MATHEMATICAL SCIENCES

Credit Points

15

Course Coordinator

Professor J Hubbuck

Pre-requisites

MA 2003

Overview

This course aims to put on a sound footing many of the results and procedures used in the Calculus. It starts by studying properties of the real number system, including suprema and infima, sequences and series. Then it considers the theory of continuous functions on closed bounded intervals, treating global extrema and intermediate values. These results are applied in the theory of differentiability and to the Riemann Integral.

Structure

12 week course - 5 one-hour lectures and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt:1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Dr S Theriault

Pre-requisites

MA 2002 and MA 2504

Overview

This is a first course in abstract algebra. The familiar, simple and useful properties of the integers places the set of integers at the core of any study of algebraic objects. But many of these properties hold for other familiar mathematical objects; for polynomials, real numbers, matrices etc. This course develops the theory of rings and fields which unifies the study of many of these objects and, at the same time, clarifies the differences between them.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Professor M Weiss

Pre-requisites

MA 2503

Overview

The course studies the Newtonian theory of the motion of a particle. Newton's laws of motion are introduced and illustrated through the study of dynamical problems such as projectile motion, air resistance and the theory of vibrations. Theoretical work is done on topics such as energy, linear and angular momentum and the role of inertial frames in Newtonian mechanics.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Professor M S Weiss

Pre-requisites

MX 3002

Overview

Numbers measure size, groups measure symmetry. Many groups occur naturally as symmetry groups of solids, patterns and other geometrical objects. This course will develop the basic ideas of group theory through such examples of groups acting on sets.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

5

Course Coordinator

Dr C Maclachlan

Pre-requisites

Available only to candidates for Honours Degrees involving Mathematics or Statistics.

Notes

The assessment of this course does not count towards Honours classification. This course is not available in 2004/05.

Overview

The student will undertake a project specified by the department. The work may be done individually or in teams. The end result of the work is to be a report and presentation by the student or team. The work will be supervised by a member of the department and will be assessed on the quality of the report and its presentation.

Structure

12 week course - Classes as appropriate.

Assessment

1st Attempt: Assessed on the report. Resit:

Credit Points

15

Course Coordinator

Dr J R Pulham

Pre-requisites

MX 3001

Overview

This is an introductory course on Complex Analysis. Holomorphic functions and power series, Cauchy’s throrem and its consequences, contour integration and the calculus of residues are discussed.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Dr G Hall

Pre-requisites

MA 2003 and MA 2503

Overview

An introduction to the vector calculus leading to the divergence theorem and some of its applications; a brief treatment of Fourier series and their applications; an introduction to partial differential equations, their behaviour and solution.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1 two-hour written examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Dr R Levi

Pre-requisites

MA 2003, MA 2504

Overview

Basis non-linear optimisation techniques for multivariable real valued functions, including the second derivative test, constrained optimisation and the method of Lagrange multipliers. Following this, the course will specialise to linear optimisation problems. The simplex algorithm will be introduced and studied, including applications to matrix games. Computations will be done using a computer-based algerbra package, as well as manually.

Structure

2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%); continuous assessment (20%).
Resit: 1 two-hour written examination (maximum of 100% resit and 80% resit with 20% continuous assessment).

Credit Points

15

Course Coordinator

Dr A Libmann

Pre-requisites

MX 3001

Overview

An introduction to metric and topological spaces, including a discussion of connectedness, compactness and the continuity of mappings.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Dr R Levi

Pre-requisites

MX 3521 or permission of Head of Mathematical Sciences.

Overview

The student will be given a mathematical topic on which to write and submit a report. The work will be supervised by a member of staff.

Structure

12 week course – Assessed on the project report and the oral presentation (the presentations are given during the second half-session).

Credit Points

15

Course Coordinator

Head of Department

Pre-requisites

MX 3521. Available only to candidates for Honours in Mathematics, Mathematics with French, Mathematics with Gaelic, Mathematics with German, and Mathematics with Spanish.

Notes

Not available in session 2004/05.

Overview

The student will undertake a part-time placement in some company, external institution or other university department to work on a project approved by the department. The placement will extend over all or part of the half-session as appropriate. Both an on-site supervisor and a departmental supervisor will be appointed to monitor the student’s progress. The assessment of the course will be based on a report written by the student and on assessments by the supervisors. The course will only be available for selected students and if suitable placements can be found.

Structure

12 week course – Classes as appropriate.

Assessment

1st Attempt: Assessed on the report and on the supervisors’ report. Resit:

Credit Points

30

Course Coordinator

Professor G Hall

Pre-requisites

MX 3521. The course is available only to students accepted into the Joint Honours Programme Mathematics-Physics (MA or BSc) or the single Honours Programmes Physics (BSc) or Natural Philosophy (MA).

Notes

This course is run over the full session.

Overview

The student will be given a Mathematical topic on which to write a report. The work will be supervised by a member of staff. The assessment of the project will be based on the report and an oral examination based on the material relevant to the assigned topic.

Structure

24 week course – 1 tutorial per week.

Assessment

1st Attempt: Assessed on the project report and on the oral examination. Resit:

Credit Points

15

Course Coordinator

Dr S Dagger

Pre-requisites

MX 3002

Overview

Number theory is the study of integers and has three main branches: Elementary, Analytical and Algebraic. This course consists of a selection of topics from these branches. The topics will include some of the following: the theory of quadratic congruences, continued fractions, pseudo-primes, primitive roots, Diophantine equations, the distribution of prime numbers, algebraic integers in quadratic number fields.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3014

Co-requisites

Available only to candidates for Honours Degrees involving Statistics.

Overview

The course covers the theory and practice of rank tests, which are widely used in statistical work. Topics covered will include: Wilcoxon signed-rank test, Wilcoxon/Mann-Whitney test, Friedman's test, Kruskal-Wallis test, tests of the Kolmogorov-Smirnov and Cramer-von Mises type, Savage's test and its relationship to the log-rank test and asymptotic efficiency. In addition some other statistical topics will be discussed.

Assessment

1 two hour written examination (100%).
Resit: No resit.

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

Either: (a) MA2003, MA 2503 and MA 2504; or: (b) MA 2003 and PX 2012.

Notes

(i) Special Option. Not available in session 2004/05.
(ii) Available only to students in programme year 3 or above.

Overview

The failure of the Newtonian model of physics. The basic principles of the Special Theory of Relativity. The Lorentz transformation and its applications, including length and time dilation. The kinematics of particles. 4-vectors and Minkowski space. The dynamics of particles, momentum, energy and force. Relativistic optics. Collision problems.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 4008

Notes

Special Option. Available in session 2004/05.

Overview

A course on some mathematical aspects of the theories of fractals and discrete dynamical processes. It will normally include a treatment of fractal dimension and the use of iterated function systems to generate fractals.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%); continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3502

Notes

Special Option. Not available in session 2004/05.

Overview

The roots of a quadratic polynomial are given by a formula involving the coefficients. Similar formulae exist for the roots of polynomial equations of degrees 3 and 4, but not for higher degrees. The precise relationship between a polynomial and the type of roots it has emerges as one of the consequences of Galois Theory, which is a unification of ideas embracing polynomials, fields and group theory. The course will also consider the classical ruler and compass constructions.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3502 and MX 4008

Notes

Special Option. Not available in session 2004/05.

Overview

An introduction to the topology associated to a variety of basic geometric spaces, including a discussion of topological invariants and applications to geometric problems.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MA 2002 and MA 2003 and 2504 and either MA 2503 or ST 2003.

Notes

(i) Special Option. Not available in session 2004/05.

(ii) Available only to students in programme year 3 or above.

Overview

An introductory course on the theory of graphs. Topics covered will include: elementary properties of graphs, Eulerian and Hamiltonian circuits, some matching theory including Hall’s theorem.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 4008

Notes

Special Option. Not available in session 2004/05.

Overview

Whereas earlier work in analysis tended to focus on single functions, this course deals with functions collectively, as elements of vector spaces or function algebras.
The course will cover topics from: normed spaces, Banach spaces, Hilbert spaces (with emphasis on sequence spaces and function spaces), linear functionals and operators, Hahn-Banach theorem, principle of uniform boundedness, open mapping and closed graph theorems, the algebra of continuous functions on a compact Hausdorff space, Stone-Weierstrass theorem and Gelfand theory.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Dr R Levi

Pre-requisites

MX 3521

Overview

The student is given a mathematical topic on which to write and submit a report. The work will be supervised by a member of staff. 12 week course - Assessed on the project report and oral presentation.

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MA 2504 and MX 4008

Notes

Special Option. Not available in session 2004/05.

Overview

A Hilbert space is a vector space which is complete with respect to the metric arising from a given inner product. This setting permits the development of geometric ideas, taken from Euclidean space, which can then be applied to spaces of functions arising naturally in the theory of differential equations. The course will cover topics from: norms, inner products and Hilbert spaces, orthogonality, orthogonal expansions and Fourier series, dual spaces, linear operators.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3014

Notes

This course will be available in session 2004/05 only.

Overview

This course will extend the material in MX 3014 and will include a selection of the following topics: generating functions, convergence in probability and in distribution, laws of large numbers, the Central Limit theorem, sufficiency, exponential family of distributions, more on maximum likelihood, hypothesis testing and interval estimation, the Bayesian and decision theory approaches to inference-utility, minimax and Bayes’ strategies, admissibility, risk and loss functions, prior and posterior distributions, estimation and hypothesis testing, credible intervals.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and continuous assessment (20%). Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3015

Notes

Special Option. Not available in session 2004/05.

Overview

The course studies computer algorithms, considering their construction, validation and effectiveness. After a general introduction to the subject a number of specific topics will be covered. These may include: the problem of sorting data sets into order, the use of abstract data types to formalise interactions, the theory of formal grammars and problems such as the parsing of arithmetic expressions, the construction and use of pseudo-random numbers. (If there is insufficient demand this course may be taught as a reading course in which case there will be no lectures and one tutorial per week).

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 4008

Notes

Special Option. Not available in session 2004/05.

Overview

The course is concerned with the analysis of functions of several variables, in particular the differentiability and integrability of such functions. Appropriate background material will be discussed in order to prove some important theorems of analysis, for instance the inverse and implicit function theorems, Fubini’s theorem and convergence theorems of integration.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3522

Notes

Special Option. Not available in session 2004/05.

Overview

A study of some important Special Functions of mathematics, providing practical illustrations of many important techniques and methods of analysis.

Structure

12 week course - 2 one-hour lectures and 1 tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3002

Notes

Special Option. Available in session 2004/05.

Overview

Traditional applied mathematics is centred in the area where calculus and its developments are used to solve problems in the physical sciences. This course looks at another and more recent set of problems deriving from such things as digital communication and the design of efficient statistical experiments. The course is primarily an introduction to the algebraic theory of error-correcting linear codes.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3001

Notes

Special Option. Not available in session 2004/05.

Overview

This course studies Fourier Series and their applications to the solution of boundary value problems associated with certain linear partial differential equations. In particular the wave equation, heat equation and Laplace’s equation will be studied using the technique of separation of variables. Various aspects of the theory of Fourier series will be discussed, for instance Bessel’s inequality, Parseval’s formula and the convergence and differentiability of Fourier series.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt:1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3012

Notes

Special Option. Not available in session 2004/05.

Overview

This course is a continuation of Mechanics A (MX 3012). The ideas and methods of that course are extended to study such topics as: Galilean transformations, systems of particles, the kinematics and dynamics of rigid bodies, analytic mechanics.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3522

Notes

Special Option. Not available in session 2004/05.

Overview

This course is concerned with the application of the Laplace and Fourier transformations to differential and integral equations. It begins with a brief discussion of differential equations. Then the theories of Laplace and Fourier transforms are developed and applied to various problems arising in the study of ordinary differential, partial differential and integral equations.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3001

Notes

Special Option. Not available in session 2004/05.

Overview

An introduction to the qualitative theory of systems of ordinary differential equations. Topics covered will include: existence and uniqueness theory, linear systems, equilibria and their stability, periodic solutions. Various particular examples will be analysed in detail.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st attempt: 1 two-hour examination. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3001

Notes

Special Option. Not available in session 2004/05.

Overview

An introduction to the differential geometry of surfaces. The emphasis will be on explicit local co-ordinate descriptions of surfaces, allowing the introduction of explicit examples throughout the course. The course will include Gauss’s Theorema Egregium, that the Gaussian Curvature, originally defined in terms of a particular embedding of the surface in space, is an intrinsic property of the surface.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination paper. Resit:

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

MX 3502 and MX 3001

Notes

Special Option. Available in session 2004/05.

Overview

An introduction to knot theory. The course will include a study of elementary invariants via Reidemeister moves and associated topological objects such as the fundamental group of a knot complement.

Structure

12 week course - 2 one-hour lectures and 1 tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination. Resit:

Credit Points

15

Course Coordinator

Dr J Pulham

Pre-requisites

MX 3526

Notes

Special Option. Available in session 2004/05.

Overview

A course on the mathematical theory of electromagnetism, including electrostatics, potential theory and applications of the wave equation. The course exploits the mathematical techniques developed in MX 3526 (Mathematical Methods).

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%) and continuous assessment (20%). Resit: