Analysis 1

15 credits
Spring teaching, Year 1

Topics include: 

• Sequences: convergence, Cauchy sequences, subsequences
• Series: proof and application of convergence/divergence criteria
• Limits of functions: definitions, examples and properties
• Continuity: intermediate value theorem, uniform continuity
• Differentiability: definition, proofs of mean value theorems

Calculus

15 credits
Autumn teaching, Year 1

Topics include: functions of one real variable: graphical representation, inverse functions, composition of functions, polynomial, trigonometric, exponential and hyperbolic functions. Limits, continuity and differentiation: one-sided limits, infinite limits, algebra of limits, continuity and the intermediate value theorem, differentiation from first principles, product rule and chain rule, Rolle’s theorem, the mean value theorem and Taylor’s theorem, stationary points of a function. Integration: indefinite and definite integrals, fundamental theorem of calculus, integration by parts and integration by substitution. Solutions to first order ODEs. Manipulations with absolute values. Quadratic forms

Geometry

15 credits
Autumn teaching, Year 1

Topics include: vectors in two and three dimensions. Vector algebra: addition, scalar product, vector product, including triple products. Applications in two- and three-dimensional geometry: points, lines, planes, geometrical theorems. Area and volume. Linear dependence and determinants. Polar co-ordinates in two and three dimensions. Definitions of a group and a field. Polynomials. Complex numbers, Argand plane, De Moivre's theorem. Matrices: addition, multiplication, inverses. Transformations in R^2 and R^3: isometries. Analytical geometry: classification and properties of conics.

Introduction to Pure Mathematics

15 credits
Autumn teaching, Year 1

Topics covered include:

• Numbers: introduction of mathematical symbols, natural numbers, integers, rationals, real numbers, basic number algebra. Ordering, inequalities, absolute value (modulus), homogeneity, triangle inequality. Concept of algebraic structure, groups. Sequences, Induction Principle, Well Ordering Principle, sums, products, factorials, Fibonacci numbers, fractions.
• Irrational roots of integers, divisibility, prime numbers, Euclidean Division, highest common factor, Euclidean Algorithm, Number Theory, Atomic Property of Primes, Coprime Factorisation, Fundamental Theorem of Arithmetic, square-free numbers.
• Logic: concept of proof, logical argument, direct proof, propositional manipulation, basic logic, and, or, not, implication, contraposition, contradiction, logical equivalence, quantifiers. 
• Axiomatic set theory: Extension Axiom, equality of sets, Specification Axiom, intersection (meet), difference (take), subset, Existence Axiom, empty set, Pairing Axiom, singletons, pairs, ordered pairs, Union Axiom, cartesian products, Power Axiom, power set. 
• Counting: maps and functions, distinguished functions, injections, surjections, bijections, one-to-one correspondences, Pigeon Hole Principle, counting the power set, counting subsets of the power set, Cherry Picking, binomial coefficients, binomial formula, combinatorics, Inclusion-Exclusion formula, permutations, counting maps.
• Functions and maps: formal definition, finite and infinite sets, Peano's Axiom (Infinity Axiom/Induction Axiom), Pigeon Hole Principle revisited, counterimage, inverse functions, partial inverses, Axiom of Choice. 
• Relations: relations, equivalence relations, modular arithmetic and quotient sets, order relations, partial ordering, total ordering, linear ordering. Rigorous extension of N to Z and Q. Rings, fields. Examples.
• Real numbers: ordering and Archimedean Property of Reals, countable vs. uncountable sets, Cantor's "Diagonal".

 

 

 

 

 

 

Linear Algebra

15 credits
Spring teaching, Year 1

You will cover:

Matrices, Elementary row and column operations, Vector spaces, Linear independence, Basis and dimension, Inner products and orthogonality, Gram-Schmidt orthonormalisation process, Linear transformations, Determinants, Eigen-values and Eigen-vectors, Polynomials, Cayley-Hamilton 
theorem, Quadratic forms.

Mathematical Modelling

15 credits
Spring teaching, Year 1

Topics include: 

• Introduction to Mathematical Models
• Dimensions and units
• Population dynamics: discrete and ordinary DEs
• First order DEs: solution methods
• First order DEs: models
• Kinematics
• Dynamics and Newton's Laws
• Particle motion in three dimentions and projectiles
• Constant-coefficient 2nd order DEs
• Simple harmonic motion, damped and forced systems.

Mathematics in Everyday Life

15 credits
Autumn teaching, Year 1

This module covers:

• Money: Rule of 72, repayments, annuities, APR, compounding, present value, tax system, Student Loans.
• Differential equations: how they arise, how they can be solved. Applications include radiocarbon dating, cooling of liquids, evaporation of mothballs, escape of water down plugholes, war, epidemics, predator-prey models.
• Applications in sports and games -- tennis, rugby, snooker, darts, athletics, soccer, ranking methods.
• Business applications: stock control, linear programming, pound-cost average, hierarchies and promotions policies, the Kelly strategy.
• Voting methods and paradoxes: Arrow's Impossibility Theorem. 
• Simpson's Paradox, disease testing (false positives etc.), gambling, TV game shows.
• Mathematical essay: eg book reviews, topic descriptions.

Numerical Analysis 1

15 credits
Spring teaching, Year 1

This module covers topics such as:

Introduction to Computing with MATLAB 

• Basic arithmetic and vectors, M-File Functions, For Loops, If and else, While statements

Introduction to Numerical Analysis

• Operating with floating point numbers, round-off error, cancellation error
• Polynomial interpolation, Basic idea of interpolation, Order of approximation, Lagrangian interpolation, Runge's example, Piecewise linear interpolation
• Numerical differentiation, finite difference quotients, order of approximation
• Numerical integration, Derive standard numerical integration scheme's and analyse, using polynomial interpolation (midpoint formula, trapezoidal rule, Simpson's formula)
• Nonlinear equations, bisection method, fixed point iteration method, Newton's method/ Secant method

Analysis 2

15 credits
Autumn teaching, Year 2

Topics covered: power series, radius of convergence; Taylor series and Taylor's formula; applications and examples; upper and lower sums; the Riemann integral; basic properties of the Riemann integral; primitive; fundamental theorem of calculus; integration by parts and change of variable; applications and examples. Pointwise and uniform convergence of sequences and series of functions: interchange of differentiation or integration and limit for sequences and series; differentiation and integration of power series term by term; applications and examples. Metric spaces and normed linear spaces: inner products; Cauchy sequences, convergence and completeness; the Euclidean space R^n; introduction to general topology; applications and examples.

Calculus of Several Variables

15 credits
Autumn teaching, Year 2

Complex Analysis

15 credits
Spring teaching, Year 2

Topics covered include:

• Holomorphic functions, Cauchy's theorem and its consequences.
• Power series, integration, differentiation and analysis of convergence.
• Taylor expansions and circle of convergence.
• Laurent expansions and classification of isolated singularities.
• Residue theorem and evaluation of integrals.
• Rouche's theorem and the fundamental theorem of algebra.

Differential Equations

15 credits
Spring teaching, Year 2

Topics include:

Ordinary differential equations:

• Solution methods: Variation of the constant formula, separation of variables
• Solution of linear ODE with constant coefficients
• Lipschitz continuity
• Existence and uniqueness (Picard-Lindeloef), maximal solutions, Gronwall
• Higher order equations into system of first order
• Boundary value problems.

Partial differential equations:

• Partial and total derivatives
• First order PDEs: Method of characteristics for semilinear and quasilinear equations, initial boundary value problems.

Groups and Numbers

15 credits
Autumn teaching, Year 2

Topics include: 

• Group axioms, examples, abelian groups
• Subgroups
• Cyclic groups; Fermat's, Euler's Theorems
• Permutations: Dihedral groups, the symmetric and alternating groups
• Homomorphisms, isomorphisms
• Direct products
• Normal subgroups, quotient groups
• Conjugacy
• Group actions, orbits and Sylows' Theorems.

Introduction to Probability and Applied Analysis

15 credits
Autumn teaching, Year 2

Numerical Analysis 2

15 credits
Spring teaching, Year 2

Topics covered include:

• Linear systems (conditioning, LU factorization, basic iterative methods, convergence analysis)
• Nonlinear systems: Newton's method
• Numerical solution of differential equations:
    - finite difference methods for first and second order initial value problems
    - finite difference/element methods for one-dimensional boundary value problems

Probability and Statistics

15 credits
Spring teaching, Year 2

Topics include: 

• Descriptive Statistics: types of data, histograms, sample mean, variance, standard deviation, quantiles;
• Statistical Inference: estimation, maximum likelihood, standard distributions, central limit theorem, model validation;
• Distribution theory: Chebychev's inequality, weak law of large numbers, distribution of sums of random variables, t,\chi^2 and F distributions;
• Confidence intervals;
• Statistical tests including z- and t-tests, \chi^2 tests;
• Linear regression;
• Nonparametric methods;
• Random number generation;
• Introduction to stochastic processes.

Advanced Numerical Analysis

15 credits
Autumn teaching, Year 4

Coding Theory

15 credits
Spring teaching, Year 4

Topics include: 

• Introduction to error-correcting codes. The main coding theory problem. Finite fields.
• Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code.
• The dual code and the parity check matrix. Hamming codes. Constructions of codes.
• Weight enumerators. Cyclic codes.

Continuum Mechanics

15 credits
Spring teaching, Year 4

Topics include: 

• Kinematics: Eulerian and Lagrangian descriptions, velocity, acceleration, rate of change of physical quantities, material derivatives, streamlines.
• Deformation: stress and strain tensors, Hooke's law, equilibrium equations.
• Conservation laws for mass, momentum and energy.
• Phase/group velocities of travelling wave solutions.
• Models of fluid and solid mechanics.

Cryptography

15 credits
Autumn teaching, Year 4

Topics covered include:

• Symmetric-key cryptosystems.
• Hash functions and message authentication codes.
• Public-key cryptosystems.
• Complexity theory and one-way functions.
• Random number generation.
• Attacks on cryptosystems.
• Cryptographic standards.

Dynamical Systems

15 credits
Spring teaching, Year 3

Topics covered include:

• General dynamical systems: semiflow, stability and attraction, omega-limit set, global attractor. 
• Ordinary differential equations: Linear systems, Lyapunov function, linearized systems around fixed points, periodic orbits, Two-dimensional systems, centre manifold, bifurcation.
• Discrete systems (iterations): Linear systems, linearized systems around fixed points.

E-Business and E-Commerce Systems

15 credits
Autumn teaching, Year 4

Topics for this module include: elementary economic theory and its interaction with e-business; alternative e-business strategies, as theories and as case studies; legal and behavioural issues; marketing, branding, and customer relationship issues; software systems for e-business and e-commerce; and commercial website management.

Financial Mathematics

15 credits
Autumn teaching, Year 4

You will study generalized cash flows, time value of money, real and money interest rates, compound interest functions, equations of value, loan repayment schemes, investment project evaluation and comparison, bonds, and project writing.

Functional Analysis

15 credits
Spring teaching, Year 4

Topics on this module include: Banach spaces; Banach fixed-point theorem; Baire's Theorem; Bounded linear operators and on Banach spaces; continuous linear functionals; Banach-Steinhaus Uniform Boundedness Principle; open mapping and closed graph theorems; Hahn-Banach Theorem; compactness of sets; Hilbert spaces; orthogonal expansions; Riesz-Fischer Theorem.

Group Theory 2

15 credits
Spring teaching, Year 3

Harmonic Analysis and Wavelets

15 credits
Autumn teaching, Year 4

You will be introduced to the concepts of harmonic analysis and the basics of wavelet theory: you will discuss the concepts of normed linear spaces and Hilbert spaces, with a focus on sequence spaces and spaces of functions, most notably the space of square-integrable functions on an interval or on the real line. You will be introduced to the ideas of best approximation, orthogonal projection, orthogonal sums, orthonormal bases and Fourier series in a separable Hilbert space.

You will then apply these concepts to the concrete case of classical trigonometric Fourier series, and both Fejer's theorem and the Weierstrass approximation theorem will be proved.

Finally, you will apply the introduced concepts for Hilbert to discuss wavelet analysis for the example of the Haar wavelet and the Haar scaling function. You will be introduced to the concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function for the case of the Haar wavelet), but will also be defined in general. The concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function) will initially be introduced for the case of the Haar wavelet, but will also be defined in general.

Introduction to Mathematical Biology

15 credits
Autumn teaching, Year 4

The module will introduce you to the concepts of mathematical modelling with applications to biological, ecological and medical phenomena. The main topics will include:

• Continuous populations models for single species;
• Discrete population models for single species;
• Phase plane analysis;
• Interacting populations (continuous models);
• Enzyme kinetics;
• Dynamics of infectious diseases and epidemics.

Linear Statistical Models

15 credits
Autumn teaching, Year 4

Topics include: full-rank model (multiple and polynomial regression), estimation of parameters, analysis of variance and covariance; model checking; comparing models, model selection; transformation of response and regressor variables; models of less than full rank (experimental design), analysis of variance, hypothesis testing, contrasts; simple examples of experimental designs, introduction to factorial experiments; and use of a computer statistical package to analyse real data sets.

Measure and Integration

15 credits
Autumn teaching, Year 4

Topics for this module include: 

• Countably additive measures, sigma-algebras, Borel sets, measure spaces.
• Outer measures and Caratheodory's construction of measures.
• Construction and properties of Lebesgue measure in Euclidean spaces.
• Measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties.
• Convergence theorems: monotone convergence, dominated convergence, Fatou's lemma.
• Application of limit theorems to continuity and differentiability of integrals depending on a parameter.
• Properties of finite measure spaces and probability theory.

Medical Statistics

15 credits
Spring teaching, Year 4

Topics include: logistic regression, fitting and interpretation. Survival times; Kaplan-Meier estimate, log-rank test, Cox proportional hazard model. Designing medical research. Clinical trials; phases I-IV, randomised double-blind controlled trial, ethical issues, sample size, early stopping. Observational studies: prospective/retrospective, longitudinal/cross-sectional. Analysis of categorical data; relative risk, odds ratio; McNemar's test, meta-analysis (Mantel-Haenszel method). Diagnostic tests; sensitivity and specificity; receiver operating characteristic. Standardised mortality rates.

Multimedia Design and Applications

15 credits
Spring teaching, Year 4

Prerequisite: Java programming skills.

Computers now manipulate many more media than simple text and numbers. This module examines how modern computing systems manage, deliver and present multimedia such as audio, video, and interactive graphics. Topics include: information coding; multimedia hardware; networked multimedia; ergonomics; interface design; and multimedia applications.

Partial Differential Equations

15 credits
Autumn teaching, Year 4

Topics include: Second-order Partial Differential Equations: wave equation, heat equation, Laplace equation. D'Alembert's solution, separation of variables, Duhamel's principle, energy method, Maximum principle, Green's identities.

Perturbation theory and calculus of variations

15 credits
Spring teaching, Year 4

The aim of this module is to introduce you to a variety of techniques primarily involving ordinary differential equations, that have applications in various branches of applied mathematics. No particular application is emphasised.

Topics covered include

• Dimensional analysis and scaling:
• physical quantities and their measurement;
• dimensions;
• change of units;
• physical laws;
• Buckingham Pi Theorem;
• scaling.

• Regular perturbation methods:
• direct method applied to algebraic equations and initial value problems (IVP);
• Poincar method for periodic solutions;
• validity of approximations.

• Singular perturbation methods:
• finding approximate solutions to algebraic solutions;
• finding approximate solutions to boundary value problems (BVP) including boundary layers and matching.

• Calculus of Variations:
• necessary conditions for a function to be an extremal of a fixed or free end point problem involving a functional of integral form;
• isoperimetric problems.

Probability Models

15 credits
Autumn teaching, Year 4

Topics include: probability spaces as models of chance experiments; axioms, conditional probability; random variables, distributions, densities, mass functions; random vectors, joint and marginal distributions, conditioning; expectation, indicator variables, laws of large numbers, moment generating functions, central limit theorem; ideas of convergence of random variables; Markov chains, including random walk; Poisson processes; and The Wiener process.

Random processes

15 credits
Spring teaching, Year 4

The aim of this module is to present a systematic introductory account of several principal areas in stochastic processes. You cover basic principles of model building and analysis with applications that are drawn from mainly biology and engineering.

Topics include:

• Poisson processes:
• Definition and assumptions.
• Density and distribution of inter-event time.
• Pooled Poisson process.
• Breaking down a Poisson process.

• Birth processes, birth- and death- processes:
• The simple birth process.
• The pure death process.
• The Kolmogorov equations.
• The simple birth-death process.
• Simple birth-death: extinction.
• An embedded process.
• The immigration-death model.

• Queues:
• The simple M/M/1 queue.
• Queue size.
• The M/M/n queue.
• The M/M/ queue.
• The M/D/1 queue.
• The M/G/1 queue.
• Equilibrium theory.
• Other queues.

• Renewal processes:
• Discrete-time renewal processes.
• The ordinary renewal process.
• The equilibrium renewal process.

• Epidemic models:
• The simple epidemic.
• General epidemic.
• The threshold in epidemic models.

Ring Theory

15 credits
Autumn teaching, Year 4

In this module we will explore how to construct fields such as the complex numbers and investigate other properties and applications of rings.

Topics covered include

• Rings and types of rings: examples.
• Special rings and special elements: unit, zero, divisor, integral domain, fraction field, irreducible element, prime element.
• Factorising polynomials: roots and multiple roots, differentiation, roots of unity, polynomials in Q[x] and Z[x], Gauss' lemma, Eisenstein's criterion.
• Manipulating roots and symmetry: coefficients of polynomials and roots, Newton's theorem.
• Euclidean domains: Gaussian integers, Euclidean algorithm, gcd's and lcm's.
• Homomorphisms and ideals: quotient rings, principal, maximal and prime ideals. 
• Finite fields.
• Unique factorisation domains: generalising Gauss' lemma.
• Special topics: Quaternions, valuations, Hurwitz ring, the four squares theorem.

Topology and Advanced Analysis

15 credits
Spring teaching, Year 4

This module will introduce you to some of the basic concepts and properties of topological spaces. The subject of topology has a central role in all of Mathematics and having a proper understanding of its concepts and main theorem is essential as part of an undergraduate mathematics curriculum.

Topics that will be covered in this module include:

• Topological spaces
• Base and sub-base
• Separation axioms
• Continuity
• Metrisability
• Completeness
• Compactness and Coverings
• Total Boundedness
• Lebesgue numbers and Epsilon-nets
• Sequential Compactness
• Arzela-Ascoli Theorem
• Montel's theorem
• Infinite Products
• Box and Product Topologies
• Tychonov Theorem. 

MMath Project

30 credits
Autumn & spring teaching, Year 4

The work for the project and the writing of the project report will have a major role in bringing together material that you have mastered up to Year 3 and is mastering in Year 4. It will consist of a sustained investigation of a mathematical topic at Masters' level. The project report will be typeset using TeX/LaTeX (mathematical document preparation system). The use of mathematical typesetting, (mathematics-specific) information technology and databases and general research skills such as presentation of mathematical material to an audience, gathering information, usage of (electronic) scientific libraries will be taught and acquired during the project.

Advanced Numerical Analysis

15 credits
Autumn teaching, Year 4

Coding Theory

15 credits
Spring teaching, Year 4

Topics covered include: 

• Introduction to error-correcting codes. The main coding theory problem. Finite fields.
• Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code.
• The dual code and the parity check matrix. Hamming codes. Constructions of codes.
• Weight enumerators. Cyclic codes. MDS codes.

Cryptography

15 credits
Autumn teaching, Year 4

You will cover the following areas: 

• Symmetric-key cryptosystems.
• Hash functions and message authentication codes.
• Public-key cryptosystems.
• Complexity theory and one-way functions.
• Primality and randomised algorithms.
• Random number generation.
• Elliptic curve cryptography.
• Attacks on cryptosystems.
• Quantum cryptography.
• Cryptographic standards.

Differential Geometry

15 credits
Autumn teaching, Year 4

This module covers: Manifolds and differentiable structures, Lie derivatives, Parallel transport, Riemannian metrics and affine connections, Curvature tensor, Sectional curvature, Scalar curvature, Ricci curvature, Bianchi identities, Schur's lemma, Complete manifolds, Hopf-Rinow theorem, Hadmard's theorem, Geodescis and Jacobi fields, Bonnet-Meyer and Synge theorems, Laplace-Beltrami operator, Heat kernels and index theorem.

E-Business and E-Commerce Systems

15 credits
Autumn teaching, Year 4

This module will give you a theoretical and technical understanding of the major issues for all large­-scale e­-business and e­-commerce systems. The theoretical component includes: alternative e-business strategies; marketing; branding; customer relationship issues; and commercial website management. The technical component covers the standard methods for large­-scale data storage, data movement, transformation, and application integration, together with the fundamentals of application architecture. Examples focus on the most recent developments in e­-business and e-commerce distributed systems. 

Financial Mathematics

15 credits
Autumn teaching, Year 4

You will study generalized cash flows, time value of money, real and money interest rates, compound interest functions, equations of value, loan repayment schemes, investment project evaluation and comparison, bonds, term structure of interest rates, some simple stochastic interest rate models, and project writing.

Financial Portfolio Analysis

15 credits
Spring teaching, Year 4

You will study valuation, options, asset pricing models, the Black-Scholes model, Hedging and related MatLab programming. These topics form the most essential knowledge for you if you intend to start working in the financial fields. They are complex application problems. Your understanding of mathematics should be good enough to understand the modelling and reasoning skills required. The programming element of this module makes complicated computations manageable and presentable.

Functional Analysis

15 credits
Spring teaching, Year 4

Topics include: Banach spaces (Banach fixed point theorem); Baire's theorem; Bounded linear operators and on Banach spaces; continuous linear functionals; Banach-Steinhaus Uniform Boundedness Principle; open mapping and closed graph theorems; Hahn-Banach theorem; Hilbert spaces; orthogonal expansions; and Riesz-Fischer theorem.

Harmonic Analysis and Wavelets

15 credits
Autumn teaching, Year 4

This module introduces you to the concepts of harmonic analysis and the basics of wavelet theory. We will discuss the concepts of normed linear spaces and Hilbert spaces, with a focus on sequence spaces and spaces of functions, most notably the space of square-integrable functions on an interval or on the real line. You will be intoroduces to the ideas of best approximation, orthogonal projection, orthogonal sums, orthonormal bases and Fourier series in a separable Hilbert space, and apply these to the concrete case of classical trigonometric Fourier series. You will also use these strategies to prove both Fejer's theorem and the Weierstrass approximation theorem. Finally you will apply the concepts for Hilbert spaces to discuss wavelet analysis using the example of the Haar wavelet and the Haar scaling function. The concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function) will initially be introduced for the case of the Haar wavelet, but will also be defined in general.

Introduction to Cosmology

15 credits
Autumn teaching, Year 4

This module covers:

• observational overview: In visible light and other wavebands; the cosmological principle; the expansion of the universe; particles in the universe.
• Newtonian gravity: the Friedmann equation; the fluid equation; the acceleration equation.
• geometry: flat, spherical and hyperbolic; infinite vs. observable universes; introduction to topology
• cosmological models: solving equations for matter and radiation dominated expansions and for mixtures (assuming flat geometry and zero cosmological constant); variation of particle number density with scale factor; variation of scale factor with time and geometry.
• observational parameters: hubble, density, deceleration.
• cosmological constant: fluid description; models with a cosmological constant.
• the age of the universe: tests; model dependence; consequences
• dark matter: observational evidence; properties; potential candidates (including MACHOS, neutrinos and WIMPS)
• the cosmic microwave background: properties; derivation of photo to baryon ratio; origin of CMB (including decoupling and recombination).
• the early universe: the epoch of matter-radiation equality; the relation between temperature and time; an overview of physical properties and particle behaviour.
• nucleosynthesis: basics of light element formation; derivation of percentage, by mass, of helium; introduction to observational tests; contrasting decoupling and nucleosynthesis.
• inflation: definition; three problems (what they are and how they can be solved); estimation of expansion during inflation; contrasting early time and current inflationary epochs; introduction to cosmological constant problem and quintessence.
• initial singularity: definition and implications.
• connection to general relativity: brief introduction to Einstein equations and their relation to Friedmann equation.
• cosmological distance scales: proper, luminosity, angular distances; connection to observables.
• structures in the universe: CMB anisotropies; galaxy clustering
• constraining cosmology: connection to CMB, large scale structure (inc BAO and weak lensing) and supernovae.

Mathematical Models in Finance and Industry

15 credits
Spring teaching, Year 4

Topics include: partial differential equations (and methods for their solution) and how they arise in real-world problems in industry and finance. For example: advection/diffusion of pollutants, pricing of financial options.

Measure and Integration

15 credits
Autumn teaching, Year 4

Topics include: 

• Countably additive measures, sigma-algebras, Borel sets, measure spaces.
• Outer measures and Caratheodory's construction of measures.
• Construction and properties of Lebesgue measure in Euclidean spaces.
• Measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties.
• Convergence theorems - monotone convergence, dominated convergence, Fatou's lemma.
• Application of limit theorems to continuity and differentiability of integrals depending on a parameter.
• Properties of finite measure spaces and probability theory.

Numerical Solution of Partial Differential Equations

15 credits
Spring teaching, Year 4

Topics covered include: variational formulation of boundary value problems; function spaces; abstract variational problems; Lax-Milgram Theorem; Galerkin method; finite element method; examples of finite elements; and error analysis.

Object Oriented Programming

15 credits
Autumn teaching, Year 4

You will be introduced to object-oriented programming, and in particular to understanding, writing, modifying, debugging and assessing the design quality of simple Java applications.

You do not need any previous programming experience to take this module, as it is suitable for absolute beginners.

Programming in C++

15 credits
Autumn teaching, Year 4

After a review of the basic concepts of the C++ language, you are introduced to object oriented programming in C++ and its application to scientific computing. This includes writing and using classes and templates, operator overloading, inheritance, exceptions and error handling. In addition, Eigen, a powerful library for linear algebra is introduced. The results of programs are displayed using the graphics interface dislin.

Random processes

15 credits
Spring teaching, Year 4

The aim of this module is to present a systematic introductory account of several principal areas in stochastic processes. You cover basic principles of model building and analysis with applications that are drawn from mainly biology and engineering.

Topics include:

• Poisson processes:
• Definition and assumptions.
• Density and distribution of inter-event time.
• Pooled Poisson process.
• Breaking down a Poisson process.

• Birth processes, birth- and death- processes:
• The simple birth process.
• The pure death process.
• The Kolmogorov equations.
• The simple birth-death process.
• Simple birth-death: extinction.
• An embedded process.
• The immigration-death model.

• Queues:
• The simple M/M/1 queue.
• Queue size.
• The M/M/n queue.
• The M/M/ queue.
• The M/D/1 queue.
• The M/G/1 queue.
• Equilibrium theory.
• Other queues.

• Renewal processes:
• Discrete-time renewal processes.
• The ordinary renewal process.
• The equilibrium renewal process.

• Epidemic models:
• The simple epidemic.
• General epidemic.
• The threshold in epidemic models.

Satellite and Space Systems

15 credits
Spring teaching, Year 4

Fundamentals of Space Missions: Evolution of space activities; Launch vehicles; orbital dynamics.

Spacecraft Systems: Attitude control of spacecraft; Telemetry and telecommand; Spacecraft thermal control; Spacecraft power systems.

Mission Environmental and Engineering Requirements: Hostile environment; System reliability; Space and Ground segments.

Applications of Satellite Space Technology: Communication satellites; Navigation satellites; GPS and tracking systems; Remote sensing from space for environmental security. The role of satellite imagery for monitoring international arms control treaties. The role of satellite imagery in the control of nuclear proliferation. The role of space systems for border security and systems.

Topology and Advanced Analysis

15 credits
Spring teaching, Year 4

Topics that will be covered in this module include:

• Topological spaces
• Base and sub-base
• Separation axioms
• Continuity
• Metrisability
• Completeness
• Compactness and Coverings
• Total Boundedness
• Lebesgue numbers and Epsilon-nets
• Sequential Compactness
• Arzela-Ascoli Theorem
• Montel's theorem
• Infinite Products
• Box and Product Topologies
• Tychonov Theorem.