UNIT CATALOGUE
MANG0069: Introduction to accounting & finance (service
unit)
Semester 1
Credits: 5
Level: Level 1
Assessment: EX50 CW50
Requisites: Co MANG0070
Aims & Learning Objectives: To provide students undertaking
any type of degree study with an introductory knowledge of accounting
and finance
Content: The role of the accountant, corporate treasurer
and financial controller
Sources and uses of capital funds
Understanding the construction and nature of the balance sheet
and profit and loss account
Principles underlying the requirements for the publication of
company accounts
Interpretation of accounts - published and internal, including
financial ratio analysis
Planning for profits, cash flow. Liquidity, capital expenditure
and capital finance
Developing the business plan and annual budgeting
Estimating the cost of products, services and activities and their
relationship to price.
Analysis of costs and cost behaviour
MANG0072: Managing human resources (service unit)
Semester 2
Credits: 5
Level: Level 2
Assessment: EX100
Requisites: Co MANG0071
Aims & Learning Objectives: The course aims to give
a broad overview of major features of human resource management.
It examines issues from the contrasting perspectives of management,
employees and public policy.
Content: Perspectives on managing human resources.
Human resource planning, recruitment and selection.
Performance, pay and rewards.
Control, discipline and dismissal.
MANG0073: Marketing (service unit)
Semester 1
Credits: 5
Level: Level 2
Assessment: EX60 CW40
Requisites: Co MANG0074
Aims & Learning Objectives: 1. To provide an introduction
to the concepts of Marketing.
2. To understand the principles and practice of marketing management.
3. To introduce students to a variety of environmental and other
issues facing marketing today.
Content: Marketing involves identifying and satisfying
customer needs and wants. It is concerned with providing appropriate
products, services, and sometimes ideas, at the right place and
price, and promoted in ways which are motivating to current and
future customers. Marketing activities take place in the context
of the market, and of competition.
The course is concerned with the above activities, and includes:
consumer and buyer behaviour
market segmentation, targetting and positioning
market research
product policy and new product development
advertising and promotion
marketing channels and pricing
MANG0074: Business information systems (service unit)
Semester 2
Credits: 5
Level: Level 2
Assessment: EX60 PR25 OT15
Requisites: Co MANG0073
Aims & Learning Objectives: Information Technology
(IT) is rapidly achieving ubiquity in the workplace. All areas
of the business community are achieving expansion in IT and investing
huge sums of money in this area. Within this changing environment,
several key trends have defined a new role for computers:
a) New forms and applications of IT are constantly emerging. One
of the most important developments in recent years has been the
fact that IT has become a strategic resource with the potential
to affect competitive advantage: it transforms industries and
products and it can be a key element in determining the success
or failure of an organisation.
b) Computers have become decentralised within the workplace: PCs
sit on managers desks, not in the IT Department. The strategic
nature of technology also means that managing IT has become a
core competence for modern organisations and is therefore an important
part of the task of general and functional managers. Organisations
have created new roles for managers who can act as interfaces
between IT and the business, combining a general technical knowledge
with a knowledge of business.
This course addresses the above issues, and, in particular, aims
to equip students with IT management skills for the workplace.
By this, we refer to those attributes that they will need to make
appropriate use of IT as general or functional managers in an
information-based age. In dealing with management issues, our
aims are to provide practical as well as theoretical knowledge.
As such, the course integrates hands-on work in the computer lab,
dealing with management problems, and practical elements of IT
practice that managers are likely to encounter when they become
involved with IT in any organisation. Thus, in addition to providing
an appreciation of the business value and opportunities stemming
from new technology, the latter includes the various issues encountered
when devising, evaluating, and managing any IT project.
Content: The course is divided into two components, to
reflect the fact that is oriented to both theoretical and practical
aspects of IT.
Section one comprises the practical element of the course. It
is primarily focused on case studies, involving the application
of selected software to management problems. It involves hands-on
work in the computer laboratory.
Section two relates to the examination of IT in its business context.
Here the focus is upon examining the value of IT in terms of why
IT is strategic and how it can affect the competitive environment,
as well as how it should be managed within the business.
The sessions will be organised as follows: IT and Corporate Strategy;
IT-Induced Transformation; Strategic Alignment of IT and Business
Strategies; Evaluation of IT Investments; Project Development
and Management: Implementation of Technology
MANG0076: Business policy (service unit)
Semester 1
Credits: 5
Level: Level 3
Assessment: EX60 CW40
Requisites:
Aims & Learning Objectives: To provide an appreciation
of how organisations develop from their entrepreneurial beginnings
through maturity and decline .
To examine the interrelationship between concepts of policy and
strategy formulation with the behavioural aspects of business
To enable students to explore the theoretical notions behind corporate
strategy
Students are expected to develop skills of analysis and the ability
to interpret complex business situations.
Content: Business objectives , values and mission; industry
and market analysis ; competitive strategy and advantage ; corporate
life cycle; organisational structures and controls .
MATH0001: Numbers
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: This course is designed
to cater for first year students with widely different backgrounds
in school and college mathematics. It will treat elementary matters
of advanced arithmetic, such as summation formulae for progressions
and will deal matters at a certain level of abstraction. This
will include the principle of mathematical induction and some
of its applications. Complex numbers will be introduced from first
principles and developed to a level where special functions of
a complex variable can be discussed at an elementary level.
Objectives: Students will become proficient in the use of mathematical
induction. Also they will have practice in real and complex arithmetic
and be familiar with abstract ideas of primes, rationals, integers
etc, and their algebraic properties. Calculations using classical
circular and hyperbolic trigonometric functions and the complex
roots of unity, and their uses, will also become familiar with
practice.
Content: Natural numbers, integers, rationals and reals.
Highest common factor. Lowest common multiple. Prime numbers,
statement of prime decomposition theorem, Euclid's Algorithm.
Proofs by induction. Elementary formulae. Polynomials and their
manipulation. Finite and infinite APs, GPs. Binomial polynomials
for positive integer powers and binomial expansions for non-integer
powers of a+b. Finite sums over multiple indices and changing
the order of summation. Algebraic and geometric treatment of complex
numbers, Argand diagrams, complex roots of unity. Trigonometric,
log, exponential and hyperbolic functions of real and complex
arguments. Gaussian integers. Trigonometric identities. Polynomial
and transcendental equations.
Students must have A-level Mathematics, normally Grade B or better,
or equivalent, in order to undertake this unit.
MATH0002: Functions, differentiation & analytic geometry
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: To teach the basic
notions of analytic geometry and the analysis of functions of
a real variable at a level accessible to students with a good
'A' Level in Mathematics. At the end of the course the students
should be ready to receive a first rigorous analysis course on
these topics. Objectives: The students should be able to manipulate
inequalities, classify conic sections, analyse and sketch functions
defined by formulae, understand and formally manipulate the notions
of limit, continuity and differentiability and compute derivatives
and Taylor polynomials of functions.
Content: Basic geometry of polygons, conic sections and
other classical curves in the plane and their symmetry. Parametric
representation of curves and surfaces.
Review of differentiation: product, quotient, function-of-a-function
rules and Leibniz rule.
Maxima, minima, points of inflection, radius of curvature. Graphs
as geometrical interpretation of functions. Monotone functions.
Injectivity, surjectivity, bijectivity.
Curve Sketching.
Inequalities. Arithmetic manipulation and geometric representation
of inequalities.
Functions as formulae, natural domain, codomain, etc. Real valued
functions and graphs.
Introduction to MAPLE.
Orders of magnitude. Taylor's Series and Taylor polynomials -
the error term. Differentiation of Taylor series.
Taylor Series for exp, log, sin etc. Orders of growth.
Orthogonal and tangential curves.
Students must have A-level Mathematics, normally Grade B or better,
or equivalent in order to undertake this unit.
MATH0003: Integration & differential equations
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: This module is designed
to cover standard methods of differentiation and integration,
and the methods of solving particular classes of differential
equations, to guarantee a solid foundation for the applications
of calculus to follow in later courses.
Objective: The objective is to ensure familiarity with methods
of differentiation and integration and their applications in problems
involving differential equations. In particular, students will
learn to recognise the classical functions whose derivatives and
integrals must be committed to memory. In independent private
study, students should be capable of identifying, and executing
the detailed calculations specific to, particular classes of problems
by the end of the course.
Content: Review of basic formulae from trigonometry and
algebra: polynomials, trigonometric and hyperbolic functions,
exponentials and logs. Integration by substitution. Integration
of rational functions by partial fractions. Integration of parameter
dependent functions. Interchange of differentiation and integration
for parameter dependent functions. Definite integrals as area
and the fundamental theorem of calculus in practice. Particular
definite integrals by ad hoc methods. Definite integrals by substitution
and by parts. Volumes and surfaces of revolution. Definition of
the order of a differential equation. Notion of linear independence
of solutions. Statement of theorem on number of linear independent
solutions. General Solutions. CF+PI. First order linear
differential equations by integrating factors; general solution.
Second order linear equations, characteristic equations; real
and complex roots, general real solutions. Simple harmonic motion.
Variation of constants for inhomogeneous equations. Reduction
of order for higher order equations. Separable equations, homogeneous
equations, exact equations. First and second order difference
equations.
Students must have A-level Mathematics, normally Grade B or better,
or equivalent in order to undertake this unit.
MATH0004: Sets & sequences
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites: Pre MATH0001
Aims & Learning Objectives: Aims: To introduce the
concepts of logic that underlie all mathematical reasoning and
the notions of set theory that provide a rigorous foundation for
mathematics. A real life example of all this machinery at work
will be given in the form of an introduction to the analysis of
sequences of real numbers.
Objectives: By the end of this course, the students will be able
to: understand and work with a formal definition; determine whether
straight-forward definitions of particular mappings etc. are correct;
determine whether straight-forward operations are, or are not,
commutative; read and understand fairly complicated statements
expressing, with the use of quantifiers, convergence properties
of sequences.
Content: Logic: Definitions and Axioms. Predicates and
relations. The meaning of the logical operators Ù,
Ú, ˜, ®,
«, ",
$. Logical equivalence and logical
consequence. Direct and indirect methods of proof. Proof by contradiction.
Counter-examples. Analysis of statements using Semantic Tableaux.
Definitions of proof and deduction.
Sets and Functions: Sets. Cardinality of finite sets. Countability
and uncountability. Maxima and minima of finite sets, max (A)
= - min (-A) etc. Unions, intersections, and/or statements and
de Morgan's laws. Functions as rules, domain, co-domain, image.
Injective (1-1), surjective (onto), bijective (1-1, onto) functions.
Permutations as bijections. Functions and de Morgan's laws. Inverse
functions and inverse images of sets. Relations and equivalence
relations. Arithmetic mod p.
Sequences: Definition and numerous examples. Convergent sequences
and their manipulation. Arithmetic of limits.
MATH0005: Matrices & multivariate calculus
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites: Pre MATH0002
Aims & Learning Objectives: Aims: The course will provide
students with an introduction to elementary matrix theory and
an introduction to the calculus of functions from IRn
® IRm and to multivariate
integrals.
Objectives: At the end of the course the students will have a
sound grasp of elementary matrix theory and multivariate calculus
and will be proficient in performing such tasks as addition and
multiplication of matrices, finding the determinant and inverse
of a matrix, and finding the eigenvalues and associated eigenvectors
of a matrix. The students will be familiar with calculation of
partial derivatives, the chain rule and its applications and the
definition of differentiability for vector valued functions and
will be able to calculate the Jacobian matrix and determinant
of such functions. The students will have a knowledge of the integration
of real-valued functions from IR2 ®
IR and will be proficient in calculating multivariate integrals.
Content: Lines and planes in two and three dimension. Linear
dependence and independence. Simultaneous linear equations. Elementary
row operations. Gaussian elimination. Gauss-Jordan form. Rank.
Matrix transformations. Addition and multiplication. Inverse of
a matrix. Determinants. Cramer's Rule. Similarity of matrices.
Special matrices in geometry, orthogonal and symmetric matrices.
Real and complex eigenvalues, eigenvectors. Relation between algebraic
and geometric operators. Geometric effect of matrices and the
geometric interpretation of determinants. Areas of triangles,
volumes etc. Real valued functions on IR3. Partial
derivatives and gradients; geometric interpretation. Maxima and
Minima of functions of two variables. Saddle points. Discriminant.
Change of coordinates. Chain rule. Vector valued functions and
their derivatives. The Jacobian matrix and determinant, geometrical
significance. Chain rule. Multivariate integrals. Change of order
of integration. Change of variables formula.
MATH0006: Vectors & applications
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites: Pre MATH0001, Pre MATH0002, Pre MATH0003
Aims & Learning Objectives: Aims: To introduce the
theory of three-dimensional vectors, their algebraic and geometrical
properties and their use in mathematical modelling. To introduce
Newtonian Mechanics by considering a selection of problems involving
the dynamics of particles.
Objectives: The student should be familiar with the laws of vector
algebra and vector calculus and should be able to use them in
the solution of 3D algebraic and geometrical problems. The student
should also be able to use vectors to describe and model physical
problems involving kinematics. The student should be able to apply
Newton's second law of motion to derive governing equations of
motion for problems of particle dynamics, and should also be able
to analyse or solve such equations.
Content: Vectors: Vector equations of lines and planes.
Differentiation of vectors with respect to a scalar variable.
Curvature. Cartesian, polar and spherical co-ordinates. Vector
identities. Dot and cross product, vector and scalar triple product
and determinants from geometric viewpoint. Basic concepts of mass,
length and time, particles, force. Basic forces of nature: structure
of matter, microscopic and macroscopic forces. Units and dimensions:
dimensional analysis and scaling. Kinematics: the description
of particle motion in terms of vectors, velocity and acceleration
in polar coordinates, angular velocity, relative velocity. Newton's
Laws: Kepler's laws, momentum, Newton's laws of motion, Newton's
law of gravitation. Newtonian Mechanics of Particles: projectiles
in a resisting medium, constrained particle motion; solution of
the governing differential equations for a variety of problems.
Central Forces: motion under a central force.
MATH0007: Analysis: Real numbers, real sequences & series
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0004, Pre MATH0005, Pre MATH0003
Aims & Learning Objectives: Aims: To reinforce and
extend the ideas and methodology (begun in the first year unit
MATH0004) of the analysis of the elementary theory of sequences
and series of real numbers.
Objectives: By the end of the module, students should be able
to read and understand statements expressing, with the use of
quantifiers, convergence properties of sequences and series. They
should also be capable of investigating particular examples to
which the theorems can be applied and of understanding, and constructing
for themselves, rigorous proofs within this context.
Content: Suprema and Infima, Maxima and Minima. The Completeness
Axiom.
Sequences. Limits of sequences in epsilon-N notation. Bounded
sequences and monotone sequences. Cauchy sequences. Algebra-of-limits
theorems.
Subsequences. Limit Superior and Limit Inferior. Bolzano-Weierstrass
Theorem.
Sequences of partial sums of series. Convergence of series. Conditional
and absolute convergence. Tests for convergence of series; ratio,
comparison, alternating and nth root tests.
Power series and radius of convergence.
Functions, Limits and Continuity. Continuity in terms of convergence
of sequences. Algebra of limits. Convergence of sequences of functions,
point-wise and uniform. Interchanging limits.
MATH0008: Algebra 1
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0004, Pre MATH0005
Aims & Learning Objectives: Aims: To teach the definitions
and basic theory of abstract linear algebra and, through exercises,
to show its applicability.
Objectives: Students should know, by heart, the main results in
linear algebra and should be capable of independent detailed calculations
with matrices which are involved in applications.
Content: Real and complex vector spaces, subspaces, direct
sums, linear independence, spanning sets,
bases, dimension. The technical lemmas concerning linearly independent
sequences.
Dimension. Complementary subspaces. Projections.
Linear transformations. Rank and nullity. The Dimension Theorem.
Matrix representation, transition matrices, similar matrices.
Examples.
Inner products, Induced norm, Cauchy-Schwarz inequality, triangle
inequality, parallelogram law, orthogonality, Gram-Schmidt process.
MATH0009: Ordinary differential equations & control
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0001, Pre MATH0002, Pre MATH0003, Pre MATH0005,
Co MATH0010
Aims & Learning Objectives: Aims: This course will
provide standard results and techniques for solving systems of
linear autonoumous differential equations. Based on this material
an accessible introduction to the ideas of mathematical control
theory is given. The emphasis here will be on stability and stabilization
by feedback. Foundations will be laid for more advanced studies
in nonlinear differential equations and control theory. Phase
plane techniques will be introduced.
Objectives: At the end of the course, students will be conversant
with the basic ideas in the theory of linear autonomous differential
equations and, in particular, will be able to employ Laplace transform
and matrix methods for their solution. Moreover, they will be
familiar with a number of elementary concepts from control theory
(such as stability, stabilization by feedback, controllability)
and will be able to solve simple control problems. The student
will be able to carry out simple phase plane analysis.
Content: Systems of linear ODEs: Normal form; solution
of homogeneous systems; fundamental matrices and matrix exponentials;
repeated eigenvalues; complex eigenvalues; stability; solution
of non-homogeneous systems by variation of parameters. Laplace
transforms: Definition; statement of conditions for existence;
properties including transforms of the first and higher derivatives,
damping, delay; inversion by partial fractions; solution of ODEs;
convolution theorem; solution of integral equations. Linear control
systems: Systems: state-space; impulse response and delta functions;
transfer function; frequency-response. Stability: exponential
stability; input-output stability; Routh-Hurwitz criterion. Feedback:
state and output feedback; root loci and high-gain feedback; servomechanisms.
Introduction to controllability and observability: definitions,
rank conditions (without full proof) and examples. Nonlinear ODEs:
Phase plane techniques, stability of equilibria.
MATH0010: Vector calculus & partial differential equations
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0002, Pre MATH0003, Pre MATH0005, Pre MATH0006,
Co MATH0009
Aims & Learning Objectives: Aims: The first part of
the course provides an introduction to vector calculus, an essential
toolkit in most branches of applied mathematics. The second part
introduces methods for the solution of linear partial differential
equations.
Objectives: At the end of this course students will be familiar
with the fundamental results of vector calculus (Gauss' theorem,
Stokes' theorem) and will be able to carry out line, surface and
volume integrals in general curvilinear coordinates. They should
be able to solve Laplace's equation, the wave equation and the
diffusion equation in simple domains, using the techniques of
separation of variables, Laplace transforms and, in the case of
the wave equation, D'Alembert's solution.
Content: Vector calculus: Work and energy; curves and surfaces
in parametric form; line, surface and volume integrals.
Grad, div and curl; divergence and Stokes' theorems; curvilinear
coordinates; scalar potential.
Fourier series: Formal introduction to Fourier series, statement
of Fourier convergence theorem; Fourier cosine and sine series.
Partial differential equations: classification of linear second
order PDEs; Laplace's equation in 2-D, including solution by separation
of variables in rectangular and circular domains; wave equation
in one space dimension, including D'Alembert's solution; the diffusion
equation in one space dimension, including solution by Laplace
transform.
MATH0011: Analysis: Real-valued functions of a real variable
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0007, Co MATH0014
Aims & Learning Objectives: Aims: To give a thorough
grounding, through rigorous theory and exercises, in the method
and theory of modern calculus. To define the definite integral
of certain bounded functions, and to explain why some functions
do not have integrals.
Objectives: Students should be able to quote, verbatim, and prove,
without recourse to notes, the main theorems in the syllabus.
They should also be capable, on their own initiative, of applying
the analytical methodology to problems in other disciplines, as
they arise. They should have a thorough understanding of the abstract
notion of an integral, and a facility in the manipulation of integrals.
Content: Functions, limits and continuity. Continuity in
terms of convergence of sequences. Weierstrass's theorem on continuous
functions attaining suprema and infima on compact interval. Algebra-of-limits.
Intermediate Value Theorem.
Functions and Derivatives. Algebra of derivatives. Leibniz Rule
and compositions. Derivatives of inverse functions. Rolle's Theorem
and Mean Value Theorem. Cauchy's Mean Value Theorem. L'Hôpital's
Rule. Monotonic functions. Maxima/Minima.
Convergence of sequences of functions, point-wise and uniform.
Interchanging limits.
Uniform Convergence. Cauchy's Criterion for Uniform Convergence.
Weierstrass M-test for series. Power series. Differentiation
of power series.
Up to the Fundamental Theorem of Calculus for the integral of
a Riemann-integrable derivative of a function. Integration of
power series. Interchanging integrals and limits. Improper integrals.
MATH0012: Algebra 2
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0008
Aims & Learning Objectives: Aims: In linear algebra
the aim is to take the abstract theory to a new level, different
from the elementary treatment in MATH0008. Groups will be introduced
and the most basic consequences of the axioms derived.
Objectives: Students should be capable of finding eigenvalues
and minimum polynomials of matrices and of deciding the correct
Jordan Normal Form. Students should know how to execute the Gram-Schmidt
process and diagonalisation of matrices, while supplying supporting
theoretical justification of the method. In group theory they
should be able to write down the group axioms and the main theorems
which are consequences of the axioms.
Content: Linear Algebra: Properties of determinants. Eigenvalues
and eigenvectors. Geometric and algebraic multiplicity. Diagonalisability.
Characteristic polynomials. Cayley--Hamilton Theorem. Minimum
polynomial and primary decomposition theorem. Statement of and
motivation for the Jordan Canonical Form. Examples.
Orthogonal and unitary transformations. Symmetric and Hermition
linear transformations and their diagonalisability. Quadratic
forms. Norm of a linear transformation. Examples.
Group Theory: Group axioms and examples. Deductions from the axioms
(e.g. uniqueness of identity, cancellation). Subgroups. Cyclic
groups and their properties. Homomorphisms, isomorphisms, automorphisms.
Cosets and Lagrange's Theorem. Normal subgroups and Quotient groups.
Fundamental Homomorphism Theorem.
MATH0013: Mathematical modelling & fluids
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX50 CW50
Requisites: Pre MATH0009, Pre MATH0010
Aims & Learning Objectives: Aims: To study, by example,
how mathematical models are hypothesised, modified and elaborated.
To study a classic example of mathematical modelling, that of
fluid mechanics.
Objectives: At the end of the course the student should be able
to· construct an initial mathematical model for a real world
process and assess this model critically· suggest alterations
or elaborations of proposed model in light of discrepancies between
model predictions and observed data or failures of the model to
exhibit correct qualitative behaviour. The student will also be
familiar with the equations of motion of an ideal inviscid fluid
(Eulers equations, Bernoullis equation) and how to solve these
in certain idealised flow situations.
Content: Modelling and the scientific method: Objectives
of mathematical modelling; the iterative nature of modelling;
falsifiability and predictive accuracy; Occam's razor, paradigms
and model components; self-consistency and structural stability.
The three stages of modelling: (1) Model formulation, including
the use of empirical information, (2) model fitting, and (3) model
validation. Possible case studies and projects include: The dynamics
of measles epidemics; population growth in the USA; prey-predator
and competition models; modelling water pollution; assessment
of heat loss prevention by double glazing; forest management.
Fluids: Lagrangian and Eulerian specifications, material time
derivative, acceleration, angular velocity. Mass conservation,
incompressible flow, simple examples of potential flow.
MATH0014: Numerical analysis
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0007, Pre MATH0008, Co MATH0011
Aims & Learning Objectives: Aims: To teach elementary
MATLAB programming. To teach those aspects of Numerical Analysis
which are most relevant to a general mathematical training, and
to lay the foundations for the more advanced courses in later
years.
Objectives: Students should have some facility with MATLAB programming.
They should know simple methods for the approximation of functions
and integrals, solution of initial and boundary value problems
for ordinary differential equations and the solution of linear
systems. They should also know basic methods for the analysis
of the errors made these methods, and be aware of some of the
relevant practical issues involved in their implementation.
Content: MATLAB Programming: handling matrices; M-files;
graphics.
Concepts of Convergence and Accuracy: Order of convergence, extrapolation
and error estimation.
Approximation of Functions: Polynomial Interpolation, error term.
Quadrature and Numerical Differentiation: Newton-Cotes formulae.
Gauss quadrature and numerical differentiation by method of undetermined
coefficients. Composite formulae. Error terms.
Numerical Solution of ODEs: Euler, Backward Euler, Trapezoidal
and explicit Runge-Kutta methods. Stability. Consistency and convergence
for one step methods. Error estimation and control. Shooting technique.
Linear Algebraic Equations: Gaussian elimination, LU decomposition,
pivoting, Matrix norms, conditioning, backward error analysis,
iterative refinement. Direct methods for 2 point Boundary Value
Problems.
MATH0015: Programming
Semester 1
Credits: 6
Topic: Computing
Level: Level 1
Assessment: EX75 CW25
Requisites: Co MATH0023
Aims & Learning Objectives: Aims: To introduce functional
programming while drawing out the similarities with abstract mathematics.
To show that the mathematical thought process is a natural one
for programming. To provide a gentle introduction to practical
functional programming.
Objectives: Students should be able to write simple functions,
to understand the nature of types and to use data types appropriately.
They should also appreciate the value and use of recursion.
Content: Expressions, choice, scope and extent, functions,
recursion, recursive datatypes, higher-order objects.
MATH0016: Information management 1
Semester 1
Credits: 6
Topic: Computing
Level: Level 1
Assessment: ES60 CW30 OT10
Requisites:
Aims & Learning Objectives: Aims: To introduce students
to the use of a workstation, to wordprocessing, spreadsheets and
relational data bases, and to the basic ideas of computing, and
to the range of applications and
misapplications of computers in science. To give students some
experience of working in small
groups.
Objectives: Students should have a practical ability to use contemporary
information
management facilities. They should be able to write a good report,
and they should have the
confidence and the language to enable criticism of the use of
computers in science.
Content: Introduction: hardware, software, networking.
Use of the workstation. Social issues. The relationship between
computing and science. Computers as calculators, as simulating
engines, and as new realtities. Mathematical and computational
models. The difficulty of validating or criticising computational
models. Example of fluid flow, and the numerical wind tunnel.
Experiment and decision making using computational models. Artificial
intelligence, expert systems, neural nets, artificial evolution.
The use and abuse of computers in science. Word processing, HTML,
Scientific journalism and scientific reports. The goals of succinctness
and clarity. Spreadsheets, organizing, exploring and presenting
numerical data. Introduction to Statistics. Mean, standard deviation,
histograms, the idea of probability density functions.
MATH0017: Principles of computer operation & architecture
Semester 1
Credits: 6
Topic: Computing
Level: Level 1
Assessment: EX75 CW25
Requisites: Co MATH0025
Aims & Learning Objectives: Aims: To introduce students
to the structure, basic design, operation and programming of conventional,
von Neumann computers at the machine level. Alternative approaches
to machine design will also be examined so that some recent machine
architectures can be introduced. In particular the course will
develop to explore the relationships between what actually happens
at the machine level and important ideas about, for example, aspects
of high-level programming and data structures, that students encounter
on parallel courses.
Objectives: Familiarity with the von Neumann model, the nature
and function of each of the main components and general principles
of operation of the machines, including input and output transfers
and basic numeric manipulations.
Understanding of the characteristics of logic elements; the ability
to manipulate/simplify Boolean functions; practical experience
of simple combinatorial and sequential systems of logic gates;
and a perception of the links between logic systems and elements
of computer processors and store.
Understanding of the role and function of an assembler and practical
experience of reading and making simple changes to small, low-level
programmes. Understanding of the test running and debugging of
programmes.
Content: Basic principles of computer operation: Brief
historical introduction to computing machines. Binary basis of
computer operation and binary numeration systems. Von Neumann
computers and the structure, nature and relationship of their
major elements. Principles of operation of digital computers;
use of registers and the instruction cycle; simple addressing
concepts; programming. Integers and floating point numbers. Input
and output; basic principles and mechanisms of data transfer;
programmed and data channel transfers; device status; interrupt
programming; buffering; devices.
Introduction to digital logic and low-level programming: Boolean
algebra and behaviour of combinatorial and sequential logic circuits
(supported by practical work). Logic circuits as building blocks
for computer hardware.
The nature and general characteristics of assemblers; a gentle
introduction to simple assembler programmes to illustrate the
major features and structures of low-level programmes. Running
assembler programmes (supported by practical work).
MATH0018: Databases/performance analysis
Semester 1
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0015, Co MATH0026
Aims & Learning Objectives: Aims: To present an introductory
account of the theory and practice of databases. To convey an
understanding of the wide variety of techniques available for
assessing the performance of programs and of computer-based systems.
Objectives: To demonstrate understanding of the basic structure
of relational database systems and to be able to make elementary
queries. Students should be able to use basic benchmark programs,
and the standard profiling tools. They should be aware of the
limitations of such techniques, and of the wide variety of possible
approaches to measuring, assessing, comparing and planning the
performance of computer-based systems.
Content: Databases: Network and relational models. Completeness
of relational models, Codd's classification of canonical forms:
first, second, third, and fourth normal forms. Keys, join, query
languages (SQL, Query-by-example). Object databases.
Performance Analysis: Benchmarking, including standard benchmarks
such as Whetstone, Dhrystone. Benchmarking suites; SPECMarks.
Contrast performance and test suites. Determining where time goes;
profiling, sampling, emulating. Use of memory. Effects of architecture.
Comparison of hardware and software monitoring. Program Comparison,
Pitfalls, Performance Engineering, Queueing Theory, Case Studies.
MATH0019: Foundations
Semester 1
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0001, Pre MATH0015
Aims & Learning Objectives: Principles of Aims: To
give the student an appreciation of the foundations of programming
by considering functions as units of computation l-calculus
and combinatory logic. To raise the issue of correctness and to
develop a critical attitude toward computing in general and logic
programming in particular. To illustrate how the various mathematical
principles discussed in this Unit are translated in practical
programming languages.
Objectives: Students should be able to perform reductions in two
reduction systems, and to prove elementary theorms in and about
these calculi. To understand enough logic so that correct logic
programming is possible. To be able to apply the theories of mathematical
logic to the development of programming languages, to contrast
pure rewriting with environment based interpretation operating
over different domains (eg. values and types). To be able to read,
understand and write programs in EuLisp.
Content: String rewriting systems, Church-Rosser ideas,
Zermelo Fraenkel set theory, types and sets, operations on types,
examples in C and ML, functions as graphs, and functions as rules
or processes; pure lambda calculus, reduction, Church Rosser again,
ordered pairs, numerals in lambda calculus, Lisp; Scott domain
theory; Logic, Logical validity, logical consequence, Conjunctive
normal form, clausal form, semantic tableau methods, Prolog, resolution
and unification.
MATH0020: Computability & decidability
Semester 1
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0001, Pre MATH0015
Aims & Learning Objectives: Aims: To extend previous
coverage of finite-state machines and Turing machines. To explore
the limitations of Turing computability.
Objectives: Students should appreciate the limitations of finite-state
machines, and the availability of different possible standard
formalisations of Turing machines. Students should understand
what can and cannot be computed using Turing machines, and the
rudiments of computational complexity theory.
Content: Finite-State Machines: Revision of the basic properties
of finite-state machines. Nondeterministic finite-state machines.
What can and cannot be computed using finite-state machines. Turing
Machines: Revision of Turing Machines. Connecting standard Turing
Machines together. Introduction to Church's Thesis. Church's Thesis:
Church's Thesis and the equivalence of different models of Turing
machine. Church's Thesis (cont): Church's thesis and the equivalence
of different models of computation - recursive functions, primitive
and general recursion. Universal Turing Machines: Universal Turing
Machines and limitations of Turing computability. Undecidability,
the Halting Problem, reduction of one unsolvable problem to another.
Computational Complexity: Philosophy of computational complexity,
upper and lower time-bounded computations, complexity classes
P and NP, NP-completeness.
MATH0021: Computer graphics
Semester 1
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0015
Aims & Learning Objectives: Aims: To provide an introduction
to the techniques of representing, rendering, and displaying computer
graphics, with assessed coursework. Objectives: Students will
be able to distinguish modelling from rendering. They will be
able to describe the relevant components of Euclidean geometry
and their relationships to matrix algebra formulations. Students
will know the difference between solid and surface modelling and
be able to describe typical computer representations of each.
Rendering for raster displays will be explainable in detail, including
lighting models and a variety visual effects and defects. Students
will be expected to describe the sampling problem and solutions
for static pictures.
Content: Background: Basic mechanisms, concepts and techniques
for creating and displaying line drawings. Output devices, input
devices. Packages. Coordinate systems, Euclidean geometry and
transformations.
Modelling: Mesh models and their representation. Constructive
solid geometry and its representation. Specialised models.
Rendering: Raster images; illumination models; meshes and hidden
surface removal; scan-line rendering. Constructive Solid Geometry;
ray-casting; visual effects and defects. Ordering dither; resolution;
aliasing; colour.
Students should have the ability to program in order to undertake
this unit.
MATH0022: Formal program development
Semester 1
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0015
Aims & Learning Objectives: Aims: To convey to students
the idea that programming can be presented as a systematic process
of calculation with mathematically secure foundations.
Objectives: Students should be able to develop modest programs
systematically with a complete understanding of the mathematical
foundations of the method advocated, and should understand the
relationship between formal and informal methods for practical
use.
Content: Programs, specifications, code, refinement. Types,
invariants and feasibility. Assignment and sequencing. Control
structures: alternatives and iteration. Introduction to data refinement.
Dijkstra's weakest precondition and language semantics in terms
of it. Basic Theorems for the Alternative and Iterative Constructs
and their relevance to program development. Use of the weakest
precondition as a basis for the refinement calculus. Proving refinement
laws from first principles; deriving one refinement law from another.
MATH0023: C Programming
Semester 2
Credits: 6
Topic: Computing
Level: Level 1
Assessment: EX75 CW25
Requisites: Co MATH0015
Aims & Learning Objectives: Aims: To ensure students
appreciate the concept of an algorithm as an effective procedure.
To introduce criteria by which algorithms may be chosen, and to
demonstrate non-obvious algorithms. To provide practical skills
at reading and writing programs in ISO Standard C.
Objectives: Students should be able to determine the time and
space complexity of short algorithms, and know 3 sorting algorithms
and 2 searching algorithms. Students should be able to design,
construct and test short programs in C, using standard libraries
as appropriate. They should be able to read and comprehend the
behaviour of programs written by others.
Content: Algorithms: Introduction: Definition of an algorithm
and characteristics of them. Basic Complexity: The efficiency
of different algorithmic solutions. Best, average and worst case
complexity in time and space. Fundamental Algorithms: Sorting.
Searching. Space-time trade-offs. Graphs. Dijkstra's shortest
path.
C Programming: Introduction: C as a simplified programming language;
ISO Standards.
Basic Concepts: Functions, variables, weak typing. Statements
and expressions.
Data Structuring: Enumeration, struct and arrays. Pointers and
construction of complex structures. The preprocessor: #include,
#if and #define Programming: Input-output. Use of standard libraries.
Multiple file programs. User interfaces. Professionalism: Coding
standards, defensive programming, documentation, testing. Ethics.
MATH0024: Information management 2
Semester 2
Credits: 6
Topic: Computing
Level: Level 1
Assessment: EX50 CW25 OT25
Requisites:
Aims & Learning Objectives: Aims: To introduce students
to the use of a workstation, to wordprocessing, spreadsheets and
relational data bases, and to the basic ideas of computing, and
to the range of applications and
misapplications of computers in science. To give students some
experience of working in small
groups.
Objectives: Students should have a practical ability to use contemporary
information
management facilities. They should be able to write a good report,
and they should have the
confidence and the language to enable criticism of the use of
computers in science.
Content: Normal and Poisson distributions. A simple introduction
to confidence intervals and
hypothesis testing. Elementary tools for dealing with non-normal
data. An introduction to
correlation. Computational experiments.
Data bases. Notations of set theory. Data types and structures.
Hierarchical, network, and relational data bases. Some natural
operations on relations: union, projection, selection, Cartesian
product, set difference. Design of relational data bases. Access
as an example of a data base system. The integrated use of word
processing, spreadsheets and relational data bases.
MATH0025: Machine architectures, assemblers & low-level
programming
Semester 2
Credits: 6
Topic: Computing
Level: Level 1
Assessment: EX100
Requisites: Co MATH0017
Aims & Learning Objectives: Aims: To introduce students
to the structure, basic design, operation and programming of conventional,
von Neumann computers at the machine level. Alternative approaches
to machine design will also be examined so that some recent machine
architectures can be introduced. In particular the course will
develop to explore the relationships between what actually happens
at the machine level and important ideas about, for example, aspects
of high-level programming and data structures, that students encounter
on parallel courses.
Objectives: Development of a critical awareness that what happens
at machine level is strongly related to the forms and conventions
developed at higher levels of programming. Reinforcement of structured
programming by practical development of low-level programming
skills that can be related to high-level practice.
Awareness of the potential advantages and disadvantages of different
architectures; appreciation of the importance of the synergistic
relationship between hardware and system software, e.g. in operating
systems. A launch point for more advanced architecture studies.
Content: Low-level programming and structures: A more detailed
examination of machine architecture and facilities, exemplified
by the 68000 series. Further exploration of different modes of
operand addressing; the implementation of program control mechanisms;
and subroutines. The relationship between the low-level and aspects
of high-level, structured programming such as decisions, loops
and modules; nested and recursive routines and conventions for
parameter transmission at high and low levels will be examined
(supported by practical programming work which may continue throughout
the semester).
Aspects of modern computer architectures: Non von Neumann architectures
and modern approaches to machine design, including , for example,
RISC (vs. CISC) architectures. Topics in contemporary machine
design, such as pipelining; parallel processing and multiprocessors.
The interaction between hardware and software.
MATH0026: Projects & their management
Semester 2
Credits: 6
Topic: Computing
Level: Level 2
Assessment: CW100
Requisites: Co MATH0018
Aims & Learning Objectives: Aims: To gain experience
of working with other people and, on a small-scale, some of the
problems that arise in the commercial development of software.
To appreciate the personal, corporate and public interest ethical
problems arising from all aspects of computer systems. To distinguish
between scientific and pseudo-scientific modes of presentation,
and to encourage competence in the scientific mode.
Objectives: To carry out the full cycle of the first phase of
development of a software package, namely; requirements analysis,
design, implementation, documentation and delivery. To know the
main terms of the Data Protection Act and be able to explain its
application in a variety of contexts. To be able to design a presentation
for a given audience. To be able to assess a presentation critically.
Content: Project Management: Software engineering techniques,
Controlling software development, Project planning/ Management,
Documentation, Design, Quality Assurance, Testing.
Professional Issues: Ethical and legal matters in the context
of information technology. Personal responsibilities: to employer,
society, self. Professional responsibilities: codes of professional
practice, Chartered Engineers. Legal responsibilities: Data Protection
Act, Computer Misuse Act, Consumer Protection Act. Intellectual
property rights. Whistle-blowing. Libel and slander. Confidentiality.
Contracts.
Presentation Skills: How to construct a good explanation. How
to construct a good presentation. Sales and manipulative techniques,
theatre, and scientific clarity. Active listening and reading.
Some items in the charlatan's toolkit: jargon, pseudo-mathematics,
ambiguity.
MATH0027: Object-oriented mechanisms
Semester 2
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0019
Aims & Learning Objectives: Aims: To provide a grounding
in the principles behind object oriented languages and how they
are realised, in order to enable the student both to use any object
oriented language and to use any language in an object oriented
way.
Objectives: To be able to classify a given object oriented language
into the categories identified above, to describe the differences
between those categories and to know the principles involved in
implementing a language belonging to any one of those categories.
Given a problem description, to be able to design suitable class
hierarchies. To be able to read, understand and write programs
in C++ and EuLisp.
Content: Introduction: definition of inheritance and identification
of the subclasses of the family of OO languages. Simple (single)
inheritance. Extending arithmetic: Complex number arithmetic in
C++ (overloading, message-passing) and EuLisp (generics). Sequence
and iterators: For classical data structures (list, vector) in
C++ and EuLisp. Polymorphism. Integration of user-defined sequence
classes. Modelling OO mechanisms: Modelling message passing and
class hierarchies. A method determination algorithm for generic
functions. Advanced topics: Multiple inheritance and the superclass
linearization problem. Meta-object protocols
MATH0028: Algorithms
Semester 2
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0020
Aims & Learning Objectives: Aims: To present a detailed
account of some fundamentally important and widely used algorithms.
To induce an appreciation of the design and implementation of
a selection of algorithms.
Objectives: To lean the general principles of effective algorithms
design and analysis on some famous examples, which are used as
fundamental subroutines in major computational procedures. To
be able to apply these principles in the development of algorithms
and make an informed choice between basic subroutines and data
structures.
Content: Algorithms and complexity. Main principles of
effective algorithms design: recursion, divide-and-conquer, dynamic
programming. Sorting and order statistics. Strassen's algorithm
for matrix multiplication and solving systems of linear equations.
Arithmetic operations over integers and polynomials (including
Karatsuba's algorithm), Fast Fourier Transform method. Greedy
algorithms. Basic graph algorithms: minimum spanning trees, shortest
paths, network flows. Number-theoretic algorithms: integer factorization,
primality testing, the RSA public key cryptosystem.
MATH0029: Compilers
Semester 2
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0015, Pre MATH0020
Aims & Learning Objectives: Aims: to give an introduction
to the processes involved in compilation and the use of C-based
compiler generation tools.
Objectives: to know the phases of the compilation process and
how to implement them. To be able to choose between different
techniques and different representations, depending on the problem
to be solved.
Content: Formal grammars, lexical analysis using lex, parsing
by recursive descent and by yacc, error handling in the parsing
process, intermediate code representations, type checking, code
generation using a code generator generator (burg).
MATH0030: History, heresy & heretics
Semester 2
Credits: 6
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites:
Aims & Learning Objectives: Aims: To inform students
of the rapid change in computing via an analysis of the history
and development of the computing industry and subject. The course
aims to do two things. First, to remove the almost mystical belief
that computers can do anything. Secondly, to encourage students
to question the appropriateness of computer systems as a solution
to any given problem.
Objectives: Describe the major trends and changes in hardware,
programming languages and software; explain the evolution of the
computing industry; extrapolate current trends in the industry,
while realising the weakness of extrapolation. Students should
be able to demonstrate reasoned arguments for and against the
use of computer technology. They should be able to compare machine
and human intelligence. They should understand the dangers of
compulsive use of computers; and the hazards that a computer solution
may introduce.
Content: The pre-history (Pascal, Babbage, Turing etc.).
1940s and 1950s: the birth of an industry and a subject. Semiconductor
technology and its evolution. 1960s and 1970s: the 'range' concept;
IBM and the Seven Dwarfs; high-level languages; operating systems;
the growth of on-line access. The rise of the mini-computer: workstations
and Unix; growth of networking. 'Professionalism'. The PC Market;
Intel and Microsoft. Where we are now. What computers do; what
programmers do. Machines: engineering a computer system. Humans:
language, understanding and reason. Human and machine problem
solving: Eliza-like systems, artificial intelligence. Programming
as a compulsion.
MATH0031: Statistics & probability 1
Semester 1
Credits: 6
Topic: Statistics
Level: Level 1
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: To introduce some
basic concepts in probability and statistics.
Objectives: Ability to perform an exploratory analysis of a data
set, apply the axioms and laws of probability, and compute quantities
relating to discrete probability distributions
Content: Descriptive statistics: Histograms, stem-and-leaf
plots, box plots. Measures of location and dispersion. Scatter
plots.
Probability: Sample space, events as sets, unions and intersections.
Axioms and laws of probability. Probability defined through symmetry,
relative frequency and degree of belief. Conditional probability,
independence. Bayes' Theorem. Combinations and permutations.
Discrete random variables: Bernoulli and Binomial distributions.
Mean and variance of a discrete random variable. Poisson distribution,
Poisson approximation to the binomial distribution, introduction
to the Poisson process. Geometric distribution. Hypergeometric
distribution. Negative binomial distribution. Bivariate discrete
distributions including marginal and conditional distributions.
Expectation and variance of discrete random variables. General
properties including expectation of a sum, variance of a sum of
independent variables. Covariance. Probability generating function.
Introduction to the random walk.
Students must have A-level Mathematics, Grade B or better in order
to undertake this unit.
MATH0032: Statistics & probability 2
Semester 2
Credits: 6
Topic: Statistics
Level: Level 1
Assessment: EX100
Requisites: Pre MATH0031
Aims & Learning Objectives: Aims: To introduce further
concepts in probability and statistics.
Objectives: Ability to compute quantities relating to continuous
probability distributions, fit certain types of statistical model
to data, and be able to use the MINITAB package.
Content: Continuous random variables: Density functions
and cumulative distribution functions. Mean and variance of a
continuous random variable. Uniform, exponential and normal distributions.
Normal approximation to binomial and continuity correction. Fact
that the sum of independent normals is normal. Distribution of
a monotone transformation of a random variable.
Fitting statistical models: Sampling distributions, particularly
of sample mean. Standard error. Point and interval estimates.
Properties of point estimators including bias and variance. Confidence
intervals: for the mean of a normal distribution, for a proportion.
Opinion polls. The t-distribution; confidence intervals
for a normal mean with unknown variance.
Regression and correlation: Scatter plot. Fitting a straight line
by least squares. The linear regression model. Correlation.
MATH0033: Statistical inference 1
Semester 1
Credits: 6
Topic: Statistics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0031, Pre MATH0032
Aims & Learning Objectives: Aims: Introduce classical
estimation and hypothesis-testing principles.
Objectives: Ability to perform standard estimation procedures
and tests on normal data. Ability to carry out goodness-of-fit
tests, analyse contingency tables, and carry out non-parametric
tests.
Content: Point estimation: Maximum-likelihood estimation;
further properties of estimators, including mean square error,
efficiency and consistency; robust methods of estimation such
as the median and trimmed mean.
Interval estimation: Revision of confidence intervals.
Hypothesis testing: Size and power of tests; one-sided and two-sided
tests. Examples. Neyman-Pearson lemma.
Distributions related to the normal: t, chi-square and
F distributions.
Inference for normal data: Tests and confidence intervals for
normal means and variances, one-sample problems, paired and unpaired
two-sample problems. Contingency tables and goodness-of-fit tests.
Non-parametric methods: Sign test, signed rank test, Mann-Whitney
U-test.
MATH0034: Probability & random processes
Semester 1
Credits: 6
Topic: Statistics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0032
Aims & Learning Objectives: Aims: Knowledge and understanding
of the statements of the three classical limit theorems of probability.
Familiarity with the main results of discrete-time branching processes.
Knowledge of the main properties of random walks on the integers.
Knowledge of the various equivalent characterisations of the Poisson
process.
Objectives: Ability to perform computations concerning branching
processes, random walks, and Poisson processes. Ability to use
generating function techniques for effective calculations.
Content: Revision of properties of expectation. Chebyshev's
inequality. The Weak Law. Martingales. Statement of the Strong
Law of Large Numbers. Random variables on the positive integers.
Branching processes. Random walks expected first passage times.
Poisson processes: inter-arrival times, the gamma distribution.
Moment generating functions. Outline of the Central Limit Theorem.
MATH0035: Statistical inference 2
Semester 2
Credits: 6
Topic: Statistics
Level: Level 2
Assessment: EX75 CW25
Requisites: Co MATH0033
Aims & Learning Objectives: Aims: Introduce the principles
of building and analysing linear models.
Objectives: Ability to carry out analyses using linear Gaussian
models, including regression and ANOVA. Understand the principles
of statistical modelling.
Content: One-way analysis of variance (ANOVA): One-way
classification model, F-test, comparison of group means.
Regression: Estimation of model parameters, tests and confidence
intervals, prediction intervals, polynomial and multiple regression.
Two-way ANOVA: Two-way classification model. Main effects and
interaction, parameter estimation, F- and t-tests.
Discussion of experimental design.
Principles of modelling: Role of the statistical model. Critical
appraisal of model selection methods. Use of residuals to check
model assumptions: probability plots, identification and treatment
of outliers.
Multivariate distributions: Joint, marginal and conditional distributions;
expectation and variance-covariance matrix of a random vector;
statement of properties of the bivariate and multivariate normal
distribution. The general linear model: Vector and matrix notation,
examples of the design matrix for regression and ANOVA, least
squares estimation, internally and externally Studentized residuals.
MATH0036: Stochastic processes
Semester 2
Credits: 6
Topic: Statistics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0034
Aims & Learning Objectives: Aims: To present a formal
description of Markov chains and Markov processes, their qualitative
properties and ergodic theory. To apply results in modelling real
life phenomena, such as biological processes, queueing systems,
renewal problems and machine repair problems.
Objectives: On completing the course, students should be able
to
* classify the states of a Markov chain, find hitting probabilities
and ergodic distributions
* calculate waiting time distributions, transition probabilities
and limiting behaviour of various Markov processes
Content: Markov chains with discrete states in discrete
time: Examples, including random walks. The Markov 'memorylessness'
property, P-matrices, n-step transition probabilities, hitting
probabilities, classification of states, symmetrizabilty, invariant
distributions and ergodic theorems.
Markov processes with discrete states in continuous time: Examples,
including the Poisson process, birth and death processes, branching
processes and various types of Markovian queues. Q-matrices, resolvents
waiting time distributions, equilibrium distributions and ergodicity.
MATH0037: Galois theory
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0012
Aims & Learning Objectives: Aims This course develops
the basic theory of rings and fields and expounds the fundamental
theory of Galois on solvability of polynomials.
Objectives At the end of the course, students will be conversant
with the algebraic structures associated to rings and fields.
Moreover, they will be able to state and prove the main theorems
of Galois Theory as well as compute the Galois group of simple
polynomials.
Content: Rings, integral domains and fields.
Field of quotients of an integral domain. Ideals and quotient
rings. Rings of polynomials. Division algorithm and unique factorisation
of polynomials over a field.
Extension fields. Algebraic closure. Splitting fields. Normal
field extensions. Galois groups. The Galois correspondence.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN
YEAR.
MATH0038: Advanced group theory
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0012
Aims & Learning Objectives: Aims This course provides
a solid introduction to modern group theory covering both the
basic tools of the subject and more recent developments.
Objectives At the end of the course, students should be able to
state and prove the main theorems of classical group theory and
know how to apply these. In addition, they will have some appreciation
of the relations between group theory and other areas of mathematics.
Content: Topics will be chosen from the following:
Review of elementary group theory: homomorphisms, isomorphisms
and Lagrange's theorem. Normalisers, centralisers and conjugacy
classes. Group actions. p-groups and the Sylow theorems.
Cayley graphs and geometric group theory. Free groups. Presentations
of groups. Von Dyck's theorem. Tietze transformations.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD
YEAR.
MATH0039: Differential geometry of curves & surfaces
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0011,Pre MATH0012
Aims & Learning Objectives: Aims This will be a self-contained
course which uses little more than elementary vector calculus
to develop the local differential geometry of curves and surfaces
in IR³. In this way, an accessible introduction is given
to an area of mathematics which has been the subject of active
research for over 200 years.
Objectives At the end of the course, the students will be able
to apply the methods of calculus with confidence to geometrical
problems. They will be able to compute the curvatures of curves
and surfaces and understand the geometric significance of these
quantities.
Content: Topics will be chosen from the following:
Tangent spaces and tangent maps. Curvature and torsion of curves:
Frenet--Serret formulae. The Euclidean group and congruences.
Curvature and torsion determine a curve up to congruence.
Global geometry of curves: isoperimetric inequality; four-vertex
theorem.
Local geometry of surfaces: parametrisations of surfaces; normals,
shape operator, mean and Gauss curvature. Geodesics, integration
and the local Gauss--Bonnet theorem.
MATH0040: Algebraic topology
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0012, Pre MATH0055
Aims & Learning Objectives: Aims The course will provide
a solid introduction to one of the Big Machines of modern mathematics
which is also a major topic of current research. In particular,
this course provides the necessary prerequisites for post-graduate
study of Algebraic Topology.
Objectives At the end of the course, the students will be conversant
with the basic ideas of homotopy theory and, in particular, will
be able to compute the fundamental group of several topological
spaces.
Content: Topics will be chosen from the following:
Paths, homotopy and the fundamental group. Homotopy of maps; homotopy
equivalence and deformation retracts. Computation of the fundamental
group and applications: Fundamental Theorem of Algebra; Brouwer
Fixed Point Theorem.
Covering spaces. Path-lifting and homotopy lifting properties.
Deck translations and the fundamental group. Universal covers.
Loop spaces and their topology. Inductive definition of higher
homotopy groups. Long exact sequence in homotopy for fibrations.
MATH0041: Metric spaces
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0011, Co MATH0043
Aims & Learning Objectives: Aims This core course is
intended to be an elementary and accessible introduction to the
theory of metric spaces and the topology of IRn for
students with both "pure" and "applied" interests.
Objectives While the foundations will be laid for further studies
in Analysis and Topology, topics useful in applied areas such
as the Contraction Mapping Principle will also be covered. Students
will know the fundamental results listed in the syllabus and have
an instinct for their utility in analysis and numerical analysis.
Content: Definition and examples of metric spaces. Convergence
of sequences. Continuous maps and isometries. Sequential definition
of continuity. Subspaces and product spaces. Complete metric spaces
and the Contraction Mapping Principle. Sequential compactness,
Bolzano-Weierstrass theorem and applications. Open and closed
sets (with emphasis on IRn). Closure and interior of
sets. Topological approach to continuity and compactness (with
statement of Heine-Borel theorem). Connectedness and path-connectedness.
Metric spaces of functions: C[0,1] is a complete metric
space.
MATH0042: Measure theory & integration
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0012, Pre MATH0041
Aims & Learning Objectives: Aims The purpose of this
course is to lay the basic technical foundations and establish
the main principles which underpin the classical notions of area,
volume and the related idea of an integral.
Objectives The objective is to familiarise students with measure
as a tool in analysis, functional analysis and probability theory.
Students will be able to quote and apply the main inequalities
in the subject, and to understand their significance in a wide
range of contexts. Students will obtain a full understanding of
the Lebesgue Integral.
Content: Topics will be chosen from the following:
Measurability for sets: algebras, s-algebras,
p-systems, d-systems; Dynkin's
Lemma; Borel s-algebras. Measure in
the abstract: additive and s-additive
set functions; monotone-convergence properties; Uniqueness Lemma;
statement of Caratheodory's Theorem and discussion of the l-set
concept used in its proof; full proof on handout. Lebesgue measure
on iRn: existence; inner and outer regularity. Measurable
functions. Sums, products, composition, lim sups, etc; The Monotone-Class
Theorem. Probability. Sample space, events, random variables.
Independence; rigorous statement of the Strong Law for coin tossing.
Integration. Integral of a non-negative functions as sup of the
integrals of simple non-negative functions dominated by it. Monotone-Convergence
Theorem; 'Additivity'; Fatou's Lemma; integral of 'signed' function;
definition of Lp and of Lp; linearity;
Dominated-Convergence Theorem - with mention that it is not the
`right' result. Product measures: definition; uniqueness; existence;
Fubini's Theorem. Absolutely continuous measures: the idea; effect
on integrals. Statement of the Radon-Nikodým Theorem. Inequalities:
Jensen, Hölder, Minkowski. Completeness of Lp.
MATH0043: Real & abstract analysis
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0011, Pre MATH0012,
Co MATH0041
Aims & Learning Objectives: Aims To introduce and study
abstract spaces and general ideas in analysis, to apply them to
examples, and to lay the foundations for the year 4 blocks in
functional analysis and Lebesgue integral.
Objectives By the end of the block, students should be able to
state and prove the principal theorems relating to uniform continuity
and uniform convergence for real functions on metric spaces, compactness
in spaces of continuous functions, and elementary Hilbert space
theory, and to apply these notions and the theorems to simple
examples.
Content: Topics will be chosen from:
Uniform continuity and uniform limits of continuous functions
on [0,1]. Abstract Stone-Weierstrass Theorem. Uniform approximation
of continuous functions. Polynomial and trigonometric polynomial
approximation, separability of C[0,1]. Total Boundedness.
Diagonalisation. Ascoli-Arzelà Theorem. Complete metric
spaces. Baire Category Theorem. Nowhere differentiable function.
Picard's theorem for c = f(c).
Metric completion M of a metric space
M. Real inner-product spaces. Hilbert spaces. Cauchy-Schwarz
inequality, parallelogram identity. Examples: l²,
L²[0,1] := C[0,1]. Separability of L²
. Orthogonality, Gram-Schmidt process. Bessel's inquality, Pythagoras'
Theorem. Projections and subspaces. Orthogonal complements. Riesz
Representation Theorem. Complete orthonormal sets in separable
Hilbert spaces. Completeness of tirgonometric polynomials in L²
[0,1]. Fourier Series.
MATH0044: Mathematical methods 1
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0009, Pre MATH0010, Pre MATH0012
Aims & Learning Objectives: Aims To furnish the student
with a range of analytic techniques for the solution of ODEs and
PDEs.
Objectives Students should be able to obtain the solution of certain
ODEs and PDEs. They should also be aware of certain analytic properties
associated with the solution e.g. uniqueness.
Content: Sturm-Liouville theory: Reality of eigenvalues.
Orthogonality of eigenfunctions. Expansion in eigenfunctions.
Approximation in mean square. Statement of completeness.
Fourier Transform: As a limit of Fourier series. Properties and
applications to solution of differential equations. Frequency
response of linear systems. Characteristic functions.
Linear and quasi-linear first-order PDEs in two and three independent
variables: Characteristics. Integral surfaces. Uniqueness (without
proof).
Linear and quasi-linear second-order PDEs in two independent variables:
Cauchy-Koivalevskii theorem (without proof). Characteristic data.
Lack of continuous dependence on initial data for Cauchy problem.
Classification as elliptic, parabolic, and hyperbolic. Different
standard forms. Constant and nonconstant coefficients.
One-dimensional wave equation: d'Alembert's solution. Uniqueness
theorem for corresponding Cauchy problem (with data on a spacelike
curve).
MATH0045: Dynamical systems
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0009, Pre MATH0011,
Pre MATH0012, Pre MATH0041, Pre MATH0062
Aims & Learning Objectives: Aims A treatment of the
qualitative/geometric theory of dynamical systems to a level that
will make accessible an area of mathematics (and allied disciplines)
that is highly active and rapidly expanding.
Objectives Conversance with concepts, results and techniques fundamental
to the study of qualitative behaviour of dynamical systems. An
ability to investigate stability of equilibria and periodic orbits.
A basic understanding and appreciation of bifurcation and chaotic
behaviour
Content: Topics will be chosen from the following:
Stability of equilibria. Lyapunov functions. Invariance principle.
Periodic orbits. Poincaré maps. Hyperbolic equilibria and
orbits. Stable and unstable manifolds. Nonhyperbolic equilibria
and orbits. Centre manifolds. Bifurcation from a simple eigenvalue.
Introductory treatment of chaotic behaviour. Horseshoe maps. Symbolic
dynamics.
MATH0046: Linear control theory
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0009, Pre MATH0011,
Pre MATH0012
Aims & Learning Objectives: Aims The course is intended
to provide an elementary and assessible introduction to the state-space
theory of linear control systems. Main emphasis is on continuous-time
autonomous systems, although discrete-time systems will receive
some attention through sampling of continuous-time systems. Contact
with classical (Laplace-transform based) control theory is made
in the context of realization theory.
Objectives To instill basic concepts and results from control
theory in a rigorous manner making use of elementary linear algebra
and linear ordinary differential equations. Conversance with controllability,
observability, stabilizabilty and realization theory in a linear,
finite-dimensional context.
Content: Topics will be chosen from the following:
Controlled and observed dynamical systems: definitions and classifications.
Controllability and observability: Gramians, rank conditions,
Hautus criteria. Controllable and unobservable subspaces. Input-output
maps. Transfer functions and state-space realizations. State feedback:
stabilizability and pole placement. Observers and output feedback:
detectability, asymptotic state estimation, stabilization by dynamic
feedback. Discrete-time systems: z-transform. Deadbeat
control and observation. Sampling of continuous-time systems:
controllability and observability under sampling.
MATH0047: Mathematical biology 1
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW12
Requisites: Pre MATH0009, Pre MATH0013
Aims & Learning Objectives: Aims The purpose of this
course is to introduce students to problems which arise in biology
which can be tackled using applied mathematics. Emphasis will
be laid upon deriving the equations describing the biological
problem and at all times the interplay between the mathematics
and the underlying biology will be brought to the fore.
Objectives Students should be able to derive a mathematical model
of a given problem in biology using ODEs and give a qualitative
account of the type of solution expected. They should be able
to interpret the results in terms of the original biological problem.
Content: Topics will be chosen from the following:
Difference equations: Steady states and fixed points. Stability.
Period doubling bifurcations. Chaos. Application to population
growth.
Systems of difference equations: Host-parasitoid systems.
Systems of ODEs: Stability of solutions. Critical points. Phase
plane analysis. Poincaré-Bendixson theorem. Bendixson and
Dulac negative criteria. Conservative systems. Structural stability
and instability. Lyapunov functions.
Prey-predator models
Epidemic models
Travelling wave fronts: Waves of advance of an advantageous gene.
Waves of excitation in nerves. Waves of advance of an epidemic.
MATH0048: Analytic mechanics
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0009, Pre MATH0010,
Pre MATH0011, Pre MATH0012, Pre MATH0013, Pre MATH0062
Aims & Learning Objectives: Aims To give a unified
presention of classical problems of particle and rigid body mechanics
in the light of deeper, more abstract methods of modern mathematics.
To present principles, concepts and examples which provide the
basis, and much of the language, for field theories in physics
(e.g. quantum mechanics, continuum mechanics, relativity) and
currently popular topics in modern applied mathematics (e.g. dynamical
systems and ergodic theory).
Objectives Students will be able to state and prove general theorems
in Lagrangian and Hamiltonian mechanics. Based on these theoretical
results and key motivating examples from classical mechanics they
will identify general qualitative properties of solutions to systems
of ordinary differential equations which have a Lagrangian or
Hamiltonian structure.
Content: Topics will be chosen from the following:
Newtonian mechanics, phase space, phase flow, Lagrangian mechanics,
variational principles, Euler-Lagrange equations, Hamilton's Principle
of least action, Legendre transform, Hamilton's equations, Liouville's
Theorem, Poincaré recurrence theorem, Poisson brackets,
Noether's Theorem. Hamilton-Jacobi theory.
MATH0049: Linear elasticity
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0010, Pre MATH0065
Aims & Learning Objectives: Aims To provide an introduction
to the mathematical modelling of the behaviour of solid elastic
materials.
Objectives Students should be able to derive the governing equations
of the theory of linear elasticity and be able to solve simple
problems.
Content: Topics will be chosen from the following:
Revision: Kinematics of deformation, stress analysis, global balance
laws, boundary conditions.
Constitutive law: Properties of real materials; constitutive law
for linear isotropic elasticity, Lame moduli; field equations
of linear elasticity; Young's modulus, Poisson's ratio.
Some simple problems of elastostatics: Expansion of a spherical
shell, bulk modulus; deformation of a block under gravity; elementary
bending solution.
Linear elastostatics: Strain energy function; uniqueness theorem;
Betti's reciprocal theorem, mean value theorems; variational principles,
application to composite materials; torsion of cylinders, Prandtl's
stress function.
Linear elastodynamics: Basic equations and general solutions;
plane waves in unbounded media, simple reflection problems; surface
waves.
MATH0050: Nonlinear equations & bifurcations
Semester 1
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX75 CW25
Requisites: Pre MATH0051, Pre MATH0041
Aims & Learning Objectives: Aims To extend the real
analysis of implicitly defined functions into the numerical analysis
of iterative methods for computing such functions and to teach
an awareness of practical issues involved in applying such methods.
Objectives The students should be able to solve a variety of nonlinear
equations in many variables and should be able to assess the performance
of their solution methods using appropriate mathematical analysis.
Content: Topics will be chosen from the following:
Solution methods for nonlinear equations: Review of Newton's method
for systems. Quasi-Newton Methods.
Theoretical Tools: Local Convergence of Newton's Method. Implicit
Function Theorem. Bifurcation from the trivial solution.
Applications: Exothermic reaction and buckling problems. Continuous
and discrete models.
Analysis of parameter-dependent two-point boundary value problems
using the shooting method. Practial use of the shooting method.
The Lyapunov-Schmidt Reduction. Application to analysis of discretised
boundary value problems.
Computation of solution paths for systems of nonlinear algebraic
equations. Pseudo-arclength continuation. Homotopy methods. Computation
of turning points. Bordered systems and their solution.
Topics from: Exploitation of symmetry or Hopf bifurcation.
MATH0051: Numerical linear algebra
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0008, Pre MATH0010, Pre MATH0012, Pre MATH0014
Aims & Learning Objectives: Aims To teach an understanding
of iterative methods for standard problems of linear algebra.
Objectives Students should know a range of modern iterative methods
for solving linear systems and for solving the algebraic eigenvalue
problem. They should be able to anayse their algorithms and should
have an understanding of relevant practical issues.
Content: Topics will be chosen from the following:
The algebraic eigenvalue problem: Gerschgorin's theorems. The
power method and its extensions. Backward Error Analysis (Bauer-Fike).
The (Givens) QR factorization and the QR method for symmetric
tridiagonal matrices. (Statement of convergence only). The Lanczos
Procedure for reduction of a real symmetric matrix to tridiagonal
form. Orthogonality properties of Lanczos iterates.
Iterative Methods for Linear Systems: Convergence of stationary
iteration methods. Special cases of symmetric positive definite
and diagonally dominant matrices. Variational principles for linear
systems with real symmetric matrices. The conjugate gradient method.
Krylov subspaces. Convergence. Connection with the Lanczos method.
Iterative Methods for Nonlinear Systems: Newton's Method. Convergence
in 1D. Statement of algorithm for systems.
MATH0052: Algebra & combinatorics
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0012
Aims & Learning Objectives: Aims This course provides
an accessible introduction to various ideas in discrete mathematics
based around the idea of counting arguments. As such, it will
give an overview of the methods of modern algebra and their application
for students who do not intend to become specialists in this area.
Objectives At the end of the course, students will be proficient
in applying a variety of algebraic techniques to solve combinatorial
problems arising in Mathematics and related disciplines.
Content: Topics will be chosen from the following:
Graphs, Trees and Forests. Philip Hall's marriage theorem. Möbius
inversion and multiplicative functions in number theory. Finite
fields and cyclotomic polynomials. Quadratic Reciprocity. Linear
recurrences over finite fields and applications of quadratic reciprocity.
Random functions and factoring methods.
MATH0053: Algebraic number theory
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0037
Aims & Learning Objectives: Aims This course will provide
a solid introduction to Algebraic Number Theory, both as a subject
in its own right and as a source of applications to Computer Science.
Objectives Students completing the course should understand algebraic
numbers, how unique factorization fails, and how it can be restored
by using "ideal numbers".
Content: Topics will be chosen from the following:
Quadratic reciprocity. Noetherian rings, Dedekind domains, algebraic
number fields and rings of algebraic integers. Primes and irreducibles.
Ramification of primes. Norms and traces. Integral bases.
Class groups and the class number formula. Dirichlet's units theorem.
Applications of Galois Theory. The method of Minkowski.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN
YEAR.
MATH0054: Representation theory of finite groups
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0038
Aims & Learning Objectives: Aims The course explains
some fundamental applications of linear algebra to the study of
finite groups. In so doing, it will show by example how one area
of mathematics and enhance and enrich the study of another.
Objectives At the end of the course, the students will be able
to state and prove the main theorems of Maschke and Schur and
be conversant with their many applications in representation theory
and character theory. Moreover, they will be able to apply these
results to problems in group theory.
Content: Topics will be chosen from the following:
Group algebras, their modules and associated representations.
Maschke's theorem and complete reducibility. Irreducible representations
and Schur's lemma. Decomposition of the regular representation.
Character theory and orthogonality theorems. Burnside's pa
qb theorem.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD
YEAR.
MATH0055: Introduction to topology
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0041
Aims & Learning Objectives: Aims To provide an introduction
to the ideas of point-set topology culminating with a sketch of
the classification of compact surfaces. As such it provides a
self-contained account of one of the triumphs of 20th century
mathematics as well as providing the necessary background for
Year 4 courses in Algebraic Topology and Functional Analysis.
Objectives To acquaint students with the important notion of a
topology and to familiarise them with the basic theorems of analysis
in their most general setting. Students will be able to distinguish
between metric and topological space theory and to understand
refinements, such as Hausdorff or compact spaces, and their applications.
Content: Topics will be chosen from the following:
Topologies and topological spaces. Subspaces. Bases and sub-bases:
product spaces; compact-open topology. Continuous maps and homeomorphisms.
Separation axioms. Connectedness. Compactness and its equivalent
characterisations in a metric space. Axiom of Choice and Zorn's
Lemma. Tychonoff's theorem. Quotient spaces. Compact surfaces
and their representation as quotient spaces. Sketch of the classification
of compact surfaces.
MATH0056: Complex analysis
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0011
Aims & Learning Objectives: Aims The aim of this course
is to cover the standard introductory material in the theory of
functions of a complex variable and to cover complex function
theory up to Cauchy's Residue Theorem and its applications.
Objectives Students should end up familiar with the theory of
functions of a complex variable and be capable of calculating
and justifying power series, Laurent series, contour integrals
and applying them.
Content: Topics will be chosen from the following:
Functions of a complex variable. Continuity. Complex series and
power series. Circle of convergence. The complex plane. Regions,
paths, simple and closed paths. Path-connectedness. Analyticity
and the Cauchy-Riemann equations. Harmonic functions. Cauchy's
theorem. Cauchy's Integral Formulae and its application to power
series. Isolated zeros. Differentiability of an analytic function.
Liouville's Theorem. Zeros, poles and essential singularities.
Laurent expansions. Cauchy's Residue Theorem and contour integration.
Applications to real definite integrals.
MATH0057: Functional analysis
Semester 2
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0041, Pre MATH0043
Aims & Learning Objectives: Aims To introduce the theory
of infinite-dimensional normed vector spaces, the linear mappings
between them, and spectral theory.
Objectives By the end of the block, the students should be able
to state and prove the principal theorems relating to Banach spaces,
bounded linear operators, compact linear operators, and spectral
theory of compact self-adjoint linear operators, and apply these
notions and theorems to simple examples.
Content: Topics will be chosen from the following:
Normed vector spaces and their metric structure. Banach spaces.
Young, Mikowski and Hölder inequalities. Examples - IRn,
C[0,1], Ip, Hilbert spaces. Riesz Lemma
and finite-dimensional subspaces. The space B(X,Y) of bounded
linear operators is a Banach space when Y is complete.
Dual spaces and second duals. Uniform Boundedness Theorem. Open
Mapping Theorem. Closed Graph Theorem. Projections onto closed
subspaces. Invertible operators form an open set. Power series
expansion for (I-T)-1. Compact operators on
Banach spaces. Spectrum of an operator - compactness of spectrum.
Operators on Hilbert space and their adjoints. Spectral theory
of self-adjoint compact operators. Zorn's Lemma. Hahn-Banach Theorem.
Canonical embedding of X in X** is isometric, reflexivity.
Simple applications to weak topologies.
MATH0058: Martingale theory
Semester 2
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0008, Pre MATH0012, Pre MATH0041, Pre MATH0042
Aims & Learning Objectives: Aims To stimulate through
theory and especially examples, an interest and appreciation of
the power of this elegant method in analysis and probability.
Applications of the theory are at the heart of this course.
Objectives By the end of the course, students should be familiar
with the main results and techniques of discrete time martingale
theory. They will have seen applications of martingales in proving
some important results from classical probability theory, and
they should be able to recognise and apply martingales in solving
a variety of more elementary problems.
Content: Topics will be chosen from the following:
Review of fundamental concepts. Conditional expectation. Martingales,
stopping times, Optional-Stopping Theorem. The Convergence Theorem.
L²-bounded martingales, the random-signs problem. Angle-brackets
process, Lévy's Borel-Cantelli Lemma. Uniform integrability.
UI martingales, the "Downward" Theorem, the Strong Law,
the Submartingale Inequality. Likelihood ratio, Kakutani's theorem.
MATH0059: Mathematical methods 2
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0044
Aims & Learning Objectives: Aims To introduce students
to the applications of advanced analysis to the solution of PDEs.
Objectives Students should be able to obtain solutions to certain
important PDEs using a variety of techniques e.g. Green's functions,
separation of variables. They should also be familiar with important
analytic properties of the solution.
Content: Topics will be chosen from the following:
Elliptic equations in two independent variables: Harmonic functions.
Mean value property. Maximum principle (several proofs). Dirichlet
and Neumann problems. Representation of solutions in terms of
Green's functions. Continuous dependence of data for Dirichlet
problem. Uniqueness.
Parabolic equations in two independent variables: Representation
theorems. Green's functions.
Self-adjoint second-order operators: Eigenvalue problems (mainly
by example). Separation of variables for inhomogeneous systems.
Green's function methods in general: Method of images. Use of
integral transforms. Conformal mapping.
Calculus of variations: Maxima and minima. Lagrange multipliers.
Extrema for integral functions. Euler's equation and its special
first integrals. Integral and non-integral constraints.
MATH0060: Nonlinear systems & chaos
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0009, Pre MATH0010,
Pre MATH0011, Pre MATH0012, Pre MATH0013, Pre MATH0014
Aims & Learning Objectives: Aims The course is intended
to be an elementary and accessible introduction to dynamical systems.
Main emphasis will be on discrete-time systems which permits the
concepts and results to be presented in a rigorous manner, within
the framework of the second year core material. Discrete-time
systems will be followed by an introductory treatment of continuous-time
systems and differential equations. Numerical approximation of
differential equations will link with the earlier material on
discrete-time systems.
Objectives An appreciation of the behaviour, and its potential
complexity, of general dynamical systems through a study of discrete-time
systems (which require relatively modest analytical prerequisites)
and computer experimentation.
Content: Topics will be chosen from the following:
Discrete-time systems. Maps from IRn to IRn
. Fixed points. Periodic orbits. a
and w limit sets. Local bifurcations
and stability. The logistic map and chaos. Global properties.
Continuous-time systems. Periodic orbits and Poincaré maps.
Numerical approximation of differential equations. Newton iteration
as a dynamical system.
MATH0061: Nonlinear & optimal control theory
Semester 2
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0046, Pre MATH0062, Pre MATH0041
Aims & Learning Objectives: Aims Four concepts underpin
control theory: controllability, observability, stabilizability
and optimality. Of these, the first two essentially form the focus
of the Year 3 course on linear control theory. In this course,
the latter notions of stabilizability and optimality are developed
within the analytical framework provided by the Year 4 courses
on metric spaces and ordinary differential equations. Together,
the courses on linear control theory and nonlinear & optimal
control provide a firm foundation for participating in theoretical
and practical developments in an active and expanding discipline.
Objectives To present concepts and results pertaining to robustness,
stabilization and optimization of (nonlinear) finite-dimensional
control systems in a rigorous manner. Emphasis is placed on optimization,
leading to conversance with both the Bellman-Hamilton-Jacobi approach
and the maximum principle of Pontryagin, together with their application.
Content: Topics will be chosen from the following:
Controlled dynamical systems: nonlinear systems and linearization.
Stability and robustness. Stabilization by feedback. Lyapunov-based
design methods. Stability radii. Small-gain theorem. Optimal control.
Value function. The Bellman-Hamilton-Jacobi equation. Verification
theorem. Quadratic-cost control problem for linear systems. Riccati
equations. The Pontryagin maximum principle and transversality
conditions (a dynamic programming derivation of a restricted version
and statement of the general result with applications). Proof
of the maximum principle for the linear time-optimal control problem.
MATH0062: Ordinary differential equations
Semester 2
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0011, Pre MATH0008, Pre MATH0013,
Pre MATH0009, Pre MATH0041
Aims & Learning Objectives: Aims To provide an accessible
but rigorous treatment of initial-value problems for nonlinear
systems of ordinary differential equations. Foundations will be
laid for advanced studies in dynamical systems and control. The
material is also useful in mathematical biology and numerical
analysis.
Objectives Conversance with existence theory for the initial-value
problem, locally Lipschitz righthand sides and uniqueness, flow,
continuous dependence on initial conditions and parameters, limit
sets.
Content: Topics will be chosen from the following:
Motivating examples from diverse areas. Existence of solutions
for the initial-value problem. Uniqueness. Maximal intervals of
existence. Dependence on initial conditions and parameters. Flow.
Global existence and dynamical systems. Limit sets and attractors.
MATH0063: Mathematical biology 2
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0013, Pre MATH0010, Pre MATH0044
Aims & Learning Objectives: Aims The aim of the course
is to introduce students to applications of partial differential
equations to model problems arising in biology. The course will
complement Mathematical Biology I where the emphasis was on ODEs
and Difference Equations.
Objectives Students should be able to derive and interpret mathematical
models of problems arising in biology using PDEs. They should
be able to perform a linearised stability analysis of a reaction-diffusion
system and determine criteria for diffusion-driven instability.
They should be able to interpret the results in terms of the original
biological problem.
Content: Topics will be chosen from the following:
Partial Differential Equation Models: Simple random walk derivation
of the diffusion equation. Solutions of the diffusion equation.
Density-dependent diffusion. Conservation equation. Reaction-diffusion
equations. Chemotaxis. Examples for insect dispersal and cell
aggregation.
Spatial Pattern Formation: Turing mechanisms. Linear stability
analysis. Conditions for diffusion-driven instability. Dispersion
relation and Turing space. Scale and geometry effects. Mode selection
and dispersion relation.
Applications: Animal coat markings. "How the leopard got
its spots". Butterfly wing patterns.
MATH0065: Viscous fluid mechanics
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0010, Pre MATH0013
Aims & Learning Objectives: Aims To introduce the general
theory of continuum mechanics and, through this, the study of
viscous fluid flow.
Objectives Students should be able to explain the basic concepts
of continuum mechanics such as stress, deformation and constitutive
relations, be able to formulate balance laws and be able to apply
these to the solution of simple problems involving the flow of
a viscous fluid.
Content: Topics will be chosen from the following:
Vectors: Linear transformation of vectors. Proper orthogonal transformations.
Rotation of axes. Transformation of components under rotation.
Cartesian Tensors: Transformations of components, symmetry and
skew symmetry. Isotropic tensors.
Kinematics: Transformation of line elements, deformation gradient,
Green strain. Linear strain measure. Displacement, velocity, strain-rate.
Stress: Cauchy stress; relation between traction vector and stress
tensor.
Global Balance Laws: Equations of motion, boundary conditions.
Newtonian Fluids: The constitutive law, uniform flow, Poiseuille
flow, flow between rotating cylinders.
MATH0066: Numerical solution of partial differential equations
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0010, Pre MATH0014
Aims & Learning Objectives: Aims To teach a broad understanding
of discretisation methods for elliptic, hyperbolic and parabolic
PDEs.
Objectives Students should be able to apply a range of standard
methods for the most important PDEs arising in applications and
should be able to perform an analysis of these methods applied
to model problems.
Content: Topics will be chosen from the following:
Introduction: examples of physically relevant PDEs and their associated
boundary conditions. Well--posed problems.
Finite difference methods for parabolic and hyperbolic PDEs. Consistency,
stability and convergence. Discrete maximum principles.
Finite element method: variational formulation of Poisson's equation.
Basis functions in one and two space dimensions. Assembly of the
stiffness matrix. Best approximation property. Convergence properties.
MATH0067: Numerical solution of boundary-value problems
Semester 2
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX75 CW25
Requisites: Pre MATH0007, Pre MATH0011, Pre MATH0051
Aims & Learning Objectives: Aims To teach the basic
notions behind the formulation and implementation of approximation
techniques for elliptic PDEs based on variational principles.
Objectives An ability to implement and analyse the finite element
method for a range of elliptic boundary value-problems.
Content: Topics will be chosen from the following:
Variational principles and weak forms of elliptic equations. Linear
and quadratic finite element approximation on triangles and quadrilaterals.
Stiffness matrix assembly. Isoparametric mapping. Quadrature.
Preconditioned conjugate gradient method.
Convergence theory for symmetric elliptic problems. Mixed boundary
conditions. Connection with the finite difference method. Discrete
maximum principles.
Extensions to be chosen from: Monotone semilinear problems, Convection-diffusion
problems, Obstacle problems.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN
YEAR.
MATH0068: Finite difference methods for evolutionary problems
Semester 2
Credits: 6
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX75 CW25
Requisites: Pre MATH0010, Pre MATH0014, Pre MATH0041
Aims & Learning Objectives: Aims To teach an understanding
of linear stability theory and its application to ODEs and evolutionary
PDEs.
Objectives The students should be able to analyse the stability
and convergence of a range of numerical methods and assess the
practical performance of these methods through computer experiments.
Content: Topics will be chosen from the following:
Solution of initial value problems for ODEs by Linear Multistep
methods: local accuracy, order conditions; formulation as a one--step
method; stability and convergence.
Introduction to physically relevant PDEs. Well--posed problems.
Truncation error; consistency, stability, convergence and the
Lax Equivalence Theorem; techniques for finding the stability
properties of particular numerical methods.
Numerical methods for parabolic and hyperbolic PDEs.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD
YEAR.
MATH0069: Programming language implementation techniques
Semester 2
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0029
Aims & Learning Objectives: Aims: To acquire an appreciation
of the suitability of different techniques for the analysis and
representations for programming languages, followed by the various
means to interpret them.
Objectives: To be able to choose suitable techniques for lexing,
parsing, type analysis, intermediate representation, transformation
and interpretation given the properties of the language to be
implemented.
Content: Construction of lexical analysers, recursive descent
parsing, construction of LR parser tables, type checking, polymorphic
type synthesis, continuation passing style, combinators, lambda
lifting, super-combinators, abstract interpretation, storage management,
byte-code interpreters, code-threaded interpreters, partial evaluation,
staging transformations.
MATH0070: Computer algebra
Semester 2
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX75 CW25
Requisites:
Aims & Learning Objectives: Aims: To show how computer
algebra can be used to solve some interesting mathematical problems
Objectives: To understand the practical possibilities and limitations
of symbolic computation, and to see how it is related to numerical
computation.
Content: Introduction to Reduce. Data representation questions.
Normal and canonical forms. Polynomials, algebraic numbers, elementary
numbers. Polynomial algebra: GCD and factorization algorithms,
modular methods. LLL algorithm. Numerical and symbolic methods
for solving systems of nonlinear equations: Newton, Wu's method,
Grbner bases. Introduction to integration.
Students must have A-level Mathematics, Grade B or better in order
to undertake this unit.
MATH0072: Safety-critical computer systems
Semester 1
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: To give an appreciation
of the current state of safe systems development. To develop an
understanding of risk in systems. To give a foundation in hazard
analysis models and techniques. To show how safety principles
may be built into all stages of the software development process.
Objectives: At the end of this course a student should be able
to demonstrate the
following skills: An understanding of the nature of risk in developing
computer-based
systems. The ability to choose and apply appropriate hazard analysis
models for simple safety-related problems. An understanding of
how to approach the design of safety-critical software
systems.
Content: The nature of risk: computers and risk; how accidents
happen; human error. System safety: historical approaches to system
safety; basic concepts and terminology. Managing the development
of safety-critical systems. Modelling human error and the accident
process. Hazard analysis: basic principles; models and techniques.
Safety principles in the software lifecycle: hazard analysis as
part of requirements analysis; designing for safety; designing
the human-machine interface; verification of safety in computer
systems.
MATH0073: Advanced algorithms & complexity
Semester 1
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0028
Aims & Learning Objectives: Aims: To present a detailed
introduction to one of the central concepts of combinatorial algorithmics:
NP-completeness; to extend this concept to real numbers computations;
to study foundations of a more general problem of proving lower
complexity bounds.
Objectives: to be able to recognise NP-hard problems and prove
the appropriate reductions. To cope with NP-complete problems.
To know some fundamental methods of proving lower complexity bounds.
Content: NP-completeness: Deterministic and Nondeterministic
Turing Machines; class NP; versions of reducibility; NP-hard and
NP-complete problems. Proof of NP-completeness of satisfiability
problem for Boolean formulae. Other NP-complete problems: clique,
vertex cover, travelling salesman, subgraph isomorphism, etc.
Polynomial-time approximation algorithms for travelling salesman
and some other NP-complete graph problems. Real Numbers Turing
machines: Definitions; completeness of real roots existence problem
for 4-degree polynomials. Lower complexity bounds: Straight-line
programs and their complexities; complexity of matrix-vector multiplication;
complexity of polynomial evaluation.
MATH0075: Advanced computer graphics
Semester 1
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX75 CW25
Requisites:
Aims & Learning Objectives: Aims: The primary aims
are to understand the ways of representing, rendering and displaying
pictures of three-dimensional objects (in particular). In order
to achieve this it will be necessary to understand the underlying
mathematics and computer techniques.
Objectives: Students will be able to distinguish modelling from
rendering. They will be able to describe the relevant components
of Euclidean and projective geometry and their relationships to
matrix algebra formulations. Students will know the difference
between solid- and surface-modelling and be able to describe typical
computer representations of each. Rendering for raster displays
will be explainable in detail, including lighting models and a
variety of visual effects and defects. Students will be expected
to describe the sampling problem and solutions for both static
and moving pictures.
Content: Euclidean and projective geometry transformations.
Modelling: Mesh models and their representation. Constructive
solid geometry and its representation. Specialised models.
Rendering: Raster images; illumination models; meshes and hidden
surface removal; scan-line rendering. CSG: ray-casting; visual
effects and defects. Rendering for animation.
Ordered dither; resolution; aliasing; colour.
MATH0076: Proposal writing
Semester 1
Credits: 6
Topic: Computing
Level: Level 3
Assessment: CW100
Requisites: Co MATH0082
Aims & Learning Objectives: Aims: To develop skills
in writing and criticquing technical proposals. To develop abilities
in requirements extraction.
Objectives: To demonstrate skills in the above aims by examination
of case-studies and the writing of the proposal for the project
to be undertaken in the following semester.
Content: Effective and ineffective written communication.
When to use graphs, diagrams and pictures. Proposal structure.
Styles of written English. Developing your own style. Interviewing.
Selecting your project and preparing your proposal.
MATH0077: Formal software development
Semester 1
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: To convey to students
the idea that software development can be presented as a systematic
process of calculation with mathematically secure foundations.
Objectives: Students should be able to develop modest programs
systematically with a complete understanding of the mathematical
foundations of the method advocated, and should understand the
relationship between formal and informal methods for practical
use.
Content: Software specification. Informal and formal development
methods and their implications for the software life-cycle. Current
status of formal development methods. Refinement methods and refinement
calculi. Refinement Calculus: Programs, specifications, code,
refinement.
Types, invariants and feasibility. Assignment and sequencing.
Control structures: alternatives and iteration. Introduction to
data refinement. Foundations of the Refinement Calculus: Dijkstra's
weakest precondition and language semantics in terms of it. Use
of the weakest precondition as a basis for the refinement calculus.
Proving refinement laws from first
principles; deriving one refinement law from another.
MATH0078: Networking
Semester 2
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0025
Aims & Learning Objectives: Aims: To understand the
Internet, and associated background and theory, to a level sufficient
for a competent domain manager.
Objectives: Students should be able to explain the acronyms and
concepts of the Internet and how they relate. Students should
be able to state the steps required to connect a domain to the
Internet, and be able to explain the issues involved to both technical
and non-technical audiences. Students should be able to discuss
the ethical issues involved, and have an "intelligent layman's"
grasp of the legal issues and uncertainties. Students should be
aware of the fundamental security issues, and should be able to
advise on the configuration issues surrounding a firewall.
Content: The ISO 7-layer model. The Internet: its history
and evolution - predictions for the future.
The TCP/IP stack: IP, ICMP, TCP, UDP, DNS, XDR, NFS and SMTP.
Berkeley Introduction to packet layout: source routing etc. The
CONS/CLNS debate: theory versus practice.
Various link levels: SLIP, 802.5 and Ethernet, satellites, the
"fat pipe", ATM. Performance issues: bandwidth, MSS
and RTT; caching at various layers. Who 'owns' the Internet and
who 'manages' it: RFCs, service providers, domain managers, IANA,
UKERNA, commercial British
activities. Routing protocols and default routers. HTML and electronic
publishing. Legal and ethical issues: slander/libel, copyright,
pornography, publishing versus carrying. Security and firewalls:
Kerberos.
MATH0079: Computer speech processing
Semester 2
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: To introduce the
essential concepts and techniques of automatic speech processing
and to use speech processing as an illustration of an area of
active research and development in computer technology that is
both novel and lies near the limits of present capability.
Objectives: Students will be able to i) outline the essential
processes of human speech production and read and write simple
phonetic transcriptions, ii) to demonstrate an understanding of
signal processing, iii) to describe, compare and contrast digital
schemes for sampling, coding and analysing speech, iv) to comprehend
the theoretical and practical issues in automatic speech processing
and v) to explain, and assess major speech synthesis and recognition
techniques.
Content: Speech production: the articulatory system; acoustic-phonetics
and prosody; phonetic transcription and co-articulation; phonemes,
phones, phonology and allophones. Speech signals: their nature,
characterisation and representation in different domains; theory
of
elementary signal processing. Speech coding and analysis: simple
PCM; sampling and quantisation errors; other coding schemes for
data compression and feature extraction. Speech synthesis: articulatory,
formant and other types of synthesis; synthesis by rule and text-to-speech
synthesis. Speech recognition: matching complex and variable patterns;
segmentation of connected and continuous speech; speaker dependence;
time variations and warping; statistically-oriented techniques
for recognition and some current methods; recognition versus understanding.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN
YEAR.
MATH0080: Computer vision
Semester 2
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0021
Aims & Learning Objectives: Aims: To present a broad
account of computer vision, with the emphasis on the image processing
required for its low level stages.
Objectives: To induce an appreciation of the processes involved
in robotic vision and how this differs from human vision.
Content: Image formation. Colour versus monochrome. Preprocessing
of the image. Edge finding: elementary methods and their shortcomings;
sophisticated methods such as those of Marr-Hildreth, Canny, and
Prager. Optical flow. Hough transform. Global and local region
segmentation techniques: histogram techniques, region growing.
Representation of the results of low level processing. Some image
interpretation methods employing probability arguments and fuzzy
logic. Hardware. Practical problems based on an image processing
package.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD
YEAR.
MATH0081: Hardware architecture & compilation
Semester 1
Credits: 6
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0029
Aims & Learning Objectives: Aims: To demonstrate the
impact that computer architecture is having on compiler design.
To explore trends in hardware development, and examine techniques
for efficient use of machine resources,
Objectives: Students should be able to describe the philosophy
of RISC and CISC architectures. They should know at least one
technique for register allocation, and one technique for instruction
scheduling. They should be able to write a simple code generator.
Content: Description of several state-of-the-art chip designs.
The implications for compilers of RISC architectures. Register
allocation algorithms (colouring, DAGS, scheduling). Global data-flow
analysis. Pipelines and instruction scheduling; delayed branches
and loads. Multiple
instruction issue. VLIW and the Bulldog compiler. Harvard architecture
and Caches. Benchmarking.
MATH0082: Double module project
Semester 2
Credits: 12
Topic: Computing
Level: Level 3
Assessment: CW100
Requisites: Co MATH0076
Aims & Learning Objectives: Aims: To satisfy as many
of the objectives as possible as set out in the individual project
proposal.
Objectives: To produce the deliverables identified in the individual
project proposal.
Content: Defined in the individual project proposal.
MATH0084: Linear models
Semester 1
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0035, Pre MATH0002, Pre MATH0003, Pre MATH0005,
Pre MATH0008
Aims & Learning Objectives: Aims To present the theory
and application of normal linear models and generalised linear
models, including estimation, hypothesis testing and confidence
intervals. To describe methods of model choice and the use of
residuals in diagnostic checking.
Objectives On completing the course, students should be able to
(a) choose an appropriate generalised linear model for a given
set of data; (b) fit this model using the GLIM program, select
terms for inclusion in the model and assess the adequacy of a
selected model; (c) make inferences on the basis of a fitted model
and recognise the assumptions underlying these inferences and
possible limitations to their accuracy.
Content: Normal linear model: Vector and matrix representation,
constraints on parameters, least squares estimation, distributions
of parameter and variance estimates, t-tests and confidence
intervals, the Analysis of Variance, F-tests for unbalanced
designs.
Model building: Criteria for use in model selection including
Mallows Cp statistic, the PRESS criterion, Akaike's information
criterion. Subset selection and stepwise regression methods with
applications in polynomial regression and multiple regression.
Effects of collinearity in regression variables. Implications
of model choice on subsequent inferential statements.
Uses of residuals: Probability plots, added variable plots, plotting
residuals against fitted values to detect a mean-variance relationship,
standardised residuals for outlier detection, masking.
Generalised linear models: Exponential families, standard form,
statement of asymptotic theory for i.i.d. samples, Fisher information.
Linear predictors and link functions, statement of asymptotic
theory for the generalised linear model, applications to z-tests
and confidence intervals, c²-tests
and the analysis of deviance. Residuals from generalised linear
models and their uses. Applications to bioassay, dose response
relationships, logistic regression, contingency tables.
MATH0085: Time series
Semester 1
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0035
Aims & Learning Objectives: Aims To introduce a variety
of statistical models for time series and cover the main methods
for analysing these models.
Objectives At the end of the course, the student should be able
to
* compute and interpret a correlogram and a sample spectrum
* derive the properties of ARIMA and state-space models
* choose an appropriate ARIMA model for a given set of data and
fit the model using the MINITAB package
* compute forecasts for a variety of linear methods and models.
Content: Introduction: Examples, simple descriptive techniques,
trend, seasonality, the correlogram.
Probability models for time series: Stationarity; moving average
(MA), autoregressive (AR), ARMA and ARIMA models.
Estimating the autocorrelation function and fitting ARIMA models.
Forecasting: Exponential smoothing, Box-Jenkins method.
Stationary processes in the frequency domain: The spectral density
function, the periodogram, spectral analysis.
Bivariate processes: Cross-correlation function, cross spectrum.
Linear systems: Impulse response, step response and frequency
response functions.
State-space models: Dynamic linear models and the Kalman filter.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD
YEAR.
MATH0086: Medical statistics
Semester 1
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0035, Pre MATH0003, Pre MATH0005
Aims & Learning Objectives: Aims To introduce students
to the statistical needs of medical research and describe commonly
used methods in the design and analysis of clinical trials.
Objectives On completing the course, students should be able to
(a) recognise the statistically important features of a medical
research problem and, where appropriate, suggest a suitable clinical
trial design; (b) analyse data collected from a comparative clinical
trial, including crossover and case-control studies, binary response
data and survival data.
Content: Drug development: Phases I to IV of drug development
and testing. Ethical considerations.
Design of clinical trials: Defining the patient population, the
trial protocol, possible sources of bias, randomisation, blinding,
use of placebo treatment, stratification, balancing prognostic
variables across treatments by "minimisation". Formulation
of clinical trials as hypothesis testing and decision problems.
Sample size calculations,
use of pilot studies, adaptive methods.
Analysis of clinical trials: Patient withdrawals, "intent
to treat" criterion for inclusion of patients in analysis,
inclusion of stratification variables in the analysis.
Interim analyses: Repeated significance tests, O'Brien and Fleming's
stopping rule, sample size calculations. Statistical analysis
following a group sequential trial, contrast between frequentist
and Bayesian analyses.
Crossover trials: Two treatment, two period design. Discussion
of more complex designs.
Case-control studies.
Binary data: Comparison of treatments with binary outcomes, inclusion
of prognostic variables in logit and probit models.
Survival data: Life tables, censoring. Parametric models for censored
survival data. Kaplan-Meier estimate, Greenwood's formula, the
proportional hazards model, logrank test, Cox's proportional hazards
regression model.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN ODD
YEAR.
MATH0087: Optimisation methods of operational research
Semester 1
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0002, Pre MATH0005
Aims & Learning Objectives: Aims To present methods
of optimisation commonly used in OR, to explain their theoretical
basis and give an appreciation of the variety of areas in which
they are applicable.
Objectives On completing the course, students should be able to
* recognise practical problems where optimisation methods can
be used effectively
* implement the simplex and dual simplex algorithms, Dantzig's
method for the transportation problem and the Ford-Fulkerson algorithm
* explain the underlying theory of linear programming problems,
including duality.
Content: The Nature of OR: Brief introduction.
Linear Programming: Basic solutions and the fundamental theorem.
The simplex algorithm, two phase method for an initial solution.
Interpretation of the optimal tableau. Duality. Sensitivity analysis
and the dual simplex algorithm. Brief discussion of Karmarkar's
method. Applications of LP. The transportation problem and its
applications, solution by Dantzig's method. Network flow problems,
the Ford-Fulkerson theorem.
Non-linear Programming: Revision of classical Lagrangian methods.
Kuhn-Tucker conditions, necessity and sufficiency. Illustration
by application to quadratic programming.
MATH0088: Data collection
Semester 1
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0035
Aims & Learning Objectives: Aims To illustrate the
principles of experimental design in randomised and factorial
designs and a variety of sample survey methods. To present components
of variance estimation in random effects models and discuss its
application in industrial quality improvement.
Objectives On completing the course, students should be able to
* identify the features of a proposed study that affect the choice
of experimental design
* choose a suitable, efficient design for a study and explain
how the data collected under this design should ultimately be
analysed
* design and analyse a components of variance experiment
* design and analyse a sample survey.
Content: Principles of experimental design: Randomisation
and the avoidance of bias. Advantages of orthogonal parameter
estimates. Efficiency and optimal designs. Practical considerations.
Observational studies: Confounding factors, reduction of bias
by matching and regression modelling. The scope of inference from
observational data.
Randomised designs: Completely randomised and randomised block
designs.
Factorial designs: Complete factorial designs, confounding and
fractional factorials, applications to modern quality improvement.
Random effects: Split plot designs, statistical models and analyses.
Sample surveys: Simple random sampling, stratified sampling, two-stage
sampling, cluster sampling, quota sampling. Inference about the
mean of a finite population. Randomised response methods for sensitive
questions.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN
YEAR.
MATH0089: Applied probability & finance
Semester 1
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0034
Aims & Learning Objectives: Aims To develop and apply
the theory of probability and stochastic processes to examples
from finance and economics.
Objectives At the end of the course, students should be able to
* formulate mathematically, and then solve, dynamic programming
problems
* describe the Capital Asset Pricing Model and its conclusions
* price an option on a stock modelled by a single step of a random
walk
* perform simple calculations involving properties of Brownian
motion.
Content: Dynamic programming: Markov decision processes,
Bellman equation; examples including consumption/investment, bid
acceptance, optimal stopping. Infinite horizon problems; discounted
programming, the Howard Improvement Lemma, negative and positive
programming, simple examples and counter-examples.
Utility theory: Risk aversion, the Capital Asset Pricing Model.
Option pricing for random walks: Arbitrage pricing theory, prices
and discounted prices as Martingales, hedging.
Brownian motion: Introduction to Brownian motion, definition and
simple properties.
Exponential Brownian motion as the model for a stock price, the
Black-Scholes formula.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN
YEAR.
MATH0090: Multivariate analysis
Semester 2
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0035, Pre MATH0008
Aims & Learning Objectives: Aims To develop facility
in the analysis and interpretation of multivariate data.
Objectives At the end of the course, students should be able to
· use graphical methods to identify possible structure in
high-dimensional data
· select appropriately among a variety of techniques for
dimensionality reduction
· combine classical inferential methods with more recent
computationally-intensive techniques to produce more in-depth
analyses than were possible before the computer era.
Content: Introduction: Graphical exploratory analysis of
high-dimensional data. Revision of matrix techniques, eigenvalue
and singular value decompositions.
Principal components analysis: Derivation and interpretation,
approximate reduction of dimensionality, scaling problems.
Factor analysis.
Multidimensional distributions: The multivariate normal distribution,
its properties and estimation of parameters. One and two sample
tests on means, the Wishart distribution, Hotelling's T-squared.
The multivariate linear model.
Canonical correlations and canonical variables: Discriminant analysis,
classification problems and cluster analysis.
Topics selected from: Metrics and similarity coefficients; multi-dimensional
scaling;
clustering algorithms; correspondence analysis, the biplot, Procrustes
analysis and projection pursuit; Classification and Regression
Trees.
MATH0091: Applied statistics
Semester 2
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0084
Aims & Learning Objectives: Aims To give students experience
in tackling a variety of "real-life" statistical problems.
Objectives During the course, students should become proficient
in
* formulating a problem and carrying out an exploratory data analysis
* tackling non-standard, "messy" data
* presenting the results of an analysis in a clear report.
Content: Formulating statistical problems: Objectives,
the importance of the initial examination of data, processing
large-scale data sets.
Analysis: Choosing an appropriate method of analysis, verification
of assumptions.
Presentation of results: Report writing, communication with non-statisticians.
Using resources: The computer, the library.
Project topics may include: Exploratory data analysis. Practical
aspects of sample surveys. Fitting general and generalised linear
models. The analysis of standard and non-standard data arising
from theoretical work in other blocks.
MATH0092: Statistical inference
Semester 2
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0033
Aims & Learning Objectives: Aims To develop a formal
basis for methods of statistical inference and decision making,
including criteria for the comparison of procedures. To give an
in depth description of Bayesian methods and the asymptotic theory
of maximum likelihood methods.
Objectives On completing the course, students should be able to
* identify and compute admissible, minimax and Bayes decision
rules
* calculate properties of estimates and hypothesis tests
* derive efficient estimates and tests for a broad range of problems,
including applications to a variety of standard distributions.
Content: Revision of standard distributions: Bernoulli,
binomial, Poisson, exponential, gamma and normal, and their interrelationships.
Sufficiency and Exponential families.
Decision theory: Admissibility and minimax decision rules; Bayes
risk and Bayes rules. Bayesian inference; prior and posterior
distributions, conjugate priors.
Point estimation: Bias and variance considerations, mean squared
error. Cramer-Rao lower bound and efficiency. Unbiased minimum
variance estimators and a direct appreciation of efficiency through
some examples. Bias reduction. Asymptotic theory for maximum likelihood
estimators.
Hypothesis testing: Hypothesis testing, review of the Neyman-Pearson
lemma and maximisation of power. Maximum likelihood ratio tests,
asymptotic theory. Compound alternative hypotheses, uniformly
most powerful tests, locally most powerful tests and score statistics.
Compound null hypotheses, monotone likelihood ratio property,
uniformly most powerful unbiased tests. Nuisance parameters, generalised
likelihood ratio tests.
MATH0093: Stochastic processes
Semester 2
Credits: 6
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0003, Pre MATH0005, Pre MATH0032
Aims & Learning Objectives: Aims To present a formal
description of Markov chains and Markov processes, their qualitative
properties and ergodic theory. To apply results in modelling real
life phenomena, such as biological processes and queueing systems,
and in controlling such systems.
Objectives On completing the course, students should be able to
* classify the states of a Markov chain and find its ergodic distribution
* calculate generating functions, waiting time distributions and
limiting behaviour of queues
* apply these results to solve OR type problems of process control.
Content: Markov chains: Definitions and examples, n-step
transition probabilities, equilibrium and stationary distributions,
classification of states and ergodic theorems, multiplicative
chains.
Markov processes with discrete states in continuous time: Properties
of the Poisson process, birth and death processes, immigration/emigration
processes, equilibrium distributions.
Queues: Kendall's classification system and examples, M/M/1 including
time dependent solution, M/M/k and other Markov queues, the method
of stages, machine interference, the queue M/G/l, priority systems.
MATH0094: Probability theory
Semester 2
Credits: 6
Topic: Statistics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0034, Pre MATH0042
Aims & Learning Objectives: Aims To teach Probability
(and Statistics) in a rigorous mathematical context.
Objectives On completing the course, students should be able to
* describe with precision distributional and sample path aspects
of long-term behaviour
* deduce the consequences of this theory in the wide range of
real-world problems to which it applies.
Content: Foundations: First and second Borel-Cantelli lemmas,
0-1 law, Weak Law of Large Numbers, Strong Law of Large Numbers
when X has finite fourth moment, Weierstrass's Theorem.
Distributions: Characteristic functions and inversion formula.
Weak convergence, Skorokhod representation. The Central Limit
Theorem and analogues. Convergence of distributions on [0,1],
[0,¥] and S¹. Weyl's Theorem.
Ergodic theory: Measure preserving transformations, ergodicity.
Riesz proof of the Ergodic Theorem. Applications to Markov chains,
Strong Law of Large Numbers and continued fractions.
MATH0105: Industrial placement
Academic Year
Credits: 60
Level: Level 2
Assessment:
Requisites:
MATH0106: Study year abroad (BSc)
Academic Year
Credits: 60
Level: Level 2
Assessment:
Requisites:
MATH0107: Study year abroad (MMath)
Academic Year
Credits: 60
Level: Level 3
Assessment:
Requisites:
MATH0115: Mathematical structures
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Aims & Learning Objectives: Aims: To provide a thorough
grounding and analysis of computational concepts and processes.
Objectives: To be able to apply mathematical techniques to the
description and analysis of computational processes and to recognise
computational concepts in practical problems.
Content: Numbers: Natural numbers, integers, prime numbers,
statement of prime decomposition theorem, complex numbers.
Algebra: Permutations and combinations, proof by induction, Binomial
Theorem.
Graphs and Trees: Node/ edge representation of graphs, adjacency
matrices, directed graphs, binary relations, decision trees, Huffman
codes, graph alogrithms, Euler and hamilton circuits.
Matrix Algebra.
Students must have A-level Mathematics, Grade C or higher or equivalent
in order to undertake this unit.
MATH0117: Project
Semester 1
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: CW100
Requisites:
Aims & Learning Objectives: Aims: To satisfy as many
of the objectives as possible as set out in the individual project
proposal.
Objectives: To produce the deliverables identified in the individual
project proposal.
Content: Defined in the individual project proposal.
MATH0117: Project
Semester 2
Credits: 6
Topic: Mathematics
Level: Level 3
Assessment: CW100
Requisites:
Aims & Learning Objectives: Aims: To satisfy as many
of the objectives as possible as set out in the individual project
proposal.
Objectives: To produce the deliverables identified in the individual
project proposal.
Content: Defined in the individual project proposal.
PHYS0030: Quantum mechanics
Semester 2
Credits: 6
Level: Level 3
Assessment: EX80 CW20
Requisites: Co PHYS0024
Aims & Learning Objectives: To develop a mathematical
model of the quantum world and to show how this may be used to
describe a wide range of physical phenomena. To show the relation
between wave functions, operators and experimental observables.
To set up and solve the Schrodinger equation in a number of standard
model systems.
Content: Introduction: Breakdown of classical concepts.
Old quantum theory.
Quantum mechanical concepts and models: The "state"
of a quantum mechanical system. Hilbert space. Observables and
operators. Eigenvalues and eigenfunctions. Dirac bra and ket vectors.
Basis functions and representations. Probability distributions
and expectation values of observables.
Schrodinger's equation: Operators for position, time, momentum
and energy. Derivation of time-dependent Schrodinger equation.
Correspondence to classical mechanics. Commutation relations and
the Uncertainty Principle. Time evolution of states. Stationary
states and the time-independent Schrodinger equation.
Motion in one dimension: Free particles. Wave packets and momentum
probability density. Time dependence of wave packets. Bound states
in square wells. Parity. Reflection and transmission at a step.
Tunnelling through a barrier. Linear harmonic oscillator.
Motion in three dimensions: Stationary states of free particles.
Central potentials; quantisation of angular momentum. The radial
equation. Square well; ground state of the deuteron. Electrons
in atoms; the hydrogen atom. Hydrogen-like atoms; the Periodic
Table.
Spin angular momentum: Pauli spin matrices. Identical particles.
Symmetry relations for bosons and fermions. Pauli's exclusion
principle.
Approximate methods for stationary states: Time independent perturbation
theory. The variational method. Scattering of particles; the Born
approximation.
SOCS0003: Introductory microeconomics 1
Semester 1
Credits: 6
Topic: Economics
Level: Level 1
Assessment: EX80 CW20
Requisites: Ex SOCS0001
Aims & Learning Objectives: The course is designed
to provide an introduction to the methods of microeconomic analysis,
including the use of simple economic models and their application.
Students should gain an ability to derive conclusions from simple
economic models and evaluate their realism and usefulness.
Content: An introduction to economic methodology; the concept
of market equilibrium; the use of demand and supply curves, and
the concept of elasticity; elementary consumer theory, indifference
curves and their relationship to market demands; elementary theory
of production, production possibilities and their relationship
to cost curves; the supply behaviour of competitive firms and
its relationship to supply curves; the idea of general competitive
equilibrium; the efficiency properties of competitive markets;
examples of market failure.
SOCS0004: Introductory microeconomics 2
Semester 2
Credits: 6
Topic: Economics
Level: Level 1
Assessment: EX80 CW20
Requisites: Pre SOCS0003
Aims & Learning Objectives: The course is designed
to provide an introduction to the methods of microeconomic analysis,
including the use of simple economic models and their application.
Students should gain an ability to derive conclusions from simple
economic models and evaluate their realism and usefulness.
Content: Course content continues from Microeconomics 1.
Equity and efficiency; the tax and benefit system; factor pricing
and the labour market; public goods and merit goods; externality,
natural resources and environmental policy; non competitive market
structures; monopoly and imperfect competition; oligopoly; regulation
of monopolies.
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Mathematical Sciences Programme Catalogue
Programme / Unit Catalogue 1997/98