Most recent update: 21 August 2014
Department of Applied Mathematics and Theoretical Physics 
Department of Pure Mathematics  

Phone  (028) 9097 6001  Phone  (028) 9097 6003 
Fax  (028) 9097 6061  Fax  (028) 9097 6060 
applied.maths@qub.ac.uk  pure.maths@qub.ac.uk 
School of Mathematics and Physics
Queen's University Belfast
University Road
Belfast BT7 1NN
This handbook is intended to provide students with information specifically on the Mathematics departments and degree programmes. It is intended to supplement School and University publications, rather than to replace them. In the event of any conflict between the contents of this handbook and School or University publications (e.g., the it is likely that we have got it wrong!
We hope that you will have an enjoyable, as well as a productive, time during your stay with us. We do all we can to provide you with the best mathematical education available. We succeed in this fairly well, as is evidenced by the 'Excellent' rating (scoring 22 out of a possible 24) that Mathematics in Queen's University Belfast (QUB) received from the QAA Subject Review in November 1999 and by the very favorable findings of the internal University Subject Review in 2005 and Education Enhancement Process in 2013. In particular, a number of Mathematics modules (Mathematical Investigations, Computer Algebra, Financial Mathematics and Mathematical Modelling in Biology and Medicine) have been picked out as examples of good practice.
For administrative purposes, there are actually two Mathematics departments at Queen's: the Department of Applied Mathematics and Theoretical Physics (which is responsible for those study modules that are coded AMA (plus a numerical identifier) and also for Statistics and Operational Research modules coded SOR), and the Department of Pure Mathematics (whose modules are coded PMA plus numerical identifier). Almost all of you will take modules from both departments initially, but will be able to specialise in later years.
Mathematics is located in the David Bates Building (DBB), which was completed in 2007 and is to the east of the McLay Library. The DBB houses most of the staff offices and some small and mediumsized teaching rooms. Applied Mathematics and Theoretical Physics and Statistics & Operational Research generally occupy the ground floor and the first floor; Pure Mathematics is on the second floor. Some of your maths classes, however, will take place in other buildings around the University.
There are notice boards in the Applied Mathematics Department (ground floor and first floor, David Bates Building) and in the Pure Mathematics Department (second floor, David Bates Building). Areas are reserved for University notices, others for departmental notices. Students are advised to consult these notice boards at regular intervals in order not to miss information that may be important to them.
A large amount of information concerning the Mathematics departments and the School of Mathematics and Physics is available online. You should be familiar with the School of Mathematics and Physics website http://www.qub.ac.uk/mp, from which the Education link can be followed.
Upon acceptance into Queen's, each student is assigned an Adviser of Studies (AoS) and a Personal Tutor (PT).
The primary role of your Advisor of Studies is to help you with module choices and to provide guidance in academic matters, e.g., requirements of your degree programme, progression between stages, resit examinations, change of programme, permissions of absence during term time, temporary withdrawal, etc. In particular, your choice of modules, as well as any changes, must always be approved by your AoS (see this Section). Students should meet with their AoS at least once a year at enrolment before the start of the academic year.
The primary role of Personal Tutors is to support students in their transition from School or other previous occupation to Higher Education. During your first year, weekly personal tutor meetings will be held throughout term. The aim of these meetings is to provide regular contact with a member of staff in a smallgroup setting. While these meetings will be centred on mathematical topics of immediate relevance to your firstyear Maths courses, feel free to raise any other issues with your PT, either during the meetings, or by contacting your PT in person or by email. For most students their PT will remain the same throught their studies, being a familiar friendly face and an important point of contact to you. (See also this Section for some general University guidance on PT.)
Students experiencing any academic difficulty should consult their Adviser of Studies or Personal Tutor at an early stage.
Your Queen's email is the official means of communication between us and you. Important information is sent to you in this way, including details of module changes and examination arrangements. It is your responsibility to check your university email account regularly and frequently  preferably on a daily basis.
When using email to contact your Adviser of Studies, Personal Tutor or lecturers, always address the person you are writing to (e.g., Dear Dr Blanco ..., Dear Prof Williams ..., Hello Gleb ..., etc.). Even more importantly, always sign your email (e.g., With best regards, John) and provide your student number. Use meaningful Subject. Be concise. Always remain courteous and polite. Please remember that in the University context email is official and formal (though often friendly), making it different from texting (which is often informal, both in style and grammatically).
The University's collection of mathematics texts is housed in the McClay Library adjacent to the David Bates Building. Although many Mathematics modules are very selfcontained compared with other University subjects, your lecturers will usually identify some textbooks as recommended or supplementary reading, with additional exercises and so on, and students should take such opportunities of developing their research and scholarship skills. Check the Library webpage for access to the catalogue and various online resources. The Library Catalogue 'QCat', can be also accessed from http://qulms.qub.ac.uk/search~S3, where past exam papers for many individual modules can be found. Information on individual modules is normally provided by the lecturers through Queen's Online (QoL).
A useful entry point to the Library resources is the Mathematics & Physics LibGuide page. Our Subject Librarian is Carol Dunlop.
All members of the Mathematics departments are expected to conduct themselves in a courteous, considerate and honest manner in all dealings with students. They are expected to be aware of their obligations and responsibilities towards their students and to meet them on time.
Students are equally expected to take a full part in their studies by attending all classes and by undertaking and handing in all the work that is set for them. The European Community believes that a fulltime student needs to devote around 40 hours per week to their degree programme, including class contact hours. (Of course, coming up to examinations, it may well be a bit more!) Given that our modules usually have 4 to 6 contact hours per week, students should be prepared to be working an additional 89 hours per week on each of three modules. This should include plenty of time to make a serious attempt at each homework. Recent studies in QUB show that examination results are closely related to the effort that is put into each homework, rather than to the mark actually attained on it. In order to allow themselves time for this work, students are expected to limit any paid employment during termtime to an absolute maximum of 12 hours per week, which should be at hours which do not interfere with their ability to attend lectures, tutorials, etc.
Students are also expected to contribute to the success of classes by keeping noise levels to a minimum: for example, by turning off mobile phones, by not eating during classes (especially crisps!) and by talking only when this is part of the educational activity of the class. Whilst occasional lateness in reaching classes may be inevitable, if you do find yourself arriving late please keep the disruption that you cause to a minimum by taking the nearest available seat.
Users of the Mathematics building are expected to make all efforts to keep it clean and hazardfree: for example, by dumping litter only in rubbish bins, by draining halffull coke cans down a sink before binning them, by keeping exits clear of obstruction and by not jamming firedoors open. Where individual students have been given an access code or card to let them into the building after hours or to work in particular rooms, the code/card is to be kept exclusively to that individual and not passed around to others; codes/cards that are abused in this fashion will be cancelled in order to maintain the security of the building.
“Students are expected to attend lectures for the modules for which they are enrolled. Failure to attend will inhibit your learning and therefore have a negative effect on assessment results. Attendance at lectures may be monitored and poor attendance could result in being interviewed by senior academic staff.”
Under our own Code of Conduct you will have seen that we go further and expect you not only to attend all classes associated with a given module but also to attempt all work set for it, even if it does not form part of the final assessment.
The University recognizes that occasionally you will be unable to carry out tasks that are expected of you, through illness, family commitments, job interviews or other similar reasons. For this reason not many of the activities that you are expected to perform are formally made compulsory. Nevertheless, you are expected to play a full part in your own education by attending those classes that have been organized for you and by handing in work that has been set for you.
You are expected to attend lectures. Copying a set of notes that someone else has made is an extremely poor substitute for the educational process that you should be undertaking by listening to a lecture and distilling its essence for yourself.
You will be set written work to undertake for each module, usually on a weekly basis. Each piece of work will have a stated deadline for handing it in and work that is handed in after this deadline will not normally be accepted. Even if you think that your attempt is poor, we still want to see it, if only to be able to advise you what to do about your problems. If you fail to hand in work on a regular basis you will be requested to attend for interview with the Head of Teaching. In extreme cases your Adviser of Studies (AoS) and the School Office will also be informed of the problem.
In return for your commitment to undertake the work set for you, we undertake to have the work marked with feedback in time for the associated tutorial and normally within one week of it being handed in.
Tutorial classes are associated with each lecture module and we take these so seriously that we actually keep records of attendance at each one. Tutorials are the means by which you can find out ways in which you can approach problems, or could have tackled the work that was set for you, or find out what was wrong with your attempt if the feedback on your marked solution was insufficiently informative.
The more preparation you put into tutorials, the more you will get out of them. If you fail to attend tutorials on a regular basis you may be requested to attend for interview with the relevant Head of Teaching. In extreme cases your Adviser of Studies and the School Office will also be informed of the problem.
Apart from monitoring of your progress by the departments, the Mathematics departments have a Student Support Group. Poor attendance and homework handin record across a number of subjects will result in you being called for interview with this group. Failure to attend at this interview may result in you being called for interview by the Director of Education.
If illness or other emergencies cause you to miss lectures, tutorials or examinations, or prevent you from completing assessed work on time, then you should send or hand in a medical certificate (or other professional documentation) to the School of Mathematics and Physics Office on the top floor of the David Bates Building, or to the Mathematics Office on the ground floor of the David Bates Building, as quickly as possible. They will forward copies to all departments concerned. It is important to do this because of the possible penalties associated with failure to attend sessions or to submit work for assessment. Promptness in handing in medical certificates is especially important at examination time as they cannot be taken into account if submitted more than three days after the last examination of the session. Please see this Section for information regarding late submission of assessed coursework.
You have a personal duty and a legal responsibility to take reasonable care for the health and safety of yourself and of other persons who may be affected by your acts or your omissions.
The Departmental Office is usually staffed by trained firstaiders. In addition, Dr C. Ramsbottom (DBB, Room 01.038, telephone 9097 6047) and Mrs J. McKee (DBB, Room 02.012, telephone 9097 6005) are trained firstaiders. First aid boxes are situated in Rooms 0G.008 and 01.040 of the Applied Mathematics Department and in Room 02.007 on the top floor of the David Bates Building. Access to these can be granted through the Mathematics Departmental Office. Students should also be aware of the location of the University Health Centre, 5 Lennoxvale (off Malone Road, just beyond Chlorine Gardens), which has fully trained nursing staff.
Any accident, no matter how small, must be reported to the lecturer or supervisor in charge as soon as possible. Accidents outside class in School buildings must also be reported as soon as possible to one of the Safety Officers or to the Mathematics Departmental Office. Details of the accident must be entered in the Accident Reporting Book and a QUB accident report form filled in. Noninjury accidents that could have caused injury should also be entered in the Accident Reporting Book.
is Dr M. Mathieu (Room 02.029, telephone 9097 6008, Email m.m@qub.ac.uk). Any student should bring any concern regarding safety to the attention of the lecturer or supervisor in charge, or a Safety Officer, or the Mathematics Departmental Office.
If you get stuck in the lift, use the emergency call button and wait to be rescued. Under no circumstances should you attempt to force your way out of the lift.
If you have a known medical condition (e.g., epilepsy, diabetes) please be advised to inform the Mathematics Disability Officer (Dr T. Todorov, telephone 9097 6030, Email t.todorov@qub.ac.uk) or the University Disability Office. The information will be treated in confidence. In case of an incident, it may be important that relevant staff have been made aware of your condition. Thus, to ensure that staff can make a swift and accurate assessment of the situation, and that they can respond appropriately to any incident, please inform the University by the route given above.
If you are pregnant, please inform us through your Personal Tutor or your Adviser of Studies. Special regulations govern safety for new or expectant mothers.
The telephone extension number of Queen's Security is 2222 for emergencies and 5099 for nonemergencies (9097 2222 and 9097 5099 if using an outside line).
The University has a No Smoking policy in place in all buildings. Designated smoking shelters are available throughout the campus. The one nearest to the David Bates Building can be found between the Library and the car park. Elsewhere on campus, smoking is not allowed, including in the vicinity of the entrance to the David Bates Building.
Emergency evacuation procedures are posted throughout the David Bates Building and other buildings in the University. Locate and read them. Also locate the nearest Fire Alarm Break Glass Point in the area where you are working.
The fire alarm is a continuously sounding bell or siren.
If you discover a fire:
If you hear the fire alarm:
In the case of an emergency, DO NOT:
DO:
If you have advised the University about a disability issue on your UCAS application form, you should have been contacted by the University's Disability Services to discuss any support needs you may have. If you have not been contacted by the Disability Services and you want specific needs in relation to your studies to be taken into account, you must register with the Disability Services: please refer to Section 5 ‘General university support and services’ of this document.
If you are not registered and have a disability which is restricting your academic studies, please contact Disability Services or your Adviser of Studies or one of the School Disability Advisers:
Dr Tchavdar Todorov (Mathematics) Tel. 9097 6030 Email: t.todorov@qub.ac.uk 
Dr Chris Watson (Physics) Tel. 9097 3175 Email: c.a.watson@qub.ac.uk 
If you are experiencing difficulty or are not receiving the support expected or required this can have a detrimental effect on your work. In such cases it is essential that you contact either you Disability Adviser or Disability Services immediately.
Each of the two departments has a StaffStudent Consultative Committee (SSCC)  a committee of staff and students that discusses and advises on matters of mutual interest. Normally each Committee looks for one or two student representatives from each of Levels 1, 2, 3 and 4 and one or more postgraduate students. Student nominations to this committee are sought in early October and, if necessary, elections are held soon after that. These committees have only a few specified tasks  their main purpose is to give students a forum through which to channel complaints or suggestions. However, most such matters can be dealt with more quickly by students (or their SSCC representatives) approaching lecturers (or the appropriate Head of Teaching/Director of Education) as soon as a problem arises, rather than waiting for the next SSCC meeting. One statutory obligation that we have is to consult the SSCC about any proposed changes to modules and degree programmes, as part of the module and programme review process.
Notices related to the SSCC will be posted on the notice boards on the ground and second floors, including requests for items for the agenda of forthcoming meetings, the agenda themselves (so that you have an opportunity to comment on items on it) and draft minutes of meetings that have taken place.
If you run into academic problems then a number of sources of help are available to you. If your problem is with your study program in general then you should consult with your Adviser of Studies. If you have a problem with an individual module, your first resort should be either the tutor for your tutorial group or the lecturer for the module. If neither of these can sort out your problem, you should next consult the relevant Head of Teaching (HoT). If you are still unable to obtain satisfaction, you should ask your SSCC representative to raise the matter at a SSCC meeting. This should be very much a last resort as this will certainly not result in a quick solution to your problem.
Where problems impact upon your performance in some assessed work or upon compulsory elements of your degree programme, it is particularly important that you follow correct procedures; please see Section 11 above and Section 19 below.
Mature students are an important part of our student body, and we are pleased to have so many people coming back to education. Returning to education after a break brings both advantages and possible drawbacks. On the plus side, you have seen more of life than a school leaver and this often gives your work an important degree of maturity. On the other hand, you may feel that students who are continuing their studies straight after school possess more finely tuned study skills. Although that is not necessarily true, we do appreciate that mature students often have different concerns from school leavers as they embark on university study. For this reason, please do not be afraid to take any concerns or queries you may have to a member of staff, e.g., your Adviser of Studies. Particularly in first year, you should feel free to see any lecturer about your studies, even if he is not your tutor.
Most difficulties can be resolved via the channels listed in Subsection Seeking help with problems above. Where you have tried these and been unable to reach a satisfactory resolution, the university has an official complaints procedure. Details of this can be found in the current version of the
Students who perform sufficiently well may be permitted to transfer from a BSc programme to the corresponding MSci programme at the end of Stage 2. In exceptional circumstances, it may also be possible to transfer at the end of Stage 3 (provided the student has the necessary prerequisites and completed the compulsory modules), but any student considering such a request should contact the Director of Education at an early stage. Students on an MSci programme are also free to transfer to the corresponding BSc programme. If you are uncertain about your intentions and have sufficiently good entrance qualifications, there is no disadvantage in enrolling as an MSci student in the first instance, and considering a switch to a BSc later.
In order to progress on an MSci programme at the end of Stage 2, your weighted average mark must be at least 55% (and 60% or over is highly desirable); likewise, a weighted average of at least 55% is required by the end of Stage 3. Students who are enrolled for an MSci degree but who fail to meet these standards will be transferred to a BSc program. See Section 22 for how the different Stages are weighted for calculating these averages.
It is very important that you make yourself aware, early in each semester, of the assessment format in each of the modules that you are studying. Normal practice is that the form of assessment to be used is announced at or near the first class meeting in each module, and again at some point later in the semester (for example, after the External Examiner's approval of the examination paper). Should you miss such an announcement (for example, because you change enrolment or enrol late), be sure to check this with your lecturer or through the QUB Student Information System (Qsis). The default form of assessment in Mathematics modules is the threehour written examination, but there are many variations on this. Please refer to the module details in Appendix 1 and Appendix 2.
Where final written examination papers are employed as assessment, they normally take place at the end of the semester in which the module is delivered, that is, January for first semester modules and May/June for second semester modules. The dates and times of examinations are determined centrally within the university, and cannot be changed to suit individual circumstances. For the students who have not been successful in the main exams, resit examinations are provided in August (see below). Should you fail a module, please contact your Adviser of Studies for further advice.
Even where a written examination is the only form of assessment, other work by the student, such as homeworks, may be taken into account by the Board of Examiners if the exam mark is a narrow fail. In such cases, the results of coursework are only ever used in a student's favour, never to their detriment.
Plagiarism: Investigations modules and Project modules are examined by means of written work (reports) submitted either several times throughout the module or at the end of the module. These modules may also involve presentations of work to staff and other students. Students must be aware that such work is intended to be their own work, except where they are explicitly told otherwise. Plagiarism, that is, presenting as your own original work, material that has been copied from other students or from published sources, or even your own earlier work, without due acknowledgement, is a serious examination offence. Any suspicion of this must be reported by the examiner(s) to the Chair of the Board of Examiners. If he/she agrees that there is any case to be answered, the matter must be referred to the Board of Examiners and possibly to the Director of Academic and Student Affairs. Penalties available to the Board of Examiners and to the Director of Academic and Student Affairs range up to and including awarding a mark of zero for the whole of the module. The Boards of Examiners in Mathematics have an obligation to treat instances of plagiarism extremely rigorously. In extreme cases the matter may be referred to the Academic Offences Committee which can impose even more severe penalties. (See for details.)
Late submissions: all coursework assessed as part of the examining process will have a deadline associated with it. Work that is handed in late will be penalised. You will be informed of the deadlines at the start of the module. According to the “Assessed work submitted after the deadline will be penalised at the rate of 5% of the total marks available for each working day late up to a maximum of five working days, after which a mark of zero shall be awarded.” If you become aware that circumstances will prevent you from meeting such a deadline, it is important that you contact the lecturer concerned (or, failing that, staff in the Departmental Office) as soon as possible  preferably three days or more before the deadline  to see if alternative arrangements can be made. For exemption from penalty or flexibility of deadline to be granted, documentation covering the circumstances will normally be expected (at the time of your request or, if that is not possible, shortly thereafter). Please also see Section 11.
Progression: in order to progress from one stage (‘year’) of study to the next, you should have passed the assessments in all of that stage's modules (and must already have passed all from any earlier stage). Students who have passed all but one module's assessment at a given stage are allowed to progress to the next stage, but will need to pass their ‘missing’ module assessment during that stage.
Resits or supplementary examinations: students who fail an AMA, PMA or SOR examination are permitted (and, in Levels 1 and 2, are expected) to register for the supplementary or ‘resit’ examination, and will be required to pay the appropriate fee at the time of registration. Resit examinations usually take place during the second half of August. Students' individual supplementary exam timetables are made available through Queen's Online at the end of July. The current charge is about £60 per module, subject to a maximum of about £180  for updates please check with the Student Guidance Centre. Resit assessment scores will appear in full on degree transcripts, but for the purposes of calculating degree classifications only the ‘bare pass’ mark of 40% is automatically used unless the Examination Board is made aware in good time of any exceptional circumstances (e.g., serious medical or personal problems) that have resulted in the student having to do a resit exam. We will not be able to take account of such circumstances unless they are professionally documented: please see also this Subsection. If, as part of your degree programme, you are taking a module from another School, be aware that their resit policy may be different from ours and  if relevant to you  find out in good time what it is.
Anticipated emergency: if, in advance of an examination or deadline for returning assessed work, you believe that your performance may be adversely affected by exceptional circumstances such as illness, then you should seek advice from your Adviser of Studies, your doctor or the University Health Service, regarding the options open to you. These may include temporary withdrawal from the University, deferral of the examination, applying for an extension to the deadline for submitting coursework or applying for a waiver to any penalty imposed for late submission. Medical certificates and documentary evidence of extenuating circumstances must be presented to the School Office (as discussed in Section 11 on documenting absence above). Note that the decision on whether to attempt the examination or submit the assessed work, and the consequences of that decision, shall remain your sole responsibility.
The external examiners for the three Mathematics areas are:
According to the External Examiner Handbook, external examiners have two main functions: to act as moderators and to act as consultants.
As moderators, external examiners should ensure that the assessment system is equitable and is fairly operated in the classification of students. This work should include:
As consultants, external examiners should ensure that the degrees awarded by Queen's are comparable in standard with those awarded in other UK or Irish universities, and are consistent with the QAA UK Quality Code for Higher Education. External Examiners are members of Subject and/or Programme Boards of Examiners. This work should include:
Only simple scientific calculators are permitted for use in examinations in Mathematics. Programmable calculators and graphics calculators are not allowed, nor any calculator that can store information prior to the examination. If you are in doubt about the acceptability of your calculator, please consult the lecturer for the module concerned.
Each module is marked on a 0100 scale. The mark is indicative of the quality of the work being assessed, according to the following categorisation used throughout the University.
Mark Band  Quality 

70100  First class (1^{st}) 
6069  Second Class, First Division (2.1) 
5059  Second Class, Second Division (2.2) 
4049  Third Class (3^{rd}) 
039  Fail 
For students on the BSc degree programmes first enrolled in 200910 and thereafter, the degree classification is determined by taking averages of the six best modules at each of Stages 1, 2 and 3, and adding these using weightings of 10%, 30% and 60%, respectively. Where this total weighted average (prior to rounding to the nearest integer) is within 3 percentage points of a higher classification and the weighted number of modules scoring in that higher classification or above is at least one half, then the higher classification shall be awarded (Predominance Rule). The award of an MSci degree is similar, except that the weightings used for the Stages 1, 2, 3 and 4 are 5%, 15%, 30% and 50%, respectively. Note that it is Stage rather than Level that counts here: for example, where a student opts to take a Level 3 module as part of the Stage 2 programme, the mark gained is used in calculating the Stage 2 average and weighted accordingly. Again, where regulations permit a student to take a Level 3 module at Stage 4, that mark is used in calculating the Stage 4 average and weighted accordingly.
Students who first enrolled before 200910 have their degree classifications determined by different weightings. Most importantly, their Stage 1 module scores are not taken into account. More precisely, for BSc pathways the weightings are 25% for Stage 2 and 75% for Stage 3; for MSci pathways they are 12.5% for Stage 2, 37.5% for Stage 3 and 50% for Stage 4.
Be aware that the rules for the classification of degrees are from time to time reconsidered by the university. However, if this occurs, you will certainly be informed about the change.
Students should consult School/University regulations concerning the possible award of other degrees such as Combined Honours or an Ordinary degree.
There are numerous international exchange programmes in Mathematics within the SOCRATES programme of the EU. We currently have partner universities in Estonia, Germany, Greece, Portugal, Spain, Sweden and Turkey. Depending on their choice of university, students can spend up to one academic year at one of these institutions. The stay will be financially supported by an EU grant.
Information on the exchange programme is available from Dr Huettemann [office DBB 02.038, email t.huettemann@qub.ac.uk] who will be happy to assist outgoing students in planning their stays. Further information is available from the website of the International Office at http://www.qub.ac.uk/ilo.
The Department of Pure Mathematics has two prizes that it may award each year. The A.C. Dixon Prize
may be awarded each year to the best first class honours student at Stage 4 in
Pure Mathematics. It is currently worth £100 in book tokens. Alfred
Cardew Dixon was Professor of Mathematics in Queen's from 1901 until
his retirement in 1930. He was elected to a Fellowship of the Royal
Society in 1904 and was President of the London Mathematical Society
between 1931 and 1933. In 1937, shortly after his death the previous
year, his niece, Mrs. Mona Woolnough, endowed the prize named after
him. The Burgess Prize was set up, in memory of Derek
Burgess who lectured and researched in Topology in this Department from
1957 until 1989, by his friends and colleagues. The prize is awarded
for the best marks in papers at Level 4 on Topology and related topics and is
currently worth £40 payable by cheque.
The Department of Applied Mathematics and Theoretical Physics each year awards the William Blair Morton Prize in Applied Mathematics. This prize was founded in 1945 to commemorate William Blair Morton, Professor of Physics in Queen's from 1897 to 1933. It is awarded to students who have distinguished themselves in the honours course in Applied Mathematics, particularly in their essay work. The annual value of the prize is approximately £300. The Raymond Flannery Prize is awarded annually “to the MSci graduate in the School of Mathematics and Physics with the best overall mark, with the condition that the student must have specialised in Applied Mathematics or Theoretical Physics and have taken a minimum of 3 AMA modules at MSci level”. Professor Flannery was pupil of St Columb's College Derry. In 1961 he graduated from Queen's University Belfast with a First Class Honours BSc degree in Mathematics, and completed his PhD at Queen's University in Theoretical Physics in 1964. At the time of establishment of the prize (2012) Professor Martin Raymond Flannery was Regents' Professor Emeritus at the School of Physics, Georgia Institute of Technology, Atlanta, Georgia. In 2012/2013 the value of the prize was £750. Another award is the Bates Prize. This prize was founded in 1982 by the friends and colleagues of Sir David Bates, Professor of Applied Mathematics and of Theoretical Physics in Queen's from 1951 to 1982. It is normally awarded annually to a student who has performed with distinction in the final honours examination in the Department of Applied Mathematics and Theoretical Physics and who subsequently pursues research in the Department. The value of the Prize is approximately £100 and should be used for the purchase of books on mathematics and theoretical physics.
Details of prizes available through the Department of Applied Mathematics and Theoretical Physics that are associated with Statistics and Operational Research modules and pathways may be obtained through the departmental office.
Here you can find more information on the awards and past recipients for the prizes in the School of Mathematics and Physics.
The alphanumeric codes which identify individual study modules consist of three letters indicating the subject area (‘AMA’, ‘PMA’ or ‘SOR’ in our case) followed by four digits, the first of which corresponds to the level at which the module is given. For example, the Level 2 Pure Mathematics module Complex Variables has the code PMA2003. Note that, prior to 2008/2009, the module codes were different, with the first three digits identifying the semester in which the module was given and its weighting. Thus, Complex Variables (a first semester module) was previously coded as 110PMA203. Sometimes it can still be useful to know both ‘versions’ of the code numbers in order to be clear which modules you have the right prerequisites for, or which past exam papers may contain material useful to your programme of study.
Nine out of ten cats prefer Mathematics
Prior to 20132014, the following rules applied:
From 20132014 onwards, “All Schools should provide an opportunity for students at all levels to resit, or exceptionally take as a first sitting, an examination or coursework which contributes to their degree classification or award, at the designated resit period before the end of the academic year.” (See 1.4.61 of the )
At stages 1 and 2 there are resit examinations in August. It is your responsibility to be available at the whole of the required time  Saturdays included  in August if you need to take a resit. Note that if you are taking modules from outside the School of Mathematics and Physics you should check the situation regarding resit examinations with your Adviser of Studies, e.g., prior to 20132014 Computer Science did not offer August resit examinations at stage 2.
The following BSc pathways involving Mathematics are available:
We will consider BSc in Mathematics with Finance, the joint degree in Applied Mathematics and Physics and the degree in Theoretical Physics later in this document.
In the following, S1 and S2 stand for semester 1 and semester 2, respectively.
At each stage students take six modules: three in each semester.
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
plus two other modules offered elsewhere in the University, subject to module prerequisites and timetabling constraints. However, it is recommended that students take SOR1001 Introduction to Probability and Operational Research (S1) and SOR1002 Statistical Methods (S2). These modules are prerequisites for higher level SOR modules, and not taking them reduces one's choice at Stages 2 and 3. Other possible pairs of modules include PHY1011 Foundation Physics 1 (S1) and PHY1012 Foundation Physics 2 (S2) (requires A or B in Alevel Physics); and CSC1011 Fundamentals of Programming and CSC1012 Programming Challenges. Note that PHY1011 is a corequisite for PHY1012 and that CSC1011 and CSC1012 run together throughout the full year. Alternatively, modules in a modern language or music may also be a possibility.
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
SOR1001 Introduction to Probability and Operational Research (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
SOR1002 Statistical Methods (S2)
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
CSC1011 Fundamentals of Programming
CSC1012 Programming Challenges
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
plus two appropriate modules in either French or Spanish.
Students should take note of the prerequisites for level 2 and level 3 modules when making their choice of modules at stage 1.
A student intending to take more than one module of Applied Mathematics at Stage 3 must pass the examinations in at least two of the AMA level 2 modules, while a student intending to take more than one module of Pure Mathematics at Stage 3 must pass the examinations in at least two of PMA2002, PMA2008 and PMA2007. A student intending to take only Pure Mathematics at Stage 3 must pass the examinations in all of these PMA modules.
Normally no student shall be permitted to take more than two of AMA2003, PMA2003 and PMA2007.
Note that SOR2002 Statistical Inference (S1) is a corequisite for SOR2004 Linear Models (S2) i.e., anyone wanting to take SOR2004 in the second semester must take SOR2002 in the first. People often overlook this!
Most students on this pathway at Stage 2 will take a mixture of Pure and Applied Mathematics, with possibly some Statistics & OR. Suggested combinations include:
Suggested combinations:
Suggested combinations:
The list of combinations is by no means exhaustive! Careful thought should be given to module choice bearing in mind: (a) any pathway requirements; (b) module prerequisites and (c) your own interests, strengths or weaknesses, and future career. Students should take note of the prerequisites for level 3 modules when making their choice of modules at stage 2.
There are a few general points to note for all students on a Mathematics pathway.
The choice of modules is very wide: students should be guided by individual interests and future career plans, as well as strengths or weaknesses, bearing in mind module prerequisites. While every effort has been made to avoid timetabling clashes, with so many modules available at this level, not all possible combinations of modules can be accommodated.
For both programmes students at Stage 1 must take AMA1001 Vector Algebra and Dynamics (S1), AMA1002 Waves and Vector Fields (S2), PHY1011 Foundation Physics 1 (S1) and PHY1022 Foundation Physics 2 (S2); plus either PHY1012 Computational Methods (S1) and PHY1024 Computational Modelling in Physics (S2) or PMA1012 Numbers, Sets and Sequences (S1) and PMA1014 Analysis and Linear Algebra (S2). Students should consider their choice carefully, e.g., taking PMA1012 and PMA1014 would allow students to satisfy the Stage 1 requirements for the BSc in Mathematics, should they wish to transfer to this programme at the end of Stage 1.
For both pathways students must take AMA2003 Methods of Applied Mathematics (S1), PHY2081 Modern Physics (S1) and PHY2084 Optics, Electricity and Magnetism (S2). TP students must in addition take AMA2001 Classical Mechanics and PHY2082 Physics of the Solid State, and one other module (preferrably AMA). AM&P students take three other modules available from AMA and PHY, so that of the six modules chosen, at least two are from AMA and at least two from PHY. Students should take note of prerequisites for level 3 modules before finalizing their choice of modules at Stage 2.
Suggested combinations:
The choice of modules is very wide: students should be guided by individual interests and future career plans, as well as strengths or weaknesses, while bearing in mind module prerequisites.
In choosing their modules, students should be guided by individual interests and future career plans, as well as strengths or weaknesses, while bearing in mind module prerequisites.
Compared with other Mathematics pathways that allow a wide choice of modules, the BSc in Mathematics with Finance (MF) pathway is much more focussed and more restrictive. It provides students with a particular set of mathematical skills that are ideal for work in software engineering and, in particular, in the financial services technology sector. The course is primarily Applied Mathematics and includes computer programming as well as a range of statistical techniques. The course is a partnership with industry and includes project work related to capital markets and capital market instruments.
In addition to the modules offered by the School of Mathematics and Physics, the students will be taking modules taught by the Queen's University Management School (module code FIN for Finance).
The Programme Director for Mathematics with Finance is Dr Jim McCann, Email j.f.mccann@qub.ac.uk.
AMA1001 Vector Algebra and Dynamics (S1)
SOR1001 Introduction to Probability and Operational Research (S1)
PHY1012 Computational Methods (S1)
AMA1002 Waves and Vector Fields (S2)
SOR1002 Statistical Methods (S2)
FIN1001 Financial Institutions and Markets (S2)
AMA2003 Methods of Applied Mathematics (S1)
SOR2002 Statistical Inference (S1)
FIN2006 Financial Decision Making (S1)
AMA2004 Numerical Analysis (S2)
SOR2003 Methods of Operational Research or SOR2004 Linear Models (S2)
FIN2008 Financial Markets Theory (S2)
AMA3006 Partial Differential Equations or SOR3001 Linear and Dynamic Programming (S1)
SOR3012 Stochastic Processes and Risk (S1)
AMA3021* Computational Finance (S1)
AMA3007 Financial Mathematics (S2)
SOR3008 Statistical Data Mining or AMA3010 Dynamical Systems** (S2)
AMA3022* Team Project: Mathematics with Finance (S2)
* Modules available only to MF students.
** AMA3010 is currently suspended; alternative AMA module choices are
AMA3013 Calculus of Variations and Hamiltonian Dynamics or
AMA3014 Mathematical Modelling for Biology and Medicine (both S2).
The following MSci programmes involving Mathematics are available:
Besides being fouryear degrees and going deeper, the main difference between these pathways and their BSc counterparts is that the students on the MSci pathways will undertake a fullyear doublemodule Project in their final year (i.e., at Stage 4), in either Pure Mathematics, or Applied Mathematics, or (for the MSOR pathway) in Statistics & Operational Research.
We will consider the joint degree in Applied Mathematics and Physics and the degree in Theoretical Physics later in this document.
In the following, S1 and S2 stand for semester 1 and semester 2, respectively.
At each stage students take six modules: three in each semester.
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
plus two other modules offered elsewhere in the University, subject to module prerequisites and timetabling constraints. However, it is recommended that students take SOR1001 Introduction to Probability and Operational Research (S1) and SOR1002 Statistical Methods (S2).
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
SOR1001 Introduction to Probability and Operational Research (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
SOR1002 Statistical Methods (S2)
AMA1001 Vector Algebra and Dynamics (S1)
PMA1012 Numbers, Sets and Sequences (S1)
AMA1002 Waves and Vector Fields (S2)
PMA1014 Analysis and Linear Algebra (S2)
CSC1011 Fundamentals of Programming
CSC1012 Programming Challenges
A student intending to take more than one module of Applied Mathematics at Stage 3 must pass the examinations in at least two of the AMA level 2 modules, while a student intending to take more than one module of Pure Mathematics at Stage 3 must pass the examinations in at least two of PMA2002, PMA2008 and PMA2007. A student intending to take only Pure Mathematics at Stage 3 must normally pass the examinations in all of these PMA modules.
Normally no student shall be permitted to take more than two of AMA2003, PMA2003 and PMA2007.
Note that SOR2002 Statistical Inference (S1) is a corequisite for SOR2004 Linear Models (S2) i.e., anyone wanting to take SOR2004 in the second semester must take SOR2002 in the first. People often overlook this!
Choice of Stage 2 modules for this pathway depends upon which ‘mode’ is being followed.
Suggested combinations:
Suggested combinations:
Suggested combinations:
In Stage 3 we seek to lay a good foundation for the doublemodule project to be undertaken at Stage 4. Students are reminded to check the prerequisites for modules to be taken in Stage 4 before finalizing their choice of level 3 modules.
Students must take an approved combination of six modules from those available at Level 3 in Applied Mathematics, Pure Mathematics and Statistics & OR, to include either AMA4020 or PMA4013; at least two modules in Pure Mathematics or at least two full modules in Applied Mathematics; and the equivalent of at least two full SOR modules.
Suggested combinations:
Students must take an approved combination of modules of total weight 120 CAT Credits (i.e., equivalent to six full modules) and must include either AMA4020 or PMA4013. At least 40 CAT Credits (i.e., one third) must be in Computer Science, and at least 40 CATS credits must be in one of AMA or PMA.
The module SOR3001 counts as an AMA module. Students on this pathway should be aware that there are two ‘modes’ at Stage 4 and should check the prerequisites for the modules available at Stage 4. The doublemodule project is only available in either AMA or PMA, not in CSC. It is strongly advised that students should consider the likely subject area (AMA or PMA) in which they wish to take their Stage 4 project, and take the appropriate ‘investigations’ module (i.e. either AMA4020 or PMA4013) and at least two other modules from this area.
Students must take the project SOR4001 which is a double module, and four other modules from Applied Mathematics, Pure Mathematics, Statistics & OR, comprising either four modules from Level 4 or three modules from Level 4 plus one not previously taken from Level 3 (other than a Level 3 Project).
Students must choose between groups A and B. Modules adding up to a total weight of 120 CATS credits must be selected.
Group A: AMA4005, CSC4002 and CSC4003 must be taken, plus a level 4 AMA module and either another level 4 AMA module or a level 3 AMA module (including SOR3001) not previously taken.
Group B: PMA4001, CSC4002 and CSC4003 must be taken, plus a level 4 PMA module and either another level 4 PMA module or a level 3 PMA module not previously taken.
For both pathways students at Stage 1 must take AMA1001 Vector Algebra and Dynamics (S1), AMA1002 Waves and Vector Fields (S2), PHY1011 Foundation Physics 1 (S1) and PHY1022 Foundation Physics 2 (S2); plus either PHY1012 Computational Methods (S1) and PHY1024 Computational Modelling in Physics (S2) or PMA1012 Numbers, Sets and Sequences (S1) and PMA1014 Analysis and Linear Algebra (S2). Students should consider their choice carefully e.g., taking PMA1012 and PMA1014 would allow students to satisfy the Stage 1 requirements for the MSci in Mathematics, should they wish to transfer to this programme at the end of Stage 1.
For both pathways students must take AMA2003 Methods of Applied Mathematics (S1), PHY2081 Modern Physics (S1) and PHY2084 Optics, Electricity and Magnetism (S2). TP students must in addition take AMA2001 Classical Mechanics and PHY2082 Physics of the Solid State, and one other module (preferrably AMA). AM&P students take three other modules available from AMA and PHY, so that of the six modules chosen, at least two are from AMA and at least two from PHY. Students should take note of prerequisites for level 3 modules before finalizing their choice of modules at Stage 2.
Suggested combinations:
“Genius is one percent inspiration, ninetynine percent perspiration.” (T. A. Edison)
The departmental secretary is:
Name  Room  Phone  

Tara Spencer  0G.006  9097 6001  t.spencer@qub.ac.uk 
One of your first tasks should be to make sure that you can find this office. Tara is thoroughly acquainted with the workings of the Department and will be able to sort out many of the common problems that students encounter.
The Associate Director of Education (ADoE, Head of Teaching, HoT) for Applied Mathematics and Theoretical Physics and Statistics & Operational Research is Dr Gleb Gribakin (term completed on 1 August 2014).
The academic staff in the department are currently:
Name  Room  Phone  

Dr A Brown  0G.018  9097 6043  andrew.brown@qub.ac.uk 
Dr K Cairns  01.033  9097 6058  k.cairns@qub.ac.uk 
Dr G De Chiara  0G.028  9097 6053  g.dechiara@qub.ac.uk 
Dr D Dundas  01.021  9097 6055  d.dundas@qub.ac.uk 
Dr A Ferraro  0G.019  9097 6018  a.ferraro@qub.ac.uk 
Dr G F Gribakin  0G.012  9097 6056  g.gribakin@qub.ac.uk 
Dr M Grüning  01.010  9097 6054  m.gruening@qub.ac.uk 
Prof J J Kohanoff  01.014  9097 6032  j.kohanoff@qub.ac.uk 
Dr I Kourakis  G047**  9097 3155  i.kourakis@qub.ac.uk 
Prof A H Marshall  01.022  9097 6033  a.h.marshall@qub.ac.uk 
Dr J F McCann  0G.032  9097 6041  j.f.mccann@qub.ac.uk 
Dr L McCrink  01.013  9097 6050  l.mccrink@qub.ac.uk 
Dr S Moutari  01.034  9097 6064  s.moutari@qub.ac.uk 
Dr V O'Neill  01.008  9097 6057  vikki.oneill@qub.ac.uk 
Dr M Paternostro  0G.036  9097 6073  m.paternostro@qub.ac.uk 
Dr C A Ramsbottom  01.038  9097 6047  c.ramsbottom@qub.ac.uk 
Dr B Reville  G022**  9097 5272  b.reville@qub.ac.uk 
Dr S F C Shearer  0G.031  9097 6049  f.shearer@qub.ac.uk 
Dr T Todorov  01.011  9097 6030  t.todorov@qub.ac.uk 
Dr G Tribello  02.009  9097 6010  g.tribello@qub.ac.uk 
Prof H W van der Hart  0G.013  9097 6048  h.vanderhart@qub.ac.uk 
Staff offices are in the David Bates building, except for those marked **, which are in the Physics building. You may also be given tutorials by some of our postgraduate students. Their contact details are available from Tara in the Mathematics Departmental Office.
AMA1001  Vector Algebra and Dynamics (1^{st} semester) 
AMA1002  Waves and Vector Fields (2^{nd} semester) 
SOR1001  Introduction to Probability and Operational Research (1^{st} semester) 
SOR1002  Statistical Methods (2^{nd} semester) 
AMA2001  Classical Mechanics (1^{st} semester) 
AMA2003  Methods of Applied Mathematics (1^{st} semester) 
AMA2004  Numerical Analysis (2^{nd} semester) 
AMA2005  Fluid Mechanics (2^{nd} semester) 
SOR2002  Statistical Inference (1^{st} semester) 
SOR2003  Methods of Operational Research (2^{nd} semester) 
SOR2004  Linear Models (2^{nd} semester) 
AMA3001  Electromagnetic Theory (1^{st} semester) 
AMA3002  Quantum Theory (1^{st} semester) 
AMA3004  Advanced Numerical Analysis (1^{st} semester) 
AMA3006  Partial Differential Equations (1^{st} semester) 
AMA3021  Computational Finance (1^{st} semester) [for Mathematics with Finance students] 
AMA3003  Tensor Field Theory (2^{nd} semester) 
AMA3007  Financial Mathematics (2^{nd} semester) 
AMA3010  Dynamical Systems (2^{nd} semester) [SUSPENDED in 20142015] 
AMA3013  Calculus of Variations and Hamiltonian Mechanics (2^{nd} semester) 
AMA3014  Mathematical Modelling in Biology and Medicine (2^{nd} semester) 
AMA3022  Team Project: Mathematics with Finance (2^{nd} semester) [for Mathematics with Finance students] 
SOR3001  Linear and Dynamic Programming (1^{st} semester) 
SOR3012  Stochastic Processes and Risk (1^{st} semester) 
SOR3008  Statistical Data Mining (2^{nd} semester) 
AMA4001  Advanced Quantum Theory (1^{st} semester) 
AMA4006  Practical Methods for Partial Differential Equations (1^{st} semester) 
AMA4009  Information Theory (1^{st} semester) 
AMA4003  Advanced Mathematical Methods (2^{nd} semester) 
AMA4004  Statistical Mechanics (2^{nd} semester) 
AMA4021  Mathematical Methods for Quantum Information Processing (2^{nd} semester) 
AMA4005  Project (full year  double module) 
AMA4020  Investigations (2^{nd} semester) [To be taken in year/stage 3] 
SOR4001  Project (full year  double module) 
SOR4007  Survival Analysis (1^{st} semester) 
The departmental secretary is:
Name  Room  Phone  

Sheila O'Brien  0G.006  9097 6003  s.obrien@qub.ac.uk 
One of your first tasks should be to make sure that you can find this office. Sheila is thoroughly acquainted with the workings of the Department and will be able to sort out many of the common problems that students encounter.
The Associate Director of Education (ADoE, Head of Teaching, HoT) for Pure Mathematics is Dr Ariel Blanco.
The academic staff in the department are currently:
Name  Room  Phone  

Dr D Barnes  02.032  9097 6020  d.barnes@qub.ac.uk 
Dr A Blanco  02.021  9097 6023  a.blanco@qub.ac.uk 
Dr T Huettemann  02.038  9097 6026  t.huettemann@qub.ac.uk 
Dr YF Lin  02.032  9097 6011  y.lin@qub.ac.uk 
Dr M Mathieu  02.029  9097 6008  m.m@qub.ac.uk 
Dr T B M McMaster  02.020  9097 6006  t.b.m.mcmaster@qub.ac.uk 
Dr S Shkarin  02.027  9097 6027  s.shkarin@qub.ac.uk 
Dr I Todorov  02.037  9097 6007  i.todorov@qub.ac.uk 
Prof A W Wickstead  02.022  9097 6004  a.wickstead@qub.ac.uk 
You may also be given tutorials by some of our postgraduate students. They are mostly located in Room 02.009 and they may be contacted by phone on 9097 6009.
PMA1012  Numbers, Sets and Sequences (1^{st} semester) 
PMA1014  Analysis & Linear Algebra (2^{nd} semester) 
PMA2003  Complex Variables (1^{st} semester) 
PMA2007  Linear Algebra (1^{st} semester) 
PMA2010  Elementary Number Theory (1^{st} semester) 
PMA2002  Analysis (2^{nd} semester) 
PMA2008  Group Theory (2^{nd} semester) 
PMA2009  Geometry (2^{nd} semester) 
The Department of Pure Mathematics is a small one, and staff losses (retirements, resignations or illness) can adversely affect our ability to offer modules at higher levels (3 and 4) as we often only have one member of staff who can teach a particular highlevel module. Although we will do all that we can to put on the modules listed, we cannot guarantee to do so. You will be notified of any unavoidable changes as soon as we know of them.
PMA3008  Computer Algebra (1^{st} and 2^{nd} semesters) 
PMA3012  Ring Theory (1^{st} semester) 
PMA3014  Set Theory (1^{st} semester) 
PMA3016  Convergence (1^{st} semester) 
PMA3017  Metric and Normed Spaces (2^{nd} semester) 
PMA3018  Algebraic Equations (2^{nd} semester) 
Note that in any given academic year the taught Level 4 modules that will be offered will depend on demand from intending Level 4 MSci students of Pure Mathematics as indicated by them during the preceding year. In the academic year 20132014 it is intended that all the listed Level 4 taught modules will be offered.
PMA4001  Project (full year  double module) 
PMA4002  Functional Analysis (1^{st} semester) 
PMA4003  Topology (1^{st} semester) 
PMA4004  Integration Theory (2^{nd} semester) 
PMA4008  Rings and Modules (2^{nd} semester) 
PMA4010  Algebraic Topology (2^{nd} semester) 
PMA4013  Mathematical Investigations (2^{nd} semester) [To be taken in year/stage 3] 
“We are all star stuff.” (Carl Sagan)
Autumn Semester  

Welcome week  Registraton, enrolment  Monday 22 September  Friday 26 September 2014 
Weeks 112  Teaching  Monday 29 September  Friday 19 December 2014 
3 weeks  Christmas Vacation  Monday 22 December 2013  Friday 9 January 2015 
Weeks 13  15  Assessment  Monday 12 January  Tuesday 27 January 2015 
InterSemester Break  Wednesday 28 January  Friday 30 January 2015  
Publication of examination results  Tuesday 10 February 2015  
Spring Semester  
Weeks 1  8  Teaching  Monday 2 February  Friday 27 March 2015 
3 weeks  Easter Vacation  Monday 30 March  Friday 17 April 2015 
Weeks 9  12  Teaching  Monday 20 April  Friday 15 May 2015 
Revision Period  Monday 18 May  Wednesday 20 May 2015  
Weeks 13  15  Assessment  Thursday 21 May  Saturday 6 June 2015 
Publication of examination results  Thursday 25 June 2015 
Other important dates:
For other years see Semester Dates.
Holidays and Key dates when University closure applies:
For other years see Holiday Dates.
Plagiarism is defined as follows: to present as new and original an idea or product derived from an existing source. This existing source may be the work of others submitted without appropriate acknowledgement, or the writer's own previously submitted work. This includes autoplagiarism (to use excerpts from your own previous work without appropriate acknowledgement) and selfplagiarism (to submit a piece of work more than once, eg one which has been previously submitted for a different assignment).
It is an academic offence for students to plagiarise. Resources about referencing and essay writing, as well as workshops and onetoone support are available from the Learning Development Service.
Definitions and procedures for dealing with academic offences can be found in the University's General Regulations:
The University takes the view that all aspects of student life offer opportunities for learning and development. Schools work closely with student support services and the Students' Union to support your personal development planning, providing a range of academic and personal support services and developmental opportunities during your time at Queen's. Support and development opportunities for students are offered through your academic School, as well as centrally in the Student Guidance Centre, International and Postgraduate Student Centre and the Students' Union.
If you are not sure where to go when you have a question about any element of University life, ask one of the Information Assistants in the Student Guidance Centre or International and Postgraduate Student Centre.
We want you to do well during your time at Queen's and all these services are here to help you. Staff and Sabbatical Officers have a long and successful history of supporting students in a range of situations, so do not hesitate to ask for help.
Every year we come across students who wish they had asked for help sooner, so take their advice and come and speak to one of the support services listed here. And if you are not sure which service to go to, come to the Student Guidance Centre and speak to one of our friendly Information Assistants who will be able to point you in the right direction.
Moving to higher education is an exciting time and a new phase in your life. At the same time, don't be surprised if you find it overwhelming at times  it's normal to feel this way. For those of you who have recently left school, studying for a degree is very different. You have much more freedom to decide what you want to learn, and you will be asked to consider and debate about the content of your course. You will be expected to be more selfdirected in how you approach your studies. Unlike school, your lecturers and tutors will guide you, but will not direct you as much as your teachers may have done in the past.
Information about making a successful transition to higher education can be found at: http://www.qub.ac.uk/welcome, including a series of Transition short films made by students about their experiences.
Within your academic School, you can discuss any issues that arise and receive guidance on approaching new situations from Academic Advisers and Personal Tutors.
Personal Development Planning (PDP) is a process of reviewing and planning your own development. You will be encouraged to take control of your learning needs by reflecting on your personal performance and the feedback you receive. An electronic portfolio (efolio) is provided in Queen's Online to help you set out plans and personal goals to improve your academic performance and enhance your employability. Research suggests that students who engage with PDP are better equipped to study and develop their professional skills and experiences than those who do not. You should also look at the PDP webpage in the Student Gateway and talk to your Personal Tutor.
PDP is a very helpful process to assist you in identifying what study and skillsrelated changes you need to make to be even more effective as an undergraduate. Our experience shows that students who use PDP do better academically than those who choose not to. Don't forget it's not just about helping you study throughout your course, but it will also help you develop skills for all aspects of your life at Queen's and beyond.
Each undergraduate student is assigned a Personal Tutor whose role it is to be a point of contact and support within your School. We strongly advise you to speak to your Tutor if you have any concerns relating to your time at Queen's, particularly if you think it will affect your progression.
Personal Tutors will support you as you reflect on your Personal Development Plan and, where appropriate, refer you to a University or Students' Union service for specialised assistance. Students will have a number of official meetings with their Personal Tutor, but if you need to speak to your Personal Tutor outside of these scheduled meetings, feel free to approach them for an appointment.
The Student Guidance Centre (SGC) is on University Road, above the Ulster Bank, with the entrance around the corner.
It brings together a number of support services that help guide and assist you throughout your time at Queen's. From managing your student record, developing your academic and employability skills and offering advice and assistance for times when you may feel under pressure, all services work together to ensure you have an enjoyable student experience.
Services located in the SGC include:
The Centre holds information and resources for a range of services both on and offcampus.
Not sure who to ask? Information Assistants on the first floor will help you with all your queries about the University, from getting a new student card, to where to submit a form, or taking advantage of specialist advice from one of the services. The Centre also offers a comfortable seating area, internet access, coffee, newspapers to read and laptops that you can borrow for use within the Centre.
Student Guidance Centre
91a University Road
Tel: +44 (0)28 9097 2727
Email: sgc@qub.ac.uk
http://www.qub.ac.uk/sgc
The International and Postgraduate Student Centre (IPSC), along with the Student Guidance Centre and the Students' Union, forms a proactive and comprehensive support and services infrastructure for the student body. Located close to The McClay Library, the Centre provides dedicated support to, and is a hub for many aspects of information, advice and guidance for international and postgraduate students. Students have the opportunity to meet and socialise with their peers from other Schools and disciplines in a welcoming and relaxed environment. Wireless access to the University network is provided throughout the Centre.
The Postgraduate Student Centre located on the first and second floors serves as a focal point for the postgraduate community, complementing the facilities and services currently provided by our academic Schools. The Centre delivers the Postgraduate Skills training programme for research students, which offers an extensive range of workshops, courses and seminars alongside numerous supported, studentled initiatives. Additionally, postgraduate research students can access onetoone guidance and advice in areas related to careers, employability and personal effectiveness, and all postgraduate students can access tailored information and advice on a range of postgraduate issues.
The second floor of the building provides dedicated study and social space for postgraduates. This includes a computer facility offering 50 networked computers with black and white and colour printing, scanning and photocopying facilities. The second floor also houses the Postgraduate Students' Association (PGSA), which represents the interests of all postgraduate students in the University and coordinates a number of researchrelated and social events throughout the year.
International Student Support is based on the ground floor in the International and Postgraduate Student Centre. The team aims to support, guide and advise international students, enabling them to maximize their student experience at Queen's. Core services include advice on studentrelated visas and immigration issues; welcome and orientation; general support and advice; and cultural awareness training.
Opening hours for postgraduate students are 9:00 am  10:00 pm Monday to Friday.
Postgraduate Centre
Tel: +44 (0) 028 9097 2585
Email: pg.office@qub.ac.uk
http://www.qub.ac.uk/postgraduate
International Student Support
Tel: +44 (0) 028 9097 3899
Email: internationstudentsupport@qub.ac.uk
http://www.qub.ac.uk/isso
All the services listed in this guide are equally available to international students and staff are happy to support you during your time at Queen's. In addition, the International Students Support Office (ISSO) has staff specifically trained to provide advice, support and guidance for international students.
The staff in the ISSO are the only staff in the University who are permitted to provide advice or guidance on immigration/visa matters. The ISSO is located in the International and Postgraduate Student Centre (IPSC). We offer a wide range of services including confidential advice on immigration, problems affecting your studies or personal concerns. We provide information on matters relating to arriving in the UK, opening a bank account, police registration, healthcare and doctors, safety and security, working in the UK, driving in the UK, activities in the University, local activities and events, travel, British culture, local shops and services, facilities for families. More general support is provided to help with settling in the UK, life in Belfast, academic life and study methods, homesickness and culture shock and schools and childcare.
It is very important that international students meet the conditions of their visa while they live in the UK. This means that if you have a job you must not work more hours than you are permitted. If you want to work (paid or unpaid) you should make an appointment with the ISSO to discuss what you are allowed to do in the UK.
The UK has introduced new immigration rules called the Points Based System. This affects both you and the University. The University has a number of obligations to meet for the UK Border Agency, which include (but are not restricted to) keeping copies of your immigration documents, monitoring your arrival/enrolment and your attendance. The International Student Handbook contains a list of the recording and reporting obligations which must be carried out by the University. If you have any questions about these please contact the ISSO.
You can contact the ISSO at internationstudentsupport@qub.ac.uk with any questions, enquiries or to make an appointment. Alternatively you can drop into the IPSC where we will be delighted to meet you and help with any problems you may be having.
When you attend an appointment at the ISSO you should always bring your passport with you.
The University has a range of accommodation for students, based mainly at the Elms Village, which is a 15minute walk from the main campus. If you would like to apply for a place in University accommodation or if you are a resident and have any queries please contact us or visit our website:
Accommodation and Hospitality
Queen's University Belfast
Elms Village
78 Malone Road
BT9 5BW
Tel: +44 (0)28 9097 4525
Email: accommodation@qub.ac.uk
http://www.stayatqueens.com
If you need assistance in searching for private accommodation or you are living in the private sector and need advice on tenancy issues or any other matter related to your accommodation please contact:
Brian Slevin
Education and Welfare Adviser
Student Advice Centre
2nd Floor, Students' Union
University Road
Tel: +44 (0)28 9097 1135/3106
Email: b.slevin@qub.ac.uk
University life can throw up all sorts of interesting situations and challenges. Sometimes you may not know exactly what to do about them and may want some advice. That's what the Students' Union Advice Centre is there for.
The Centre employs Advisers dedicated to providing all Queens' students with free, confidential, independent and accurate advice.
Connie Craig
Financial Adviser
Tel: +44 (0)28 9097 1049
Email: connie.craig@qub.ac.uk
Connie advises on grants, loans, fees, Support & Hardship Funds, the financial aspects of repeating years and course changes, Social Security Benefits and other general financial issues.
Education & Welfare Issues
Email: studentadvice@qub.ac.uk
For advice on accommodation (including QUB accommodation), private landlords, deposits, repairs, checking leases/contracts. Also academic issues including Progress Committees, complaints and appeals.
Debbie Forsey
Money Management Adviser
Tel: +44 (0)28 9097 1166
Email: d.forsey@qub.ac.uk
Debbie advises on debt  this includes overdrafts, credit cards, loan agreements, arrears of payments, negotiating with creditors and any other debt issues. Guidance on budgeting and money management is available and students do not need to be in a ‘crisis situation’ to seek advice!
Careers, Employability and Skills offers a range of facilities to help students develop their career potential. These facilities include:
It is never too early to start thinking about how you can use your time at Queen's to enhance your employability and develop your career. Come in and speak to us.
Careers, Employability and Skills
Student Guidance Centre
91a University Road
BT7 1NN
Tel: +44 (0)28 9097 2727
Email: careers@qub.ac.uk
http://www.qub.ac.uk/careers
Degree Plus is a unique and innovative programme which allows Queen's students to gain accreditation for skills and experiences developed outside of their academic programme.
Students may register for the programme at any time during their academic career but must have submitted evidence for the award of Degree Plus by 1 April in the year they hope to graduate. Successful completion of the programme provides students with the award of Degree Plus on their transcript. Any extracurricular activities which enables the development of academic, personal, career or employability skills may be included, eg volunteering, involvement in a club or society, completion of an additional course or programme, summer experience or working parttime. The award may be obtained in two different ways: either by undertaking a programme which is fully accredited through Queen's, or by combining two experiences and presenting evidence on a Degree Plus application form. Visit the website for further details: http://www.qub.ac.uk/degreeplus.
A high percentage of students work parttime whilst completing their degree. There are a number of good reasons for doing this as parttime work can help you:
Register with the Student Jobshop to access a range of opportunities and fairly paid parttime jobs with Oncampus Jobs.
Please note: the University strongly recommends that students do not exceed 15 hours parttime work per week as there is strong evidence to show that significant levels of parttime work can affect degree outcomes.
International Students may have prohibitions or restrictions on working in the UK. It is very important that you confirm you have a legal right to work and if you do have the right to work, that you don't exceed the permitted hours. If you want to work, you must bring your visa to the International Student Support Office (see above) where your visa will be checked to confirm whether or not you can work and if you can, how many hours you are allowed to work.
The University is committed to supporting the attainment and success of its students. Through its Student Care Protocol, staff work with students to identify appropriate actions and support either within or external to the University, to help students through the array of difficulties they may encounter during their time as a student.
No issue is too small to raise with the helpful support team. If you are worried about your studies, or a personal issue affecting your studies, you can contact any of the University's support team, including:
For information about the range of support available to students, visit the Student Gateway website. Any queries may be sent to the Student Guidance Centre by emailing: sgc@qub.ac.uk.
The University's Student Charter, its Policy on Equality and Diversity and its Student Antibullying and Harassment Policy make reference to working in a learning environment which is free from harassment including discrimination, victimisation and bullying, and expects individuals to treat fellow students, staff and visitors equally and respectfully. In line with its regulations the University will take disciplinary action against students who cause distress by comments made about others, whether said or in writing. This includes comments written in the public domain, for example on social networking sites.
Currently 17 faiths and denominations are represented at the University. Our work is varied and farreaching, but we always hope to offer a warm welcome, support and advice (spiritual and otherwise) to all members of the University community. Each of us is committed to playing a constructive and beneficial role in the building up of the individual person and of the University community. The Chaplaincies website is the best source of information; however, the four main chaplaincy centres and points of contact are:
Catholic  28 Elmwood Ave  Rev Fr Gary Toman 
 
Church of Ireland  22 Elmwood Ave  Rev Barry Forde 
 
Methodist  24 Elmwood Ave  Rev John Alderdice 
 
Presbyterian  12 Elmwood Ave  Rev Karen Mbayo 

Email: http://www.qub.ac.uk/chaps
Whilst we hope your time at Queen's is trouble free, there may be times when you find things difficult for a range of reasons. If that is the case then please speak to the Counselling Service. Staff are friendly, approachable and experienced in dealing with a wide range of issues that students have to face at University and in their personal lives. Don't leave things until the problem escalates; speak to them at the earliest opportunity. Emotional distress and difficulty can seriously impede your ability to study effectively. Counselling can support you in managing your difficulty so that your studies do not suffer unduly. Counselling is free and confidential to any student of the University, and can range from a fiveminute chat to a series of 50minute sessions. Counsellors are professionally trained and accredited and are bound by the Code of Ethics of their professional body, the BACP.
Counselling Service
Student Guidance Centre
91a University Road
Belfast BT7 1NN
Freephone: 0808 800 0016* (During office hours)
Email counsellingappointments@qub.ac.uk (for an appointment)
http://www.qub.ac.uk/counselling
(*also free from mobiles)
During outofoffice hours students can avail of 24 hour telephone counselling support on 0808 800 0002.
Disability Services provides support to students with a wide range of disabilities including mental health difficulties and dyslexia. If you have a disability or acquire a disability whilst studying at Queen's, we can help arrange academic and personal support to meet your individual needs.
Disability Services
Student Guidance Centre
91a University Road
Belfast BT7 1NN
Tel: +44 (0)28 9097 2727
Email: disability.office@qub.ac.uk
http://www.qub.ac.uk/disability
The Income and Student Finance Office provides advice on course tuition fees, including the assessment and collection of fees. If you have any concerns about your fee assessment speak to staff in the office who have experience in advising student on these matters. They administer a range of bursaries and student support and hardship funds, to help students in financial difficulty, which do not need to be repaid. They also provide a finance function for the University's Clubs and Societies.
Student Finance and Fees
Student Guidance Centre
91a University Road
Belfast BT7 1NN
Tel: +44 (0)28 9097 2767
Email: IncomeOffice@qub.ac.uk
http://www.qub.ac.uk/directorates/sgc/finance
The Students' Union Advice Centre also has two members of staff who provide advice and guidance on personal finance, debt management, income maximisation and applying for bursaries. They can see students between 9.30am  4.30pm, Monday  Thursday and 9am  3pm on Friday. If you can only come outside of these hours, they will do their best to accommodate you.
Connie Craig
Financial Adviser
Tel: +44 (0)28 9097 1049
Email: connie.craig@qub.ac.uk
Debbie Forsey
Money Management Adviser
Tel: +44 (0)28 9097 1166
Email: d.forsey@qub.ac.uk
Students with a Belfast address, even if they only live there during the week, are strongly encouraged to register with a General Practice (GP) surgery close to the University  although it must be within a 10 mile radius of your address. If you are ill and need a doctor's note relating to your studies, you must see a GP as soon as possible  your School will have a policy indicating the length of time after your absence that a GP note must be submitted. It is also very important to be registered with a local GP surgery if you are suddenly and unexpectedly ill and require GP (nonemergency) attention. Students from within the United Kingdom can switch back to their ‘home’ GP during summer break.
The University Health Centre (UHC) at Queen's offers studentfocused NHS services and University funded nonNHS services for Queen's students. The UHC has extensive experience in the health needs of young adults and is made up of a friendly team who understand university life. International students in the UK for six months or more on a student visa are entitled to free NHS care and can also register with the practice. Visit our website or phone the Health Centre for more information on how to register.
University Health Centre
5 Lennoxvale
Belfast BT9 5BY
Tel: +44 (0)28 9097 5551
Email: reception.157@uhcq.gp.ni.nhs.uk
http://www.universityhealthcentreatqueens.co.uk
The Learning Development Service is available to help you with academic skills. You can have a onetoone appointment and/or attend a range of workshops on topics including essay writing, referencing, time management, presentation skills and preparation for exams. You can find out more on their website or by calling into the Student Guidance Centre to make a free appointment.
Learning Development Service
Student Guidance Centre
91a University Road
Belfast BT7 1NN
Tel: +44 (0)28 9097 3618
Email: lds@qub.ac.uk
http://www.qub.ac.uk/sgc/learning
Speaking to your Personal Tutor or Supervisor and using some of the material on the Student Gateway site can be helpful ways of supporting your studies. You may also benefit from more specific help. See the section on Learning Development Service for details on the support they can offer you. The Learning Development website also offers excellent resources on referencing, essay writing, time management and stress management, as well as a range of other topics that will help you in your studies.
The goal of Information Services at Queen's is to provide the highest quality information resources and services to students and staff of the University. This commitment to quality is well illustrated by the building of The McClay Library, which blends the best features of a traditional library with the latest learning technologies to create a truly 21stcentury environment for students and staff. There are also further libraries: the Medical and Healthcare Library (across four sites) and the AgriFood and Biosciences Institute Library (in the main building on the AFBI Headquarters site in Newforge Lane), as well as extensive online resources.
Information Services also supports student computing, with student computing areas across the campus. The student computing web pages provide a range of information to support the use of computing in your studies; information includes the status of key computing services and computers currently free on campus, as well as information about accessing the wireless network, training and the virtual learning environment: http://www.qub.ac.uk/student.
The facilities at Queen's Sport are second to none. Whatever your interest  performance sport or just recreation  you are sure to find an activity that suits you!
We have four main sites; our flagship PEC Sports Centre in Botanic Park, Malone Playing Fields, the Boat House at Stranmillis, and our cottage at the base of the Mourne Mountains.
The PEC has a stateoftheart gym, squash courts, climbing wall, swimming and diving pool plus much more. Flexible membership packages and rates are available to all students. Our Activity Programme also has much to offer, with an unrivalled choice of classes and courses, all of which are delivered by qualified instructors.
Queen's has over 50 Student Sports Clubs catering for all sporting interests.
The Malone site has grass pitches for both competition and training, sanddressed and waterbased hockey pitches and a 3G playing area. This site is currently undergoing a multimillionpound investment, and is due for completion in Summer 2011. For further information, please see information on the facilities.
For information on our memberships, please visit: http://www.queenssport.com/sites/QueensSport/Membership.
Queen's University Belfast
Physical Education Centre
Botanic Park
Belfast BT9 5EX
Tel: +44 (0)28 9068 1126
Email: sport@qub.ac.uk
http://www.queenssport.com
The Students' Union offers a range of membership services including entertainment venues, food and other retail outlets, nonalcoholic study space in The SPACE, a student enterprise centre, the Students' Union Advice Centre, clubs and societies, student volunteering, campaigns and representative work and much more.
Every student of the University is automatically a member (which means there are about 20,000 members). Open 18 hours a day during term time, the Students' Union welcomes over one million visitors every year. It is recognised by the University as the representative body of students and is run by elected fulltime Student Officers (Sabbaticals) and studentcentred staff.
The Sabbatical Officers, management and staff, work with the student body to ensure the improvement of facilities and support services for students of Queen's.
Queen's Students' Union
University Road
Belfast BT7 1NF
Tel: +44 (0)28 9097 3106
Email: studentsunion@qub.ac.uk
http://www.qubsu.org
If you do have financial pressures that mean you have to work more hours than is advisable, please come and talk to us. Both the Student Income and Finance Department in the Student Guidance Centre and the Students' Union can give you advice on funds that are available to help students in your position. The Learning Development Service can also offer advice on time management.
The Language Centre provides a wide range of language courses for all students. Classes, which usually last two hours, are held weekly and usually run for most of the academic year. Languages currently offered are: Arabic, Bengali, Chinese (Mandarin), Czech, Dutch, Finnish, French, German, Greek (Modern), Hindi, Irish, Italian, Japanese, Korean, Latin, Polish, Portuguese, Russian, Sign Language, Spanish, Swedish, Turkish and Urdu. These are all taught at various levels ranging from beginners in all languages to advanced level in the more popular languages. These courses are accredited in line with the Common European Framework of Reference and a Language Centre Certificate is awarded for over 70 per cent attendance. Language Centre courses are included in the Degree Plus Award programme. A small administration fee is levied for each 16week course.
Apart from the courses for nonspecialists, the Language Centre now offers courses leading to a Certificate in Languages for Special Purposes. Specially designed computerbased courses are available in: French, German and Spanish for Business, Practical Irish, French, German, Spanish and Italian for Tourism and Leisure. These fullyaccredited courses are available at beginners, intermediate and advanced levels. (French commences at postGCSE level). Courses can be accessed using the Language Centre's CAN8 multimedia online system affording students the opportunity for guided autonomous learning alongside tutorled sessions. The nature of these online courses means that students can study in their own time and at their own pace allowing them to ‘catch up’ or to progress at their chosen pace. Upon successful completion of the course, students will be awarded a Queen's University Certificate in Languages for Special Purposes. An administration fee is levied for each module leading to the Certificate.
Apart from the formal language learning opportunities, the Centre also provides the opportunity for selfstudy language courses for use in its private study area. There is a growing library of resources in over 30 languages, available at various levels and in various formats ie books, videos, DVDs, tapes, CDROMs and selfstudy online courses. Carefully designed selfstudy packs are available in most languages. Software installed on the PCs includes dictionaries, grammar packages and interactive CDROM courses for all levels and in many languages.
The Language Centre is open for private study and class teaching from 9am  9pm Monday to Thursday and 9am  5pm on Friday. Staff are available for guidance and assistance during opening hours. The selfstudy facilities are open during normal library hours.
For further information contact:
The Language Centre
The McClay Library
Tel: +44 (0)28 9097 6178
Email: langcent@qub.ac.uk
http://www.qub.ac.uk/lc
Prerequisite: This course is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The modules begins with a revision of basic
calculus and elementary functions and introduces Maclaurin series and
complex variables. The module then introduces two essential
techniques used in applied mathematics: namely, differential equations
and vectors. These techniques find many applications. One of the oldest
applications is to dynamics. Following in Newton's steps, we will study the
motion of a particle subject to various forces.
Contents
Basic calculus: numbers; definition of a function and limit;
continuous and discontinuous functions; examples of elementary functions;
radians and trigonometric functions; derivatives and differentials;
integration (indefinite and definite integral, methods of integration);
Maclaurin series; complex numbers and Euler's formula.
Differential equations (DE); firstorder DE: variable
separable, linear; secondorder linear DE with constant coefficients:
homogeneous and inhomogeneous.
Vectors: definitions and notation, operations on vectors,
application to geometry (ratio theorem, centroid of a triangle), basis and
components, scalar and vector products, triple products, equations of a plane
and straight line.
Newtonian mechanics: position, velocity and acceleration
(tangential and normal), motion under gravity, Newton's laws, motion
under gravity with resistance,
oscillatory motion, forced oscillations, damped oscillations.
Collisions: momentum, kinetic energy, centre of mass, restitution.
Planetary motion: central forces, angular momentum, plane polar coordinates, orbital and escape velocities, Kepler's problem and Kepler's laws.
Assessment
During the course there will be a onehour test which will count as 10%
of the total final mark. The remaining 90% comes from a 3hour written
examination. The paper will consist of two sections: there will be 4 questions
in each section, and 3 questions in each section must be attempted.
Corequisite: AMA1001
Introduction
The aim of this module is to develop
skills in the calculus of several variables and to apply this theory to
solve differential equations such as the diffusion and wave equation,
as well as investigate the properties and applications of scalar and
vector fields. Module AMA1001 introduces calculus for a function of a
single variable. Essentially, this theory teaches us how to solve
onedimensional problems. However, the world is a threedimensional
environment and many of the interesting and useful applications of
mathematics are formulated in two, three and fourdimensional calculus.
In this module, we consider functions of several variables. We learn
how to define integrals and derivatives in 2D and 3D and apply this
knowledge to solve problems of practical interest in acoustics,
biology, engineering, physics and chemistry.
Contents
Series: Convergence of infinite series  D'Alembert ratio test;
Rolle's theorem and mean value theorem; Taylor series; L'Hôpital's rule.
Functions of several variables: Partial derivatives and multiple integrals. The chain rule for differentiation. Taylor series in two independent variables.
Stationary points in 2D and 3D. Functions of space and time.
Diffusion and Wave equations: Partial differential equations:
diffusion and wave equations in one space and one time dimension.
Separation of variables method.
Periodic functions and Fourier series. Applications to random processes, heat transfer and wave motion. Mathematics of music.
Functions in 2D and 3D space: Vector algebra. Vector equations
of lines and planes. Cartesian and plane polar coordinates. Jacobians.
Double and triple integrals.
Cylindrical and spherical coordinates, volumes of 3D shapes. Curves in
3D and tangent vectors. Equation of normal vectors and tangent planes
to surfaces.
Vector fields: Scalar and vector fields. Partial differential
operators. Div, grad and curl. Directional derivatives, normal
derivative. Tangent line integrals, circulation, work by a force, potential
energy, conservative fields. Surfaces in 3D, surface integrals. Gauss's
divergence theorem. Green's theorem and Stokes' theorem.
Assessment
There is one 1hour Class Test worth 10% of
the final mark and a Computational Mathematics exercise worth 10% of the final
mark. The remaining 80% comes from a 3hour written examination. The paper
will consist of two sections: there will be 4 questions in each section, and
3 questions in each section must be attempted.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Probability is an important branch
of statistics that deals with chance and how to predict whether a
result is likely or unlikely. The applications of probability are
diverse, occurring in industry, mathematics, science, technology,
medicine, etc. Probability models can be described in terms of random
variables (which are functions of the outcomes of a random experiment),
usually described as being discrete or continuous. The probability
distribution of a discrete random variable is a list of probabilities
associated with each of its possible values. An example of a discrete
random variable is the outcome of throwing a dice. Alternatively, there
are other situations which are better described by a continuous random
variable such as the measure of height. The probability density
function of a continuous random variable is a function which can be
integrated to obtain the probability that the random variable takes a
value in a given interval.
Operational Research (OR) helps to improve operations in business and governments through the use of scientific methods and the development of specialised techniques. OR provides systematic and general approaches to problem solving and decision making, regardless of the nature of the system, product, or service. The approaches used in OR are based on one or more of mathematical methods, simulation, and qualitative or logical reasoning. OR originated in Great Britain during World War II to bring mathematical or quantitative approaches to bear on military operations. Since then OR has evolved to be applicable to the management of all aspects of a system, product, or service, and hence is often referred to as Systems Science or Management Science. It has now become recognised as an important input to decisionmaking in a wide variety of applications in business, industry, and government.
The course provides an introduction to probability theory and operational research, the foundation of further courses in Statistics and OR. No prior knowledge of Probability, Statistics or OR is assumed.
Contents
Probability: Probability theory, probability axioms,
conditional probability. Random variables, discrete and continuous
probability distributions. Properties and applications of standard
discrete distributions: geometric, binomial, negative binomial,
Poisson, hypergeometric; and standard continuous distributions:
uniform, exponential, normal (Gaussian). Bivariate distributions.
Operational Research: Elementary linear programming: simplex method. Decision making under uncertainty. Random sampling and simulation.
Assessment
During the course there will be a one hour test which will count as 10%
of the total final mark. The remaining 90% comes from a 3hour written
examination with two sections, A and B. Section A, accounting for 60 %
of the examination mark, is compulsory and comprises a number of short
questions of variable length designed to test the student's basic
understanding of the module. Section B, accounting for 40% of the
examination mark, comprises 6 longer questions of equal length, with 4
questions to be attempted.
Corequisite: SOR1001
Introduction
When we wish to know something about
a particular population it is usually impractical, especially in large
populations, to collect data from every unit of that population. It is
more efficient to collect data from a sample of the population under
study and, from that sample, make estimates of the population
parameters. The method of using samples to estimate population
parameters is known as statistical inference. The course provides an
elementary introduction to statistical inference; the body of
principles and methods underlying the statistical analysis of data. The
methods draw on the probability distributions previously discussed in
SOR1001.
Contents
Data Collection and Preliminary Analysis: Statistical models.
Sampling. Initial data analysis: discrete and continuous variables.
Descriptive statistics
and graphical/diagrammatic representations.
Descriptive statistics for ungrouped data; boxplot.
Frequency table for grouped discrete data: relative frequency,
cumulative frequency, bar diagram; sample mean, variance, standard
deviation and percentiles.
Frequency table for grouped continuous data: stemandleaf plot,
histogram, cumulative percentage frequency plot; sample mean, variance,
standard deviation and percentiles. Linear transformations.
Bivariate data; scatter diagram, sample correlation coefficient.
Estimation: Properties of an estimator, point estimates, unbiasedness, efficiency and the three methods of estimation.
Methods of moments, maximum likelihood and least squares. Regression.
Hypothesis Testing: Parametric methods: tests based on the Normal distribution, tdistribution, Fdistribution and χ^{2} distribution.
Nonparametric methods: tests on contingency tables, testing for association, homogeneity,
Sign test, Wilcoxon signedrank test, MannWhitney test and the run test.
Statistical Quality Control: Shewart control charts; upper and
lower control and warning limits. Analysis of patterns. Control charts
for attributes and for variables.
Assessment
During the course there will be a one
hour test which will count as 10% of the total final mark. The
remaining 90% comes from a 3hour written examination with two
sections, A and B. Section A, accounting for 60% of the examination
mark, is compulsory and comprises a number of short questions of
variable length designed to test the student's basic understanding of
the module. Section B, accounting for 40% of the examination mark,
comprises 6 longer questions of equal length, with 4 questions to be
attempted.
Prerequisite: AMA1001 and AMA1002
Introduction
Mechanics has always been a source
of inspiration for mathematics and mathematicians. Calculus, as well as
the famous Newton's laws, were “invented” by Newton largely because he
wanted to solve a single but very important mechanical problem, the
problem of planetary motion. Mechanics was at the beginning of such
branches of mathematics as theory of functions, calculus of variations,
differential equations and more recently, theory of chaos. Mechanics
and its mathematical methods are important in optics, electromagnetic
theory, statistical mechanics, quantum mechanics, theory of relativity,
quantum field theory and many ‘nonphysical’ applications such as
theory of optimisation and control.
For students, besides giving a comprehensive picture of mechanical phenomena and teaching how to solve a wide variety of problems, Classical Mechanics offers a unique opportunity to see the mathematical methods they have learned at work and to practice their mathematical skills. Vectors, partial derivatives, single and multiple integrals, differential equations, stationary points, complex numbers, as well as matrices and determinants are all among the tools used in the course.
For those who have studied mechanics before this course offers a completely new look on familiar facts, and shows a more universal approach to solving mechanical problems. For those who have not had much experience with mechanics the course will provide a comprehensive introduction into the subject, as it does not require any previous physics knowledge. Finally, for those interested in theoretical physics, the course will serve as a first step into this wide and extremely interesting area.
Contents
Introduction: Basic revision of Newtonian mechanics: conservation of mechanical energy and angular momentum.
Revision of key concepts in linear algebra (matrices): eigenvalues and eigenvectors.
Revision of basic multivariable calculus (partial differentiation and multiple integration).
Motion of a single particle in a central potential: Attractive forces: planetary motion and transfer orbits.
Repulsive forces: Rutherford scattering.
Rotating frames of reference: Angular velocity, effect of the Earth's rotation; apparent gravity, cyclones and anticyclones,
Foucault's pendulum, Larmor precession.
Conservation laws for a system of particles: internal and external forces, conservation of energy, centre of mass as the origin.
Rigid body motion: Moments and products of inertia, parallel axis theorem, Euler's equations of motion, Euler angles and rotation matrices.
Lagrange's equations: Constraints, equations for holonomic constraints; examples.
Motion of a top: Precession and nutation; stability of a sleeping top; gyroscopes and the gyrocompass.
Lagrange's equations for impulses: Applications to systems of rods.
Small oscillations: Normal modes of oscillation.
Lagrange's equations with nonholonomic constraints: Constrained optimisation; Lagrange multipliers.
Assessment
One threehour written examination with 7 questions: all questions carry equal marks; the best 5 answers are credited.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 2 of either an MSci or a BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This is a core Level 2 Applied
Mathematics module. It provides an introduction to important techniques
of Applied Mathematics. Students completing the module will be able to
understand and apply these methods to solve problems. These skills are
required for further studies in science and mathematics. Often physical
problems are framed in terms of linear equations, requiring their
numerical solution with modern computers. Students will have the basic
knowledge of linear algebra necessary to study such highlydeveloped
numerical and computational methods. Functions of a complex variable
and associated calculus lead to a range of techniques that enable
analytical solutions to physical problems. These techniques link up
also with other modules later on. One of the most
powerful and elegant applications of complex analysis is in the
calculation of integrals, and the module places an emphasis on that.
Contents
PART A, Linear Algebra: Matrix algebra; Gaussian elimination;
LUfactorization; matrix inversion; vector spaces; solution of m
equations in n unknowns; linear independence, null spaces, row and
column spaces; inner products of vectors; fundamental theorem of linear
algebra; projections and the leastsquares approximation; determinants;
complex matrices; eigenvalues and eigenvectors; matrix diagonalisation,
positive definite matrices.
PART B, Complex Functions: Algebra; analytic functions;
CauchyRiemann equations; Cauchy's theorem; infinite series; Taylor's
theorem; Laurent expansions; theory of residues; evaluation of
integrals using the residue theorem.
Assessment
One 3hour written examination.
The paper has 8 questions divided into two sections:
Section A  linear algebra,
Section B  complex functions.
In each section 3 questions must be answered from a choice of 4.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 2 of either an MSci or a BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Numerical Analysis is concerned with
devising methods for finding approximate, numerical solutions to
mathematically expressed problems. The methods are analysed for their
accuracy, efficiency and robustness. As a simple example, consider
solving the equation f(x) = 0, where f(x) is a specified function. If f(x)
is of even moderate complexity, we will not be able to solve this
equation analytically. In Numerical Analysis we develop different
procedures, or algorithms, to solve this problem. Finding the most
suitable requires an appreciation of the methods. For example, some
will guarantee convergence to a solution, but may require much effort,
while other methods may converge quickly with less effort, but may also
diverge, depending on the function and initial guess. Faced with such
differing behaviour of the methods, what is the ‘best’ strategy to
adopt? In AMA2004 we cover the basic introductory material of Numerical
Analysis. We investigate the solution of equations, interpolation,
function approximation, differentiation, integration and the solution
of ordinary differential equations.
An important element of this module are the practical classes, in which algorithms developed during the lectures are implemented using MATLAB, which is an interactive and programmable software package. The practical work culminates in a numerical project, in which a MATLAB program is written to implement a particular algorithm, or to investigate the behaviour of a method. Although prior familiarity with computers is helpful, it is not essential.
A third element of the module is the oral presentation. You are required to give a 5minute presentation to a small group of students and staff.
Contents
Introduction and basic properties of errors: Introduction;
Review of basic calculus; Taylor's theorem and truncation error;
Storage of nonintegers; Roundoff error; Machine accuracy; Absolute
and relative errors; Richardson's extrapolation.
Solution of equations in one variable: Bisection method; Falseposition method; Secant method; NewtonRaphson method; Fixed point and onepoint iteration; Aitken's δ^{2} process; Roots of polynomials.
Solution of linear equations: Gaussian elimination; Pivoting
strategies; Calculating the inverse; LU decomposition; Norms; Condition
number; Illconditioned linear equations; Iterative refinement;
Iterative methods.
Interpolation and polynomial approximation: Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.
Approximation theory: Norms; Leastsquares approximation;
Linear leastsquares; Orthogonal polynomials; Error term; Discrete
leastsquares; Generating orthogonal polynomials.
Numerical quadrature: NewtonCotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature.
Numerical solution of ordinary differential equations:
Boundaryvalue problems; Finitedifference formulae for first and
second derivatives; Initialvalue problems; Errors; Taylorseries
methods; RungeKutta methods.
Assessment
The module is assessed by: an
examination of 3 hours duration; a project undertaken during the
semester; and an oral presentation given during the semester. The
examination is worth 60% of the final mark, the project is worth 30%,
and the presentation is worth 10%. The project is assessed through a
report to be handed in at the end of week 12. The project and the
presentation are compulsory, and a score of at least 40% is required
for both the project report (i.e., 12 out of 30 marks) and the
presentation (i.e., 4 out of 10 marks) to pass the module. In the
written examination 5 questions must be attempted from a choice of 7.
Prerequisite: AMA1002
Introduction
This course provides an elementary
introduction to the mechanics of nonviscous and viscous fluids. The
course uses the methods of vector field theory that have been
introduced at level 1 in course AMA1002, and it is thus a natural
followon from AMA1002. It also provides a useful link to several
level3 courses, such as AMA3001 Electromagnetic Theory and AMA3003
Tensor Field Theory.
The course starts with a revision of the main aspects of vector field theory that are needed later in the course. This requires several extensions of the material covered in AMA1002.
The main body of the course is concerned with fluid mechanics. The fundamental equations are derived. The theory is then applied to situations in which the equations simplify to the extent that analytic solutions are possible. This course does not consider computational solutions. It is hoped that, in addition to some reasonably challenging mathematics, the course contains enough simple treatments of everyday phenomena that all level2 students will find manageable, stimulating and rewarding.
Contents
I. Review of Vector Field Theory (AMA1002)
Vector
algebra: scalar product; vector product; triple products. Coordinates:
Cartesian; general orthogonal curvilinear coordinates; spherical polar;
cylindrical polar. Scalar Fields: gradient; timedependent fields.
Vector fields: flux and divergence; Gauss divergence theorem;
lineintegrals and curl; Stokes theorem. Miscellaneous topics: ∇
notation; ∇^{2}; identities; Green's Theorem.
II. Fluid Mechanics
1) General: Basic definitions and properties: materials; solids,
plastics and fluids; density; pressure; equation of state; body forces;
surface forces; viscosity. Velocity: pathlines; streamlines; boundary
conditions. Continuity equation; incompressible fluids; streamtubes.
Euler's Momentum Equation.
2) Applications: Hydrostatics; Sound waves.
3) Vorticity and circulation: Definitions; Vortex lines and tubes; Vorticity equation; Kelvin's Circulation Theorem.
4) Bernoulli's equation: Derivation and conditions; Simple examples; Class demonstrations; Open channel flows.
5) Twodimensional flow of incompressible Fluid: Stream function:
properties; flux; vorticity; solid boundaries. Some model flows. Steady
flow of an inviscid liquid past a cylinder.
6) Irrotational flow in two dimensions: Review of complex variables.
Complex potential: definition; some model flows; Shifts and rotations;
examples. Image theorems for walls and circles. Conformal mapping:
general theory; particular mappings; the Joukowski transformation; the
cambered aerofoil.
7) Irrotational flow in three dimensions: Method of separation of
variables. Spherical polar solution; Cylindrical polar solution;
Cartesian solution. Gravity surfacewaves.
8) Viscous fluids: The stress tensor; surface forces revisited;
deformations and the strain tensor; Newtonian fluids. NavierStokes
equation: general derivation; incompressible fluids; boundary
conditions. Examples. Laminar and turbulent flow; Reynolds and Froude
numbers; dimensional analysis.
Assessment
One 3hour examination with 5 questions to be attempted from a choice of 7.
Prerequisite: SOR1002
Introduction
The course builds upon the
probability theory in SOR1001 and the statistical inference methods in
SOR1002 to provide a secondlevel account of the principles and most
important methods of estimation and hypothesis testing.
An important element of this module will be a weekly practical data analysis lab using the SAS software package. SAS is probably the leading statistical package used in industry. The lab sessions, approximately 2 hours long, will give students an opportunity to put the theory of lectures into practice. During the lab sessions students will be guided through practical tutorials on reading, accessing and editing data in SAS, descriptive statistical analyses, cross tabulations, graphs, charts, frequency tables, χ^{2} tests, ttests and analysis of variance tests. Some of the lab time will also be allocated for project work.
The practical work in the lab sessions culminates in a group project. During the semester the student will work as a team member on a specific statistical investigation using the methods described in lectures and performed in the SAS lab sessions. This provides a valuable insight into the applications of statistics in industry allowing students to gain the necessary skills required for such a working environment. Near completion of the group project, each group will be required to give an oral presentation (approximately 5 mins per student) on the results of their analysis. This provides an opportunity for the group to receive feedback on their progress and helps consolidate their findings before submitting a final written report on their investigation.
Contents
Statistical Investigations: Understanding the problem. Collecting the data. Initial data analysis. Definitive analysis: modelling. Conclusions.
Initial Data Analysis: Data structure. Processing data: coding, input, screening, editing and modifying. Data quality.
Preliminary analysis: measures of location, dispersion; tables; graphs.
Sample Diagnostics: Testing for independence  nonparametric
tests, serial correlations. Testing for normality  skewness and
kurtosis, goodnessoffit tests, probability plotting. Identification
of outliers. Transformations.
Point Estimation of Parameters: Definitions of estimate, estimator,
sampling distribution. Unbiasedness. Relative efficiency. Bias. Mean
squared error.
Sufficiency: FisherNeyman factorization theorem. Regular exponential class of distributions.
Maximum Likelihood: Likelihood function. Calculation of MLE. Log relative likelihood function. Asymptotic properties of MLE. Applications.
Least Squares Estimation and Linear Regression: Standard
linear model: matrix notation. Properties of LSE. Weighted least
squares. Fitting a straight line. Multiple regression. Goodnessoffit:
residuals. Hypothesis tests and confidence intervals using the tdistribution.
Experimental Design and Comparative Studies: Principles of
design  experimental unit, treatment, replication, randomization;
factorial design. Analysis of variance. Completely randomized design.
Randomized block design. Dichotomous treatment/risk and outcome
studies. Sampling schemes  crosssectional; longitudinalcohort,
casecontrol study.
Measures of association: rates, relative risk, odds ratio.
Significance Tests and Hypothesis Testing: NeymanPearson
approach  critical region, Type I and Type II errors, significance
level, power function. Best critical region. Generalized likelihood
ratio test.
Computer intensive methods: randomization tests; Monte Carlo sampling.
Confidence Intervals: Construction  pivotal quantity; MLE procedure. Confidence region. Prediction interval.
Bayesian Methods: Prior and posterior distributions. Conjugate
families. Point estimates, confidence regions, hypothesis testing.
Prediction. Improper and noninformative priors.
Assessment
One three hour written examination with 5 questions to be attempted out
of a choice of 7. The examination is worth 70% of the final mark, the
project report is worth 20%, and the oral presentation is worth 10%.
All aspects of the module assessment are compulsory: each student must
sit the examination, submit a written report and give an oral
presentation.
Prerequisite: SOR1001
Introduction
This course applies mathematical
analysis to a series of problems which occur in business and industry.
The analysis can be more far reaching if we use a deterministic model
but a degree of uncertainty (e.g. about future events) is often an
important feature of the situation and a stochastic model has to be
used. The statistical knowledge assumed is that contained in SOR1001.
Although novel ways of setting out the work may be used in some topics,
the mathematical techniques required on this course are no more
advanced than simple calculus and algebra and most practical problems
require only arithmetic and the use of tables.
The aim of the course is to teach a range of simple techniques illustrating the application of mathematics and probability theory to the problems of business and industry. Apart from the first two chapters each chapter is a distinct and separate topic. Some topics (e.g., Forecasting) involve lengthy calculation and students are taught how to use a spreadsheet for the computation. Students who do not have access to a spreadsheet on a personal computer can use the Open Access Areas. Specific instructions on the use of the Excel spreadsheet is given on the course and there is a practical session in an Open Access Area.
Homeworks are an essential part of the learning process, but there is no continuous assessment.
Emphasis is placed on choosing the correct model for the circumstances and on presenting answers in a form intelligible to management. If a question is posed in words then the final answer should be in words and not left in algebra or in a table. The practical problems associated with obtaining data are discussed. The answer should be to a number of significant figures consistent with the accuracy of the original data, or rounded to an integer if that is appropriate.
Contents
Deterministic inventory including
quantity discounts, common cycle production, constrained inventory and
the use of Lagrange multipliers.
Stochastic inventory models including service levels. Simple and
adaptive forecasting. The use of spreadsheets and their application to
forecasting and equipment replacement. The replacement of deteriorating
equipment and the replacement of equipment liable to sudden failure.
Acceptance sampling by attribute and variable. Network planning
including PERT, speeding up, the use of LP, Gantt charts and resource
smoothing. Decision analysis including utility curves, decision trees
and Bayesian statistics.
Assessment
One 3hour written examination in which 5 questions are to be attempted out of a choice of 7.
Corequisite: SOR2002. Make sure you are enrolled for this module in the 1st semester. This corequisite is frequently overlooked!
Introduction
The aim of this module is to cover
linear models encompassing multiple linear regression and analysis of
variance (ANOVA). These models are the workhorses of statistical data
analysis and are found in virtually all branches of the sciences as
well as in the industrial and financial sectors.
Multiple linear regression is concerned with modelling a measured response as a function of explanatory variables. For example, a pharmaceutical company might use a a regression model to relate the effectiveness of a new cancer drug to the patients age, gender, weight, diet, tumour size, etc. ANOVA is concerned with the analysis of data from designed experiments. A materials manufacturer for example, may wish to analyse the results from an experiment to compare the heat resisting properties of four different polymers.
Regression and ANOVA will be initially developed using a classic least squares approach and later the correspondence between least squares and the method of maximum likelihood will be examined. After a thorough development of linear models the groundwork will have been laid to allow an extension to the broader class of Genealized Linear Models (GLM). These permit regression models to be applied to situations where the recorded response is not normally distributed. One famous example of the use of GLM was the analysis of Oring failures on the space shuttle Challenger.
An important element of this module will be a weekly practical data analysis class using the SAS software package. SAS is probably the leading statistical package used in industry. These classes, lasting up to three hours, will introduce the student to elementary data entry in SAS, elementary matrix manipulation using the SAS Interactive Matrix Language (IML) and analysis of data using linear and generalized linear models. Each week the student will complete a data analysis task using SAS and is required to submit a report the following week.
Contents
Multiple linear regression: ordinary least squares, model selection and diagnostics, weighted least squares.
Analysis of variance: Nonsingular and singular cases; extra
sum of squares principle, analysis of residuals, generalized inverse
solution, estimable functions, testable hypotheses.
Experimental designs: completely randomized, randomized block, factorial, contrasts, analysis of covariance.
Generalised linear model: maximum likelihood and least squares, exponential family, Poisson and logistic models, model selection for GLM.
Assessment
One three hour written examination with 5 questions to be attempted out
of a choice of 7, comprising 80%. Ten weekly assignments worth a total
of 20%.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The ideas of electromagnetic theory
are developed at a comparatively sophisticated mathematical level
making extensive use of the methods of vector calculus. The module
develops the subject from the simplest situation  charges at rest
(electrostatics), through charges in uniform motion (magnetostatics) to
charges in nonuniform motion (electrodynamics). The full Maxwell
equations appropriate to the latter general situation are built up from
simpler forms, with their origins in experimental findings.
Students completing this module will understand the mathematical treatment of electromagnetism and through solving the problems, will have experience in mathematical modelling. They will be fluent in the use of vector calculus. The methods used here find application in more advanced studies of theoretical physics.
Contents
Electrostatics: electric field; Gauss's law; Poisson's
equation; Laplace's equation; polarisation; electric displacement;
boundary conditions in electrostatics; methods for solving Laplace's
equation; electrostatic energy density.
Steady currents: electric current; equation of continuity.
Magnetostatics: magnetic scalar and vector potentials;
magnetic dipole; magnetic field; magnetisation; boundary conditions in
magnetostatics; potential problems.
Electromagnetic induction: electromotive force; magnetic energy; energy density of the magnetic field.
Maxwell's equations: electromagnetic energy; electromagnetic potentials; the wave equation; plane waves; electromagnetic radiation; waveguides.
Assessment
One 3 hour written examination with 4 questions to be attempted out of a choice of 6.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Together with relativity, quantum mechanics defines our understanding of nature
gained throughout the 20th century and beyond. Though the development of
quantum theory is one of the greatest intellectual achievements this
introductory course does not require advanced mathematical skills or
advanced knowledge of physics.
After a brief historical survey, the discussion of Young doubleslit experiment and the PlanckEinsteinde Broglie relations, we introduce the mathematical foundations of quantum mechanics, introducing Dirac notation and the state space. We then discuss to the Postulates of quantum theory, giving particular emphasis to the Heisenberg uncertainty principle and the Schrödinger equation, which is derived and discussed in depth. To illustrate its key role in quantum theory, we address some exactly solvable simple problems, including the motion of a particle in various potentials, the spin, a simple harmonic oscillator, a particle in a central potential field and the hydrogen atom. The last part of the module is dedicated to the introduction of the perturbative technique to address problems that cannot be solved exactly (they are the majority, actually). We thus develop both the timeindependent and the timedependent perturbation theory, and apply it to a number of important (yet simple) problems.
Although the course is largely selfcontained and has no specific prerequisites from Level 2, advantage would be gained by taking AMA2001 Classical Mechanics and AMA2003 Methods of Applied Mathematics.
Contents
Introduction to the fundamentals of quantum mechanics: Historical
context; light quanta and PlanckEinstein relations; Young's double slit
experiment; de Broglie relations; wave functions (introduction).
Dirac notation and state space: notion of ket and bra; linear
operators; Hermitian conjugation and Hermitian operators; representation of
kets/bras and operators; position and momentum representations; tensor
products of state spaces (with applications).
Postulates of quantum mechanics: statement of the postulates (including
measurement process); physical interpretation of the postulates (measurement
of observables, mean and standard deviation, compatibility of observables,
Heisenberg uncertainty principle, timeenergy uncertainty); derivation of
Schrödinger equation and its physical interpretation (determinism of the
unitary evolution, Ehrenfest principle, constants of motion, stationary
states); superposition principle and physical predictions (probability
amplitudes and interference, linear superpositions, degenerate
eigenvalues).
Applications of the postulates I: free particle and particle in a
box (infinite potentials; finitewell potential; step potential, including a
quick introduction to scattering).
Applications of the postulates II: Spin1/2 particle.
Applications of the postulates III: Quantum harmonic oscillator.
Applications of the postulates IV: Angular momentum in quantum
mechanics.
Applications of the postulates V: Central potential (hydrogen atom).
Timeindependent perturbation theory and application to simple problems.
Timedependent perturbation theory and application to simple problems.
Assessment
One 3 hour written examination with 4 questions to be attempted out of a choice of 6. Best 4 questions are counted.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The notion of scalar and vector
quantities is essential in the application of mathematics to physical
phenomena. For example it plays a pivotal role in dynamics: Newton's
2nd law relates the vector force to the rate of change of the vector
momentum. A vector quantity such as a particle's position is often
given as 3 components referred to a rectangular set of axes, the
familiar Cartesian components. However the physics does not depend on
the choice of reference axes. Thus the equations that describe the
motion are independent of coordinate system. The vector components in
one coordinate system will transform in a prescribed way to another
coordinate system so as to ensure that this independence is preserved.
The key idea is that of invariance with respect to coordinate
transformations.
A tensor is a set of quantities that transform in a prescribed manner
when a coordinate transformation is made. It is a generalisation of the
notion of a vector and a vector is in fact a tensor of rank 1. A scalar
is a tensor of rank 0. Tensors are the appropriate objects for
describing many physical phenomena, such as solid and fluid mechanics,
elasticity, special and general relativity.
General tensors allow coordinate transformations of any type. Cartesian tensors axe restricted to orthogonal transformations. The possible transformations are then rotations and/or reflections so that the reference axes remain orthogonal. While the theory of Cartesian tensors is simpler than that of general tensors, it still has wide application in physics, including special relativity. However general tensors are required for a study of general relativity.
In this module you will study the algebra and calculus of Cartesian and general tensors with applications in special and general relativity. The pure mathematician will delight in the tensors and the differential geometry while the theoretical physicist can wallow in the relativity. There is something here for everyone. You should be confident with vectors.
Contents
Cartesian tensors (6 lectures): Notation, Kronecker delta,
permutation symbol, determinants, Euclidean space, orthogonal
transformations, definition of a Cartesian tensor, tensor algebra,
addition and subtraction, outer product, inner product, contraction,
symmetry and skewsymmetry, quotient rule, tensor equations, tensor
densities, isotropic tensors, tensor fields and calculus, vectors in E_{3}, vector identities.
Special relativity (9 lectures): Newtonian mechanics, Galilean
transformation, Lorentz transformation, Minkowski spacetime, Minkowski
diagram, causality LorentzFitzgerald contraction, time dilation.
length paradox, relativistic mechanics, 4velocity, transformation of
4velocity, 4acceleration, transformation of 4acceleration,
4momentum and 4force, massenergy relationship, energymomentum
relationship, transformation of 4momentum, conservation of 4momentum,
potential energy, electromagnetism, Maxwell's equations, Lorentz force,
scalar and vector potentials, Lorentz gauge and waveequations, tensor
formulation of electromagnetism, electromagnetic tensor, dual
electromagnetic tensor density, invariants, transformation of fields.
General tensors (9 lectures): Notation, Riemann space,
contravariant vectors, covariant vectors, tensors, Cartesian tensors,
tensor algebra, raising and lowering indices, relative tensors, tensor
calculus, covariant derivative, constant vector fields, geodesics,
covariant derivative of tensors, geodesic coordinates, Bianchi
identities, metric affinity, 3dimensional vectors, symmetries of
curvature tensor, Ricci tensor, divergence and Laplacian, Einstein
tensor.
General relativity (6 lectures): General relativity, special
relativity, Newtonian gravitation, field equations, cosmology,
RobertsonWalker metric, Friedmann equation, models, gravitational
redshift, Schwarzschild solution, timelike geodesics, orbit equation,
advance in the perihelion of planets, null geodesics, deflection of a
light ray near the sun, blackholes, Eddington form, radial motion.
Assessment
One 3 hour examination with 4 questions to be attempted out of a choice of 6.
Prerequisite: AMA2004
Introduction
The aim of this course is to develop further the ideas introduced in
the level 2 course on Numerical Analysis. In particular, it aims to
give further insight into modern approximation techniques; to explain
how, why and when they work; and to provide a firm basis for future
study in numerical analysis and scientific computing.
The course covers three areas: approximation theory, numerical integration, eigenvalue and eigenvector approximation. In the first of these we attempt to approximate a continuous, realvalued function by a polynomial (e.g. Chebyshev polynomials) or by a rational function in order to provide computationally economical ways of evaluating the particular function. We also discuss polynomial interpolation techniques such as the Lagrange and Newton forms of interpolation and cubic splines  a technique widely used in computer aided design.
Numerical integration, or quadrature, is a hugely enjoyable subject. Like many other areas in numerical analysis, the development of good approximate integration techniques requires a wonderful mixture of mathematical rigour and intuition. One needs numerical integration when faced with definite integrals which have no analytical solution. For example, while the sine function is one of the most common mathematical functions, calculation of its length gives rise to an elliptic integral of the second kind, which cannot be evaluated analytically. In such cases approximation techniques provide the only way forward.
The solution to many physical problems requires the calculation of eigenvalues and eigenvectors of a matrix. Theoretically, the eigenvalues of a matrix A, can be obtained by finding the roots of the characteristic polynomial p(λ) = det(A−λI). In practice, p(λ) may be hard to obtain and its roots even harder to compute directly. Various elegant and powerful approximation techniques have therefore been developed to find the eigenvalues and eigenvectors of a matrix.
Contents
Approximation theory: Norms, polynomial minimax approximation,
Chebyshev polynomials, summation of Chebyshev series, Chebyshev
expansions, rational Padé approximation, rational minimax
approximation, numerical calculation of the rational minimax
approximation, piecewise polynomial approximation, polynomial
interpolation, divided differences and the Lagrange and Newton forms of
interpolation, cubic splines.
Numerical integration techniques: Review of basic techniques;
interpolatory quadrature; Gaussian quadrature  types, properties and
ways of constructing the nodes and weights, error estimation; automatic
adaptive quadrature; integration over infinite intervals.
Eigenvalues/eigenvector analysis: Definitions, vector and
matrix norms, matrix algorithms, rounding errors and error analysis,
Givens' and Householder zeroising transformations, power methods and
inverse iteration, methods for symmetric matrices, Jacobi's method,
tridiagonalisation using Householder transformations, general matrices
and the QR method.
Practical assignments: you will be asked to write and run Matlab programs for some of the algorithms discussed in lectures. You will be required to prepare a written report demonstrating your efforts in carrying out the work.
Assessment
One 3hour written examination
contributes 70% of the final mark for the module, with the additional
30% coming from the practical assignment portfolio (see above). The
examination paper consists of two sections with three questions in
each, and you will be asked to answer two questions from each section.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This module develops further some
ideas and methods initially introduced in AMA1002. It concerns those
problems in Applied Mathematics which can be formulated as differential
equations involving functions of more than one variable (e.g., position
and time, or several coordinates). These are partial differential
equations (PDE). Besides AMA1002, students deal with examples of such
equations in AMA2005, AMA3001, AMA3002, AMA3007, and SOR3012. This
course studies properties of these equations, and develops methods for
solving them.
Contents
Introduction: Terminology: order, linear and nonlinear
equations, initial and boundary conditions. Derivation of the wave
equation (in one dimension, 1D) and heat equations. Further examples:
Laplace's equation, Poisson's equation, Schrödinger equation. Method of
Separation of Variables: for the wave equation and heat equation in 1D,
and for the circular membrane. Bessel functions. Eigenvibrations
(modes).
Fourier Method: Fourier expansion of a function. Piecewise
continuous functions. Convergence of Fourier series (FS). FS for even
and odd functions. Halfrange FS. FS near discontinuities  the Gibbs
phenomenon. Application of FS to solving PDE. Laplace's equation for a
disk: Poisson's integral.
Integral Transform Methods: Fourier transform (FT) as a limit
of the complex Fourier series for the infinite interval. Notion of the
Dirac delta function. FT of even and odd functions, sine and cosine FT,
and FT of partial derivatives. Application to PDE (heat, wave,
Laplace). Laplace transform (LT). LT of some common functions,
convolution and shift theorems. Applications of LT to ordinary and
partial differential equations.
Orthogonal expansions: SturmLiouville Theory. Inner product
and norm, orthogonal systems of functions. GramSchmidt process.
Expansion of functions in orthogonal systems. Convergence in the mean
and completeness. Selfadjoint differential operators and Green's
formula. Singular, periodic and homogeneous boundary conditions.
SturmLiouville theory: properties of the eigenfunctions and
eigenvalues; degeneracy. Generalised Fourier series. FourierBessel
expansion.
Green's Functions: Dirac delta function. Green's function of
the SturmLiouville equation. Green's functions in several dimensions
(Dirichlet problem for the Laplace equation; heat and wave equations
with source terms).
Normal Forms of 2ndorder PDE in Two Variables: Linear and
quasilinear PDE. Hyperbolic, parabolic and elliptic types. Reduction
to the normal form, and use of this method for solving PDE.
Assessment
One threehour examination with the mark based on the 4 best questions out of total of 6.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Mathematical skills are highly
sought after in the financial services industries, and this employment
sector remains a favoured destination for graduates. Around 40% of
Mathematics graduates entering employment across the UK (see
www.prospects.ac.uk for recent data) go into financial services, which
includes, accountancy, retail and investment banking, mergers and
acquisitions, insurance and actuarial work, capital market trading, and
hedge fund employment, and so on.
At the low end of this sector, retail banking for example, a degree in mathematics is certainly not essential. This work is mainly concerned with simple arithmetic operations. However, at the high end of financial services, in a hedge fund for example, employers expect to see PhDlevel qualifications in mathematics from their applicants along with excellent software skills. These mathematicians are involved in the business of derivative pricing and trading and earn salaries well over 100k. Derivatives are financial products (instruments as they are called in the trade) derived from assets that have an unpredictable price. The total outstanding notional value of derivatives contracts today has grown beyond a quadrillion dollars (that's 10^{15} to you and me). It is a perilous and lucrative business!
Derivatives were originally devised to avoid risk by providing an insurance on a risky asset. Nowadays, they are an essential part of risk taking in capital markets. Indeed the speculation in buying and selling these instruments, specifically credit derivatives, precipitated the current credit crunch. Of course, this trade relies upon knowing the fair price of a derivative. Pioneering work by Black, Merton and Scholes, showed that, under certain assumptions for the unpredictability of the asset, the price of the derivative obeys a partialdifferential equation. The construction of such equations and their solution is where mathematicians come in!
The objective of the course is to provide an introduction to the mathematical techniques which can be applied to pricing problems for financial derivatives. Specifically, our focus is on stochastic calculus and the theory and practice of pricing simple derivatives such as contracts and options.
We are grateful to First Derivatives plc for their support of this course and the provision of prizes for the best examination performance.
Contents
Introduction to financial derivatives:
forwards, futures and options. Future markets and prices. Option
markets. Binomial models and the riskfree portfolio. Stochastic
calculus and random walks. BlackScholes equation. Pricing European
options. Various option pricing models. Interestrate derivatives.
Credit derivatives. Swaps.
Assessment
One 3hour written examination with 4 questions to be attempted out of a choice of 6.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
A picture is sometimes worth a thousand words.
Dynamical systems theory gives a graphical, qualitative understanding of differential equations, which arise ubiquitously in nature when properties such as position, density, or temperature change in time. Frequently these equations are nonlinear  the sum of two solutions is not again a solution  and are impossible to solve in closed form. Also, many of the quantitative details of the solutions are not of interest to us. To gain qualitative insight the graphical approach is then useful.
Dynamical systems is a new area of mathematics and has many applications: e.g., weather forecasting, how Google works, the solar system's future. This course teaches you how to draw the phase portrait of a differential equation and how to use the phase portrait to understand the long and shortterm behaviour of the solutions. Bifurcations, the butterfly effect, chaos and fractals are also introduced.
As an example, take the damped pendulum ml d^{2}θ/dt^{2} + B dθ/dt + mg sinθ = 0, where a mass m at the end of a light rod of length l rotates under the influence of gravity g and friction B. The sinθ term makes this difficult to solve: routinely sinθ ≈ θ approximation is made to give a linear differential equation. To deal with the exact nonlinear equation you will learn how this equation can be represented by the graph below: in particular why we use two dimensions for the graph and not one or three, what the curved lines mean and why the points towards which the lines swirl matter.
Contents
Flows; Fixed points and stability;
Phasespace solutions; Vector fields; Linear flows; One, Two and
Threedimensional nonlinear flows; Linearisation about fixed points;
Models of competition and predation; Limit cycles; PoincareBendixson
theorem; Bifurcations; Lorenz equations; Chaos on a strange attractor;
Iterative maps; Google; The logistic map; The Mandelbrot set; Fractals;
Fractional dimensions.
Assessment
One 3hour written examination with 4 questions to be attempted out of a choice of 6.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The aim of this course is to develop
the essential ideas and mathematics of the Calculus of Variations, with
applications to various problems.
In its simplest form the Calculus of Variations is concerned with finding a function which makes a given integral a maximum or minimum, or, more generally, ‘stationary’. A famous historical example (solved by the Bernoullis, Newton and others) is the following. A particle, which is initially at rest, slides under gravity along a smooth wire connecting two points: find the shape of the wire, which minimizes the time taken for the particle to travel between the two points. This corresponds to finding the function which minimizes the time integral. More generally, ‘variational’ problems may involve integrals containing more than one function or higher derivatives of the functions than the first, or involve constraints on the functions, which may be given in integral form or pointwise. In addition, the variational integral may be multidimensional.
In the formulation of physical laws variational methods have a much deeper significance. This was first suggested by the elegant work of Lagrange and the Irish mathematician Hamilton who looked at the basic mathematical structure of classical mechanics. The formalism that they developed is not only essential for a full understanding of quantum mechanics and statistical mechanics but turns out to have a much wider application in that it can be extended to systems that are not normally considered in dynamics, e.g., the Electromagnetic Field. In this sense variational methods provide a ‘unifying’ principle of physics.
The first part of this course concentrates on the basic mathematics of the Calculus of Variations. The second part deals with Lagrangian and Hamiltonian mechanics and their variational basis. While the course is more or less selfcontained, it does requires a knowledge of basic firstyear calculus, including functions of several variables and a smattering of ordinary differential equations. Newton's laws of motion in vector form and the elementary ideas of kinetic and potential energy are also needed.
Contents
Part I: The Calculus of Variations (approximately 15
lectures): Motivation: the Brachistochrone and Isoperimetric problems.
Functional; extremum, stationary point. Function classes. Weak and
strong variations. The simplest variational problem: necessary
condition for an extremum; EulerLagrange lemma; Euler's equation.
Several unknown functions. Fermat's principle. Geodesics. Functionals
depending upon higher order derivatives. Variational problems with
subsidiary conditions: Lagrange multipliers; finite subsidiary
conditions. Variable endpoint theorem: broken extremals;
WeierstrassErdmann corner condition. Second variation of a functional:
Legendre's necessary condition for a minimum. Direct methods: the Ritz
method; the method of finite differences; the SturmLiouville problem.
Part II: Analytical Mechanics (approximately 15 lectures):
Constraints and generalized coordinates; holonomic constraints. Virtual
displacement; D'Alembert's principle; Lagrange's equations. Action
integral. Hamilton's principle. Generalized momentum; cyclic
coordinates; conservation laws. The Hamiltonian; Hamilton's equations;
derivation of Hamilton's equations from a variational principle.
Principle of least action; Jacobi's form of the principle of least
action. Canonical transformations: generating function; symplectic
matrices and canonical transformations; Hamilton's equations in
symplectic form. Poisson brackets: Jacobi's identity; canonical
transformations of Poisson brackets; Hamilton's equations in Poisson
bracket form; Poisson's theorem. The HamiltonJacobi equation;
Hamilton's characteristic function. Actionangle variables. Phasespace
diagram.
Assessment
One threehour exam. Two questions to
be answered from Section A and two questions to be answered from
section B. There are three questions in each section.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Mathematics has always benefitted
from its involvement with developing sciences. Biomedical science is
clearly the premier science of the forseeable future. With the example
of how mathematicians have have benefited from and influenced physics,
it is clear that if mathematicians become involved in the biosciences
they will be part of a field of important and exciting scientific
discoveries.
In this module no previous knowledge of biology is assumed. With each topic a brief description of the biological background will be provided sufficient to understand the models being studied. The emphasis throughout the course is on the practical application of mathematical models in helping unravel the underlying mechanisms involved in the biological processes. The mathematics in the module is dictated by the biology and not viceversa.
By the end of the course the student should have a good appreciation of the art of modelling which relies on: (i) a sound understanding and appreciation of the biological problem; (ii) a realistic mathematical representation of the important biological phenomena; (iii) finding useful solutions, preferably quantitative; and finally what is crucially important (iv) a biological interpretation of the mathematical results in terms of predictions and insights.
Contents
The module will cover models used in population biology and ecology. These are listed below.
1. Continuous population models for single species.
2. Discrete population models for single species.
3. Models for interacting populations.
4. Biochemical genetics.
5. Biological motion including reaction diffusion models and chemotaxis.
6. Biological waves for single species models.
Assessment
One three hour exam. Two questions to
be answered from Section A and two questions to be answered from
section B. There are three questions in each section.
Prerequisite: Only available to students on the BSc Mathematics with Finance programme.
Introduction
This module should equip the students with the knowledge of mathematical,
analytical and numerical techniques applied to capital markets. It will allow
them to develop the ability to work with market data to visualise, capture and
use software to analyse financial data. It will also provide the students
with knowledge of random process, Markov chains and timeseries and will give
them capacity to develop software and perform regression analysis of
timeseries. Upon completion the students will have skills in the application
of numerical methods to analyse financial data
Contents
Capital Financial Markets: Financial institutions. Money markets and
foreign exchange. Bond market and spreads. Equity Markets. Forwards and
Futures. Swaps and Options. Trading platforms.
Financial data analysis: Discrete random walks and Markov processes.
Centrallimit theorem. Timeseries analysis. Stationarity. Autocorrelation and
regression. Curve fitting and goodnessoffit. ARIMA. Volatility and GARCH.
Simulation: Theory of arbitrage. Monte Carlo simulation. Bond pricing.
ValueatRisk. Default. Modern portfolio theory and optimisation.
Capital asset pricing model.
Derivatives: Fundamental theorem of asset pricing. Options and
Futures. Binomial models of pricing European options.
Assessment
Assessment is based on one 2hour written
examination paper (contributing a maximum of 50% to the overall module mark),
assignments (contributing up to 15% to the module mark), and project (35%).
Prerequisite: Only available to students on the BSc Mathematics with Finance programme.
Introduction
As a result of taking this module, students will learn to respond to a briefing
on a problem by a client. They will be able to work successfully as part of a
team to address the problem. They will also be able to make a final
presentation on the outcome of the work.
Contents
Business skills workshop. Presentation skills. Negotiation skills. Customer
relationship. Project management/team building.
Teams required to negotiate, plan, develop and deliver a completed task working
as a group, commissioned by the `client' company.
The project will require software development skills.
Assessment
This module has no written exam. The final mark is based on Skills (20%),
Continuous assessment (20%), Presentation (20%) and Writeup (40%).
Prerequisite: AMA1001 or PMA1012
Introduction
Operational Research is the
application of quantitative analysis to problems outside of the
physical sciences and in particular to the problems of business,
industry and administration. However, all the techniques can be used
outside of Operational Research and emphasis is given to formulation of
problems and recognition of the appropriate technique. Blackboard
examples are given for each technique and the homework problems are
drawn from a wide field. Most techniques are in the form of algorithms
and for hand calculations the problems have to be fairly simple, but
indications are given how computers could be applied to bigger problems
and computer outputs are examined.
The aim of this course is to develop competence in two of the most important mathematical techniques used in Operational Research, in formulating problems and expressing answers clearly.
This course covers the two main generalpurpose mathematical techniques used in Operational Research together with some specialized applications. These are all optimisation techniques, although they also give insight into the problems and the interpretation of answers is stressed. Calculus cannot be used because either (i) the objective function and the constraints cannot be expressed simply, or (ii) the constraints dominate the problem and the optimal solution will be at the edge of the feasible region, or (iii) the variables are not continuous (e.g., they are integer).
Some students may have met some of the techniques in courses at other levels, but no previous knowledge is assumed and more mathematical rigour and understanding is required than at those levels. However, the mathematical knowledge assumed does not extend beyond that in Level 1. The main knowledge assumed is elementary linear algebra (bases, linear dependence of vectors, matrix notation, partitioning of matrices). No knowledge of economics is required but the economic interpretation of some results is explored. Apart from a passing reference to stochastic Dynamic Programming this course is purely deterministic and no knowledge of statistics is required.
Contents
The scope of Operational Research, formulating a problem from a verbal description.
Dynamic programming:
formulation, principle of optimality, value iteration, applications
including equipment replacement, allocation, production planning and
optimal routes. Special algorithms for production planning and optimal
routes.
Linear programming: formulation, theory, Primal Simplex
Method, interpretation of the final tableau, Revised Simplex Method,
duality theory including economic interpretation, Dual Simplex Method,
Postoptimal Analysis, Transportation and Assignment problems. Network
flows: maximal flow problem, minimal cost flows, the outofkilter
algorithm. A wide variety of practical problems and applications is
discussed.
Assessment
One 3hour written examination in which 4 questions are to be attempted out of a choice of 6.
Prerequisite: SOR2004
Introduction
In the 1990's there was an explosive
growth in both the generation and collection of data due mainly to the
advancement of computing technology in processing and storage of data
and the ease of scientific data collection. As a result, overwhelming
mountains of data are being generated and stored. For example, in the
business world large supermarket chains such as WalMart and
Sainsbury's collect data amounting to millions of transactions per day.
In the US all healthcare transactions are stored on computers yielding
terabyte databases which are constantly being analysed by insurance
companies. There are huge scientific databases as well. Examples
include the human genome database project and NASA's Earth Observatory
System. This has brought about a need for vital techniques for the
modelling and analysis of these large quantities of data: data mining.
Data Mining is the process of selection, exploration, and modelling of large quantities of data to discover previously unknown regularities or relations with the aim of obtaining clear and useful results for the owner of the database. The application of data mining includes many different areas, such as market research (customer preferences), medicine, epidemiology, risk analysis, fraud detection and more recently within bioinformatics for modelling DNA.
This module will focus on data mining techniques which have evolved from and are strongly based on statistical theory.
Contents
Introduction to Data Mining.
Exploratory data analysis: Principal Component Analysis; handling big
data  reformatting data; Multiple Imputation.
Cluster analysis: Hierarchical clustering; Partitioning algorithms.
Classification: Nearest neighbour algorithms; Classification trees;
Naïve Bayes Classifier; Bayesian networks; Ordinal Regression; Multinomial
Logit; Techniques for comparing classifiers  including bagging and boosting
in ensemble methods.
Prediction (continuous targets): Regression Trees; Random Forests;
Neural Networks; Support Vector Machines; Regression splines.
Association Rule mining.
Assessment
Assessment is based on one written examination paper (contributing a maximum
of 85% to the overall module mark) and coursework (contributing a maximum of
15% to the overall module mark). The 3hour written examination paper comprises
6 questions: 4 questions are to be attempted out of the choice of 6.
Coursework is linked with the computing sessions, assessing students' use of
statistical software to produce and interpret results.
Prerequisite: SOR1001
Introduction
Uncertainty or risk is a natural
part of life. Continually faced with choices, we are required to make
decisions based on the possible consequences or outcomes. We often use
the phrase ‘a calculated risk’ to describe how we analyse a
problem and come up with our decision. Now the question is: how do we calculate
risk  mathematically.
Statistical analysis of a random process helps determine whether there are any underlying patterns. If there are, we can use this to our advantage in predicting the future  or at least make an educated guess! Not surprisingly, this topic is primarily recognised for its career opportunities where ‘risk’ assessment and planning matter: for example the provision of power stations, telecoms, transport planning, the spread of infectious diseases (swine flu, for example). A major application of this kind of mathematics is the financial services sector: insurance and investment, credit risk, capital market trading etc. Many students that have taken this course are currently employed as highlypaid actuaries.
Aside from the educational and commercial value of this subject, it leads into very interesting and fun topics such as measure theory, martingales and potential theory, noise, fractals, etc.
Contents
Introduction to stochastic processes and
some applications. Counting and combinatorics. Set algebra.
Inclusionexclusion theorem. Mutually exclusive events. De Morgan Laws.
Philosophy and axioms of probability. Events and probability spaces.
σfields.
Simple random walk  gambler's ruin. Moments and generating functions.
Binomial and Poisson distributions. Laws of large numbers.
Centrallimit theorem. Correlation and covariance.
Discrete Markov chains. ChapmanKolmogorov relation. Transient and
absorbing states. Ergodic theorem. Martingales. Hitting times.
MonteCarlo Markov chains.
Continuous Markov processes. Poisson processes. Queueing theory. Random
walks in continuous time. Chronic and infectious diseases.
Assessment
One 3hour written examination with a choice of 4 questions from 6.
Prerequisite: AMA3002 or, subject to the approval of HoT (AMA), PHY3011
Introduction
The course is intended as a follow on from AMA3002 Quantum Theory.
Contents
Assessment
One 3hour written examination with a choice of 4 questions from 6.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 4 of an MSci Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This course falls into two equal
parts. The first part is devoted to a study of partial differential
equations of the first order. Its orientation is very much geometrical.
Here the solution of a partial differential equation is envisaged as a
surface in the space of the dependent and independent variables. It is
first shown how solution surfaces of the quasilinear Lagrange equation
may be generated from families of curves. Knowledge of how to solve
this equation enables us to find, by methods due to Charpit and Jacobi,
socalled ‘complete integrals’ of the general firstorder partial
differential equation. A complete integral may be viewed as a family of
surfaces. Envelopes of this family, and of subfamilies of this family,
also solve the differential equation and give solutions known as
‘singular integrals’ and ‘general integrals’.
The second half of the course deals with linear integral equations. Often many problems can be formulated in terms of an integral equation rather than the more familiar differential equation. This can have some advantages, for example, usually the boundary conditions are automatically included in the integral equation formulation and do not have to be imposed afterwards as is usual with differential equations. In addition, linear integral equation theory represents a generalization of matrix theory in which the discrete indices of a matrix element become continuous variables.
There are many analogies between matrix theory and linear integral equation theory. For example, a linear matrix equation is solved by constructing the inverse matrix. Similarly, a solution of a linear integral equation is obtained if we can find the resolvent kernel. In matrix theory Hermitian matrices, with their eigenvalues and eigenvectors, play an important role. Analogously, Hermitian kernels, with their characteristic values and characteristic functions, play an important part in the HilbertSchmidt theory of linear integral equations.
While the course is more or less selfcontained, some basic knowledge of other areas of mathematics is required, in particular, competence at solving ordinary differential equations, a clear understanding of partial differentiation, matrix theory, and, for the section on integral transform methods for solving integral equations, some knowledge of complex contour integration (complex variable theory).
Contents
Partial Differential Equations of the First Order: Integral surface. Curves parallel to the vector (P,Q,R).
Families of curves. Surfaces generated by curves. Envelope of a family
of curves. Surfaces: normals, envelopes.
Lagrange's equation: a solution; the most general solution; integral
surface passing through a given curve; characteristic curves;
generalization to n independent variables.
The general firstorder equation: classification of integrals, complete
integral, singular integral, general integral; integral surface passing
through a given curve; Charpit's method for finding a complete
integral; Cauchy's method of characteristics; characteristic strips;
special types of firstorder equations, equations only involving the
derivatives, equations not involving the independent variables,
separable equations, Clairaut equations; Jacobi's method for finding a
complete integral, generalization to n independent variables.
Integral Equations: Fredholm and Volterra equations of the
first and second kinds; kernel. Operator notation and scalar product.
Solving integral equations : general solution; characteristic values
and characteristic functions; resolvent kernel; separable kernel;
conversion to differential equation. Connection with matrix theory.
Square integrable functions and kernels. Singular kernel. Singular
integral equation.
Use of Integral Transforms to Solve Integral Equations:
Convolution kernels. Laplace and Fourier Transforms; convolution
theorems. Application to Fredholm and Volterra equations with
convolution kernels. Abel's integral equation.
Solution of Integral Equations by the Method of Successive Approximations: Neumann series; iterated kernels; resolvent kernel; convergence properties of Neumann series.
Degenerate Kernels: Fredholm Formulae for Continuous Kernels; Fredholm determinant; recurrence relations; characteristic values.
HilbertSchmidt Theory: Hermitian kernel  characteristic
values and functions; orthonormality. HilbertSchmidt theorem. Solution
of Fredholm equation of the second kind.
Assessment
One 3 hour written examination,
divided into 2 sections containing 3 questions each and corresponding
to the first (partial differential equations) and second (integral
equations) halves of the course, with the requirement of answering 2
questions out of each section.
Prerequisite: AMA3002 or PHY3011
Introduction
Statistical mechanics is a formalism
that aims at explaining the physical properties of matter at the
macroscopic level in terms of the dynamical behavior of its microscopic
constituents. The scope of the formalism is almost unlimited as it is
applicable to matter in any state of aggregation, ranging from gases,
liquids and solids to matter in equilibrium with radiation and
biological specimens. The aim of this course is to introduce the basic
principles and methods of statistical mechanics and to apply them to a
number of model systems in order to illustrate their use and potential
in a systematic manner.
We start by introducing the fundamentals of thermodynamics and then proceed to develop the concepts and techniques needed to evaluate probability distributions and partition functions. This later quantity establishes the link between the microscopic description of a system, based on quantum states or positions in phase space, and the macroscopic characterization provided by a small set of independent thermodynamic variables.
Among the applications we consider ideal gases made up of classical and quantum particles, vibrations in solids (phonons) and electromagnetic radiation quanta (photons), the behavior of electrons in metals and BoseEinstein condensation. We then move to interacting classical systems, where we consider the treatment of real gases and liquids and introduce the concept of phase transition. The Ising model and some of its variants are solved in the meanfield approximation. Time permitting, we provide a general introduction to Monte Carlo methods, a set of powerful numerical techniques used to investigate manybody interacting systems.
Contents
Fundamentals of thermodynamics: Systems, phases and state quantities.
Equilibrium and temperature. Equations of state. Reversible and
irreversible processes. Work and heat. The laws of thermodynamics.
Entropy and the second law. Global and local equilibrium. Homogeneous
functions: Euler's equation and GibbsDuhem relations. Thermodynamic
potentials. The principle of maximum entropy. Entropy and energy as
thermodynamic potentials. Legendre transformations. Maxwell relations.
Jacobi transformations. Phase and chemical equilibrium. Response
functions. Thermodynamic stability.
Equilibrium statistical mechanics: Quantum states and phase space.
Ensemble theory. The microcanonical ensemble. Connection with
thermodynamics: density of states and entropy. The canonical ensemble.
Canonical partition function and free energy. Internal energy and energy
fluctuations. Microscopic description of heat and mechanical work. The
grand canonical ensemble. Grand canonical partition function and grand
potential. Density and energy fluctuations. Entropy maximisation: a
general method to derive distribution functions. Fluctuations and
response functions. Equivalence between ensembles. Final considerations
on Boltzmann statistics. Classical statistical mechanics: phasespace and
partition function. MaxwellBoltzmann distribution. Equipartition and
virial theorems.
Applications of Boltzmann statistics to ideal systems: Factorisation
approximation. Monoatomic gases. Gases with internal degrees of
freedom: vibrations and rotations. Chemical equilibrium and Saha's
ionisation formula.
Quantum statistical mechanics: Indistinguishable quantum particles and
symmetry requirements. FermiDirac and BoseEinstein statistics:
derivation of partition functions. Recovering the classical limit.
Statistical mechanics of quasiparticles: phonons and photons. Ideal
Fermi gas: low and highdensity limits. Electrons in metals. Ideal Bose
gas: low and highdensity limits. BoseEinstein condensation.
Statistical mechanics of interacting systems: Interaction potentials.
Perturbation theory using a control parameter. Cluster and virial
expansions. The van der Waals equation of state and the liquidvapor
phase transition. Models of Ising and Heisenberg. Exact solution and
Mean field theories. Introduction to simulation methods in statistical
physics.
Assessment
One 3hour written examination with 4 questions to be attempted out of a choice of 6.
Prerequisite: This twosemesterlong double module is only available to students on the Mathematics/Applied Mathematics and Physics/Theoretical Physics MSci pathways.
Introduction
The project involves a substantial
investigation of a research problem incorporating literature survey,
development of appropriate theoretical models and when necessary the
construction of computer programs to solve specific stages of the
problem, presentation of the work in the form of a technical report, a
sequence of oral presentations culminating in a 30minute presentation
which is assessed. Each student will work under individual supervision
of a member of staff.
Contents
The mathematical contents of the project will depend on the nature of the research problem.
Assessment
80% dissertation, 20% oral presentation.
Prerequisite: While there are no specific prerequisites, this course is intended for students at stage 4 of an MSci Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Many problems in applied mathematics
reduce to solving either ordinary or, more usually, partial
differential equations subject to certain boundary conditions. In many
cases of practical interest exact analytical solutions are not
available.
In nonrelativistic quantum mechanics for example the problems involve finding numerical solutions to the Schrödinger equation. In fluid mechanics solutions of the NavierStokes equation are needed in the context for example of weather systems, or of hypersonic flow around a space shuttle, or the flow of blood through arteries. In financial mathematics solutions of the BlackScholes equation that describes the pricing of options are required. A familiar partial differential equation arising in many problems in physics and engineering is the diffusion equation, which describes how heat flows from hot to cold regions and many other processes. Another familiar example is the wave equation that governs phenomena all around us: electromagnetic waves, the vibrations of guitar strings, and the propagation of the sound they produce.
In order to solve these or other problems we need to understand first the conditions that give rise to a unique solution. Since analytical solutions are only rarely available we examine techniques that have been developed to acquire a numerical approximation to the solution of a particular problem. The third feature of the course is to actually find numerical solutions to a set of representative model problems, by writing computer programs to implement the requisite methods.
Contents
1. Introduction. Examples of partial differential equations in mathematical physics.
2. Firstorder linear and quasilinear equations. Method of characteristics.
3. Secondorder linear equations and their classification. Canonical forms.
4. The method of separation of variables. Applications.
5. Introduction to numerical methods. Explicit and implicit finitedifference schemes. Applications.
6. Practical assignment. Numerical solutions to model problems.
7. SturmLiouville problem. Eigenfunctions and eigenvalues. Eigenfunction expansion.
8. Elliptic equations.
9. Dirac delta function. Green's functions and integral representation.
Assessment
The assessment consists of two parts.
The written examination accounts for 75% credit. Three questions may be
attempted from a choice of five. There is also a practical assignment,
in which candidates are asked to solve numerically a set of model
problems by implementing the requisite methods on the computer, and to
submit a short written report of their solutions and associated codes.
The emphasis of the written account should be on the practical aspects
of assessing the accuracy of the solution and understanding its
properties. The practical part accounts for 25% credit.
Prerequisite: While there are no specific prerequisites, this course is intended primarily for students at stage 4 of the MSci Mathematics/MSOR/M&CS pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Information is a message received and understood. It was Claude Shannon who,
in 1948, came up with an idea to quantify the amount of information with a
welldefined means of mathematics. Shannon's creation of the subject of
information theory was one of the great intellectual achievements of the
20th century and it is an important basis of the development of information
technology in the 21st century. Theory of information, which may come in
various forms including a sound pattern, a page of a book and a TV message,
is one of the important applications of mathematics for the modern society.
In fact, information theory is not merely an application of mathematics.
Information theory has had an important and significant influence on
mathematics and has given profound contribution to pure mathematics.
Information transfer occurs through "codes". These codes can take various forms, ranging from the English language to the Morse code to the ASCII code for computer data storage. In this module, we will start by defining what we mathematically mean with a code. To measure the amount of information stored within a message expressed in the code, we will then introduce the key quantity of information theory, entropy. We will also introduce the basics of coding theory: if a message is received with errors, how can we retrieve the original information content.
Some of the basic concepts of probability theory are discussed. When the information is transmitted from one point to another, we quantify how much information is transmitted through a noisy communication channel. We discuss how the information content of a random variable can be measured. How to infer the original message from the distorted data received is considered.
Contents
Introduction to information theory. Random variables and some concepts of
probabilities. Basic modular arithmetic and factoring. Secret coding. Uniquely
decipherable and instantaneous codes. Optimal codes and Huffman coding. Code
extensions. Entropy, conditional entropy, joint entropy and mutual information.
Shannon noiseless coding theorem. Noisy information channels. Binary symmetric
channel. Decision rules. The fundamental theorem of information theory. Basic
coding theory. Linear codes.
Assessment
One 3hour written examination with 4 questions to be attempted out of a choice of 6.
Prerequisite: Only available to students on the MSci pathways. This module is taken in stage/year 3.
Introduction
Problems in real life come in various shapes and sizes, usually with no
answer “at the back of the book”. The purpose of this module is to
equip the MSci students with problemsolving and reportwriting skills
that they will need for the yearlong Project (AMA4005 or SOR4001) that
they do in their final year. The problems that the students are offered
to study will often be openended, and in many cases will require the
student to set the problem in mathematical terms before attempting its
solution.
Contents
Students conduct a short practice
investigation, followed by two short investigations (one in pairs  two
weeks, and one solo  three weeks) in a range of problems in Applied
Mathematics and Theoretical Physics. This is followed by a long
investigation (five weeks), which is a literature study of a
Mathematical or Theoretical Physics topic not covered in the offered
(or chosen) lecture modules. The two short and the long investigation
are typed up in reports and submitted for assessment. [The practice
report is also typed up.] Feedback on reports is given to assist
students with improving their writing skills. A short introduction to
LaTeX, possibly the best tool for typesetting mathematical papers and
reports, will be provided near the beginning of the solo investigation.
Assessment
The final mark will be based on the two short investigation reports
(pairs and solo), each contributing 25%, and long investigation report
(50%).
Prerequisite: Knowledge of linear algebra as in AMA2003 or PMA2007. AMA3002 is NOT required.
Introduction
We are currently witnessing an information revolution: digital devices are
everywhere around us. If the incredible level of miniaturisation of electronic
devices continues at the current pace, in a few years the elementary components
will be made of a few atoms. At this level, physical effects ruled by quantum
mechanics will start playing a major role. Inspired by this change of
perspective, quantum information processing has been developed as a new
framework for future computers. The logic of these quantum computers is
different from the traditional computers, as the elementary unit of
information, the quantum bit, can be in a superposition of two states 0 and 1.
The aim of this module is to give a broad overview of this emerging field by introducing the most important applications: quantum computing, quantum communication (including teleportation) and entanglement. Quantum mechanics is not a requisite and will be introduced at the beginning of the module as an abstract framework in linear algebra.
Students completing this module will acquire knowledge of the mathematical concepts of quantum information processing with possible applications in theoretical physics, applied and pure maths and computer science.
Contents
1. Operatorial quantum mechanics: review of linear algebra in Dirac notation;
basics of quantum mechanics for pure states.
2. Density matrix and mixed states; Bloch sphere; generalised measurements.
3. Maps and operations: complete positive maps; Kraus operators; master
equations.
4. Quantum Communication protocols: quantum cryptography; cloning;
teleportation; dense coding.
5. Quantum computing: review of classical circuits and logic gates; quantum
circuits and algorithms; physical implementation of quantum computers.
6. Theory of entanglement: entanglement basics; detection of entanglement;
measures of entanglement; entanglement and nonlocality, Bell's inequality;
multipartite entanglement.
Assessment
One 3hour written examination with a choice of 4 questions from 6.
Textbooks
1. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information,
Cambridge University Press.
2. J. Preskill, Quantum Information and Computation, Lecture Notes for Physics 229, available at:
http://theory.caltech.edu/~preskill/ph229/#lecture.
Prerequisite: This twosemesterlong double module is only available to students on the Mathematics and Statistics & Operational Research MSci pathway.
Introduction
A substantial investigation of a
statistical research problem incorporating literature survey, use of
relevant statistical packages and when necessary the construction of
computer programs to solve specific stages of the problem, presentation
of the work in the form of a technical report, a sequence of oral
presentations culminating in a 30minute presentation which is
assessed. Each student will work under individual supervision of a
member of staff.
Contents
The mathematical contents of the project will depend on the nature of the research problem.
Assessment
80% dissertation, 20% oral presentation.
Prerequisite: SOR2004
Introduction
Survival analysis is an important tool for research in medicine and
epidemiology. It is that part of statistics that deals with timetoevent data.
For example, in a clinical study the data might consist of the posttreatment
survival times of patients with hypernephroma (i.e., a malignant tumour of the
kidney). Survival analysis might address questions such as:
The module introduces the student to the special features of survival data such as censoring (e.g. where a patient is lost to follow up but is known to have survived to a particular time) and positive skew in the distribution of survival times. Fundamental concepts of survival analysis will be introduced including the survivor function, the hazard function and the hazard ratio. The course will build from some elementary nonparametric techniques such as the KaplanMeier estimate of the survival curve to the Cox proportional hazards model  one of the most flexible and widely used tools for the analysis of survival data.
An important element of this module will be two hours a week of survival data analysis classes using statistical software packages, in particular SAS and R. These will be used to demonstrate the theory taught in the lectures. Previous experience of SAS is required.
Contents
Introduction to survival data: Features of survival data, distribution of survival times, survivor function, hazard function, cumulative hazard function.
Some nonparametric procedures: Estimating the survivor
function  lifetable, KaplanMeier, NelsonAalen, confidence
intervals. Estimating the hazard function, estimating median and
percentile survival and confidence intervals. Comparing two groups of
survival data, the logrank and Wilcoxon tests. Comparison of kgroups
of survival data.
The Cox proportional hazards model: The Cox proportional
hazard model (Cox model), baseline hazard function, hazard ratio,
including variates and factors, maximum likelihood for the Cox model.
Treatment of ties in the Cox model. Confidence intervals for the Cox
model regression parameters and hypothesis testing. Estimating the
baseline hazard. Model building, Wald tests, likelihood ratio tests and
nested models.
Evaluating the proportional hazards assumptions.
The stratified Cox procedure.
Extending Cox proportional hazards models for time dependent variables.
Recurrent events survival analysis.
Competing risks survival analysis.
Design issues for randomised trials.
Parametric models for survival data, timedependent variables and
nonproportional hazards, acceleratedfailuretime models. Fitting parametric
distributions.
Assessment
Assessment is based on one written examination paper (contributing a maximum
of 85% to the overall module mark) and coursework (contributing a maximum of
15% to the overall module mark. The 3hour written examination paper comprises
6 questions: 4 questions are to be attempted out of the choice of 6.
Coursework is linked with the computing sessions, assessing students' use
of statistical software to produce and interpret results.
Prerequisite: A strong background in Alevel Mathematics or equivalent
This module, together with its companion module PMA1014, acts as a link between school mathematics and degreelevel work in Pure Mathematics and in other mathematical disciplines.
In sixthform mathematics, the primary emphasis is on applying techniques and on getting correct answers. University level mathematics goes a step further and aims at understanding the structure of a problem by concentrating on the essential or "universal" aspects rather than only on specifics. This abstraction allows us to deal with a variety of more complicated problems and lies at the very heart of modern mathematics.
Generally speaking, the aim of this module is to understand the nature of mathematical arguments and to be able to analyse and, even more importantly, to construct proofs of mathematical statements. To this end, you will be introduced to the very basics of mathematical language and methodology. This includes familiar subjects like number systems (integers, rationals) and elementary combinatorics, and these will be discussed throughout using the allimportant language of set theory.
Attending lectures is not enough to develop understanding of mathematics. It is even more important to apply mathematics and to solve problems. This will be encouraged by weekly homework exercises, offering the opportunity for independent work. Exercises will be discussed in weekly tutorials. Exercises and tutorials form an integral part of the module.
Syllabus Content: Elementary logic and set theory, number systems (including integers, rational numbers, real numbers), sequences of numbers, convergence, completeness, basic combinatorics.
Assessment
One 3hour written examination, with some choice of questions.
Prerequisite: A strong background in Alevel Mathematics or equivalent
The distinction between calculus and analysis is roughly this: calculus deals with differentiation and integration and focuses largely on techniques of calculation; analysis is wider in its scope (it includes, for example, sequences and series) and is concerned with the precise formulation of definitions, results and proofs. Students embarking on this module will already have met some calculus, but here we shall focus on the analytic foundation of that discipline. The second aspect of the module is devoted to an introduction to linear algebra, which aims to give an equally strong foundation to the study of linear equations, determinants and matrices which will already be partly familiar to many students. The module's contents will include:
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
No particular book will be followed. You may find the following helpful:
Thomas and Finney, Calculus and Analytic Geometry, 9th ed. (AddisonWesley, 1996).
Hirst, Numbers, Sequences and Series (Arnold).
Prerequisite: PMA1014 Analysis and Linear Algebra
The main topics are:
SEQUENCES AND SERIES: Supremum and infimum, Cauchy sequences,
convergent sequences. The BolzanoWeierstrass theorem. Infinite series,
convergence tests.
LIMITS AND CONTINUITY: Limit of a function at a point.
Continuity. Intermediate value theorem. Bounds of a continuous function
on a bounded closed interval.
DIFFERENTIATION: Definition of derivative. Basic results on
the derivative. Rolle's theorem. Mean value theorems. L'Hôpital's rule.
Taylor's theorem. Local maxima and minima.
RIEMANN INTEGRATION: Definition of the Riemann integral and study of its main properties. Differentiation of the indefinite Riemann integral.
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis (Wiley & Sons)
R. Haggarty, Fundamentals of Mathematical Analysis (AddisonWesley)
K. E. Hirst, Numbers, Sequences and Series (Arnold)
P. E. Kopp, Analysis (Arnold)
K. G. Binmore, Mathematical Analysis (Cambridge University Press)
Prerequisite: PMA1014 Analysis and Linear algebra
In the sixteenth century, problems involving the solution of algebraic equations naturally gave rise to imaginary numbers, now called complex numbers. Though they were originally viewed as rather mysterious, the theory of complex numbers and functions was given a sound foundation in the nineteenth century by the four great mathematicians Cauchy, Gauss, Riemann and Weierstrass. The theory of complex variables is one of the most beautiful and useful branches of mathematics, containing striking theorems which have no counterpart in the theory of real variables.
The main topics are:
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
The module is not based on any one book, but useful references are:
R. V. Churchill, Complex Variables and Applications (McGrawHill)
M. R. Spiegel, Theory and Problems of Complex Variables (Schaum Publ. Co.)
H. A. Priestley, Introduction to Complex Analysis (Oxford University Press)
P. L. Walker, An Introduction to Complex Analysis (Adam Hilger)
Prerequisite: PMA1014 Analysis and Linear Algebra
The emphasis throughout this module will be on developing a rigorous approach to Mathematics. This involves precise definitions (you need to understand what they say and what they don't say), logical proofs, solving problems and communicating the solutions. You will also learn techniques and when they may be used. The content follows on from and develops the linear algebra component of the prerequisite module. The provisional syllabus comprises:
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
Almost any text on linear algebra should cover the material of this module. In particular:
Grossman, Elementary Linear Algebra (Wadsworth)
Prerequisite: PMA1012 Numbers, Sets and Sequences, and PMA1014 Analysis and Linear Algebra
Equivalence relations, binary operations, groups, examples and properties, groups of maps, countability, homomorphisms, subgroups, equivalence relations, permutation groups, normal subgroups, quotient groups, structure of finite abelian groups, composition series and solvable groups.
Assessment
One 3hour written examination, with some choice of questions.
Prerequisite: PMA1012 Numbers, Sets and Sequences
This module aims to examine the legacy of Euclid's work as a style of argument and as a body of factual information. The outline programme is:
Assessment
90% by threehour written examination with some choice of questions,
10% by presentations to the class [two per student] of short prepared
sections of geometric material.
Prerequisite: There is no specific prerequisite for this module.
The great mathematician C. F. Gauss stated that “Mathematics is the queen of the sciences and the Theory of Numbers is the queen of Mathematics”. The distinguished mathematician G. H. Hardy wrote: “The elementary theory of numbers should be one of the very best subjects for early mathematical instruction. It demands very little previous knowledge; its subject matter is tangible and familiar; the processes of reasoning which it employs are simple, general and few; it is unique among the mathematical sciences in its appeal to natural human curiosity.”
Number theory is an attractive subject partly because it contains many propositions which are easy to state but very difficult to prove. Among the conjectures which have so far defied proof are those that every even number is the sum of two primes and that there is an infinite number of ‘twin primes’ (i.e. primes which differ by two: 3, 5; 11, 13; 41, 43; etc.)
Topics treated in the module will include:
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
Some books are, in approximately decreasing order of relevance:
The theory of numbers, A. Adler and J. E. Coury (Jones & Bartlett)
Elementary number theory and its applications, Kenneth H. Rosen (AddisonWesley)
Introduction to number theory with computing, R. Allenby & E. Redfern (Edward Arnold)
Elementary number theory, Donald M. Burton (Allyn & Bacon)
A concise introduction to the theory of numbers, Alan Baker (C.U.P.)
In
addition, the Libraries contain books with titles similar to those
above, most of which include at least some relevant material.
The Department of Pure Mathematics is a small one, and staff losses (retirements, resignations or illness) can adversely affect our ability to offer modules at higher levels (3 and 4) as we often only have one member of staff who can teach a particular high level module. Although we will do all that we can to put on the modules listed, we cannot guarantee to do so. You will be notified of any unavoidable changes as soon as we know of them.
Prerequisite:
Because of demand, this module is taught in both semesters. Students may only obtain credit for passing it once. If you enrol for the first semester version and disenrol after the fifth week of the module then you may not subsequently enrol for the second semester version. Even though the module is offered in both semesters, the demand in the second semester is so high that it may be necessary to refuse to accept late enrolments into this module.
When most people think of computers in the context of mathematics, they think of some numerical calculation being performed thousands of times in order to approximate something. There is much more to the mathematical use of computers than this. Apart from their use for sophisticated wordprocessing, access to databases and remote collaboration by electronic mail, their use to manipulate symbols is becoming increasingly important. There have been, for several decades, computer programs that can, for example, expand (1 + x)(1 + x^{2}) to obtain the answer 1 + x + x^{2} + x^{3}. More impressively they can evaluate the integral of (1 + x + x^{2})/(1 + x + x^{2} + x^{3}) from 0 to 1 (using suitable notation) as π/8 + 3 log(2)/4. Whilst any calculation that such a program can perform could also be done by a human, they can ‘remember’ a large number of rules for you and not make silly mistakes in copying an expression from one line to the next. Those who want numerical answers should not feel left out either as these programs can evaluate numerical expressions to as many significant figures as you like (within physical limitations  it is certainly possible to work with a thousand significant figures).
These programs are finally becoming available to working mathematicians as the programs become more efficient and the computers needed to run them become cheaper. This module will provide a practical introduction to the use of such programs by teaching you to use one such program (Sage Math) in order to solve mathematical problems. Whilst some of the module will necessarily involve having the program demonstrated to you, the emphasis is on you using the program, so the module is very much practically oriented. You will be set assignments to carry out on a weekly basis, and assessment is all based on practical work.
The contents of the lectures in the first six weeks of the module are approximately as follows:
Using Sage; simple calculations; manipulating expressions; lists; strings; and printing; graph drawing; defining functions; calculus; Boolean expressions and conditional statements; loops; solving equations; vectors and matrices.
The remainder of the module consists of case studies from many areas of pure mathematics.
The mathematical background needed for this module is approximately Alevel mathematics. You should know what expanding or factorizing an algebraic expression means, what the differential and integral calculus are all about, what vectors and matrices are etc. Recent practice at performing calculations in these topics is not needed as Sage will do that for you.
This module is taught in 2 two hour sessions from 4pm to 6pm (Tuesday and Thursday in the first semester; Monday and Tuesday in the second semester). Three of these hours consist of lectures/demonstration whilst the fourth hour will be a tutorial, preparing you for the coming week's assignment. There will also be a one hour laboratory session (Tuesday, 1pm to 2pm in the first semester, Monday 2pm to 3pm in the second semester, at which the lecturer will be available for consultation, except for weeks when this time is used for assessments).
Assessment
This module is examined by means of
three fortyminute practical tests during the laboratory sessions
around weeks 6 to 8. Students may undertake two of these and the best
mark will be counted. There will also be a twohour computerbased
practical examination arranged by the department (usually this takes
place immediately after the end of the lecture term). This examination
will not appear on the university examination timetable. The midcourse
practical mark and the final exam mark will each count for 50% of the
assessment.
Prerequisite: PMA2007 Linear Algebra
The purpose of this module is to give a general introduction to the theory of rings, which is a subject of central importance in algebra. Historically, some of the major discoveries have helped to shape the course of developments of modern abstract algebra. Today, ring theory is a possible meeting ground for many algebraic subdisciplines such as group theory, representation theory, Lie theory, algebraic geometry, homological algebra, to name but a few.
Main topics:
Rings, subrings, ideals, quotient
rings, homomorphisms, canonical factorisation, isomorphism theorems,
integral domains, principal ideal rings, fields, simple rings,
Noetherian rings, polynomial rings, Hilbert's basis theorem.
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
J. Beachy, Introductory lectures on rings and modules (London Math. Soc. Student Text No. 47, Cambridge University Press)
T. W. Hungerford, Algebra (Springer GTM, 1971)
Prerequisite: PMA2007 Linear Algebra. This module is a compulsory component of the MSci course in Pure Mathematics.
Set theory is the language in which most of contemporary Pure Mathematics is most readily expressed. It is also a subject of study in its own right, whose techniques and insights find application across the entire discipline and whose unresolved/unresolvable issues compel us to question our “intuitive expectation of certainty” in many areas. This module will seek to teach fluency in the language of elementary set theory, facility in the use of key techniques such as transfinite induction and maximality principles, and basic arithmetic of cardinal and ordinal numbers (the ‘arithmetic of infinity’). It will also develop an axiomatic description of set theory to allow some discussion of the issues of completeness and consistency.
The chapters and their approximate numbers of lectures are as follows:
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
No prescribed text will be followed in detail. The following references may be of use:
Lipschutz, Set Theory and Related Topics (Schaum).
Simmons, Topology and Modern Analysis (McGrawHill).
Rotman and Kneebone, Theory of Sets and Transfinite Numbers (Oldbourne).
Stewart, Foundations of Mathematics (OUP).
In addition, the first (or zeroeth!) chapter of almost any book on
modern algebra, analysis or topology will provide some discussion of
set theory.
Prerequisite: PMA2002 Analysis [or PMA2003 and appropriate summer reading]
Convergence of series of nonnegative real numbers: tests for convergence including lim sup/lim inf versions of the ratio and n'th root tests and the integral test.
Convergence of general series of reals, including convergence of absolutely convergent series, alternating series test, power series and their radius of convergence.
Bracketing and rearrangement of series. Product series.
Pointwise and uniform convergence of a sequence of realvalued functions; uniform convergence of sequences of continuous, integrable and differentiable functions; Weierstrass' approximation theorem; uniform convergence of series; Weierstrass' Mtest; uniform convergence of power series; manipulation of power series.
Taylor's theorem.
Convergence in other settings.
Assessment
One 3hour written examination, with some choice of questions.
Prerequisite: PMA2002 Analysis
Definition and examples of metric spaces; open sets, closed sets, closure points, sequential convergence, compactness; completeness; continuous mappings between metric spaces; Banach's fixedpoint theorem and applications; Baire category theorem and applications.
Normed spaces; Banach spaces; finite dimensional normed spaces; subspaces and quotient spaces; linear operators; boundedness; compact operators; dual spaces.
Assessment
One 3hour written examination, with some choice of questions.
Prerequisite: PMA2007 Linear Algebra and PMA2008 Group Theory
The theory of algebraic equations is the study of solutions of polynomial equations. Although the problem originates in explicit manipulations of polynomials, the modern treatment is in terms of field extensions and groups of ‘symmetries’ of fields.
The content includes: review of polynomial rings and characteristic of rings and fields, factoring polynomials, extension fields, construction of some extension fields, algebraic and transcendental elements, constructions with straightedge and compass, splitting fields, the fundamental theorem of Galois theory, groups of automorphisms of fields, separable, normal and Galois extensions, examples.
Assessment
One 3hour written examination, with some choice of questions.
The Department of Pure Mathematics is a small one, and staff losses (retirements, resignations or illness) can adversely affect our ability to offer modules at higher levels (3 and 4) as we often only have one member of staff who can teach a particular high level module. Although we will do all that we can to put on the modules listed, we cannot guarantee to do so. You will be notified of any unavoidable changes as soon as we know of them.
Note that in any given academic year the taught Level 4 modules that will be offered will depend on demand from intending Level 4 MSci students of Pure Mathematics as indicated by them during the preceding year. In the academic year 20122013 it is intended that all the listed Level 4 taught modules will be offered.
Prerequisite: This project is a compulsory component of the MSci pathway in Pure Mathematics. There is no specific prerequisite for this module, but the student will need enough Level 3 background in Pure Mathematics to undertake an extended project at this level in some area of Pure Mathematics for which supervision can be offered.
This is an extended project designed to test the student's ability to work independently at a high level for a prolonged period of time with a restricted amount of supervision. This will give a taste of the kind of work expected of a mathematician in the commercial or academic world, unlike the relatively short bursts of work expected in most undergraduate modules. It will also provide an opportunity to develop those transferable skills that are sought by employers, including IT (both wordprocessing and database access), presentational and personal ones.
The project takes place during the first two terms of Level 4. It will normally involve study and exposition of a piece of mathematical work beyond the normal undergraduate syllabus and which will probably not be available in easily assimilated form. Originality of exposition will be expected, but not necessarily much in the way of original results. The main part of the assessment will consist of a wordprocessed report, but 20% of the marks for the project are awarded for an oral presentation of the work which will take place just after Easter. As preparation for this assessed oral presentation, the student will be expected to give two oral progress reports around the middle of each of the first two terms to a small group of staff and any other students undertaking this module. Constructive advice on these presentations will be provided after each one.
Near the beginning of the first semester, there will be computerbased workshops for students taking this module. These will cover topics including the use of LaTeX (the internationally accepted standard language for mathematical typesetting which is accepted by the majority of mathematical publishers, using the internet to access sources of mathematical information (including the use of MathSciNet, the online version of Mathematical Reviews which reviews almost every published paper in Pure Mathematics) and using the computer algebra package Sage to produce mathematical diagrams.
Students intending to take this module should seek advice and think about their choice of project during the summer. The selection of a project should be finalized no later than the start of the academic year, and it would be helpful to all involved if students actually did this even earlier.
Assessment
80% by final wordprocessed report, 20% by oral presentation.
Prerequisite: PMA3017 Metric and Normed Spaces, and PMA3014 Set Theory
Functional analysis arose in the early twentieth century when the need became apparent to study whole classes of functions rather than individual ones. For example, differential equations may be regarded as concerning maps from a set of functions into itself and (after some reformulation) looking for a solution of a differential equation is asking for a function that is left fixed under the action of a certain map. The proof that, in certain circumstances, such a function always exists also shows how to approximate such a function numerically when an analytic solution cannot be found. Further impetus to the development of functional analysis came when quantum mechanics was found to be describable within its ambit. This has been an active area of research ever since and remains so to this day.
To some extent you can regard linear functional analysis, to which this module is restricted, as an attempt to place linear algebra on a firm foundation within an infinitedimensional context. The module will emphasize the topological tools (metric and nonmetric ones) which are necessary for this. Familiarity with the Level 4 module Topology is desirable.
The topics will be chosen from:
Assessment
One 3hour written examination, with some choice of questions.
Prerequisite: PMA3017 Metric and Normed Spaces, and PMA3014 Set Theory
Topology (rather like Algebra or Analysis) is not so much a single branch of mathematics but a loose confederation of subject areas differing widely in their origins, techniques and motivation but united by sharing a common core of basic concepts and constructions. Problems of a topological nature include: how can we describe and classify knots? how can we describe and classify surfaces? to what extent is it possible to extend the ideas of analysis into sets that don't have metrics defined on them? what can be meant by saying that two objects are “fundamentally the same shape”, and how do we decide whether they are or not? what ‘models’ are available to describe certain aspects of theoretical computer science? Rather than attempting to supply answers to any such major questions, this module will concentrate on developing enough of the ‘common core’ to allow students to begin to appreciate how such issues can be tackled topologically.
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
The libraries contain several texts on topology which include sections
relevant to this module, but be aware that a lot of them are written at
postgraduate level. More accessible are:
Mansfield, M. J., Introduction to Topology (Van Nostrand).
Mendelson, B., Introduction to Topology (Allyn & Bacon).
Moore, T. O., Elementary General Topology (Prentice Hall).
Simmons, G. F., Introduction to Topology and Modern Analysis (McGrawHill).
Lipschutz, S., General Topology (Schaum).
Prerequisite: PMA2002 Analysis and PMA3014 Set Theory
The theory of integration, developed by Lebesgue in the early part of the twentieth century in the context of the real line and subsequently extended to more general settings, is indispensable in modern analysis. The Lebesgue theory allows a very wide class of functions to be integrated and includes powerful convergence theorems which are not available in Riemann integration. In this module the theory is developed in the context of a general σalgebra of sets. Special attention is given to the case of Lebesgue measure on the reals, and some applications of the integral to Fourier series are given.
Contents: σalgebras of sets, measurable spaces, measurable functions. Measures. Integrals of nonnegative measurable functions: properties including Fatou's lemma and monotone convergence theorem. Integrable functions: Lebesgue dominated convergence theorem. Lebesgue integral on intervals: comparison with Riemann integral. L^{p}spaces: inequalities of Hölder and Minkowski; Fourier series in L^{2}.
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
R. G. Bartle, The Elements of Integration and Lebesgue Measure (Wiley, 1995).
Prerequisite: PMA3012 Ring Theory
Historically, ring theory as we know it was born with the classification result (due to Wedderburn) of certain finitedimensional algebras over fields. This was later extended by Artin to classify the socalled semisimple rings, and it is one of the most fundamental results (and one of the nicest too) that will keep us occupied for the first part of the module.
In order to develop this theory, techniques arising from the need to generalize certain aspects of Linear Algebra  such as the concept of a ‘module’  will be introduced. The second part of the module follows the historical path of developments. We will study the Jacobson radical, introduced by Jacobson in 1945. This is a tool that has proved fundamental in the development of the theory, e.g., of Artinian rings, primitive rings, local and semilocal rings.
Our approach will be pedestrian in nature, and we will therefore focus on a number of examples in order to motivate the different concepts that will be used.
Contents: Basics: rings and modules, definitions and examples. Direct sums and short exact sequences. The structure of semisimple rings. Radicals: the Jacobson and the prime radicals. Structure of primitive rings. Local and semilocal rings.
Assessment
One 3hour written examination, with some choice of questions.
Textbooks
There are a number of books that can
be used for an introductory module such as this. The basic bibliography
upon which the lecture notes will be based is:
T. Y. Lam, A First Course in Noncommutative Rings (2nd ed.) GTM 131, (SpringerVerlag, 2001).
T. Y. Lam, Exercises in Classical Ring Theory, Problem Books in Mathematics, (SpringerVerlag, 1995).
R. Pierce, Associative Algebra, Graduate Texts in Mathematics (SpringerVerlag, 1982).
Prerequisite: PMA4003 Topology
Topological spaces, compact and connected spaces; continuous maps, homeomorphisms, retractions; product topologies; examples (spheres and disks, tori, Moebius strip); main problems in algebraic topology: invariance of domain, invariance of dimension, retractions, distinguishing between nonhomeomorphic spaces; example: is S^{1} homeomorphic to D^{2}? Is S^{1} a retract of D^{1}?; connected and path connected spaces, path components, π_{0}(X); homotopy of maps, homotopy as equivalence relation, π_{0}(X) the set of homotopy classes [pt, X]; contractible spaces; homotopy equivalences; π_{0}(X) as a functor; applications to nonhomeomorphic spaces; [S^{1},S^{1}] = Z; application: fundamental theorem of algebra; pointed sets and spaces, pointed homotopies, π_{0}(X) = [S^{0},S^{1}] as pointed set; pointed contractions and homotopy equivalences; fundamental group π_{1}(X) for X = S^{1}, X = Q, X = contractible; π_{1} as a functor; application: S^{1} is not a retract of D^{2}, Brower fixed point theorem for D^{2}.
Assessment
One 3hour written examination, with some choice of questions.
Prerequisite: This module is a compulsory component of the MSci course in Pure Mathematics.
The way in which we teach mathematics, whether in schools, universities or anywhere else, is only very distantly related to the way mathematicians actually do creative work. Creative need not only mean highlevel research, any attempt to do a standard tutorial question is likely to be creative. In general we are set a question which reads “Find the value of...” or “Prove that...” and are expect to give an answer which sets out a polished final exposition of the result but not including any indication of how you reached that final result. For a discussion of the false starts that a mathematician has made in the solution of a problem, of what examples finally led to a solution or of what impossible feature of a soughtfor counterexample eventually led to a proof you must look, not in mathematical text books or research papers but in the few books where mathematicians have tried to analyse their own creative processes. Nor will you find in their finished expositions any discussion of what they were hoping to prove before they started work. That may well not have been what they ended up proving.
To be sure, many applications of mathematics in commerce or industry, such as estimation of quantities or pricing, will require exact or reasonably exact answers. However, many problems that a mathematician working in industry is confronted with are more on the lines “What can you tell me about ... by Friday?” The answer to “How long will that job take?” will not be “3 months” but “Not less than 10 weeks and not more than 15 weeks”. Even when an exact answer is required this need not mean producing a formula, more often it will involve computation of some kind. One of the most difficult things a working mathematician has to cope with is the fact that whoever is asking for the answer probably doesn't know exactly what he/she does want  even if they think they do. As some training in this kind of real life problem, our problems will be openended ones: “What can you tell me about this situation?” or “Such and such happens here, how typical or special is this?”
Once you have solved your problem, you must communicate what you have done. You will need to explain why what you have done is the best you could do in the time available, to what extent more could be done and how long it would take  all this in sufficient detail so that it is clear and understandable while still being concise. The preparation of your reports should give you some practice in writing coherent English  an activity that is ignored in most traditional mathematics courses. You will also have to give oral presentations of parts of your work, which will be assessed as part of the examining process. In summary, we hope that this module will help you to perform meaningful investigations using the mathematics that you know as efficiently as possible and then to communicate your findings to others.
While some of the investigations require little more than GCSE as a background, students will be required to undertake at least one investigation which needs knowledge of Mathematics at Level 2 or Level 3 standard and/or some background reading.
Assessment
90% by written reports submitted at different times during the semester, 10% by oral presentation.
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