examiners’ report: final honour school of mathematics part

Examiners’ Report: Final Honour School of

Mathematics Part C Trinity Term 2009

October 29, 2009

Part I

A. STATISTICS

• Numbers and percentages in each class.See Table 1, page 1. When drawing comparison with historic data itshould be noted that from 2007 classification for Part C was based onPart C alone. There were 96 candidates entered for the examination;1 withdrew during the Easter vacation after a dissertation had beensubmitted.

Table 1: Numbers in each class

Number Percentages %(2009) (2008) (2007) (2006) (2005) (2009) (2008) (2007) (2006) (2005)

I (48) (44) (38) (52) (34) (50.53) (46.3) (45.8) (58.4) (44.2)II.1 (30) (45) (35) (31) (37) (31.58) (47.4) (42.2) (34.8) (48.1)II.2 (13) (6) (9) (6) (4) (13.68) (6.3) (10.8) (6.7) (5.2)III (3) (0) (1) (0) (2) (3.16) (0) (1.2) (0) (2.6)P (0) (0) (0) (0) (0) (0) (0) (0)F (1) (0) (0) (0) (0) (1.05) (0) (0) (0) (0)Total (95) (95) (83) (89) (77) (100) (100) (100) (100) (100)

• Numbers of vivas and effects of vivas on classes of result.As in previous years there were no vivas conducted for the FHS ofMathematics Part B.

• Marking of scripts.The number of scripts double marked : twenty whole unit Disserta-tions, and two half unit dissertations. One was also read by a third

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reader. The remaining scripts were all single marked according to apre-agreed marking scheme which was strictly adhered to. For detailsof the extensive checking process, see Part II, Section A.

• Numbers taking each paper.See Table 6 on page 11 and continued in Table 7 on page 12.

B. New examining methods and procedures

This year we extended the way we reconciled dissertation marks. We invitedsupervisors to propose a class for the dissertations they had assessed andthen discussion followed between the two assessors and the supervisor. Theensuing discussion was intended to focus on the reasons for a difference inthe classes. Although this took some time, this was useful as some classesand USMs were modified based on this discussion and the insight of thesupervisor was helpful to assessors. It is hoped we can conduct future dis-cussions using a secure electronic medium, rather than the need for meetingsand conference calls. Feedback was again given to supervisors and both as-sessors on the final assigned USM. These modifications are all intended tobuild greater confidence and knowledge of the process of assessing projectwork.

There were two changes made to Part C Examinations in Trinity term2009 (the changes to regulation had been proposed by the MathematicsTeaching Committee in 2007/08). Firstly the number of questions on eachhalf unit was reduced from 4 to 3. The was to try to ensure candidates knowa larger proportion of both Michaelmas and Hilary terms’ work.

The second change was a change to the duration of a half unit examina-tion. Candidates had been allowed 1.75 hours; in 2009 this was reduced to1.5 hours. In part this was to be fair to candidates taking the whole unitwho would have 3 hours whilst a candidate taking two half units would have3.5 hours (a difference in the performance of candidates had been observed).This also draws us into line with the Computer Laboratory. These changesmake the examination process more challenging for the candidates and ex-aminers took this into account, however this should be noted particularlywhen drawing comparisons with historic data.

C. Changes in examining methods and procedures currentlyunder discussion or contemplated for the future

No future changes are planned in the short term. A period of consolidationis necessary to allow the revisions to be firmly embedded and monitor theirfunction.

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D. Notice of examination conventions for candidates

The first Notice to Candidates was issued on 6th March 2009 and the secondnotice on the 19th May.

These can be found at http://www.maths.ox.ac.uk/node/9012, and con-tain details of the examinations and assessments. The course Handbookcontains the full examination conventions and all candidates are issued withthis at Induction in their first year. All notices and examination conventionsare on-line at http://www.maths.ox.ac.uk/notices/undergrad.

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Part II

A. General Comments on the Examination

The examiners would like to thank in particular Yan-Chee Yu, WaldemarSchlackow and Helen Lowe for their commitment and dedication in runningthe examinations systems; without their tireless efforts examiners wouldbear a much heavier load and we are indebted to them. We would also liketo thank Charlotte Rigdon, Sandy Patel and Margaret Sloper for all theirsterling work in keeping track of the scripts and marks and everything elsethey do during the busy exam period. We also thank the assessors for theirprompt setting of questions and for the care in checking their own and theother half unit. All the assessors and the internal examiners would like tothank the external examiners Professors Chris Budd and Peter Gibblin fortheir prompt and careful reading of the draft papers and insightful commentsthroughout the year.

Examiners also warmly thank Prof Jon Chapman for his assistance andalso Prof Sir John Ball.

Finally we thank Prof Philipp Podsiadlowski from the Physics Depart-ment for the prompt return of scripts for C7.4 (ahead of the deadlines Physicsworks to).

Timetable

Examinations began on Monday May 25th and finished on Saturday 13thJune.

Medical certificates and other special circumstances

The examiners were not presented with any medical notes but there werecandidates for special consideration. There were minor misprints in severalpapers which required a change to the marking scheme. One was correctedduring the examination, three others were corrected at the time of markingand assessors gave special consideration of this during the marking andreported to examiners on how they had compensated candidates. Examinersaccepted these proposals. These included those candidates taking C7.2b. Amisprint was noted during the examination but it had not been possibleto make a correction whilst the paper was being sat. The assessor gavecareful consideration when marking and proposed an additional number ofmarks (4) be added to each candidate affected by the error in question 6.Examiners accepted this proposal. Physics candidates sitting this paperwere the larger group (28 candidates) affected by this so examiners reportedto the Physics Examination Board on their actions.

In summary, each special case was given careful regard following a scrutinyof their marks (and scripts where necessary).

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Setting and checking of papers and marks processing

As is our usual practice, the questions were initially set by the course lec-turer, with the lecturer of the other half of the course and the Subject PanelConvenor involved as checkers before the first drafts of the questions werepresented to the examiners. The course lecturers also acted as assessors,marking the questions on their course(s).

The internal examiners met in early January to consider the questions onMichaelmas term courses, and changes and corrections were agreed with thelecturers. The revised questions were then sent to the external examiners.Feedback from external examiners was given to examiners and the relevantassessor for each paper who responded to internal examiner for their nextmeeting. Internal examiners met a second time to consider the externalexaminers’ comments and the assessor responses making further changesas necessary before finalising the questions. The same cycle was repeatedtowards the end of Hilary term for the Hilary term courses, although theschedule here was much tighter. The Camera Ready Copy was preparedfor Michaelmas term courses and each assessor invited to sign this off soas to reduce the workload in early April when deadlines are much tighter.Following the preparation of the Camera Ready Copy for HT courses, eachassessor signed off their paper in time for submission to Examination schoolsin week 1 of Trinity term.

This year all examination scripts were collected from the MathematicalInstitute rather than Examination Schools, which all worked smoothly. As-sessors had a short time period to return the marks on standardised marksheets. More time was allowed this year as the deadlines had been too tightlast year which seems to have worked well.

A team of graduate checkers under the supervision of Yan Chee Yu, andassisted by Helen Lowe and Sandy Patel, sorted all the scripts for each paperof this examination, carefully cross checking against the marks scheme tospot any unmarked questions or part of questions, addition errors or wronglyrecorded marks. Also sub-totals for each part were checked against themarks scheme, noting correct addition. In this way a number of errors werecorrected, each change signed by one of the examiners who were presentthroughout the process. A check-sum is also carried out to ensure thatmarks entered into the database are correctly read and transposed from themarks sheets.

Determination of University Standardised Marks

This year the Mathematics Teaching Committee issued each examinationboard with broad guidelines on the proportion of candidates that might beexpected in each class. This was based on the average in each class overthe last four years, together with recent historic data for Part C, and the

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MPLS Divisional averages. It should be noted that at Part C there aresignificantly more first class classifications awarded. There are two reasonsfor this : firstly candidates need at least a 2.ii to proceed from Part B toPart C, and so only the stronger candidates are entered at Part C, thuscandidates are self-selecting. Secondly, those who attain a first at Part Bare highly likely to be awarded a first at Part C.

Examiners this year also looked at the overall distribution of classes forthe cohort at Part B as a guide to the overall quality of the group. This datacould have been extracted from OSS but examiners asked for this during themeeting. We appreciate the work of Mr Schlackow in running this databasequery. The results were informative and are given in the table below,for the combined cohort of 115 Mathematics and Mathematics & Statisticscandidates.

Table 2: Part C candidates Percentage in each class at Part B

Class Percentage of candidates at Part BI 36.5%

II.1 44.3%II.2 17.4%III 2.6%P 0

Examiners may recalibrate the raw marks to arrive at university stan-dardised marks reported to candidates, adopting the procedures outlinedbelow, similarly to previous years. Examiners also take into account reportson each question from the examiner/assessor who marked this work, tak-ing into account the standard of work, comparison with previous years, theoverall level of work presented for each question in each examination. Rawmarks for each paper were converted to USM using the algorithm establishedfor Mathematics final examinations in previous years. Calibration uses dataon the Part B performances of the candidates taking that paper. As inprevious years it seemed to work well for papers with a reasonable numberof candidates. Adjustments had to be made for some courses, particularlywhere the number of candidates was below 10.

The table below gives the corners of each piecewise linear transformationused. Each half-unit paper was considered separately and the algorithm foreach half-unit was combined to give the scaling for a whole unit paper.Generally we adopted 3 corner points for our transformation, aiming toavoid locating a corner at a class boundary. Generally default corners werechosen in a Raw 7→ USM mapping as (x1, 37), (x2, 57) and (x3, 72). The

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algorithm takes into account the proportion of each class taking the paperbased on their Part B performances. These indicated the proportion foreach class of the Part C papers, and the algorithm calculated the valuesfor x1, x2 and x3. As these values are “corners” we will denote these by(C37, 37), (C57, 57) and (C72, 72) where C37, C57 and C72 are the rawmarks which were mapped to USMs of 37, 57 and 72, respectively.

The final position of the corners of the algorithm for courses with atleast 5 candidates is given in the table below, Table 3, page 8.

For each paper, N denotes the population of candidates which for ouralgorithm is the collection of Mathematics and Mathematics & Statisticscandidates taking that paper. Similarly N1, N2 and N3 are the numbersof those candidates whose incoming averages USM for Part B were in theranges (69, 100], (59, 69] and (0, 59] respectively. That is, for calibration atPart C, we use the incoming results of candidates.

Half unit papers are usually indicated with (a) or (b) and total rawmarks are 50, whilst the whole unit papers are reported by section (and soalso have a total raw mark of 50). Here all marks are reported as USMs (iescaled out of 100). In this table raw marks are reported as the final rawmark of each corner. These are not always the default raw mark assignedby the algorithm but an adjustment may have been made. No Mathematicsor Mathematics & Statistics candidates took MS1b so no data is recordedhere.

In the Corners Table, Table 3 on page 8 the following key is used :

• ∗ indicates that a corner has been moved;

• ∗∗ indicates an extra corner has been inserted;

• † denotes the use of no corners or corners have been inserted by hand;

• ‡ denotes the removal of one corner.

Table 4 on page 9 gives the overall performance of candidates based ontheir rank within the whole cohort together with the number and percentageof candidates with the given overall USM or greater value.

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Table 3: Position of corners of piecewise linear function

Paper N N1 N2 N3 C72 C57 C37 Added cornerC1.1 (section 1) 8 0 6 2 71.2 17.2 7.9C1.1 (section 2) 8 0 6 2 82 21.6 13C1.2 (section 1) 15 8 7 0 68.8 24 8.2C1.2 (section 2) 15 8 7 0 68 42 19.4C2.1 (section 1) 7 3 3 1 62.4 44.4 20.4C2.1 (section 2) 7 3 3 1 70 44 25C3.1 (section 1) 9 6 2 1 77.2 32 10.6C3.1 (section 2) 9 6 2 1 68 34 14C4.1 (section 1) 16 10 5 1 72.4 39.4 18.1C4.1 (section 2) 16 10 5 1 76 26 7.2C5.1 †C5.1a 7 4 3 0 70 52 27.4C6.1a 18 3 10 5 70 31 24C6.2b 23 5 13 5 86 56 19.2C6.3 (section 1) 23 9 12 2 72 (38,56)∗ 10.9C6.3 (section 2) 23 9 12 2 70 (34,52)∗ 9.8C6.4a 28 8 17 3 80 44.2 20.3C7.1b 8 1 3 4 80 64 30C7.4 †C8.1 (section 1) 35 8 21 6 82 44 19.4C8.1 (section 2) 35 8 21 6 78 42 20.9C9.1 (section 1) 7 3 4 0 80 32 9.2C9.1 (section 2) 7 3 4 0 66 38.4 17.6C10.1 †C10.1a 10 4 4 2 70 42.4 19.4C11.1 (section 1) 22 6 10 6 78 48 22C11.1 (section 2)∗∗ 22 6 10 6 58 (40,62) (24,50)∗ (16,32)∗C12.1 (section 1) 8 0 5 3 84 58 28.4C12.1 (section 2) 8 0 5 3 (82,75)∗ 37.2 (16,30)∗C12.2 †C12.2a 5 1 3 1 78 66 34.5MS2b 14 5 7 2 72 44 20.2MS3b 23 7 15 1 61 (36,55)∗ 12.6

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Table 4: Rank and percentage of candidates with this orgreater overall USM

Av USM Rank Candidates with this USM or above %94 1 1 1.0593 2 2 2.1188 3 4 4.2186 5 6 6.3285 7 8 8.4284 9 9 9.4783 10 10 10.5381 11 11 11.5880 12 15 15.7979 16 19 2078 20 20 21.0577 21 23 24.2176 24 27 28.4275 28 30 31.5874 31 33 34.7473 34 34 35.7972 35 35 36.8471 36 37 38.9570 38 48 50.5368 49 53 55.7967 54 56 58.9566 57 60 63.1665 61 63 66.3264 64 64 67.3763 65 67 70.5362 68 72 75.7961 73 76 8060 77 78 82.1159 79 82 86.3258 83 85 89.4757 86 87 91.5855 88 88 92.6354 89 89 93.6851 90 91 95.7949 92 92 96.8448 93 94 98.9514 95 95 100

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B. Equal opportunities issues and breakdown of the resultsby gender

Table 5, page 10 shows the performances of candidates broken down bygender. Examiners note that no discernable difference between the gendersis apparent although this does vary from year to year. Our processes remainunchanged. Some years the Student Union requests a Finals Forum and DrGiovanna Scataglini-Belghitar usually runs this although this year we havenot been asked to.

Table 5: Breakdown of results by gender

Class Total Male FemaleNumber % Number % Number %

I 48 50.53 34 50.75 14 50II.1 30 31.58 21 31.34 9 32.14II.2 13 13.68 10 14.93 3 10.71III 3 3.16 1 1.49 2 7.14P 0 0 0 0 0 0F 1 1.05 1 1.49 0 0Total 95 100 67 100 28 100

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C. Detailed numbers on candidates’ performance in each partof the exam

The Figure 1 in the appendix on page 41 shows the quartiles for the rawmarks on each section and half unit paper.

The following table 6 on page 11 and continued in table 7 on page 12gives the details numbers and performance of candidates on each half unitpaper.

Table 6: Numbers taking each paper

Paper Section Number of Avg StDev Avg StDevCandidates RAW RAW USM USM

C1.1 1 8 15.12 10.27 55 13.08C1.1 2 8 20.5 8.23 61.12 6.01C1.1a 1 2 25.5 9.19 66.5 4.95C1.1b 1 2 37.5 0.71 70.5 0.71C1.2 1 15 26.33 9.55 67.33 7.42C1.2 2 15 31.87 8.94 71.13 12.91C1.2a 1 11 27.45 11.09 68.73 10.51C1.2b 1 4 29.5 6.81 67.5 9.47C2.1 1 7 26.43 8.16 63.71 13.34C2.1 2 7 34.57 8.96 73.86 13.89C2.1a 1 4 36.5 7.59 79.75 11.79C2.1b 1 3 38 6.93 78.33 12.7C3.1 1 9 33.22 12.59 71.56 13.45C3.1 2 9 32.44 15.39 74.11 21.16C3.1a 1 3 28 16.52 66.33 18.5C4.1 1 16 36.12 9.14 76.06 13.6C4.1 2 16 33.38 10.61 72.06 11.75C4.1a 1 1 37 74C4.1b 1 1 14 58C5.1 1 4 42 2.94 85 5.89C5.1 2 4 40.5 4.12 81 8.25C5.1a 1 7 38.14 5.7 78 10.33C5.2b 1 4 36.5 10.97 74 17.8C6.1a 1 18 25.33 7.75 64.28 8.68C6.2b 1 23 34.87 12.01 65.52 20.22C6.3 1 23 29.7 10.43 65.35 17.56C6.3 2 23 31.3 11.68 67.57 19.8C6.3a 1 20 28.45 8.47 64.35 9.72C6.3b 1 3 22.33 3.51 57.33 3.51C6.4a 1 28 33.64 6.75 66.93 6.88C7.1b 1 8 35.62 9.72 68 16.33

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Table 7: Numbers taking each paper CTD

Paper Section Number of Avg StDev Avg StDevCandidates RAW RAW USM USM

C8.1 1 35 35 10.26 68.63 15.35C8.1 2 35 33.97 9.04 69 12.08C8.1b 1 16 32.12 8.82 67.25 10.56C9.1 1 7 31.71 12.72 68 14.47C9.1 2 7 34.86 6.77 74.71 9.81C9.1a 1 5 37.6 6.23 73.2 10.31C9.1b 1 2 32 16.97 73.5 21.92C10.1 1 2 49.5 0.71 99 1.41C10.1 2 2 45 1.41 88.5 3.54C10.1a 1 7 34.86 10.37 74.43 15.02C11.1 1 14 31.36 6.98 65.71 10.36C11.1 2 14 20.36 9.32 60.07 13.96C11.1a 1 20 30 7.33 63.35 9.39C12.1 1 5 33 3 66.6 4.51C12.1 2 5 29.6 6.66 66 5.15C12.1a 1 8 32.88 7.08 66.75 11.26C12.1b 1 1 41 75C12.2 1 4 38.75 9.57 72.75 22.37C12.2 2 4 32.75 11 64.25 19.09C12.2a 1 3 34.33 10.79 63.67 18.34C12.2b 1 4 32 9.06 62.25 17.63MS2b 1 3 26.33 9.29 60.67 12.1MS3b 1 12 25.5 10.23 64.75 15.81

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The tables that follow give the Question statistics for each paper forMathematics candidates.

Paper C1.1: Godel’s Incompleteness Theorems and Model Theory

Question Mean Mark Std Dev Number of AttemptsAll Used Used Unused

Q1 6.43 6.43 4.08 7 0Q2 9.00 9.00 6.73 7 0Q3 6.50 6.50 4.95 2 0Q4 12.38 12.38 4.27 8 0Q5 7.57 7.57 5.29 7 0Q6 6.50 12.00 7.78 1 1

Paper C1.1a: Godel’s Incompleteness Theorems


Q1 8.00 8.00 4.24 2 0Q2 17.50 17.50 4.95 2 0

Paper C1.1b: Model Theory


Q1 19.00 19.00 1 0Q2 20.00 20.00 1.41 2 0Q3 16.00 16.00 1 0

Paper C1.2: Analytic Topology and Axiomatic Set Theory


Q1 5.00 10.00 3.74 1 3Q2 14.57 14.57 6.10 14 0Q3 12.07 12.07 4.46 15 0Q4 17.33 17.33 3.99 15 0Q5 9.38 12.50 7.27 4 4Q6 15.27 15.27 4.78 11 0

Paper C1.2a: Analytic Topology


Q1 3.00 3.00 1 0Q2 15.27 15.27 6.25 11 0Q3 13.10 13.10 4.63 10 0

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Paper C1.2b: Axiomatic Set Theory


Q1 17.00 17.00 4.55 4 0Q2 10.00 11.00 1.41 1 1Q3 13.00 13.00 2.65 3 0

Paper C2.1: Lie Algebras and Representation Theory of Symmet-ric Groups


Q1 6.50 8.75 6.66 4 2Q2 17.14 17.14 4.67 7 0Q3 8.00 10.00 5.48 3 1Q4 18.50 18.50 3.51 6 0Q5 18.50 18.50 3.54 2 0Q6 15.67 15.67 6.47 6 0

Paper C2.1a: Lie Algebras


Q1 16.00 16.00 2.65 3 0Q2 19.00 19.00 5.89 4 0Q3 22.00 22.00 1 0

Paper C2.1b: Representation Theory of Symmetric Groups


Q1 19.67 19.67 2.52 3 0Q2 14.00 14.00 1 0Q3 20.50 20.50 4.95 2 0

Paper C3.1: Lie Groups and Differentiable Manifolds


Q1 18.00 18.00 4.16 7 0Q2 15.57 15.57 8.10 7 0Q3 14.20 16.00 10.26 4 1Q4 14.00 15.29 9.13 7 1Q5 17.89 17.89 8.36 9 0Q6 12.00 12.00 4.24 2 0

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Paper C3.1a: Lie Groups


Q1 15.00 15.00 7.55 3 0Q2 18.00 18.00 4.24 2 0Q3 3.00 3.00 1 0

Paper C4.1: Functional Analysis and Banach and C∗-Algebras


Q1 16.23 16.23 5.67 13 0Q2 20.82 22.30 5.76 10 1Q3 15.20 16.00 3.85 9 1Q4 16.50 16.50 7.19 16 0Q5 17.00 17.00 4.22 12 0Q6 16.50 16.50 3.42 4 0

Paper C4.1a: Functional Analysis


Q1 17.00 17.00 1 0Q3 20.00 20.00 1 0

Paper C4.1b: Banach and C∗-Algebras


Q1 5.00 5.00 1 0Q2 9.00 9.00 1 0

Paper C5.1: PDEs for Pure and Applied Mathematicians andCalculus of Variations


Q1 20.67 20.67 0.58 3 0Q2 21.50 21.50 2.52 4 0Q3 20.00 20.00 1 0Q4 18.75 18.75 4.35 4 0Q5 22.67 22.67 1.15 3 0Q6 19.00 19.00 1 0

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Paper C5.1a: PDEs for Pure and Applied Mathematicians


Q1 20.33 20.33 2.52 3 0Q2 17.86 17.86 3.63 7 0Q3 20.25 20.25 3.40 4 0

Paper C5.2b: Fixed Point Methods for Nonlinear PDEs


Q1 18.00 18.00 7.75 4 0Q2 17.67 17.67 4.73 3 0Q3 21.00 21.00 1 0

Paper C6.1a: Solid Mechanics


Q1 12.20 12.64 5.85 14 1Q2 12.38 13.00 4.03 15 1Q3 12.00 12.00 4.04 7 0

Paper C6.2b: Elasticity and Plasticity


Q1 16.94 16.94 5.26 17 0Q2 15.74 16.28 8.31 18 1Q3 20.09 20.09 4.74 11 0

Paper C6.3: Perturbation Methods and Applied Complex Vari-ables


Q1 11.53 14.00 7.56 13 4Q2 12.17 13.19 6.47 16 2Q3 16.28 17.06 4.59 17 1Q4 16.26 16.26 6.08 23 0Q5 15.10 15.60 5.88 20 1Q6 11.33 11.33 12.06 3 0

Paper C6.3a: Perturbation Methods


Q1 13.64 13.64 3.72 11 0Q2 13.31 13.31 5.06 13 0Q3 15.38 15.38 5.21 16 0

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Paper C6.3b: Applied Complex Variables


Q1 12.67 12.67 3.06 3 0Q2 9.67 9.67 5.51 3 0

Paper C6.4a: Topics in Fluid Mechanics


Q1 15.91 16.10 4.36 21 1Q2 17.61 17.61 3.69 23 0Q3 16.58 16.58 4.25 12 0

Paper C7.1b: Quantum Theory and Quantum Computers


Q1 15.88 19.17 7.72 6 2Q2 18.75 18.75 3.85 8 0Q3 10.00 10.00 2.83 2 0

Paper C8.1: Mathematics and the Environment and MathematicalPhysiology


Q1 10.25 10.25 8.30 4 0Q2 15.33 15.72 6.06 32 1Q3 20.03 20.03 3.93 34 0Q4 17.65 18.16 4.30 19 1Q5 15.74 15.74 5.51 27 0Q6 17.20 18.22 7.43 23 2

Paper C8.1b: Mathematical Physiology


Q1 14.22 14.22 2.59 9 0Q2 14.31 15.17 5.81 12 1Q3 18.55 18.55 6.02 11 0

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Paper C9.1: Analytic Number Theory and Elliptic Curves


Q1 15.00 15.00 6.32 6 0Q2 13.00 13.00 8.29 4 0Q3 20.00 20.00 2.45 4 0Q4 18.00 18.00 3.46 5 0Q5 17.20 17.20 4.82 5 0Q6 17.00 17.00 3.16 4 0

Paper C9.1a: Analytic Number Theory


Q1 20.20 20.20 2.95 5 0Q2 16.50 16.50 6.36 2 0Q3 18.00 18.00 3.61 3 0

Paper C9.1b: Elliptic Curves


Q1 15.50 15.50 9.19 2 0Q2 22.00 22.00 1 0Q3 11.00 11.00 1 0

Paper C10.1: Stochastic Differential Equations and Brownian Mo-tion in Complex Analysis


Q2 24.50 24.50 0.71 2 0Q3 25.00 25.00 0.00 2 0Q5 22.00 22.00 1.41 2 0Q6 23.00 23.00 2.83 2 0

Paper C10.1a: Stochastic Differential Equations


Q1 15.50 15.50 7.05 4 0Q2 17.14 17.14 6.52 7 0Q3 15.75 20.67 10.50 3 1

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Paper C11.1: Graph Theory and Probabilistic Combinatorics


Q1 14.79 14.79 5.18 14 0Q2 16.07 16.69 3.79 13 1Q3 12.50 15.00 3.54 1 1Q4 9.14 9.14 5.32 14 0Q5 18.00 18.00 6.24 3 0Q6 9.07 9.36 3.54 11 3

Paper C11.1a: Graph Theory


Q1 14.80 14.80 4.29 20 0Q2 14.63 14.89 4.62 18 1Q3 13.40 18.00 7.70 2 3

Paper C12.1: Numerical Linear Algebra and Continuous Opti-mization


Q1 16.00 16.00 1.41 4 0Q2 17.00 17.00 1.87 5 0Q3 10.50 16.00 7.78 1 1Q4 17.60 17.60 0.89 5 0Q5 11.50 11.50 7.78 2 0Q6 12.33 12.33 6.43 3 0

Paper C12.1a: Numerical Linear Algebra


Q1 18.86 18.86 1.86 7 0Q2 13.38 13.38 5.53 8 0Q3 24.00 24.00 1 0

Paper C12.1b: Continuous Optimization


Q1 20.00 20.00 1 0Q3 21.00 21.00 1 0

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Paper C12.2: Approximation Theory and Finite Element Methods


Q1 14.50 14.50 0.71 2 0Q2 19.75 19.75 4.57 4 0Q3 23.50 23.50 0.71 2 0Q4 14.50 14.50 6.95 4 0Q5 18.25 18.25 4.92 4 0Q6 17.00 0 1

Paper C12.2a: Approximation Theory


Q1 15.00 14.00 1.41 1 1Q2 16.00 16.00 7.55 3 0Q3 20.50 20.50 2.12 2 0

Paper C12.2b: Finite Element MEthods for Partial DifferentialEquations


Q1 11.67 11.67 2.08 3 0Q2 17.75 17.75 3.86 4 0Q3 22.00 22.00 1 0

Paper MS2b: Stochastic Models in Mathematical Genetics


Q1 15.50 15.50 7.78 2 0Q2 12.33 12.33 6.03 3 0Q3 11.00 11.00 1 0

Paper MS3b: Levy processes and Finance


Q1 15.75 15.75 5.17 12 0Q2 8.83 8.83 5.98 6 0Q3 9.43 10.67 6.40 6 1

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D. Recommendations for Next Year’s Examiners and Teach-ing Committee

We have some minor recommendations.First that there are no changes to the current examining method and

procedures to allow for a period of consolidation.Second we recommend that a secure electronic system is used to facilitate

discussion between the three faculty who participate in the reconciliationof each project mark (we believe weblearn2 will enable this and hope thedepartment will develop a link to this). Third that the timeframe betweenreceipt of external examiner comments and the next examiners meeting canbe reduced. Finally we recommend that Teaching Committee guidance onclass percentages might also include the table for the Part B results for thecandidates under review in Part C or at least base their recommendationsin the light of this.

We also recommend that a prize is instituted for dissertations.One external examiner made a suggestion about how to ensure examina-

tion papers are transferred between staff by hand and suggests “BY HANDONLY” folders. We would endorse his suggestion. The other external exam-iner suggests that it is common practice in some universities to re-work theexamination a week before it is set. Looking at the paper afresh after sometime has lapsed might iron out the misprints that have occurred. Examinerswould endorse this suggestion as good practice.

This year Examiners did look at dissertations which were borderline orfor borderline candidates. Examiners did not routinely look at borderlinescripts this year and for the future, it might be worthwhile identifying bor-derline candidates for consideration. We can usually identify borderlinescripts per half unit before the meeting for scrutiny by the board. To beable to replicate this for the class list would require a break in the meetingas we would not know which candidates were borderline until the maps hadbeen assigned. This would seem good practice.

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E. Comments on sections and on individual questions

The following comments were submitted by the assessors. Examiners thankassessors for their comments on where the possible borderlines might lie andhave used this in helping them to determine the maps. For this report suchstatements have been removed.

C1.1a: Godel’s Incompleteness Theorems

Twenty-three candidates in TT 2009 took C1.1/C1.1a Godel’s Incomplete-ness Theorems, 10 from Mathematics, 11 from Mathematics and Philosophy,2 from Mathematics and Computer Science. The average mark per questionoverall was 13.3. The average mark per question for each school was: Math-ematics 8.6, Mathematics and Philosophy 16.2, Mathematics and ComputerScience 20.5. Five candidates (four from Maths & Phil, one from Maths &Comp) scored above 20 on each of their two answers. Six candidates (fivefrom Maths—half the cohort, one from Maths & Phil.) scored a total ofless than 15 marks on their two answers. The questions this year turnedout to be more taxing than last year. The difficulty of the difficult partsof the questions does not explain the unprecedented low marks of thosecandidates who were unable to answer basic book work parts of questions.Each question received at least one completely or very nearly completelycorrect solution. There were 21 answers to question 1 (on the First and Sec-ond Incompleteness Theorems), 22 answers to question 2 (Rosser’s Theoremand the relationship between the Godel and Rosser sentences), 3 answers toquestion 3 (on Lob’s Theorem and provability logic).

C1.1b: Model Theory

Of the 19 students who wrote the exam, all but two did two questions, 14did questions 4 and 5 (resp. 1 and 2 on the half-unit counting), only 7 tookquestion 6 (resp. 3).

In question 4, only 3 students managed to do part (c)(ii)(β) and part(d) despite the fact that this was in the notes as exercise. In question 5(c),at least half the students didn’t read carefully (they overlooked the word”infinite”).

The two best students were the two Maths and Computer Science stu-dents, the two next were Maths and Philosophy , positions 4,5, 6 and 7were the three MMath students taking half-units and another Maths andPhilosophy student followed.

There is an average of 29 points fairly evenly distributed through thefull range.

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C1.2a: Analytic Topology

There was a wide spread of marks, from 11/50 to 47/50 (a M&C candidate).My impression is that there were more relatively weak candidates (and per-haps fewer outstanding candidates) taking this option than last year.

Q1 was about a topology they had not seen before, but which has prop-erties similar to things studied in lectures and on problem sheets. Strongercandidates, and those who had prepared the bookwork preferred to opt forthe safety of the more standard questions 2 and 3.

Q2 consisted of bookwork plus a rider that most people had a go at.Hardly anyone suggested the Stone–Cech compactification of the naturalnumbers as an example in the last part.

Q3 combined something about filters with something about compactifi-cations. It was (at least) very similar to material covered in the course andon problem sheets. In giving a one-point compactification of R, the majorityof candidates wrote out (at great length) the general Alexandroff construc-tion in this special case, rather than giving a concrete homeomorphism fromR to a dense subset of a circle. This made the question very long for them,thought a few did see how to modify the Alexandroff construction to givea 2-point compactification (again a concrete homeomorphism from R onto(-1,1) is much easier.)

C1.2b: Axiomatic Set Theory

27 students sat the exam, the number slightly bigger than in previous years.Practically all students demonstrated sufficient understanding of the mate-rial and scored at least 11 marks in two out of three questions, but overallresults were a bit below the expectations. Only two students scored morethan 20 marks in two questions.

Question 4. This was attempted by all 27 candidates and all of themscored at least 11 marks. In fact 11-14 marks one could get from doing justa bookwork, and the bookwork in this question was quite basic - standardproperties of the cumulative hierarchy. But the unseen part (iv) of the ques-tion proved to be rather difficult, so only 4 candidates scored 21 or more forthe question.

Question 5. The least popular, attempted by only 13 students and 8 ofthese scored below 10 marks. The question was about cardinal arithmetic,including Koenig’s Lemma. The proof of the Lemma itself wasn’t a majorproblem, although some made serious technical mistakes in the proof. But,

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rather unexpectedly, only few of the candidates demonstrated reasonableskills in direct cardinal arithmetic calculations.

Question 6. Here candidates demonstrated rather good understanding ofthe idea of absoluteness, though with quite a few technical errors. The mostcommon were mistakes in writing down statements in the AST-language.

C2.1a: Lie Algebras

The performance of the candidates overall was marginally less good than Iexpected, but much better on one question.

Question 1. A rather disappointing response. The first part, a slightlydisguised version of an early result (but with one of the conceptually hardestproofs in the course) was done well by only four candidates. Of the threeapplications, two were anticipated in problem sheets and were well done;in the third, which was easier, no-one thought of using the Jacobi identity.Average mark 9.7.

Question 2. This question, concerning a range of ideas central to thecourse, and with a final part remote from problem sheets, was encouraginglywell done: average mark 17.6, one perfect solution and four others withmarks over 20.

Question 3. The topic came from near the end of the course, and thosewho had studied the material properly seemed to enjoy this relatively easyquestion. But some students who could not remember the relevant defini-tions brought the average mark down to 10.3.

C2.1b: Representation Theory of Symmetric Groups

Qu. 1 [Hook formula and Murnaghan-Nakayama formula]There was a mistake in the question: in part (3) one needs to add ...and

λk = 1. Several candidates noticed this (or if not, assumed it anyway).Almost everybody attempted the question, and received good marks.

Qu. 2 [Specht modules in characteristic zero]This was the least popular question. The solutions were of good stan-

dard.

Qu. 3 [Simple modules over prime characteristic]Almost everybody attempted the question, and most attempts obtained

good results.

My impression is that the questions were of appropriate difficulty, andgenerally candidates showed good understanding (and some ingenuity).

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C3.1a: Topology and Groups

There were 24 candidates for this course, which is perhaps slightly lowerthan in previous years. The overall standard of the scripts was extremelyhigh. My judgement of the paper is that its standard was about right, andthat the questions successfully assessed a broad range of the course.

As an aside, I feel that the move to setting only 3 questions on eachof the half course worked very well, since it meant that students could notabandon large parts of the course without a substantial penalty.

Question 117 attempts; Average mark: 17.9; 7 alphas (in mark range 20-25); 6

betas (in mark range 13-19).This question was based on a central part of the course, dealing with

interplay between the topological and combinatorial definitions of the fun-damental group. The first part was bookwork and required a statement andapplication of the simplicial approximation theorem. The second half of thequestion was a fairly straightforward calculation. The quality of the answerswas, on average, very high. Question 2

21 attempts Average mark: 17.4 11 alphas 6 betasThis question required an understanding of both the group-theoretic and

the topological parts of the course. The students were required to definepush-outs of groups and their universal property. They then had to statethe Seifert - van Kampen theorem and apply it to give a presentation for thefundamental group of the Klein bottle. The range of marks for this questionwas very broad: one student got zero, another only got two marks, whereassomeone received 25 marks.

Question 312 attempts; Average mark: 16.6; 12 alphas; 6 betas.This was the least popular question, because it was on the final third of

the course. The bookwork part required the students to prove that coveringmaps are pi 1-injective. The second part, and the bulk of the question, askedthe student to construct covering spaces and thereby find certain subgroupsof the free product of Z and Z/2. Unsurprisingly, this divided studentsconsiderably. Those who understood covering spaces gave some excellentanswers (including three 25s), but those students who had not really got togrips with covering spaces floundered.

C3.1b: Algebraic Topology

Generally the exam went reasonably well though the distribution of markswas closer to an inverse bell curve. It was disappointing that most studentsconcentrated on the first two questions. I expect that the distribution of thecourse material over the three questions where none of the questions couldonly be answered from one third of the syllabus made it quite hard for those

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to score high who were still struggling with some parts of the conceptuallydemanding course.

Half-unit: There were no candidates taking this as a half unit.Third fourth year option: 4 third year students and 12 fourth year stu-

dents took the exam. The two groups could not be distinguished by theirperformance. It was pleasing to see that two out of the four third year stu-dents achieved (scaled) raw marks in the 90s.

Q4: (15 attempts) Students had problems defining the local degree andproving the formula relating the sum of local degrees to the degree. Pleas-ingly, the majority of candidates knew how to attack the application in (c).

Q5: (16 attempts) This question was done very well with 12 candidatesachieving alpha marks. In part (a) points were lost because the definitionof a CW complex was not precise, or because a proof for ∂2 = 0 was notremembered. Part (b) was generally well- done, including the statement ofthe universal coefficient theorem and its application here.

Q6: (2 attempts) Only one serious attempt really. The last part wasreally an application of the Mayer-Vietoris Theorem. This was spotted butnot carried out.

Marks achieved: ≥ 45: 4; ≥ 40: 3; ≥ 35: 1; ≥ 30: 1;≥ 25: 2; ≥ 20: 3;≥ 0: 2.

C4.1a: Functional Analysis

The standard of the scripts was high, considerably better than 2007 andcomparable overall with 2008 (there were a few more really excellent scriptsthis year, but also a few more mediocre ones). The presentation was againgenerally good, with proper sentences and accurate use of quantifiers.

Q.1 and Q.3 both had stings in the tail—deliberately, in order to avoid anembarrassment of high marks- while Q.2’s sting was milder. This was partlydeliberate, because questions involving weak and weak* topologies had beenunpopular in the past and the reduction to 3 questions on the paper thisyear meant that there was less scope to avoid that part of the syllabus. Thiswas successful in generating a good number of attempts at Q.2 and can-didates being rewarded for understanding the topic. Some account of thestrength of the stings was taken when marks were allocated. The marks onQ.2 ended up higher than on Q.1 and Q.3, but Q.2 was far from being a gift.

In Q.1, a few candidates did not notice that the proof in lectures thatcompactness of the unit ball implies finite-dimensionality works under the

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weaker assumption of total boundedness. One or two thought they couldquote the compactness result, and a decent partial mark was given if theythen dealt properly with the passage to the totally bounded case. One can-didate claimed that one part of the question made no sense, making an(implausible) interpretation of it which he answered correctly for no credit,but he then wrote down the relevant fact and so received the one mark al-located to it. No other candidate was confused by the wording.

In Q.3, surprisingly many candidates had difficulty showing that‖x‖ ≤ supp ‖Pkx‖ < ∞ and supn |φn(x)| < ∞. Each of the three in-

equalities is a simple fact from Mods analysis of series, with norms replacingabsolute values. Although few marks were available for this, failure thereusually caused problems with some of the easy implications in the last part.It was much less surprising that few candidates made much progress withthe sting in the implications involving IMT/UBT/CGT.

C4.1b: Banach and C∗-Algebras

General Comments Eighteen candidates engaged in this part of PaperC4.1. That the average mark was somewhat lower than in previous yearscan be accounted for by the fact that the questions were not quite as pre-dictable as they have sometimes been. It is clear that there were severalcandidates of a very high standard.

• 4. Every candidate attempted this question which produced sevenpieces of work of α standard. Whilst the piece of bookwork was rea-sonably well done, the problem, which was similar to some on theproblem sheets, was not well done by most candidates.

• 5. Fourteen candidates attempted this question which produced fivesolutions of α standard. The first part of the question required can-didates to describe the functional calculus for normal elements of aC∗-algebra and most were able to do this. Although the propertiesof faces of convex sets had been covered in the lectures none of thecandidates were able to use their knowledge to prove that the elementsof a proper face of the order interval [0, 1A] not containing 0 must beof norm one.

• 6. Probably because this question concerned material in the latterstages of the course, it attracted only four attempts, one of which wasof α standard. The main obstruction was that the candidates did not

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seem familiar with the formula

Tr(a) = 〈aξ1, ξ1〉+ 〈aξ2, ξ2〉for the trace of a 2 × 2 matrix a, where ξ1, ξ2 forms an orthonormalbasis for C2.

C5.1a: PDEs for Pure and Applied Mathematicians

The paper was very well done with an average raw mark of 38/50. Everycandidate displayed a good grasp of the material and many produced ex-tremely good solutions. All but one candidate tackled question 2 with sevendoing question 1 and eight doing question 3. The average mark for eachquestion lay between 20 and 21.

Question 1: Only the last part of (b) caused trouble. Several candidateshad the right idea but no one completed it successfully. The remainder waswell done.

Question 2: Only one candidate fell into the trap of using the Poincareinequality. In general candidates were clear about which spaces they wereusing and could produce the basic arguments. The chief difficulties came inproving that I(v) is bounded below in H1(Ω) and in producing the correctinequality in which to take limits using weak convergence.

Question 3: This was also well done and there were some very elegantsolutions. The part that caused most trouble was showing that B∗[u, v] =B[v, u] and that the associated K∗ is indeed the adjoint of K.

C5.1b:Fixed Point Methods for Nonlinear PDEs

Question 1 was attempted by all candidates. The first part on Banach’sFixed Point Theorem was done very well by everybody. The application toan elliptic PDE was done well by three candidates, one did not attempt itseriously. Some candidates lost a few marks by forgetting to argue why thethe fixed point problem is well-defined, i.e. why for given square integrableu, f(u) is square integrable. The last part proved to be difficult but onecandidate presented a nice and complete solution.

Question 2 was attempted by three candidates and well done throughout.Again, few marks were missed sometimes by not arguing that the set-up ofthe fixed-point problem is well-defined.

Question 3 was only attempted by one candidate. The parts of thequestion which were repetition from the lecture notes and problem sheetswere done well, but the new parts posed some problems.

C5.2b: Calculus of Variations

Question 1. All candidates attempted this question. With minor excep-tion, it was well done. The most difficult part was the construction of a

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minimizing sequence in part (b)(ii). Well done by the best students.Question 2. Half of students attempted this question. Surprisingly the

bookwork part turned out to be difficult. Candidates simply reproduced theproof of Legendre’s necessary condition for weak local minimizers. However,one should apply such a proof to the difference between the second variationand its lower bound.

Question 3. This was attempted by the rest of students. Bookwork wasreasonably well done. Unfortunately, all of them overlooked in (b)(ii) thepossible application of Legendre’s necessary condition. This would immedi-ately rule out u = 0 being a weak local minimizer. In (b)(iii), the relativelydifficult part was to show violation of the Weierstrass necessary conditionfor a strong local minimizer.

C6.1a: Solid Mechanics

The average mark on all three questions was about the same (about 12).Q1. There was one essentially perfect, and three good solutions to this.There was some trouble in proving that eK orthogonal implies K skew, eventhough this was essentially in the notes for the course. A lot of people usedeA+B = eAeB which is only true if A,B commute. Q2. There was onlyone good solution to this. Q3. The least popular question and only tworeasonable solutions. I did not find the reduction to three questions helpful,at least in setting the exam. The course naturally split into 4 parts, and soI struggled to find a question (Q3) that simultaneously covered two of thesetogether.

One candidate entered the wrong title for the exam on the scripts - Iguess he/she must have been surprised!

C6.2b: Elasticity and Plasticity

Question 1: Problems with (b)(i) - very few students managed to write downthe number of independent material-dependent constants in Cijkl correctly.I explicitly gave this information in lectures.

(b)(ii) Generally well-attempted.(c) Problems with using the symmetry of the strain tensor correctly and

carrying out a Taylor expansion of the strain energy density. A few studentsforgot to say that the first variation of the energy necessarily vanishes foran energy minimizer.

Question 2: (a) Well-attempted.(b) (ii) Some students had problems working out the Fourier transforms

of the derivatives of the Airy Stress function. This material was covered inthe course and should be familiar to them.

(iii) Some students made calculation errors with inverse Fourier trans-forms and using the concept of convolutions.

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Generally, most of the students attempted Question 2 well and under-stood the underlying concepts.

Question 3: (b) Some problems in the last part of (b) involving theCoulomb yield criterion. Some students did not spot the trigonometric re-lations involved.

(c) Generally very well-attempted.Some students had problems solving the quadratic equation

4k = (1 + k)2 cos2 φ

but otherwise, Question 3 was very well-attempted.

C6.3a: Perturbation Methods

I purposely made this exam harder than the 07-08 one, which I felt wasa bit too easy. I think the 08-09 exam was at the right level of difficulty,though of course one would also expect scores to be depressed a bit by theexamination rule change (choose 2 of 3 questions instead of 2 of 4). Themean exam score was just under 29/50. The students did much better onthings like scaling and steepest descents this year than last. Importantly, Ipurpose made problem #2 harder than the steepest descents problem (#1)so that people who skipped the hardest conceptual problem (#1) would befaced with a question that was in fact harder. The individual scores on allthree questions were reasonable, and the distribution of attempts on the 3questions was roughly even (or at least close enough to even). This examconveyed a handful of students (who achieved scores of 40+) who clearlyknow the material much better than the others.

Problem 1 was on steepest descents. The students did pretty well onthis, which pleased me. In this problem (and in the others), I was amazedby students’ inability even in their 4th year to read directions and state theiranswers to precisely the questions asked. Most students got a lot of points onpart (i). Almost everybody got part (ii) completely, which is unsurprisinggiven that it is trivial. Many people did part (iii), though some peoplegave this estimate via stationary phase instead of following instructions tocompute the integral exactly. Only the best students got to the end in (iv),which was by design.

Problem 2, a dominant balance + nonlinear boundary layer problem, wasthe hardest question on the exam (in my opinion). I noticed throughout thisproblem that students used notation (apparently introduced by one of thetutors) without ever actually defining it. I eventually figured out what theymeant from context, but as with the precise answering of questions, I find itabsurd for a student to introduce notation that was not the standard for theclass without defining it. Neither I nor any other assessor is going to havemuch sympathy for that. Most students did reasonably on part (i), whichcovered dominant balance, but many students forgot to discuss the outer

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solution and all but a handful of students considered the case of keeping allthree terms in the balance (which I purposely included in the problem as achallenge). Students had difficulty with part (ii). Many of them seem to notknow what a phase plane is, which is disappointing given how often studentshave seen this throughout their time at Oxford. Many of the students usedthe Lienard plane instead of the canonical way of writing (2) as a dynamicalsystem. (I hadn’t anticipated that, but I should have. It was certainlypossible to get things to work correctly that way, so marking this part ofthe problem was quite a chore.) None of them did it quite correctly, because(and again I find this very disappointing) they completely ignored the ideaof a constant of integration. In general the students had trouble here. Moststudents were able to do quite well on part (iii) and some achieved perfectscores on it.

Problem 3 was on WKB theory (the simplest type of turning point prob-lem) with a little bit of boundary layer theory thrown in. There was an in-consequential typo in the question—the provided expression for Bi(0) had alatex error—but this did not affect things in any way because it only meantthe constant obtained in part (ii) was different than what I had in the an-swer sheet. Hence, this literally had no effect on anybody’s score because Ijust marked based on the constant one should get with the given Bi(0). Themajority of candidates were able to score most of the marks here (and I waspleased to see that they mostly understood dominant balance). Almost ev-erybody missed the fact that both c1 and c2 can take any value, as epsilon isfixed as one takes x− > ∞. (Students could realise this based on the resultsfor part (iii), but I think only 1 or two people did.) Again in (i), I also sawthat students seem not to realize that they have to consider constants ofintegration. They do get absorbed in other constants, but I required eitherthat process to be done or to state that that would happen (as a reason fornot including constants of integration at appropriate intermediate stages) toget full credit. Part (ii) was trivial and nearly every student got all of thepoints for it. A couple of people were completely confused and then a fewdidn’t follow directions on the last part to give β = β(α) (and instead gave,say, α = α(β). Again, I have no sympathy for not following instructions.Students had some trouble with part (iii) and many of them assumed thatcanonical boundary-layer matching would be the best thing to do, which isdifferent from what I discussed with them about turning points for WKB inthe lectures. I think students in general have trouble with WKB. Finally,some students seem to think that “write down the final formula” doesn’tneed to be followed. It’s so trivial to do that after the previous parts ofthe questions, but one has to answer the precise question that is asked. It’samazing that so many people will lose several points over the course of anexam on things they clearly understand just by not following instructions.

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C6.3b: Applied Complex Variables

No errors on the paper, and the questions proved good at distinguishingbetween classes. A few very good answers. Almost all people chose to doQs 1 and 2: there were only 2 answers to Q3.

C6.4a: Topics in Fluid Mechanics

There we’re no problems with the paper.

The first question was long but straightforward. Nobody managed toget it all correct, but there were some reasonable answers.

The second question was also fairly well done on the whole. Nobody atall managed to get the last big right.

I think the raw marks should pretty much reflect the USMs. First classborder is probably about 70-75.

Question 3 was a problem on Rayleigh-Benard instability in a rotatingframe. Students for the most part performed quite well on the first twoparts of the question, while the last part was more difficult.

On part (a) of the question the answer sheet gives the Taylor number asTa = f2d4ρ2

0/µ2. However, I also accepted Ta = fd2ρ0/µ as correct (theywere not introduced to the Taylor number in class and the latter definition ismore intuitive upon nondimensionalisation). Other than that the markingconformed to the answer sheet.

C7.1b: Quantum Theory and Quantum Computers

A course which falls evenly into two halves is not easily tested in threequestions. Apart from that, these questions seemed to work reasonablywell, though the first was probably a bit too long. The third question (onthe second half of the course) was least popular, but the attempts includeda good solution. There were some perfect and near perfect solutions to theother two questions, and a good spread of marks.

C7.2b: General Relativity I

Three mathematics candidates attempted the paper, with one result verypoor, and too very average.

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Twenty six physics candidates, and two physics and philosophy candi-dates, attempted the paper, with five very satisfactory results, five ratherpoor, the remaining average.

Q4: Attempted by almost all candidates, with only one of them almostdoing the last, unseen problem.

Q5: Attempted only by three candidates, with two average and one goodresult.

Q6: There was an unfortunate misprint in part (b) of that question: asubstitution u = 2m/r was proposed, but the standard substitution in thiscontext is u = m/r, and the answer was given assuming that substitution.The proposed answer was thus incorrect with the substitution proposed,which led to a loss of time and confusion for some candidates.

The background to question Q6(b) is the following: the bulk of the workwas to to derive the constants of motion following from the invariance ofthe Lagrangian. The calculation required was done both in the lectures,and in problem sheets; at the end of the calculation it is usual to make thesubstitution u = m/r to obtain a simple first order equation of motion. Thefinal equation is sensitive to the choice u = m/r or u = 2m/r. The proposedmarks for that question were intended to be given for the correct analysisof the Lagrangian.

All candidates who attempted Q6 attempted Q6(b) as well, did the es-sential part of the work correctly, and were awarded the originally intendedmarks regardless of their handling of the question whether u = 2m/r oru = m/r which was a minor issue in this context.

After careful inspection of the answers, the following moreover emerges:

- the three candidates who did Q5 did not leave any visible trace ofattempting question Q6, and therefore can not have been affected

- one candidate complained about losing 20 minutes of time to resolvethe issue

- five candidates ignored the instruction to set u = 2m/r, used imme-diately the substitution u = m/r, and continued from there without anyfurther comments

- three candidates derived the proposed equation using the proposedsubstitution

- three candidates do not seem to have noticed the difference- the remaining candidates noticed that the substitution u = m/r would

have led to the suggested answer, and continued the problem using thesubstitution u = m/r.

It is clear that some loss of time has resulted from the error on the paperfor most candidates attempting Q6. Because this might have impacted onthe time available for all questions, I suggest to add 4 points to all candidateswho gave evidence of attempting Q6.

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C7.4: Theoretical Physics I

Our markers finished marking in record time by Tuesday evening. Unfortu-nately, because our Physics computer system and, in particular, the accessto the database broke down completely yesterday, we are only now able togive you the marks for the Mathematics candidate taking C6.

The average of all the Physics candidates was 72.6 with a standard de-viation of 11.5.

This average was essentially identical with the target mark, so that norescaling had to be done. The were no problems with the paper.

C8.1a: Mathematics and the Environment

No report for Question 1.All but three out 35 of the students did both my questions (2 & 3) on

the exam. The average for each questions was 61% and 78% respectivelywith standard deviations just over 10 for both. The better performance onthe second question was most likely to this one being the most similar innature to previous exams. The average for the exam is approx 70% with amedian of 76%.

C8.1b: Mathematical Physiology

Question 4:There was an obvious typo in d(i) which was picked up immediately and

did not cause problems. By and large, parts (a) through to (d) were donevery well. Candidates struggled a bit after that.

Question 5:(a) done well and (b) reasonably well, although very few candidates

really appreciated the subtleties of the argument. Many candidates did notrealise a linear stability analysis was required for (c). The phase plane workwas sloppy. (d) was done reasonably well. Few got to (e).

Question 6:Very well done indeed. Some candidates simply described the function

in (b) rather than explain physiologically why it should take this form. Inthe first equation, the ”T” should have read ”t” but this caused no problemsfor the nondimensionalisation in (d) because either candidates realised thisor they had learnt the bookwork off by heart and ignored the question.

C9.1a: Analytic Number Theory

A reasonably successful paper with raw marks ranging from 16 to 46.

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Q1 was most popular, with all but the last part done well. Only onecandidate gave a really crisp solution to the unseen part at the end.

Q2 proved more difficult, with some poor treatments of analytic contin-uation. No one said that L(1) = 1−1/3+1/5−1/7+ ... = pi/4 > 0, thoughone candidate did prove correctly that L(1) > 0.

Q3 was generally well done, with the first part successfully sorting outthose who really understood the bookwork.

C9.1b: Elliptic Curves

The exam went smoothly with no problems. There were 9 students who satthese questions (some as the half unit C9.1b “Elliptic Curves”, and someas part of the unit C9.1 “Analytic Number Theory and Elliptic Curves”).Overall, the exam questions seemed to do a good job of spreading the stu-dents out, and the marks correspond fairly well to my general impression oftheir standard of work during the term.

C10.1a: Stochastic Differential Equations

Overall the exam proved difficult for a number of the candidates althoughthose who took both parts of the paper were clearly very strong and havelittle difficulty.

One of the more disturbing difficulties displayed by most of the candi-dates related to their difficulty in producing something that was precise andrigorous - difficulty which is certainly not related to the course in hand.They also found very great difficulty (as exposed in Question one) in basicanalysis and the simple estimate where they were required to control thevalue of an integral by splitting the domain of integration into two partsand using a separate estimate on each part defeated many even though theywere lead through the problem and one could see that they had the infor-mation in front of them. Basic examples of this type of approach are seenin the first year when for example one looks at approximate identities andthe convergence of Fourier series.

Stochastic Analysis builds on a strong foundation in real analysis andis quite quantitative at times. Last week I was a a workshop in Leicesterwhere these sort of techniques were widely used by senior people the Quantcommunity at Merrell Lynch/Bank of America. Employment in these aresasrequires fluency in these basic tools.

C10.1b: Brownian Motion in Complex Analysis

The two candidates sitting this examination both produced excellent per-formances and showed an authoritative grasp of the course material.

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C11.1a: Graph Theory

The first two questions were by far the most popular this year, althoughmany candidates found 1(b) difficult and few candidates were successful at2(d). Both of these had fairly straightforward proofs, and a few candidatesspotted a very quick proof of 2(d). The third question was relatively un-popular, although it was the shortest of the three; many of those who madea significant attempt at question 3 produced excellent solutions.

C11.1b Probabilistic Combinatorics

Comments on individual questions below. Overall, the scripts were disap-pointing: the bookwork was generally well done, but it seems many can-didates did not know what to do with the problem parts, which involvedslightly unfamiliar settings. With hindsight, the paper was a bit too hardin that anything even slightly different from what they are used to seems tothrow them!

Question 4: part (a) was generally done well. (b) not so well; at least thestart of this is simple (the variance bound is perhaps a little hard), but theunfamiliar setting seemed to throw people off. Very few candidates seriouslyattempted part (c).

Question 5: not that many candidates attempted this question. Of thosethat did, almost all gave very good answers.

Question 6 was attempted poorly. Candidates could usually correctlystate the general local lemma, but could not define a dependency graph.The answers to (a)(iii) (how to apply the lemma in practice, in terms of un-derlying independent variables) were poor, though this has been emphasizedin lectures and classes.

As a consequence, when applying the lemma, some candidates tried tocount the number of events that are not independent of a given event, whichlead to counting paths with 2 edges in common rather than the (needed andeasier) counting of paths with 1 edge in common. Only two candidatesanswered the last part; essentially all that was required was ‘two types ofevents, so use general form of local lemma’.

C12.1a : Numerical Linear Algebra and Approximation

All 17 candidates on this course showed some understanding of the material,with the majority having good to very good performances. These questionsset seem to been a fair test.

Q1 on Givens Rotations was attempted by most with many good an-swers, though none were perfect.

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Q2 on Gauss Seidel iteration was also popular with a slightly wider rangeof scores.

Q3 on Lanczos Method was attempted by 5 candidates with the highestscore for any question on the exam being achieved by one candidate.

C12.1b : Continuous Optimisation

I have not encountered any difficulties in that it was fairly straightforwardto assess the quality of the scripts. One paper is a clear fail; the personclearly had no knowledge of the subject. Otherwise the marks are spreadmore or less continuously from 41 down to 22. The top script was very solidbut not so outstanding that it is a clear first class grade.

C12.2a Approximation of Functions

Question 1. 4 attempts, average 14.75Interpolation, barycentric representation, difference between uniform dis-

tribution of interpolating points and those based on roots of Chebyshevpolynomials.

Those parts the students had seen before were done very well, unseenpart was too difficult for all.

Question 2. 9 attempts, average 19.22L2-approximation, orthogonal polynomials, calculating a best approxi-

mation and a Pade approximant.Mostly very good solutions to parts based on L2 approximation theory,

Pade approximants too hard for some, only two complete answers.Question 3. 6 attempts, average 21.83Proof of equi-oscillation theorem, calculating a simple best approxima-

tion and a part on cubic splines.Uniformly good proof of equi-oscillation theorem, good determination of

best linear approximation and good understanding of cubic B-spline shownby most.

C12.2b: Finite Element Methods for Partial Differential Equations

The set questions seem to have been a fair test of the material of the course.

Q1 on piecewise linear finite elements in 2 dimensions was attempted bymost candidates with a range of marks from 8 to 21.

Q2 on Cea’s lemma and convergence analysis was also popular and manygood scores were obtained.

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Q3 on finite element semideserialization for the heat equation was at-tempted only by two candidates who both achieved high scores.

Dissertations

There were 20 whole unit Dissertations, and two half unit dissertations sub-mitted for assessment; two additional dissertations had been approved butwere incomplete.

The process of reconciliation was overseen by an examiner. Each dis-sertation was read by the supervisor and two independent assessors. Allsubmitted a report; assessors completed the standard form with proposedmarks and supervisors generally submitted a short report with a suggestedclass. The marks were submitted to the administrator overseeing projects.Where there were small differences and the marks and class were all withinthe same class boundaries, the examiner proposed an averaging of the twomarks and assessors agreed to this as the final USM. Where differences werelarger or a difference in class was proposed by the supervisor or there wasdisagreement between assessors, the three parties were advised to discussthis, focusing on their differences and reach agreement (namely, for the classinitially with the supervisor, then the USM for the two assessors). Discus-sion confirmed marks; four at the higher class and 5 at the lower class ofthe three; two were borderline classes highlighted for consideration. Somecommunications could be conducted on e-mail but the examiner reportedthat he was hampered by the new reconciliation process and requests somebetter means of communication. One dissertation was also read by a thirdreader.

USMs for dissertations were recorded as : 88, 83, 79, 78, 78, 77, 73, 72,71, 71, 70, 69, 69, 69, 68, 67, 65, 65, 64, 63, 62, 61.

See the Mathematics & Statistics Report for Assessor comments on thefollowing courses:

MS1b: Statistical Data Mining

MS2a : Bioinformatics and Computational Biology

MS2b : Stochastic Models in Mathematical Genetics

MS3: Levy Processes and Finance

F. Comments on performance of identifiable individuals

Removed from public version.

G. Names of members of the Board of Examiners

• Examiners:Prof Chris Budd (External Examiner)

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Dr A Curnock (Chair)Prof P ChruscielProf V FlynnProf P Gibblin (External Examiner)Dr M PorterProf J Wilson

• Assessors for Papers C1.1–C12.2Dr IssacsonDr J KoenigsmannProf R HaydonProf B ZilberProf J WilsonDr K ErdmannProf M LackenbyProf U TillmannProf C BattyDr M EdwardsDr J DysonProf B NiethammerProf G SereginProf J BallDr A MajumdarProf S ChapmanDr S PeppinDr K HannabussDr H GrambergDr G SanderProf P MainiProf R Heath-BrownProf T LyonsDr T CassProf A ScottProf O RiordanDr WathenDr D OrtnerDr I Sobey

• Assessors for dissertationsDr J OliverDr G SanderDr H JinDr C Reisinger

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Dr D AllwrightProf J ChapmanProf T LyonsDr M EdwardsDr B WardhaughDr R FloodProf R Heath-BrownProf R HaydonDr M MonoyiosDr D StirzakerDr M WinkelProf P MainiDr E GaffneyDr A DancerProf U TillmannDr J NorburyDr T CassDr A CurnockDr G Vincent-SmithDr I SobeyDr S PeppinDr B StewartProf J WilsonDr K ErdmannDr J GrabowskiDr K HannabussProf C McDiarmidProf M Vaughan-LeeProf M Collins

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Figure 1: Table of quartiles for raw marks of Examination papers

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examiners’ report: final honour school of mathematics part

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