DURHAM UNIVERSITYmaths.dur.ac.uk/Ug/coursebooks/2H/2007_2008/booklet.pdf · 2009. 8. 30. · DURHAM UNIVERSITY Department of Mathematical Sciences Second-Year Mathematics 2007-2008

DURHAM UNIVERSITYDepartment of Mathematical Sciences

Second-Year Mathematics2007-2008

Science LaboratoriesSouth RoadDurham Email: [email protected] 3LE Web: www.maths.dur.ac.uk

mailto:[email protected]://www.maths.dur.ac.uk

Contents

1 General Information 4

1.1 The Department . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Useful Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Course Director . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Natural Sciences Co-ordinator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Departmental Adviser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Registration for 2H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.7 Consultation with Members of Staff . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.8 Staff-Student Consultation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.9 Students with Special Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.10 Illness and Absence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.11 Course Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.12 Computers, ICT and DUO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.13 Private Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.14 Books and Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.15 Student Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.16 Smoking and Mobile Phones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.17 Durham University Mathematical Society . . . . . . . . . . . . . . . . . . . . . . 10

1.18 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Examinations and Assessment 11

2.1 Regulations for B.Sc., M.Math. and M.Sci . . . . . . . . . . . . . . . . . . . . . . 11

2.2 University Assessment Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Board of Examiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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2.4 Plagiarism, Cheating and Collusion . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Monitoring of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Submission of Formative Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Submission of Summative-Assessed Work . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.9 Examinations and Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.10 Illness and Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Teaching and Learning 15

3.1 The Mathematics Modules and Degrees . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Honours Natural Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Further Remarks on Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Booklists and Descriptions of Courses . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Algebra and Number Theory II – MATH2061 . . . . . . . . . . . . . . . . 18

3.4.2 Analysis in Many Variables II – MATH2031 . . . . . . . . . . . . . . . . 20

3.4.3 Codes and Actuarial Mathematics II – MATH2131 . . . . . . . . . . . . . 22

3.4.4 Codes and Geometric Topology II – MATH2141 . . . . . . . . . . . . . . 24

3.4.5 Complex Analysis II – MATH2011 . . . . . . . . . . . . . . . . . . . . . 26

3.4.6 Contours and Actuarial Mathematics II –MATH2171 . . . . . . . . . . . . 28

3.4.7 Linear Algebra II – MATH2021 . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.8 Mathematical Physics II – MATH2071 . . . . . . . . . . . . . . . . . . . 32

3.4.9 Numerical Analysis II – MATH2051 . . . . . . . . . . . . . . . . . . . . . 34

3.4.10 Probability and Actuarial Mathematics II – MATH2161 . . . . . . . . . . 36

3.4.11 Probability and Geometric Topology II – MATH2151 . . . . . . . . . . . . 38

3.4.12 Statistical Concepts II – MATH2041 . . . . . . . . . . . . . . . . . . . . . 40

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http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2061&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2031&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2131&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2141&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2011&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2171&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2021&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2071&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2051&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2161&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2041&ayear=2006

A Details of Modules and Programmes 42

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1 General Information

Welcome to 2H! About 1,200 undergraduates take modules provided by the Department. Thisbooklet has been written for second-year students registered for Single Honours degrees in Math-ematics and also for second-year Natural Sciences students and those registered for CombinedStudies in Arts or Social Sciences.

This booklet contains information specific to your programme of study within the Department. Forinformation concerning general University regulations, examination procedures etc., you shouldconsult the Faculty Handbooks and the University Calendar, which provide the definitive ver-sions of University policy (www.dur.ac.uk/faculty.handbook). The web address of the Teachingand Learning Handbook which contains information about assessment procedures amongst otherthings, is www.dur.ac.uk/teachingandlearning.handbook/.

You may find it useful to keep this booklet for future reference. For instance, prospective employersmight find it of interest.

An on-line version of this booklet may be found by following the link ‘Teaching > 2H Mathe-matics’ from the Department’s home page (www.maths.dur.ac.uk). If you intend to take mathsmodules at Level 1 or 3 this year, do consult the relevant booklets on-line.

It is usual for the second year to be regarded as harder than the first year. Time spent during thevacation in catching up on first-year work which was not understood, or in browsing through someof the recommended 2H books and following the suggestions made at the end of each moduledescription, will be well rewarded. Have a good vacation!

1.1 The Department

Besides undergraduate teaching, the Department has a second main function — research. Just likeyou, lecturers and tutors are building on their existing expertise and trying to solve mathematicalproblems. Together with various administrative tasks, it is their main occupation outside the class-room. One difference, however, is that the problems you are asked to tackle should actually besolvable! The important thing is that Mathematics in Durham is living, developing and growing —and you are joining in.

1.2 Useful Contacts

The first point of contact for issues referring to a particular course or module should be the relevantlecturer. For more general questions or difficulties you are welcome to consult the Course Directoror the Head of Department. For issues involving University registration for mathematics modules,please see the Registration Co-ordinator.

Head of Department:Prof. M. Goldstein, (room CM207), Email: [email protected] Director:Dr. C. Kearton,(room CM318), Email: [email protected] Co-ordinator:Dr. S. Borgan, (room CM216), Email: [email protected] of Support Teaching:Dr. S. Borgan, (room CM216), Email: [email protected]

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For each Joint Honours degree there is a designated member of staff from each participating de-partment whom you may contact if you wish to discuss any aspect of your joint degree course. Therelevant contacts in the Department are as follows:

Joint degrees with Physics:Prof. P. R. W. Mansfield, (room CM210), Email: [email protected] degree with Chemistry:Dr D. J. Smith, (room CM231a), Email: [email protected]

We may also wish to contact you! Please keep the Mathematics Office informed of your currentterm-time residential address and e-mail address.

1.3 Course Director

Dr. C. Kearton,(room CM318), Email: [email protected] Second-Year Course Director is responsible for overseeing the academic progress of second-year students. If at any time you would like to discuss aspects of your course, or if there arequestions about the Department which this booklet leaves unanswered, please contact him.

1.4 Natural Sciences Co-ordinator

If you are a Natural Sciences student, and wish to discuss specific aspects of your programme,you may contact the liaison officer with Natural Sciences, Dr. J. Bolton, (room CM314), Email:[email protected]

1.5 Departmental Adviser

Each Single Honours Mathematics, Natural Sciences, Combined Arts and Social Sciences studentwho takes mathematics modules has a designated Departmental adviser who is a member of staffwho acts as an academic presence throughout the period of study of each advisee. The advisercontacts advisees at the beginning of each academic year to remind them of the opportunity theyhave to come and discuss any aspect of their academic life. In many cases, the adviser will be theperson to contact if you need a reference letter, you will meet your adviser at least once during theyear, when you discuss your exam results and register (if not final year) for the subsequent year.

If you need to contact your advisor and he/she is not available, please contact Dr. S. Borgan, (roomCM216), Email: [email protected]

1.6 Registration for 2H

You will register for the required number of modules in June. You may attend additional mod-ules during the first few weeks of the Michaelmas Term. If you then decide that you wantto change one or more of your modules you must see Dr. S. Borgan, (room CM216), Email:[email protected] Any such change must be completed during the first four weeks ofthe Michaelmas Term.

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1.7 Consultation with Members of Staff

If you have any questions on the subject or are experiencing difficulties with a lecture course, youshould consult the member of staff giving the course as soon as possible. The consultation may be:

- immediately after the lecture,- by calling on the lecturer concerned in his or her office,- by email.

1.8 Staff-Student Consultation

The Board of Studies has two Staff-Student Consultative Committees ((A) for Honours Mathemat-ics and Natural Sciences and (B) for Auxilliary level 1 modules) to provide an effective means ofcommunication between staff and students.

The Committees first meet in the Michaelmas Term. They have student representatives from eachyear (each module for Committee (B) and if you have issues you wish to raise, you should contactyour year (or module) representatives.

The Staff-Student Consultative Committees also seek feedback from all students on all aspectsof Mathematics courses by way of a questionnaire during the penulimate week of Michaelmasand Epiphany terms. These are considered by the lecturers concerned, the Head of Departmentand the members of the Department’s Monitoring Committee. The Staff-Student ConsultativeCommittees report to the Board of Studies, the main decision-making body of the Department, andplay an active role in promoting high quality teaching in the Department. There are also studentrepresentatives to the Board of Studies, who act in an advisory capacity and also provide directfeedback to the student body.

Summary results of the questionnaires and minutes of the Staff-Student Committees are posted onthe relevant noticeboards (first floor corridor of the Department).

If you have concerns about teaching which are not covered by these meetings and questionnaires,contact can be made directly with the Staff-Student Consultative Committee Chairmen:Dr N. Martin, (room CM304), Email: [email protected] for Committee (A).Dr. S. Borgan, (room CM216), Email: [email protected] for Committee (B).

1.9 Students with Special Needs

The University is committed to full compliance with the aims of the Special Educational Needs andDisability Act 2001. Once a student has been accepted for a course of study, the University acceptsa responsibility to ensure appropriate provision for that student throughout his/her course. Studentswith disabilities can expect to be integrated into the normal University environment. They areencouraged and helped to be responsible for their own learning and so achieve their full academicpotential.

Durham University Service for Students with Disabilities (DUSSD) aims to provide appropriatecare and support for all Durham students with a disability, dyslexia, medical or mental healthcondition which significantly affects study. DUSSD can advise you and organise special academicfacilities if you have a disability and need some help. They will try to provide whatever support isnecessary to enable you to study effectively and to make full use of your opportunities at University.This help will be specific and appropriate to you and relevant to the courses you choose.

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Special arrangements and facilities may well be required by disabled students when taking ex-aminations. These might include extra reading time or a separate quiet room and are intended tominimise the effects of disability, which are often exacerbated by examination conditions. DUSSDorganises all the requisite examination concessions for hearing-impaired, visually-impaired anddyslexic students. DUSSD also makes recommendations to departments for students with otherdisabilities who have regular support from the Service.

For further advice, or to obtain a copy of the University’s Disability Statement, please contactDurham University Service for Students with Disabilities (DUSSD), Pelaw House, Leazes Road,Durham, DH1 1TA, Tel: 0191 334 8115 (Voice and Minicom),Email: [email protected].

1.10 Illness and Absence

If you miss tutorials or fail to hand in written work because of illness you must ask your College toinform the Department. If this work is summatively assessed (i.e., counts towards your final markfor a module), you must complete a self-certification of illness form.

If your academic performance is significantly affected by illness or other difficulties at any time,you should obtain documentary evidence as described in the Teaching and Learning Handbook ofthe University of Durham.The relevant section is: 6.1.4.14 Student Absence and Illness. This is accessible on-lineunder the address www.dur.ac.uk/teachingandlearning.handbook/6-1-4-14.pdf and con-tains links to downloadable self-certification forms and requests for a doctor’s certificate.

A member of the Department is liaising with the Colleges regarding illness and absences related toillness. Feel free to contact her if you feel it might be beneficial for you to discuss matters withinthe department.

Liaison officer (Colleges - Dept of Mathematical Sciences):Mrs F. Giblin ([email protected])

If you miss tutorials or fail to hand in written work for other reasons you should contact the CourseDirector as soon as possible.

1.11 Course Information

Term time in Durham is Michaelmas (10 weeks), Epiphany (9 weeks) and Easter (9 weeks). Thereare 22 teaching weeks, and the last seven weeks are dedicated to private revision, examinationsand registration for the subsequent academic year.

Timetables giving details of places and times of your commitments are available on Departmentalweb pages and noticeboards in the first floor corridor of the Department. It is assumed that youread them!You may access your own Maths timetable at www.maths.dur.ac.uk/teaching/ and then click-ing on the ‘My Maths timetable’ link.

Also, teaching staff often send you important information by e-mail to your local ‘@dur.ac.uk’address, and so you should scan your mailbox regularly (see below). Note that in October it takestime to sort out groups for tutorials and practicals, and so these classes start in week 2.

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1.12 Computers, ICT and DUO

You are expected to use the internet — i.e., e-mail and the World-Wide Web (WWW) — andfacilities are provided by the Information Technology Service (ITS). You should take advantageof ITS instruction courses to make sure that you have a basic acquaintance with computers. Theweb-address is www.dur.ac.uk/ITS.

The Maths Department web-address is www.maths.dur.ac.uk and a valuable link is ‘Teaching’.Here besides lecture and tutorial timetables you will find material provided by lecturers. For thisthey may use ‘Durham University Online’ (DUO).

DUO is a virtual learning environment which is a collection of on-line resources including linksto web pages, lecture notes and exercise sheets/solutions, communication tools like email andassessment features such as formative quizzes. Your login area on DUO is where you can accessall on-line course materials offered by your lecturers.

Soon after your registration details have been entered onto the University’s student records system(Banner), you will automatically be enrolled by the Learning Technologies Team at the IT Serviceon the DUO courses related to the mathematics modules that you are taking. Details of how tologon to the DUO system are given at duo.dur.ac.uk and in the IT Service publication ‘Computingat Durham’. Individual lecturers will inform you about its use for their courses from time to timeduring the year. The Department will also make use of the Announcements area in DUO to passon important information to you so please get into the habit of logging in every day.

1.13 Private Study

‘An undergraduate module with effect from October 1998 is defined as a study unit comprising200 hours of SLAT (Student Learning Activity Time) per annum and lasts one academic year’(University of Durham Teaching and Learning Handbook). The total ‘contact time’ that a studentspends in lectures, tutorials, etc. amounts to around 30% of the total SLAT. You would be wise toplan how best to use the remaining 70% (140 hours for a 20-credit module, i.e., 6.4 hours per weekof a 22-week academic year per 20-credit module). Your tutor will be able to help you with this.This time is allocated within the module to be spent, not only in preparing submitted work (e.g.essays, assessed problems), but in private study of the lecture course material and in revision. Youare advised to organise your time in such a way that you are able to devote a number of hours eachweek to reviewing your lecture notes, reading around the subject and working through exercisesextra to those which have been set by the lecturer. By so doing you will be developing yourstudy and personal management skills and be giving yourself the best opportunity to gain a firmunderstanding of the topics as they unfold. By attending to any difficulties or misconceptions youhave as the course progresses you will be in an excellent position at the end of the course to makethe most of your revision time. Planning and preparation are the key to reducing examination stress.It was within this framework that the University developed PDPs (Personal Development Plans)which is a structured and supported process which will enable you to reflect on the mathematicalprocess and to discover, at a personal level, the best way for you to learn the mathematics you aretaught. To find out more visit duo.dur.ac.ukDUO.

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1.14 Books and Libraries

The main collection of mathematics books, including all the texts recommended in this booklet, ishoused in the University Library. Each College has its own library.

Here are a few books which convey the excitement of modern mathematics — a good way to spendyour book tokens! —

Timothy Gowers, Mathematics. A Short Introduction, OUP 2003.Simon Singh, Fermat’s Last Theorem, Fourth Estate 1997.Tom Körner, The Pleasures of Counting, CUP 1996.Philip J. Davis & Reuben Hersch, The Mathematical Experience, Pelican 1988.

For mathematical sites on the Internet, visit www.yahoo.co.uk/Science/Mathematics/

1.15 Student Records

Your Department record file contains some or all of the following:

-Your UCAS form,-Annual Department registration forms,-Your plagiarism forms,-Annual examination results,-Copies of letters received from you and sent to you.

Lists of student names are used in the preparation of registers for tutorials, practicals and examplesclasses and in the examinations. All such Departmental computer files are registered under theData Protection Act. Each student’s marks for all examinations and assessed work are confidentialto the members of the Board of Examiners of the Department; aggregate marks are known to themembers of the Faculty Board of Examiners, College Principals and Senior Tutors, the Examina-tions Department, and the individual student.

1.16 Smoking and Mobile Phones

Please note that (i) smoking is not allowed in the Department, and (ii) mobiles must always beswitched off in teaching rooms.

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1.17 Durham University Mathematical Society

∃MathSoc∀The society makes it possible to meet people away from lectures in an informal environment.

We arrange a variety of events to suit everyone throughout the year. These include bowling,Christmas meal, bar crawls, paintballing, the infamous quiz at the AGM and of course, thehighlight of the year is the trip to see Countdown being filmed.

MathSoc also helps to arrange guest lectures given by lecturers within the mathematical depart-ment at Durham as well as lecturers from other universities.

We have our own website at www.durham.ac.uk/mathematical.society with up to date informationon what is happening along with the second hand book list. You can contact us via our emailaddress [email protected]

TO JOIN:

See us at the freshers’ fair or send us an email.

Price:- 6 for life membership or 3.50 per year.

This year’s Exec. is:President - Steve Wilson ([email protected]) TrevelyanTreasurer - Matthew Jones ([email protected]) TrevelyanSecretary - Mat Coultas ([email protected]) - St CuthbertsSocial Secretary - Laura Cavanagh ([email protected]) - St MarysPublicity Officer - Susan Nelmes ([email protected]) - St Chads

1.18 Disclaimer

The information in this booklet is correct at the time of going to press in May 2007 . The Univer-sity, however, reserves the right to make changes without notice to regulations, programmes andsyllabuses. The most up-to-date details of all undergraduate modules can be found in the FacultyHandbook on-line at www.dur.ac.uk/faculty.handbook/.

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2 Examinations and Assessment

2.1 Regulations for B.Sc., M.Math. and M.Sci

The General Regulations for the B.Sc. and M.Math./M.Sci. degrees and the special regulations forthe courses described in this booklet are printed in Volume II of the current version of the DurhamUniversity Calendar, which is available for consultation in the main library or in the Departmentof Mathematical Sciences Office. An offprint of the B.Sc. and M.Sci./M.Math. regulations maybe obtained from the Science Faculty Office.

2.2 University Assessment Process

Full details of the University procedures for Examinations and Assessment may be found in theTeaching and Learning Handbook (www.dur.ac.uk/teachingandlearning.handbook/)

2.3 Board of Examiners

The Board of Examiners is responsible for all assessment of Mathematics courses.Chair: Prof. A. Taormina (room CM302) Email: [email protected] Chair: Dr. S. Borgan, (room CM216), Email: [email protected]: Dr. P.S. Craig (room CM317) Email: [email protected]

2.4 Plagiarism, Cheating and Collusion

Working with your fellow students is perfectly acceptable, but joint work should be declared assuch. The University has a strict policy against plagiarism and other forms of cheating, a state-ment of which may be found in the Teaching and Learning Section in Volume I of your Faculty’sUndergraduate Handbook.

Plagiarism includes• The verbatim copying of another’s work without acknowledgement.• The close paraphrasing of another’s work by simply changing a few words, or altering the orderof the presentation, without acknowledgement.• Unacknowledged quotation of phrases from another’s work.• The deliberate and detailed presentation of another’s concept as one’s own.

Cheating includes• Communication with or copying from any other student during an examination.• Communication during an examination with any person other than a properly authorised invigi-lator or another authorised member of staff.• Introducing any written or printed material into the examination room unless expressly permittedby the Board of Examiners in Mathematical Sciences or course regulations.• Introducing any electronically stored information into the examination room, unless expresslypermitted by the Board of Examiners in Mathematical Sciences or course regulations.• Gaining access to unauthorised material during or before an examination.• The provision or assistance in the provision of false evidence or knowledge or understanding inexaminations.

Collusion includes• The collaboration, without official approval, between two or more students in the preparation and

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production of work which is ultimately submitted by each in an identical, or substantially similar,form and/or represented by each to be the product of his or her individual efforts.• The unauthorised co-operation between a student and another person in the preparation andproduction of work which is presented as the student’s own.

2.5 Monitoring of Work

Under the general regulations of the University with regard to the keeping of terms, you are re-quired to complete written work to a standard satisfactory to the Chairman of the Board of Studies.In practice this means that you will be expected to hand in at least 75% of written work on time at astandard of grade C or better (see table below). To encourage this, your performance is monitoredby the course director.

Formative assessment1 of coursework occurs at all Levels, while summative continuous assessmentof coursework occurs for the auxiliary2 Level I Mathematics modules.

The purpose of formative and summative continuous assessment of coursework is to help the stu-dent at each stage of the learning process. It is designed to encourage effort all year long andprovides manageable milestones, in preparation for the summative assessment of end of year exam-inations. Course lecturers provide problems of an appropriate standard and length to the students,as well as assessment templates (model solutions) to the markers.

Each script is returned to the student with the grade written on it. The interpretation of grades is asin the table below.

The returned scripts should indicate clearly where errors and gaps in arguments occur, and the na-ture of errors. They should give brief indications as to the approach required, bearing in mind thatmodel solutions for all set problems will be provided to students by lecturers shortly after the mark-ing has occurred. The lecturer makes relevant model solutions available to students via the coursewebpage or/and Durham On-Line (DUO) shortly after they have submitted their assignments.

Remark: A grade C is deemed acceptable. D/E or a failure to hand in work is a demerit. If say 4questions of equal standard are set and 2 are answered very well and 2 are not tackled at all thenthere is close to 50% attainment, resulting in grade C.

In all cases, performance at marked written work can provide useful evidence for the Board ofExaminers if examination performance is adversely affected by illness or other circumstances.

1‘Summative’ assessment counts towards the overall mark for the module. ‘Formative’ assessment does not.2MATH1031, MATH1541, MATH1551, MATH1561, MATH1571, MATH1711

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Grade Equivalent Mark Quality

A ≥ 80% Essentially complete and correct workB 60%—79% Shows understanding,

but contains a small number of errors or gapsC 40%—59% Clear evidence of a serious attempt at the work,

showing some understanding, but with important gapsD 20%—39% Scrappy work, bare evidence of understanding

or significant work omittedE Exam info’ from the Department’s home-page.

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2.9 Examinations and Progression

Details of examination papers structure may be found by following the links ‘Teaching > Examinfo’ from the Department’s home-page.

For all degrees, you must satisfy the University rules on progression. You should refer to Sec-tion 3 of the Undergraduate Handbook Volume II-Core (Final Honours) for the precise rules forprogression from Year 2 to Year 3. Flowcharts have been designed to help you understand the im-plications of the progression regulations. They may be found at www.dur.ac.uk/faculty.handbookby following the link ‘Student survival guide > Flowchart of Progression Regulations’. You willbe eligible to proceed to the third year if you obtain at least 40% in each of your six modules. As asecond-year student, if you have failed no more than 3 modules, you have the opportunity to resitno more than 2 of those failed modules 3.

The rules to proceed to Level 3 of your current degree programme or transfer to the next Level ofan alternative degree programme (e.g. 3H Honours to 3O Ordinary or vice-versa) are summarisedin the table below. The number of modules passed refers to the minimum number which must bepassed at Level 2, after any allowed resits. Any transfer to an alternative programme is subject tohaving suitable pre-requisites to enable you to have a possible route to complete the programme.Note that in addition you require a Level 2 average of at least 50% to progress to Level 3 of aMasters programme.

Current Level Modules passed Next Level2H 6 3H2H 5 3O2O 6 3H2O 5 3O

The following summarises the minimum requirements for an undergraduate student to leave theuniversity with some award. The numbers refer to the minimum number of modules passed in totaland where applicable at specific Levels. These numbers refer to passes after any allowed resits orcompensation 4.

Award Total Additional Level requirementsCertificate 5Diploma 10 Minimum 3 modules above Level 1Ordinary degree 15 Minimum 3 modules above Level 2Bachelors Honours degree 17 A Level 1 failed module onlyMasters Honours degree 23 A Level 1 failed module only

Note that the award of an Honours degree allows one module to be failed at Level 1 but all 6modules must be passed at Levels 2 and 3, and also at Level 4 for a Masters degree.

The final degree awarded depends on your performance in each year after your first one. The2H, 3H and 4H results are weighted in the ratios 2:3:4 to produce the final mark on which thedegree classification is decided. The Department operates Scheme 1 (Mean/Weighted) for eachof its degrees. The year abroad in the B.Sc./B.A. (European Studies) degree is assessed using the

3If you failed three modules, the two resit modules are chosen with the help of the Chair of the Board of Examiners.4Compensation may occur at Level 3 in some specific circumstances summarized in the 3H/4H Booklet.

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University model B.

Exam results will be available to you on-line before a Departmental interview held early in the lastweek of term. For times and places of interviews, see Departmental noticeboards after results arepublished. The interview is an opportunity to discuss your progress and plans. It also deals withformal Registration for the third year, and cannot be missed.

Official written statements are available from the Examinations Office, Old Shire Hall, after thesecond full week of July, on payment of the appropriate fee. Students must authorise, in writing,the Chairman of the Board of Examiners if they wish their marks to be disclosed in reply to writtenrequests from outside bodies such as EPSRC and other universities.

2.10 Illness and Examinations

If your academic performance is significantly affected by circumstances beyond your control – forinstance, illness or bereavement – at any time during your programme of study, and especially inthe period leading up to or during the examination period, you might wish to bring these mitigatingcircumstances to the attention of the Board of Examiners.

The Board of Examiners has discretion to take mitigating circumstances into account when makinga final decision on a student’s progression to the next year of study or on his/her class of degree.Students must inform the Board of Examiners before they meet, using the Mitigating Circum-stances form, which can be obtained from Colleges or downloaded fromwww.durham.ac.uk/teachingandlearning.handbook/6a21.doc. Supporting evidence suchas a doctor’s certificate, or other evidence from an independent professional such as a counselloror members of DUSSD, should be submitted with the form if available and appropriate.

Students considering claiming Mitigating Circumstances are advised to read Section 6 of theTeaching and Learning Handbook of the University of Durham, accessible on-line under the ad-dress www.durham.ac.uk/teachingandlearning.handbook/index6.pdf. The relevant sec-tion is:6.1.4.14.4. Evidence for Boards of Examiners.

3 Teaching and Learning

Each level II course has fortnightly problems classes where the lecturer demonstrates solutions toproblems for the entire class. In the intervening weeks each level II course has tutorials in groupsof around fifteen students. Certain modules also have practical classes. Details of tutorial groupsand practicals will be displayed on the tutorial notice-board on the first floor of the MathematicsDepartment. Written work is set regularly for each module together with problems to be preparedfor and discussed in tutorials. All these are an integral part of the course and you are expected toattend all tutorials and hand in your attempts at all written work on time. Your solutions are markedand graded A-E, reflecting your attainment as a proportion of the total work set. Model solutions towork set are then provided for reference on the appropriate website or in the Department Library. Itis important to devote time and effort to your solutions. Attempting problems adds to the enjoymentand excitement of mathematics as well as developing your mathematical and problem-solvingskills. It also enhances your ability to explain your ideas and methods to others in writing. This isone of the most valuable transferable skills you can gain from your studies.

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www.durham.ac.uk/teachingandlearning.handbook/6a21.docwww.durham.ac.uk/teachingandlearning.handbook/index6.pdf

3.1 The Mathematics Modules and Degrees

In Durham, students take the equivalent of six modules in each year, each single module beingworth 20 credits. The flowchart at the centre of the booklet shows the pre-requisites of all modulesoffered by the Department of Mathematical Sciences to students at level 2 in October 2007 . Itshould be read in conjunction with the information in the Appendix where excluded combinationsare spelt out. Each degree programme requires you take a core of compulsory modules in year 2,but allows you to choose extra optional modules. The flexibility increases as you progress throughyour studies. Your Departmental Adviser and the relevant lecturers will be happy to discuss yourproposed choice of modules.

As a second-year Honours Mathematics student, you will have chosen one of the three degreeprogrammes in Table 1 of the Appendix. These are first degrees, eligible for local authority fundingin the normal manner. The latest possibility to change from a 3-year to a 4-year degree is by theend of the second year, but you are strongly advised to make up your mind as early as possible,as this must be negotiated with your Local Education Authority. The Department will stronglydiscourage you to change from a B.Sc. to an M.Math. degree if your Level 2 average is less than55%.

Each of the above programmes will qualify you for a wide range of employment and also for furtherstudy in programmes such as the PGCE, specialised M.Sc. courses which make use of mathematicsin other areas (Operational Research, Applied Statistics, Computation etc.) or research leading toa Ph.D. The overall 3- or 4-year structure of the degree programmes is described in your Faculty’sFirst-Year Undergraduate Handbook under ’Degree Programme Frameworks’.

3.2 Honours Natural Sciences

Natural Sciences students who wish to include mathematics modules at level II and above as partof their programme must have taken Core Mathematics A (MATH1012) during the first year. Inaddition Core Mathematics B1 (MATH1051) must be taken in year 1 or in year 2. Subject tothese constraints students may choose any mathematics module for which they satisfy the pre-requisites other than Independent Study III (MATH3161) and Communicating Mathematics III(MATH3131). Take note of the minimal requirements listed on the first page of the Appendixregarding level 3 and level 4 mathematics modules.

Example routes for progression for NS students who have taken Core Mathematics A and CoreMathematics B1 (though not Core Mathematics B2 (MATH1041)) during the first year are shownin Table 2 (Appendix).

Example routes for progression for NS students who have taken Core Mathematics A (thoughnot Core Mathematics B1 or Core Mathematics B2) during the first year are shown in Table 3(Appendix).

A student who has taken Core Mathematics A in year 1 and is unable to take Core MathematicsB1 in year 2 (because he/she needs to take two Level I modules to begin a new subject), may beallowed to continue with Linera Algebra II (MATH2021), Statistical Concepts II (MATH2041)and one of Codes and Actuarial Mathematics II (MATH2131), Codes and Geometric TopologyII (MATH2141), Probability and Actuarial Mathematics (MATH2161) and Probability and Ge-ometric Topology II (MATH2161) followed by Decision Theory III (MATH3071), OperationsResearch III (MATH3141) and Statistical Methods III (MATH3051) .

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http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH1012&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH1051&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH3161&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH3131&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH1041&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2021&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2041&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2131&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2141&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2161&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2161&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH3071&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH3141&ayear=2006http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH3051&ayear=2006

The BSc/MSci degrees in Mathematics and Physics and the MSci degree in Chemistry and Math-ematics have fixed programmes which are shown in Table 4 (Appendix).

The Natural Sciences mathematics adviser is:Dr. J. Bolton, (room CM314), Email: [email protected] .

3.3 Further Remarks on Degrees

The Master of Mathematics, the Master in Science Joint Honours in Mathematics and Physics orin Chemistry and Mathematics are particularly suitable degrees if you wish to study your subjectin greater depth, for example with the intention of doing research. They aim to provide a math-ematical education comparable with that which may be gained elsewhere in Europe, in terms ofdepth and breadth. One third of the final year will be spent on project work, allowing an in-depthstudy of a particular topic.

For all joint honours degrees, timetable clashes may prevent a free choice in your third year (andfourth year if relevant) from all available modules in each department.

3.4 Booklists and Descriptions of Courses

The following pages contain brief descriptions of the Level 2 modules available to you.Each module description is followed by a list of recommended books and a syllabus. For somemodules you are advised to buy a particular book, indicated by an asterisk; for others a choice oftitles is offered or no specific recommendation is given. There are also suggestions for preliminaryreading and some time spent on this during the summer vacation may well pay dividends in thefollowing years.

Syllabuses, timetables, handbooks, exam information, and much more may be found atwww.maths.dur.ac.uk/teaching/, or by following the link ‘teaching’ from the Department’s homepage (www.maths.dur.ac.uk). The syllabuses are intended as guides to the modules. No guaranteeis given that additional material will not be included and examined nor that all topics mentionedwill be treated.

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mailto:[email protected]://www.maths.dur.ac.uk/teaching/http://www.maths.dur.ac.uk

3.4.1 ALGEBRA AND NUMBER THEORY II – MATH2061 (38 lectures)

Dr. M. Belolipetsky/Dr H. Gangl

Term 1: The first part of the module is an introduction to the theory of rings and we look atvarious examples of rings and fields (integers, rationals, complex numbers, integers modulo n,polynomials. . .). Among the topics covered are various properties of polynomials (factorizability,for example). The study of commutative rings and fields is the basis for much of mathematics (and,consequently, several third and fourth year topics including Number Theory, Galois Theory andAlgebraic Geometry).In the last few lectures we return to the group theory begun in Core A with some revision and somenew examples, and more details on general properties of groups.Term 2: The central idea of this term is that of a group acting on, or permuting, the elements ofa set. For example, the group of rotational symmetries of a cube acts on the set of edges of thecube (each rotation permutes the edges). By exploiting this idea we can obtain results in grouptheory (a partial converse to Lagrange’s theorem), geometry (the classification of the finite rotationgroups) and combinatorics (there are 9,099 essentially different ways of labelling the faces of adodecahedron using three labels).An important related theme is that of homomorphisms, normal subgroups and quotient groups,including the description of finitely generated abelian groups as quotients of Zn. In the last fewlectures we complete the discussion of finitely generated abelian groups showing that each is iso-morphic to just one direct product of cyclic groups of a particular form.

Recommended Books*R.B.J.T. Allenby, Rings, Fields and Groups, Arnold 1991, ISBN:0-340-544-406.M.A. Armstrong, Groups and Symmetry, Springer-Verlag, Undergraduate Texts in Mathematics,1988, ISBN 0-387-96675-7.I.N. Herstein, Topics in Algebra, Wiley 1977.T.S. Blyth and E.F. Robertson, Groups, rings and fields: Algebra through practice, Book 3,CUP, ISBN:0-521-27288-2.T.S. Blyth and E.F. Robertson, Rings, fields and modules: Algebra through practice, Book 6,CUP, ISBN:0-521-27291-2.P.J. Cameron, Introduction to Algebra, Oxford University Press, 1998, ISBN 0-19-850194-3.The book by Allenby covers most of the main topics. It contains many historical references andpotted biographies of famous mathematicians which you may well find interesting. Complemen-tary material on group theory may be found in the book by Armstrong. Herstein covers simi-lar ground to Allenby but is somewhat leaner and less comprehensive. The books by Blyth andRobertson contain many examples and problems. Chapter 2 in Cameron is nicely written and veryclear.

Preliminary ReadingTerm 1: Read Chapter 1 of Allenby and, if time allows, the first half of Chapter 3. Dip into thePrologue as well. Alternatively, read Chapter 1.3-1.4 and the beginning of Chapter 2 in Cameron’sbook.Term 2: Consolidate first year group theory by reading the relevant sections in Allenby or Arm-strong (above), and look at H. Weyl, Symmetry, Princeton University Press.

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http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2061&ayear=2006

Outline of course Algebra and Number Theory II

Aim: To introduce further concepts in abstract algebra, develop their theory, and apply them tosolve problems in number theory and other areas.Term 1 (20 lectures)

Rings and Fields: Definitions and examples of rings (Z/nZ), matrix rings, quaternions, polyno-mial rings), homomorphism of rings, integral domains and fields, units of a ring. Mention linearalgebra over arbitrary fields. Polynomials over a field, gcd of polynomials (division algorithm,Euclidean algorithm), factorisation, roots, irreducibility (Gauss’s lemma, Eisenstein’s criterion).

Ideals and Quotient Rings: Ideals (e.g., kernels of ring homomorphisms), quotient rings, firstisomorphism theorem for rings, Chinese remainder theorem for rings, prime/maximal ideals andtheir characterisation in terms of quotient rings.

Examples of Groups: Cyclic and dihedral groups, rotational symmetry groups of regular polyhe-dra, matrix groups, symmetric and alternating groups.

Term 2 (18 lectures)

Operations on Groups: Direct product, product of cyclic groups. Isomorphisms between variousgroups (including groups of platonic solids), Cayley’s theorem.

Group actions: Action of a group on a set. Orbits, stabilisers and the orbit-stabiliser theorem.Cauchy’s theorem. Conjugacy, conjugacy classes in Sn. Centre, centre of a p-group, groups oforder p2. Burnside’s theorem, applications to combinatorial problems. Finite subgroups of O(2)and SO(3) (outline proofs).

Homomorphisms and Quotient Groups: Homomorphism of groups, kernel and image. Nor-mal subgroups. First isomorphism theorem for groups and applications. [Automorphisms, innerautomorphisms, InnG ≡ G/Z(G). Simple groups: An for n ≥ 5.]

Finitely generated abelian groups: Classification, uniqueness of rank and torsion coefficients.Systems of linear equations in integers, recognition of groups from a finite presentation.

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3.4.2 ANALYSIS IN MANY VARIABLES II – MATH2031 (38 lectures)

Prof. P. R. W. MansfieldThe description of most natural phenomena is based on models that involve functions of severalvariables. The highlight of this module is the study of two prominent partial differential equa-tions, namely Laplace’s equation and the heat diffusion equation. These equations relate two ormore partial derivatives of an unknown function of several variables, and play an enormous rolein science. The methods to solve these partial differential equations are cornerstones of appliedmathematics.The first quarter of the module deals with functions depending on n real variables. The mathe-matical properties and procedures are simply the natural extensions of those for the one-variablecase you are already familiar with. We then move on to study ordered triples of functions of 3real variables and their generalisations, which are called vector functions or vector fields. Vectoralgebra is so prodigiously rich in applications that it plays a crucial role in many areas of science.But vector calculus goes much beyond vector algebra. Differential and integral vector calculusopens the door to three great integration theorems: Green’s theorem, Gauss’s theorem (commonlyknown as the divergence theorem) and Stokes’s theorem. All three theorems can be cast in thesame general form: an integral over a region S is equal to a related integral over the boundary of S.Vector calculus was actually invented to provide an elegant formulation of the laws of electrostat-ics but it has applications in many other scientific contexts. When you master it, you will be fullyprepared for many courses in year 3 and 4, such as Electromagnetism, Continuum Mechanics andQuantum Mechanics.

Recommended BooksMost topics in this module are covered in:M.L. Boas, Mathematical Methods in the Physical Sciences, Wiley, 1983 (hardback), ISBN 0-471-04409-1 (paperback is only available second hand: ISBN 0-471-09960-0.We also recommend the following American textbook, which has numerous illustrations andsolved problems:Salas, Hille and Etgen, One and several variables calculus, 9th edition, Wiley, 2003 (hardback),ISBN 0-471-23120-7.A useful source of worked examples for the first and the second terms of the module, with sum-maries of the theory, are respectivelyR.C. Wrede and M.R. Spiegel, Advanced Calculus, 2nd edition, Schaum’s Outline Series, McGraw-Hill, ISBN 0-07-137567-8.F. Ayers, Differential Equations, Schaum’s Outline Series, McGraw-Hill, ISBN 0-07-09906-9.P. Duchateau and D.W. Zachmann, Partial Differential Equations, Schaum’s Outline Series,McGraw-Hill, ISBN 0-07-017897-6.Preliminary Reading: Revise partial derivatives and integration by reading e.g. chapters 4 and 5of the book by Boas, or chapters 3, 5 and Section 14.4 of the book by Salas et al.

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Outline of course Analysis in Many Variables II

Aim: To provide an understanding of calculus in more than one dimension, together with an under-standing of and facility with the methods of vector calculus. To understand the application of theseideas to a range of forms of integration and to solutions of a range of classical partial differentialequations.

Term 1 (20 lectures)Vector Algebra: Suffix notation and use for scalar and vector products.Functions on Rn: Open subsets of Rn, continuity and differentiability, gradient. Continuous par-tials imply differentiability.Functions on Rn to Rm: Differentiation, chain rule, inverse and implicit function theorems (with-out proof). Curves given by f (x,y) = c, curvature. Local diffeomorphisms, orientation and relationwith Jacobian.Vector Calculus and Integral Theorems : Div, curl, Laplacian (in Cartesian coordinates). Mul-tiple integration. Line, surface and volume integrals. Change of variables. Stokes’ and divergencetheorems. Conservative field and scalar potential.

Term 2 (18 lectures)Solution of Poisson’s and Laplace’s Equations: Uniqueness of solution of Laplace’s and Pois-son’s equations. General solution of Poisson’s equation. Green’s function. Simple examples ofsolution of Laplace’s equation by separation of variables.Fourier Transform, Heat Kernels and Green’s Functions: Fourier transform and inverse, con-volution theorem. Solution of heat equation on Rn using Fourier transform and construction ofheat kernel. Connection between heat kernel and Green’s function.

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3.4.3 Codes and Actuarial Mathematics II – MATH2131 (38 lectures)

Dr. S. K. Darwin/Prof. F. P. A. CoolenTerm 1 - Codes:Error-correcting codes form a link between abstract mathematics and very concrete applications:in essence we will do the mathematics that is behind the capability of a CD-player to cope witha scratch on a CD. But error-correcting codes are used more widely, e.g., for data transmissionover noisy channels. We start with the basics of error-correcting codes, working with vectors andmatrices with coefficients in Zp (p prime), and discuss the error-detection and error-correctioncapabilities of various codes. In order to understand the special error-correcting capabilities ofthe Reed-Solomon code used on CD’s we have to introduce finite fields and finite field extensionstowards the end.A good capability of working in Zn in essential. Also, some of the homework will be computerbased.

Recommended BooksThe course is based on ‘A First Course in Coding Theory’ by Raymond Hill (Oxford Univer-sity Press, ISBN 0198538030, about £25), but ‘Coding theory: a first course’ by San Ling andChaoping Xing (Cambridge University Press, ISBN 0521529239, about £24) also has large partsin common with it. Only the very last topic will be taken from ‘Error-Correcting Codes and FiniteFields’ by Oliver Pretzel (Oxford University Press, ISBN 0192690671, about £28), which as awhole is more substantial than the first book.Term 2 - Actuarial Mathematics:This course provides an introduction to applications of mathematics in actuarial work, focussingon aspects of life insurance. The emphasis is on mathematics of compound interest and probabilis-tic models for lifetime, including the use of life tables. As insurance is a topic of great practicalinterest, this course will be of benefit not only to those who may opt for a career in insurance, butto all students as many of the concepts introduced are quite common in every day life. Studentstaking this course should have a good understanding of basic probability theory.

Recommended BooksStudents are not required to buy a book for this course, but an excellent concise book is ‘LifeInsurance Mathematics’ by H.U. Gerber and S.H. Cox (3rd edition, 1997, about £34.50 (Ama-zon.co.uk)).

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Outline of course Codes and Actuarial Mathematics IIAim: To study two separate topics in mathematic at least one of which will demonstrate how math-ematics can be applied to real world situations.

Term 1 (20 lectures)Recall of modular arithmetic, in particular for Z/pZ with p prime; matrices over Z/pZ and matrixmultiplication. Basic linear algebra over Z/pZ (determinant, linear independence, basis, dimen-sion).Block codes, Hamming distance, error detection/correction procedure, error correction and detec-tion capabilities. Sphere packing bound, Singleton bound, notion of perfect code. Linear code,rank.Equivalence of linear codes, decoding algorithm (generator matrix, parity matrix, standard array,syndromes). Probability of undetected errors/wrong correction.Binary Hamming code (including decoding), BCH codes over Z/pZ.Z/pZ[x], division algorithm, working modulo f (x), notion of irreducible polynomial.Reed-Solomon codes over a field with 2n elements. Application: the code used on CDs (bursterror, interleaving, interpolation).Term 2 (18 lectures)Introduction to Life Insurance: What is life insurance; the role of mathematics, in particularprobability and statistics.The Mathematics of Compound Interest: Effective and nominal interest rates; continuous pay-ments; perpetuities; annuities; repayment of a debt.The Future Lifetime: Models and notation; force of mortality; analytical distributions; curtatefuture lifetime; life tables.Life Insurance: Elementary insurance types, including whole life and term insurance and endow-ments; more general types of life insurance.Life Annuities: Elementary life annuities; variable life annuities.Net Premiums: Random total loss to insurer; equivalence principle; net premiums for elementaryforms of insurance; exponential utility.Further Topics: There is a wide variety of possible further topics, a selection of which can be in-cluded. Examples are: stochastic interest; net premium reserves; multiple decrements; multiple lifeinsurance; portfolios; expense loading; estimating probabilities of death; commutation functions,family income insurance.

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3.4.4 Codes and Geometric Topology II – MATH2141 (38 lectures)

Dr. S. K. Darwin/Dr D. SchuetzTerm 1 - Codes:Error-correcting codes form a link between abstract mathematics and very concrete applications:in essence we will do the mathematics that is behind the capability of a CD-player to cope witha scratch on a CD. But error-correcting codes are used more widely, e.g., for data transmissionover noisy channels. We start with the basics of error-correcting codes, working with vectors andmatrices with coefficients in Zp (p prime), and discuss the error-detection and error-correctioncapabilities of various codes. In order to understand the special error-correcting capabilities ofthe Reed-Solomon code used on CD’s we have to introduce finite fields and finite field extensionstowards the end.A good capability of working in Zn in essential. Also, some of the homework will be computerbased.

Recommended BooksThe course is based on ‘A First Course in Coding Theory’ by Raymond Hill (Oxford Univer-sity Press, ISBN 0198538030, about £25), but ‘Coding theory: a first course’ by San Ling andChaoping Xing (Cambridge University Press, ISBN 0521529239, about £24) also has large partsin common with it. Only the very last topic will be taken from ‘Error-Correcting Codes and FiniteFields’ by Oliver Pretzel (Oxford University Press, ISBN 0192690671, about £28), which as awhole is more substantial than the first book.Term 2 - Geometric Topology:This course gives an introduction to topology in an intuitive, visual way by studying mainly knotsand links. Apart from everyday life, where knots are used to tie ropes and shoelaces, this subjecthas applications in many sciences (for instance in quantum physics or molecular biology) and invarious branches of mathematics. Knots and links give rise to exciting geometric objects and thecourse provides tools to study them. Starting with combinatorial moves we introduce sophisticatedinvariants, like the Conway- and Jones-polynomials, which are effectively computable and carryimportant topological information.Arguments will often be based on pictures and constructions in ‘rubber sheet geometry’ in orderto develop a geometric intuition valuable for many further courses.

Recommended BooksThe course is based on the book ‘The Knot Book’ by C. Adams (American Mathematical So-ciety, Providence, ISBN 0821836781). Other books which cover the material are ‘Knots andSurfaces’ by N.D. Gilbert and T. Porter (Oxford University Press, ISBN 0198533977) and ‘Knotsand Links’ by P. Cromwell (Cambridge University Press, ISBN 0521839475) and ‘On Knots’ byL. Kauffman (Princeton University Press, ISBN 0691084351).

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Outline of course Codes and Geometric Topology IIAim: To study two separate topics in mathematic at least one of which will demonstrate how math-ematics can be applied to real world situations.

Term 1 (20 lectures)Recall of modular arithmetic, in particular for Z/pZ with p prime; matrices over Z/pZ and matrixmultiplication. Basic linear algebra over Z/pZ (determinant, linear independence, basis, dimen-sion).Block codes, Hamming distance, error detection/correction procedure, error correction and detec-tion capabilities. Sphere packing bound, Singleton bound, notion of perfect code. Linear code,rank.Equivalence of linear codes, decoding algorithm (generator matrix, parity matrix, standard array,syndromes). Probability of undetected errors/wrong correction.Binary Hamming code (including decoding), BCH codes over Z/pZ.Z/pZ[x], division algorithm, working modulo f (x), notion of irreducible polynomial.Reed-Solomon codes over a field with 2n elements. Application: the code used on CDs (bursterror, interleaving, interpolation).Term 2 (18 lectures)Work entirely in Euclidean space. Open sets, continuity, homeomorphism, very brief discussion ofcompactness and connectedness.

Knots, equivalence, knot diagram, Reidemeister moves, 3-colouring, spanning surface, genus,knots cannot cancel, Jones polynomial, trefoil not isotopic to mirror image, alternating knots.Surfaces, triangulation surface symbols, classification via scissors and paste, Euler characteristic,classification via surgery, topological invariance of Euler characteristic.

Plane vector fields, index round a closed curve, non-zero index means field vanishes somewhereinside curve. Brouwer Fixed Point Theorem. Degree of map of circle into punctured plane, vectorfields on surfaces, Poincare-Hopf theorem.

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3.4.5 COMPLEX ANALYSIS II – MATH2011 (38 lectures)

Dr J. Bolton/Dr. F. Tari

Term1: The module develops the theory of functions of a single complex variable. By this timeyou will have already met several examples of complex functions such as polynomials in z,ez,coszand sinz. Functions of a complex variable behave in a very different way from their real coun-terparts. For example, a complex function which is differentiable once everywhere is differen-tiable any number of times; even more striking, if f (z) is differentiable everywhere and bounded,then it must be constant. In this term we shall investigate the properties of complex differen-tiable functions, show how to integrate them along curves in the complex plane, and apply ourresults to produce simple algorithms for the evaluation of real integrals of the form

R ∞−∞ f (x)dx

andR 2π

0 R(cosθ,sinθ)dθ where R is a rational function.

Term 2: In this term, the course covers conformal mappings and other applications of complexfunction theory. A conformal mapping is a mapping in the real plane that preserves angles betweencurves. They turn out to be represented by complex analytic functions if one views the real planeas complex numbers. After that we give further applications of complex analysis. This is followedby a discussion of uniform convergence.

Recommended BooksH.A. Priestley, Introduction to Complex Analysis, OUP, ISBN 0-198-525621.E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable, OUP.L. Ahlfors, Complex Analysis, Tata McGraw Hill.G.J.O. Jameson, A First Course on Complex Functions, Chapman and Hall.D.O. Tall, Functions of a Complex Variable, (Parts I & II), Routledge and Kegan Paul.M.R. Spiegel, Complex Variables, Schaum (for worked examples).

Preliminary ReadingFind out as much as you can about(i) complex functions(ii) the Cauchy-Riemann equations(iii) Cauchy’s theoremby reading the relevant sections in one of the above references.

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Outline of course Complex Analysis II

Aim: To introduce the student to the theory of complex analysis.

Term 1 (20 lectures)Complex Differentiation: Complex functions, differentiation, Cauchy-Riemann equations. Ele-mentary functions: ez, logz, zα, cosz, coshz, etc. Cuts and branches.Power Series: Circle of convergence, term by term differentiation, Taylor series. Application topower series of ez, sinz etc.Contour Integrals: Curves in the complex plane. Mention of Jordan curve theorem, simply con-nected domain. Contours, integral of complex valued functions along a contour. Cauchy’s theorem.Morera’s theorem. Cauchy’s integral formula, Taylor’s theorem. Singularities, Laurent’s theorem.Residue theorem.Calculus of Residues: Evaluation of integrals (and series) by calculus of residues. Jordan’slemma.

Term 2 (18 lectures)Conformal Mappings: Angles between arcs, conformal mappings and Cauchy-Riemann equa-tions. Examples of conformal mappings, e.g. zα, coshz, ez etc. Group of Möbius transformations,point at infinity, Riemann sphere. Three points determine a Möbius transformation. Möbius trans-formations send circles to circles, preservation of cross ratio. Möbius transformations of unit circleto itself. Statement of Riemann mapping theorem illustrated by mappings of half-disc and quarter-disc to the unit disc. Application to finding solutions of Laplace’s equation in two dimensions withgiven boundary conditions (with mention of fluid flow, electrostatic potential, heat flow)Applications: Principle of the argument. Application to the conformal mapping of a simple closedcontour. Rouche’s theorem. Liouville’s theorem, Casorati-Weierstrass theorem. Fundamentaltheorem of algebra, partial fraction theorem. Uniqueness of analytic continuation.Uniform Convergence: Motivation for uniform convergence (simple counterexamples). Point-wise and uniform convergence of sequences. Continuity of the limit. Term-by-term integrationand differentiation of sequences. Uniform convergence of series. Weierstrass M-test.

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3.4.6 CONTOURS AND ACTUARIAL MATHEMATICS II – MATH2171 (38 lectures)

Dr J. Bolton/Prof F.P.A. CoolenTerm 1 - Basic Complex Analysis:The module develops the theory of functions of a single complex variable. By this time you willhave already met several examples of complex functions such as polynomials in z,ez,cosz and sinz.Functions of a complex variable behave in a very different way from their real counterparts. Forexample, a complex function which is differentiable once everywhere is differentiable any numberof times; even more striking, if f (z) is differentiable everywhere and bounded, then it must be con-stant. In this module we shall investigate the properties of complex differentiable functions, showhow to integrate them along curves in the complex plane, and apply our results to produce simplealgorithms for the evaluation of real integrals of the form

R ∞−∞ f (x)dx and

R 2π0 R(cosθ,sinθ)dθ

where R is a rational function.

Recommended BooksFor term 1, see the references given for the module ‘Complex Analysis II’.Term 2 - Actuarial Mathematics:This course provides an introduction to applications of mathematics in actuarial work, focussingon aspects of life insurance. The emphasis is on mathematics of compound interest and probabilis-tic models for lifetime, including the use of life tables. As insurance is a topic of great practicalinterest, this course will be of benefit not only to those who may opt for a career in insurance, butto all students as many of the concepts introduced are quite common in every day life. Studentstaking this course should have a good understanding of basic probability theory.


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Outline of course Contours and Actuarial Mathematics II

Aim: To study two separate topics in mathematic at least one of which will demonstrate how math-ematics can be applied to real world situations.

Term 1 (20 lectures) Complex Differentiation: Complex functions, differentiation, Cauchy-Riemannequations. Elementary functions: ez, logz, zα, cosz, coshz, etc. Cuts and branches.Power Series: Circle of convergence, term by term differentiation, Taylor series. Application topower series of ez, sinz etc.Contour Integrals: Curves in the complex plane. Mention of Jordan curve theorem, simply con-nected domain. Contours, integral of complex valued functions along a contour. Cauchy’s theorem.Morera’s theorem. Cauchy’s integral formula, Taylor’s theorem. Singularities, Laurent’s theorem.Residue theorem.Calculus of Residues: Evaluation of integrals (and series) by calculus of residues. Jordan’slemma.

Term 2 (18 lectures)Introduction to Life Insurance: What is life insurance; the role of mathematics, in particularprobability and statistics.The Mathematics of Compound Interest: Effective and nominal interest rates; continuous pay-ments; perpetuities; annuities; repayment of a debt.The Future Lifetime: Models and notation; force of mortality; analytical distributions; curtatefuture lifetime; life tables.Life Insurance: Elementary insurance types, including whole life and term insurance and endow-ments; more general types of life insurance.Life Annuities: Elementary life annuities; variable life annuities.Net Premiums: Random total loss to insurer; equivalence principle; net premiums for elementaryforms of insurance; exponential utility.Further Topics: There is a wide variety of possible further topics, a selection of which can be in-cluded. Examples are: stochastic interest; net premium reserves; multiple decrements; multiple lifeinsurance; portfolios; expense loading; estimating probabilities of death; commutation functions,family income insurance.

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3.4.7 LINEAR ALGEBRA II – MATH2021 (38 lectures)

Dr C. Kearton/Dr. J. R. Parker

Term 1: This term we shall generalise the notion of vectors in R2 and R3 that you met in thefirst year. We shall begin by considering Rn, consisting of all n-tuples (x1,x2, . . . ,xn) with xi ∈ R.These can be added, or multiplied by a real number, just as elements of R3 can. We shall go onto list the axioms for a vector space over a field and analyse the basic properties of vector spaces.We shall define linear maps, those maps from one vector space to another which respect additionand scalar multiplication, and check their properties. Also, we shall see how these ideas can berepresented in terms of coordinates by the use of matrices. Finally we shall look at eigenvectorsand their properties and applications.

Recommended BooksThere are many texts on linear algebra. The following book contains a good account.S. Lipschutz, M. Lipson Linear Algebra, 3rd edition, Schaum’s Outlines, McGraw Hill, ISBN0071362002.

Preliminary ReadingChapters 4, 5 and 6 of Lipschutz and Lipson.

Term 2: In this term we considerably expand the area of applicability of ideas of linear algebra byintroducing new notions related to the classical concepts of “length” and “angle”. These conceptshave been already treated via the technique of dot product in the first year. There follows the studyof arbitrary symmetric bilinear and quadratic forms and the problem of finding a special basis of alinear space where the matrix of a given linear operator has the simplest form. This technique willbe generalised in the context of arbitrary linear spaces and will be then illustrated by the study ofthe famous systems of orthogonal polynomials: the Legendre, Chebyshev, Laguerre and Hermitepolynomials and their use to solve second order differential equations.

Recommended BooksS. Lipschutz, M. Lipson Linear Algebra, 3rd edition, Schaum’s Outlines, McGraw Hill, ISBN0071362002 – this book describes the basic ideas that you probably remember from the first term.Howard Anton & Robert C. Busby, Contemporary Linear Algebra, Wiley. ISBN 0-471-16362-7

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Outline of course Linear Algebra II

Aim: To develop Linear Algebra (vector spaces, inner product spaces) providing a foundation forstudy in a number of other areas.

Term 1 (20 lectures)Vector Spaces: Generalisation from R2 and R3. Axioms [over a general field]. Examples. Sub-spaces, linear independence, spanning sets, bases. Dimension and independence of basis. Coordi-nates with respect to a basis. Intersections and sums of subspaces. Direct sum.Linear Mappings: Definition. Image and kernel. Isomorphisms, monomorphisms, epimorphisms,automorphisms, endomorhpisms. Rank + nullity = dimension. Representation by matrices [sum-

mation convention]. Composition of linear maps. Change of basis. Canonical form(

Ir 00 0

)for

maps of one space to another. Determinant. Adjugate matrix. Row rank = column rank. Vectorspace of linear maps between two spaces (Hom(V,W )). Dual space V ∗, dual basis. Canonicalisomorphism of V ∗∗ with V .Eigenvalues and Eigenspaces: Eigenvalues and eigenvectors. Characteristic polynomial. Similarmatrices. Eigenspaces, diagonalisation of a diagonalisable matrix by change of basis. Solutionsof first order linear differential equations x′ = Ax where A is diagonalisable. Cayley-HamiltonTheorem. Symmetric matrices, orthogonal diagonalisation. Rotations in R3 have an axis.Term 2 (18 lectures)Inner Product Spaces: Real inner product spaces. Cauchy-Schwarz inequality. Orthogonal-ity, orthogonal subspaces, orthogonal complement, orthogonal projection. Orthonormal bases,Gram-Schmidt othogonalisation process. Linear isometries. Complex inner product spaces. Com-plexification of real inner product spaces and linear operators. Hermitian and unitary operators.Skew-symmetric operators.Sturm-Liouville Theory: Trigonometric polynomials as eigenfunctions of d2/dx2. Self-adjointnessof Sturm-Liouville operators. Realisation of special (Legendre, Laguerre, Hermite, Chebyshev)systems of orthogonal functions as eigenfunctions of such operators.Forms: Symmetric bilinear forms and quadratic forms. The Sylvester law. Classification of centralquadrics by signature.Jordan Normal Forms : Endomorphisms. Eigenvalues and eigenvectors revisited. Invariantsubspaces. Jordan normal blocks.

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3.4.8 MATHEMATICAL PHYSICS II – MATH2071 (38 lectures)

Prof. P.M. Sutcliffe /Dr P. BowcockThe concept of symmetry is the basis of many modern developments in fundamental physics. Thismodule explores the power of symmetry in Special Relativity and revisits Classical Dynamicsusing the Lagrangian/Hamiltonian formalism, which is designed with symmetry in mind. It pro-vides the necessary foundations for a sound understanding of recent developments in MathematicalPhysics where more abstract symmetries are encountered in Nature’s modelling.Term 1: The astounding physical consequences of Special Relativity are worked out on Einstein’sprofound appreciation of the power of symmetry: mass is equivalent to energy and time is marriedto space. We show how many concepts based on common sense must be jettisoned in SpecialRelativity, like the absolute character of time, which is deeply rooted in us. We introduce themathematical setting appropriate for the description of the kinematics and the dynamics of SpecialRelativity, and make an incursion in the world of 4-dimensional vectors and tensors. Once we havedone this, it is easy to see for instance that the famous equation E = mc2 is just the consequenceof the invariant length of a vector in four dimensions. The term concludes with a brief discussionof collision processes among elementary particles.Recommended BooksA. P. French, Special Relativity, Stanley Thornes 1971, ISBN 0-412-34320-7, £27.25; W. Rindler,Introduction to Special Relativity, 2nd edition, Oxford 1991,IBSN 0-19-853952-5, £17.95; N.M.J.Woodhouse, Special Relativity, Springer 1992, ISBN 3-540-55049-6 £23.00.

Term 2: In the first part, we make use of the calculus of variations and Hamilton’s Principleto set up an elegant and powerful formulation of classical mechanics. This allows us to quicklyanalyse the motion of a wide range of systems, which would appear to be complicated by usingthe more Newtonian ‘forces-on-a-body’ approach. This formulation also allows us to identify thesymmetries of a system quite readily, and we discover the link between symmetries and moreadvanced algebraic approaches like group theory through the concept of ‘Poisson Brackets’. Otherimportant applications we consider are small oscillations of systems with many degrees of freedom.This naturally leads us to the second part, where we study a familiar system of an oscillator with aninfinite number of degrees of freedom, a stretched string. We therefore revisit the wave equation,using our new language, and study some of its properties and related physical phenomena such asdispersion, reflection, refraction, and transmission of energy.Recommended BooksT.W.B. Kibble and F.H. Berkshire, Classical Mechanics, Longman (4th ed) (1996, ISBN 0-582-25972-X, £30.99; J. Marion, Classical Dynamics of Particles and Systems, Holt, Rinehart andWinston Inc. 1995, ISBN 0-030-97302-3, £28.95; H. Goldstein, Classical Mechanics, AddisonWesley (2nd ed) 1980, ISBN 0-201-02918-9; £36.99; L.D. Landau and E.M. Lifschitz, MechanicsPergamon 1991, ISBN 0-08-029141-4.Preliminary Reading: The introductory chapter and Appendix A of the book by Kibble andrelevant chapters of your favourite physics textbook.

32


Outline of course Mathematical Physics II

Aim: To appreciate the conceptual framework of classical physics, both discrete and continuous,as well as the mathematical foundation of Special Relativity.

Term 1. Special Relativity (20 lectures)Relativity of simultaneity, time and space: Inertial frames. Speed of light. Events. Spacetime.Time dilation and length contraction. Applications and false paradoxes.Kinematics: Lorentz transformations. Standard Lorentz boosts. Composition of velocities. Dopplereffect. Group structure of standard Lorentz boosts.Four vectors: Minkowski spacetime. Lorentz and Poincaré groups. Worldline and light cone.Causality. Proper time, velocity, acceleration. Spacetime vectors and tensors.Dynamics: Fundamental equation. Mass-energy equivalence. Einstein’s relation. Zero massparticles.Systems of free particles: Conservation of 4-momentum. Centre of mass frame. Collision pro-cesses.

Term 2 (18 lectures)Lagrangian and Hamiltonian Dynamics: Hamilton’s principle in dynamics. Generalised coordi-nates and momenta. Derivation of Lagrange’s equations. Generalised forces. Conservative forces.Ignorable coordinates and conservation laws. Noether’s theorem. Systems with constraints. Im-pulses. Hamilton’s equations. Formulation in terms of Poisson brackets.Small oscillations of systems of particles: Positions of equilibrium and stability. Normal modesof oscillation and normal coordinates. Stationary properties of frequencies of systems with con-straints.Waves: Review one-dimensional wave equation. Energy, energy density, energy carried by wave.Boundaries and junctions: reflection of a monochromatic wave at a fixed boundary, reflection andtransmission at a junction, energy flow. Wave equation in two or more dimensions. Reflection andrefraction at a boundary.

33

3.4.9 NUMERICAL ANALYSIS II – MATH2051 (38 lectures)

Dr. J.F. Blowey/Dr M. Jensen

It is easy to find the roots of the equation x2 − 3x + 2 = 0. We also think of the evaluation ofR 10 sinxdx as an easy problem because we can express it as 1− cos 1, and a calculator can give a

numerical value, to a certain accuracy. Such simplicity is unusual. For example, it is not possibleto express the integral

R 10

sin xx dx in terms of a finite number of elementary functions, and there is

no simple formula for the root of the equation x− cosx = 0; numerical methods allow us to solvethese, and other more interesting problems, to any required accuracy.The ability to compute numerical answers to mathematical problems has always been an importantpart of mathematics. For example, an effective method for evaluating square roots was discoveredmore than 3700 years ago. Now numerical methods are more important than ever because math-ematical models are widely used in science, engineering, finance and other areas, to formulatetheories, to interpret data and to make predictions. Indeed, modern areas of mathematics such asMathematical Biology hinge on numerical techniques. While one has to develop a realistic modelconsistent with the Biology, the equations which arise are rarely solvable and without numericaltechniques are of very little use. Numerical Analysis is concerned with the development of nu-merical methods, and the “Analysis” in the title refers to the study of the accuracy, reliability andefficiency of the resulting algorithms.Numerical Analysis shares some of the attractions of both pure and applied mathematics. For itsderivations and analysis it draws on many areas of pure mathematics, yet its objective is practical— to produce reliable numerical approximations, and to do so efficiently. Practical experience ofnumerical computation is essential for a full understanding of the successes and failures of par-ticular numerical methods. The practical sessions for this course form part of the assessment andare designed to allow you to experiment with a variety of numerical methods through the softwarepackage Maple. The computer will do all the calculations for you, leaving you free to investigateand to use your knowledge of the theory to explain and interpret the numerical results.

Recommended Books*R.L. Burden and J.D. Faires, Numerical Analysis (Brooks Cole)W. Cheney and D. Kincaid, Numerical Mathematics and Computing (Brooks Cole)P. Henrici, Essentials of Numerical Analysis (Wiley)R. Plato, Concise Numerical Mathematics (AMS)J. Stoer and R. Burlisch, Introduction to Numerical Analysis (Springer)E. Süli and D. Mayers, An Introduction to Numerical Analysis (CUP)All of the recommended books cover the necessary material for this module. Burden and Faireswill be used extensively in classes and covers the majority of material for future numerical anal-ysis modules. It is less mathematical than Stoer and Burlisch but is more mathematical than theexcellent book by Cheney and Kincaid.Preliminary Reading: Burden and Faires, Chapter 2.

34

http://www.maths.dur.ac.uk/php/ug.php3?module_code=MATH2051&ayear=2006http://library.dur.ac.uk/search/i0534392016http://www.as.ysu.edu/~faires/Numerical-Analysis/http://library.dur.ac.uk/search/i0534351840http://library.dur.ac.uk/search/i0471059048http://www.ams.org/bookpages/gsm-57/http://library.dur.ac.uk/search/i038795452xhttp://library.dur.ac.uk/record=b1918126a

Outline of course Numerical Analysis IIAim: To introduce the basic framework of the subject, enabling the student to solve a variety ofproblems and laying the foundation for further investigation of particular areas in the Levels 3 and4.Term 1 (20 lectures)Introduction (1): The need for numerical methods. Statement of some problems which can besolved by techniques described in this course. What is Numerical Analysis?Linear Equations (7): Triangular systems. Positive definite matrices, Cholesky factorisation.Gaussian elimination, pivoting and row scaling, LU factorisation. Operation counts. Vector andmatrix norms. Condition numbers.Errors (2): Rounding error and truncation error. Computer arithmetic. Loss of significance. Well-conditioned and ill-conditioned problems.Non-Linear Equations (6): Bracketing and bisection. Fixed-point theorem and convergence offixed-point iteration. Error analysis and order. The Newton-Raphson formula. Aitken’s method.Sets of non-linear equations. Newton iteration, without analysis of convergence.Term 2 (18 lectures)Polynomial Interpolation (8): The Lagrange form with truncation error. Uniqueness. Divided dif-ferences, interpolating polynomial in Newton’s form. Forward and backward differences (briefly).Chebyshev polynomials - definition, recurrence relation, zeros and extrema, minimax approxima-tion property. Choice of interpolation nodes. Runge’s example of divergence of an interpolationsequence. Inverse interpolation. Hermite interpolation.Least Squares Approximation (6): Discrete least squares approximation, normal equations, somenon-linear problems reduced to linear form. Ill-condition of normal equations and how to avoid it.Continuous least squares approximation, weight functions. Orthogonal polynomials, particularlyChebyshev and Legendre polynomials.Numerical Differentiation (1): Finite difference approximations and their truncation errors. Richard-son extrapolation. Effects of rounding errors.Numerical Integration (7): Interpolatory formulae, recalling examples from first-year. TheNewton-Cotes formulae. Local and global truncation errors. Adaptive methods. Statement ofthe Euler-MacLaurin formula, Romberg’s method, cases where the trapezium rule is very success-ful, approximate evaluation of coefficients of Chebyshev series. Formulae using derivative values.Gaussian formulae. Methods for improper integrals.Practical Sessions: Each student will have a weekly one-hour practical session in a computerclassroom. The Maple package will be used to implement the numerical methods introduced in thelectures.

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3.4.10 Probability and Actuarial Mathematics II – MATH2161 (38 lectures)

Dr O. Hryniv/Prof. F. P. A. CoolenTerm 1 - Probability:Modern probability theory began around 1930 with A.N.Kolmogorov, and continues to flourish.The theory has many applications, probabilistic reasoning being essential to statistics, or whenevera model for an unpredictable future is needed (in physics, chemistry, financial and actuarial math-ematics etc). At the same time, the subject appeals to pure mathematicians with a taste for realanalysis.In this course we shall cover topics such as Markov Chains, random walks, generating functions,elements of convergence and integration.Recommended BooksThe course is based on ‘Probability and Random Processes’ 3rd Edn by Geoffrey Grimmett,David Stirzaker, OUP (2001). Price about £28.50 from Amazon (This very large book is also therecommended text for the 3H/4H courses Probability and Stochastic Processes.)The book ‘Markov Chains’ by James R. Norris, published by CUP (1998) extends and comple-ments the Markov Chain part of the course.Term 2 - Actuarial Mathematics:This course provides an introduction to applications of mathematics in actuarial work, focussingon aspects of life insurance. The emphasis is on mathematics of compound interest and probabilis-tic models for lifetime, including the use of life tables. As insurance is a topic of great practicalinterest, this course will be of benefit not only to those who may opt for a career in insurance, butto all students as many of the concepts introduced are quite common in every day life. Studentstaking this course should have a good understanding of basic probability theory.


36

Outline of course Probability and Actuarial Mathematics IIAim: To study two separate topics in mathematic at least one of which will demonstrate how math-ematics can be applied to real world situations.

Term 1 (20 lectures)Introduction to probability, revision of Core A Probability;Markov chains – Markov property, classification of states, convergence to equilibrium;Random walks, hitting times, infinite collections of events, Borel-Cantelli lemmas;Real and complex generating functions with simple applications;Introduction to convergence and integration – definition of integral via simple functions, statementof limit theorems, introduction to convergence in function spaces and to Lp spaces (mostly p = 1,2);Properties of generating functions and applications (some of - continuity and inversion theorems,weak forms of LLN & CLT, random walks and

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DURHAM UNIVERSITYmaths.dur.ac.uk/Ug/coursebooks/2H/2007_2008/booklet.pdf · 2009. 8. 30. · DURHAM UNIVERSITY Department of Mathematical Sciences Second-Year Mathematics 2007-2008