Useful Dates and Deadlines - Chicks Dig X-Pecclux.x-pec.com/files/mathstuff/1styear/PYDC2006-2007/green.pdf · follow the module but are strongly encouraged to do some background

Useful Dates and Deadlines

Term 1 starts Mon 2nd Oct 2006Online module registration Weeks 1–3

Term 1 ends Sat 9th Dec 2006Term 2 starts Mon 8th Jan 2007

Online module registration Weeks 15–17Deregistration deadline, unusual options Fri 2nd March 2007

Deregistration deadline, April Exams Fri 16th March 2007Term 2 ends Sat 17th Mar 2007

Term 3 starts Mon 23rd Apr 2007April Exams First week of Term 3

Deregistration deadline, May/June Exams Fri 27th April 2007Second Year Exams Mon 28th May–Sat 23rd June 2007

Term 3 ends Sat 30th Jun 20072007–2008 Term 1 starts Mon 1st Oct 2007

Contents

Introduction 4

Objectives 4

Mathematics 5

Terms 1–2MA247 Mathematical Excursions . . . . . . . . . . . . . . . . . . 5MA240 Modelling Nature’s Nonlinearity . . . . . . . . . . . . . . 7MA213 Second Year Essay . . . . . . . . . . . . . . . . . . . . . 8

Term 1MA242 Algebra I . . . . . . . . . . . . . . . . . . . . . . . . . . 9MA244 Analysis III . . . . . . . . . . . . . . . . . . . . . . . . . 10MA241 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . 11MA243 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 12MA231 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . 13

Term 2MA249 Algebra II . . . . . . . . . . . . . . . . . . . . . . . . . . 15MA225 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 16MA235 Introduction to Mathematical Biology . . . . . . . . . . . 17MA222 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 18MA228 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . 19MA250 Introduction to Partial Differential Equations . . . .. . . 19MA117 Programming for Scientists . . . . . . . . . . . . . . . . . 20

Term 3MA246 Number Theory . . . . . . . . . . . . . . . . . . . . . . . 21MA209 Variational Principles . . . . . . . . . . . . . . . . . . . . 23

Computer Science 24

Term 1CS242 Formal Specification and Verification . . . . . . . . . . . . 24CS252 Fundamentals of Relational Databases . . . . . . . . . . . 24CS253 Topics in Database Systems . . . . . . . . . . . . . . . . . 25

Term 2CS245 Automata and Formal Languages . . . . . . . . . . . . . . 25CS246 Further Automata and Formal Languages . . . . . . . . . . 26CS243 Data Structures and Algorithms . . . . . . . . . . . . . . . 26CS244 Algorithm Design . . . . . . . . . . . . . . . . . . . . . . 27

2 Green (Second Year) PYDC 2006–2007

Economics 27

Term 1EC220 Mathematical Economics IA . . . . . . . . . . . . . . . . . 27

Term 2EC221 Mathematical Economics IB . . . . . . . . . . . . . . . . . 28

Institute of Education 29

Term 1Development of Mathematical Concepts . . . . . . . . . . . . . . 29

Term 2IE2A6 Introduction to Secondary School Teaching . . . . . . . . .30IE420 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . 31

Engineering 31

Term 2ES312 Modelling and Simulation . . . . . . . . . . . . . . . . . . 31

Language Centre 33

Terms 1–3LL201 Russian for Scientists I . . . . . . . . . . . . . . . . . . . . 33

Philosophy 34

Terms 1–3PH201 History of Modern Philosophy . . . . . . . . . . . . . . . . 34

Physics 35

Terms 1–2PX262 Quantum Mechanics and its Applications . . . . . . . . . . 35

Term 1PX266 Geophysics . . . . . . . . . . . . . . . . . . . . . . . . . . 35PX267 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 36PX273 Physics of Electrical Power Generation . . . . . . . . . . . 37PX268 Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Term 2PX263 Electromagnetic Theory and Optics . . . . . . . . . . . . . 38PX269 Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . 39PX272 Global Warming . . . . . . . . . . . . . . . . . . . . . . . 40PX261 Mathematical Methods for Physicists II . . . . . . . . . . . 41PX264 Physics of Fluids . . . . . . . . . . . . . . . . . . . . . . . 42

Green (Second Year) PYDC 2006–2007 3

Statistics 43Terms 1–2

ST217 Mathematical Statistics A or A+B . . . . . . . . . . . . . . 43Term 1

ST217A Mathematical Statistics A . . . . . . . . . . . . . . . . . 43ST202 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 44

Term 2ST213 Mathematics of Random Events . . . . . . . . . . . . . . . 45ST217B Mathematical Statistics B . . . . . . . . . . . . . . . . . 46

Warwick Business School 47Terms 1–2

IB109 Foundations of Accounting and Finance . . . . . . . . . . . 47Term 1

IB207 Mathematical Programming II . . . . . . . . . . . . . . . . 48Term 2

IB211 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 48IB3A7 The Practice of Operational Research . . . . . . . . . . . . 49

Term 3IB206 Introduction to Business Studies . . . . . . . . . . . . . . . 50


INTRODUCTIONThis Green section of PYDC 2006–2007 describes core modulesand op-

tions available to WarwickSecond YearMathematics students. Course regula-tions (core modules, normal load, options, etc.) for Mathematics and Masterof Mathematics along with joint degrees are given in the White PYDC booklet.First, third and fourth year modules are listed in their corresponding booklets,copies of which are available for consultation.

Modules are listed according to department, with Mathematics first and theremainder in alphabetical order. The main data for each module is: when ithappens (which term, or which weeks within a term for moduleswhich don’tstart and end at the beginning and end of the term, e.g. Term 2 (15–19) meansthe first five weeks of Term 2), the CATS credit it carries, and its status (core,option List A, or List B, or unusual option) for the mathematics degree courses.The second year normal load is 120 CATS. Weeks are numbered from week 1 ofTerm 1, and numbering now continues through the vacations, so Term 2 is weeks15–24 and Term 3 is weeks 30–39. As with any changeover to a newsystem,some information will still be in the old format, so be careful with the startingweeks of options — refer to the timetable to be sure.

This booklet does not claim complete and absolute infallibility on all mat-ters; in case of doubt, check the information given about modules in other de-partments (availability, which term, credit weighting, etc.) with that department.The details may have changed after this booklet went to press. Updates to moduledescriptions will appear on the Department’s web pages at

www.maths.warwick.ac.uk/pydc/

It is advisable to check the timetable as early possible (if possible before thecourse starts) to guard against clashes. Some will be inevitable, but others maybe avoided if noticed sufficiently well in advance. This is particularly importantif you take a slightly unusual combination of modules, and ifyou intend to takeoptions outside the Science Faculty. It is possible for a module advertised hereas in Term 2 to be switched to Term 1 after this book has been sent for printing.You will only be able to tell this from the Timetable. So checkall the modulesyou are interested in at the start of Term 1.

OBJECTIVESAfter completing the second year the students will have

1. covered the foundational core;2. had the opportunity to follow options which build on theircore knowledge;3. acquired sufficient knowledge and understanding to be in aposition to

make an informed choice of options in their final years;4. (joint degrees) acquired their core mathematical knowledge and been pre-

pared, through their choice of options, for their final year in the department


of their second specialism.

MATHEMATICSMost mathematics modules have online resources available through Math-

Stuffmathstuff.maths.warwick.ac.uk

Some second year modules MA209 Variational Principles, MA228 Numer-ical Analysis, MA240 Modelling Nature’s Nonlinearity, MA241 Combinatorics,MA243 Geometry and MA246 Number Theory are available as third year List Aoptions worth 6 or 12 CATS. However, some of these do not take place everyyear.

Some of the third year maths modules may be suitable for well-preparedsecond year students, but MA3E5 History of Mathematics may not be taken inYear 2. See the Pink (third year) PYDC booklet for the module descriptions.

MA247 Terms 1–2Mathematical Excursions 12 CATSStatus: List A for Maths.Commitment: One 90-minute seminar per week in weeks 1-20. No lectures. Inmore detail, this module is likely to be organised as follows:

1. Homework will be set on weeks 1, 3, 5, 7, 9 and 10; it will be due oneweek after it is set, except week 10 homework which will be dueon week11 (i.e. week 1 of Term 2). Homework will be discussed in class, one weekafter it is handed in.

2. On week 1 and even weeks, there will be some instruction on the use ofMathematica and other in-class work.

3. Presentation of projects will take place in weeks 13–20. Attendance atproject presentations is required.

The project has to be word-processed or, preferably, typeset in TEX. Theprojects, which may arise from the problems set weekly, would involve indepen-dent reading and, if appropriate, mathematical experimentation using Mathemat-ica.Prerequisites: 1st year modules in MA106 Linear Algebra and MA131 Analy-sis. Partial differentiation, critical points and maxima and minima for functionsof two variables from MA127 3D-Geometry & Motion. Solution of constantcoefficient ordinary differential equations. Most important, students must be en-thusiastic about, and committed to, mathematics and willing to work hard andregularly on the coursework, especially in Term 1.Content: A broad range of topics will be explored through problem solvingand researching the literature, using material covered in the first and second year


Mathematics core modules as the point of departure. The module will provide anopportunity to explore some of the deeper and more beautifulaspects of mathe-matics. The unity of the subject will be emphasized.

Extracts from SSLC evaluations of the module: The students were almost unan-imously pleased with the module. There were some difficult problems, but theywere by far my favourite ones. Assignment problems were found to be help-ful with the understanding of some other modules. Projects were popular. Themodule required me to make substantial use of the library, more so than in oth-er modules. A team spirit evolved among most students on the module. Morestudents should take this module; it is not as daunting as it sounds and it is veryrewarding and beneficial.

The following is a short selection of possible topics:Proof of Fermat’s last theorem forn = 3 andn = 4.Approximation of rationals by irrationals, transcendental numbers.Quaternions.Commutation relations for matrices.Zeroes of truncated power series.Critical points and level sets.

Aims:

(i) To provide an opportunity to explore some of the deeper and more beautifulaspects of mathematics and to exhibit its unity.

(ii) To foster and maintain enjoyment of mathematics and inspire some stu-dents to consider seriously a career in mathematical teaching or research.

(iii) To teach how to use the computer to solve mathematical problems and todiscover, formulate and test mathematical conjectures.

Objectives:

(i) To develop problem solving skills.(ii) To teach how to write mathematics well.(iii) To give practice in presenting mathematics to a group.(iv) To develop research skills, including use of library, the internet, group dis-

cussions, etc.

Leads to: MA395 Third Year Essay, MA469 Project.

Books: A home produced problem list. The project will involve consulting anumber of books, all of which are available from the library or from the moduleorganiser. You need not purchase any books.

Assessment:100% assessed: 50% for solutions to problems, 35% for the writtenproject and 15% for quality of presentation of homework solutions and project tothe class.

Organiser: Oleg Kozlovski


MA240 Terms 1–2 (6–10 & 15–19)Modelling Nature’s Nonlinearity 12 CATS

Status: List A for Maths.

Commitment: 30 one-hour lectures.

Prerequisites: This module leads on directly from MA132 Differential Equa-tions, or MA128 Differential Equations B. For those students who took onlyMA113 Differential Equations A in their first year, you should still be able tofollow the module but arestronglyencouraged to do some background reading.I will assume at least a rudimentary memory of coupled lineardifferential equa-tions and competence with linear second order differentialequations. A goodbook is James Robinson’s “An Introduction to Ordinary Differential Equations”.

Content: This module is designed to be a gentle introduction to the area of non-linear dynamical systems by way of its application to the “Natural World”. Somequite deep ideas are introduced to help explain or describe natural phenomenasuch as evolutionary theory, species diversity, weather forecasting, animal loco-motion and epidemics. The mathematics considered will cover the full spectrumof nonlinear dynamical systems theory including game theory, nonlinear oscilla-tions, symmetry, sensitive dependence upon initial conditions (chaos) and (if timepermits) fractals. In many cases these ideas are introducedoutside of a rigoroussetting so that the beauty and power of the techniques can be explored. Therewill be occasional reference to numerical solutions of someproblems, and someof the assessed work may require use of a computer, but no previous experience(or love) of computing will be assumed.

Aims: To provide a general introduction to the many aspects of dynamical sys-tems theory through its application to the “Natural World”.

Objectives: At the end of the module you should be familiar with the ideas ofstable/unstable equilibria and periodic orbits, strange attractors, Poincare maps,bifurcations, catastrophes, nonlinear oscillations, chaos and fractals.

Leads to: Although not leading directly onto another course, this module shouldprovide a useful introduction/motivation/complement to MA235 Introduction toMathematical Biology.

Books: There is no one textbook which adequately covers the whole module, butJ.D. MurrayMathematical Biologyor it’s recent revisionMathematical BiologyI: An Introductionis recommended for many aspects. Other suggestions will bemade during the course.Lecture Notes: Printed lecture notes for the module will be available, but theseshould be seen to complement the lectures rather than replace them since therewill be additional material (including examples) covered during lectures. Thisadditional material will almost certainly form the basis ofa significant amount ofthe assessments.


Assessment:The module is 100% assessed, through two assessments and whathas on many previous years proved to be a popular mini project. Expect the as-sessments to be quite demanding, and make sure that you understand the univer-sity rules on plagiarism. In previous years there have been some harsh penaltiesimposed for breach of these regulations. Ignorance is not anexcuse.

Lecturer: Dave Wood.

MA213 Terms 1–2Second Year Essay 6 CATS

Status: Corefor all Maths students.

Organisation: A list of essay topics will be offered by the end of week 6 ofTerm 1. You can pick a topic on this list. Alternatively, you can choose your owntopic in consultation with your tutor who must submit it to the second year essaycoordinator for approval by the end of week 8 of Term 1.

Students may, and are strongly advised to, submit a draft of their essay totheir tutor by the end of the first week of Term 2. You are expected to haveconsulted the web pages in Mathstuff on essay writing prior to submission of thedraft. The tutor will provide written comments and discuss the draft, normallyby week 4 of Term 2.

Students have to give a 15-minute oral presentation of the essay to their tutorand a small group of other second year students, normally in week 19. Thispresentation is a compulsory requirement and 15% of the essay mark is allocatedto the quality of the presentation. Students should seek advice, e.g. from theirtutor, on how to convey the content of their essay within sucha short period oftime; they must not get bogged down in technicalities but they should not bevague.

Aims:

(i) To provide an opportunity for students to learn some mathematics directlyfrom books and other sources.

(ii) To develop written and oral exposition skills.

Objectives:

(i) To learn how to write mathematics well.(ii) To practice presenting mathematics orally to a group.(iii) To develop research skills, including planning, use of library and the inter-

net.

Deadline The essay must be submitted to the Undergraduate Office by 12:00noon on Tuesday of the first week of Term 3.This deadline is enforcedby themechanism described in White PYDC.


It is the students’ responsibility to choose their essay topic, to prepare thedraft on time, to seek advice where necessary, to prepare thepresentation ontime and to submit the final version of the essay on time.

The essay will be marked by your tutor and a second marker. Your tutorwill also award the mark for the oral presentation. Instructions about the essayand information on the marking scheme will be given out by theend of Term 1.Students are advised to read the instructions carefully, since failure to follow oneof the University Regulations (on plagiarism, for example)could result in a markof zero.

Organiser: Dave Wood

MA242 Term 1Algebra I: Advanced Linear Algebra 12 CATS

Status: Corefor Maths.This module will be examined in Week 30, the first week of Term 3.

Commitment: 30 one-hour lectures plus six assignments.

Prerequisites: MA106 Linear Algebra and MA132 Foundations.

Content: This module is a continuation of First Year Linear Algebra. In thatcourse we studied conditions under which a matrix is similarto a diagonal matrix,but we did not develop methods for testing whether two general matrices aresimilar. Our first aim is to fill this gap for matrices overC. Not all matrices aresimilar to a diagonal matrix, but they are all similar to one in Jordan canonicalform; that is, to a matrix which is almost diagonal, but may have some entriesequal to 1 on the superdiagonal.

We next study quadratic forms. Aquadratic formis a homogeneous quadra-tic expression

∑aijxixj in several variables. Quadratic forms occur in geometry

as the equation of a quadratic cone, or as the leading term of the equation ofa plane conic or a quadric hypersurface. By a change of coordinates, we canalways writeq(x) in the diagonal form

∑aix

2

i . For a quadratic form overR,the number of positive or negative diagonal coefficientsai is an invariant of thequadratic form which is very important in applications.

Finally, we study matrices over the integersZ, and investigate what happenswhen we restrict methods of linear algebra, such as elementary row and columnoperations, to operations overZ. This leads, perhaps unexpectedly, to a completeclassification of finitely generated abelian groups.

Aims: To develop further and to continue the study of linear algebra, which wasbegun in Year 1.To point out and briefly discuss applications of the techniques developed to otherbranches of mathematics, physics, etc.

Objectives: By the end of the module students should be familiar with: thetheo-ry and computation of the the Jordan canonical form of matrices and linear maps;


bilinear forms, quadratic forms, and choosing canonical bases for these; the the-ory and computation of the Smith normal form for matrices over the integers, andits application to finitely generated abelian groups.

Leads to: third year algebra modules, such as MA3D5 Galois Theory, MA377Rings and modules. Some of the theory is also needed in MA371 QualitativeTheory of ODEs.

Books: P M Cohn,Algebra, Vol. 1, WileyI N Herstein,Topics in Algebra, Wiley.Neither is essential, but are a good idea if you are intendingto study further

algebra modules.

Assessment:Assignments (15%), two-hour examination (85%).

Lecturer: Derek Holt.

MA244 Term 1Analysis III 12 CATS

Status: Corefor Maths.This module will be examined in the first week of Term 3.

Commitment: 30 lectures.

Prerequisites: MA131 Analysis, MA106 Linear Algebra.

Content: This covers three topics: (1) integration, (2) convergenceof sequencesand series of functions, (3) Norms.

The idea behind integration is to compute the area under a curve. The fun-damental theorem of calculus gives the precise relation between integration anddifferentiation. However, integration involves taking a limit, and the deeper prop-erties of integration require a precise and careful analysis of this limiting process.This module proves that every continuous function can be integrated, and provesthe fundamental theorem of calculus. It also discusses how integration can beapplied to define some of the basic functions of analysis and to establish theirfundamental 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 Fouri-er series is the sum of a trigonometric series with coefficients given by certainintegrals. The second part of the module develops methods for deciding when afunction defined as the limit of a sequence of other functionsis continuous, dif-ferentiable, integrable, and for differentiating and integrating this limit. Normsare used at several stages and finally applied to show that a Differential Equationhas a solution.

Aims:

1. To develop a good working knowledge of the construction ofthe integralof regulated functions;


2. to study the continuity, differentiability and integralof the limit of a uni-formly convergent sequence of functions;

3. to use the concept of norm in a vector space to discuss convergence andcontinuity there.

Objectives:

1. Understand the need for a rigorous theory of integration,and that this canbe developed for regulated functions by approximating the area under thegraph by rectangles;

2. understand uniform and pointwise convergence of functions together withproperties of the limit function;

3. be able to prove the main results of integration: any continuous functioncan be integrated on a bounded interval and the Fundamental Theorem ofCalculus;

4. prove and apply the Contraction Mapping Theorem.

Leads to: MA222 Metric Spaces, MA225 Differentiation, MA359 MeasureThe-ory, MA3F4 Linear Analysis and MA3G1 Theory of PDE’s.

Books: No book covers the module although the MathSoc Revision Guide isrecommended. A list of books to consult is given on Mathstuff.

Assessment:Two-hour examination (85%), assignments (15%).

Lecturer: Anthony Manning

MA241 Term 1Combinatorics 12 CATSStatus: List A for Mathematics.


Prerequisites: No formal prerequisites. The module follows naturally fromfirstyear core modules and/or computer science option CS128 Discrete Mathematics.

Content: Despite the title of the module, we will concentrate on enumerativecombinatorics, which is that branch of combinatorics whereone counts things.Questions of counting are often important in computer science and statistics.Moreover such questions naturally lead to manipulations ofalgebraic expressionswhich go beyond secondary school techniques, and which willbe essential in allparts of algebra and many more parts of mathematics. Emphasis lies on examplesrather than theory.

Detailed contents:

1. Introductory material and background: Recurrence problems such asthe Tower of Hanoi. Sums and term rearrangement. Floor, ceiling, andmod functions.


2. Binomial coefficients: Counting of permutations and combinations. Var-ious definitions of binomial coefficients. Identities. Binomial theorem.Inversion formula. Derangements. Multinomial coefficients.

3. Special numbers:Definitions, properties, applications. Stirling numbers;Harmonic numbers; Bernoulli numbers and Fibonacci numbers.

4. Generating functions: Partial fractions. Applications to counting prob-lems. Domino theory. Mutually recursive sequences. Convolutions. Cata-lan numbers. Spanning Trees. Exponential generating functions. Generat-ing functions in more variables.

Book: Lecture Notes will be available at cost price. An excellent but expensivebook is Graham, Knuth and Patashnik,Concrete Mathematics, Addison-Wesley.

Assessment:10% by 5 fortnightly assignments during the term, 90% by a two-hour written examination.

Lecturer: Daan Krammer.

MA243 Term 1Geometry 12 CATS

Status: List A for Mathematics.

Commitment: 30 lectures plus weekly worksheets.

Prerequisites: None, but an understanding of MA125 Introduction to Geometryor MA130 From Geometry to Groups will be helpful.

Content: Geometry is the attempt to understand and describe the worldaroundus and all that is in it; it is the central activity in many branches of mathematicsand physics, and offers a whole range of views on the nature and meaning of theuniverse.

Klein’s Erlangen program describes geometry as the study ofproperties in-variant under a group of transformations. Affine and projective geometries con-sider properties such as collinearity of points, and the typical group is the fulln × n matrix group. Metric geometries, such as Euclidean geometry and hyper-bolic geometry (the non-Euclidean geometry of Gauss, Lobachevsky and Bolyai)include the property of distance between two points, and thetypical group is thegroup of rigid motions (isometries or congruences) of 3-space. The study of thegroup of motions throws light on the chosen model of the world.

The module includes a diversity of topics, such as the rules of life and self-consistency of the non-Euclidean world, symmetries of bodies both Euclideanand otherwise, tilings of Escher and the regular solids, andthe geometric rules ofperspective in photography and art.

Aims: To introduce students to various interesting geometries via explicit ex-amples; to emphasize the importance of the algebraic concept of group in the


geometric framework; to illustrate the historical development of a mathematicalsubject by the discussion of parallelism.

Objectives: Students at the end of the module should be able to give a full anal-ysis of Euclidean geometry; discuss the geometry of the sphere and the hyper-bolic plane; compare the different geometries in terms of their metric properties,trigonometry and parallels; concentrate on the abstract properties of lines andtheir incidence relation, leading to the idea of affine and projective geometry.

Leads to: MA3D9 Geometry of Curves and Surfaces, MA4E0 Lie Groups,MA473 Reflection Groups.

Books: M Reid and B Szendroi, Geometry and Topology, CUP, 2005 (someChapters will be available from the General office).

E G Rees,Notes on Geometry, SpringerHSM Coxeter,Introduction to Geometry, John Wiley & Sons

Assessment:the weekly worksheets carry 15% assessed credit; the remaining85% credit by 2-hour examination.

Lecturer: Miles Reid.

MA231 Term 1Vector Analysis 12 CATS

Status: Corefor Maths.


Prerequisites: MA127 3D Geometry and Motion or PX129 (Maths/Physics)Worksheets.

Content: The first part of the module provides an introduction to vector calcu-lus which is an essential toolkit for differential geometryand for mathematicalmodelling. After a brief review of line and surface integrals, div, grad and curlare introduced and followed by the two main results, namely,Gauss’ DivergenceTheorem and Stokes’ Theorem. These theorems will be proved only in simplecases; complete proofs are best deferred until one has learned about manifoldsand differential forms. The usefulness of these results in applications to flowproblems and to the representation of vector fields with special properties bymeans of potentials will be emphasized. This leads to Laplace’s and Poisson’sequations which will be discussed briefly. The solution of these equations arediscussed more fully in modules on partial differential equations. Cartesian co-ordinates are in many cases not well suited to a particular problem: for example,polar coordinates yield simpler equations for the flow of water in a cylindricalpipe. We will show how to represent div, grad and curl in general curvilinearcoordinates, paying particular attention to spherical andcylindrical geometries.

The second part of the module introduces the rudiments of complex analysisleading up to the calculus of residues. The link with the firstpart of the moduleis achieved by considering a complex valued function of one complex variable


as a vector field in the plane. This idea is particularly useful in the study of two-dimensional fluid flow. Complex differentiability leads to the Cauchy-Riemannequations which are interpreted as conditions for the vector field to have both zerodivergence and zero curl. Cauchy’s theorem for complex differentiable functionsis then established by means of the main integral theorems ofvector calculus.Cauchy’s integral formula which expresses the value of a complex differentiablefunction at a point as a line integral of the function on a contour surrounding thepoint is the key result from which the stunning properties ofcomplex differen-tiable functions follow.

Aims: This module aims to

1. Teach a practical ability to work with functions of two or three variablesand vector fields;

2. Present the theorems of Gauss and Stokes as generalisations of the funda-mental theorem of calculus to higher dimensions;

3. Establish Cauchy’s theorem in complex analysis as a consequence of theCauchy-Riemann equations and the divergence theorems;

4. Teach those rudiments of complex analysis which follow from Cauchy’stheorem, namely, the Cauchy integral formula, Taylor expansions and resi-due calculus.

Objectives: On successful completion of this module, a student should

1. Be able to calculate line, surface and volume integrals ingeneral curvilin-ear coordinates;

2. Be familiar with and use in a variety of contexts the fundamental results ofvector calculus, namely, the divergence theorem and Stokes’ theorem;

3. Understand the relation between the existence of a scalaror vector potentialof a vector field and the vanishing of the curl or divergence ofthat vectorfield and be able to calculate the potential when it exists,

4. Be able to establish the Cauchy-Riemann equations for a complex differen-tiable function and establish Cauchy’s theorem from the integral theoremsof vector calculus;

5. Be able to prove Cauchy’s integral formula from Cauchy’s theorem, and touse the integral formula to establish differentiability and series propertiesof complex differentiable functions;

6. Be able to calculate Taylor expansions, residues and use them in the eval-uation of definite integrals and summation of series.

Leads to: MA3D1 Fluid Dynamics, MA3G1 Theory of PDEs, MA3D9 Ge-ometry of Curves and Surfaces, MA3B8 Complex Analysis MA390Topics inMathematical Biology, Various 400 level courses.

Books: There are a huge number of books that cover Vector and ComplexAnal-ysis at roughly the right level for this course. Comments on aselection of books


that are useful for this module will be distributed at the first lecture and postedon the Mathstuff website for this module.

Assessment:2-hour examination (85%) and coursework (15%)

Lecturer: Jochen Voss

MA249 Term 2Algebra II: Groups and Rings 12 CATS

Status: Core for Year 2 mathematics students. It could be suitable as a usual orunusual option for non-maths students.

If you are a Year 3 mathematics student who delayed taking Algebra II untilthis year, you must take MA245 Algebra II instead.


Prerequisites: First year MA129 Foundations, MA106 Linear Algebra, andMA242 Algebra I: Advanced Linear Algebra.

Content: This is an introductory abstract algebra module. Abstract algebra is abit like solfeggio. The latter is an abstract language that is served to preserve andcommunicate beautiful music. The former is an abstract language to preserveand communicate beautiful mathematics. And both require anessential mentaleffort to learn.

As the title suggests, the two main objects of the study are groups and rings. Youalready know that a group is a set with one binary operation. But you have al-so seen examples of rings which are sets with two binary operations. The mostnotable example is the set of integers with addition and multiplication. We willdevelop the theories of groups and rings. Theorems discussed include the Orbit-Stabiliser Theorem, the Chinese Remainder Theorem, and Gauss’ theorem onunique factorisation in polynomial rings. We will also enjoy some beautifulmathematics by seeing examples and applications such as RSA, game 15 and,maybe, the discrete Fourier transform.

Aims: To study abstract algebraic structures, their examples andapplications.

Objectives: By the end of the module the student should know several resultsabout groups and rings as well as be able to manipulate with them.

Leads to: The results of this module are used in several modules including:MA377 Rings and Modules, MA453 Lie Algebras, MA362 Non-commutativeRings, MA3D5 Galois Theory, MA3E1 Group and Representations and MA3G0Modern Control Theory.

Books: This is a new module, so a printed study guide will be available from thegeneral office toward the end of the Spring term.

The study guide will be updated during the term and will be available onlineat www.maths.warwick.ac.uk/ ˜rumynin .

The only recommended book is


Niels Lauritzen,Concrete Abstract Algebra, Cambridge University Press.You should seriously consider buying it for a number of reasons including:

• it is relatively cheap:£22 for the paperback on Amazon or£6 on the Ama-zon marketplace;

• it will be the only book I will use to prepare lectures (disclaimer: I can usemy head, so not everything in the course will be in the book);

• it will be used as a source of exercises.

Assessment:Three example sheets will be assessed and are worth 5% each.Optional support classes are available.

The two-hour examination in June is worth 85%.

Lecturer: Dmitriy Rumynin

MA225 Term 2Differentiation 12 CATSStatus: This module isCore for all Maths students.

Commitment: Three one-hour lectures per week.

Prerequisites: MA131 Analysis, MA244 Analysis III.

Content: There are many situations in maths where one has to consider the con-tinuity and differentiability of a functionf : Rm

→ Rn (e.g., the determinant ofan n × n matrix as a function of its entries, or the wind velocity as a functionof space and time). The derivative is interpreted as a lineartransformation, ormatrix, and basic properties which generalise those of ordinary calculus are es-tablished, including finding maxima and minima and Taylor expansions. Theinverse and implicit function theorems are proved - these have many applicationsin both geometry and the study of solutions of nonlinear equations.

We will also study norms on infinite dimensional vector spaces and someapplications.

Aims:

1. To extend the results on differentiation of functions of 1-variable to func-tions between higher dimensional linear spaces.

2. To develop the theory of the derivative as a linear map and study its rela-tionship with partial derivatives.

3. To introduce the basic theory of normed vector spaces as needed for thistheory and to provide a basis for later modules.

4. To show how different branches of mathematics, in this instance linearalgebra and analysis, combine to give an aesthetically satisfying and pow-erful theory.

5. To encourage self-motivated study of mathematics.


Objectives: At the end of this module the student should have a basic workingknowledge of higher dimensional calculus. The student should understand thisin the context of normed spaces and appreciate the role this level of abstractionplays in the theory. They should understand basic linear analysis to the extent ofbeing able to follow it up in the relevant third year modules.They should also bein a position to make use of more advanced textbooks if they wish to go furtherinto these theories.

Books: J Marsden and A Tromba,Vector Calculus, McGraw Hill.T Apostol,Mathematical Analysis, Addison-Wesley.W Rudin,Principles of Mathematical Analysis, McGraw Hill.M Spivak,Calculus on manifolds, Benjamin Cummings.

Assessment:Two-hour examination.

Lecturer: Vassili Gelfreich.

MA235 Term 2 (20–24)Introduction to Mathematical Biology 6 CATS



Prerequisites: MA113 Differential Equations A.

Content: Basic modelling methodology will be introduced and illustrated withexamples drawn from a wide range of biological systems. Mostexamples thoughwill be drawn from population ecology. The mathematics required will be re-viewed and expanded as necessary.

Aims:

• to introduce some basic modelling methodology in a practical manner,based on a series of simple case studies;

• to give a taste of applications of differential equations inthe life sciences;

• to rehearse and reinforce basic material on ordinary differential equationsin an applied context.

Objectives: At the end of the module you should be able:

• to translate a simple biological problem into a mathematical model

• to give a biological interpretation for simple systems of ODEs and theirbehaviour

• to non-dimensionalise a model

• to apply well-known techniques for solving small systems ofODEs.

• to apply the same modelling methodology to simple problems in other dis-ciplines such as engineering, computer science or economics.


Leads to: MA390 Topics in Mathematical BiologyBooks: A good book which will also be useful in later years for those whocontinue in Mathematical Biology isJ. Murray’s “Mathematical Biology, VolumeI”, Springer.Assessment:One-hour examination.Lecturer: Hugo van den Berg.

MA222 Term 2Metric spaces 12 CATS

Status: List A for Maths.Commitment: Three one hour lectures per week.Prerequisites: MA129 Foundations, MA131 Analysis and MA244 Analysis III.Content: Roughly speaking, a metric space is any set provided with a sensiblenotion of the “distance” between points. The ways in which distance is measuredand the sets involved may be very diverse. For example, the set could be thesphere, and we could measure distance either along great circles or along straightlines through the globe; or the set could be New York and we could measuredistance “as the crow flies” or by counting blocks. Or the set might be the setof real valued continuous functions on the unit interval, inwhich case we couldtake as a measure of the distance between two functions either the maximum oftheir difference, or alternatively its “root mean square”.

This module examines how the important concepts introducedin first yearanalysis, such as convergence of sequences, continuity of functions, complete-ness, etc, can be extended to general metric spaces. Applying these ideas wewill be able to prove some powerful and important results, used in many partsof mathematics. For example, a continuous real-valued function on acompactmetric space must be bounded. And such a function on aconnectedmetric spacecannot take both positive and negative values without also taking the value zero.Continuity is readily described in terms of open subsets, which leads us natural-ly to study the above concepts also in the more general context of a topologicalspace, where, instead of a distance, it is declared which subsets are open.Aims: To introduce the theory of metric and topological spaces; toshow how thetheory and concepts grow naturally from problems and examples.Objectives: To be able to give examples which show that metric spaces are moregeneral than Euclidean spaces, and that topological spacesare yet more generalthan metric spaces. To be able to work with continuous functions, and to recog-nize whether spaces are connected, compact or complete.Leads to: The module is a vital prerequisite for most later (especially Pure)Mathematics modules, including MA3F1 Introduction to Topology, MA3D9 Ge-ometry of Curves and Surfaces, MA3F4 Linear Analysis, MA359Measure The-ory, MA3B8 Complex Analysis, MA371 Qualitative Theory of ODEs, MA3G1


Theory of PDEs, MA424 Dynamical Systems, MA4E0 Lie Groups, MA475 Rie-mann Surfaces.

Books: W A Sutherland,Introduction to Metric and Topological Spaces, OUP.You are strongly recommended to have your own copy.Other books worth consulting:

E T Copson,Metric Spaces, CUP.W Rudin,Principles of Mathematical Analysis, McGraw Hill.G W Simmons,Introduction to Topology and Modern Analysis, McGraw

Hill. (More advanced, although it starts at the beginning; helpful for several thirdyear and MMath modules in analysis).

A M Gleason,Fundamentals of Abstract Analysis, Jones and Bartlett.D Epstein, Metric Spaces Lecture Notes, 1999–2000, Mathstuff.

Assessment:Two-hour examination 85%, class tests 15%.

Lecturer: David Preiss

MA228 Term 2 (15–19)Numerical Analysis 6 CATS


Commitment: 15 lectures and 3 or 4 computing exercises.

Prerequisites: MA117 Programming for Scientists or equivalent, MA113 Dif-ferential Equations A, MA127 3D Geometry and Motion.

Content: This module focuses on basic numerical methods for problemsaris-ing in mathematics and the physical sciences. Through selected examples suchas multi-dimensional zero-finding and the solution of ordinary differential equa-tions, the important concepts of iteration, convergence, cost, accuracy and stabil-ity will be covered.

Aims: To introduce the numerical methods used in tackling mathematical equa-tions which do not yield to exact forms of analysis.

Assessment:By reports from computing exercises.

Lecturer: Sergei Mazarenko

MA250 Term 2Introduction to Partial Differential Equations 12 CATS

Status: List A

Commitment: 30 lectures

Prerequisites: MA131 Analysis, MA244 Analysis III, MA135 Vectors and Ma-trices, MA133 Differential Equations, MA231 Vector Analysis


Content: The theory of partial differential equations (PDE) is important bothin pure and applied mathematics. On the one hand they are usedto mathemat-ically formulate many phenomena from the natural sciences (electromagnetism,Maxwell’s equations) or social sciences (financial markets, Black-Scholes mod-el). On the other hand since the pioneering works on surfacesand manifolds byGauss and Riemann partial differential equations have beenat the centre of manyimportant mathematical developments (geometry, Poincare-conjecture).

In this module I will classify the most important equations and discuss thequalitative behaviour of the solutions. I will develop several approaches to con-struct solutions: Method of characteristics, Green’s functions and Fourier seriesto solve the classical equations

1. Laplace equation (elliptic),

2. Heat equation (parabolic),

3. Wave equation (hyperbolic).

The module will build upon the Analysis courses, Vector Analysis and Dif-ferential Equations. In particular different notions of convergence of a sequenceof functions will be discussed, and an introduction to the theory of Fourier-serieswill be given.

Aims: To introduce the basic phenomenology of partial differential equationsand their solutions. To construct solutions using classical methods.

Objectives: At the end of the course you will be able to classify partial differ-ential equations and know which types of boundary conditions can be used. Youwill understand that the solutions of PDEs depend in a very sensitive way on thetype of the equation and you will be able to solve the most important equations.

Leads to: MA3G1 Theory of Partial Differential Equations, MA4A2 AdvancedPDEs, MA4A7 Quantum Mechanics of Atoms and Molecules, MA433FourierAnalysis and MA592 Topics in PDE.

Books: W. StraussPartial Differential Equations. An introduction.John Wiley(1992).

M. Renardy and R.C. Rogers,An introduction to partial differential equa-tions, Springer TAM 13 (2004).

Assessment:2 hour exam.

Lecturer: Florian Theil

MA117 Term 2Programming for Scientists 9 CATS

Status: List B for Maths. MA117 may not be taken before or after CS118Programming for Computer Scientists.

Commitment: 10 lectures plus lab sessions/tutorials.


Prerequisites: No previous computing experience will be assumed, but studentsshould have obtained a code to use the IT Services work area systems prior to thismodule. Information and assistance is available in the Student Computer Centrein the Library Road.

Content: Aspects of software specification, design, implementationand testingwill be introduced in the context of the Java language. The description of basicelements of Java will include data types, expressions, assignment and compound,alternative and repetitive statements. Program structuring and object orienteddevelopment will be introduced and illustrated in terms of Java’s method, classand interface. This will enable the development of softwarethat reads data in avariety of contexts, performs computations on that data anddisplays results intext and graphical form. Examples of iterative and recursive algorithms will begiven. The importance of Java and Java Virtual Machine in networked computingwill be described. The majority of examples will be standardapplications butthe development of Java Applets to be delivered by web browsers will also becovered.

Aims: To provide an understanding of the process of scientific software devel-opment and an appreciation of the importance of data vetting, sound algorithmsand informative presentation of results.

Objectives: To enable the student to become confident in the use of the Javalanguage for scientific programming.

Leads to: MA228 Numerical Analysis and modules given by the Computer Sci-ence Department that are based upon the Java language, including CS223 Intro-duction to Software Engineering, CS236 Data Structures andAlgorithms, andCS237 Concurrent Programming.

Books: Books are not essential for this module as use will be made of on-linetutorial and reference material. An informative, optionaltext is

H M Deitel & P J Deitel,Java How to Program(2nd or 3rd Ed), PrenticeHall.

Assessment:Three programming assignments.

Lecturer: Petr Plechac.

MA246 Term 3Number Theory 6 CATS

Status: List A for Maths. Students may take this module in any year (but notmore than once!). Third years will find it difficult to fit this module aroundtheir April and June examinations, and so will need to have studied the materialbeforehand. They may also find that some finals examinations conflict with themodule tests. For this reason it is best to take this module inYears 1 or 2.

Commitment: There are five workbooks for this course. Each contains notes,examples and questions. Solutions to the questions are available on Mathstuff.


You study the workbooks on your own, or with the help of friends. The work-books are self-contained, but you may also wish to refer to the recommendedtexts. An e-mail or forum support address will be announced.Each workbookleads to a 50 minute multiple-choice test; these are held once a week in the firsthalf of term 3.

Prerequisites:MA129 Foundations, MA106 Linear Algebra, MA130 From Ge-ometry to Groups, and second year Algebra courses are also useful in under-standing the material. First year students should be able totackle the module ifthey are prepared to do a little reading around some of the topics (and if they areconvinced it will not interfere unduly with their revision for other examinations).

Content:Workbook 1The arithmetic of congruence classes, solving linear congruences,the multiplicative structure ofZn (Euler’s Theorem and Fermat’s Little Theo-rem).Workbook 2Primitive roots and finite logarithms, Euler’s phi-function, decimalrepresentation of rational numbers.Workbook 3The greatest integer function, de Polignac’s formula, standard mul-tiplicative functions, perfect numbers and Mersenne primes, Mobius’s functionand inversion formula.Workbook 4Finite continued fractions and Euclid’s algorithm, infinite continuedfractions for irrational numbers.Workbook 5Periodic continued fractions, Pell’s equation.

Aims:

1. To introduce students to the delights of elementary number theory.2. To encourage independent study through using specially-prepared work-

books which develop abstract theory through sequences of concrete exer-cises, problems, and calculations.

Objectives:

1. To give students an easy facility with modular arithmetic, continued frac-tions, and the elementary functions of number theory; in particular, to de-velop their ability to do serious calculations with these objects.

2. To stress the role of problem-solving in developing mathematical under-standing.

3. To stimulate the use of pocket calculators in investigative mathematics (butnote that calculators will not be needed and will not be allowed in theexamination).

4. To provide an incentive for cooperative study.

Leads to: MA3D5 Galois Theory, MA426 Elliptic Curves.

Books: Prices are fromamazon.co.uk (2004) where available.


First Choice: Harry Davenport,The Higher Arithmetic, 7th Edn. (CUP,1999), ISBN 0521634466,£19.95.

A Problems-Based Approach:R.P. Burn,A Pathway into Number Theory(CUP, 2nd edition, 1997), ISBN 0521575400.

Also Recommended:Joseph Silverman,A Friendly Introduction to NumberTheory(Prentice-Hall, 2nd edition, 2001), ISBN 0130309540,£40.99.

Also Recommended: James J. Tattersall,Elementary Number Theory inNine Chapters(CUP, 1999), ISBN 0521585317,£18.95.

The Classic: G.H.Hardy & E.M.Wright,An Introduction to the Theory ofNumbers(OUP),£28.50.

A Number Theorist’s World View: G.H.Hardy,A Mathematician’s Apolo-gy (CUP, 1992), ISBN 0521427061,£7.69.

Assessment:Four or five weekly multiple-choice tests (depending on timetabling).Your final test score will be based on your best 4 results out of5 (or 3 out of 4),and contributes 25% to final credit. A 90-minute final examination makes up theremaining 75% of the module credit.

Organiser: Trevor Hawkes

MA209 Term 3Variational Principles 6 CATS



Prerequisites:MA131 Analysis and a module on differential equations (MA225Differentiation is also helpful).

Content: This module consists of a study of the mathematical techniques of vari-ational methods, with applications to problems in physics and geometry. Criticalpoint theory for functionals in finite dimensions is developed and extended tovariational problems. The basic problem in the calculus of variations for contin-uous systems is to minimise the integral

I(y) =

∫ b

a

f(x, y, yx)dx

on a suitable set of differentiable functionsy: [a, b] → R. The Euler–Lagrangetheory for this problem is developed and applied to dynamical systems (Hamil-tonian mechanics and the least action principle), shortesttime (path of light raysand Fermat’s principle), shortest length and smallest areaproblems in geome-try. The theory is extended to constrained variational problems using Lagrangemultipliers.

Aims: To introduce the calculus of variations and to see how central it is to theformulation and understanding of physical laws.


Objectives: To show you how to set up and solve minimisation problems withand without constraints, to derive Euler–Lagrange equations and to have youappreciate how the laws of mechanics fit into this framework.

Books: A useful introduction is:R Weinstock,Calculus of Variations with Applications to Physics and Engi-

neering, 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 ofany book.

Assessment:One-hour examination.

Lecturer: John Rawnsley

COMPUTER SCIENCESee the White Book for information on the Joint Degree in Mathematics with

Computing.

CS242 Term 1Formal Specification and Verification 15 CATS

Status: List B

Commitment: 20 one-hour lectures and 10 seminars.

Content: Propositional logic: proofs, semantics, normal forms, SATsolvers.Predicate logic: proofs, semantics.Specifying and modelling software.Verification by model checking.Proof calculi for program verification.

Books: Huth M and Ryan M,Logic in Computer Science: Modelling and Rea-soning about Systems(2nd ed), Cambridge University Press, 2004.

Peled D,Software Reliability Methods, Springer-Verlag, 2001.Potter B, Sinclair J and Till A,An Introduction to Formal Specification and

Z (2nd ed), Prentice-Hall, 1996.

Assessment:Three-hour examination (80%), coursework (20%).

Lecturers: Ranko Lazic and Jane Sinclair.

CS252 Term 1Fundamentals of Relational Databases 7.5 CATSStatus: List B


Commitment: 5 one-hour lectures and 8 one-hour seminars.Content: Overview of databases and database management systems.

Concepts and language for relational theory (including values, variables,types, operators, propositions and predicates).

Relations and predicates; relational and logical operators.Relational algebra.Integrity constraints.Database design issues (including functional dependency and normalisa-

tion).SQL for table definition, selection and queries; constraints, joins and aggre-

gation.Books: Date C J,Database in Depth: Relational Theory for Practitioners, O’-Reilly, 2005.

Date C J,Introduction to Database Systems(8th ed), Addison Wesley, 2004.Begg C, Connolly T,Database Systems: A Practical Approach to Design,

Implementation and Management(4th ed), Addison Wesley Longman, 2004.Assessment:One and a half-hour examination (70%), coursework (30%)Lecturer: Alexandra Cristea, Hugh Darwen and Tim Heron

CS253 Term 1Fundamentals of Relational Databases 7.5 CATSStatus: List BContent: Derivation of database design from an EER diagram.

Transactions, recovery from failures, concurrency issues.Decision support systems (including data warehouses and OLAP).Java database connectivity (JDBC).SQL security (including injection attacks and permissions).Databases on the web.3-tier architecture.OO and object-relational databases.(Not all the above topics would be taught each year.)

Books: Date C J,Introduction to Database Systems(8th ed), Addison Wesley,2004.

Riccardi G,Principles of Database Systems with Internet and Java Applica-tions, Addison Wesley, 2001.Assessment:Two hour examination (100%).Lecturer: Alexandra Cristea and Tim Heron

CS245 Term 2Automata and Formal Languages 7.5 CATSStatus: List B .


Commitment: 15 one-hour lectures plus one revision class.

Content: Regular languages: finite automata, non-determinism, regular expres-sions, non-regular languages.

Context-free languages: context-free grammars (formal definition and ex-amples, ambiguity, Chomsky normal form).

Introduction to parsing.Turing machines (formal definition and examples).Introduction to computability (The Church-Turing thesis,decidability, the

halting problem).

Books: Sipser M,Introduction to the Theory of Computation (2nd ed), ThomsonCourse Technology, 2005.

J Hopcroft & J Ullman,Introduction to Automata Theory, Languages andComputation, Addison-Wesley, 1979.

Assessment:1.5-hour examination (70%), programming assignment (30%).

Lecturer: Raja Nagarajan.

CS246 Term 2Further Automata and Formal Languages 7.5 CATS

Status: List B .

Prerequisites: CS245 Automata and Formal Languages.


Content: Regular languages: pumping lemmas, minimization of automata, trans-lating automata to regular expressions.

Context free languages: pumping lemma, push-down automata, closure prop-erties.

Introduction to computability: reducibility, proofs of undecidability.Omega-regular languages: Buechi automata and their closure properties.

Books: Sipser M,Introduction to the Theory of Computation (2nd ed), ThomsonCourse Technology, 2005.

J Hopcroft & J Ullman,Introduction to Automata Theory, Languages andComputation, Addison-Wesley, 1979.

Assessment:1.5-hour examination (100%).

Lecturer: Raja Nagarajan.

CS243 Term 2Data Structures and Algorithms 7.5 CATS

Status: List B

Commitment: 15 one-hour lectures and 1 group seminar.


Content: Analysis of running time of algorithms: asymptotic notation, analysisof recursive algorithms.

Efficient algorithms for sorting and selection: selection sort, merge sort, in-sertion sort, quick sort, binary search.

Efficient data structures: sets, lists, queues and stacks.Dictionary data structures: hash tables, binary search trees.Elementary tree and graph algorithms: depth first and breadth first search.

Book: Brassard& Bratley,Fundamentals of Algorithmics, Prentice-Hall, 1996

Assessment:.5-hour examination (80%), class test (20%).

Lecturer: Marcin Jurdzinski and Artur Czumaj.

CS244 Term 2Algorithm Design 7.5 CATS

Status: List B

Commitment: 15 one-hour lectures and 1 group seminar.

Content: Algorithms are fundamental to programming and to understandingcomputation. The purpose of this module is to provide students with a coherentknowledge of techniques for designing algorithms, and withthe tools for apply-ing these techniques to computational problems. Teaching and learning methodsinclude lectures and reading material which describe algorithmic techniques andapplications of these techniques to specific problems. A problem sheet givesstudents an opportunity to practice problem solving.

Book: Brassard& Bratley,Fundamentals of Algorithmics, Prentice-Hall, 1996

Assessment:.5-hour examination (80%), class test (20%).

Lecturer: Marcin Jurdzinski and Artur Czumaj.

ECONOMICSThe Economics 2nd and 3rd Year Handbook is available on request from

the Economics Department and contains details of their modules and prerequi-sites, including information on which will actually run during 2006–2007. Thisinformation is also available from the Economics web pages

http://www.warwick.ac.uk/economics/ug/index.html

See the Economics Handbooks for information on the Joint degree in Math-ematics and Economics.

Once you have consulted the Economics handbook, Dr Cave in Economicsshould be consulted if you have questions about the joint degree, or about eco-nomics options for the maths degrees.


EC220 Term 1Mathematical Economics IA 12 CATSStatus: List B for Maths.

Commitment: Two lectures each week, one problem class per fortnight and onetutorial per fortnight.

Prerequisites: Students must have taken EC106 Introduction to QuantitativeEconomics.

Content: The focus of this module is an introduction to game theory andthesyllabus comprises:

• games in strategic form: Nash equilibria and its applications to votinggames, oligopoly, provision of public goods

• games in extensive form: sub game perfect equilibria and itsapplicationsto voting games, repeated games

• static games with incomplete information: Bayesian equilibria and its ap-plications to auctions, contracts and mechanism design

• dynamic games of incomplete information: perfect Bayesianequilibria,sequential equilibria and its application to signalling games

• bargaining theory: Nash bargaining, non-cooperative bargaining with al-ternating offers and applications to economic markets.

Aims: This module aims to provide a basic understanding of pure game theoryand also introduce the student to a number of applications ofgame theory toeconomic problems of resource allocation. Strategic, normative and bargainingapproaches to resource allocation are treated.

Books: PK Dutta,Strategies and Games: Theory and Practice, 1999, MIT Press.For supplementary reading and for the student seeking a deeper understand-

ing:Fudenberg & Tirole,Game Theory, 1996.

Assessment:3-hour examination (80%) and tests (20%).

Lecturer: Herakles Polemarchakis.

EC221 Term 2Mathematical Economics IB 12 CATSStatus: List B for Maths.

Commitment: Two lectures per week, one problem class and one tutorial perfortnight.

Prerequisites: EC220 Mathematical Economics 1A

Content: The focus of this module is introduction to general equilibrium and thesyllabus comprises:


• Elements of Static Optimisation: a) Equality Constraints,b) InequalityConstraints. Barro and Sala-I-Martin, mathematical appendix.

• Theory of incentives in regulation: a) The Classical theory, b) Cost-Reim-bursement Rules, c) Pricing, d) Regulation of Quality. Laffont and Tirole.

• Introduction to the differential equations. Barro and Sala-I-Martin, mathe-matical appendix.

• The Solow Model. Barro and Sala-I-Martin.

• Introduction to the dynamic optimisation. Barro and Sala-I-Martin, math-ematical appendix.

• Introduction to the dynamic optimisation. Barro and Sala-I-Martin, math-ematical appendix.

• The Ramsey Model.Barro and Sala-I-Martin.

Aims: This module aims to provide a basic understanding of concepts and tech-niques of general equilibrium theory to include asset markets, externalities, dif-ferential information and inter-temporal trade.

Books: R. J. Barro and X. Sala I Martin,Economic Growth, 2003.J.J Laffont and J. Tirole,A theory of incentives in procurement and Regula-

tion, MIT press, 1993.A. C. Chang,Fundamental Methods of Mathematical Economics.A. C. Chang,Elements of Dynamic Optimization, McGraw Hill.

Assessment:3-hour examination (80%) and tests (20%).

Lecturer: Herakles Polemarchakis.

INSTITUTE OF EDUCATION

IE419 Term 1Development of Mathematical Concepts 12 CATS

Status: List B for Maths.

Commitment: 30 hours, but see below.

Content: This module aims to enhance students’ understanding of the ways inwhich mathematical concepts are developed by learners. A study of some aspectsof the research literature will enable the module to build animage of how themind reacts in different mathematical situations. In addition, students will beexpected to draw on their own experiences of mathematics learning (both positiveand negative) to put these theoretical aspects into context. The module addressesquestions such as

• how does the mind construct abstract objects – from the earliest encounterswith the counting numbers to abstract group theory?


• how do people solve mathematical problems?

• how does mathematics learning at school differ from that at university?

• how do we reason about mathematical objects and procedures,and howdoes such reasoning relate to proving?

The module is designed for students who are interested in thepsychologicalaspects of learning, may provide a useful background for those thinking aboutteaching and may help students to understand more fully how they themselvescome to build (or fail to build) mathematical knowledge. Each week the modulewill have two hours of whole group teaching, with a one-hour seminar based onsome set reading.

Assessment:One 2000 word assignment and one 2-hour examination.

Lecturer: Peter Johnston-Wilder ([email protected])

IE2A6 Term 2Introduction to Secondary School Teaching 24 CATS

Status: List B for Maths.THIS MODULE IS NOT AVAILABLE TO STUDENTS IN THE FINAL

YEAR OF THEIR DEGREE.All students taking this module are expected to have access to sustained

experience of working with teachers of mathematics and their pupils in a sec-ondary school.The module is intended for students who participate in theStudents Associates Scheme at Warwick. The module provides an opportunityfor students to undertake academic study in mathematics education, and to relatetheir studies directly to their experiences of observing teachers and working withpupils in secondary school mathematics classes.

Commitment: 1 hour per week for 10 weeks plus school experience.

Content: The module will be structured in five parts. Each part will consist oftwo taught sessions and will focus on a single theme relatingto teaching mathe-matics in secondary schools. The taught sessions will include workshop activi-ties in many of which you will work together in small groups toconsider issuesraised. For each theme there will be one or two required readings from relevantacademic texts or from research papers in education.

The five themes are:

• The National Curriculum for Mathematics

• Learning

• Planning to Teach

• Mathematics

• Assessment


Assessment:Written portfolio (100%).

Lecturer: Peter Johnston-Wilder ([email protected])

IE420 Term 2Problem Solving 12 CATS


Commitment: 30 Contact hours.

Content: Students on this module will have the opportunity to reflect on theirown mathematical thinking and to identify and develop problem-solving stra-tegies. The module is not like many other mathematics modules. There are nolecture notes and getting other people’s notes if a session is missed will be of littleuse. In lectures, students will be expected to think, to workon problems and todiscuss experiences. Occasionally students may be asked tolead a discussionbased on their work.

As the lectures involve working in mathematics, students are advised to comeprepared with plenty of paper and a calculator: even a ruler and coloured pensmay be useful.

The module code may be confusing - this is a module for second and thirdyear undergraduates.

Students are advised that it is essential to have access to the set book,Think-ing Mathematically, but it is preferable NOT to read it before the module starts.

Books: J Mason, L Burton and K Stacey, (1985)Thinking Mathematically, Wok-ingham: Addison Wesley (ISBN 0201 10238 2)

Assessment:One 2000-word problem solving assignment (50%) and a 2-hourexamination (50%).

Lecturer: Adrian Simpson ([email protected]).

ENGINEERINGFurther information is available from the Engineering web pages

www.eng.warwick.ac.uk/courses

ES312 Term 2/3Systems Modelling and Simulation 15 CATS

Previous description

Status: Optional module for students on the Maths with Computing degree.


Prerequisites: Knowledge of transforms, basic computer programming and ele-mentary probability theory.


Content: A wide variety of processes behave as dynamic systems where the sys-tem states vary in time, often in response to external stimuli. This module willintroduce techniques for analysing, predicting and understanding such dynamicbehaviour in engineering and other systems. The ability to engage in mathemat-ical modelling and use computer simulation tools is centralto the application ofthese techniques.

Aims: The module aims to present techniques available for the modelling andsimulation of physically based dynamical systems. It will cover the followingareas:

• Physically based modelling.

• Models of dynamical systems as initial value ordinary differential and dif-ference equations.

• Numerical integration and handling of discrete events.

• Modelling as a structured, systematic process and the role of empirical datain model validation.

• Computer-aided modelling and simulation.

• Continuous system simulation tools, eg. Simulink.

• Interactive matrix analysis, eg. MATLAB.

• Symbolic computation.

Objectives: The course is aimed at introducing the student to:

• Procedures for developing physically based dynamic modelsof processes.

• The role and use of continuous system simulation in dynamic system mod-elling.

• Analytical techniques for assessing the qualitative behaviour of dynamicsystem models.

• Methods for deriving dynamic system models from experimental data.

Books: This is a selection of books useful for the module. For more detailssee the entry for ES312 Systems Modelling and Simulation on the School ofEngineering’s website.

CM Close, DK Frederick & JC Newell,Modelling and Analysis of DynamicSystems(3rd ed), Wiley, 2002.

KG Godfrey, Compartmental Models and Their Applications, AcademicPress, 1983.

K Ogata,System Dynamics(3rd ed), Prentice Hall International, 1997.

Assessment:Examination (70%) Coursework (30%), and two assignments

Lecturer: R Jones & M Chappell


LANGUAGE CENTREThe Language Centre offers academic modules in French, German, Russian

and Spanish at a wide range of levels. These modules are available for examcredit as unusual options to mathematicians in all years. Pick up a leaflet listingthe modules from the Language Centre, on the ground floor of the HumanitiesBuilding by the Central Library. Full descriptions are available on request. Notethat you may only take one language module (coded LL, FR, GE orIT whetheras an Unusual Option or from List B) for credit in each year. These modules maycarry 24 or 30 CATS and that is the credit you get. But, where a language moduleis offered at a choice of 24 or 30 CATS, you MUST choose the 24 CATS version.

There is also an extensive and very popular programme of ‘leisure’ languageclasses provided by the Centre to the local community, with discounted fees forWarwick Students. Enrolment is from 9am on Wednesday of week1. Theseclasses donotcount as credit towards your degree.

The Language Centre also offers audiovisual and computer assisted self-access facilities, with appropriate material for individual study at various levels inArabic, Chinese, Dutch, French, German, Greek, Italian, Japanese, Portuguese,Russian and Spanish. (This kind of study may improve your mind, but does notcount as exam credit.)

A full module listing with descriptions is available on the Language Centreweb pages

www.warwick.ac.uk/LanguageCentre/Important note for students who pre-register for Language Centre modules

It is essential that you confirm your module pre-registration by coming tothe Language Centre as soon as you can during week one of the new academicyear. If you do not confirm your registration, your place on the module cannot beguaranteed. If you decide, during the summer, NOT to study a language moduleand to change your registration details, please have the courtesy to inform theLanguage Centre of the amendment.

LL201 Terms 1–3Russian for Scientists I 24 CATSStatus: List B for Maths.Commitment: The weekly load is 2 consecutive hours of teacher contact. Inaddition at least 1 hour of private study in the Open Access Suite are necessary.The teaching extends from week 2 of Term 1 to week 5 of Term 3.Content: This module is offered to students with no prior knowledge ofRussian.Students will acquire basic communication skills with a view to being able tocope in a variety of everyday contexts. This module is specially devised to enablethe students to read and translate texts of general scientific interest relevant totheir degree course, which will give them the opportunity tocombine learning


Russian with expanding their awareness of the current international scientificissues relevant to their professional fields.

Aims: The aim of the module is to acquire basic Russian speaking andcom-prehension skills, both spoken and written, and to provide basic knowledge ofgeneral scientific terminology and the ability to read and translate scientific textswith the aid of a dictionary. The module is delivered using modern teachingmethods, including computer-based learning. As a project during the module,each student will prepare a translation of a general scientific paper (two pages).

Leads to: The prerequisite for the 3rd year Russian II option (LL301) is thatstudents should have satisfactorily completed this optionor have an equivalentstandard of Russian.

Assessment:50% for the translation project, and 50% for the aural-oral exam inthe Language Centre.

Tutor: TBA

PHILOSOPHYStudents following modules in Philosophy are required to complete a De-

partmental record card and to return it to the Philosophy Department Secretaryat the start of Term 1. Students who fail to register correctly with the PhilosophyDepartment will not be sent details concerning assessment procedures, or otherimportant information.

See the White book for details of the joint degree.

PH201 Terms 1–3History of Modern Philosophy 30 CATS

Status: List B for Maths,Core for Mathematics and Philosophy

Prerequisites: Previous study of Philosophy

Content: The first part, taught in the Autumn term and the first two weeksofthe Spring term, covers key texts and arguments of Berkeley and Hume. Topicswill include: Berkeley on abstraction, the external world,and science; Hume onideas and impressions, relations of ideas and matters of fact, induction, causation,liberty and necessity, and moral judgement.

The second part aims to give students an understanding of some central ar-guments of Kant’s Critique of Pure Reason. Topics typicallyinclude space andtime, objectivity, self-awareness, causation, the self, freedom.

Lecturer: Bill Brewer/Stephen Houlgate.


PHYSICSStudents from the Department of Mathematics may take any combination of

the modules listed below. All exams are one hour per 6 CATS. Julie Staunton(Room PS132) will be glad to answer any queries concerning the second yearPhysics modules.

Module Seminars for Physics Options.Certain physics modules are supportedby module seminars which start one week after the start of themodule. Theseare 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 theSecond YearPhysics Laboratory.

PX262 Terms 1–2Quantum Mechanics and its Applications 15 CATS


Prerequisites: PX101 Quantum Phenomena.

Content: Revision of wavefunctions, probability densities and the Schrdingerequation in 1 dimension. The Hydrogen atom: orbital angularmomentum, quan-tum numbers, probability distributions. Atomic spectra and Zeeman effect. Elec-tron spin: Stern-Gerlach, spin quantum numbers, spin-orbit coupling, exclusionprinciple and periodic table. X-ray spectra. The development of formal quantummechanics. The quantum harmonic oscillator, creation and annihilation oper-ators. Angular momentum. Molecules: bonding and spectra. Solids: crystallattices and the amorphous state. Classical free electron model; mean free path;electrical and thermal conductivity. Fermi-Dirac distribution. Band theory ofsolids; scattering of electron waves by a periodic lattice;conductors, insulatorsand semiconductors. Semiconductor devices, e.g. diode, LED’s.

Leads to: PX359 Qyantum Physics I.

Books: HD Young & RA Freedman,University Physics, 11th Edition, Pearson.S.M. McMurry,Quantum Mechanics, Addison-Wesley 1994.Other useful books: P.C.W. Davies and D.S. Betts,Quantum Mechanics,

Chapman and Hall 1994.F. Mandl,Quantum Mechanics, John Wiley 1992.

Assessment:2 hour exam (85%); assessed work (15%).

Lecturers: Jim Robinson, Don Paul, Nicholas d’Ambrumenil.

PX266 Term 1Geophysics 7.5 CATS



Content:

1. Introduction: Basic characteristics of Earth: size, shape, mass, structure,age. Earth geometry, spherical co-ordinates.

2. Geochronology: Geological time. Radiometric dating.3. Gravity: Consequences of spherical geometry, geoid. Gravity measure-

ments and anomalies. Isostasy and mountain heights.4. Seismology: Types of seismic waves. Elasticity and elastic waves. Earth-

quake location and magnitudes. Seismology and Earth’s interior.5. Plate tectonics: Divergent, convergent and conservative plate boundaries.

Plate movement on Flat Earth. Rotation poles and present dayplate mo-tions. Past plate movements, role of Earth’s magnetic field.

6. Heat: Overview of heat budget and Earth. Heat flow and depthof oceans.Convection in the mantle. Thermal structure of the core. Earth’s magneticfield.

Assessment:1 hour exam.Books: William Lowrie, Fundamentals of Geophysics, CUP.

C.M.R Fowler,The Solid Earth - An Introduction to Global Geophysics,CUP.Lecturer: Gavin Bell.

PX267 Term 1Hamiltonian Mechanics 7.5 CATS

Replaces PX242.Status: List B for Maths.Prerequisites: PX132 Mechanics A.Content:

1. Introduction from the ‘Theory of Everything’: Quantum Mechanics as asum over paths. Phase as the integral of a Lagrangian over thepath. Anal-ogy with Optics and constructive interference. Stationaryphase for domi-nant=classsical paths . Examples of L.

2. Euler Lagrange Equations: 1-d trajectory, T-V case, worked examples.T+V as a constant of the motion; multiple coordinates with examples.

3. Generalised Coordinates and Canonical Momenta: polar coordinates; an-gular momentum; treatment of constraints; examples.

4. Symmetry and Conservation Laws.5. Small Oscillations and Normal Modes.6. Hamiltonian Formulation: Hamilton’s Equations; examples. Integrability:

Phase Plane Analysis; Hidden constants of the motion. Principle of LeastAction: and optics again. Overview.


Books: A good text going well beyond the module is H Goldstein,ClassicalMechanics.

A helpful reference for the beginning of the module is: Feynmann, Leighton& Sands,The Feynmann Lectures on Physics, Vol. 2, Chapter 19.

Assessment:One-hour examination.

Lecturer: Matthew Turner.

PX273 Term 1Physics of Electrical Power Generation 7.5 CATS


Prerequisites: PX121 Thermal Physics I, PX120 Electricity and Magnetism.

Content:

1. Energy resources, estimated reserves and current consumption.

2. The Carnot cycle to determine maximum possible efficiency.

3. Coal power stations, transformers, power transmission lines and three phase.

4. Nuclear power both thermal and fast breeder. Reprocessing.

5. Hydro-electric power. Conventional dams and Dinorwic.

6. Wind and wave generators and estimates of maximum possible resource.

7. Photo-voltaic cells.

8. Fusion power, Lawson criteria, design concepts, inertial and magnetic con-finement.


Books: There are no suitable texts.

Lecturer: Tony Arber.

PX268 Term 1Stars 7.5 CATSStatus: List B for Maths.

Prerequisites: PX144 Introduction to Astronomy.

Content:

1. The celestial sphere - coordinate systems: horizon system, equatorial sys-tem, galactic system - what is observed during a night, a month, a year -culmination, circumpolar stars.

2. Fundamental properties of stars - colour, luminosity, distance.

3. Trigonometric Parallax. The parsec and parallax angles.Statistical paral-lax.


4. Apparent and absolute magnitudes - blackbody radiation.5. Observational facilities - the optical/IR window - spacebased astronomy.6. Different types of stars - spectral classification - the Hertzsprung-Russel

diagram - population I and II - open and globular clusters - the age of stars.7. Stellar atmospheres - where does the light that we observeoriginate - in-

teraction between radiation and matter - radiation transfer.8. The structure of stars - basic equations - nuclear energy production - mass-

radius-luminosity relation - understanding the observed Hertzsprung-Russ-ell diagram.

9. Stellar evolution - main sequence life time - from birth todeath - youngstellar objects, stellar remnants: white dwarfs, neutron stars, black holes.

10. Binary stars - mass transfer - progenitors of supernovae- cosmologicaldistance scale.

Leads to: PX269 Galaxies, PX381 Astrophysics from Space, PX311 RelativisticCosmology.Books: BW Carroll & DA Ostlie, An Introduction to Modern Astrophysics,Addison-Wesley.

Prialnik, D,An introduction to the theory of stellar structure and evolution,CUP.Assessment:1 hour examination.Lecturer: Boris Gansicke.

PX263 Term 2Electromagnetic Theory and Optics 7.5 CATS

Replaces PX207.Status: List B for Maths.Prerequisites: PX120 Electricity and Magnetism.Commitment: About Eighteen lectures plus four one-hour problem classes.Content:

1. Vector Calculus (div, grad, curl and all that!)2. The fundamental laws of electricity and magnetism in integral form:

• Coulomb’s Law in surface integral form• No magnetic monopoles in surface integral form• Amphere’s Law (without displacement current term) in line integral

form• Faraday/Lenz law in line integral form, the definition of thevector

fields E, B, H, D, P and M. Emphasis is placed on the microscopicorigin of P and M, and the distinction between bound and free chargesand currents.


3. The Divergence Theorem. Conservation of charge and the equation of con-tinuity.

4. Stokes’ Theorem and the meaning of the curl. Simple examples of curl incylindrical polar co-ordinates.

5. The displacement current. Capacitor argument and compatibility with theconservation of charge.

6. Maxwell’s equations in differential form.

7. The wave equations for the E and B fields in a vacuum and a dielectric.The refractive index.

8. The general three dimensional plane wave and the wave vector k. Therelationship between E, B and k. Intrinsic impedance.

9. Polarisation of light. Linear eliptic and circular states.

10. The Poynting Vector and its properties.

11. The wave equations in an ohmic conductor, specialising to a good conduc-tor. Skin depth.

12. Amplitude and phase relations at a dielectric interface. Laws of reflectionand refraction, including Fresnel’s equations, and consequences.

13. Introduction to optical materials, including dichroism, birefringence andoptoelectric effects.

14. The behaviour of wavefronts at plane and spherical surfaces.

15. The lens equation, lens power and characteristics of images.

16. The basics of optical instrumentation (including lightsources).

17. Image resolution.

18. Temporal and spacial coherence.

19. Introduction to holographic image formation.

20. Quantum optics. The particle nature of light.

Book: IS Grant & WR Phillips,Electromagnetism, Wiley.E Hecht,Optics.H D Young and R A Freedman,University Physics 11th Edition, Pearson.ER Dobbs,Basic Electricity and Magnetism, Chapman and Hall (out of

print).R Feynman,Feynman Lectures Vol II, Addison Wesley.

Lecturer: Robert Pettifer.

PX269 Term 2Galaxies 7.5 CATS

Replaces PX252.



Prerequisites: PX268 Stars.Content: Brief review of our assumptions about the state of the universe: Olber’sParadox, the Cosmological Principal, General Relativity.

Parameters and Measurement: Distance, Luminosity, ColourIndex, the H-RDiagram, stellar clusters, spectroscopy.

Hierarchical structures: stellar clusters, galaxies, clusters, superclusters, walls,voids, filaments and sheets, gravitational lensing.

Spiral galaxy: morphology, constituents (stars, the ISM and ?), size andmass, stellar populations and their dynamics, the LSR, statistical astronomy, sim-ple models.

Classification of Galaxies: The Hubble and other classification schemes, re-view of galactic types and their structure, elliptical galaxies.

Dark Matter and galactic evolution: Missing gravity (localand general), ob-servation of mass, fundamental particles, baryonic and leptonic dark matter, MA-CHOS, WIMPS, exotic particles - Axions, topological defects and singularities,stellar mass black holes, cold and hot dark matter.

Exotic galaxies and AGN’s: Seyfert galaxies, Starburst galaxies, Quasars -Remote objects, red shift, age of universe

Leads to: PX311 Relativistic Cosmology and PX381 Astrophysics from Space.Books: RJ Taylor,Galaxies: Structure and Evolution, CUP.

B Jones, RJA Lambourne & DA Rothery,Images of the Cosmos, The OpenUniversity.

B Jones,Galaxies, The Open University.


Lecturer: Peter Wheatley.

PX272 Term 2Global Warming 7.5 CATS

Status: List B for Maths.Content:

1. The Greenhouse Effect and the main gases responsible.An essentiallyqualitative account of why there might be a problem.

2. CO2 production and accumulation.You will be challenged to work throughthe key numbers for yourself.

3. Evidence for Climate Change.Qualitative survey of what has been mea-sured and whether it is significant.

4. The role of Ice Caps, and sea level rise.Temperature stabilization by theTriple Point, observed rates of melting and breakup, and thewater volumesin question.

5. Albedo effects.Ice-caps, deserts, clouds and global dimming.


6. Estimated hazards to man and Environment.Warming, sea level rise,changing ocean currents, thermal runaway.

7. The political response to date.Kyoto, UK and EU targets.

8. Available options.Conservation; Sequestration; Nuclear Energy; Renew-able Energy Sources.

9. Cost-benefit analysis.Case studies: free market adaptation vs internation-al action.

10. The likelihood of action.Politics of decisions, prospective winners andlosers.

Leads to: PX350 Weather and the Environment.

Books: There are few relevant up to date printed sources. Instead the internetwill be relied upon quite heavily.

Assessment:1 hour exam (80%), assessed work (20%).

Lecturer: Robin Ball.

PX261 Term 2Mathematical Methods for Physicists II 7.5 CATS


Prerequisites: PX260 Mathematical Methods for Physicists I or PX253 PartialDifferential Equations.

Content: The module covers:

1. Fourier Series: Representation for function f(x) definedon -L to L; briefmention of convergence issues; real and complex forms; differentiation,integration; periodic extensions.

2. Fourier Transforms: Fourier series whenL → ∞ . Definition of Fouriertransform and standard examples: Gaussian, exponential and Lorentzian.Domains of application: (Time t — frequencyω), (Space x — wave vec-tor k). Delta function and properties, Fourier’s Theorem. Convolutions,example of instrument resolution, convolution theorem.

3. Interference and diffraction phenomena: Huygens-Fresnel principle. Cri-teria for Fraunhofer and Fresnel diffraction. Fraunhofer diffraction forparallel light. Fourier relationship between an object andits diffractionpattern. Convolution theorem demonstrated by diffractionpatterns. Fraun-hofer diffraction for single, double and multiple slits. Fraunhofer diffrac-tion at a circular aperture; the Airy disc. Image resolution, the Rayleighcriterion and other resolution limits. Fresnel diffraction, shadow edges anddiffraction at a straight edge.


4. Lagrange Multipliers: Variation of f(x,y) subject to g(x,y) = constant im-plies grad f parallel to grad g. Lagrange multipliers. Example of quadraticform.

5. Vectors and Coordinate Transformations: Summation convention, Kro-necker delta, permutation symbol and use for representing vector products.Revision of cartesian coordinate transformations. Diagonalizing quadraticforms.

6. Tensors: Physical examples such as mass, current, conductivity electricfield.

Books: KF Riley, MP Hobson & SJ Bence,Mathematical Methods for Physicsand Engineering; a Comprehensive Guide, Cambridge University Press, 1997.

HD Young & RA Freedman,University Physics, 11th Edition, Pearson.

Assessment:1 hour exam (85%); assessed work (15%).

Lecturer: Mark Newton.

PX264 Term 2Physics of Fluids 7.5 CATS

Replaces PX244.


Prerequisites: PX132 Mechanics A; PX253 Partial Differential Equations andMA231 Vector Analysis or PX260 Mathematical Methods for Physicists I.

Content: Introduction: Fluids as materials which do not support shear. Idea of aNewtonian fluid.

Equations of Motion: Hydrostatics: forces due to pressure and gravity. Hy-drodynamics: acceleration, continuity and incompressibility. Euler equation.

Streamlined Flow: Streamlines: Integrating Euler for steady flow along astreamline to give Bernoulli. Derivation of Bernoulli via conservation of energy.Applications of Bernoulli: flux through a hole, Pitot-static tube, aerofoil.

Hydrodynamics of Viscous Flow: Forces due to viscosity, Navier-Stokesequation. Derivation of Poiseuille’s formula for laminar flow between plates.

Turbulence: Laminar flow only one possibility. Turbulent slugs. Need fordimensionless number, Re, Pressure gradient as a function of Re. 2 Regimes:Physical interpretation of Re as Inertial forces/Viscous forces. Poiseuille workswhen Re small.

Irrotational Flow: Definition of vorticity and circulation. Importance of ir-rotational flow, Kelvin’s circulation theorem. Examples ofirrotational flow: uni-form flow, flow past a cylinder. Derivation of lift on thin aerofoil, as example forMagnus Effect. Circulation around a cylinder. The vortex. Circulation constantround 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 dragcrisis.

Leads to: PX350 Weather and the Environment, MA394 Waves, MA3D1 FluidDynamics.

Books: LD Landau & EM Lifshitz,Fluid Mechanics, Pergamon.DJ Tritton,Physical Fluid Dynamics, OUP.TE Faber,Fluid Dynamics for Physicists, Cambridge University Press.


Lecturer: Boris Muzykantskii.

STATISTICS

Joint degree in Mathematics and Statistics.Entry. Students who have suc-cessfully completed the first year in Maths and have taken statistics options intheir first year may apply to the Department of Statistics fortransfer to the jointdegree. Alternatively, transfer may be made at the beginning of the third year ifthe appropriate second year modules have been taken. Further information maybe obtained from the Department of Statistics.

ST217 Terms 1–2Mathematical Statistics A or A+B 12 or 24 CATSStatus: Part A isList A and Part B isList B . Students may take A or A+B.

ST217A Term 1Mathematical Statistics A 12 CATSStatus: List A for Maths.

Commitment: Three lectures per week and one tutorial per fortnight.

Prerequisites: ST111 Probability A and ST112 Probability B.

Content: This module is a key module for all students wishing to study statis-tics beyond the introductory level, and a prerequisite for all further statistics andeconometric modules.

The module develops the main ideas of mathematical statistics, with an em-phasis on probabilistic inference and the basic concept of likelihood. Topics in-clude empirical probability models, random variables and expectations, the Cen-tral Limit Theorem and applications, parametric statistical models and graphicalmethods, likelihood functions, estimation and asymptoticdistributions, hypothe-sis testing and confidence intervals. Practical examples and case studies will beused to illustrate all of these topics.

This module is strongly recommended to students wishing to take opera-tional research modules, numerical business modules and any modules involving


uncertainty, whether concerned with data analysis, forecasting, finance, systemmodelling, marketing, quality management or decision making. It is essential forstudents who wish to become actuaries and highly recommended to those whowish to gain exemption from professional accountancy statistical examinations.Aims: To introduce the main ideas of mathematical statistics and how they areused in practical applications.Objectives: To understand the concept of a statistical model. To understand, useand interpret the statistical methods discussed in the module.

Leads to: ST217 Mathematical Statistics B, ST215 Forecasting and Control,ST301 Bayesian Statistics and Decision Theory, ST304 Time Series and Fore-casting, ST305 Designed Experiments, ST323 Multivariate Statistics, ST327Applied Statistical Modelling, ST329 Topics in Statistics, ST332 Medical Statis-tics, ST323 Multivariate Statistics, IB320 Simulation, IB321 Forecasting, EC3xxEconometrics modules, Postgraduate MSc in Mathematical Finance, Institute ofActuaries paper CT3.Books: For both ST217A & B:

G Casella & RL Berger,Statistical Inference, Duxbury.DS Moore & GP McCabe,Introduction to the Practice of Statistics, WH

Freeman.M.H. DeGroot, M.J. Schervish,Probability and Statistics, 3rd Ed., 2002,

Addison-Wesley.

Assessment:100% by examination in Week 1 of Term 2.Lecturer: David Firth

ST202 Term 1Stochastic Processes 12 CATSStatus: List A for Maths.Prerequisites: ST111 Probability A & B and MA131 Analysis.

Content: Loosely speaking, a stochastic or random process is something whichdevelops randomly in time. Only the simplest models will be considered in thiscourse, namely those where the process moves by a sequence ofjumps in discretetime steps. We will discuss: Markov chains, which use the idea of conditionalprobability to provide a flexible and widely applicable family of random process-es; random walks, which serve as fundamental building blocks for constructingother processes as well as being important in their own right; and renewal the-ory, which studies processes which occasionally “begin allover again”. Suchprocesses are common tools in economics, biology, psychology and operationsresearch, so they are very useful as well as attractive and interesting theories.

Aims: To introduce the idea of a stochastic process, and to show howsimpleprobability and matrix theory can be used to build this notion into a beautiful anduseful piece of applied mathematics.


Objectives: At the end of the module students will:

• understand the notion of a Markov chain and how simple ideas of condi-tional probability and matrices can be used to give a thorough and effectiveaccount of discrete-time Markov chains.;

• understand notions of long-time behaviour, including transience, recur-rence and equilibrium;

• be able to apply these ideas to answer basic questions in several appliedsituations, including genetics, branching processes and random walks.

Leads to: ST333 Applied Stochastic Processes.

Assessment:80% by examination, 20% by coursework.

Lecturer: Saul Jacka.

ST213 Term 2Mathematics of Random Events 12 CATSStatus: List B for Maths.

Prerequisites: ST111 Probability A and MA131 Analysis.

Content: Imagine picking a real numberx between 0 and 1 at “random” and withperfect accuracy. Writex out in decimal form and count the number of times thedigit 5 appears in the firstn decimal places. Since there are ten possibilitiesfor each digit we would expect that roughly one tenth of the digits are 5s. Letfn(x) = (number of 5s in the firstn places)/n. Is it always true that

limn→∞

fn(x) =1

10?

To answer this question rigorously we need to develop a mathematical frame-work in which we can model the notion of picking a real number “at random”.The mathematics we need, called measure theory, permeates through much ofmodern mathematics, probability and statistics. The aim ofthe module is to pro-vide an introduction to this theory, concentrating on examples and applications.

Aims: This module aims to provide an introduction to the mathematical ideasand language underlying the notion of randomness, which permeates throughmuch of modern mathematics as well as statistics and probability theory. It willconcentrate on the applications and examples of these ideas, rather than formalproofs (these are left to the third-year Mathematics moduleMA359 MeasureTheory).

Objectives: By the end of the module the students will be able to:

• use and understand the language of measure theory and probability;

• compute the probabilities of complicated events using countable additivity;


• understand the proper formulation of the notion of statistical independence;

• understand the basic theory of integration, particularly as applied to expec-tation of random variables, and be able to compute expectations from firstprinciples in simple cases;

• understand and identify convergence in probability and almost sure conver-gence of sequences of random variables, and use and justify convergencein the computation of integrals and expectations.

Leads to: ST318 Probability Theory, MA359 Measure Theory

Assessment:80% by examination, 20% by coursework.

Lecturer: Anastasia Papavasiliou

ST217B Term 2Mathematical Statistics B 12 CATSStatus: List B for Maths.

Prerequisites: ST217 Mathematical Statistics A.

Commitment: 3 lectures per week; 1 tutorial per fortnight.

Content: This module builds on Mathematical Statistics A to study theinterrela-tionship between unknown quantities, enabling better predictions and decisions.The main topics covered are: Bivariate and multivariate distributions, condition-al expectations, the multivariate normal distribution, likelihood ratio and relatedhypothesis tests, statistical concepts and techniques formulti-parameter models,the linear statistical model, inference for model parameters, residuals and theanalysis of variance.

Aims: To review, expand and apply the ideas from ST217A (Mathematical Statis-tics A). In particular to analyse interrelationships between unknowns such as ran-dom variables, rather than just one unknown at a time.

Objectives: After completing this module, students should be able to do thefollowing:

• Quote and prove important simple results such as propertiesof conditionalexpectations and variances, least squares estimates, and others related tothe syllabus.

• Know and understand more advanced results such as asymptotic proper-ties of likelihood ratios, and some simple formulae appearing in multipleregression and analysis of variance.

• Apply their knowledge to derive estimators, hypothesis tests etc. in unfa-miliar situations.

• Apply theoretical results when analysing data, and discussthe results ob-tained.


Books: G Casella & R L Berger,Statistical Inference, Duxbury.Morris H DeGroot,Probability & Statistics, Addison-Wesley.D S Moore & G P McCabe,Introduction to the Practice of Statistics, W H

Freeman.

Leads to: ST301 Bayesian Statistics and Decision Theory, ST304 Time Seriesand Forecasting, ST305 Designed Experiments, ST323 Multivariate Statistics,ST327 Applied Statistical Modelling, ST329 Topics in Statistics, ST332 Medi-cal Statistics, proposed new module to cover the actuarial syllabus, IB320 Sim-ulation, IB321 Forecasting, EC306 Econometric Theory, EC322 Topics in Ap-plied Econometrics, other Econometrics modules, and Institute of Actuaries pa-per CT3.

Assessment:100% by examination.

Lecturer: Jim Smith

WARWICK BUSINESS SCHOOLStudents intending to transfer at the end of the second year to the joint degree

Mathematics and Business Studiesrun by the Warwick Business School shouldnote that a second class mark (at least 50%) in one of the options IB217 Starting aBusiness or IB206 Introduction to Business Studies is a formal prerequisite (seeWhite PYDC).

IB109 Terms 1–2Foundations for Accounting and Finance 24 CATS


Commitment: Two one-hour lectures and a one-hour seminar each week.

Content: The syllabus comprises:

1. balance sheets, profit and loss accounts and cash flow statements;2. accounting conventions and creative accounting;3. forecasting financial statements;4. financial statement analysis;5. cost behaviour;6. cost tracing (direct and indirect);7. product costing;8. budgeting;9. elements of finance.

Aims: This module is designed to provide students with a broad introduction toaccounting and finance, with a user’s or manager’s perspective rather than that ofan accounts preparer or specialist.


Objectives: On completion of this module students will be able to:

1. understand the relation between the principal financial statements;2. understand the major assumptions and limitations employed in convention-

al financial reporting;3. forecast financial statements for simple cases and adjustfinancial state-

ments for transactions;4. understand the interpretation of financial statements, the analysis of prof-

itability, of solvency and gearing;5. understand cost behaviour and develop product costs under competing as-

sumptions;6. understand the construction and use of budgets inside organisations;7. value investments and capital projects under the certainty case;8. understand the trade-offs between risk and return;9. understand the principles of valuation of basic instruments of debt and eq-

uity;10. identify the issues in setting a rate of return and how this is influenced by

borrowing policy.

Books: McLaney & Atrill, Accounting: An Introduction.

Assessment:Two-hour examination (80%), test (20

Lecturer: Louise Gracia.

IB207 Term 1Mathematical Programming II 12 CATS


Commitment: One two-hour lecture and one one-hour seminar each week.

Prerequisites: IB104 Mathematical Programming I.

Content: This module includes coverage of theoretical and practicalaspects ofmathematical programming. In particular it covers linear programming problemswith integer variables; the branch and bound algorithm; dynamic programming;network optimisation including project management problems; stochastic linearprogramming; convex sets and functions and their role in optimisation; simpleoptimality conditions for non-linear programming problems; the use of spread-sheets for the solution of optimisation problems.

Leads to: IB352 Mathematical programming III.

Assessment:Two-hour open book examination plus 15 minutes reading time(70%), assessed exercise (30%).

Lecturer: Nalan Gulpinar


IB211 Term 2Simulation 12 CATSStatus: List B . Students taking this module may not later take the third yearcourse IB320 Simulation.

Commitment: Eight 2 hour lectures and 1 tutorial per week for 7 weeks.

Content: Topics covered will be: introduction to simulation methods, the dis-crete-event simulation method, software for discrete-event simulation (with useof a specific package e.g. Simul8 or Witness), performing a simulation study(conceptual modelling, data collection and analysis, experimentation and verifi-cation and validation).

The tutorials provide the opportunity for supervised exercises and help stu-dents develop their own computer based simulation programmes.

Books: Robinson, S.Simulation: The Practice of Model Development and Use,Wiley, 2003.

Assessment:100% assessed. A report on the development of a computer modelfollowed by suitable experiments.

Tutor: Ruth Davies

IB3A7 Term 2The Practice of Operational Research 12 CATS

Status: List B . Students can take IB3A7 The Practice of Operational Researchin their second or third years.

This module replaces IB212 and IB317 Nature and Method of OperationalResearch.

Commitment: 9 1 hour lectures and 9 seminars/tutorials.

Content:

• The nature and methods of operational research (OR).

• Models and modelling in OR studies.

• Issues in problem structuring and data collection.

• OR model validation and verification.

• Multi-methodology.

Books: Daellenbach, H.G. and McNickle D.C.Management Science: decisionmaking through systems thinking, Palgrave MacMillan (2005).

Mitchell, G. The Practice of Operational Research, Wiley (1993).Pidd, M. Tools forThinking: modelling in Management Science (2nd ed.),

Wiley (2003).Rivett, B.H.P.The Craft of Decision Modelling, Wiley (1994).


Rosenhead, J. and Mingers, J.Rational Analysis for a Problematic WorldRevisited: problem structuring methods for complexity, uncertainty and conflict(2nd ed.), Wiley (2001).

A reading pack will be provided.

Assessment:3 hour exam

Lecturer: Alberto Franco

IB206 Term 3Introduction to Business Studies 6 CATSStatus: List B for Maths.

Commitment: Three one-hour lectures per week.

Content: There are four main subject topics:

• marketing;

• human resources;

• accounting & finance;

• operations management.

Objectives: The module is intended as a general introduction to key disciplinesin business and management.

Book: A good up-to-date text is:R Pettinger,Introduction to management, 1994.An additional text is:EC Eyre,Mastering Basic Management, MacMillan Master Series, 1994.Additional materials and some cases and exercises will be given out during

the module. These will form the basis of the module for revision and assessmentpurposes.

Assessment:Two-hour examination (100%). The examination will consistof amultiple choice question paper together with extended questions.

Lecturer: Jonathan Freeman.

Documents

Useful Dates and Deadlines - Chicks Dig X-Pecclux.x-pec.com/files/mathstuff/1styear/PYDC2006-2007/green.pdf · follow the module but are strongly encouraged to do some background