical model may have on the behaviour of the whole system.

Learning outcomes

At the end of the course the students should be able to • derive the freezing-in of vortex lines for incompressible fluids; • use complex variable theory to derive the formula for lift on an infinite cylinder; • explain in broad terms how an aircraft is able to fly; • understand the role of Coriolis and centrifugal forces in a rotating fluid; • describe how rotation causes various phenomena in fluids; • solve the simple equations for motion in an Ekman layer.

Content

Vortex dynamics: freezing-in of vortex lines, why vorticity can be treated as a pollutant. Examples. Flow past wing sections: two-dimensional flow, flow at sharp corners, generation of lift. Blasius’ formula. Three-dimensional flows, trailing vortices, induced drag. Supersonic flow past wing sections. Rotating fluid systems: equation of motion of a rotating fluid. Geostrophic flow and simple properties. Secondary flow and examples (e.g. meanders, tea leaves in a cup). Inertial waves. Viscosity-rotation interactions: Ekman layers and boundary fluxes. The atmosphere and oceans: large-scale motions and the role of Coriolis forces. Tornado generation. Effects of the earth’s curvature and induced waves.

Indicative text

Fluid Mechanics – P K Kundu and I M Cohen (Academic Press 2002) Library ref. 532 KUN

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MT422 Advanced Electromagnetism and Special Relativity (Term 1: Dr T J Osborne)

Prerequisite: PH242 (=MT324) preferred Teaching: 33hr lectures Assessment: 2hr written examination

Aims

• To show how Maxwell’s equations lead to electromagnetic waves and indirectly to the special theory of relativity;

• To show how electromagnetic fields propagate with the speed of light; • To derive the laws of optics from Maxwell’s equations; • To show how the laws of special relativity lead to time dilation and length

contraction.

Learning outcomes

On completion of the course students should be able to • use Maxwell’s equations to demonstrate the polarization, reflection and refraction

of electromagnetic waves; • understand the fundamental ideas of electromagnetic radiation; • demonstrate the Galilean non-invariance and Lorentz invariance of Maxwell’s

equations; • derive the fundamental properties of relativistic optics.

Content

Electromagnetic theory: electromagnetic waves, reflection and refraction with both normal and oblique incidence, total internal reflection, waves in conducting media, wave guides. Radiation: the Hertz vector and related field strengths, fields of moving charges, Lienhard-Wiechart potentials, motion of charged particles. Special relativity: the Lorentz transformation. Relativistic invariance, the Fitzgerald contraction, time dilation. Relativistic electromagnetic theory: Lorentz invariance of Maxwell’s equations, the transformation of E and B . Relativistic mechanics: mass, momentum, energy. Relativistic optics: aberration, the Doppler effect.

Indicative text

Foundations of Electromagnetic Theory (Fourth Edition) – J R Reitz, F J Milford and R W Christy (Addison-Wesley 1993) Library reference 538.141 REI.

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MT441 Channels (Term 1: Dr C Elsholtz)

Prerequisite: MT361; MT232 and MT282 recommended Teaching: 33hr lectures Assessment: 2hr examination

Aims

To investigate the problems of data compression and information transmission in both noiseless and noisy environments.

Learning outcomes

On completion of the course, students should be able to: • state and derive a range of information-theoretic equalities and inequalities; • explain data-compression techniques for ergodic as well as memoryless sources; • explain the asymptotic equipartition property of ergodic systems; • understand the proof of the noiseless coding theorem; • define and use the concept of channel capacity of a noisy channel; • explain and apply the noisy channel coding theorem; • evaluate and understand a range of further applications of the theory.

Content

Entropy: Definition and mathematical properties of entropy, information and mutual information. Noiseless coding: Memoryless sources: proof of the Kraft inequality for uniquely decipherable codes, proof of the optimality of Huffman codes, typical sequences of a memoryless source, the fixed-length coding theorem. Ergodic sources: entropy rate, the asymptotic equipartition property, the noiseless coding theorem for ergodic sources. Lempel-Ziv coding. Noisy coding: Noisy channels, the noisy channel coding theorem, channel capacity. Further topics, such as hash codes, or the information-theoretic approach to cryptography and authentication.

Indicative Texts Codes and Cryptography − D Welsh (Oxford UP). Library Ref. 001.5436 WEL Elements of Information Theory − T M Cover and J A Thomas (Wiley). Library Ref. 001.539 COV Information Theory, Inference and Learning Algorithms – D J C MacKay (Cambridge UP). Library Ref. 001.539 MAC

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MT447 Advanced Financial Mathematics (Term 2: Dr A F Sheer)


Aims

• To investigate the validity of various linear and non-linear time series occurring in finance;

• To extend the use of stochastic calculus to interest rate movements and credit rating;

Learning outcomes

On completion of the course, students should: • make use of some of the ARCH (autoregressive conditionally heteroscedastic)

family of models in time series; • appreciate the ideas behind the use of the BDS test and the bispectral test for time

series. • understand the partial differential equation for interest rates and the assumptions

that lead to it; • be able to model forward and spot rates; • understand how a Poisson process can be included to model the possibility of

default on a bond or similar asset.

Content

Financial time series: Linear time series: ARMA and ARIMA models, stationarity, autoregressions. Testing of linearity, using spectral analysis. ARCH and GARCH models. Structure of financial series: The random walk model, trend and volatility, moments. Comparison with chaotic systems, dimensionality and memory effects in financial series. The nearest neighbour algorithm and the BDS test. Interest rate analysis: Revision of ideas in stochastic calculus. Modelling of interest rates, the bond pricing equation. Bond derivatives. The Heath-Jarrow-Morton model. Credit risk: Modelling of default probabilities. The equation for a risky bond.

Indicative Texts

Paul Wilmott Introduces Quantitative Finance – P Wilmott (Wiley 2001) Library reference 332.632 WIL The Econometric Modelling of Financial Time Series – T C Mills (Cambridge UP 1999) Library reference 330.0151 MIL Market Models – C Alexander (Wiley 2001) Library reference 332.6 ALE

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MT454 Combinatorics (Term 1: to be arranged)

Prerequisite: MT261 Teaching: 33hr lectures and seminars, 67hr private study, including problem sheets Assessment: 2hr written examination

Aims: To introduce the standard techniques of combinatorics, including

• methods of counting: generating functions, induction, subdivision; • Principle of Inclusion and Exclusion; • partitions, Ramsey and Polya Theory.

Learning Outcomes:

On completion of the course, students should be able to: • find small partition numbers; • perform simple calculations with generating functions;. • understand Ramsey numbers and calculate upper bounds for these (where

practical); • calculate sets by inclusion and exclusion and understand the applications to

number theory; • calculate cycle indexes for the standard groups and the numbers of distinct

configurations of symmetrical objects.

Content

Enumeration: Binomial identities. The Principle of Inclusion-Exclusion with applications to number theory. Rook polynomials. Posets and lattices. The Möbius function of a lattice. Generating functions: Linear recursion. Power series and ordinary generating functions. Partitions and partition identities. Ramsey Theory: Monochromatic subsets, Ramsey numbers and Ramsey's Theorem. Polya Theory: Automorphisms of graphs. The Orbit-Stabiliser Theorem, and the Orbit Counting Lemma. Cycle index of a permutation group. Polya's Theorem.

Indicative Texts Discrete Mathematics − N L Biggs (Oxford UP 2002); Library reference 510 BIG. Combinatorics: Topics, Techniques, Algorithms – P J Cameron (Cambridge UP 1994). Library reference 512.23 CAM.

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MT466 Public key cryptography (Term 2: Dr S D Galbraith)


Aims

• To introduce some of the mathematical ideas essential for an understanding of public key cryptography, such as discrete logarithms, integer factorisation, lattices and elliptic curves;

• To introduce several important public key cryptosystems, such as RSA, Rabin, ElGamal, Diffie-Hellman, Schnorr signatures;

• To discuss modern notions of security and attack models for public key cryptosystems.

Learning outcomes

On completion of the course, students should: • be familiar with the RSA and Rabin cryptosystems, the hard problems on which

their security relies and certain attacks on them; • have a basic knowledge of finite fields and elliptic curves over finite fields, and

the discrete logarithm problem in these groups; • be familiar with cryptosystems based on discrete logarithms, and some

algorithms for solving the discrete logarithm problem; • know the definition of a lattice and be familiar with the LLL algorithm and some

applications of lattices in cryptography and cryptanalysis; • be able to define security notions and attack models relevant for modern

theoretical cryptography, such as indistinguishability and adaptive chosen-ciphertext attack.

Content

Background: Integers modulo n; Chinese remainder theorem; finite fields; fast exponentiation; public key cryptography and security; complexity theory; primality testing and certificates. RSA/Rabin: Key generation; implementation; encryption and signatures with OAEP; the RSA problem and relationship with factoring; square roots modulo a prime; Hastad attack; Wiener attack; smooth numbers; survey of integer factorisation methods such as

1!p method and index calculus.

Discrete logarithms: Diffie-Hellman; ElGamal encryption; Schnorr signatures; Diffie-Hellman problem and decision Diffie-Hellman; methods to solve discrete logarithms such as baby-step-giant-step, Pollard rho and lambda, index calculus. Lattices: Definition of a lattice; GGH cryptosystem; LLL algorithm; lattice attacks on RSA with small public or private exponents. Elliptic curves: Group law; Hasse bound; group structure; ECC protocols; elliptic curve factorisation and primality certificates; Maurer equivalence of DH and DL.

Indicative Texts

Cryptography: an introduction – Nigel Smart (McGraw Hill) Library Ref. 001.5436 SMA Cryptography theory and practice – Doug Stinson (CRC press, 2nd ed.) Library Ref. 001.5436 STI

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MT481 Symmetry (not given 2006-7)

Prerequisite: MT283 Teaching: 33hr lectures, Assessment: 2hr written examination

Aims

To develop the properties of groups with emphasis on non-commutative finite groups. The subgroup structure of such groups is analysed, significant examples are constructed and the simplicity of certain standard groups is established.

Objectives

At the end of the course a student should be able to • Recognize groups which are direct products of smaller groups • Understand all aspects of cyclic groups, including their automorphism groups • Construct groups with a complementary pair of subgroups, one of which is normal • Classify all groups of certain orders • Prove that alternating groups and the projective special linear groups are in general

simple.

Content

Groups: A reminder of the axioms of a group, order of a group element, cosets, conjugacy, especially in

nS and

nA . Normal subgroups, the first isomorphism theorem,

commutator subgroup, direct product characterization, complementary subgroups. The second isomorphism theorem, subgroups of cyclic groups, automorphism groups of cyclic groups, the semi-direct product.

Permutation groups: Permutation representations, orbits and stabilizers, Cayley’s

theorem, the Sylow theorems, groups of order pq, the transfer, Burnside’s theorem. Simplicity of

nA (when 5!n ) and )(FPSL

n.

Other topics: The symmetry groups of the Platonic bodies, the wreath product, the outer automorphism of

6S .

Indicative Texts Groups and Symmetry − M A Armstrong (Springer). Library Ref. 512.51 ARM Groups: a Path to Geometry − R P Burn (Cambridge UP). Library Ref. 512.51 BUR Geometric Algebra − E Artin (Interscience). Library Ref. 512.8 ART The theory of groups: an introduction − J.J Rotman (Allyn & Baron). Library Ref. 512.51 ROT

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MT485 Applications of Field Theory (Term 1: Dr R M Damerell)

Prerequisite: MT282 and MT283 Teaching: 33hr lectures and seminars, 67hr private study, including problem sheets Assessment: 2hr written examination Aims: To introduce some of the basic theory of extension fields, with special emphasis on finite fields and their applications.

Learning Outcomes:

On completion of the course, students should be able to: • understand simple field extensions of finite degree; • classify finite fields and determine the number of irreducible polynomials over a

finite field; • state the fundamental theorem of Galois theory for finite fields; • compute in a finite field; • understand some of the applications of fields.

Content (Other than the first section, all topics refer to finite fields only, unless otherwise specified.)

Extension theory: Polynomial factorisation. Field extensions. Simple extensions. The degree of an extension. Applications to ruler and compass constructions. Classifying finite fields: The number of irreducible polynomials. Existence and uniqueness of finite fields. Concrete representations of a finite field. The structure of finite fields: Roots of irreducible polynomials and the Frobenius automorphism. Cyclotomic polynomials. The Galois correspondence for finite fields. An indication of Galois correspondence for general fields. The norm and trace of an element. Applications to m-sequences. Dual and self-dual bases. Normal bases and the normal basis theorem. Applications to multiplication in finite fields. Discrete logarithms: The discrete log problem and its applications. The Pohlig-Hellman and baby step, giant step algorithms.

Indicative Texts Finite Fields: Structure and Arithmetics − D Jungnickel (Bibliographisches Institut Wiss. Verlag, 1993); Library reference 512.4 INT. Introduction to Finite Fields and their Applications – R Lidl and H Niederreiter (Cambridge UP, 1994); Library reference 512.4 LID. Galois Theory – I Stewart (Chapman and Hall, 1989); Library reference 512.4 STE.

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MT492 Integration and Function Spaces (Given in 2007-8)

Prerequisite: MT294 Teaching: 33 hours lectures. Assessment: two hour written examination.

Aims

To provide a unified introduction to integration and to a functional analytic account of the most important function spaces.

Learning outcomes

On completion of the course, students should be able to: • understand and be able to apply the basic ideas of the Lebesgue integral; • understand how the basic ideas of functional analysis apply to the classical function

spaces.

Content

Measure: Lebesgue outer measure on sets in the real line; it is sub-additive and gives an interval its length. Measurable sets: these form a sigma algebra on which outer measure is additive on disjoint countable unions. Hence Lebesgue measure. Regularity of the measure; approximation by finite sets of intervals. Existence of non-measurable sets Measurable functions and the Lebesgue integral: Lebesgue integral for measurable simple functions, hence for non-negative measurable functions. Fatou’s lemma; Lebesgue Monotone Convergence Theorem. Non-negative measurable functions are monotone limits of simple functions. Integration of general measurable functions. Lebesgue’s Dominated Convergence Theorem, with examples. Comparison of Riemann and Lebesgue integrals. Differentiation of indefinite integrals, the Lebesgue Density Theorem. Integration of series of functions. Spaces of functions: Lp space and its norm. Jensen, Holder and Minkowski’s inequalities ( 1!p ). Lp spaces are complete normed spaces )1( !"" p . Continuous and step

functions are dense in Lp. Normed space: linear transformations and their continuity and norms. Dual spaces. The dual of Lp is Lq where 1/p + 1/q = 1. The dual of L1. L2 as a Hilbert space. The least distance theorem and orthogonal complements. Orthonormal

sets. )1,1(2 !L and Legendre Polynomials. Fourier series coefficients; in L2 , Fourier series

converge and obey Parseval’s identity.

Indicative texts

G de Barra – Measure Theory and Integration (Ellis Horwood 1981) Library reference 515.52 DEB W Rudin – Real and Complex Analysis (McGraw Hill 1966) Library reference 515.23 RUD W Rudin – Functional Analysis (McGraw Hill 1974) Library reference 515.61 RUD

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12. The Greek alphabet alpha Α α Usually used as a constant. beta Β β Another constant. gamma Γ γ Some special uses, for example )!1()( !=" nn delta Δ δ Often used for small quantities or changes. epsilon Ε ε Often used for small numbers – ε is much less than 1. zeta Ζ ζ ),,( !"# may be used as an alternative to ),,( zyx . eta Η η See zeta. theta Θ θ Often used for angles. iota Ι ι Rarely used, but ι may be used as a vector of 1s. kappa Κ κ An alternative to k. lambda Λ λ Often used for coefficients, and for eigenvalues. mu Μ µ Means the mean in statistics, but many other uses. nu Ν ν A few special uses, for example in fluid dynamics. xi Ξ ξ See zeta. Don't confuse them. omicron Ο ο Never used. pi Π π π is familiar; Π is used when quantities are multiplied. rho Ρ ρ Various uses, for example density of a fluid. sigma Σ σ Σ is commonest in a sum, σ as standard deviation. tau Τ τ An alternative to t, for example in an integral. upsilon Υ υ Never used. phi Φ φ Generally used as an angle, and in number theory. chi Χ χ Most common in statistics, when it is always squared. psi Ψ ψ Another one used for angles. Don't confuse it with phi. omega Ω ω Used for angular frequency. The end.

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If you have a problem … with a topic in a particular course unit: see the lecturer concerned, or your tutor, or your workshop leader with the running of a particular course unit: see the lecturer concerned with general queries about Departmental facilities or organization see the Academic Coordinator with course selection: see your adviser with your degree programme: discuss it with your adviser and then the Academic Coordinator with a personal problem see your adviser, or the Welfare Services (Counsellors, Health Centre, Chaplaincy, Student Union … )

and you think it is a problem common to other students: see your Student-Staff Committee representative with College registration, enrolment, fees, accommodation, loans etc.: go to the Student Service Point, Founders W141

Documents

C O N TE N TSma.rhul.ac.uk/files/undergradhandbook.pdf · 3.1 A c a d e m ic W e lf a re 3.2 S ta ff -S tu d e n t C o m m itte e 3.3 Q u e stio n n a ire s 3.4 S tu d y A b ro a