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

Further Programming

15 credits
Spring teaching, Year 1

This module follows on from "Introduction to Programming" and provides an introduction to more advanced programming concepts and techniques. This module covers Java programming, including the use of subclasses and library classes to create well-organised programs, the choice and implementation of appropriate algorithms and data structures (e.g. arrays, lists, trees, graphs, depth- and breadth-first search, the minimax and A* algorithms), and the construction of graphical user interfaces for Java programs.

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 Programming

15 credits
Autumn teaching, Year 1

The module introduces you to a collection of basic programming concepts and techniques, including designing, testing, debugging and documenting programmes.  

For both absolute beginners and those with prior computing experience, the module introduces the programming language Java, a language used for other components of undergraduate modules. Java will be the primary language used for programming assignments in nearly all first year modules taught by the department of Informatics.  

You do not need previous experience of programming to take this module, but you will need basic knowledge of NT/Windows2000/XP.

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.

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

Computer Networks

15 credits
Spring teaching, Year 2

This module provides an introduction to the basics of packet switching technologies as used in the Internet. Emphasis is placed on core Internet protocols such as IP and TCP. Subjects covered include: network access technologies; design of network protocols using layering, local area networks, TCP/IP routing and switching, congestion control. This module introduces the Internet with a top-down view: the accent is on layer abstractions and the associated protocols (and how to program with them). Particular attention is paid to questions of network security.

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.

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.

Program Analysis

15 credits
Autumn teaching, Year 2

Part 1: Foundations

The first part of the module introduces the idea of the asymptotic analysis of algorithms, and in particular we will consider the following: specifying a problem; the notion of an algorithm and what it means for an algorithm to solve a problem; the upper, lower and tight asymptotic bounds associated with an algorithm; the best-, worst- and expected-case analysis of an algorithm; the lower bound for a problem.

In the remainder of Part 1 we consider a number of important data structures, with particular emphasis on priority queues and the generic graph data structure. Several basic graph algorithms will be considered, in particular: depth-first search of graphs; breadth-first search of graphs; and topological sorting of directed acyclic graphs.

Part 2: Generic Design Paradigms

In Part 2 we will consider four of the most important methods used as the basis for algorithm design: greedy methods; divide and conquer approaches; dynamic programming; and network flow.

In considering these generic design paradigms we will look at a number of well-known problems, including: interval scheduling; single source shortest path; minimum spanning tree; Huffman codes construction; weighted interval scheduling; subset sum; sequence alignment; network flow; and bipartite matching.

Limits of Computation

15 credits
Spring teaching, Year 3

This module is all about fundamental questions like 'what is computable?' and 'what is feasibly computable?'. The following topics are covered: what is a universal program? What is program specialisation? (partial evaluation, also known as s-m-n theorem). What is self-application? (boot-strapping). How can it be used to speed-up programs? How can an unsolvable problem be defined using WHILE? How can this be generalised? (Rice's theorem). Are there decidable but unfeasible problems? What are typical examples? What does feasible mean? How can one measure resource-usage of (time, space, non-determinism) of WHILE programs? What are asymptotic complexity classes and what are their limitations? What do we know about existence of optimal solutions?

Advanced Numerical Analysis

15 credits
Autumn teaching, Year 3

Coding Theory

15 credits
Spring teaching, Year 3

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 3

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 3

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 3

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 3

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 3

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.

Harmonic Analysis and Wavelets

15 credits
Autumn teaching, Year 3

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.

Human-Computer Interaction

15 credits
Autumn teaching, Year 3

Human computer interaction (HCI) is concerned with understanding and designing interactive technologies from a people-centred perspective. This HCI module will give an introduction to the basic principles, methods and developments in HCI, with the objective of getting you to think constructively and analytically about how to design and evaluate interactive technologies, with opportunities to apply the principles and methods in practice. Topics include: principles of design, evaluating interactive technologies, understanding users, generating requirements, prototyping and iterative evaluation.

Introduction to Mathematical Biology

15 credits
Autumn teaching, Year 3

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 3

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.

Mathematical Research Project

15 credits
Spring teaching, Year 3

The project aims to introduce you under the guidance of a supervisor into a mathematical topic 

Measure and Integration

15 credits
Autumn teaching, Year 3

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 3

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.

Partial Differential Equations

15 credits
Autumn teaching, Year 3

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 3

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 3

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 3

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 3

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 3

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. 

Web Computing

15 credits
Autumn teaching, Year 3

This module provides an introduction to the models and technologies used to provide services over the Internet and, in particular, the World Wide Web. Topics covered include: XML, including DTD, Schema, DOM, XPATH and XSLT, client-side programming (embedded scripting languages, style sheets), server-side programming (Java Servlets, JSP), and applications.