Mathematical Sciences Unit Catalogue
ECOI0006: Introductory microeconomics
Semester 1
Credits: 6
Contact:
Topic: Economics
Level: Level 1
Assessment: ES40 EX40 OT20
Requisites:
Aims & learning objectives:
Aims: The Unit 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. Learning Objectives: By the end of
the course unit students should be able to understand and apply basic microeconomic
principles to the economic decisions of households and firms under a variety
of market conditions. They should be able to use these principles both to describe
and to appraise these decisions. They should be competent in the verbal, diagrammatic
and basic mathematical concepts and techniques used in introductory microeconomics.
Additional provision is made for those students without A Level Economics. The
Unit is supported by the CAL package WinEcon.
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 output decisions of perfectly and imperfectly competitive
firms and industries; supply curves; the idea of general competitive equilibrium;
the efficiency properties of competitive markets; examples of market failure.
Key texts: Richard G. Lipsey and K. Alec Chrystal 'An Introduction to Positive
Economics'. Jean Soper and Phil Hobbs (eds) 'The WinEcon Workbook'. M. L. Katz
and H.S. Rosen 'Microeconomics'. Alan Griffiths and Stuart Wall 'Applied Economics:
An Introductory Course'.
ECOI0007: Introductory macroeconomics
Semester 2
Credits: 6
Contact:
Topic: Economics
Level: Level 1
Assessment: ES40 EX40 OT20
Requisites:
Aims & learning objectives:
Aims: The Unit is designed to provide an introduction to the methods of macroeconomic
analysis, including the use of simple macroeconomic models and their application
in a UK policy context. Students should gain an ability to derive conclusions
from simple economic models and evaluate their realism and usefulness in policy
making. Learning Objectives: By the end of the course unit students should be
able to understand and apply basic macroeconomic principles to the economic
decisions of the policy-maker. They should be able to use these principles to
both describe and appraise these decisions as well as to understand how macroeconomic
problems arise. They should be competent in the verbal, diagrammatic and basic
mathematical concepts used in introductory macroeconomics, providing a suitable
platform for the more advanced study of this subject in future years. Additional
provision is made for those students without A Level Economics. The Unit is
supported by the CAL package WinEcon.
Content:
The circular flow of income and expenditure; national income accounting; aggregate
demand and supply; the components and determinants of private and public sector
aggregate expenditure in closed and open economies; output and the price level
in the short- and long-run; monetary institutions and policy; inflation and
unemployment; the balance of payments and exchange rates; economic growth, the
Kondratieff. Key texts: Richard G. Lipsey and K. Alec Chrystal 'An Introduction
to Positive Economics'. Jean Soper and Phil Hobbs (eds) 'The WinEcon Workbook'.
M.J. Artis (ed) 'The UK Economy: a Manual of Applied Economics'. Alan Griffiths
and Stuart Wall 'Applied Economics: An Introductory Course'.
ECOI0010: Intermediate microeconomics
Semester 1
Credits: 6
Contact:
Topic: Economics
Level: Level 2
Assessment: EX50 OT50
Requisites:
Aims & learning objectives:
This course unit covers the core concepts and methods of microeconomic analysis,
using some mathematics in modelling and explication, in conformity with modern
intermediate micro texts. It is supported by a course unit in Mathematical Economics,
where single honours Economics students will acquire a more rigorous mathematical
approach. The aim of this unit is to enable students to deepen their analytical
ability in microeconomics so that they can use theory to generate predictions
and explanations with respect to economic phenomena. The learning objectives
are that by the end of the course unit, students should be able to tackle economic
problems with the sustained application of (mainly neo-classical) economic principles
and be familiar with recent contributions to the subject. Manipulation of short
problems under test conditions allows the demonstration of economic insight.
The course unit is essential for anyone wishing to undertake further study of
the economics of industry, labour, environment and other sectoral economic issues.
Content:
The course will cover the theory of consumer behaviour, the theory of the firm
in a competitive situation, industrial organisation and imperfect competition,
the theory of factor markets, the economics of uncertainty and information,
welfare economics and general equilibrium theory. Key texts: H. Varian,'Intermediate
Microeconomics'. D. Laidler and S. Estrin,'Introduction to Microeconomics'.
ELEC0047: Design & realisation of integrated circuits
Semester 2
Credits: 6
Contact:
Topic:
Level: Undergraduate Masters
Assessment: EX100
Requisites:
Aims & learning objectives:
This course covers all aspects of the realisation of integrated circuits, including
both digital, analogue and mixed-signal implementations. Consideration is given
to the original specification for the circuit which dictates the optimum technology
to be used also taking account of the financial implications. The various technologies
available are described and the various applications, advantages and disadvantages
of each are indicated. The design of the circuit building blocks for both digital
and analogue circuits are covered. Computer aided design tools are described
and illustrated and the important aspects of testing and design for testability
are also covered. After completing this module the student should be able to
take the specification for an IC and, based on all the circuit, technology and
financial constraints, be able to determine the optimum design approach. The
student should have a good knowledge of the circuit design approaches and to
be able to make use of the computer aided design tools available and to understand
their purposes and limitations. The student should also have an appreciation
of the purposes of IC testing and the techniques for including testability into
the overall circuit design.
Content:
Design of ICs: the design cycle, trade-offs, floorplanning, power considerations,
economics. IC technologies: Bipolar, nMOS, CMOS, BiCMOS, analogue, high frequency.
Transistor level design: digital gates, analogue components, sub-circuit design.
IC realisation: ASICs, PLDs, gate arrays, standard cell, full custom. CAD: schematic
capture, hardware description languages, device and circuit modelling, simulation,
layout, circuit extraction. Testing: types of testing, fault modelling, design
for testability, built in self test, scan-paths.
ESML0208: Chinese stage 3A (advanced beginners) (6 credits)
Semester 1
Credits: 6
Contact:
Topic: Chinese
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0209
Aims & learning objectives:
This course builds on the Chinese covered in Chinese Stage 2 A and B in order
to enhance the student's abilities in the four skill areas.
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary relating to China,
Singapore and Taiwan. There will be discussion in the target language of topics
derived from teaching materials, leading to small-scale research projects based
on the same range of topics and incorporating the use of press reports and articles
as well as audio and visual material. Students are encouraged to devote time
and energy to developing linguistic proficiency outside the timetabled classes,
for instance by additional reading and/or participating in informally arranged
conversation groups and in events at which Chinese is spoken.
ESML0209: Chinese stage 3B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: Chinese
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0208
Aims & learning objectives:
A continuation of Chinese Stage 3A
Content:
A continuation of Chinese Stage 3A
ESML0214: French stage 9A (further advanced) (6 credits)
Semester 1
Credits: 6
Contact:
Topic: French
Level: Level 2
Assessment: EX45 CW40 OR15
Requisites: Co ESML0215
Aims & learning objectives:
A continuation of the work outlined in French 8A and 8B
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary. Teaching materials
used cover a wide variety of sources and cover aspects of cultural political
and social themes relating to France. Works of literature or extracts may be
included, as well as additional subject-specific material, as justified by class
size. This may encompass scientific and technological topics as well as materials
relevant to business and industry. There will be discussion in the target language
of topics relating to and generated by the teaching materials, with the potential
for small-scale research projects and presentations. Audio and video materials
form an integral part of this study, along with newspaper, magazine and journal
articles. Students are actively encouraged to consolidate their linguistic proficiency
outside the timetabled classes, by additional reading, links with native speakers
and participating in events at which French is spoken. Audio and video laboratories
are available to augment classroom work.
ESML0215: French stage 9B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: French
Level: Level 2
Assessment: EX45 CW40 OR15
Requisites: Co ESML0214
Aims & learning objectives:
A continuation of French Stage 9A
Content:
A continuation of French Stage 9A
ESML0220: French stage 6A (advanced intermediate) (6
credits)
Semester 1
Credits: 6
Contact:
Topic: French
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0221
Aims & learning objectives:
This course concentrates on the more advanced aspects of French with continued
emphasis on practical application of language skills in a relevant context,
in order to refine further the student's abilities.
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary. There is continued
further development of the pattern of work outlined in French Stage 5A and 5B
ESML0221: French stage 6B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: French
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0220
Aims & learning objectives:
A continuation of course French Stage 6A
Content:
A continuation of course French Stage 6A
ESML0226: German stage 3A (advanced beginners) (6 credits)
Semester 1
Credits: 6
Contact:
Topic: German
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0227
Aims & learning objectives:
This course builds on the German covered in German Stage 2A and 2B in order
to enhance the student's abilities in the four skill areas.
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary relating to a selection
of topics. Teaching materials cover a wide range of cultural, political and
social topics relating to German speaking countries and may include short works
of literature. There will be discussion in the target language of topics derived
from teaching materials, leading to small-scale research projects based on the
same range of topics and incorporating the use of press reports and articles
as well as audio and visual material. Students are encouraged to devote time
and energy to developing linguistic proficiency outside the timetabled classes,
for instance by additional reading and/or participating in informally arranged
conversation groups and in events at which German is spoken. Audio and video
laboratories are available to augment classroom work.
ESML0227: German stage 3B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: German
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0226
Aims & learning objectives:
A continuation of German Stage 3A
Content:
A continuation of German Stage 3A
ESML0238: German stage 6A (advanced intermediate) (6
credits)
Semester 1
Credits: 6
Contact:
Topic: German
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0239
Aims & learning objectives:
This course concentrates on the more advanced aspects of German with continued
emphasis on practical application of language skills in a relevant context,
in order to refine further the student's abilities.
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary. There is continued
further development of the pattern of work outlined in German Stage 5A and 5B
ESML0239: German stage 6B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: German
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0238
Aims & learning objectives:
A continuation of German Stage 6A
Content:
A continuation of German Stage 6A
ESML0244: Italian stage 3A (advanced beginners) (6 credits)
Semester 1
Credits: 6
Contact:
Topic: Italian
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0245
Aims & learning objectives:
This course builds on the Italian covered in Italian Stage 2A and 2B in order
to enhance the students abilities in the four skill areas.
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary relating to a selection
of topics. Teaching materials cover a wide range of cultural, political and
social topics relating to Italy and may include short works of literature. There
will be discussion in the target language of topics derived from teaching materials,
leading to small-scale research projects based on the same range of topics and
incorporating the use of press reports and articles as well as audio and visual
material. Students are encouraged to devote time and energy to developing linguistic
proficiency outside the timetabled classes, for instance by additional reading
and/or participating in informally arranged conversation groups and in events
at which Italian is spoken. Audio and video laboratories are available to augment
classwork
ESML0245: Italian stage 3B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: Italian
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0244
Amis & Learning Objectives: A continuation of Italian Stage 3A.
Content:
A continuation of Italian Stage 3A.
ESML0262: Spanish stage 6A (advanced intermediate) (6
credits)
Semester 1
Credits: 6
Contact:
Topic: Spanish
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0263
Aims & learning objectives:
This course concentrates on the more advanced aspects of Spanish with continued
emphasis on practical application of language skills in a relevant context,
in order to refine further the student's abilities.
Content:
This unit contains a variety of listening, reading, speaking and writing tasks
covering appropriate grammatical structures and vocabulary. There is continued
further development of the pattern of work outlined in Spanish Stage 5A and
5B
ESML0263: Spanish stage 6B (6 credits)
Semester 2
Credits: 6
Contact:
Topic: Spanish
Level: Level 1
Assessment: EX45 CW40 OR15
Requisites: Co ESML0262
Aims & learning objectives:
A continuation of Spanish Stage 6A
Content:
A continuation of Spanish Stage 6A
MANG0069: Introduction to accounting & finance
Semester 2
Credits: 5
Contact:
Topic:
Level: Level 1
Assessment: EX50 CW50
Requisites:
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
MANG0071: Organisational behaviour
Semester 1
Credits: 5
Contact:
Topic:
Level: Level 1
Assessment: EX60 CW40
Requisites:
Aims & learning objectives:
To develop the student's understanding of people's behaviour within work organizations
Content:
Topics of study will be drawn from the following: The meaning of organising
and organisation Socialisation, organisational norms and organisational culture
Bureaucracy, organisational design and new organisational forms Managing organisational
change Power and politics Business ethics Leadership and team work Decision
-making Motivation Innovation Gender The future of work
MANG0072: Managing human resources
Semester 1
Credits: 5
Contact:
Topic:
Level: Level 1
Assessment: EX100
Requisites:
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
Semester 2
Credits: 5
Contact:
Topic:
Level: Level 1
Assessment: EX100
Requisites: Ex MANG0016
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
Semester 1
Credits: 5
Contact:
Topic:
Level: Level 1
Assessment: EX60 CW25 OT15
Requisites:
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.
Content:
Following on from the learning aims and objectives, the course is divided into
two main parts: Part I considers why IT is strategic and how it can affect the
competitive environment, taking stock of the opportunities and problems it provides.
It consists of lectures, discussion, case studies. The objective is to investigate
the business impact of IS. For example: in what ways are IS strategic? what
business benefits can IS bring? how does IS transform management processes and
organisational relationships? how can organisations evaluate IS? how should
IS, which transform organisations and extend across functions, levels and locations,
be implemented? Part II examines a variety of technologies available to the
manager and examines how they have been used in organisations. A number of problem-oriented
case studies will be given to project groups to examine and discuss. The results
may then be presented in class, and are open for debate. In summary, the aim
of the course is to provide the knowledge from which students should be able
to make appropriate use of computing and information technology in forthcoming
careers. This necessitates some technical understanding of computing, but not
at an advanced level. This is a management course: not a technical computing
course.
MANG0076: Business policy
Semester 2
Credits: 5
Contact:
Topic:
Level: Level 1
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
Contact:
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Students must have A-level Mathematics, normally Grade B or better, or equivalent,
in order to undertake this unit. 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.
MATH0002: Functions, differentiation & analytic geometry
Semester 1
Credits: 6
Contact:
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Students must have A-level Mathematics, normally Grade B or better, or equivalent,
in order to undertake this unit. 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.
MATH0003: Integration & differential equations
Semester 1
Credits: 6
Contact:
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Students must have A-level Mathematics, normally Grade B or better, or equivalent,
in order to undertake this unit. 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.
MATH0004: Sets & sequences
Semester 2
Credits: 6
Contact:
Topic: Mathematics
Level: Level 1
Assessment: EX100
Requisites:
Pre MATH0115 or 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,
˜, (r), «, ",
$. 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
Contact:
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 (r)
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 (r) 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
Contact:
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
Contact:
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0006, Pre MATH0004, Pre MATH0005
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 and to extend these ideas to sequences of functions.
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
Contact:
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0006, 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. Students
should know how to execute the Gram-Schmidt process.
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
Contact:
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0001, Pre MATH0002, Pre MATH0003, Pre MATH0005
Aims & learning objectives:
Aims: This course will provide standard results and techniques for solving systems
of linear autonomous 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; 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
Contact:
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
Contact:
Topic: Mathematics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0007
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:
Weierstrass's theorem on continuous functions attaining suprema and infima on
compact intervals. 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. Uniform Convergence. Cauchy's Criterion
for Uniform Convergence. Weierstrass M-test for series. Power series.
Differentiation of power series. Reimann integration 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
Contact:
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
diagonalise 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 Hermitian 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
Contact:
Topic: Mathematics
Level: Level 2
Assessment: EX75 CW25
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
Contact:
Topic: Mathematics
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0007, Pre MATH0008
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 by 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.
MATH0017: Systems I: architecture & operating systems
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: CW25 EX75
Requisites:
Aims & learning objectives:
Aims: To introduce students to the structure, basic design, operation and programming
of conventional, von Neumann and non-von Neumann computers at the machine level.
To explore the correspondence between high level programming language control
and data structures and what happens at the machine level. Objectives: To understand
how the forms and conventions of high level languages are related to the machine
level. To experience how structured programming can be applied in low as well
as high level languages. To be able to assess the potential advantages and disadvantages
of different architectures and how these may affect system software such as
operating systems. To understand the basic functions and possible organizations
of operating system software.
Content:
Principles of digital computer operation: use of registers and the instruction
cycle; simple addressing concepts; Integers and floating point numbers. Input
and output. Introduction to digital logic. Aspects of modern computer architectures:
Von Neumann and 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. Prototypical operating systems
and the history of operating systems. Program loaders (e.g. DOS, Windows), operating
systems (e.g. Windows, NT, Unix).
MATH0019: Computation III: introduction to formal logic
& semantics
Semester 1
Credits: 6
Contact:
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0139
Aims & learning objectives:
Aims: To give the student an understanding of formal logic.. To illustrate how
these ideas are implemented or partly implemented in logic programming . To
introduce semantics of programming languages and domain theory.. Objectives:
Students should be able to use semantic tableaux and the sequent calculus for
proof in the predicate calculus. They should understand the Gödel completeness
theorem. Students should be able to write and to understand the behaviour of
programs in a logic programming language, such as Prolog . They should have
a basic understanding of semantics.
Content:
Formal grammar and languages, term algebras, unification. Predicate calculus,
first order languages, translating from informal to formal language, .logical
validity, conjunctive normal form, disjunctive normal form, prenex normal form,
Skolem form, clausal form. Semantic tableaux, sequent calculus, Gödel completeness
theorem. Logic programming Introduction to Scott domains and denotational semantics.
MATH0020: Computation II: computability & decidability
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0139
Aims & learning objectives:
Aims: To introduce the capabilities of different kinds of machines, to explore
the relationship between Turing machines and algorithms, and to explore the
limitations of Turing computability. To introduce the Lambda calculus. 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 relationship between Turing machines and algorithms. In the lambda calculus,
students should be able to find normal forms, when these exist, using alpha
and beta reduction.
Content:
Languages and regular expressions. The basic properties of finite-state machines.
Nondeterministic finite-state machines. What can and cannot be computed using
finite-state machines.Turing Machines. Connecting standard Turing Machines together.
Grammars, languages and the Chomsky classification. Introduction to Church's
Thesis. Universal Turing Machines and limitations of Turing computability. Undecidability,
the Halting Problem, reduction of one unsolvable problem to another. Lambda
calculus. Alpha and Beta reduction. Confluence. Church-Rosser Theorem.
MATH0026: Projects & their management
Semester 2
Credits: 6
Contact:
Topic: Computing
Level: Level 2
Assessment: CW100
Requisites:
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 1
Credits: 6
Contact:
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites:
Some knowledge of programming, as approved by the Director of Studies
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: Computation IV: Algorithms
Semester 2
Credits: 6
Contact:
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre MATH0139
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. Complexity
classes P and NP. NP completeness.
MATH0029: Applications IV: compliers
Semester 2
Credits: 6
Contact:
Topic: Computing
Level: Level 2
Assessment: EX75 CW25
Requisites:
Some knowledge of programming, as approved by the Director of Studies
Aims & learning objectives:
Aims: To give an introduction to the processes involved in compilation and the
use of compiler generation tools and compiler support. 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, simple code generation. The interface to the operating system.
Design of run-time systems and issues in storage management, including garbage
collection.
MATH0030: History of computing and its industry 2
Semester 2
Credits: 3
Contact:
Topic: Computing
Level: Level 2
Assessment: EX75 ES20 OT5
Requisites: Pre MATH0131
Aims & learning objectives:
Aims: The aims remain the same as those in MATH0131 with the additional aim
of giving students experience of a formal presentation of their work. Objectives:
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 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
Contact:
Topic: Statistics
Level: Level 1
Assessment: EX100
Requisites:
Students must have A-level Mathematics, Grade B or better in order to undertake
this unit. 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.
MATH0032: Statistics & probability 2
Semester 2
Credits: 6
Contact:
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
Contact:
Topic: Statistics
Level: Level 2
Assessment: EX100
Requisites: 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
Contact:
Topic: Statistics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0002, Pre MATH0032, Pre MATH0004
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
Contact:
Topic: Statistics
Level: Level 2
Assessment: EX75 CW25
Requisites: Pre 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
Contact:
Topic: Statistics
Level: Level 2
Assessment: EX100
Requisites: Pre MATH0034, Pre MATH0003, Pre MATH0005
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
Contact:
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
Contact:
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
Contact:
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
Contact:
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
Contact:
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0011
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
Contact:
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
Contact:
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0007, Pre MATH0008, Pre MATH0011, Pre MATH0012
Aims & learning objectives:
Aims: To introduce and study abstract spaces and general ideas in analysis,
to apply them to examples, to lay the foundations for the Year 4 unit in Functional
analysis and to motivate the Lebesgue integral. Objectives: By the end of the
unit, 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 trigonometric polynomials in L² [0,1]. Fourier Series.
MATH0044: Mathematical methods 1
Semester 1
Credits: 6
Contact:
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-Kovalevskaya 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
Contact:
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
Contact:
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
Contact:
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW25
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: Analytical & geometric theory of differential
equations
Semester 1
Credits: 6
Contact:
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 systems of ordinary differential equations
that have a Hamiltonian or Lagrangian structure. Geomtrical and analytical insights
will be used to prove qualitative properties of solutions. These ideas have
generated many developments in modern pure mathematics, such as sympletic geometry
and ergodic theory, besides being applicable to the equations of classical mechanics,
and motivating much of modern physics. Objectives: Students will be able to
state and prove general theorems for Lagrangian and Hamiltonian systems. Based
on these theoretical results and key motivating examples they will identify
general qualitative properties of solutions of these systems.
Content:
Lagrangian and Hamiltonian systems, phase space, phase flow, variational principles
and Euler-Lagrange equations, Hamilton's Principle of least action, Legendre
transform, Liouville's Theorem, Poincaré recurrence theorem, Noether's
Theorem.
MATH0049: Linear elasticity
Semester 2
Credits: 6
Contact:
Topic: Mathematics
Level: Level 3
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
Contact:
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. Exploitation of symmetry. Hopf
bifurcation.
MATH0051: Numerical linear algebra
Semester 1
Credits: 6
Contact:
Topic: Mathematics
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0008, 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 analyse 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
Contact:
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
Contact:
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
Contact:
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 can 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
Contact:
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 the Year 4 unit in Algebraic
Topology. 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
Contact:
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
Contact:
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, Minkowski and Hölder inequalities. Examples
- IRn, C[0,1], l, 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
Contact:
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites: Pre MATH0041, Pre MATH0042, Pre MATH0031, Pre MATH0032
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
Contact:
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
Contact:
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
Contact:
Topic: Mathematics
Level: Undergraduate Masters
Assessment: EX100
Requisites:
Pre (MATH0046 or 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 3/4 course on linear control theory. In this course, the latter
notions of stabilizability and optimality are developed. 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
Contact:
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
Contact:
Topic: Mathematics
Level: Level 3
Assessment: EX100
Requisites:
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 1
Credits: 6
Contact:
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
Contact:
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
Contact:
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
Contact:
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
Contact:
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
Contact:
Topic: Computing
Level: Level 3
Assessment: EX75 CW25
Requisites:
Students must have A-level Mathematics, normally Grade B or better, or equivalent,
in order to undertake this unit. 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, Gröbner bases. Introduction to integration.
MATH0071: Application of logic
Semester 2
Credits: 6
Contact:
Topic: Computing
Level: Level 3
Assessment: EX75 CW25
Requisites: Pre MATH0019
Aims & learning objectives:
Aims: To explore the world of knowledge representation and knowledge manipulation.
To gain an overview of ways of approaching problems that may be incompletely
or inaccurately defined. To gain experience of different kinds of logics. Objectives:
Students completing this course will have written some programs that represent
and manipulate knowledge. They will appreciate the problems that are unique
to this subject, and will have an overview of the techniques that are available
to tackle them.
Content:
LISP Programming. Knowledge Representation: Predicates, semantic networks, slots
and frames, objects. The Problems of Natural Language: top down, bottom up parsing,augmented
transition networks. Searching: Breadth and depth first, backtracking,goal searching,
alpha-beta pruning and games, GPS. Deduction: Predicate calculus, forward chaining
and unification, backward chaining, non-monotonic reasoning, resolution. Reasoning
under uncertainty: abduction, causality and evidence, problem solving, binary
and Bayesian deduction. Production Systems and Toolkits.
MATH0072: Safety-critical computer systems
Semester 1
Credits: 6
Contact:
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
Contact:
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
Contact:
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: Project preparation
Semester 1
Credits: 6
Contact:
Topic: Computing
Level: Level 3
Assessment: OT100
Requisites:
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. GANTT charts. Proposal structure. Styles of written English. Developing
your own style. Interviewing. Selecting your project and preparing your proposal.
MATH0078: Networking
Semester 2
Credits: 6
Contact:
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0138
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.
MATH0080: Computer vision
Semester 2
Credits: 6
Contact:
Topic: Computing
Level: Level 3
Assessment: EX100
Requisites:
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; more 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.
MATH0081: Hardware architecture & compilation
Semester 1
Credits: 6
Contact:
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
Contact:
Topic: Computing
Level: Level 3
Assessment: OT100
Requisites: Pre MATH0076, Pre 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
Contact:
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
Contact:
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.
MATH0086: Medical statistics
Semester 1
Credits: 6
Contact:
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0003, Pre MATH0005, Pre MATH0035
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, ncluding 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.
MATH0087: Optimisation methods of operational research
Semester 1
Credits: 6
Contact:
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.
MATH0089: Applied probability & finance
Semester 2
Credits: 6
Contact:
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0036
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.
MATH0090: Multivariate analysis
Semester 2
Credits: 6
Contact:
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
Contact:
Topic: Statistics
Level: Level 3
Assessment: CW100
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
Contact:
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.
MATH0105: Industrial placement
Academic Year
Credits: 60
Contact:
Topic:
Level: Level 2
Assessment:
Requisites:
MATH0106: Study year abroad (BSc)
Academic Year
Credits: 60
Contact:
Topic:
Level: Level 2
Assessment:
Requisites:
MATH0107: Study year abroad (MMath)
Academic Year
Credits: 60
Contact:
Topic:
Level: Undergraduate Masters
Assessment:
Requisites:
MATH0117: Project (MMath)
Semester 1
Credits: 6
Contact:
Topic: Mathematics
Level: Undergraduate Masters
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.
MATH0118: Management statistics
Semester 2
Credits: 5
Contact:
Topic:
Level: Level 3
Assessment: EX60 CW40
Requisites:
Pre MATH0097 or MATH0035 Aims & learning objectives:
This unit is designed primarily for DBA Final Year students who have taken the
First and Second Year management statistics units but is also available for
Final Year Statistics students from the Department of Mathematical Sciences.
Well qualified students from the IMML course would also be considered. It introduces
three statistical topics which are particularly relevant to Management Science,
namely quality control, forecasting and decision theory. Aims: To introduce
some statistical topics which are particularly relevant to Management Science.
Objectives: On completing the unit, students should be able to implement some
quality control procedures, and some univariate forecasting procedures. They
should also understand the ideas of decision theory.
Content:
Quality Control: Acceptance sampling, single and double schemes, SPRT applied
to sequential scheme. Process control, Shewhart charts for mean and range, operating
characteristics, ideas of cusum charts. Practical forecasting. Time plot. Trend-and-seasonal
models. Exponential smoothing. Holt's linear trend model and Holt-Winters seasonal
forecasting. Autoregressive models. Box-Jenkins ARIMA forecasting. Introduction
to decision analysis for discrete events: Revision of Bayes' Theorem, admissability,
Bayes' decisions, minimax. Decision trees, expected value of perfect information.
Utility, subjective probability and its measurement.
MATH0125: Markov processes & applications
Semester 1
Credits: 6
Contact:
Topic: Statistics
Level: Level 3
Assessment: EX100
Requisites: Pre MATH0036
Aims & learning objectives:
Aims: To study further Markov processes in both discrete and continuous time.
To apply results in areas such as genetics, biological processes, networks of
queues, telecommunication networks, electrical networks, random walks and elsewhere.
Objectives: On completing the course, students should be able to
* formulate appropriate Markovian models for a variety of real life problems
and apply suitable theoretical results to obtain solutions
* classify a birth process as explosive or non-explosive
* find the Q matrix for a time reversed chain
Content:
Topics covering both discrete and continuous time Markov chains will be chosen
from: Genetics, the Wright-Fisher and Moran models. Epidemics. Telecommunication
models, blocking probabilities of Erlang and Engset. Models of interference
in communication networks, the ALOHA model. Open and closed migration processes,
birth-death processes. Explosions. Resource management. Electrical networks.
Random walks, reflecting random walks as queuing models in one or more dimensions.
The strong Markov property. The Poisson process in time and space. Other applications.
MATH0126: Introduction to contemporary computing
Semester 1
Credits: 6
Contact:
Topic: Computing
Level: Level 1
Assessment: CW100
Requisites:
Aims & learning objectives:
Aims: To survey the diversity of contemporary computing practice, to give the
students confidence in the use of systems, and to foster a critical attitude
towards computing. To introduce programming in Matlab. To give the students
an opportunity for collaborative work, and to give presentations. To encourage
clear explanation. Objectives: To develop competence in the use of a wide variety
of computing systems, and a basis for intelligent criticism of them. To be able
to write simple programs in Matlab. To have some experience of collaborative
work. To sharpen verbal and written presentation skills.
Content:
A brief history of computing: from automated calculation to systems of interacting
processes. Modern systems and packages, e.g. for word processing and spreadsheets.
Scientific report writing, and bibliographic search. Using the internet to access
information. Web design. Operating systems (like UNIX and MSDOS) and utilities
to support programming (e.g. editors like Emacs); programming languages; compilers
and interpreters. Matlab as an example of a high-level package for scientific
programming. Programming in Matlab. Operators and control. Loops. Scripts and
functions. Group projects. Presentations.
MATH0128: Project
Semester 2
Credits: 6
Contact:
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.
MATH0129: Programming laboratory A
Semester 1
Credits: 3
Contact:
Topic:
Level: Level 2
Assessment: OT100
Requisites: Pre MATH0023
Aims & learning objectives:
Aims: This unit aims to give the student confidence and competence in C programming,
shell programming and problem solving using tools. It does this through a combination
of laboratory sessions, project work and supporting lectures. Objectives: To
have students demonstrate a mastery of C at a level above single file simple
I/O programs, experience interfacing to programs whose inner workings are not
known, learn to adapt or combine existing tools to build solutions and practice
in a professional approach to programming and presentation.
Content:
Further C: Multiple file programs. User interfaces. The C library. Storage allocation
mechanisms (malloc, calloc, realloc, alloca). Using standards. Make. Installing
packages. String processing: grep, egrep, fgrep, sesd, awk, prl. Shell variables
and programs. Unix utilities such as tr, tar, In, find, sort, uniq sargs, etc.
General utilities such as gnuplot, Tcl/tk, LaTeX, HTML etc.
MATH0130: Programming laboratory B
Semester 2
Credits: 3
Contact:
Topic:
Level: Level 2
Assessment: PR80 OT20
Requisites: Pre MATH0129
Aims & learning objectives:
Aims: This unit aims to build on the programming skills developed in MATH0129,
extending the scope of the demonstrations and project work. The aims remain
the same as in MATH0129.
Content:
Continuation of the topics listed for MATH0129. Project: an independent or small
group project, to simulate the processes of researching, planning, performing,
analysing and reporting a small-scale experimental investigation. This is to
be reported in writing and in the form of a Poster Presentation, in the style
of conference posters. This will be viewed by staff and students on an open
evening.
MATH0131: History of computing and its industry 1
Semester 1
Credits: 3
Contact:
Topic:
Level: Level 2
Assessment: OT100
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.
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. Economic factors driving
computing: high-level languages: operating systems.
MATH0134: Programming 1
Semester 1
Credits: 12
Contact:
Topic:
Level: Level 1
Assessment: CW40 EX60
Requisites:
Aims & learning objectives:
Aims: To introduce students to the development of computer software, including
problem analysis, establishing requirements, designing, implementing and evaluating.
To provide practical skills at reading and writing programs and producing programs
to solve real world problems. Objectives: Students should be able to design,
construct and test short programs. They should be able to defend design decisions.
To understand the idea of type and to use data types appropriately. To be able
to develop iterative and recursive programs. To be able to read, and comprehend
the behavior of, programs written by others. To be able to assess the complexity
of simple algorithms.
Content:
Introduction to computers and programming. Introduction to system development:
problem analysis, requirements synthesis; system design; evaluation. Scenario
based design. Algorithms. Control structures: sequence, selection and iteration.
Scope and extent. Simple data types. Testing. Object-orientation: reuse, inheritance,
classes, objects and methods. Recursion. Complexity.
MATH0135: Programming II
Semester 2
Credits: 12
Contact:
Topic:
Level: Level 1
Assessment: CW60 EX40
Requisites: Pre MATH0134
Aims & learning objectives:
Aims: To continue the practice of the programming process begun in Programming
I. To extend the notion of object-oriented software development. To increase
practical skills at reading and writing programs and producing programs to solve
real world problems. Objectives: Students should be able to design, construct
and evaluate substantial programs, using libraries as appropriate. They should
be able to read, and comprehend the behavior of, programs written by others.
Given a problem description, they should be able to design suitable software
systems.
Content:
Task based design. User interface design. Evaluation. Data structures. Algorithms
and complexity. Exception handling. Abstract data types and classes. Inheritance
vs composition. Abstract vs concrete classes. Self-referential classes. Event
handling. Graphics. Multithreading. Network programming.
MATH0136: Software engineering I
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: CW25 EX75
Requisites:
Aims & learning objectives:
Aims: To give the students an introductory understanding of requirements analysis,
design modelling and specification, evaluation and testing. To raise students
awareness of usability and human factors along with software engineering methods.
Objectives: The students should be able to carry out analysis of domain, user,
task and software requirements using a taught method of analysis. They should
be able to produce specifications of designs and to set evaluation and test
criteria with the ability to use simple forms of evaluation and testing.
Content:
Software design process models, task analysis, requirements analysis, domain
analysis. Domain modelling, task modelling, conceptual design model, prototyping.
Empirical evaluation methods, and simple analytical evaluation and performance
testing methods.
MATH0137: Software engineering II
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: CW25 EX75
Requisites: Pre MATH0136
Some knowledge of programming, as approved by the Director of Studies
Aims & learning objectives:
Aims: To give the students an understanding and practical knowledge of software
engineering and human factors approaches to analysis, design and evaluation
of software systems. To develop an approach to software design and development
focussed upon analytical, and predictive modelling of domains, tasks, users
and software. To introduce appropriate forms of abstraction for representing
and reasoning about design issues. Objectives: To enable students to analyse
design problems choosing between analytical approaches. To construct abstract
models using appropriate formal and informal languages. To select and use correctly
appropriate forms of evaluation criteria and methods. To use appropriate forms
of design and prototyping approaches.
Content:
Analytical methods for gathering user, task, domain and software requirements,
including participatory and traditional approaches. Formal and informal modelling
languages, including process and object oriented approaches. Descriptive, predictive
and prescriptive models. Experimental methods and formal methods of testing.
Prototyping and redesign including paper and runnable forms.
MATH0138: Systems II: low-level programming & C
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: CW25 EX75
Requisites:
Some knowledge of programming, as approved by the Director of Studies
Aims & learning objectives:
Aims: To provide practical skills in low-level programming and basic computer
services Objectives: Students should be able to write short assembler-level
programs, and call basic (Chapter 1) Unix service using C. They should be able
to identify, use and package operating services. They should be able to write
short programs in ANSI C.
Content:
Assembler Programming: low level programming and structures. Registers, memory
and addressing modes. Subroutine structures, calling standards. Interrupts and
system calls. C Programming: C programming structures. String handling in C.
Input and output (both chapter 2 and chapter 3). Questioning the operating system
and filing system. Binary structures, relocation concepts.
MATH0139: Computation I: numbers & structures
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: CW25 EX75
Requisites:
Aims & learning objectives:
Aims: To introduce logic and set theory, as used in computer science, and to
give elementary explanations of the classical computational structures, such
as the integers and the real numbers, and to indicate some of the applications.
Objectives: Students should be able to understand and use the language of mathematics.
Students should be able to write simple proofs, using, for example, induction.
Content:
Natural numbers, integers, rationals, reals, complex numbers. Set theoretic
language. Relations, functions, predicates, equivalence relations. Standard
logical operators, including quantifiers. Connection with Data base techniques.
Proof. Induction. Binomial coefficients. Partially ordered sets. Well ordered
sets. Induction and recursion in general. Proof of termination of algorithms,
using well ordering. Modular arithmetic. Connection with cryptography. Iterative
solutions of equations by Newton's method. Convergence. Connection with fractals.
Arithmetic of polynomials and matrices. Concept of a ring.
MATH0140: Introduction to programming in Java
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: CW50 EX50
Requisites:
Aims & learning objectives:
Aims: To introduce object oriented programming in Java Objectives: To be able
to write programs which solve simple problems of the sort which may occur in
scientific applications. To understand the basic concepts of object oriented
programming
Content:
Basic programming concepts. How Java works. Operators and control. Arrays, references.
Methods, objects, classes, inheritance.. Standard libraries. Scientific applications.
MATH0141: Advanced human computer interaction
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 3
Assessment: CW25 EX75
Requisites:
Some knowledge of programming, as approved by the Director of Studies
Aims & learning objectives:
Aims: To give students an advanced level understanding of current research issues
in human computer interaction. To focus upon HCI research methods, HCI theory
covering topics of user interface design, evaluation and modelling. Application
areas such as safety and dependable systems, collaborative systems, virtual
environments and agent interaction are examples of current application topics.
Issues of HCI in mobile and embedded contexts. Objectives: The students should
obtain an in-depth understanding of HCI theory and methods in state of the art
research. Particular focus will be placed on the interdisciplinary nature of
HCI and on the relationship between theory, and design practice. The students
should be able to contribute to both HCI theory and HCI practice as a result..
Content:
Psychological theories of human behavior, contextual analysis, frameworks for
HCI, model based design. Topics in agent-agent and human-agent interaction.
Modelling collaborations, group work and domains of high safety or dependency
requirements. HCI and creativity. Evolutionary theories of design. The role
of formal methods in HCI. Advanced forms of interaction technologies..
MATH0142: Music & digital signal processing
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 3
Assessment: EX100
Requisites:
Some knowledge of programming; complex numbers; some knowledge of sine/cosine
functions, integration and elementary calculus, as approved by the Director
of Studies Aims & learning objectives:
Aims: To introduce the basic ideas of DSP programming and the ways in which
musical signals can be treated as data. Objectives: Students should be able
to code simple digital filters, and construct simple oscillators. They will
be able to control a frequency domain analysis and resynthesis, and use 3 synthesis
methods.
Content:
Introduction: Musical signals: their nature, chacterisation and representation.
Pitch, amplitude and timbre. PCM representation: sampling and quantisation errors.
MIDI representation and its limitations. Software Systems: Music5 family,: Csound.
Additive Synthesis: Simple oscillators and their coding; wavetable synthesis.
Helmholz theory and Fourier analysis. Subtractive Synthesis: Noise, and digital
filters. Filter types, IIR and FIR. Issues in filter design. Psycho-acoustics:
Basic ideas and Shepard tones as an example. Lossy compression. MPEG level 2
and MPEG-4. Time and frequency domains: Phase vocoding. FFT and IFFT; analysis
and resynthesis. Pitch changing. Physical Models: The wave equation. Delay lines
and wave guides. The plucked string. FM and non-linear synthesis: Analysis and
coding of FM. Introduction to Granular Synthesis, formants and FOF. Pitch changing.
Spacialisation: Stereo panning, reverberation, localisation and audio clues.
Composition: Process based, algorithmic composition. Pitch and Tuning: ET and
Just; introduction to Sethares theory of consonance.
MATH0143: User interface programming
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 2
Assessment: CW40 EX60
Requisites:
Some knowledge of programming, as approved by the Director of Studies
Aims & learning objectives:
Aims: To give the students knowledge, understanding and experience of designing,
constructing and evaluating user interfaces. To introduce principles, methods
and tools for user interface design. To focus upon user interaction. Objectives:
The students should learn how to program user input, application output, and
user interface input/output event handling. Dialogue design. To design user
interfaces to optimise usability and efficiency for the users' tasks. To be
aware of relevant principles, guidelines and tools to support user interface
design.
Content:
User interface display guides. Principles of user interface design, user interface
management systems, user interface tool-sets, Java user interface programming,
event handling, special features such as "help" and "undo". Analysis of usability
of user interface designs.
MATH0145: Applications II: databases
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 2
Assessment: CW40 EX60
Requisites:
Aims & learning objectives:
Aims: To introduce students to database concepts. To provide practical skills
at using database management systems, designing, using and managing databases.
To teach database theories. Objectives: Given a description of user requirements,
students should be able to design and build a database using a database management
system. They should be able to answer non-trivial theories using databases produced
by others. They should understand the importance of data protection and be able
to implement secure databases.
Content:
Introduction to databases and database management systems (DBMSs). User interaction
with databases. Functions of a DBMS. Data models and conceptual modelling. Sets
and relations. Union, intersection, relative complement, cross product, projection,
selection. Relations as subsets of cross products. Connection between logical
operations on ideas and set theoretic operations on relations. Logical database
design, physical database design. Entity-relationship modelling. Constraints.
Network and relational models. Completeness of relational models. Codd's classification
of canonical forms. : first, second, third and fourth normal forms. Keys, join,
SQL query language. Transaction management and database security. Data protection
legislation.
PHYS0002: Properties of matter
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: EX80 CW20
Requisites:
Students must have A-level Physics or Chemistry and A-level Mathematics to undertake
this unit. Aims & learning objectives:
The aims of this unit are to gain insight into how the interplay between kinetic
and potential energy at the atomic level governs the formation of different
phases and to demonstrate how the macroscopic properties of materials can be
derived from considerations of the microscopic properties at the atomic level.
After taking this unit the student should be able to - use simple model potentials
to describe molecules and solids - solve simple problems for ideal gases using
kinetic theory - describe the energy changes in adiabatic and isothermal processes
- derive thermodynamic relationships and analyse cycles - derive and use simple
transport expressions in problems concerning viscosity, heat and electrical
conduction.
Content:
Balance between kinetic and potential energy. The ideal gas - Kinetic Theory;
Maxwell- Boltzmann distribution; Equipartition. The real gas - van der Waals
model. The ideal solid - model potentials and equilibrium separations of molecules
and Madelung crystals. Simple crystal structures, X-ray scattering and Bragg's
law. First and second laws of thermodynamics, P-V-T surfaces, phase changes
and critical points, thermodynamic temperature and heat capacity of gases. Derivation
of mechanical (viscosity, elasticity, strength, defects) and transport properties
(heat and electrical conduction) of gases and solids from considerations of
atomic behaviour. Qualitative understanding of viscosity (Newtonian and non-Newtonian)
in liquids based on cage models.
PHYS0004: Relativity & astrophysics
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment: EX80 CW20
Requisites:
Students must have A-level Physics and Mathematics to undertake this unit. Aims
& learning objectives:
The aims of this unit are to introduce the concepts and results of special relativity
and to provide a broad introduction to astronomy and astrophysics. An additional
aim is that the student's appreciation of important physical phenomena such
as gravitation and blackbody radiation should be reinforced through their study
in astrophysical contexts. After taking this unit, the student should be able
to - write down the essential results and formulae of special relativity - describe
the important special relativity experiments (real or thought) - solve simple
kinematic and dynamical special relativity problems - give a qualitative account
of how the sun and planets were formed - describe how stars of differing masses
evolve - give a simple description of the expanding Universe and its large-scale
structure - solve simple problems concerning orbital motion, blackbody radiation,
cosmological redshift, stellar luminosity and magnitude.
Content:
Special Relativity: Galilean transformation. Speed of light - Michelson-Morley
experiment; Einstein's postulates. Simultaneity; time dilation; space contraction;
invariant intervals; rest frames; proper time; proper length. Lorentz transformation.
Relativistic momentum, force, energy. Doppler effect. Astrophysical Techniques:
Telescopes and detectors. Invisible astronomy : X-rays, gamma-rays, infrared
and radio astronomy. Gravitation: Gravitational force and potential energy.
Weight and mass. Circular orbits; Kepler's Laws; planetary motion. Escape velocity.
Solar System: Earth-Moon system. Terrestrial planets; Jovian planets. Planetary
atmospheres. Comets and meteoroids. Formation of the solar system. The interstellar
medium and star birth. Stellar distances, magnitudes, luminosities; black-body
radiation; stellar classification; Hertzsprung-Russell diagram. Stellar Evolution:
Star death: white dwarfs, neutron stars. General Relativity: Gravity and geometry.
The principle of equivalence. Deflection of light; curvature of space. Gravitational
time dilation. Red shift. Black holes. Large scale structure of the Universe.
Galaxies: Galactic structure; classification of galaxies. Formation and evolution
of galaxies. Hubble's Law. The expanding universe. The hot Big Bang. Cosmic
background radiation and ripples therein.
PHYS0024: Contemporary physics
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 3
Assessment: ES100
Requisites:
Students should have taken an appropriate selection of Year 1 and Year 2 Physics
units in order to undertake this unit. Aims & learning objectives:
The aim of this unit is to enable students to find out about some of the most
exciting developments in contemporary Physics research. While taking this unit
the student should be able to - demonstrate good time management skills in allocating
appropriate amounts of time for the planning, research and writing of reports
- carry out literature searching methods for academic journals and computer-based
resources in order to research the topics studied - develop the ability to extract
and assimilate relevant information from extensive sources of information -
develop structured report writing skills - write a concise summary of each seminar,
at a level understandable by a final year undergraduate unfamiliar with the
subject of the seminar - write a detailed technical report on one of the seminar
subjects of the student's choice, displaying an appropriate level of technical
content, style and structure.
Content:
This unit will be based around 5 or 6 seminars from internal and external speakers
who will introduce topics of current interest in Physics. Students will then
choose one of these subjects on which to research and write a technical report.
Topics are likely to include recent developments in: Astrophysics and Cosmology;
Particle Physics; Medical Physics; Laser Physics; Semiconductor Physics; Superconductivity;
Quantum Mechanical Simulation of Matter.
PHYS0029: Thermodynamics & statistical mechanics
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 3
Assessment: EX80 CW20
Requisites: Pre PHYS0002, Pre PHYS0008
Aims & learning objectives:
The aims of this unit are to develop an appreciation of the concepts of classical
thermodynamics and their application to physical processes and to introduce
the concepts of statistical mechanics, showing how one builds from an elementary
treatment based on ways of arranging objects to a discussion of Fermi-Dirac
and Bose systems, simple phase transitions, and more advanced phenomena. After
taking this unit, the student should be able to - define terms such as isobaric,
isothermal, adiabatic, etc. and state and apply the 1st and 2nd Laws - calculate
work done and heat interchanges as various paths are followed on a PV diagram
- explain the operation of, and carry out calculations for, heat engines and
refrigerators - write down the Clausius -Clapeyron equation and describe its
applications - carry out simple calculations on various Virial equations of
state - solve problems using Maxwell's relations in various contexts - define
entropy, temperature, chemical potential in statistical terms - derive the Boltzmann,
Planck, Fermi-Dirac and Bose-Einstein distribution functions and apply them
to simple model systems - outline the mean-field approach to phase transitions
in strongly interacting systems, and appreciate its limitations.
Content:
Classical thermodynamics; First and second laws of thermodynamics. Isothermal
and adiabatic processes. Thermodynamic temperature scale, heat engines, refrigerators,
the Carnot cycle, efficiency and entropy. Thermodynamic functions, Maxwell's
relations and their applications. Specific heat equations, phase changes, latent
heat equations and critical points. Statistical Mechanics; Basic postulates.
Systems in thermal contact and thermal equilibrium. Statistical definitions
of entropy, temperature and chemical potential. Boltzmann factor and partition
function illustrated by harmonic oscillator and two-state system. Planck distribution:
photons, radiation, phonons. Fermions and Bosons: Fermi-Dirac and Bose-Einstein
distribution functions. Properties of Fermi systems: ground state of a Fermi
gas, density of states; Fermi gas at non-zero temperature; electrons in solids,
models of white dwarf and neutron stars. Properties of Bose systems: Bose-Einstein
condensation, superfluidity and superconductivity. Applications of Statistical
Mechanics to classical and quantum systems such as non-reacting and reacting
mixtures of classical gases; equilibrium of two-phase assemblies; models of
magnetic crystals, the Ising model; mean-field and other approaches to phase
transitions in ferromagnets and binary alloys; elementary kinetic theory of
transport processes; transport theory using the relaxation-time approximation:
electrical conductivity, viscosity; propagation of heat and sound.
PHYS0030: Quantum mechanics
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 3
Assessment: EX80 CW20
Requisites:
Students must have A-level Physics in order to undertake this unit and must
have undertaken appropriate maths units provided by either the Departments of
Physics or Mathematical Sciences. Aims & learning objectives:
The aims of this unit are to show how a mathematical model of considerable elegance
may be constructed, from a few basic postulates, to describe the seemingly contradictory
behaviour of the physical universe and to provide useful information on a wide
range of physical problems. After taking this unit the student should be able
to: - discuss the dual particle-wave nature of matter - explain the relation
between wave functions, operators and experimental observables - justify the
need for probability distributions to describe physical phenomena - set up the
Schröödinger equation for simple model systems - derive eigenstates of
energy, momentum and angular momentum - apply approximate methods to more complex
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.
PHYS0031: Simulation techniques
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 3
Assessment: EX80 CW20
Requisites: Pre PHYS0020
Aims & learning objectives:
The aims of this unit are to identify some of the issues involved in constructing
mathematical models of physical processes, and to introduce major techniques
of computational science used to find approximate solutions to such models.
After taking this unit the student should be able to - dedimensionalise an equation
representing a physical system - discretise a differential equation using grid
and basis set methods - outline the essential features of each of the simulation
techniques introduced - give examples of the use of the techniques in contemporary
science - use the simulation schemes to solve simple examples by hand - describe
and compare algorithms used for key processes common to many computational schemes.
Content:
Construction of a mathematical model of a physical system; de-dimensionalisation,
order of magnitude estimate of relative sizes of terms. Importance of boundary
conditions. The need for computed solutions. Discretisation using grids or basis
sets. Discretisation errors. The finite difference method; review of ODE solutions.
Construction of difference equations from PDEs. Boundary conditions. Applications.
The finite element method; Illustration of global, variational approach to solution
of PDEs. Segmentation. Boundary conditions. Applications. Molecular Dynamics
and Monte-Carlo Methods; examples of N-body problems, ensembles and averaging.
The basic MD strategy. The basic MC strategy; random number generation and importance
sampling. Applications in statistical mechanics. Simulated annealing. Computer
experiments. Solving finite difference problems via random walks. Other major
algorithms of computational science; the Fast Fourier Transform, matrix methods,
including diagonalisation, optimisation methods, including non-linear least
squares fitting.
XXXX0001: Any other units approved by the Director of
Studies
Semester 1
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment:
Requisites:
This pseudo-unit indicates that you are allowed to choose other units from around
the University subject to the normal constraints such as staff availability,
timetabling restrictions, and minimum and maximum group sizes. You should make
sure that you indicate your actual choice of units when requested to do so.
Details of the University's Catalogue can be seen on the University's Home Page.
XXXX0001: Any other units approved by the Director of
Studies
Semester 2
Credits: 6
Contact:
Topic:
Level: Level 1
Assessment:
Requisites:
This pseudo-unit indicates that you are allowed to choose other units from around
the University subject to the normal constraints such as staff availability,
timetabling restrictions, and minimum and maximum group sizes. You should make
sure that you indicate your actual choice of units when requested to do so.
Details of the University's Catalogue can be seen on the University's Home Page.