Course Regulations for Year 2
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MATHEMATICS BSC. G100
Normal Load = 120 CATS. Maximum Load = 150 CATS.
Students must take the 6 core modules (total 66 CATS), plus options. List A modules have a high mathematical content. The Core modules are: MA259 Multivariable Calculus, MA244 Analysis III, MA251 Algebra I, MA249 Algebra II, MA260 Norms, Metrics and Topologies, MA213 Second Year Essay.
MASTER OF MATHEMATICS MMATH G103
Normal Load = 120 CATS. Maximum Load = 150 CATS.
The first two years are in common with the BSc Mathematics degree course G100 except that in Year 2 students must take at least 90 CATS credits from the core and List A combined.
To remain on the G103 course at the second year exam board students must have achieved a weighted average on their best 90 CATS of maths modules (Core and List A modules starting with an MA2 code) of a good 2.1 standard. The department strictly interprets this to mean 65.0% or above (if a student has less than 90 CATS of such modules the average is taken over the number of such CATS they have been examined for). Experience has shown that students who do not achieve this threshold struggle with the four year degree, and by being transferred to the BSc. have a better chance of achieving a good 2.1 or first class degree and can plan their future better. For students who take 90 CATS of List A, but less than 90 CATS of MA2 modules, we would take the average over the MA2 modules that have been taken, and then look at the overall mark profile, invcluding the other List A modules taken, to make a progression decision on a case by case basis.
Please note:4th year MMath students are not be able to take second year modules except as unusual options. It is highly unlikely that MA2 modules would be allowed as unusual so choose your modules this year to take this into account.
MATHEMATICS WITH BUSINESS STUDIES G1NC
Normal Load = 120 CATS. Maximum Load = 150 CATS.
Students must take the 6 core modules for G100 students (total 66 CATS), plus one of the List B Warwick Business School modules below (coded IBxxx). To transfer to the Business School at the end of the second year students must get at least 50% in one of these modules, gain an overall honours mark (40% end of year) and be successfully interviewed by WBS.
MATHEMATICS AND ECONOMICS GL11
Normal Load = 120 CATS. Maximum Load = 150 CATS.
Year 2 core consists of 60 CATS of Mathematics and 60 CATS of Economics. The Economics modules are EC204 Economics 2 (30 CATS), plus either EC226 Econometrics 1 (30 CATS) or EC220/221 Mathematical Economics 1a and 1b (30 CATS). The Mathematics modules are MA251 Algebra I, MA244 Analysis III, MA259 Multivariable Calculus, MA260 Norms, Metrics and Topologies, plus 12 CATS from option list A/Core for the second year of the Mathematics BSc (G100). Students taking EC226 as a core module should consider, as recommended options, ST202 Stochastic Processes and/or ST213 Mathematics of Random Events. Students taking EC220/1 as a core module should consider MA209 Variational Principles.
Note, in year 3 GL11 students transfer to the Economics department where overcatting is not permitted and level 1 modules are also not allowed as options.
For a full list of available modules see the relevant course regulation page.
Maths Modules
Term  Code  Module  CATS 
List GL11 
List Others 
Term 1  MA241  Combinatorics  12  List A  List A 
MA243  Geometry  12  List A  List A  
MA244  Analysis III  12  Core  Core  
MA251  Algebra I: Advanced Linear Algebra  12  Core  Core  
MA259  Multivariable Calculus  12  Core  Core  
Terms 1 & 2  MA213  Second Year Essay  6  List A  Core 
Term 2  MA117  Programming for Scientists  12  List B  List B 
MA249  Algebra II: Groups and Rings  12  List A  Core  
MA250  Introduction to Partial Differential Equations  12  List A  List A  
MA252  Combinatorial Optimization  12  List A  List A  
MA254  Theory of ODEs  12  List A  List A  
MA257  Introduction to Number Theory  12  List A  List A  
MA260  Norms, Metrics and Topologies  12  Core  Core  
MA261  Differential Equations: Modelling and Numerics  12  List A  List A  
Term 3  MA209  Variational Principles  6  List A  List A 
MA256  Introduction to Systems Biology  6  List A  List A 
Maths Modules for External Students
These modules are not available to Maths students.
Term  Code  Module  CATS 
Term 1  MA258  Mathematical Analysis III  12 
Term 2  MA222  Metric Spaces  12 
Interdisciplinary Modules (IATL and GSD)
Second, third and fourthyear undergraduates from across the University faculties are now able to work together on one of IATL's 1215 CAT interdisciplinary modules. These modules are designed to help students grasp abstract and complex ideas from a range of subjects, to synthesise these into a rounded intellectual and creative response, to understand the symbiotic potential of traditionally distinct disciplines, and to stimulate collaboration through group work and embodied learning.
Maths students can enrol on these modules as an Unusual Option, you can register for a maximum of TWO IATL modules but also be aware that on many numbers are limited and you need to register an interest before the end of the previous academic year. Contrary to this is IL006 Challenges of Climate Change which replaces a module that used to be PX272 Global Warming and is recommended by the department, form filling is not required for this option, register in the regular way on MRM (this module is run by Global Sustainable Development from 2018 on).
Please see the IATL page for the full list of modules that you can choose from, for more information and how to be accepted onto them, but some suggestions are in the table below:
Term  Code  Module  CATS  List 
Term 1  IL005  Applied Imagination  12/15  Unusual 
GD305  Challenges of Climate Change  7.5/15  Unusual  
Term 2  IL016  The Science of Music  7.5/12/15  Unusual 
IL023  Genetics: Science and Society  12/15  Unusual 
Statistics Modules
Students who have successfully completed the first year in Maths and have taken statistics options in their first year may apply to the Department of Statistics for transfer to the joint degree. Alternatively, transfer may be made at the beginning of the third year if the appropriate second year modules have been taken. Further information may be obtained from the Department of Statistics.
Term  Code  Module  CATS  List 
Term 1  ST222  Games, Decisions and Behaviour  12  List A 
ST220  Introduction to Mathematical Statistics  12  List A  
Term 2  ST202  Stochastic Processes  12  List A 
Economics Modules
The Economics 2nd and 3rd Year Handbook is available on request from the Economics Department and contains details of their modules and prerequisites, including information on which will actually run during the year. This information is also available from the Economics web pages.
See the Economics Handbooks for information on the Joint degree in Mathematics and Economics.
Once you have consulted the Economics handbook, the Economics department should be consulted if you have questions about the joint degree, or about economics options for the maths degrees.
Term  Code  Module  CATS 
List GL11 
List Others 
Term 1  EC220  Mathematical Economics 1A  15  Op Core 
List B but must have taken EC106 or EC107 
Term 2  EC221  Mathematical Economics 1B  15  Op Core 
List B but must have taken EC106 or EC107 
Terms 1,2,3  EC204  Economics 2  30  Core  N/A 
EC226  Econometrics 1  30  Op Core  N/A 
Computer Science
Term  Code  Module  CATS  List 
Term 1  CS260  Algorithms  15  List B 
Term 2  CS262  Logic and Verification  15  List B 
CS254  Algorithmic Graph Theory  15  List B 
Physics
Students from the Department of Mathematics may take any combination of the modules listed below. All exams are one hour per 6 CATS. Please contact the Physics department to answer any queries concerning their second year modules.
Module Seminars for Physics Options: Certain physics modules are supported by module seminars which start one week after the start of the module. These are timetabled locally and details will be announced at the start of each module.
Model solutions to past weeks examples are kept in a file in the Second Year Physics Laboratory.
Term  Code  Module  CATS  List 
Term 1  PX266  Geophysics  7.5  List B 
PX267  Hamiltonian Mechanics  7.5  List B  
PX277  Computational Physics  7.5  List B  
Terms 1 & 2  PX262  Quantum mechanics and its Applications  15  List B 
Term 2  PX263  Electromagnetic Theory and Optics  7.5  List B 
PX264  Physics of Fluids  7.5  List B  
PX268  Stars  7.5  List B  
PX276  Methods of Mathematical Physics  7.5  List B 
Philosophy Modules
Students following modules in Philosophy should register for them as normal on the module registration system, but are also encouraged to check with the Philosophy department to ensure that the module still has places available in case it is oversubscribed.
Term  Code  Module  CATS  List 
Term 1  PH210  Logic II: Metatheory  15  List B 
Terms 1 & 2  PH201  History of Modern Philosophy  30  List B 
Warwick Business School
Students intending to transfer at the end of the second year to the joint degree Mathematics and Business Studies run by the Warwick Business School should note at the end of the second year students must get at least 50% in any IB coded module, gain an overall honours mark (40% Seymour) and be interviewed by WBS. Information for all WBS modules can be found from here.
PLEASE NOTE: from 2019/20 3rd years will NOT be allowed to take IB132 or IB133 as unusual options (or any other IB1xx module), so if you wish to take one/both of these during your degree you must do so this year.
Term  Code  Module  CATS  List 
Term 1  IB133  Foundations of Accounting  12/15  List B 
IB207  Mathematical Programming II  12  List B  
Term 2  IB132  Foundations of Finance  12/15  List B 
IB320  Simulation  12  List B  
IB217  Starting a Business  6  List B  
IB3A7  The Practice of Operational Research  12  List B 
Centre for Education Studies
Note: we advise students to take this module in their second year rather than third since it involves teaching practice over the Easter vacation which may interfere with revision for final year modules examined immediately after that vacation.
Term  Code  Module  CATS  List 
Term 2  IE3E1  Introduction to Secondary School Teaching  24  List B 
Film and Television Studies
Back on List B after being absent for a number of years. In the past this has been a popular choice for Maths students looking for something a bit different.
Term  Code  Module  CATS  List 
Term 1 and 2  FI101  Discovering Cinema  12/24 (TBC)  List B 
Languages
The Language Centre offers academic modules in Arabic, Chinese, French, German, Japanese, Russian and Spanish at a wide range of levels. These modules are available for exam credit as unusual options to mathematicians in all years. Pick up a leaflet listing the modules from the Language Centre, on the ground floor of the Humanities Building by the Central Library. Full descriptions are available on request. Note that you may only take one language module (as an Unusual Option) for credit in each year. Language modules are available as whole year modules, or smaller term long modules; both options are available to maths students. These modules may carry 24 (12) or 30 (15) CATS and that is the credit you get. We used to restrict maths students to 24 (12) if there was a choice, but we no longer do this.
Plan ahead! Note 3rd and 4th year students cannot take beginners level (level 1) Language modules.
There is also an extensive and very popular programme of lifelong learning language classes provided by the centre to the local community, with discounted fees for Warwick students. Enrolment is from 9am on Wednesday of week 1. These classes do not count as credit towards your degree.
The Language Centre also offers audiovisual and computer selfaccess facilities, with appropriate material for individual study at various levels in Arabic, Chinese, Dutch, English, French, German, Greek, Italian, Portuguese, Russian and Spanish. (This kind of study may improve your mind, but it does not count for exam credit.)
A full module listing with descriptions is available on the Language Centre web pages.
Important note for students who preregister for Language Centre modules
It is essential that you confirm your module preregistration by coming to the Language Centre as soon as you can during week one of the new academic year. If you do not confirm your registration, your place on the module cannot be guaranteed. If you decide, during the summer, NOT to study a language module and to change your registration details, please have the courtesy to inform the Language Centre of the amendment.
Information on modules can be found at
http://www2.warwick.ac.uk/fac/arts/languagecentre/academic/
Objectives
After completing the second year the students will have
 covered the foundational core;
 had the opportunity to follow options which build on their core knowledge;
 acquired sufficient knowledge and understanding to be in a position to make an informed choice of options in their final years;
 (joint degrees) acquired their core mathematical knowledge and been prepared, through their choice of options, for their final year in the department of their second specialism.
MA209 Variational Principles
Lecturer: Vassili Gelfreich
Term(s): Term 3
Status for Mathematics students: List A for Maths
NOTE: To avoid clashes with April exams this module starts in the 2nd week of Term 3 and is lectured 4 times a week. It overlaps with the 3rd/4th year examination periods in April and May so these students should be aware that they may miss examinable material.
Commitment: 15 lectures
Assessment: Onehour examination
Prerequisites: MA131 Analysis, a module on solving ordinary differential equations (and it is probably a good idea to revise at least separation of variables and linear constant coefficient ODEs) and MA259 Multivariable Calculus is also helpful.
Leads To: MA4L3 Large Deviation Theory.
Content: This module consists of a study of the mathematical techniques of variational methods, with applications to problems in physics and geometry. Critical point theory for functionals in finite dimensions is developed and extended to variational problems. The basic problem in the calculus of variations for continuous systems is to minimise an integral of the form
$$ I(y)=\int_a^b f(x,y,y_x)\,dx $$
on a suitable set of differentiable functions $ y\colon[a,b]\to\mathbb{R} $ where $y_x$ denotes the derivative of $y$ with respect to $x$. The EulerLagrange theory for this problem is developed and applied to dynamical systems (Hamiltonian mechanics and the least action principle), shortest time (path of light rays and Fermat's principle), shortest length and smallest area problems in geometry. The theory is extended to constrained variational problems using Lagrange multipliers.
Aims: To introduce the calculus of variations and to see how central it is to the formulation and understanding of physical laws and to problems in geometry.
Objectives: At the conclusion of the course you should be able to set up and solve minimisation problems with and without constraints, to derive EulerLagrange equations and appreciate how the laws of mechanics and geometrical problems involving least length and least area fit into this framework.
Books:
A useful and comprehensive introduction is:
R Weinstock, Calculus of Variations with Applications to Physics and Engineering, Dover, 1974.
Other useful texts are:
F Hildebrand, Methods of Applied Mathematics (2nd ed), Prentice Hall, 1965.
IM Gelfand & SV Fomin. Calculus of Variations, Prentice Hall, 1963.
The module will not, however, closely follow the syllabus of any book.
Additional Resources
MA257 Introduction to Number Theory
Lecturer: Dr. Adam Harper
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 one hour lectures
Assessment: 2 hour Exam 85%, Homework Assignments 15%
Prerequisites: MA136 Introduction to Abstract Algebra
Corequisite: MA249 Algebra II: Groups and Rings
Leads To: MA3A6 Algebraic Number Theory, MA426 Elliptic Curves
Content:
• Factorisation, divisibility, Euclidean Algorithm, Chinese Remainder Theorem.
• Congruences. Structure on Z/mZ and its multiplicative group. Theorems of Fermat and Euler. Primitive roots.
• Quadratic reciprocity, Diophantine equations.
• Elementary factorization algorithms.
• Introduction to Cryptography.
• padic numbers, Hasse Principle.
• Geometry of numbers, sum of two and four squares.
Aims:
To introduce students to elementary number theory and provide a firm foundation for later number theory and algebra modules.
Objectives:
By the end of the module the student should be able to:
 work with prime factorisations of integers
 solve congruence conditions on integers
 determine whether an integer is a quadratic residue modulo another integer
 apply padic and geometry of numbers methods to solve some Diophantine equations
 follow advanced courses on number theory in the third and fourth year
Books:
H. Davenport, The Higher Arithmetic, Cambridge University Press.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, SpringerVerlag, 1990.
Additional Resources
Archived Pages: 2015
MA257 Forum
MA257 Forum

1 post, started by John Cremona, 15:57, Fri 2 Jan 2015
MA213 Second Year Essay
Organiser: Andrew BrendonPenn
Term(s): Terms 12
Status for Mathematics students: Core for all Maths students except GL11 (for whom it is List A).
Commitment: Independent study with guidance from Personal Tutor.
Assessment: Essay 80%, presentation 20%.
Organisation: You can choose your own topic in consultation with your tutor (who must approve it) or base an essay on one of the Maths at Work topics after attending the talks.
Students may, and are strongly advised to, submit a draft of their essay to their tutor by the end of the first week of Term 2. You are expected to have consulted the web pages in the additional resources page on essay writing prior to submission of the draft. The tutor will provide written comments and discuss the draft, normally by Week 4 of Term 2.
Students have to give a 15minute oral presentation of the essay to their tutor and a small group of other second year students, normally in week 9 of Term 2. This presentation is a compulsory requirement and 20% of the essay mark is allocated to the quality of the presentation. Students should seek advice, e.g. from their tutor, on how to convey the content of their essay within such a short period of time; they must not get bogged down in technicalities but they should not be vague.
Aims:
 To provide an opportunity for students to learn some mathematics directly from books and other sources.
 To develop written and oral exposition skills.
Objectives:
 To learn how to write mathematics well.
 To practice presenting mathematics orally to a group.
 To develop research skills, including planning, use of library and the internet.
Deadline The essays should be submitted electronically online through Tabula and in the form of two hard copies and two cover sheets, to the Undergraduate Office by 12:00 noon on Thursday 23rd April 2020. This deadline is enforced by the mechanism described in the Course Handbook section on Assessment.
It is the students' responsibility to choose their essay topic, to prepare the draft on time, to seek advice where necessary, to prepare the presentation on time and to submit the final version of the essay on time.
The essay will be marked by your tutor and a second marker. Your tutor will also award the mark for the oral presentation. Instructions about the essay and information on the marking scheme will be given out by the end of Term 1. Students are advised to read the instructions carefully, since failure to follow one of the University Regulations (on plagiarism, for example) could result in a mark of zero.
Additional Resources
MA241 Combinatorics
Lecturer: Roman Kotecký
Term(s): Term 1
Status for Mathematics students: List A for Mathematics.
Commitment: 30 lectures.
Assessment: 10% by 4 fortnightly assignments during the term, 90% by a twohour written examination.
Prerequisites: No formal prerequisites. The module follows naturally from first year core modules and/or computer science option CS128 Discrete Mathematics.
Leads To: MA3J2 Combinatorics II
Content:
I Enumerative combinatorics

Basic counting (Lists with and without repetitions, Binomial coefficients and the Binomial Theorem)
 Applications of the Binomial Theorem (Multinomial Theorem, Multiset formula, Principle of inclusion/exclusion)

Linear recurrence relations and the Fibonacci numbers

Generating functions and the Catalan numbers

Permutations, Partitions and the Stirling and Bell numbers
II Graph Theory

Basic concepts (isomorphism, connectivity, Euler circuits)

Trees (basic properties of trees, spanning trees, counting trees)

Planarity (Euler's formula, Kuratowski’s theorem, the Four Colour Problem)

Matching Theory (Hall's Theorem and Systems of Distinct Representatives)

Elements of Ramsey Theory
III Boolean Functions
Book:Edward E. Bender and S. Gill Williamson, Foundations of Combinatorics with Applications, Dover Publications, 2006. Available online at the author's website: http://www.math.ucsd.edu/~ebender/CombText/
John M. Harris, Jeffry L. Hirst and Michael J. Mossinghoff, Combinatorics and graph theory, SpringerVerlag, 2000.
Additional Resources
MA243 Geometry
Lecturer: Emanuele Dotto
Term(s): Term 1
Status for Mathematics students: List A for Mathematics
Commitment: 30 lectures plus weekly worksheets
Assessment: The weekly worksheets carry 15% assessed credit; the remaining 85% credit by 2hour examination.
Prerequisites: None, but an understanding of MA125 Introduction to Geometry will be helpful.
Leads To: Third and fourth year courses in Algebra and Geometry, including: MA3D9 Geometry of Curves and Surfaces, MA3E1 Groups and Representations, MA4A5 Algebraic Geometry, MA4E0 Lie Groups, MA473 Reflection Groups, MA4H4 Geometric Group Theory, MA448 Hyperbolic Geometry and others
Content: Geometry is the attempt to understand and describe the world around us and all that is in it; it is the central activity in many branches of mathematics and physics, and offers a whole range of views on the nature and meaning of the universe.
Klein's Erlangen program describes geometry as the study of properties invariant under a group of transformations. Affine and projective geometries consider properties such as collinearity of points, and the typical group is the full matrix group. Metric geometries, such as Euclidean geometry and hyperbolic geometry (the nonEuclidean geometry of Gauss, Lobachevsky and Bolyai) include the property of distance between two points, and the typical group is the group of rigid motions (isometries or congruences) of 3space. The study of the group of motions throws light on the chosen model of the world.
Aims: To introduce students to various interesting geometries via explicit examples; to emphasize the importance of the algebraic concept of group in the geometric framework; to illustrate the historical development of a mathematical subject by the discussion of parallelism.
Objectives: Students at the end of the module should be able to give a full analysis of Euclidean geometry; discuss the geometry of the sphere and the hyperbolic plane; compare the different geometries in terms of their metric properties, trigonometry and parallels; concentrate on the abstract properties of lines and their incidence relation, leading to the idea of affine and projective geometry.
Books:
M Reid and B Szendröi, Geometry and Topology, CUP, 2005 (some Chapters will be available from the General office).
E G Rees, Notes on Geometry, Springer
HSM Coxeter, Introduction to Geometry, John Wiley & Sons
Additional Resources
MA244 Analysis 3
Lecturer: Professor Jose Rodrigo
Term(s): Term 1
Status for Mathematics students: Core for Maths.
Commitment: 30 lectures
Assessment: Twohour examination (85%), assignments (15%)
This module will be examined in the first week of Term 3.
Prerequisites: MA131 Analysis (MA137 Mathematical Analysis for nonmaths students), MA106 Linear Algebra
Leads To: MA260 Norms, Metrics and Topologies, MA250 Introduction to PDE's, MA359 Measure Theory and MA3G7 Functional Analysis I
Content: This covers three topics: (1) Riemann integration, (2) convergence of sequences and series of functions, (3) introduction to complex valued functions.
The idea behind integration is to compute the area under a curve. The fundamental theorem of calculus gives the precise relation between integration and differentiation. However, integration involves taking a limit, and the deeper properties of integration require a precise and careful analysis of this limiting process. This module proves that every continuous function can be integrated, and proves the fundamental theorem of calculus. It also discusses how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties.
Many functions can be written as limits of sequences of simpler functions (or as sums of series): thus a power series is a limit of polynomials, and a Fourier series is the sum of a trigonometric series with coefficients given by certain integrals. The second part of the module develops methods for deciding when a function defined as the limit of a sequence of other functions is continuous, differentiable, integrable, and for differentiating and integrating this limit. Norms are used at several stages and finally applied to show that a Differential Equation has a solution.
The final part of module focuses on complex valued functions, starting with the notion of complex differentiability. The module extends the results from Analysis II on power series to the complex case. The final section focuses on contour integrals, where a complex valued function is integrated along a curve. Cauchy's integral formula will be developped and a series of applications presented (to compute integrals of real valued functions, Liouville's Theorem and the Fundamental Theorem of Algebra).
Learning outcomes:
 To develop a good working knowledge of the construction of the Riemann integral;
 to understand the fundamental properties of the integral; main ones include: any continuous function can be integrated on a bounded interval and the Fundamental Theorem of Calculus (and its applications);
 to understand uniform and pointwise convergence of functions together with properties of the limit function;
 to study the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions;
 to study complex differentiability (CauchyRiemann equations) and complex power series;
 to study contou integrals: Cauchy's integral formulas and applications.
Books:
Additional Resources
MA249 Algebra 2: Groups and Rings
Lecturer: Martin Orr
Term(s): Term 2
Status for Mathematics students: Core for Year 2 mathematics students. It could be suitable as a usual or unusual option for nonmaths students
Commitment: 30 lectures.
Assessment: Assignments (15%), twohour examination (85%)
Prerequisites: MA132 Foundations (MA138 Sets and Numbers for nonmaths students), MA106 Linear Algebra, and MA251 Algebra I: Advanced Linear Algebra
Leads To: The results of this module are used in several modules including: MA377 Rings and Modules, MA3A6 Algebraic Number Theory, MA453 Lie Algebras, MA3G6 Commutative Algebra, MA3D5 Galois Theory, MA3E1 Group and Representations, and MA3J3 Bifurcations Catastrophes and Symmetry, although unfortunately not all of these modules are offered every year.
Content: This is an introductory abstract algebra module. As the title suggests, the two main objects of study are groups and rings. You already know that a group is a set with one binary operation. Examples include groups of permutations and groups of nonsingular matrices. Rings are sets with two binary operations, addition and multiplication. The most notable example is the set of integers with addition and multiplication, but you will also be familiar already with rings of polynomials. We will develop the theories of groups and rings.
Some of the results proved in MA242 Algebra I: Advanced Linear Algebra for abelian groups are true for groups in general. These include Lagrange's Theorem, which says that the order of a subgroup of a finite group divides the order of the group. We defined quotient groups for abelian groups in Algebra I, but for general groups these can only be defined for certain special types of subgroups H of G, known as normal subgroups. We can then prove the isomorphism theorems for groups in general. An analogous situation occurs in rings. For certain substructures I of rings R, known as ideals, we can define the quotient ring , and again we get corresponding isomorphism theorems.
Other results to be discussed include the OrbitStabiliser Theorem for groups acting as permutations of finite sets, the Chinese Remainder Theorem, and Gauss' theorem on unique factorisation in polynomial rings.
Aims: To study abstract algebraic structures, their examples and applications.
Objectives: By the end of the module the student should know several fundamental results about groups and rings as well as be able to manipulate with them.
Books:
One possible book is
Niels Lauritzen, Concrete Abstract Algebra, Cambridge University Press.
Additional Resources
MA250 Introduction to Partial Differential Equations
Lecturer: Dr. Björn Stinner
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 lectures
Assessment: 2 hour exam.
Prerequisites: Analytical knowledge as obtained in MA131 Analysis is required. Some techniques on ordinary differential equations as seen in MA133 Differential Equations, on uniform convergence of series as taught in MA244 Analysis III, and on the divergence theorem as presented in MA259 Multivariable Calculus will be needed and only briefly introduced in the lectures.
Leads To: MA254 Theory of ODEs, MA3G7 Functional Analysis I, MA3D1 Fluid Dynamics, MA3G1 Theory of Partial Differential Equations, MA3G8 Functional Analysis II, MA3H0 Numerical Analysis and PDEs, MA3H7 Control Theory, MA3J4 Mathematical modelling with PDE and MA4L3 Large Deviation theory.
Content:
The theory of partial differential equations (PDE) is important both in pure and applied mathematics. On the one hand they are used to mathematically formulate many phenomena from the natural sciences (electromagnetism, Maxwell's equations) or social sciences (financial markets, BlackScholes model). On the other hand since the pioneering work on surfaces and manifolds by Gauss and Riemann partial differential equations have been at the centre of many important developments on other areas of mathematics (geometry, Poincareconjecture).
Subject of the module are four significant partial differential equations (PDEs) which feature as basic components in many applications: The transport equation, the wave equation, the heat equation, and the Laplace equation. We will discuss the qualitative behaviour of solutions and, thus, be able to classify the most important partial differential equations into elliptic, parabolic, and hyperbolic type. Possible initial and boundary conditions and their impact on the solutions will be investigated. Solution techniques comprise the method of characteristics, Green's functions, and Fourier series.
Aims:
To introduce the basic phenomenology of partial differential equations and their solutions. To construct solutions using classical methods.
Objectives:
At the end, you will be familiar with the notion of wellposed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose. You will have studied some techniques which enable you to solve some simple PDE problems. You will also understand that properties of solutions to PDEs sensitively depend on the its type.
Books:
A script based on the lecturer's notes will be provided. For further reading you may find the following books useful (sections of relevance will be pointed out in the script or in the lectures):
S Salsa: Partial differential equations in action, from modelling to theory. Springer (2008).
A Tveito and R Winther: Introduction to partial differential equations, a computational approach. Springer TAM 29 (2005).
W Strauss: Partial differential equations, an introduction. John Wiley (1992).
JD Logan: Applied partial differential equations. 2nd edt. Springer (2004).
MP Coleman: An introduction to partial differential equations with MATLAB. Chapman and Hall (2005).
M Renardy and RC Rogers: An introduction to partial differential equations, Springer TAM 13 (2004).
LC Evans: Partial differential equations. 2nd edt. American Mathematical Society GMS 19 (2010).
Additional Resources
Archived Pages: 2013
MA251 Algebra 1: Advanced Linear Algebra
Lecturer: Professor Derek Holt (weeks 15) Professor Gavin Brown (weeks 610)
Term(s): Term 1
Status for Mathematics students:
Core for Maths.
This module will be examined in Week 30, the first week of Term 3.
Commitment: 30 onehour lectures plus six assignments
Assessment: Assignments (15%), twohour examination (85%).
Prerequisites: MA106 Linear Algebra and MA132 Foundations (MA138 Sets and Numbers for nonmaths students)
Leads To: third year algebra modules, such as MA3D5 Galois Theory, MA377 Rings and modules.
Content: This module is a continuation of First Year Linear Algebra. In that course we studied conditions under which a matrix is similar to a diagonal matrix, but we did not develop methods for testing whether two general matrices are similar. Our first aim is to fill this gap for matrices over . Not all matrices are similar to a diagonal matrix, but they are all similar to one in Jordan canonical form; that is, to a matrix which is almost diagonal, but may have some entries equal to 1 on the superdiagonal.
We next study quadratic forms. A quadratic form is a homogeneous quadratic expression in several variables. Quadratic forms occur in geometry as the equation of a quadratic cone, or as the leading term of the equation of a plane conic or a quadric hypersurface. By a change of coordinates, we can always write in the diagonal form . . For a quadratic form over , the number of positive or negative diagonal coefficients is an invariant of the quadratic form which is very important in applications.
Finally, we study matrices over the integers , and investigate what happens when we restrict methods of linear algebra, such as elementary row and column operations, to operations over . This leads, perhaps unexpectedly, to a complete classification of finitely generated abelian groups.
Aims: To develop further and to continue the study of linear algebra, which was begun in Year 1.
To point out and briefly discuss applications of the techniques developed to other branches of mathematics, physics, etc.
Objectives: By the end of the module students should be familiar with: the theory and computation of the the Jordan canonical form of matrices and linear maps; bilinear forms, quadratic forms, and choosing canonical bases for these; the theory and computation of the Smith normal form for matrices over the integers, and its application to finitely generated abelian groups.
Books:
P M Cohn, Algebra, Vol. 1, Wiley
I N Herstein, Topics in Algebra, Wiley.
Neither is essential, but are a good idea if you are intending to study further algebra modules.
Additional Resources
MA252 Combinatorial Optimisation
Lecturer: Vadim Lozin
Term(s): Term 2
Status for Mathematics students: List A for mathematics
Commitment: 30 lectures.
Assessment: 2 hour exam
Prerequisites: No formal prerequisites. Students who have taken CS126 Design of Information Structures, CS137 Discrete Mathematics and its Applications 2, MA241 Combinatorics or CS260 Algorithms will find it helpful, but no background from these modules will be assumed.
Leads To: This module may be useful for students interested in taking MA241 Combinatorics, MA3J2 Combinatorics II or MA4J3 Graph Theory, but it is not a formal prerequisite for them.
Content: The focus of combinatorial optimisation is on finding the "optimal" object (i.e. an object that maximises or minimises a particular function) from a finite set of mathematical objects. Problems of this type arise frequently in real world settings and throughout pure and applied mathematics, operations research and theoretical computer science. Typically, it is impractical to apply an exhaustive search as the number of possible solutions grows rapidly with the "size" of the input to the problem. The aim of combinatorial optimisation is to find more clever methods (i.e. algorithms) for exploring the solution space.
This module provides an introduction to combinatorial optimisation. Our main focus is on several fundamental problems arising in graph theory and algorithms developed to solve them. These include problems related to shortest paths, minimum weight spanning trees, matchings, network flows, cliques, colourings and matroids. We will also discuss "intractible" (e.g. NPhard) problems.
Main Reference:
 B. Korte and J. Vygen, Combinatorial Optimization: Theory and Algorithms, Springer, 6th Edition, 2018. Ebook available through the Warwick Library; click the link.
Other Resources:
 A. Bondy and U. S. R. Murty, Graph Theory with Applications, Elsevier, 1976. Available through the link.
 W.J. Cook, William H. Cunningham, W. R. Pulleybank, and A. Schrijver, Combinatorial Optimization, WileyInterscience Series in Discrete Mathematics, 1998.
 C.H. Papadimitriou and K. Steiglitz Combinatorial Optimization: Algorithms and Complexity Optimization: Algorithms and Complexity, Dover Publications, 1998.
Additional Resources
MA254 Theory of ODEs
Lecturer: Claude Baesens
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 one hour lectures
Assessment: Two hour exam (100%)
Prerequisites: MA133 Differential Equations (MA113 Differential Equations A for Stats students, although additional reading may be required), MA131 Analysis III, MA244 Analysis III, MA106 Linear Algebra and MA251 Algebra 1
Leads to: MA3G1 Theory of PDEs, MA3H0 Numerical Analysis and PDEs, MA3J3 Bifurcations, Catastrophes and Symmetry and other modules on modelling, theory and numerics of ODEs and PDEs.
Content:
Many fundamental problems in the applied sciences reduce to understanding solutions of ordinary differential equations (ODEs). Examples include the laws of Newtonian mechanics, predatorprey models in Biology, and nonlinear oscillations in electrical circuits, to name only a few. These equations are often too complicated to solve exactly, so one tries to understand qualitative features of solutions.
Some questions we will address in this course include:
When do solutions of ODEs exist and when are they unique? What is the long time behaviour of solutions and can they "blowup" in finite time? These questions culminate in the famous PicardLindelof theorem on existence and uniqueness of solutions of ODEs.
The main part of the course will focus on phase space methods. This is a beautiful geometrical approach which often enables one to understand the behaviour of solutions near critical points  often exactly the regions one is interested in. Different trajectories will be classified and we will develop techniques to answer important questions on the stability properties (or lack thereof) of given solutions.
We will eventually apply these powerful methods to particular examples of practical importance, including the LotkaVolterra model for the competition between two species and to the Van der Pol and Lienard systems of electrical circuits.
The course will end with a discussion of the SturmLiouville theory for solving boundary value problems.
Aims:
To extend the knowledge of first year ODEs with a mixture of applications, modelling and theory to prepare for more advanced modules later on in the course.
Objectives:
1) Determine the fundamental properties of solutions to certain classes of ODEs, such as existence and uniqueness of solutions.
2) Sketch the phase portrait of 2dimensional systems of ODEs and classify critical points and trajectories.
3) Classify various types of orbits and possible behaviour of general nonlinear ODEs.
4) Understand the behaviour of solutions near a critical point and how to apply linearization techniques to a nonlinear problem.
5) Apply these methods to certain physical or biological systems.
Books:
(Complete Lecture Notes will be made available)
Ordinary Differential Equations and Dynamical Systems, Gerald Teschl, [Available online]
Elementary Differential Equations and Boundary Value Problems, Boyce DiPrima 1997
Differential Equations, Dynamical Systems, and an Introduction to Chaos, Hirsch, Smale 2003
Nonlinear Systems, Drazin 1992
Additional Resources
MA256 Introduction to Systems Biology
Lecturer: Mike Tildesley
TA:
Term(s): Term 3
Status for Mathematics students: List A
Commitment: 15 one hour lectures
Assessment: One hour exam
Prerequisites: MA133 Differential Equations, ST111 Probability A, ST112 Probability B [Recommended: MA254 Theory of ODEs]
Course content:
1. General introduction to the course
2. Introduction to Systems Biology
3. Introduction to Epidemiology
Aims:
Introduction to Mathematical Biology and Systems Biology. Modelling techniques (based on core module material).
Objectives:
To develop simple models of biological phenomena from basic principles.
To analyse simple models of biological phenomena using mathematics to deduce biologically significant results.
To reproduce models and fundamental results for a range of biological systems.
To have a basic understanding of the biology of the biological systems introduced.
Books:
H. van den Berg, Mathematical Models of Biological Systems, Oxford Biology, 2011
James D. Murray, Mathematical Biology: I. An Introduction. Springer 2007
Christopher Fall, Eric Marland, John Wagner, John Tyson, Computational Cell Biology, Springer 2002
L. Edelstein Keshet, Mathematical Models in Biology, SIAM Classics in Applied Mathematics 46, 2005.
Keeling, M.J. and Rohani, P. Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2007.
Anderson, R. and May, R. Infectious Diseases of Humans, Oxford University Press, 1992.
Glendinning, P. Stability, Instability and Chaos, Cambridge Texts in Applied Mathematics, 1994.
Additional Resources
Archived Pages: 2012 2013 2014 2015 2016 2017
MA259 Multivariable Calculus
Lecturer: Dr. Mario Micallef
Term(s): 1
Status for Mathematics students: Core
Commitment: 30 lectures
Assessment: 85% by examination, 15% by assignments
Prerequisites: MA131 Analysis I and II OR MA137 Mathematical Analysis, MA106 Linear Algebra, MA134 Geometry and Motion OR PX129 Tutorial
Leads To: MA209 Variational Principles, MA3D9 Geometry of Curves and Surfaces, MA3G7 Functional Analysis I, MA3G8 Functional Analysis II, MA3H5 Manifolds, MA3J3 Bifurcations Catastrophes and Symmetry.
Content:
• Continuous VectorValued Functions
• Some Linear Algebra
• Differentiable Functions
• Inverse Function Theorem and Implicit Function Theorem
• Vector Fields, Green’s Theorem in the Plane and the Divergence Theorem in $\mathbb{R}^3$
• Maxima, minima and saddles
Learning Outcomes:
 Demonstrate understanding of the basic concepts, theorems and calculations of multivariate analysis.
 Demonstrate understanding of the Implicit and Inverse Function Theorems and their applications.
 Demonstrate understanding of vector fields and Green’s Theorem and the Divergence Theorem.
 Demonstrate the ability to analyse and classify critical points using Taylor expansions.
Books:
1. R. Abraham, J. E. Marsden, T. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer, second edition, 1988.
2. T. M. Apostol. Mathematical Analysis. AddisonWesley Publishing Co., Reading, Mass.LondonDon Mills, Ont., second edition, 1974.
3. R. Coleman. Calculus on normed vector spaces, Springer 2012. [available online via Warwick's library]
4. J. J. Duistermaat, J. A. C. Kolk. Multidimensional Real Analysis I : Differentiation, CUP, 2004 [available online via Warwick's library]
5. T. W. Körner. A Companion to Analysis: A Second First and First Second Course in Analysis, volume 62 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2004.
6. J. E. Marsden and A. Tromba. Vector Calculus. Macmillan Higher Education, sixth edition, 2011.
Additional Resources
MA260 Norms Metrics and Topologies
Lecturer: Professor James Robinson
Term(s): 2
Status for Mathematics students: Core
Commitment: 30 lectures
Assessment: 100% by examination.
Prerequisites: MA131 Analysis I and II OR MA137 Mathematical Analysis, MA224 Analysis III
Leads To: The module is a vital prerequisite for most later (especially Pure) Mathematics modules, including MA3F1 Introduction to Topology, MA3D9 Geometry of Curves and Surfaces, MA359 Measure Theory, MA3B8 Complex Analysis, MA3G1 Theory of PDEs, MA3H5 Manifolds, MA424 Dynamical Systems, MA4E0 Lie Groups, MA475 Riemann Surfaces.
Content:
To introduce the notions of Normed Space, Metric Space and Topological Space, and the fundamental properties of Compactness, Connectedness and Completeness that they may possess. Students will gain knowledge of definitions, theorems and calculations in
• Normed, Metric and Topological spaces
• Open and closed sets and their relation to continuity
• Notions of Compactness and relations to continuous maps
• Notions of Connectedness and relations to continuous maps
• Notions of Completeness and relations to previous topics in the module.
The module comprises the following chapters:
• Normed Spaces
• Metric Spaces
• Open and closed sets
• Continuity
• Topological spaces
• Compactness
• Connectedness
• Completeness
Learning Outcomes:
 Demonstrate understanding of the basic concepts, theorems and calculations of Normed, Metric and Topological Spaces.
 Demonstrate understanding of the openset definition of continuity and its relation to previous notions of continuity, and applications to open or closed sets.
 Demonstrate understanding of the basic concepts, theorems and calculations of the concepts of Compactness, Connectedness and Completeness (CCC).
 Demonstrate understanding of the connections that arise between CCC, their relations under continuous maps, and simple applications.
Books:
1. W A Sutherland, Introduction to Metric and Topological Spaces, OUP.
2. E T Copson, Metric Spaces, CUP.
3. W Rudin, Principles of Mathematical Analysis, McGraw Hill.
4. G W Simmons, Introduction to Topology and Modern Analysis, McGraw Hill. (More advanced, although it starts at the beginning; helpful for several third year and MMath modules in analysis).
5. A M Gleason, Fundamentals of Abstract Analysis, Jones and Bartlett.
Additional Resources
MA261 Differential Equations: Modelling and Numerics
Lecturer: Andreas Dedner
Term(s): 2
Status for Mathematics students:
Commitment: 10 x 3 hour lectures + 9 x 1 hour support classes
Assessment: Coursework (50%) and 1 hour written exam (50%)
Prerequisites: MA133 Differential Equations and MA124 Maths by Computer (or equivalent)
Leads To: MA398 Matrix Analysis and Algorithms, MA3H0 Numerical Analysis and PDE's
Content:
Concepts of Mathematical Modelling, e.g. conservation and dissipation principle, dimensional analysis, non dimensionalization, asymptotic expansion, introduction to calculus of variations, minimization, Hamiltonian dynamics, Lagrange multipliers, inverse and optimal control problems, gradient flow
Numerical approximations
Derivation of explicit and implicit Runge Kutta and multistep methods, Butcher tableau, Newton’s method, polynomial interpolation and quadrature, stability, consistency, and convergence analysis,
This module focuses on fundamental concepts of mathematical modelling involving ordinary differential equations and their numerical solution. Modelling concepts such as conservation and dissipation principles, calculus of variations, and non dimensionalisation will be covered using typical examples from physics, biology, and other areas of science and engineering. Basic numerical approximation methods will be presented for solving the resulting systems of differential equations like RungeKutta and multistep methods. Concepts like stability, consistency, and convergence will be covered in this module, with the aim of introducing the approximation techniques used in tackling mathematical problems which do not yield to closed form analytic formulae.
Aims:
By the end of the module the student should be able to:
 Understand the central concepts of mathematical modelling
 Be able to derive and analyse fundamental numerical methods
 Implement and test numerical methods using a scripting language
Books:
 F. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differental Equations: Inital Value Problems, Springer (2010)
 Witlski, M.Brown, Methods of Mathematical Modelling: Continous System and Differential Equations, Springer (2015)
3. R. L. Burden and J. D Faires, Numerical Analysis, 8th edition, BrooksCole Publishing (2004).
Additional Resources
FI101 Discovering Cinema
EC204 Economics 2
Principal Aims
The module aims to enable students to develop a deeper understanding of economic concepts introduced in first–year analysis and to introduce new concepts in both micro and macroeconomic analysis. New concepts include material drawn from general equilibrium, welfare economics, game theory, rational expectations and time consistency. The module aims to introduce students to the analysis of public policy issues such as market failure and counterinflation policy.
Principal Learning Outcomes
By the end of the module, the student will be expected to be familiar with a range of tools for the analysis of both micro and macroeconomic problems. The student will have a rigorous knowledge of the theoretical models which underlie economic analysis and an understanding of both the applicability and the limitations of particular models and approaches.
Syllabus
The module will typically cover the following topics:
Microeconomics The analysis of general equilibrium and welfare economics. Consideration of the economics of public policy issues such as externalities and public goods. Game theoretic approaches to oligopoly, entry and other strategic areas in industrial and business economics.
Macroeconomics The unemploymentinflation relationship. The effect of monetary policy. Expectations, financial markets and the Macroeconomy. Political business cycles. The Time inconsistency problem. The open economy.
Context
 Core Module
 LM1D (LLD2)  Year 2, V7ML  Year 2, GL11  Year 2, L1L8  Year 2, R9L1  Year 2, R3L4  Year 2, R4L1  Year 2, R2L4  Year 2, R1L4  Year 2, V7MR  Year 2, V7MP  Year 2
 Optional Module
 LA99  Year 2
 Pre or Corequisites
 EC106 (for MORSE students) or EC107 or EC131 and EC229 with a mark of 60% in each plus passes in IB121 and IB122
 Restrictions
 May not be taken by L100 and L116 students or WBS students in their second year. May not be combined with EC201 or EC202.
 Partyear Availability for Visiting Students
 Available in the Autumn term only (1 x 2000 word essay – 12 CATS) and in the Spring term only (1 x 2000 word essay – 12 CATS) and in the Autumn and Spring terms together (2 x 2000 word essays  24 CATS)
Assessment
 Assessment Method
 Coursework (20%) + 3 hour exam (80%)
 Coursework Details
 Two assignments (2000word essays) (worth 10% each)
 Exam Timing
 May/June
Exam Rubric
Time Allowed: 3 hours.
Answer ALL FOUR questions from Section A (40 marks total), ONE question from Section B (30 marks total) and ONE question from Section C (30 marks total). Answer Section A questions in one booklet, Section B questions in a separate booklet; and Section C questions in a separate booklet.
Approved pocket calculators are allowed.
Read carefully the instructions on the answer book provided and make sure that the particulars required are entered on each answer book. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in the answer book(s): answers beyond that number will not be considered.
Previous exam papers can be found in the University’s past papers archive. Please note that previous exam papers may not have operated under the same exam rubric or assessment weightings as those for the current academic year. The content of past papers may also be different.
Reading Lists
EC220 & 221 Mathematical Economics I
Principal Aims
Mathematical Economics 1a, “Introduction to Game Theory”, aims to provide a basic understanding of pure game theory and also introduce the student to a number of applications of game theory to economic problems of resource allocation.
Principal Learning Outcomes
12 CATS  By the end of the module the student should be able to acquire a sense of the importance of strategic considerations in economic problem solving and the normative significance of competitive markets in obtaining Pareto optimal allocations via appropriate extensions of the commodity space. Learn that a few simple, intuitive principles, formulated precisely, can go a long way in understanding the fundamental aspects of many economic problems.
15 CATS  By the end of the module the student should be able to understand the importance of strategic considerations in economic problem solving and the normative significance of competitive markets in obtaining Pareto optimal allocations via appropriate extensions of the commodity space. Learn that a few simple, intuitive principles, formulated precisely, can go a long way in understanding the fundamental aspects of many economic problems.
Syllabus
12 CATS  The module will typically cover the following topics:
Games in strategic form: Nash equilibrium and its applications to voting games, oligopoly, provision of public goods.
Games in extensive form: sub game perfect equilibrium and its applications to voting games, repeated games.
Static games with incomplete information: Bayesian equilibrium and its applications to auctions, contracts and mechanism design.
Dynamic games of incomplete information: Perfect Bayesian equilibrium, Sequential equilibrium and its application to signalling games.
Bargaining theory: Nash bargaining, noncooperative bargaining with alternating offers and applications to economic markets.
Evolutionary Game Theory
Evolutionary game theory.
15 CATS  The module will typically cover the following topics:
Games in strategic form: Nash equilibrium and its applications to voting games, oligopoly, provision of public goods.
Games in extensive form: sub game perfect equilibrium and its applications to voting games, repeated games.
Static games with incomplete information: Bayesian equilibrium and its applications to auctions, contracts and mechanism design.
Dynamic games of incomplete information: Perfect Bayesian equilibrium, Sequential equilibrium and its application to signalling games.
Bargaining theory: Nash bargaining, noncooperative bargaining with alternating offers and applications to economic markets.
Evolutionary Game Theory
Context
 Core Module
 G300  Year 2, Y602  Year 2
 Optional Core Module
 GL11  Year 2, GL12  Year 2
 Optional Module
 LM1D (LLD2)  Year 2, V7ML  Year 2, V7ML  Year 3, G100  Year 2, G100  Year 3, G103  Year 2, G103  Year 3, LA99  Year 2, LA99  Year 3, L100  Year 2, L1L8  Year 2, L1L8  Year 3, R9L1  Year 4, R3L4  Year 4, R4L1  Year 4, R2L4  Year 4, R1L4  Year 4, V7MM  Year 4, V7MP  Year 2, V7MP  Year 3, L1P5  Year 1, L1PA  Year 1, V7MR  Year 2, V7MR  Year 3
 Pre or Corequisites
 EC120 or EC107 for GL11 students
 Prerequisite for
 EC301, EC341
 Restrictions
 MORSE students must take 12 CAT version.
 Partyear Availability for Visiting Students
 12 CATS  Not available on a partyear basis
15 CATS  Available in the Autumn term only (1 x test – 12 CATS)
Assessment
 Assessment Method
 12 CATS  2 hour exam (100%)
15 CATS  Coursework (20%) + 2 hour exam (80%)  Coursework Details
 One 50 minute test (20%)
 Exam Timing
 May/June
Exam Rubric
Time Allowed: 2 hours.
Answer TWO questions ONLY. All questions carry equal weight (50 marks each). Answer each question in a separate booklet.
Approved pocket calculators are allowed.
Read carefully the instructions on the answer book provided and make sure that the particulars required are entered on each answer book. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in the answer book(s): answers beyond that number will not be considered.
Previous exam papers can be found in the University’s past papers archive. Please note that previous exam papers may not have operated under the same exam rubric or assessment weightings as those for the current academic year. The content of past papers may also be different.
Reading Lists
EC220 & 221 Mathematical Economics I
Principal Aims
Mathematical Economics 1a, “Introduction to Game Theory”, aims to provide a basic understanding of pure game theory and also introduce the student to a number of applications of game theory to economic problems of resource allocation.
Principal Learning Outcomes
12 CATS  By the end of the module the student should be able to acquire a sense of the importance of strategic considerations in economic problem solving and the normative significance of competitive markets in obtaining Pareto optimal allocations via appropriate extensions of the commodity space. Learn that a few simple, intuitive principles, formulated precisely, can go a long way in understanding the fundamental aspects of many economic problems.
15 CATS  By the end of the module the student should be able to understand the importance of strategic considerations in economic problem solving and the normative significance of competitive markets in obtaining Pareto optimal allocations via appropriate extensions of the commodity space. Learn that a few simple, intuitive principles, formulated precisely, can go a long way in understanding the fundamental aspects of many economic problems.
Syllabus
12 CATS  The module will typically cover the following topics:
Games in strategic form: Nash equilibrium and its applications to voting games, oligopoly, provision of public goods.
Games in extensive form: sub game perfect equilibrium and its applications to voting games, repeated games.
Static games with incomplete information: Bayesian equilibrium and its applications to auctions, contracts and mechanism design.
Dynamic games of incomplete information: Perfect Bayesian equilibrium, Sequential equilibrium and its application to signalling games.
Bargaining theory: Nash bargaining, noncooperative bargaining with alternating offers and applications to economic markets.
Evolutionary Game Theory
Evolutionary game theory.
15 CATS  The module will typically cover the following topics:
Games in strategic form: Nash equilibrium and its applications to voting games, oligopoly, provision of public goods.
Games in extensive form: sub game perfect equilibrium and its applications to voting games, repeated games.
Static games with incomplete information: Bayesian equilibrium and its applications to auctions, contracts and mechanism design.
Dynamic games of incomplete information: Perfect Bayesian equilibrium, Sequential equilibrium and its application to signalling games.
Bargaining theory: Nash bargaining, noncooperative bargaining with alternating offers and applications to economic markets.
Evolutionary Game Theory
Context
 Core Module
 G300  Year 2, Y602  Year 2
 Optional Core Module
 GL11  Year 2, GL12  Year 2
 Optional Module
 LM1D (LLD2)  Year 2, V7ML  Year 2, V7ML  Year 3, G100  Year 2, G100  Year 3, G103  Year 2, G103  Year 3, LA99  Year 2, LA99  Year 3, L100  Year 2, L1L8  Year 2, L1L8  Year 3, R9L1  Year 4, R3L4  Year 4, R4L1  Year 4, R2L4  Year 4, R1L4  Year 4, V7MM  Year 4, V7MP  Year 2, V7MP  Year 3, L1P5  Year 1, L1PA  Year 1, V7MR  Year 2, V7MR  Year 3
 Pre or Corequisites
 EC120 or EC107 for GL11 students
 Prerequisite for
 EC301, EC341
 Restrictions
 MORSE students must take 12 CAT version.
 Partyear Availability for Visiting Students
 12 CATS  Not available on a partyear basis
15 CATS  Available in the Autumn term only (1 x test – 12 CATS)
Assessment
 Assessment Method
 12 CATS  2 hour exam (100%)
15 CATS  Coursework (20%) + 2 hour exam (80%)  Coursework Details
 One 50 minute test (20%)
 Exam Timing
 May/June
Exam Rubric
Time Allowed: 2 hours.
Answer TWO questions ONLY. All questions carry equal weight (50 marks each). Answer each question in a separate booklet.
Approved pocket calculators are allowed.
Read carefully the instructions on the answer book provided and make sure that the particulars required are entered on each answer book. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in the answer book(s): answers beyond that number will not be considered.
Previous exam papers can be found in the University’s past papers archive. Please note that previous exam papers may not have operated under the same exam rubric or assessment weightings as those for the current academic year. The content of past papers may also be different.
Reading Lists
EC226 Econometrics 1
Principal Aims
The course aims to provide students with important skills, which are of both academic and vocational value, being an essential part of the intellectual training of an economist and also useful for a career. In particular the course aims to equip students with the following competencies: 1. An awareness of the empirical approach to economics; 2. Experience in the analysis and use of empirical data in economics; 3. Understanding the nature of uncertainty and methods of dealing with it; 4. The use of econometric software packages as tools of quantitative and statistical analysis.
Principal Learning Outcomes
By the end of the module students will have acquired the necessary skills and knowledge to be able to critically appraise work in the area of applied economics. They will have a good intuitive and theoretical grasp of the dangers, pitfalls and problems encountered in doing applied modelling. The module will also equip students with the necessary background material so that they are able to go on to study more advanced and technical material in the area of econometrics.
Syllabus
The module will typically cover the following topics:
Linear regression model. Least squares estimation. Dummy variables. Linear Restrictions. Classical Linear Regression Model Assumptions. Breakdown of CLRM assumptions. Errors in variables. Heteroscedasticity and implications for OLS. Structural change. Incorrect functional form and implications for OLS. Instrumental variable estimation. Dynamic models with lagged dependent variable. Serial Correlation and implications for OLS. Types of autocorrelation. Nonstationarity and Cointegration. Panel data models. Limited dependent variable models.
Context
 Core Module
 L100  Year 2, L116  Year 2, L1P5  Year 1, L1PA  Year 1
 Optional Core Module
 LM1D (LLD2)  Year 2, R9L1  Year 2, R3L4  Year 2, R4L1  Year 2, R2L4  Year 2, R1L4  Year 2, V7MR  Year 2, GL11  Year 2, GL12  Year 2
 Optional Module
 V7ML  Year 2, V7ML  Year 3, V7MM  Year 4, V7MP  Year 2, V7MP  Year 3, GL12  Year 4, GL11  Year 3
 Pre or Corequisites
 EC121 or EC123 and EC124 or IB122 for WBS students. EC106 or EC107 for GL11, MORSE and other students from Mathematics/Statistics Departments.
 Prerequisite for
 EC306, EC338
 Restrictions
 May not be combined with EC203.
 Partyear Availability for Visiting Students
 Available in the Autumn term only (1 x test, 1 assignment 12 CATS) and in the Autumn and Spring term together 2 x test, 2 x assignments and problem sets 24 CATS)
Assessment
 Assessment Method
 Coursework (35%) + 3 hour exam (65%)
 Coursework Details
 One 50 minute test (worth 15%) and one assignment (worth 15%) and problem sets worth (5%)
 Exam Timing
 May/June
Exam Rubric
Time Allowed: 3 Hours, plus 15 minutes reading time during which notes may be made (on the question paper) BUT NO ANSWERS MAY BE BEGUN.
Answer ALL EIGHT questions from Section A (52 marks total), and THREE questions from Section B (16 marks each). Answer Section A questions in one booklet and Section B questions in a separate booklet.
Approved pocket calculators are allowed. Statistical Tables and a Formula Sheet are provided.
Read carefully the instructions on the answer book provided and make sure that the particulars required are entered on each answer book. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in the answer book(s): answers beyond that number will not be considered.
Previous exam papers can be found in the University’s past papers archive. Please note that previous exam papers may not have operated under the same exam rubric or assessment weightings as those for the current academic year. The content of past papers may also be different.
Reading Lists
CS254 Algorithmic Graph Theory
Note: This module is only available to students in the second year of their degree and is not available as an unusual option to students in other years of study.
Academic Aims
The module is concerned with studying properties of graphs and digraphs from and algorithmic perspective. The focus is on understanding basic properties of graphs that can be used to design efficient algorithms. The problems considered will be typically motivated by algorithmic/computer science/IT applications.
Learning Outcomes
On completion of the module the student shoud be able to:
 understand the basics of graphs, directed graphs, weighted graphs and be able to relate them to practical examples.
 use effectively algorithmic techniques to study basic parameters and properties of graphs.
 design efficient algorithms for various optimisation problems on graphs.
 use effectively techniques from graph theory to approach practical problems in networking and communication.
Content
Typical topics include:
 Introduction to graphs: undirected graphs, directed graphs, weighted graphs, graph representation and special classes of graphs (trees, planar graphs etc.).
 Applications of graphs (in telecommunications, networking etc.).
 Basic algorithmic techniques for graph problems: graph traversals (DFS and BFS), topological sorting, Eular tours.
 Further algorithmic problems on graphs: minimum spanning trees, shortest path problems, matching problems.
 Planar graphs and their properties. Eular's formula, planar separateor theorem and their algorithmic applications.
 Further optimization problems on graphs including graph colouring and graph questions in distributed systems.
 Discussing practical applications of graphs and efficient algorithms for such practical problems. Approximation algorithms and heuristic algorithms. Applications to searching in massive graphs (e.g. page ranking); use of structural properties and algebraic properties.
CS260 Algorithms
Academic Aims
Data structures and algorithms are fundamental to programming and to understand computation. The purpose of this module is to provide students with a coherent introduction to techniques for using data structures and some basic algorithms, and with the tools for applying these techniques to computational problems. Teaching and learning methods include lectures and reading materials which describe algorithmic techniques and applications of these techniques to specific problems. Problem sheets give students opportunities to practice problem solving.
Learning Outcomes
On completion of the module the student will be able to:

Understand a variety of data structures and be able to use them effectively in design and implementation of algorithms.
 Understand a variety of techniques for designing efficient algorithms, proving their correctness, and analyzing their efficiency.
 Understand some fundamental algorithmic problems and algorithms for solving them.
Content
 Basics of algorithm analysis.
 Elementary graph algorithms.
 Greedy algorithms.
 Divideandconquer algorithms.
 Dynamic programming.
 Network flows.
 NP and computational intractability.
CS262 Logic and Verification
Note: This module is only available to students in the second year of their degree and is not available as an unusual option to students in other years of study.
Academic Aims
To give students an understanding of the basics of mathematical logic, and its applications to specifying and verifying computing systems. Algorithms and proof calculi for verification, as well as associated tools, will be studied. Theory and practice relating to reliability of systems form a vital part of computer science.
Learning Outcomes
On completion of the module the student will be able to:

Construct and reason about proofs in a variety of logics.
 Understand and compare the semantics of a variety of logics.
 Apply logic to specify and verify computing systems.
 Understand basic algorithms for formal verification.
 Use formal verification tools.
Content

Propositional logic: proofs, semantics, normal forms, SAT solvers.
 Predicate logic: proofs, semantics.
 Specifying and modelling software.
 Verification by model checking.
 Proof calculi for program verification.
PX262 Quantum Mechanics and its Applications
Lecturers: Gavin Bell and Julie Staunton
Weighting: 15 CATS
The first part of this year's module uses ideas, introduced in the first year module, to explore atomic structure. The module also covers the mathematical tools needed in quantum mechanics and outlines the fundamental postulates that form the basis of the theory. The module discusses the timeindependent and the timedependent Schrödinger equations for spherically symmetric and harmonic potentials, angular momentum and hydrogenic atoms.
The second half of the module looks at manyparticle systems and aspects of the Standard Model of particle physics. The module introduces the quantum mechanics of free fermions and discusses how it accounts for the conductivity and heat capacity of metals and the state of electrons in white dwarf stars. Introducing the effect of the ionic lattice and the scattering of electrons off ions then leads to a description of the properties of semiconductors and insulators. The Standard Model of particle physics is a quantum field theory and beyond simple quantum mechanics. However, using ideas from quantum mechanics, it is possible to explain a number of phenomena in particle physics such as antiparticles and particle oscillations.
Aims:
To introduce the mathematical structure of quantum mechanics and to explain how to compute expectation values for observable quantities of a system. To show how quantum theory accounts for properties of atoms, elementary particles, nuclei and solids.
Objectives:
To develop the foundations of quantum mechanics. At the end of the module you should:
 know the origin of the n,l,m and s quantum numbers and be able to use the Pauli exclusion principle to explain the periodic table.
 understand the significance of Hermitian operators and eigenvalue equations and be able to use the correspondence principle to find the form of a quantum mechanical operator.
 be able to use quantum mechanics to derive a description of the electron states of the hydrogen atom.
 be familiar with the freeelectron model of a metal
 be aware of the different crystal lattice types and how waves in a crystal are scattered by the ions
 be able to describe the elements of the standard model and to apply simple ideas from quantum theory to explain phenomena observed in particles and nuclei
Syllabus:
Revision of wavefunctions, probability densities and the Schrödinger equation in 1 dimension. The hydrogen atom: orbital angular momentum, quantum numbers, probability distributions. Atomic spectra and Zeeman effect. Electron spin: SternGerlach, spin quantum numbers, spinorbit coupling, exclusion principle and periodic table. Xray spectra.
Formal Quantum Mechanics
The first postulate  the wavefunction to describe the state of a system; the principle of superposition of states; Operators and their rôle in quantum mechanics; the correspondence principle; measurement, Hermitian operators and eigenvalue equations; the uncertainty principle  compatibility of measurements and commuting of operators; the time dependent Schrödinger equation.
The quantum harmonic oscillator, creation and annihilation operators.
Angular momentum
The angular momentum operators and their commutators; the eigenvalues of the angular momentum operators, the l and m quantum numbers; the eigenfunctions of the angular momentum operators, the Spherical Harmonics. The hydrogen atom revisited.
Models of Matter
Statement of the many body problem. Why do molecules, nuclei and solids form? The free fermion model model, the Fermi surface, density of states. FermiDirac distribution. Heat capacity, magnetic susceptibility, Pauli paramagnetism, ferromagnetism. Current in quantum mechanics and conductivity in a metal. Fermion degeneracy in white dwarf and neutron stars, gravitational collapse. The liquid drop model of the nucleus, energy release in fission. The crystal lattice: Lattices as repeated cells, unit cell. Lattice types in 3D. Reciprocal lattice vectors, relation to material on Fourier series. Planes and indices. Xray diffraction. The nearly free electron model, scattering of electron waves by a periodic lattice and band structure. Insulators and semiconductors. doping. Semiconductor devices, e.g. diode, LED.
The Standard Model
The constituents of the standard model and the use of natural units. KleinGordon and Dirac equations. Solution to Dirac for particle in its rest frame and for particles with zero rest mass. Antiparticles and the origin of spin, W± exchange and Fermi's contact interaction.
Constructing Models
Relation between quantum mechanics and linear algebra. Dirac's braket notation. Modelling the ammonia clock. Neutrino oscillations, kaon decay.
Commitment: about 40 Lectures + problems classes
Assessment: 2 hour examination (85%) + assessed work (15%).
Recommended Text: H D Young and R A Freedman, University Physics, Pearson, AIM Rae, Quantum Mechanics, IOP
Other useful books: P.C.W. Davies and D.S. Betts, Quantum Mechanics, Chapman and Hall 1994; F. Mandl, Quantum Mechanics, John Wiley 1992, S McMurry, Quantum Mechanics, AddisonWesley.
This module has a home page.
Leads from: PX101 Quantum Phenomena
Leads to: PX382 Quantum Physics of Atoms, PX395 The Standard Model, PX385 Condensed Matter Physics
PX263 Electromagnetic Theory and Optics
Lecturer: Tom Marsh
Weighting: 7.5 CATS
The module develops the ideas of first year electricity and magnetism into Maxwell's theory of electromagnetism. Maxwell's equations pulled the various laws of electricity and magnetism (Faraday's law, Ampere's law, Lenz's law, Gauss's law) into one unified and elegant theory. Establishing a complete theory of electromagnetism has proved to be one the greatest achievements of physics. It was the principal motivation for Einstein to develop special relativity, it has served as the model for subsequent theories of the forces of nature and it has been the basis for all of electronics (radios, telephones, computers, the lot...). The module shows that Maxwell's equations in free space have timedependent solutions, which turn out to be the familiar electromagnetic waves (light, radio waves, Xrays etc), and studies their behaviour at material boundaries (Fresnel Equations). The module also covers the basics of optical instruments and light sources.
Aims:
The module should study Maxwell's equations and their solutions.
Objectives:
By the end of the module you should:
 understand Maxwell's equations and quantities like the Poynting vector and refractive index
 be able to manipulate these equations in integral or differential form and derive the appropriate boundary conditions at boundaries between linear isotropic materials
 be familiar with planewave solutions to these equations in free space, dielectrics and ohmic conductors
 have an understanding of geometrical optics, polarisation of light, the behaviour of light in lenses and prisms, and the properties of different light sources (including lasers)
Syllabus:
Refresher on vector calculus
Ampere's law, Faraday/Lenz's law, Gauss's law in differential form. Need for the displacement current. Statement of Maxwell's equations.
Maxwell equations in vacuum and in matter. Magnetisation and polarization of materials. Relation of E and P, B and M.
Solutions to Maxwell equations in vacuum. Electromagnetic waves, Poynting vector, intrinsic impedance, polarisation.
Boundary conditions. Interfaces between dielectrics, separation into perpendicular and parallel components. Refractive index.
Ohm's law. Interface with a metal, skin effect.
Optics: reflection and refraction. Wavefronts at plane and spherical surfaces. Lenses. Basics of optical instruments, resolution.
Commitment: about 18 Lectures + problems classes
Assessment: 1 hour examination(85%) + assessed work (15%)
This module has a home page.
Recommended Text: IS Grant and WR Phillips, Electromagnetism, Wiley, E Hecht, Optics, H D Young and R A Freedman, University Physics, Pearson also ER Dobbs, Basic Electricity and Magnetism, Chapman and Hall (out of print). R Feynman, Feynman Lectures Vol II, Addison Wesley
Leads from: PX120 Electricity and Magnetism
Leads to: PX384 Electrodynamics
PX264 Physics of Fluids
Lecturer: Tony Arber
Weighting: 7.5 CATS
The field of fluids is one of the richest and most easily appreciated in physics. Tidal waves, cloud formation and the weather generally are some of the more spectacular phenomena encountered in fluids. The module establishes the basic equations of motion for a fluid  the NavierStokes equations  and shows that in many cases they can yield simple and intuitively appealing explanations of fluid flows. The module concentrates on incompressible fluids.
Aims:
The module should explain why PDEs (with associated boundary conditions) are an appropriate model for fluids. To show how physical ideas and limiting cases can help analyse these PDEs which, in general, cannot be solved. These include the role of the Reynolds number, laminar viscous flow, the boundary layer concept and irrotational flow.
Objectives:
At the end of the module you should be able to
 Recognise and write down the equations of motion for incompressible fluids (the Navier Stokes equations) and understand the origin and physical meaning of the various terms including the boundary conditions
 Derive Poiseuille's formula and understand the conditions for it to be a valid description of fluid flow
 Use dimensional analysis to analyse fluid flows. In particular, you should appreciate the relevance of the Reynolds number.
 Simplify the equations of motion in the case of incompressible irrotational flow and solve them for simple cases including vortices
 Explain the boundary layer concept
Syllabus:
 Introduction
 Fluids as materials which do not support shear. Idea of a Newtonian fluid. "Plausibility of τ = μ ∂u/∂y from assumption of a relaxation time for stress.
 Equations of Motion
 Hydrostatics: forces due to pressure and gravity. Hydrodynamics: acceleration, continuity and incompressibility. Euler equation.
 Streamlined Flow
 Streamlines: Integrating Euler for steady flow along a streamline to give Bernoulli. Derivation of Bernoulli via conservation of energy. Applications of Bernoulli: flux through a hole, Pitotstatic tube, aerofoil, waves on shallow water.
 Hydrodynamics of Viscous Flow
 Forces due to viscosity, NavierStokes equation. Derivation of Poiseuille's formula for laminar flow between plates.
 Turbulence
 Laminar flow only one possibility. Turbulent slugs. Need for dimensionless number, Re, Pressure gradient as a function of Re. 2 Regimes: Physical interpretation of Re as Inertial forces/Viscous forces. Poiseuille works when Re small.
 Irrotational Flow
 Definition of vorticity and circulation. Importance of irrotational flow, Kelvin's circulation theorem.

Examples of irrotational flow: uniform flow, flow past a cylinder. Derivation of lift on thin aerofoil, as example for Magnus Effect.
Circulation around a cylinder. The vortex. Circulation constant round vortex line, need to close or end on surfaces. Advection of unlike vortices. The vortex ring. Circling of like vortices.
Vortices at edges of wings.
 Real Flows
 Idea of boundary layer; Boundary layer separation and drag crisis.
Commitment: about 18 Lectures
Assessment: 1 hour examination
This module has a home page.
Recommended Texts: DJ Tritton, Physical Fluid Dynamics, OUP; TE Faber Fluid Dynamics for Physicists, CUP
Leads from: PX148 Classical Mechanics and relativity
Leads to: PX350 Weather and the Environment, MA3D1 Fluid Dynamics
PX266 Geophysics
Lecturer: Matthew Broome
Weighting: 7.5 CATS
This introductory module describes the behaviour of the solid Earth using physical principles. The topics covered to some extent include: the age of the Earth, plate tectonics, seismology, gravity and the shape of the Earth, oceanic and continental heat, the Earth's core and magnetic field.
Aims:
To present an understanding of the Earth in terms of simple physical principles
Objectives:
The module will develop your understanding of Earth based on the physical principles of radioactivity, gravity, waves, heat and magnetism. You should obtain an overview of the structure of the Earth, of the largescale processes affecting the Earth, and of the experimental and observational techniques used to probe them.
Syllabus:
 Introduction. Basic characteristics of Earth: size, shape, mass, structure, age. Earth geometry, spherical coordinates
 Geochronology. Geological time, radiometric dating
 Gravity. Consequences of spherical geometry, geoid, gravity measurements and anomalies, isostasy and mountain heights
 Seismology. Types of seismic waves, elasticity and elastic waves, earthquake location and magnitudes, seismology and Earth's interior
 Plate tectonics. Divergent, convergent and conservative plate boundaries, plate movement on flat earth, rotation poles and present day plate motions, past plate movements, role of Earth's magnetic field
 Heat. Overview of heat budget and Earth, heat flow and depth of ocean, convection in the mantle, thermal structure of the core, earth's magnetic field.
Commitment: about 18 Lectures
Assessment: 1 hour examination.
This module has a home page.
Recommended Texts: William Lowrie, Fundamentals of Geophysics, CUP; C.M.R Fowler, The Solid Earth  An Introduction to Global Geophysics, CUP
Leads from: 1st year physics modules
PX267 Hamiltonian Mechanics
Lecturer: James LloydHughes
Weighting: 7.5 CATS
This module introduces the Hamiltonian formulation of classical mechanics. This elegant theory has provided the natural framework for several important developments in theoretical physics including quantum mechanics. The module starts by covering the general "spirit" of the theory and then goes on to introduce the details. The module uses a lot of examples. Many of these should be familiar from earlier studies of mechanics while others, which would be much harder to deal with using traditional techniques, can be dealt with quite easily using the language and methods of Hamiltonian mechanics.
Aims:
To revise the key elements of Newtonian mechanics and use this to motivate and then develop Lagrangian and Hamiltonian mechanics
Objectives:
At the end of the module, you should
 Understand the significance of the Lagrangian. You should be able to derive and solve the EulerLagrange equations for simple models.
 (Working from the Lagrangian) be able to find the canonical momenta and to construct the Hamiltonian function
 Be able to derive and solve Hamilton's equations for simple systems
 Appreciate the role of (and relations between) constraints, conserved quantities and generalised coordinates
Syllabus:
1. Introduction. Analogy with Optics and constructive interference; principle of least action; examples of L: TV, mc2/γ
2. Euler Lagrange Equations. 1d trajectory, TV case, worked examples; T+V as a constant of the motion; multiple coordinates with examples
3. Generalised Coordinates and Canonical Momenta. Polar coordinates; angular momentum; moment of inertia of rigid bodies; treatment of constraints; examples
4. Symmetry and Conservation Laws
5. Hamiltonian Formulation. Hamilton's Equations, phase space, examples
6. Normal Modes and Small Oscillations. Inertial and stiffness matrices, diatomic and Triatomic molecules
Commitment: about 18 Lectures
Assessment: 1 hour examination
This module has a home page.
Recommended Text: A good text going well beyond the module is H Goldstein, Classical Mechanics; A helpful reference for the beginning of the module is: Feynmann, Leighton & Sands, The Feynmann Lectures on Physics, Vol 2, Chapter 19
Leads from: PX148 Classical Mechanics and Relativity
PX268 Stars
Lecturer: Don Pollacco
Weighting 7.5 CATS
People have been studying stars for as long as anything else in science. Yet the subject is advancing faster now than almost every other branch of physics. With the arrival of spacebased instruments, the prospects are that the field will continue to advance and that some of the most exciting discoveries reported in physics during your lifetime will be in astrophysics.
The module deals with the physics of the observation of stars and with the understanding of their behaviour and properties that the observations lead to. The module covers the main classifications of stars by size, age and distance from the earth and the relationships between them. It also looks at what the observations of stars' behaviour tell us about the evolutionary history of galaxies and of the Universe as a whole.
Aims:
The module should introduce the methods used to measure the distances between stars, their brightness and colour and provide evidence for the large variability of stars found in our Galaxy. It should show how fundamental concepts of physics are used to quantitatively describe the structure and evolution of stars. The module should also explain how observational methods, such as imaging and spectroscopy, can be used to test our understanding of the origin, life, and death of stars.
Objectives:
At the end of the module, you should:
 Be able to define the position of a star and to describe the techniques used to determine their distance from us
 Be able to relate quantities such as apparent magnitude, absolute magnitude, flux, luminosity, stellar radius, effective temperature and distance
 Be able to identify the main features of the HertzsprungRussell diagram and explain the characteristics of stellar spectra along the main sequence
 Understand the mechanisms of interaction between photons and matter occurring in the atmosphere of a star
 Understand the physical principles behind the structure and evolution of stars
 Be able to describe the processes of nuclear fusion that powers almost all stars
Syllabus:
 Observational facilities  the optical/IR window  space based astronomy
 Coordinate systems: how to define the position of a star. What stars are visible during a night, a month, a year
 Trigonometric Parallax. The parsec and parallax angles. Statistical parallax
 Fundamental properties of stars  colour, luminosity, apparent and absolute magnitude, stellar radius
 Blackbody radiation, thermal equilibrium, effective temperature
 Different types of stars  spectral classification  the Harvard spectral classification
 Stellar atmospheres  where does the light that we observe originate  interaction between radiation and matter  radiation transfer
 The structure of stars basic equations  nuclear energy production  mass/radius/luminosity relation  understanding the observed HertzsprungRussell diagram
 Stellar evolution  main sequence life time  from birth to death  young stellar objects, stellar remnants: white dwarfs, neutron stars, black holes
 Using stellar populations as test beds for stellar evolution open and globular clusters

Exoplanets: discovery and characteristics. Equilibrium temperature and the habitable zone
Commitment: about 18 Lectures
The questions on the problem sheets relate principally to the techniques presented in the module and working through them will help you to understand the material. Please feel free to approach me if you have any difficulties with the questions.
Assessment: 1 hour examination
This module has a home page.
Recommended Texts: B.W. Carroll and DA Ostlie, An Introduction to Modern Astrophysics, AddisonWesley; Prialnik, D, An introduction to the theory of stellar structure and evolution., CUP.
Leads to: PX397 Galaxies, PX387 Astrophysics, PX389 Cosmology
PX269 Galaxies
PX272 Global Warming
PX276 Methods of Mathematical Physics
Lecturer: Gareth Alexander
Weighting: 7.5 CATS
The module starts with the theory of Fourier transforms and the Dirac delta function. Fourier transforms are used to represent functions on the whole real line using linear combinations of sines and cosines. Fourier transforms are a powerful tool in physics and applied mathematics. A Fourier transform will turn a linear differential equation with constant coefficients into a nice algebraic equation which is in general much easier to solve.
The module explains why diffraction patterns in the farfield limit are the Fourier transforms of the "diffracting" object. It then looks at diffraction generally. The case of a repeated pattern of motifs illustrates beautifully one of the most important theorems in the business  the convolution theorem. The diffraction pattern is simply the product of the Fourier transform of the repeated delta functions and the Fourier transform for a single copy of the motif.
The module also introduces Lagrange multipliers, coordinate transformations and cartesian tensors illustrating them briefly with examples of their use in physics.
This module is not available to Physics students (F300, F303, F3N1).
Aims:
To teach mathematical techniques needed by second, third and fourth year physics modules.
Objectives:
Students should:
 Be able to represent simple functions in terms of Fourier series and Fourier transforms
 Possess a good understanding of diffraction and interference phenomena
 Be able to minimise/maximise simple functions subject to constraints using Lagrange multipliers
 Be able to express vectors in different coordinate systems, recognise some physical examples of tensors
 Be familiar with the derivation, and with some applications in physical contexts, of Stokes’s theorem
Syllabus:
1. Fourier Series (Revision):
Representation for function f(x) defined L to L; brief mention of convergence issues; real and complex forms; differentiation, integration; periodic extensions
2. Fourier Transforms:
Fourier series when L tends to infinity. Definition of Fourier transform and standard examples: Gaussian, exponential and Lorentzian. Domains of application: (Time t  frequency w), (Space x  wave vector k). Delta function and properties, Fourier's Theorem. Convolutions, example of instrument resolution, convolution theorem.
3. Interference and diffraction phenomena:
Interference and diffraction: the HuygensFresnel principle. Criteria for Fraunhofer and Fresnel diffraction. Fraunhofer diffraction for parallel light. Fourier relationship between an object and its diffraction pattern. Convolution theorem demonstrated by diffraction patterns. Fraunhofer diffraction for single, double and multiple slits. Fraunhofer diffraction at a circular aperture; the Airy disc. Image resolution, the Rayleigh criterion and other resolution limits. Fresnel diffraction, shadow edges and diffraction at a straight edge.
4. Lagrange Multipliers
Variation of f(x,y) subject to g(x,y) = constant implies grad f parallel to grad g. Lagrange
multipliers. Example of quadratic form.
5. Vectors and Coordinate Transformations:
Summation convention, Kronecker delta, permutation symbol and use for representing vector products. Revision of cartesian coordinate transformations. Diagonalizing quadratic forms.
6. Tensors:
Physical examples of tensors: mass, current, conductivity, electric field.
Worksheet
7. Stokes’ Theorem: Line integrals, circulation; curl in Cartesians; statement and proof of Stokes’ theorem for triangulations; dependence on region of integration and vector field; gradient, irrotational, solenoidal and incompressible vector fields; applications drawn from electromagnetism, fluid dynamics, condensed matter physics, differential geometry
Commitment: 20 Lectures + 10 Examples Classes
Assessment: 1 hour written examination (80%) + inclass tests/assessed coursework (10%) + 1 worksheet on Stokes's Theorem (10%)
This module has a home page.
Recommended texts: KF Riley, MP Hobson and SJ Bence, Mathematical Methods for Physics and Engineering: a
Comprehensive Guide, CUP; H D Young and R A Freedman, University Physics (11th
Edition), Pearson.
Leads to: Second, third and fourth year physics modules.
PX277 Computational Physics
Lecturer: Yorck Ramachers
Weighting: 7.5 CATS
This module develops programming in the Python programming language and follows from PX150 Physics Programming Workshop.
Aims:
To acquire programming skills necessary to solve physics problems with the help of the Python programming language, a language widely used by physicists.
Objectives:
Students should:
 Understand how computers can be used to solve physics problems
 Be able to translate physics problems into a form suitable for solution using a computer program
 Be able to design algorithms and implement them
 Be able to handle and analyse physics data.
Syllabus:
1. Handling, processing and analysing physics data: plotting distributions, least square and maximum likelihood fit.
2. Monte Carlo simulation for physics modelling. Different types of random numbers, quality of random number generators. Generation of random numbers according to specific distributions. Brownian motion and diffusion.
3. Numerical integration and differentiation. Mass and centre of mass of object with variable density. Electric fields generated by distributed charge.
4. Numerical solutions of ordinary differential equations. Mechnical oscillations, motion with resistance.
Commitment: 5 lectures and 10 x 1 hour workshops
Assessment: 3 assignments
This module has a home page.
Recommended texts: M. Newman, Computational Physics, CreateSpace Independent Publishing Platform; H.P. Langtangen, A Primer on scientific programming with Python, Springer (ebook).
Leads from: PX150 Physics Programming Workshop
Leads to: PX390 Scientific Programming