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Mathematical Sciences

This edition of the University of Nottingham Catalogue of Modules went to press on 7th September 2011. It wasderived from information held on the database. The Catalogue is also published on the Web athttp://winster.nottingham.ac.uk/modulecatalogue/. Circumstances may arise which cause a module to be modifiedor withdrawn and the database will be updated to reflect this. Thus, if you find a discrepancy between theinformation printed here and that published on the Web, you should regard the latter as definitive.

Autumn Semester

Level 1G11FPM Foundations of Pure Mathematics

Credits 10 Level 1

Target students Single Honours and Joint Honoursstudents from the School of Mathematical Sciences.Includes 'study abroad'

Semester Autumn

Co-requisite

Code TitleG11ACF Analytical and Computational Foundations

Description This module provides a foundation for allsubsequent modules in Pure Mathematics. The first sectiongives an informal introduction to some basic countingprinciples. The most important number systems areintroduced. All Pure Mathematics is written in the languageof sets, functions and relations and a large part of themodule is devoted to gaining familiarity with both readingand writing this language. Along the way, a variety ofuseful and interesting facts will be discussed. While themodule will be fairly informal in style, it will include formalproofs and students will be given practice in writing proofsthemselves. Topics to be covered will include:

counting problems, binomial coefficients;the language of set theory;relations and functions;countable and uncountable sets.

Method and frequency of class:

Activity DurationNo. ofSessions

Lecture 1 2hr0minper wk.

Workshop 1 1hr0minper wk.

Two lectures per week. A third hour will be timetabled andused, normally fortnightly, for problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment

Assessment Type Requirements

Exam 1 (90%) 2-hour written examination

Coursework 1 (5%)

Coursework 2 (5%)

Dr JPH ZachariasConvenor

G11PRB Probability

Credits 10 Level 1

Target students Single Honours and Joint Honoursstudents from the School of Mathematical Sciences.Mathematical Physics students.Includes 'study abroad'

Semester Autumn

Co-requisite


G11CAL Calculus

G11LMA Linear Mathematics

Description This module provides an introduction toprobability by developing a mathematical framework forthe logic of uncertainty. The language of set theory is usedto describe random events. An axiomatic definition ofprobability is introduced and used as a basis fordeveloping material such as conditional probability,independence and Bayes' Theorem. Random variables areintroduced, including definitions and manipulationsinvolving mass, density and distribution functions, andexpectation and variance. Standard discrete andcontinuous random variables are presented. A number ofadditional topics are considered, such as sums of randomvariables, simple transformations of random variables,bivariate discrete random variables, and an introduction tothe central limit theorem. This module is a prerequisite forG12PMM and G11STA.





Two lectures per week. A third hour will be timetabled andused, normally fortnightly, for problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (90%) 2 hour exam

Coursework 1 (5%)

Coursework 2 (5%)

Professor FG BallConvenor

HG1DMA Discrete Mathematics and its Applications

Credits 10 Level 1

Target students Students with home School outside theFaculty of Engineering

Semester Autumn

Description This module provides an introduction to topicsin discrete mathematics. A brief description of the topicscovered is as follows:

sets and counting: this chapter will introduce the notion ofwhat a set is and some of the elementary rules of countingincluding the well-known binomial theorem. This will beapplied to questions such as: how many 4-digit evennumbers are there whose digits are all different? In howmany different ways can one arrange the letters of theword MATHEMATICS?graphs: here we study some basic graph theory includingan introduction to trees and amongst other things we willgive Euler's solution to the famous K&oumlnigsberg bridgeproblem of 1736.modular arithmetic and cryptography: in this final chapterwe first cover some elementary number theory includingthe notion of congruence and the Euclidean algorithm. Thiswill then be applied to code breaking including the RSAkey cryptosystem (which is probably what is used toprotect your credit card!). BJZ NTOO WR PWOR SJKRDTQMRE SMTL.





Two lectures per week, plus fortnightly examples classes.There will be regular formative coursework assignments.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr A VishikConvenor

HG1EM1 Environment Engineering Mathematics 1

Credits 10 Level 1

Target students BEng and MEng students in the School ofBuilt Environment within the University of NottinghamCampuses.Includes 'study abroad'

Semester Autumn

Description This module introduces the algebra ofcomplex numbers to provide a key mathematical tool foranalysis of linear mathematical and engineering problems.The complexity of solving general (large) systems ofequations is introduced and their study using matrixtechniques. The calculus of a single variable is reviewedand extended to develop techniques used in the analysis ofengineering problems:

algebra of complex numbers;matrix algebra and its applications to systems of equationsand eigenvalue problems;functions and their properties;advanced differential and integral calculus of one variable.





Tutorial classes;example classes;common recommended text to support student learning;MELEES - web-based support for modules to include copiesof key module specific learning materials and additionalsupport materials.clinic session: optional weekly one-hour sessions forstudents to obtain additional individual support.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%)

Coursework 2 (20%)

Inclass Exam 1 (10%)

Professor Y FyodorovConvenor

HG1M01 Calculus and its Applications

Credits 10 Level 1

Target students BSc, BA students from the BusinessSchool or other Schools outside Science or Engineering.Includes 'study abroad'

Semester Autumn

Description This module provides key mathematicalanalysis and tools for advanced modelling for a wide rangeof applications used in business, finance and economics.Elementary calculus of a single variable is reviewed andextended and used to give insight to modelling throughuse of first-order differential equations. Differentialcalculus of functions of several variables is introduced andtheir applications for multi-dimensional models.





Seminar 1 1hr0minper wk.

Each week there will normally be two lecture hours and aseminar hour of worked examples or tutorial problemsupport.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (80%) 2-hour written examination - toinclude a computer-marked

multiple-choice section

Coursework 1 (10%)

Coursework 2 (10%)

Dr H SusantoConvenor

HG1M11 Engineering Mathematics 1

Credits 10 Level 1

Target students BSc, BEng and MEng students studying anEngineering course within the University of NottinghamCampuses.Includes 'study abroad'

Semester Autumn

Description This module introduces the algebra ofcomplex numbers to provide a key mathematical tool foranalysis of linear mathematical and engineering problems.The complexity of solving general (large) systems ofequations is introduced and their study using matrixtechniques. The calculus of a single variable is reviewedand extended to develop techniques used in the analysis ofengineering problems:

algebra of complex numbers;matrix algebra and its applications to systems of equationsand eigenvalue problems;functions and their properties;advanced differential and integral calculus of one variable.





Tutorial classes;example classes;common recommended text to support student learning;MELEES - web-based support for modules to include copiesof key module specific learning materials and additionalsupport materials.clinic session: optional weekly one-hour sessions forstudents to obtain additional individual support.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%)


Dr M KurthConvenor

HG1MC1 Mathematics for Chemistry 1

Credits 10 Level 1

Target students BSc and MSci students in the School ofChemistry.Includes 'study abroad'

Semester Autumn

Description This module provides definition, manipulationand graphical representation of important functions. Thecalculus of one variable is reviewed and then extended todevelop techniques of differential and integral calculustogether with solution of first-order differential equations.Basic elements of probability and statistics are introduced.Examples in the context of chemistry are used throughout.Topics are:

functions of single variable;differential calculus of a single variable;integral calculus of a single variable;first-order ordinary differential equations;elementary probability and statistics.





Each week there will normally be two lecture hours and afurther hour of worked examples or problem workshops.There will be regular formative coursework assignmentsand a weekly mathematics 'clinic'.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%) Assignment


Dr A A HillConvenor

Level 2G12MAN Mathematical Analysis

Credits 10 Level 2


Semester Autumn

Prerequisite Knowledge of the basic elements ofdifferential and integral calculus, as provided by G11CAL;and the basic concepts of analysis, as provided byG11ACF.

Code Title

G11ACF Analytical and Computational Foundations

G11CAL Calculus

Description This module provides an introduction tomathematical analysis building upon the experience oflimits of sequences and properties of real numbers gainedin G11ACF and calculus studied in G11CAL. It will includelimits and continuity of functions between Euclideanspaces, differentiation and integration.





Two 1-hour lectures per week. One problem class perfortnight.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (5%)

Coursework 2 (5%)

Dr JF FeinsteinConvenor

G12VEC Vector Calculus

Credits 10 Level 2

Target students Single Honours and Joint Honoursstudents in the School of Mathematical Sciences.Mathematical Physics students.Includes 'study abroad'

Semester Autumn

Prerequisite Knowledge and understanding of vectors,calculus and limits as provided by the the coremathematics modules G11CAL, G11ACF and G11LMA..

Code Title


G11CAL Calculus


Description This module provides a grounding in vectorcalculus methods that are widely used in AppliedMathematics and Mathematical Physics. The moduleintroduces the vector differentiation operations ofgradient, divergence and curl, develops integrationmethods of scalar and vector quantities over paths,surfaces and volumes, and relates these operations toeach other via the integral theorems of Green, Stokes andGauss. The methods are then applied to the solution ofLaplace's equation under simple boundary conditions byseparation of variables. This module covers materialfundamental to applied mathematics modules at levels 2,3 and 4.






Assessment



Coursework 1 (5%)

Coursework 2 (5%)

Dr SM CoxConvenor

HG2M03 Advanced Calculus and DifferentialEquation Techniques

Credits 10 Level 2

Target students MSci, BSc, BEng and MEng students in theFaculties of Engineering or Science or the School of theBuilt Environment.Includes 'study abroad'

Semester Autumn

Description This module introduces the differentialcalculus of functions of several variables and differentialvector operators. The remaining part of the module isassociated with development of techniques for the solutionof boundary and initial value problems for ordinarydifferential equations. The module will cover:

differential calculus of functions of two variables;ordinary differential equations;basic Laplace transform techniques;introduction to Fourier series.





Each week there will normally be two lecture hours and anhour of worked examples or a tutorial/problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%)


Dr S GnutzmannConvenor

HG2M13 Differential Equations and Calculus forEngineers

Credits 10 Level 2

Target students BSc, BEng and MEng students in theFaculty of Engineering and the School of the BuiltEnvironment.Includes 'study abroad'

Semester Autumn

Description The majority of the module is concerned withproviding techniques for solving selected classes ofordinary differential equations (ODEs) relevant to theanalysis of engineering topics. This module also providesthe basic calculus to help analyse engineering problems intwo- or three-dimension and special solutions of partialdifferential equations relevant to engineering applications.The module will cover:

ordinary differential equations;Fourier series;vector calculus;partial differential equations;multiple integrals;Laplace transform techniques.





Each week there will normally be two lecture hours and anhour of worked examples or a tutorial/problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (5%)

Coursework 2 (5%)

Dr S HibberdConvenor

HG2ME1 Mathematical Techniques for Electrical andElectronic Engineers 1

Credits 10 Level 2

Target students BEng and MEng students in the School ofElectrical and Electronic Engineering.

Semester Autumn

Description The solution of the equations arising from themathematical modelling of engineering problems mayrequire special analytical techniques or may require theapplications of numerical methods to solve them. Thismodule presents advanced mathematical approaches tosuch problems. Problems where there is a degree ofuncertainty may need to be modelled using the probabilitytheory developed in this module. The module also providesthe basic calculus to help analyse engineering problems intwo and three dimensions. The module topics are:

second-order ordinary differential equations;numerical techniques for ordinary differential equations;Laplace transform techniques;Fourier transformsvector calculus;probability theory.





Each week there will normally be two one-hour lecturesessions and an hour session of worked examples or atutorial problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (75%) 2-hour written exam

Coursework 1 (15%)

Coursework 2 (10%)

Dr A A HillConvenor

Level 3G13AQT Advanced Quantum Theory

Credits 20 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Mathematical Physicsstudents. Available to JYA Erasmus students.Includes 'study abroad'

Semester Autumn

Prerequisite Students should be familiar with the basicoperations of vector calculus from G12VEC, such as theform of the Laplacian in spherical polar coordinates, withsecond order linear differential equations from G12DEF,and with the basic formalism of quantum mechanics asdescribed in G12IMP.

Code Title


G12DEF Differential Equations and Fourier Analysis

G12IMP Introduction to Mathematical Physics

Description This module builds on the foundations ofquantum mechanics introduced in the module G12IMP. Itfurther develops the fundamental theory so that it appliesto more general problems, such as those involving spin,and introduces key calculational approaches, such as thoseunderlying angular momentum, the hydrogen atom,scattering problems and approximation methods such asperturbation theory.The module begins with a description of the quantumtheory of angular momentum, using ladder operators andintroducing the concept of spin. The quantum theory of theHydrogen atom is then described, incorporating aspects ofangular momentum such as spin. The fundamentalformalism of quantum mechanics is set out in a moregeneral settting than considered in G12IMP, introducingconcepts such as bra-ket notation, symmetries, unitaryoperators and the Heisenberg picture. Approximationmethods such as perturbation theory and variationalapproaches are described and scattering theory isintroduced in the context of three-dimensional wavepropagation in a central potential.





One two-hour class and 2 one-hour classes per week,some of which may be used for example and/or problemclasses.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (100%) 3 hour written examination

Dr A OssipovConvenor

G13DIF Differential Equations

Credits 20 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Available toJYA/Erasmus Students.Includes 'study abroad'

Semester Autumn

Prerequisite Students should be familiar with elementarymethods for the analytical solution of ordinary differentialequations (separable equations, integrating factors,constant coefficient second order equations), and the basicideas of phase plane analysis (stationary points and theirclassification, sketching phase portraits).

Code Title

G12MDE Modelling with Differential Equations

Description Mathematical models based on systems ofordinary or partial differential equations are used in a vastrange of disciplines, ranging from classical fields such asfluid and solid mechanics to more recent applications inmathematical biology and finance. The complexity of thesemodels is often so great that numerical methods are theonly ones available to construct solutions. However, in thismodule we will learn how to make analytical progress inthe presence of a small parameter using asymptoticmethods, and to obtain qualitative information using thetechniques of dynamical systems theory. Topics willinclude:

Asymptotic expansions and order symbols.Asymptotic solutions of algebraic equations.Laplace’s method and the method of stationary phase.The method of matched asymptotic expansions.The method of multiple scales.The Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) expansion.The centre manifold theorem.Lyapunov’s theorems.Bifurcation theory for first order ODEs.Hopf bifurcations.Global bifurcations.Chaos.





One two-hour class and two one-hour classes per weektimetabled centrally, some of which may be used forexample classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (100%) One 3-hour written examination(100%).

Professor J BillinghamConvenor

G13FNT Further Number Theory

Credits 20 Level 3

Target students Single and Joint Honours students andPure Mathematics MSc students from the School ofMathematical Sciences. Available to JYA/Erasmus students.Includes 'study abroad'

Semester Autumn

Prerequisite Knowledge of the basic definitions, notionsand theorems concerning elementary number theory,including primes and primality testing, methods offactorization, modular arithmetic and congruences,Chinese Remainder Theorem, primitive roots, Euler'sphi-function as covered in G12ALN. Some basic knowledgeof complex analysis can be useful but is not essential.

Code Title

G12ALN Algebra and Number Theory

Description Number theory concerns the solution ofpolynomial equations in whole numbers, or fractions. Forexample, the cubic equationx3 + y3 = z3 with x, y, z non-zerohas infinitely many real solutions yet not a single solutionin whole numbers. Equations of this sort are calledDiophantine equations, and were first studied by theGreeks. What makes the study of these equations sofascinating is the seemingly chaotic distribution of primenumbers within the integers. We shall apply themysterious properties of the Riemann zeta-function to findout how evenly these primes are distributed in nature. Thismodule will cover some of the most important andinteresting questions addressed by classical numbertheory. It will present several methods to solveDiophantine equations including analytic methods usingzeta-functions (surprisingly calculus can be used to studynumber theory but in an indirect way), p-adic methods(these mimic the use of power series in calculus and cansolve congruence problems efficiently), Gaussian integers(using a more general type of n umbers), and quadraticreciprocity (a first glimpse of "global method" ).

Möbius inversionQuadratic reciprocityBernoulli numbersThe distribution of primesDirichlet's theorem on primes in arithmetic progressionsPythagorean triples and Gaussian integersp-adic numbers.





One two-hour class and two one-hour classes per weektimetabled centrally, some of which may be used forexamples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (80%) 2 hour 30 minute writtenexamination

Coursework 1 (10%) written coursework


Dr C WuthrichConvenor

G13GRA Graph Theory

Credits 10 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Available toJYA/Erasmus students.Includes 'study abroad'

Semester Autumn

Prerequisite Familiarity with proof and, in particular, proofby induction, such as provided by G11ACF.

Code Title


Description A ‘graph’ (in the sense used in Graph Theory)consists of vertices and edges, each edge joining twovertices. Graph Theory has become increasingly importantrecently through its connections with computer scienceand its ability to model many practical situations. Topicscovered in the module include paths and cycles, theresolution of Euler’s Königsberg Bridge Problem,Hamiltonian cycles, trees and forests, labelled trees, thePrüfer correspondence, planar graphs, Demoucron et al.algorithm, Kruskal's algorithm, the Travelling Salesman'sproblem, the statement of the four-colour map theorem,colourings of vertices, chromatic polynomial and colouringsof edges.




Two one-hour classes per week, including lectures andexamples classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr L PrevidiConvenor

G13GTH Group Theory

Credits 20 Level 3

Target students Single and Joint Honours students andPure Mathematics MSc students from the School ofMathematical Sciences. Available to JYA/Erasmus students.

Semester Autumn

Prerequisite Knowledge of the basic definitions, notionsand theorems concerning groups, including the notions ofcyclic, symmetric, alternating and dihedral groups, abeliangroups, subgroups, normal subgroups, quotient groups,group homomorphisms, group automorphisms and themain homomorphism theorems as covered in G12ALN.

Code Title


Description This module builds on the basic ideas of grouptheory and covers a number of key results such as thesimplicity of the alternating groups, the Sylow theorems(of fundamental importance in abstract group theory), andthe classification of finitely generated abelian groups(required in algebraic number theory, combinatorial grouptheory and elsewhere). Other topics to be covered aregroup actions, used to prove the Sylow theorems, andseries for groups, including the notion of solvable groupsthat will be used in Galois theory.






Assessment



Dr K ArdakovConvenor

G13INF Statistical Inference

Credits 20 Level 3


Semester Autumn

Prerequisite Students should have a basic knowledge ofstatistical models; likelihood; parameter estimation,including maximum likelihood estimation; confidenceintervals; and hypothesis testing, including the concepts ofsize and power of a lest. They should also have somefamiliarity with the notions of conditional probability andconditional probability densities.

Code Title

G12SMM Statistical Models and Methods

Description This module is concerned with the two maintheories of statistical inference, namely classical(frequentist) inference and Bayesian inference. Theclassical inference component of the module builds on theideas of mathematical statistics introduced in G12SMM.Topics such as sufficiency, estimating equations, likelihoodratio tests and best-unbiased estimators are explored indetail. There is special emphasis on the exponential familyof distributions, which includes many standarddistributions such as the normal, Poisson, binomial andgamma. In Bayesian inference, there are three basicingredients: a prior distribution, a likelihood and aposterior distribution, which are linked by Bayes' theorem.Inference is based on the posterior distribution, and topicsincluding conjugacy, vague prior knowledge, marginal andpredictive inference, decision theory, normal inversegamma inference, and categorical data are pursued.Common concepts, such as likelihood and sufficiency, areused to link and contrast the two approaches to inference.Students will gain experience of the theory and conceptsunderlying much contemporary research in statisticalinference and methodology.





Two one-hour and one two-hour classes per weektimetabled centrally, some of which may be used forexample classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Professor ATA WoodConvenor

G13MMB Mathematical Medicine and Biology

Credits 20 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. MSc students inMathematical Medicine and Biology. MSc students inScientific Computation with Mathematical Medicine andBiology. Available to JYA/Erasmus students.Includes 'study abroad'

Semester Autumn

Prerequisite G12MDE is a pre-requisite for BSc andMMaths students. Students should be familiar with mainconcepts and techniques of modelling with differentialequations. They should be able to use linearisation andphase plane techniques to analyse systems of ordinarydifferential equations;(and to solve the partial differentialequation for heat transfer/diffusion in one spacedimension.

Code Title


Description Mathematics can be usefully applied to a widerange of applications in medicine and biology. Withoutassuming any prior biological knowledge, this moduledescribes how mathematics helps us understand topicssuch as population growth, biological oscillations andpattern formation, the spread of disease, and the growthof tumours. There is considerable emphasis on modelbuilding and development. Specific topics covered include:

Model building: scaling and nondimensionalisation,reaction kinetics, conservation laws, principle of massbalance;ODE models: single and multiple species populationmodels, simple biochemical reaction networks,epidemiology;Delay-differential equations;PDE models: reaction-diffusion systems, travelling waves,pattern formation, tumour growth.






Assessment



Dr M OwenConvenor

G13MTS Metric and Topological Spaces

Credits 20 Level 3

Target students Single and Joint Honours students andPure Mathematics MSc students from the School ofMathematical Sciences. Students of Mathematical Physics.Available to JYA/Erasmus students.Includes 'study abroad'

Semester Autumn

Prerequisite Students should be familiar with mainconcepts, theorems and examples of real valued analysis,such as closed/open intervals, continuity of real-valuedfunctions, the Intermediate Value Theorem, and theHeine-Borel Theorem; they should also be familiar with thenotion of n-dimensional real linear space and Euclideandistance.

Code Title

G12MAN Mathematical Analysis

Description Metric space generalises the concept ofdistance familiar from Euclidean space. It provides anotion of continuity for functions between quite generalspaces. This allows us to consider fundamental conceptssuch as completeness, compactness and connectedness,and to prove key results concerning them. These in turnthrow new light on staples of real-variable theory (such asthe Intermediate Value Theorem), and set the stage formore abstract applications (for example in FunctioinalAnalysis). With topological spaces we are even able toremove the reliance on distance, placing the above ideasin a context which is much more general still. The modulecovers metric spaces, topological spaces, compactness,separation properties like Hausdorffness and normality,Urysohn's lemma, quotient and product topologies, andconnectedness. Finally, Borel sets and measurable spacesare introduced.






Assessment




Dr M GrantConvenor

G13STM Stochastic Models

Credits 20 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Only available tostudents who are not taking or have not taken G13ST2.Available to JYA/Erasmus students.Includes 'study abroad'

Semester Autumn

Prerequisite Students should be familiar with mainconcepts and examples of probability theory and Markovchains.

Code Title

G12PMM Probability Models and Methods

Description In this module the ideas of discrete-timeMarkov chains, introduced in the module G12PMM, areextended to include more general discrete-state spacestochastic processes evolving in continuous time andapplied to a range of stochastic models for situationsoccurring in the natural sciences and industry. The modulebegins with an introduction to Poisson processes andbirth-and-death processes. This is followed by moreextensive studies of epidemic models and queueingmodels, and introductions to component and systemreliability. The module finishes with a brief introduction toStochastic Differential Equations. Students will gainexperience of classical stochastic models arising in a widevariety of practical situations. In more detail, the moduleincludes:

homogeneous Poisson processes and their elementaryproperties;birth-and-death processes - forward and backwardequations, extinction probability;epidemic processes - chain-binomial models, parameterestimation, deterministic and stochastic general epidemic,threshold behaviour, carrier-borne epidemics;queueing processes - equilibrium behaviour of singleserver queues;queues with priorities;component reliability and replacement schemes;system reliability;Stochastic differential equations and Ito's lemma.





Two one-hour and one two-hour classes per weektimetabled centrally, some of which may be used forexamples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr S UtevConvenor

HG3MMM Mathematics for Engineering Management

Credits 10 Level 3

Target students BEng and MEng students in the Faculty ofEngineering.Includes 'study abroad'

Semester Autumn

Description A manager of a company is normally requiredto arrange its operations so as to maximise profit. Theseoperations must be planned within the constraints of plantcapacity, estimated sales, raw material availability, etc.The module concentrates on non-statistical operationsresearch problems such as linear programming, dynamicprogramming and nonlinear programming problems. Theformulation and solution of such management andoperations research problems will be presented.





Each week there will normally be two 1-hour lectures perweek and one 1-hour worked example or problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (5%)

Coursework 2 (5%)

Dr CD LittonConvenor

HG3MOD Advanced Mathematical Techniques inOrdinary Differential Equations for

EngineersCredits 10 Level 3


Semester Autumn

Description This module covers advanced mathematicaltechniques used to provide exact or approximate solutionsto certain classes of ordinary differential equations (ODEs).Techniques covered are:

exact solution methods for linear (non-constantcoefficient) ODEs;series method for linear (non-constant coefficient) ODEs;perturbation methods for nonlinear ODEs.





Each week there will normally be two 1-hour lectures andone 1-hour worked example or problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr GP ParryConvenor

Level 4G14ADE Advanced Techniques for Differential

EquationsCredits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Mathematical Physicsstudents.Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title


Description The development of techniques for the studyof nonlinear differential equations is a major worldwideresearch activity to which members of the School havemade important contributions. This module will cover anumber of state-of-the-art methods, both analytical andnumerical, namely:

techniques for model formulation;analytical and asymptotic procedures for the analysis ofnonlinear evolution equations;modern numerical methods, with error and stabilityanalysis, for elliptic, parabolic and hyperbolic equations.

These will be illustrated by applications in the biologicaland physical sciences.




Three 1-hour lectures per week some of which may beused for example classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr SC CreaghConvenor

G14AGE Algebraic Geometry

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences.Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title

G13RIM Rings and Modules

Description This module describes geometrical structuresusing the language of algebra. Algebraic geometry is oneof the great twentieth century achievements inmathematics. This module presents and discusses affineand projective algebraic varieties over algebraically closedfields, and the associated algebraic structures: co-ordinaterings and function fields. Various notions are illustrated onalgebraic curves, including elliptic curves. The modulefurther introduces the concept of the Zariski topology andthe spectrum of rings. The structure sheaf on the spectrumof rings and its properties are motivated and explained.Birational geometry notions are discussed.





Assessment



Dr S PumpluenConvenor

G14CO2 Functions of a Complex Variable

Credits 10 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Only available tostudents who are not taking or have not taken G14COA.Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title

G12COF Complex Functions

Description Complex analysis is one of the central areasof pure mathematics, with many remarkable, oftensurprising, theorems. Whereas the introductory moduleG12COF emphasizes applications and computationaltechniques, differentiable functions of a complex variabledisplay striking properties, and the module will investigatethese. Topics should normally include the following:

the Riemann sphere;Möbius transformations and their properties;the topological properties of analytic functions, such as thetheorem that open sets are mapped to open sets;the remarkable theorem of Picard, that a function omitting2 values and differentiable in the whole complex planemust e a constant.




Two 1-hour lectures per week some of which may be usedfor example classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Professor JK LangleyConvenor

G14COA Complex Analysis

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences.Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title


Description Complex analysis is one of the central areasof pure mathematics, with many remarkable, oftensurprising, theorems. Whereas the introductory moduleG12COF emphasizes applications and computationaltechniques, differentiable functions of a complex variabledisplay striking properties, and the module will investigatethese. Topics should normally include the following:

the Riemann sphere;Moebius transformations and their properties;sequences and series of analytic functions;the topological properties of analytic functions, such as thetheorem that open sets are mapped to open sets;the Riemann mapping theorem, which establishes theexistence of analytic mappings between regions;the remarkable theorem of Picard, that a function omitting2 values and differentiable in the whole complex planemust be a constant;the connection between Riemann’s result and Picard’s;the elements of the theory of complex dynamics, in whichfractals arise, such as the famous Julia and Mandelbrotsets. This has been a very active research area in recentyears.





Assessment




G14DGE Differential Geometry

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Mathematical Physicsstudents. MSc students on the Gravity, Particles and Fieldsprogramme.Includes 'study abroad'

Semester Autumn

Prerequisite Pre-requisites G13REL or F34AG1(co-requisite for MSc students only) and G12MAN.

Code Title


G13REL Relativity

Co-requisite

Code TitleF34AG1 Gravity

Description The modern study of general relativityrequires familiarity with a number of tools of differentialgeometry, including manifolds, symmetries, Lie Groups,differentiation and integration on manifolds. These areintroduced using examples of curved space-times whosecontext is familiar from the study of general relativity inG13REL. The presentation of geometric concepts will besignificantly more abstract and powerful than in therelativity module G13REL.





Assessment



Professor J W BarrettConvenor

G14NWA Nonlinear Waves

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Mathematical Physicsstudents.Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title


Description This module describes techniques for solvingnon-linear wave equations. The following topics arecovered:

a review of methods from linear wave theory;the mathematical theory of integrable systems and novelsolution techniques for integrable equations (Lax pairs,hierarchy of conservation laws, Backlund transforms);analysis of the effects of perturbations (such as loss termsand forcing terms) by the use of asymptotic methods andideas of energy balancing;applications of such equations to, for example, fibre-opticcables, electrical transmission lines and Josephsonjunctions;advanced applications at current research frontiers, suchas water waves, nonlinear diffusion and wave propagationin biological systems.





Assessment



Dr JAD WattisConvenor

G14PTH Probability Theory

Credits 20 Level 4

Target students Single Honours students.

Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title



Description In this module the theory of probability andrandom variables, introduced in the module G12PMM, isdeveloped further to provide a fully rigorous mathematicalframework for probability theory. The module begins withan introduction to measure theoretic probability, dealing ina rigorous manner with such topics as probability space,random variable, distribution, expectation andindependence. The concept of independence plays a largeand important part in probability theory and importantconsequences of independence, such as the Borel-Cantellilemmas, the zero-one law, the weak and strong laws oflarge numbers, and the central limit theorem are studied.This requires discussion of modes of convergence forinfinite sequences of random variables, and ofcharacteristic functions and their properties. The moduleends with an introduction to conditional expectation.Students will gain experience of several important,classical results in probability theory.





Three hours of lectures per week some of which may beused for example classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Professor PD O'NeillConvenor

G14QIS Introduction to Quantum InformationScience

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Mathematical Physicsstudents. MSc students on the Gravity, Particles and FieldsProgramme.Includes 'study abroad'

Semester Autumn

Prerequisite

Code Title


Description The paradigm of Quantum InformationScience (QIS) is that quantum devices made of systemssuch as atoms and photons, can outperform the presentday technology in key applications ranging from computingpower and communication security to precisionmeasurements. Quantum information processing and themeasurement and control of individual quantum systemsare central topics in QIS, lying at the intersection ofquantum mechanics with "classical" disciplines such asinformation theory, probability and statistics, computerscience and control engineering.This module gives an introduction to QIS, emphasising thedifferences and similarities between the classical and thequantum theories. After a short review of the necessaryprobabilistic notions, the first part introduces theoperational framework of quantum theory involving thefundamental concepts of states, measurements, quantumchannels, instruments. This includes some of theinfluential results in the field such as entanglement andquantum teleportation, Bell's theorem and the quantumno-cloning theorem. The second part covers at least twotopics from: quantum Markovian evolutions, quantumstatistics, continuous variable systems.






Assessment



Dr M GutaConvenor

G14VMS Variational Methods

Credits 20 Level 4

Target students This module forms part of the 60 creditcore of the MSc in Scientific Computation, and is alsoavailable to students taking any of the MSc in ScientificComputation 'with' strands. Additionally, this module willbe available to students from Part III of the MMathprogrammes offered by the School of MathematicalSciences.

Semester Autumn

Prerequisite

Code Title

G12INM Introduction to Numerical Methods

Description Partial differential equations arise in themathematical modelling of many physical, chemical andbiological phenomena, and play a crucial role in subjects,such as fluid dynamics, electromagnetism, materialscience, astrophysics and financial modelling, for example.Typically, the equations under consideration are socomplicated that their solution may not be determined bypurely analytical techniques; instead one has to resort tocomputing numerical approximations to the unknownanalytical solution.This course is devoted to a general class of numericaltechniques for determining the approximate solution ofpartial differential equations, referred to as VariationalMethods. This class of schemes includes the standardGalerkin finite element method, Petrov-Galerkin methods,finite volume methods, and boundary element methods.Here, we will provide an introduction to their mathematicaltheory, with special emphasis on theoretical and practicalissues such as accuracy, reliabili




Three one hour lectures per week, some of which may beused for examples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (100%) One 3-hour written examination.

Dr E HallConvenor

Spring Semester

Level 1G11MSS Mathematical Structures

Credits 10 Level 1


Semester Spring

Prerequisite A knowledge of basic number systems, andthe concepts of sets, functions and relations, as providedby G11FPM/G1AFPM.

Code Title

G11FPM Foundations of Pure Mathematics

Co-requisite


Description Building on G11FPM Foundation of PureMathematics, this module is an introduction to axiomaticsystems in mathematics including the basic concepts ofsome key mathematical structures from algebra (groups)and analysis (metric spaces), together with the basicarithmetic properties of integers and polynomials.






Assessment



Inclass Exam 1 (5%) Class test

Inclass Exam 2 (5%) Class test


G11STA Statistics

Credits 10 Level 1


Semester Spring

Prerequisite Familiarity with basic rules of probability,basic distributions (eg binomial and normal) and theconcept of expectation, as covered by G11PRB/G1APRB orA-level Mathematics courses with a substantial statisticalcomponent. Competence in differentiation and integration,as provided by a grade B or above in A-level Mathematics.

Code Title

G11PRB Probability

Description In this module a range of statistical ideas andskills are developed, building on the foundations ofprobability covered in G11PRB. It describes mathematicalconcepts and techniques for modelling situations involvinguncertainty and for analysing and interpreting data. Inparticular, exploratory data analysis, point estimation,confidence intervals, hypothesis testing, linear regressionand analysis of categorical data are covered. Use is madeof an appropriate statistical package to apply the principlesand methods described in the lectures. This module is aprerequisite for G12SMM.






Two 1-hour lectures per week. A third hour will betimetabled and used, normally fortnightly, for eithercomputer workshops or problems classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%) Exercise 1


Dr C J BrignellConvenor


Credits 10 Level 1

Target students BEng and MEng students in the School ofBuilt Environment within the University of NottinghamCampuses.Includes 'study abroad'

Semester Spring

Prerequisite A study of mathematics which providesfluency in the topics of algebra, geometry, functions -including logarithm and exponential, trigonometry,sequences and series, elementary differential and integralcalculus and vectors as provided by a pass in a GCEA-level Mathematics.

Code Title


Description This module introduces the techniques forsolving selected first-order and second-order differentialequations relevant to the analysis of generic engineeringproblems. The module also provides mathematical tools interms of advanced differential calculus and vectors formodelling of generic engineering situations given in termsof multi-dimensional models:

first-order ordinary-differential equations;second-order linear constant coefficient ordinarydifferential equations;differential calculus of functions of several variables;vector spaces and their applications.





Tutorial classes;example classes;common recommended text to support student learning;MELEES - web-based support for modules to include copiesof key module specific learning materials and additionalsupport materials;clinic session: optional weekly one-hour sessions forstudents to obtain additional individual support.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (5%)

Coursework 2 (5%)

Coursework 3 (20%)


Dr K SoldatosConvenor


Credits 10 Level 1

Target students BSc, BEng and MEng students studying anEngineering course within the University of NottinghamCampuses.Includes 'study abroad'

Semester Spring

Prerequisite A study of mathematics which providesfluency in the topics of algebra, geometry, functions -including logarithm and exponential, trigonometry,sequences and series, elementary differential and integralcalculus, and vectors as provided by a recent good pass ina GCE A-level Mathematics or pass in the Foundation Yearat UNUP, UNMC or UNNC, or equivalent.

Code Title


Description This module introduces the techniques forsolving selected first-order and second-order differentialequations relevant to the analysis of generic engineeringproblems. The module also provides mathematical tools interms of advanced differential calculus and vectors formodelling of generic engineering situations given in termsof multi-dimensional models:

first-order ordinary-differential equations;second-order linear constant coefficient ordinarydifferential equations;differential calculus of functions of several variables;vector spaces and their applications.





Tutorial classes;example classes;common recommended text to support student learning;MELEES - web-based support for modules to include copiesof key module specific learning materials and additionalsupport materials;clinic session: optional weekly one-hour sessions forstudents to obtain additional individual support.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (5%)

Coursework 2 (5%)


Dr G AdessoConvenor

HG1MC2 Mathematics for Chemistry 2

Credits 10 Level 1

Target students BSc and MSci students in the School ofChemistry.Includes 'study abroad'

Semester Spring

Description This module is a continuation of HG1MC1providing ancillary mathematics knowledge and skills forstudents majoring in chemistry-based courses. Complexnumber is introduced and used with a study of solutions oflinear second-order differential equations. Matrix algebrais developed to solving systems of equations and to studyeigenvalue problems. The differential calculus of severalvariables is introduced. An introduction is provided toalgebra of matrices and their applications in chemistry.Topics are:

complex numbers;solution of second-order ODEs;differential calculus of several variables;vector algebra;matrix algebra.





Each week there will normally be two lecture hours and afurther hour of worked examples or problem workshops.There will be regular formative coursework assignmentsand a weekly mathematics 'clinic'.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment






Level 2G12COF Complex Functions

Credits 10 Level 2


Semester Spring

Prerequisite Knowledge of elementary analysis, calculusof real functions, and complex numbers, as provided bythe modules listed below.

Code Title


G11CAL Calculus


Description This module provides an introduction to thetheory and applications of functions of a complex variable,using an approach oriented towards methods andapplications. The elegant theory of complex functions isdeveloped and then used to evaluate certain real integrals.Topics to be covered will include: analytic functions andsingularities; series expansions; contour integrals and thecalculation of residues; applications of contour integration.






Assessment



Coursework 1 (10%)



Credits 10 Level 2

Target students Single Honours and Joint Honoursstudents from the School of Mathematical Sciences.Mathematical Physics Students.Includes 'study abroad'

Semester Spring

Prerequisite Knowledge and understanding of calculus tothe level provided by G11CAL.

Code Title

G11CAL Calculus

Description This module is an introduction to Fourierseries and integral transforms and to methods of solvingsome standard ordinary and partial differential equationswhich occur in applied mathematics and mathematicalphysics. The module describes the solution of ordinarydifferential equations using series and introduces Fourierseries and Fourier and Laplace transforms, withapplications to differential equations and signal analysis.Standard examples of partial differential equations areintroduced and solution using separation of variables isdiscussed. The module covers material fundamental toapplied mathematics modules at levels 2, 3 and 4.





One problem class per fortnight.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (5%)

Coursework 2 (5%)

Dr R S GrahamConvenor

HG2M02 Applied Algebra

Credits 10 Level 2

Target students BSc or BA students from the BusinessSchool or other Schools outside Science or Engineering.Includes 'study abroad'

Semester Spring

Description To provide students with analytical capabilityin a range of key applied algebra techniques as typicallyused in the quantitative study of problems in business,finance and economics. The complexity of solving general(large) systems of equations is examined in terms ofmatrix techniques. Matrix algebra is extended to identifycharacteristics of matrix systems in term of eigenvaluesand eigenvectors. Techniques are developed to solvedifference equations and systems of equations subject toconstraints. Optimisation of management and operationsresearch type problems will be addressed with elementarylinear programming techniques.






Each week there will normally be two lecture hours and aseminar hour of worked examples or tutorial problemsupport.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%)

Dr L PrevidiConvenor

HG2ME2 Mathematical Techniques for Electrical andElectronic Engineers 2

Credits 10 Level 2

Target students BEng and MEng students in the School ofElectrical and Electronic Engineering.Includes 'study abroad'

Semester Spring

Description The solution of the equations arising from themathematical modelling of engineering problems mayrequire special analytical techniques or may require theapplications of numerical methods to solve them. Thismodule presents advanced differential and integralapproaches to problems involving several variables. Themodule also provides the basic vector algebra to helpanalyse engineering problems in two and threedimensions. The module topics are:

vector calculus;introduction to differential equations;numerical techniques for differential equations;multiple integrals;vector integral theorems.





Each week there will normally be two one-hour lecturesessions and an hour session of worked examples or atutorial problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (15%)

Coursework 2 (10%)

Dr E HallConvenor

HG2MPN Probabilistic and Numerical Techniques forEngineers

Credits 10 Level 2

Target students BEng and MEng students on courses inthe Faculty of Engineering or in the School of the BuiltEnvironment.Includes 'study abroad'

Semester Spring

Description The module is divided into two sections:

Numerical techniques for ordinary differential equations.Probability theory and Introduction to statistical inference.

Problems in engineering may be formulated in terms ofODEs that cannot be solved analytically. This modulestudies the use of approximate methods that can be usedto obtain numerical solutions to ODEs.Situations where there is a degree of uncertaintyassociated with quantities of engineering importance needto be analysed and quantified using data values. In thismodule the mathematical formulation of such problems interms of probability theory is developed and relatedstatistical techniques to utilise data values are introduced.





Each week there will normally be two lecture hours and afurther hour of worked examples or a tutorial class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment




Dr RH TewConvenor

HG2MPS Probabilistic and Statistical Techniques forEngineers

Credits 10 Level 2

Target students BEng and MEng students in the Faculty ofEngineering or in the School of the Built Environment.Includes 'study abroad'

Semester Spring

Description In many engineering situations it is impossibleto be in possession of precise information about allrelevant factors. In the face of such uncertainty it isnecessary to derive probabilistically based models of theproblems and to use statistical methods to interpret thesolutions. This module introduces the mathematics neededfor such situations. The module topics are:

Introduction to Data Analysis.Probability Theory.Statistical Inference.





Each week there will normally be two lecture hours and anhour of worked examples or tutorial problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%)

Dr CD LittonConvenor

Level 3G13CCR Coding and Cryptography

Credits 10 Level 3


Semester Spring

Prerequisite A first module in linear algebra, such asG11LMA, and a familiarity with mathematical prooftechniques, as in G11ACF. Some further experience ofpure mathematics (especially algebra) would be anadvantage, and some familiarity with basic modulararithmetic is helpful .

Code Title



Description This module consists of two main topics ofcoding theory: error-correction codes and cryptography.In digit transmission (as for mobile phones), noise thatcorrupts the message can be very harmful. The idea oferror-correcting codes is to add redundancy to themessage so that the receiver can recover the correctmessage even from a corrupted transmission. The modulewill concentrate on linear error-correcting codes (such asHamming codes), where encoding, decoding and errorcorrection can be done efficiently through syndromelook-up tables. We will also discuss cyclic codes, which arethe ones most frequently used in practice.In cryptography, the aim is to transmit a message suchthat an unauthorised person cannot read it. The messageis encrypted and decrypted using some method, called acipher system.There are two main types of ciphers: private and publickey ciphers. Among the private key ciphers, we willdiscuss basic monoalphabetic and polyalphabetic ciphers(and attacks on them), as well as the more advanced DESand AES. For public key cryptography, the module willcover RSA and Elgamal and the elementary propertiesfrom number theory needed for them. Key exchangeprotocols and digital signatures (DSA) are included.




Two one-hour lectures per week some of which may beused for example classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr C WuthrichConvenor

G13EMA Electromagnetism

Credits 20 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Mathematical PhysicsStudents. Available to JYA/Erasmus Students.Includes 'study abroad'

Semester Spring

Prerequisite The divergence and Stokes’ theorems will beused extensively as an aid to calculation and as a tool fortheoretical development.

Code Title


Description The module provides an overview ofelectrostatic, magnetostatic and electrodynamicalphenomena and their mathematical description.

The concept of an electric field is used to characterisestationary charge distributions. The effects of differentensembles of charges, configured as conductors ordielectrics, on the field are obtained microscopically andmacroscopically. The concept of a magnetic field isintroduced to characterise steady current flow, andgeneralized to account for different macroscopic magneticphenomena. The Lorentz force is introduced. The solutionsof electro- and magneto-static problems are obtained,including superposition and potential theory. The energyassociated with the fields is obtained.Time dependent problems and electromagnetic induction.Conservation of charge.Maxwell’s equations and the wave nature of light. Thebehaviour of electromagnetic fields in dielectric andconductors.Poynting’s theorem, electromagnetic momentum, Maxwellstress tensor.Electrodynamics, the behaviour of test charges inelectromagnetic fields. Exact and approximate solutions.






Assessment


Exam 1 (100%) 2.5 hour written examination

Dr KI HopcraftConvenor

G13FLU Fluid Dynamics

Credits 20 Level 3


Semester Spring

Prerequisite Understanding of inviscid fluid flow, asprovided by G12MDE (Modelling with DifferentialEquations). Thorough understanding of vector calculus, asprovided by G12VEC (Vector Calculus)

Code Title



Description The dynamics of fluids is important in manydifferent areas, including weather forecasting, engineering,and biology. This module includes solutions of the full,nonlinear equations describing fluid motion, and severalexamples of approximate solution techniques incircumstances where full analytical solutions are notavailable. Topics include:

Inviscid fluid motion and wave propagation.Understanding of and solutions to the Navier-Stokesequations.Boundary layers, jets and wakes.Slow flow.Lubrication theory.Rotating flows.





One two-hour and two one-hour sessions per weektimetabled centrally, some of which will be used forexample classes or problems classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (90%) One 3-hour written examination



Professor OE JensenConvenor

G13GAM Game Theory

Credits 10 Level 3


Semester Spring

Description Game theory contains many branches ofmathematics (and computing); the emphasis here isprimarily algorithmic. The module starts with aninvestigation into normal-form games, including strategicdominance, Nash equilibria, and the Prisoner’s Dilemma.We look at tree-searching, including alpha-beta pruning,the ‘killer’ heuristic and its relatives. It then turns tomathematical theory of games; exploring the connectionbetween numbers and games, including Sprague-Grundytheory and the reduction of impartial games to Nim.




Two one-hour lectures per week some of which may beused for example classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr M KurthConvenor

G13LNA Linear Analysis

Credits 20 Level 3

Target students Single and Joint Honours students andPure Mathematics MSc students from the School ofMathematical Sciences. Mathematical Physics students.Available to JYA/Erasmus students.Includes 'study abroad'

Semester Spring

Prerequisite Good knowledge of fundamental concepts inreal analysis, including convergence of sequences of pointsand functions, continuity, basic topological conceptsincluding open and closed subsets and sequentialcompactness of subsets of the reals and of n-dimensionalreal linear space as taught in G12MAN.

Code Title


Description This module gives an introduction into somebasic ideas of functional analysis with an emphasis onHilbert spaces and operators on them. Many concepts fromlinear algebra in finite dimensional vector spaces (e.g.writing a vector in terms of a basis, eigenvalues of a linearmap, diagonalisation etc.) have generalisations in thesetting of infinite dimensional spaces making this theory apowerful tool with many applications in pure and appliedmathematics. The module aims at a presentationassuming minimal technical background.

Recap of metric spaces and basic topological concepts;normed, Banach and Hilbert spaces, basic examples;bases in Hilbert spaces, orthonormal bases;dual spaces;linear operators and the operator norm;spectral theory;special classes of linear operators, some operator theory;compact operators;the Spectral Theorem for compact self-adjoint and normaloperators;a glimpse into more general Spectral Theorems andapplication.






Assessment


Exam 1 (90%) Three-hour written examination

Coursework 1 (10%) Written coursework

Convenor

G13MAF Mathematical Finance

Credits 20 Level 3


Semester Spring

Prerequisite Students should be familiar with mainconcepts and results of probability and Markov chains,such as conditional probabilities, conditional expectationsand central limit theorem. They should also be familiarwith the construction of Markov chains.

Code Title


Description In this module the concepts of discrete timeMarkov chains studied in the module G12PMM areextended and used to provide an introduction toprobabilistic and stochastic modelling for investmentstrategies, and for the pricing of financial derivatives inrisky markets. The probabilistic ideas that underlie theproblems of portfolio selection, and of pricing, hedging andexercising options, are introduced. These includestochastic dynamic programming, martingales andBrownian motion. The capital asset pricing model isdescribed and two Nobel Prize winning theories areobtained: the Markowitz mean-variance efficient frontierfor portfolio selection and the Black-Scholes formula forarbitrage-free prices of European type options on stocks.Students will gain experience of a topic of considerablecontemporary importance, both in research and inapplications.





Two one-hour and one two-hour classes per weektimetabled centrally, some of which may be used forexamples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr S UtevConvenor

G13NGA Number Fields and Galois Theory

Credits 20 Level 3


Semester Spring

Prerequisite Knowledge of the basic definitions, notionsand theorems concerning groups, commutative rings (inparticular fields), and polynomials as covered by G12ALN.

Code Title


Description Galois Theory concerns the study of the groupof symmetries associated with polynomial equations (inmodern language, the group of automorphisms of fieldextensions). It shows the impossibility of solving thegeneral quintic equation by radicals, and it can be used toprove that one cannot square the circle, duplicate a cube,nor trisect an angle using only ruler and compass!We start by laying a theoretical foundation to theconstruction of splitting fields, and thence the factorizationof polynomials. The Fundamental Theorem of GaloisTheory lies at the heart of the module, and shows a closeconnection between field theory and group theory: weshow how the question of solvability of a polynomial canbe completely settled using group theory. Another topicwill concern finite fields, which are of central importance inmany applications including coding and cryptography.Number fields form another large class of fields which, onthe one hand, are sufficiently simple so as to allow aninitial study by elementary methods, and, on the otherhand, are sufficiently rich in structure to represent allmajor aspects of Galois theory. Number fields will be usedfor illustration purposes throughout, and some of them,e.g. cyclotomic fields, are studied in detail. The principalobjectives of studying Galois Theory are not only itsapplications, but also the appreciation of the beauty of asingle brilliant idea.






Assessment



Dr A VishikConvenor

G13REL Relativity

Credits 20 Level 3

Target students Single and Joint Honours students fromthe School of Mathematical Sciences and MathematicalPhysics students. Available to JYA/Erasmus students.Includes 'study abroad'

Semester Spring

Prerequisite Students should be familiar with classicalmechanics, the basic formalism of Lagrangian mechanics,and elementary calculus of several variables.

Code Title


Description The module is an introduction to Einstein'stheory of special and general relativity. When velocitiesare a significant fraction of the speed of light, the conceptsof spatial distance and elapsed time need to be modified;they become relative to the observer. In this module therelativistic laws of mechanics are described in a unifiedframework of space and time and some implications, suchas Einstein’s famous equation E=mc2, are explained.Gravitational effects require that space-time is warped orcurved. The relevant mathematical machinery to describethis curvature is introduced and is used to discuss itsphysical effects. Topics covered:

Lorentz transformations.Minkowski space.Relativistic particle mechanics.Special relativity continuum mechanics.Elementary differential geometry.Newtonian gravitation.General relativity.Einstein field equations.Examples of spacetimes, including Schwarzschildgeometry.





One two-hour class and two one-hour classes per weektimetabled centrally, some of which may be used forexample classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (100%) One 3-hour written examination

Dr GA JaroszkiewiczConvenor

G13RIM Rings and Modules

Credits 20 Level 3


Semester Spring

Prerequisite Knowledge of the basic definitions, notionsand theorems concerning commutative rings, including thenotions of ideal, quotient ring, prime and maximal ideals,principal ideals, ring homomorphisms, integral domains,fields and polynomial rings, irreducibility, as covered inG12ALN.

Code Title


Description Commutative rings and modules over themare the fundamental objects of what is often referred to ascommutative algebra. Already encountered key examplesof commutative rings are polynomials in one variable overa field and number rings such as the usual integers or theGaussian integers. There are many close parallels betweenthese two types of rings, for example the similaritiesbetween the prime factorization of integers and thefactorization of polynomials into irreducibles. In thismodule, these ideas are extended and generalized tocover polynomials in several variables and power series,and algebraic numbers. Topics for this module include:

modules, noetherian modules and rings;unique factorization domains and principal ideal domains;

structure of modules over a a principal ideal domain;tensor products and multilinear maps;basic properties of free modules, flat modules;injective modules, projective modules;localization, completion, local rings, regular rings.






Assessment



Professor IB FesenkoConvenor

G13TSC Topics in Scientific Computation

Credits 20 Level 3

Target students This module is available to studentsstudying within the School of Mathematical Sciences.Available to JYA/Erasmus students.Includes 'study abroad'

Semester Spring

Prerequisite In particular, students should be familiar withMATLAB programming, iterative methods for solvingsystems of linear equations, and numerical differentiationand integration.

Code Title


Description Differential equations play a crucial role insubjects, such as fluid dynamics, electromagnetism,material science, astrophysics and financial modelling.Typically, the equations under consideration are socomplicated that their solution may not be determined bypurely analytical techniques; instead one has to resort tocomputing numerical approximations to the unknownanalytical solution. In this module we study two generalclasses of numerical techniques for determining theapproximate solution of both ordinary and partialdifferential equations: finite difference methods andspectral methods. A detailed list of topics covered by thismodule is given below.

Initial Value Problems for ODEs. One-step methods,consistency and convergence, implicit one-step method,Runge-Kutta methods, linear multistep methods, zerostability, consistency, Dahlquist’s theorems, systems ofequations.Boundary value problems. Finite difference methods forODEs, error analysis, extension to PDE problems.Introduction to Spectral Methods. Fourier series, the fastFourier transform, second-order PDEs, Chebyshevmethods.Eigenvalues and Eigenvectors for Symmetric Matrices. Thecharacteristic polynomial, Jacobi’s method, Householder’smethod, eigenvalues of a tridiagonal matrix, the QRalgorithm, inverse iteration for eigenvectors, the Rayleighquotient, perturbation analysis.Krylov Subspace Methods. The conjugate gradient method,preconditioning.





Two one-hour lectures and one two-hour computerworkshop per week timetabled centrally, some of whichmay be used for examples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (70%) One 2.5-hour written examination

Coursework 1 (10%) Assessed piece of courseworkinvolving the understanding,

implementation and application ofnumerical algorithms outlined in the

module within MATLAB.







Dr M KurthConvenor

G13TST Topics in Statistics

Credits 20 Level 3


Semester Spring

Prerequisite Students should be familiar with definitions,concepts and theorems relating to univariate distributions,such as the normal and chi-squared distributions;elementary matrix algebra and operations; momentgenerating functions, parameter estimation and hypothesistesting, such as the likelihood ratio test; linear regressionand analysis of variance models.

Code Title


Description In this module three distinct topics instatistics will be introduced. These topics will build on thetheory and methods introduced in the module G12SMM. Itis anticipated that the topics to be covered will besequential analysis, multivariate analysis and designedexperiments. These topics may be briefly summarised asfollows:

sequential analysis - hypothesis testing for situationswhere the data are collected sequentially over time, eg ona factory production line;multivariate analysis – investigation of the structure ofmultivariate data using principal components analysis,inference based on vectors of observations, andclassification using discriminant analysis;designed experiments – methods for planning andanalysing carefully designed experiments for investigatingrelationships between factors and a response.

An equal amount of time will be spent on each topic.





Four hours of lectures per week, some of which may beused as problem classes, examples classes or computerworkshops.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Professor FG BallConvenor

HG3MCE Computerised Mathematical Methods inEngineering

Credits 10 Level 3

Target students BEng and MEng students in the Faculty ofEngineering. Numbers may be restricted by the availabilityof computer laboratory places. Not available to JYA andErasmus students.There is a limit to the number of places on this module. Students arereminded that enrolments which are not agreed by the Offering School inadvance may be cancelled without notice.

Semester Spring

Description This module covers a selection of numericaltechniques that can be implemented on a computer andused to evaluate problems that cannot be solvedanalytically. Topics include:

introduction to concepts of Numerical Analysis;quadrature and curve fitting;numerical linear algebra;qualitative and finite-difference methods for ODEs;numerical methods for solving PDEs.

MATLAB will be introduced within computer-basedworkshops and used to supplement and illustrate thetheoretical aspects.




Tutorial 1 1hr0minper wk.


Each week there will normally be two 1-hour lectures andone 1-hour worked example/problem class orcomputational workshop.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment





Convenor

Level 4G14AFM Advanced Fluid Mechanics

Credits 20 Level 4


Semester Spring

Prerequisite

Code Title

G13FLU Fluid Dynamics

Description In this module, you will study the flow offluids in a variety of interesting and important situations.The module will begin with a discussion of the topic ofimportant linear stability theory for fluid flow. Then twotopics of active research in fluid mechanics from the listbelow will be covered:

nonlinear hydrodynamic stability;convection, the Lorenz equations and chaos;physiological fluid dynamics: internal flows in complexgeometries;interfacial fluid mechanics: surface tension, surfactants,flows of films, jets and drops;fluid-solid interactions: moving contact lines, flows overflexible surfaces, flutter instabilities;geophysical fluid dynamics;magnetohydrodynamics.




Some of the lectures may be used for problem and/orexamples classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%) Linear Stability theory coursework

Professor J BillinghamConvenor

G14ANT Algebraic Number Theory

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences.

Semester Spring

Prerequisite

Code Title

G13FNT Further Number Theory

G13NGA Number Fields and Galois Theory

Description This module presents the fundamentalfeatures of algebraic number theory, the theory in whichnumbers are viewed from an algebraic point of view. Sonumbers are often treated as elements of rings, fields andmodules, and properties of numbers are reformulated interms of the relevant algebraic structures. This approachleads to understanding of certain arithmetical properties ofnumbers (in particular, integers) from a new point of view.For example, various diophantine problems (eg sums ofsquares) can be interpreted in terms of certain algebraicrelations and then successfully solved.The module discusses some of the central results in thetheory which were obtained by several generations ofmathematicians in the nineteenth and the first twodecades of the twentieth century: the theory of finiteextensions of the field of rational numbers, the structure oftheir rings of integers, ideal classes and uniquefactorisation, Dirichlet’s unit theorem, the splitting ofprime ideals in field extensions, and applications to variousclassical problems of number theory.





Assessment



Professor IB FesenkoConvenor

G14ASP Advanced Stochastic Processes

Credits 20 Level 4

Target students Single, Joint Honours and MSc studentsfrom the School of Mathematical Sciences.Includes 'study abroad'

Semester Spring

Prerequisite

Code Title


Description This module builds on the theory ofdiscrete-time Markov chains, introduced in the moduleG12PMM, by considering three advanced topics instochastic process theory: martingales, Brownian motionand renewal processes. The main properties of andtheorems for martingales are developed. The importanceof martingale theory is illustrated by applications to avariety of examples, including random walks, branchingprocesses and sequential analysis. Brownian motion is theclassical example of a diffusion process, i.e. a continuoustime Markov process with a continuous state space. Thebasic properties of Brownian motion are investigated andmethods for calculating probabilities and expectationsassociated with Brownian motion are described. Methodsfor calculating properties of renewal processes, such as theexpected number of renewals by a given time, are derived.Students will gain experience of a range of stochasticmodels and techniques for their analysis that areimportant for research in applied probability.





Three hours of lectures per week some of which may beused for example classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Dr H LeConvenor

G14BLH Black Holes

Credits 20 Level 4


Semester Spring

Prerequisite

Code Title

G14DGE Differential Geometry

Description General relativity predicts the existence ofblack holes which are regions of space-time into whichobjects can be sent but from which no classical objects canescape. This module uses techniques learnt in G14DGE tosystematically study black holes and their properties,including horizons and singularities. Astrophysicalprocesses involving black holes are discussed, and there isa brief introduction to black hole radiation discovered byHawking.





Assessment



Dr JMT LoukoConvenor

G14CLA Computational Linear Algebra

Credits 20 Level 4

Target students This module forms part of the 60 creditcore of the MSc in Scientific Computation, and is alsoavailable to students taking any of the MSc in ScientificComputation 'with' strands. Additionally, this module willbe available to students from Part III of the MMathprogrammes offered by the School of MathematicalSciences.

Semester Spring

Prerequisite Basic numerical analysis: rounding errors;Gaussian elimination; simple iterative methods for linearequations, Jacobi, Gauss-Siedel, SOR; methods for findingthe eigenvalues of symmetric dense matrices.Programming skills and familiarity with at least one ofMatlab, Fortran or C.

Code Title

G14VMS Variational Methods

Description Solving large, sparse, linear systems ofequations and computing the eigenvalues of large, sparsematrices are key elements in many scientific and industrialproblems. Frequently they are the most computationallyintensive part of a numerical simulation. It is veryimportant not only to choose good algorithms for thesetasks but also to make sure that they are implementedefficiently on modern computer architectures.This module will introduce students to recentdevelopments in computational linear algebra, withparticular emphasis on implementation issues relevant tomodern high performance parallel computers.A detailed list of key topics covered by this module isgiven below:

Review of basic linear algebra and numerical analysis.Norms, sensitivity analysis, rounding errors.Solving sparse linear equations. Direct methods. Iterativemethods, including Krylov subspace methods andmultilevel methods. Preconditioning.Computation of eigenvalues. Symmetric eigenvalueproblems. Power method, inverse iteration. Methods forlarge sparse matrices. Arnoldi methods. Generalizedeigenvalue problems.Modern computer architectures. Vector and parallelprocessors. Hierarchical memory.Programming parallel machines. Message passing. MPI.Parallel algorithms. Optimizing performance. Parallelefficiency. Ahmdahl’s Law. Scalability.Parallel software. PLBAS, BLACS, ScaLAPACK, Trilinos,PETSc.





Two one-hour lectures per week plus one one-hourcomputer workshop.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (70%) One 2.5-hour written examination

Project 1 (30%) One assessed computing project

Professor KA CliffeConvenor

G14FTA Further Topics in Analysis

Credits 20 Level 4


Semester Spring

Prerequisite

Code Title

G13MTS Metric and Topological Spaces

Description This module builds on the fundamental theoryof metric and topological spaces, as introduced inG13MTS, and develops this to some more advancedsettings. The module will include the study of the mainconcepts of mathematical analysis in these settings byinvestigating the properties of numerous examples, and bydeveloping the associated theory, with a strong emphasison problem-solving and rigorous, axiomatic proof. Themodule will also include a selection of more advancedtopics in analysis. Topics covered will include thefollowing:

normed spaces, Banach spaces and bounded linearoperators;measurable spaces and measurable functions;the Baire Category Theorem for complete metric spaces;linear functionals and dual spaces of normed spaces.

At least two of the following topics will be covered:the construction of Lebesgue measure;the Lebesgue integral and the main integral convergencetheorems;basic properties of Banach algebras and/or C*-algebras;Fréchet differentiation for maps between normed spaces.




Activities may take place every teaching week of theSemester or only in specified weeks. It is usually specifiedabove if an activity only takes place in some weeks of aSemester. Some of the lectures may be used for problemclasses or examples classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (90%) 3-hour written examination (end ofSpring Semester)

Coursework 1 (10%) One piece of coursework

Dr JF FeinsteinConvenor

G14TBM Topics in Biomedical Mathematics

Credits 20 Level 4

Target students Single and Joint Honours students fromthe School of Mathematical Sciences. Available toJYA/Erasmus students. Students taking the MSc inMathematical Medicine and Biology.Includes 'study abroad'

Semester Spring

Prerequisite Students should be familiar with the use ofdifferential equations in biology and medicine, andtechniques such as phase planes, linear stability analysis,and the analysis of travelling waves and pattern formationin partial differential equation models.

Code Title


Description This module illustrates the applications ofadvanced techniques of mathematical modelling usingordinary and partial differential equations. A variety ofmedical and biological topics are treated bringing studentsclose to active fields of mathematical research. Topics tobe investigated will be drawn from the following areas:

theoretical neuroscience;biomechanical modelling (surface tension effects in thelung, muscle mechanics and peristalsis, physiologicalflow-structure interactions;multiphase models for growing tissues);biomedical transport processes (boundary layers,facilitated transport);spiral waves in reaction-diffusion systems.





Assessment



Dr M OwenConvenor

G14TFG Time Series and Forecasting

Credits 20 Level 4

Target students Single Honours students, and studentstaking MSc in Statistics and MSc in Statistics and AppliedProbablity in the School of Mathematical Sciences.Includes 'study abroad'

Semester Spring

Prerequisite G14FOS is required as a pre-requisite forMSc students only instead of G12SMM

Code Title


Description This module will provide a generalintroduction to the analysis of data that arise sequentiallyin time. Several commonly occurring models will bediscussed and their properties derived. Methods for modelidentification for real time series data will be described.Techniques for estimating the parameters of a model,assessing its fit and forecasting future values will bedeveloped. Students will gain experience of using astatistical package and interpreting its output. The modulewill cover:

concepts of stationary and non-stationary time-series;philosophy of model building in the context of time seriesanalysis;simple time series models and their properties;the model identification process;estimation of parameters;assessing the goodness of fit;methods for forecasting;use of a statistical package.





One two-hour lecture and one one-hour lecture per weektimetabled centrally, some of which may be used forexamples classes, problem classes and/or computer labs.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%)

Coursework 2 (10%)

Dr T KypraiosConvenor

G14TNS Theoretical Neuroscience

Credits 20 Level 4

Target students MSc students in Mathematical Medicineand Biology. MMath students. Available to JYA/Erasmusstudents.Includes 'study abroad'

Semester Spring

Prerequisite Students should be familiar with the use ofFourier analysis, linear algebra, differential equations inbiology, the qualitative theory of ordinary differentialequations including phase planes, linear stability analysisand bifurcation theory.

Code Title


Description The module will provide a general introductionto theoretical neuroscience and neuronal networks usingthe techniques of dynamical systems theory. The emphasiswill be on the modern approach to modelling the singleneuron and to introduce the mathematical techniquesappropriate for analysing the behaviour of systems ofinteracting neurons. In more detail the module will cover:The single neuron:Models of the single neuron;analysis of excitable and oscillatory behaviour usingdynamical systems techniques;phase response curves and isochronal coordinates;mode-locking to periodic stimuli.Neural systems:Models of the synapse and dendrite;networks of interacting phase oscillators;firing rate models, population models and mean fielddescriptions for large networks;waves and pattern formation in neural systems.




Activities may take place every teaching week of theSemester or only in specified weeks. It is usually specifiedabove if an activity only takes place in some weeks of aSemester.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (75%) 2 hour 30 min written examination

Coursework 1 (12%) Assessed problem sheet

Coursework 2 (13%) Assessed problem sheet

Professor S CoombesConvenor

HG4MPD Mathematical Techniques in PartialDifferential Equations for Engineers

Credits 10 Level 4


Semester Spring

Description Many mathematical models developed for theanalysis of complex physical and engineering problemsrequire the solution of partial differential equations. Thismodule describes a variety of analytic techniques forsolving partial differential equations. Techniques coveredare:

characteristic methods;separation of variables;transform methods (Fourier and Laplace);similarity methods;d'Alembert's solution and Duhamel's principle.





Each week there will normally be two 1-hour lectures andone 1-hour worked example or problem class.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Inclass Exam 1 (10%) 1-hour inclass test

Dr H SusantoConvenor

Full Year

Level 0HG0FAM Foundation Algebra and Mathematical

TechniquesCredits 20 Level 0

Target students Students on Y120 or H100

Semester Full Year

Co-requisite

Code TitleHG0FCA Foundation Calculus

Description This module provides a basic course inalgebra and core mathematical techniques and skillsrequired in elementary quantitative analysis and modellingof problems in engineering and physical sciences. A keyelement is to provide basic algebra, trigonometric andgeometrical mathematical skills. Application to solving reallife problems is included with the development of selectedmathematical topics and skills. The module will cover:

algebraic manipulation;sequences and series;co-ordinate geometry;vectors;complex numbers;matrices;elementary probability and statistics.





Each week a three hour session will be used flexiblybetween lecture activities, example and supervised tutorialsession. Workshop activities will involve problem solvingexercises and assessment activities. Use will be made ofe-learning courseware, computer-assisted assessment andsoftware packages will be completed by each studentthrough directed self-study.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Assignment (15%) Workshop assignments (in-class ortake home)

Test (25%) In-module test (OMR orcomputer-based)

Dr S Barton

Dr R J Symonds

Convenor

HG0FCA Foundation Calculus

Credits 20 Level 0


Semester Full Year

Co-requisite

Code TitleHG0FAM Foundation Algebra and Mathematical Techniques

Description This module provides a basic course indifferential and integral calculus. Initially key elements ofdefinition, manipulation and graphical representation offunctions are introduced prior to establishing calculustechniques used in the analysis of problems in engineeringand physical sciences. Application to solving real lifeproblems is developed. The module will cover:

functions;techniques of differentiation;applications of differentiation;techniques of integration;applications of integration;differential equations.





Each week a three hour session will be used flexiblybetween lecture activities, example and supervised tutorialsessions. Workshop activities will involve problem solvingexercises and assessment activities. Use will be made ofe-learning courseware, computer assisted assessment andsoftware packages to be completed by each studentthrough self-directed study.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment




Test (25%) In-module tests (OMR orcomputer-based)

Mr F HobbsConvenor

HG0FNB Foundation Mathematics for BiologicalSciences

Credits 20 Level 0

Target students Foundation year students following aBiological Sciences pathway.Includes 'study abroad'

Semester Full Year

Description The module covers basic topics in algebra andcalculus together with applications to Biological Sciences:

basic algebraic manipulation;simultaneous equations and polynomials;elementary differentiation and integration (polynomial,exponential and logarithm functions);simple modelling of kinetic reactions;statistical data analysis;further differentiation;further integration;solution of separable first-order ODEs;maxima and minima including elementary curve sketching;matrices;complex numbers.





Each week there will normally be a two-hour workshopincorporating theory and examples. In addition, there is aone-hour application workshop integrating the use ofmathematics in a Biological Sciences context.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%) Workshop assignments



Inclass Exam 1 (10%) Inclass test

Dr JPH Zacharias

Dr A Paulin

Convenor

Level 1G11ACF Analytical and Computational Foundations

Credits 20 Level 1


Semester Full Year

Co-requisite

Code TitleG11CAL Calculus


Description This module is one of three linked 20-credityear-long modules introducing students to a broad rangeof core mathematical concepts and techniques thatunderpin all the School of Mathematical Sciences' degreeprogrammes. It has three components.

Mathematical reasoning (the language of mathematics, theneed for rigour, and methods of proof).The computer packages MAPLE and MATLAB and theirapplications.Elementary analysis.








Two lectures per week, plus problem classes, tutorialsupport, and computer workshops.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment



Coursework 1 (10%) MAPLE Exercise 1

Coursework 2 (10%) MATLAB Exercise 2



Dr M GrantConvenor

G11APP Applied Mathematics

Credits 20 Level 1


Semester Full Year

Co-requisite


G11CAL Calculus


Description The module gives an introduction to classicalmechanics and modelling in applied mathematics andillustrates how methods learned in G11CAL and G11LMA(such as differential equations and vectors) providepowerful tools for solving real-world problems. Thismodule provides a foundation for a broad range of moreadvanced topics across applied mathematics.





Further Activity Detail: Two lectures per week. A third hourwill be timetabled and used, normally fortnightly, forproblem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (80%) 2 hours 30 mins written examination



Inclass Exam 1 (5%) In-class test 1

Inclass Exam 2 (5%) In-class test 2

Dr SM CoxConvenor

G11CAL Calculus

Credits 20 Level 1


Semester Full Year

Co-requisite



Description In the autumn semester the moduleintroduces the basic elements of differential and integralcalculus of functions of one variable. The syllabusconsolidates A-level work and, to some extent, goesbeyond. Basic concepts of functions, limits, continuity,differentiation, integration are studied and applied todifferential equations, Taylor series, numerical integrationand Laplace transforms.In the spring semester the calculus of functions of severalvariables is studied and topics include partial derivatives,chain rules, the vector operator grad, Lagrange multipliersand multiple integrals.The focus is on the development of skills and confidence inapplying the methods of calculus. The module does,however, illustrate the need for mathematical rigour.




Two lectures per week plus problem classes and tutorialsupport.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment





Dr PC MatthewsConvenor


Credits 20 Level 1


Semester Full Year

Co-requisite


G11CAL Calculus

Description The module introduces students to manyconcepts and techniques of mathematics that will be usedin subsequent modules. Firstly the basic concepts ofcomplex numbers, vector algebra and matrix algebra areestablished. Then these ideas are extended to vectorspaces, linear transformations and inner product spaces.Throughout the emphasis is on developing techniques thatare widely applicable.




Two lectures per week plus problem classes and tutorialsupport.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment





Dr G Adesso

Dr A Paulin

Convenor

HG1CLA Calculus and Linear Algebra

Credits 20 Level 1

Target students Students on the Natural SciencesProgramme.Includes 'study abroad'

Semester Full Year

Description The module consolidates core GCEmathematical topics in the differential and integral calculusof a function of a single variable and used to solving someclasses of differential equations. Basic theory is extendedto more advanced topics in the calculus of severalvariables. In addition, the basic concepts of complexnumbers, vector and matrix algebra are established andextended to provide an introduction to vector spaces. Anemphasis in the module is to develop general skills andconfidence in applying the methods of calculus anddeveloping techiniques and ideas that are widelyapplicable and used in subsequent modules. Major topicsare:

differential and integral calculus of a single variable;differential equations;differential calculus of several variables;multiple integrals;complex numbers;matrix algebra;vector algebra and vector spaces.





Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (10%) 1 hour 30 minutes writtenexamination

Coursework 1 (5%)


Coursework 2 (5%)

Coursework 3 (5%)

Coursework 4 (5%)

Dr M KurthConvenor

HG1FNC Foundation Mathematics

Credits 20 Level 1

Target students First-year students in the Faculty ofScience without a recent pass in A-level Mathematics orequivalent.Includes 'study abroad'

Semester Full Year

Description The module covers basic topics in algebra andcalculus together with applications to chemistry:

basic algebraic manipulation;simultaneous equations and polynomials;elementary differentiation and integration (polynomial,exponential and logarithm functions);simple modelling using calculus;statistical data analysis;further differentiation;further integration;solution of separable first-order ODEs;maxima and minima including elementary curve sketching;matrices;complex numbers.





Each week there will normally be two lecture classesincorporating theory and examples. In addition, there is aone-hour application workshop integrating the use ofmathematics in Chemistry.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment






Inclass Exam 1 (10%) Inclass test

Dr JPH Zacharias

Dr A Paulin

Convenor

HG1MPA Mathematics for Physics and Astronomy

Credits 20 Level 1

Target students BSc and MSci students within the Schoolof Physics and Astronomy.Includes 'study abroad'

Semester Full Year

Description A selection of mathematical techniques thatare useful for analysing physical behaviour. The moduletopics are:

complex numbers;calculus of a single variable (differentiation andintegration);plane geometry and conic sections;ordinary differential equations;calculus of several variables (partial derivatives and doubleintegrals);matrices and matrix algebra.





Use will be made of duplicated notes, handouts andproblem sheets. Self-study will be supported by access toE-based materials.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (10%) Mid-session 1 hour writtenexamination

Coursework 1 (5%) Example problems






Level 2G12ALN Algebra and Number Theory

Credits 20 Level 2

Target students Single Honours and Joint Honours fromthe School of Mathematical Sciences. Available toJYA/Erasmus students.Includes 'study abroad'

Semester Full Year

Prerequisite

Code Title



G11FPM Foundations of Pure Mathematics

G11MSS Mathematical Structures

Description

Building upon the core material of G11LMA and G11ACFand further basic pure mathematics concepts introduced inG11FPM and G11MSS, this module will develop in moredetail the fundamental concepts in algebra such as groupsand rings; the fundamental concepts in linear algebra suchas vector spaces, linear transformations and matrices; andwill also provide an introduction to elementary numbertheory.

In algebra, basic concepts concerning groups and rings willbe developed, in particular their substructures (subgroups,subrings, ideals), their quotient structures (quotientgroups and quotient rings), and maps between them(group and ring homomorphisms). It will also be shownhow general algebraic concepts can be applied in concretesituations in linear algebra number theory.

In linear algebra, the important concepts of eigenvaluesand eigenvectors will be studied. This will lead to a studyof the minimum polynomial and diagonalisation. Quotientspaces will also be introduced.

In number theory, after a review of primes, integerfactorization and module arithmetic, the focus will be onclassical problems and modern applications also incryptography. This includes Fermat’s Little Theorem andits application to primality testing; methods offactorization; primitive roots; discrete logarithms; someclassical Diophantine equations (linear and polynomial).





One two-hour class and one one-hour class per weektimetabled centrally, one of which may be used forexamples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment





Inclass Exam 1 (20%) Class test 1

Dr M Edjvet

Dr L Previdi

Convenor


Credits 20 Level 2

Target students Single Honours and Joint Honours fromthe School of Mathematical Sciences includingMathematical Physics students. Available to JYA/Erasmusstudents.Includes 'study abroad'

Semester Full Year

Prerequisite G11ACF, G11CAL, G11LMA, G11APP orF31CO1. Knowledge of the material taught in theMathematics core Level 1 modules (G11LMA, G11CAL,G11ACF) or equivalent, and G11APP or F31CO1.

Code Title


G11CAL Calculus



Co-requisite

Code TitleG12VEC Vector Calculus


Description

This module explores the classical and quantummechanical description of motion. The laws of classicalmechanics are investigated both in their originalformulation due to Newton and in the mathematicallyequivalent but more powerful formulations due toLagrange and Hamilton. Applications are made toproblems such as planetary motion, rigid body motion andvibrating systems. Quantum mechanics is developed interms of a wave function obeying Schroedinger's equation,and the appropriate mathematical notions of Hermitianoperators and probability densities are introduced.Applications include problems such as the harmonicoscillator and a particle in a three-dimensional centralforce field. The module is the foundation for MathematicalPhysics modules available at levels 3 and 4.





One two-hour class and one one-hour class per weektimetabled centrally, some of which may be used forexamples classes and/or problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment







Dr GA Jaroszkiewicz

Dr JMT Louko

Convenor


Credits 20 Level 2

Target students Single Honours and Joint Honoursstudents from the School of Mathematical Sciences.Available to JYA/Erasmus students.Includes 'study abroad'

Semester Full Year

Prerequisite Knowledge of core mathematical concepts,methods and techniques as taught in G11ACF, G11CALand G11LMA.

Code Title


G11CAL Calculus


Description

This module introduces basic techniques in numericalmethods and numerical analysis which can be used togenerate approximate solutions to problems that may notbe amenable to analysis. Specific topics include:

MatlabIterative methods for nonlinear equations (simpleiteration, bisection, Newton, convergence);Discussion of errors (including rounding errors);Gaussian elimination, matrix factorisation, and pivoting;Iterative methods for linear systems, matrix norms,convergence, Jacobi, Gauss-Siedel, successive overrelaxation;Polynomial Interpolation;Orthogonal polynomials;Numerical Integration;Numerical Differentiation





Three one-hour classes per week timetabled centrally ofwhich one hour per week will be used for computerpractical sessions.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment







Professor P HoustonConvenor


Credits 20 Level 2


Semester Full Year

Prerequisite Knowledge of applied mathematics ascovered in G11APP (or G11MOD) and knowledge of coremathematical concepts, methods and techniques as taughtin G11ACF, G11CAL and G11LMA.

Code Title


G11CAL Calculus


Co-requisite

Code TitleG12VEC Vector Calculus



Description The success of applied mathematics indescribing the world around us arises from the use ofmathematical models, often using ordinary and partialdifferential equations. This module continues thedevelopment of such models, building on the modulesG11CAL and G11APP. It introduces techniques for studyinglinear and nonlinear systems of ordinary differentialequations, using linearisation and phase planes. Partialdifferential equation models are introduced and analysed.These are used to describe the flow of heat, the motion ofwaves and traffic flow. Continuum models are introducedto describe the flow of fluids (liquids and gases, such asthe oceans or the Earth's atmosphere).






Assessment







Dr G TannerConvenor


Credits 20 Level 2


Semester Full Year

Prerequisite Knowledge of probability as covered byG11PRB and knowledge of core mathematical concepts,methods and techniques as taught in G11ACF, G11CALand G11LMA.

Code Title

G11PRB Probability


G11CAL Calculus


Description

In the first part of this module, the ideas of probabilityintroduced in G11PRB are extended to provide a moreformal introduction to the theory of probability andrandom variables, with particular attention being paid tocontinuous random variables. Fundamental concepts, suchas independence, conditioning, moments, jointdistributions, transformations and generating functions arediscussed in detail. This part concludes with anintroduction to some famous limit theorems and themultivariate normal distribution.

The second part of the module gives an introduction tostochastic processes, i.e. random processes that evolvewith time. The focus is on discrete-time Markov chains,which are fundamental to the wider study of stochasticprocesses. Topics covered include transition matrices,recurrence and transience, irreducibility, periodicity,equilibrium distributions, ergodic theorems, absorptionprobabilities, mean passage times and reversibility.Discrete-time renewal processes and branching processesare considered as applications. The module finishes withan introduction to simple one-dimensional random walks,including sample path diagrams, the reflection principle,recurrence and transience, first passage probabilities andarcsine laws.







Assessment



Coursework 1 (5%) Exercises 1




Dr H LeConvenor

G12PSM Professional Skills for Mathematicians

Credits 10 Level 2

Target students Single and Joint Honours taking Part I oncourses G100, G103, G104, G1H1, G1HD, G1T1, GN12and GV15. Not available to JYA/Erasmus students

Semester Full Year

Prerequisite Knowledge of core mathematical concepts,methods and techniques as taught in G11ACF, G11CALand G11LMA.

Code Title


G11CAL Calculus


Description Activities taking place include:

Employer contributions to motivate the need to developcommunication and presentation skills, teamwork andcommercial awareness;Groupwork on the two projects including posterpreparation. Projects are on non-standard open-endedmathematical topics;How to summarise a technical document as a pressrelease for a general audience (assessed coursework in theAutumn Semester);Students will have the opportunity to demonstrate skillslike planning and time-management by keeping areflective group log during the autumn semester;Group oral presentations of the second project to whichevery member of the group must contribute.





One two-hour workshop per week, timetabled centrally.Not all of the workshop time will be contact time.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Report (10%) Group reflective log

Assignment (15%) Press release

Presentation 1 (15%) Poster on Group Project I

Presentation 2 (20%) Oral Presentation Group Project II

Report 2 (40%) Written Group Report on Project II



Credits 20 Level 2


Semester Full Year

Prerequisite Knowledge of probability as covered byG11PRB, knowledge of statistics as covered by G11STAand knowledge of core mathematical concepts, methodsand techniques as taught in G11ACF, G11CAL andG11LMA.

Code Title

G11PRB Probability


G11CAL Calculus


G11STA Statistics

Co-requisite

Code TitleG12PMM Probability Models and Methods

Description

The first part of this module provides an introduction tostatistical concepts and methods. A wide range ofstatistical models will be introduced to provide anappreciation of the scope of the subject and todemonstrate the central role of parametric statisticalmodels. The key concepts of inference including estimationand hypothesis testing will be described. Special emphasiswill be placed on maximum likelihood estimation andlikelihood ratio tests. While numerical examples will beused to motivate and illustrate, the content will emphasisthe mathematical basis of statistics. Topics includemaximum likelihood estimation, confidence intervals,likelihood ratio tests, categorical data analysis andnon-parametric procedures.

The second part of the module introduces a wide class oftechniques such as regression, analysis of variance,analysis of covariance and experimental design which areused in a variety of quantitative subjects. Topics coveredinclude the general linear model, least squares estimation,normal linear models, simple and multiple regression,practical data analysis, and assessment of modeladequacy. As well as developing the theory, practicalexperience will be obtained by the use of a statisticalcomputer package.







Assessment







Dr T KypraiosConvenor

Level 3G13CMM Communicating Mathematics

Credits 20 Level 3

Target students Third-year G100/G103 in the School ofMathematical Sciences. Not available to JYA/Erasmusstudents.

Semester Full Year

Description Summary of Content: This module providesan opportunity for third-year students taking G100 andG103 to gain first-hand experience of being involved withproviding mathematical education. Students will work atlocal schools alongside practising mathematics teachers ina classroom environment and will improve their skills atcommunicating mathematics. Typically, each student willwork with a class (or classes) for half a day a week forabout sixteen weeks. Students will be given a range ofresponsibilities from classroom assistant to leading aself-originated mathematical activity or project. Theassessment is carried out by a variety of means: on-goingreflective log, contribution to reflective seminar, oralpresentation and a final written report.The closing date for applications is the first Tuesday of thesummer term and selection is made during the followingtwo weeks.




Students will spend 1) a minimum of 16 half-day workingsessions in a school with a similar amount of time beingspent on preparation ; 2) about 18 hours maintaining areflective record of their classroom role and preparing forreflective seminar; 3) a further 40 hours writing theirinterim and final reports; 4) 8 hours preparing and makingtheir presentation.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Coursework 1 (20%) On-going reflective log (350 wordsper week max)

Report (50%) Final project report (4,500 wordsmax)

Seminar (10%) Reflective seminar (1 hour)

Presentation 1 (20%) Oral presentation (20 mins),including handout

Dr GP ParryConvenor

G13MCD Modelling Chaos and Disorder

Credits 20 Level 3

Target students Single Honours and Joint Honoursstudents from the School of Mathematical Sciencesincluding Mathematical Physics students. Available toJYA/Erasmus students.There is a limit to the number of places on this module. Students arereminded that enrolments which are not agreed by the Offering School inadvance may be cancelled without notice.


Semester Full Year

Prerequisite Students should be familiar with mainconcepts of advanced classical dynamics (in theLagrangian and Hamiltonian formulation) and quantumtheory. They should be able to solve simpleone-dimensional problems such as the harmonic oscillator.Students should also be familiar with the concepts ofsymmetry and constants of motion – and the relationbetween these concepts.

Code Title


Description The module introduces and explores modelsof classical and quantum mechanical systems with randomparameters (“disorder”) or irregular dynamical patterns(“chaos”). In classical dynamics we start with simpleconcepts like random walk and diffusion. We then discussthe associated differential equations outline a modernmethods such as functional integration for the solution. Inquantum mechanics we recall Schrödinger equation andshow that the concept of functional integrals can beadopted to this framework as well. This will allow us tointroduce the semiclassical approximation and discussfundamental differences between quantum systems whoseclassical counterparts show regular behaviour(“integrability”) and those who are not regular (“hamiltonchaos”). Finally we will discuss the statistical properties ofspectra and eigenfunctions typical for regular and chaoticspectra and eigenfunctions typical for regular and chaoticquantum systems. In addition, a training session coveringthe assessment criteria for the summary and the oralpresentation and elements of good practice will beincluded.




One two-hour session per week timetabled centrally, someof which may be used for examples classes, problemclasses and presentations.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (70%) One 2 hour 30 minute writtenexamination

Report (10%) Written summary of presentation,1000 words

Presentation 1 (20%) 15 minute presentation

Professor Y FyodorovConvenor

G13MED Medical Statistics

Credits 20 Level 3

Target students Single Honours, Joint Honours and MScstudents from the School of Mathematical Sciences.Available to JYA/Erasmus students.Includes 'study abroad'

Semester Full Year

Prerequisite G12SMM or G14FOS. Knowledge of leastsquares, maximum likelihood estimation, the constructionof confidence intervals and the use of hypothesis testingas covered in G12SMM and G14FOS. Familiarity withprobability distributions and their properties, particularlythe normal and associated distributions as covered inG12PMM and G14FOS and used in G12SMM and G14FOS.

Code Title


Co-requisite

Code TitleG14FOS Fundamentals of Statistics

Description Medical statistics is one of the largest singleareas of application of statistical methodology. Initially thismodule extends the theoretical aspects of statisticalinference (first met in G12SMM or G14FOS) by developingthe theory of the generalised linear model and its practicalimplementation. Then several application areas areconsidered beginning with an examination of the designand analysis of the prospective clinical trial. Thereafter,the module extends the understanding and application ofstatistical methodology established in previous modules tothe analysis of discrete data and survival data, twofrequently occurring types of data in the medical field. Inthe module students will be trained in the use of anappropriate high-level statistical package.





In the autumn semester, one two-hour class and oneone-hour class per week timetabled centrally. In the springsemester, one two-hour class and one one-hour class perweek for the first three weeks and then one one-hour classper week until the end of the spring term. Some of theclasses may be used for examples classes, computerworkshops and problem classes.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (50%) One 1.5 hour written examination

Coursework 1 (20%) Individual coursework. Assessed inthe autumn semester (two pieces of

coursework, each worth 10%,consisting of a mixture of theoretical

questions and those involving theuse of a statistical package)

Report (25%) Group report submitted in the springsemester (about 25 pages inclusive

of figures and tables) on project

Presentation 1 (5%) Twenty minute group presentationon project in spring semester

Convenor

Level 4G14CST Computational Statistics

Credits 20 Level 4

Target students Single Honours students. MSc students.


Semester Full Year

Prerequisite

Code Title

G13INF Statistical Inference

Description The increase in speed and memory capacity ofmodern computers has dramatically changed their use andapplicability for complex statistical analysis. This moduleexplores how computers allow the easy implementation ofstandard, but computationally intensive, statisticalmethods and also explores their use in the solution ofnon-standard analytically intractable problems byinnovative numerical methods. The material builds on thetheory of the module G13INF to cover several topics thatform the basis of some current research areas incomputational statistics. Particular topics to be coveredinclude a selection from simulation methods, Markov chainMonte Carlo methods, the bootstrap and nonparametricstatistics, statistical image analysis, and wavelets.Students will gain experience of using a statistical packageand interpreting its output.




Two hours of lectures per week, although some slots willnot be used (eg at times when students are working onassessed coursework). Additional computing labs will bearranged as appropriate. Total nominal contact time is 36hours.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment





Professor PD O'NeillConvenor

G14DIS Mathematics Dissertation

Credits 40 Level 4

Target students This module is compulsory for Part III forstudents registered for the MMath degree in Mathematics(G103) or Mathematics with Engineering (G1H1). Notavailable to JYA/Erasmus students.

Semester Full Year

Description This module will consist of self-directed butsupervised study of an appropriate area of mathematicsfor the whole year. The study should result in a sustainedpiece of work assessed by an interim report, an oralpresentation and a dissertation. A list of possible topicswill be supplied by the School. Students choose a topic ofinterest to them, work under the supervision of a memberof staff, and write a dissertation on their work. Thestudents give an oral presentation of their work. Furtheradvice and information is available from the convener andwill be given to students at appropriate stages of theircourse.




Students should arrange regular meetings, normallyweekly, with their supervisor. Supervision arrangementsare flexible, but the student should expect to see thesupervisor for at least 12 hours in total. An oralpresentation training session (1 hour) is run by theconvenor in the Autumn term and the student gives apractice oral presentation in the Spring term.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Dissertation (85%) Word-processed. Normally between60 and 120 pages.

Oral (10%) Oral presentation - 15 minutes inlength

Report (5%) Progress report - word-processed.Normally between 3 and 6 pages.

Dr JPH ZachariasConvenor

G14QFT Quantum Field Theory

Credits 20 Level 4


Semester Full Year

Prerequisite

Code Title


G13AQT Advanced Quantum Theory

G13REL Relativity

Description Quantum Field Theory is the study of thequantum dynamics of relativistic particles. The modulegives the quantum description of the electrons, photonsand other elementary particles, including a discussion ofspin, and bosons and fermions. Lectures will provide anintroduction to functional integrals, Feynman diagrams,and the standard model of particle physics.





Assessment



Project 1 (30%) Presentation and essay


Professor J W BarrettConvenor

Intensive Block

Level 0HG0FAJ Foundation Algebra and Mathematical

Techniques (J)Credits 20 Level 0


Semester Jan Year

Co-requisite

Code TitleHG0FCJ Foundation Calculus (J)

Description This module provides a basic course inalgebra and core mathematical techniques and skillsrequired in elementary quantitative analysis and modellingof problems in engineering and physical sciences. A keyelement is to provide basic algebra, trigonometric andgeometrical mathematical skills. Application to solving reallife problems is included with the development of selectedmathematical topics and skills. The module will cover:

algebraic manipulation;sequences and series;co-ordinate geometry;vectors;complex numbers;matrices;elementary probability and statistics.





Three hour sessions will be used flexibly between lectureactivities, example and supervised tutorial sessions.Workshop activities will involve problem solving exercisesand assessment activities. Use will be made of e-learningcourseware, computer-assisted assessment and softwarepackages will be completed by each student through directstudy.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (60%)


Test (25%) In-module test (OMR orcomputer-based)

Dr S Barton

Dr R J Symonds

Convenor

HG0FCJ Foundation Calculus (J)

Credits 20 Level 0


Semester Jan Year

Prerequisite Grade B GCSE mathematics or equivalent.

Code Title

HG0FAJ Foundation Algebra and Mathematical Techniques (J)

Co-requisite

Code TitleHG0FAM Foundation Algebra and Mathematical Techniques

Description This module provides a basic course indifferential and integral calculus. Initially key elements ofdefinition, manipulation and graphical representation offunctions are introduced prior to establishing calculustechniques used in the analysis of problems in engineeringand physical sciences. Application to solving real lifeproblems is developed. The module will cover:

functions;techniques of differentiation;applications of differentiation;techniques of integration;applications of integration;differential equations.





Three hour sessions will be used flexibly between lectureactivities, example and supervised tutorial sessions.Workshop activities will involve problem solving exercisesand assessment activities. Use will be made of e-learningcourseware, computer assisted assessment and softwarepackages to be completed by each student throughself-directed study.Activities may take place every teaching week of the Semester or only inspecified weeks. It is usually specified above if an activity only takesplace in some weeks of a Semester

Assessment


Exam 1 (60%) 2.5 hour written examination


Test (25%) In-module tests (OMR orcomputer-based)

Mr F HobbsConvenor

Index by code

G11ACF

G11APP

G11CAL

G11FPM

G11LMA

G11MSS

G11PRB

G11STA

G12ALN

G12COF

G12DEF

G12IMP

G12INM

G12MAN

G12MDE

G12PMM

G12PSM

G12SMM

G12VEC

G13AQT

G13CCR

G13CMM

G13DIF

G13EMA

G13FLU

G13FNT

G13GAM

G13GRA

G13GTH

G13INF

G13LNA

G13MAF

G13MCD

G13MED

G13MMB

G13MTS

G13NGA

G13REL

G13RIM

G13STM

G13TSC

G13TST

G14ADE

G14AFM

G14AGE

G14ANT

G14ASP

G14BLH

G14CLA

G14CO2

G14COA

G14CST

G14DGE

G14DIS

G14FTA

G14NWA

G14PTH

G14QFT

G14QIS

G14TBM

G14TFG

G14TNS

G14VMS

HG0FAJ

HG0FAM

HG0FCA

HG0FCJ

HG0FNB

HG1CLA

HG1DMA

HG1EM1

HG1EM2

HG1FNC

HG1M01

HG1M11

HG1M12

HG1MC1

HG1MC2

HG1MPA

HG2M02

HG2M03

HG2M13

HG2ME1

HG2ME2

HG2MPN

HG2MPS

HG3MCE

HG3MMM

HG3MOD

HG4MPD

Index by title

Advanced Calculus and Differential EquationTechniques

Advanced Fluid Mechanics

Advanced Mathematical Techniques inOrdinary Differential Equations forEngineers

Advanced Quantum Theory

Advanced Stochastic Processes

Advanced Techniques for DifferentialEquations

Algebra and Number Theory

Algebraic Geometry

Algebraic Number Theory

Analytical and Computational Foundations

Applied Algebra

Applied Mathematics

Black Holes

Calculus

Calculus and its Applications

Calculus and Linear Algebra

Coding and Cryptography

Communicating Mathematics

Complex Analysis

Complex Functions

Computational Linear Algebra

Computational Statistics

Computerised Mathematical Methods inEngineering

Differential Equations

Differential Equations and Calculus forEngineers

Differential Equations and Fourier Analysis

Differential Geometry

Discrete Mathematics and its Applications

Electromagnetism

Engineering Mathematics 1

Engineering Mathematics 2

Environment Engineering Mathematics 2

Environment Engineering Mathematics 1

Fluid Dynamics

Foundation Algebra and MathematicalTechniques

Foundation Algebra and MathematicalTechniques (J)

Foundation Calculus

Foundation Calculus (J)

Foundation Mathematics

Foundation Mathematics for BiologicalSciences

Foundations of Pure Mathematics

Functions of a Complex Variable

Further Number Theory

Further Topics in Analysis

Game Theory

Graph Theory

Group Theory

Introduction to Mathematical Physics

Introduction to Numerical Methods

Introduction to Quantum InformationScience

Linear Analysis

Linear Mathematics

Mathematical Analysis

Mathematical Finance

Mathematical Medicine and Biology

Mathematical Structures

Mathematical Techniques for Electrical andElectronic Engineers 1

Mathematical Techniques for Electrical andElectronic Engineers 2

Mathematical Techniques in PartialDifferential Equations for Engineers

Mathematics Dissertation

Mathematics for Chemistry 1

Mathematics for Chemistry 2

Mathematics for Engineering Management

Mathematics for Physics and Astronomy

Medical Statistics

Metric and Topological Spaces

Modelling Chaos and Disorder

Modelling with Differential Equations

Nonlinear Waves

Number Fields and Galois Theory

Probabilistic and Numerical Techniques forEngineers

Probabilistic and Statistical Techniques forEngineers

Probability

Probability Models and Methods

Probability Theory

Professional Skills for Mathematicians

Quantum Field Theory

Relativity

Rings and Modules

Statistical Inference

Statistical Models and Methods

Statistics

Stochastic Models

Theoretical Neuroscience

Time Series and Forecasting

Topics in Biomedical Mathematics

Topics in Scientific Computation

Topics in Statistics

Variational Methods

Vector Calculus

ug mathematical sciences - mod cat sch · complex numbers to provide a key mathematical tool for...

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