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Department of Mathematical Sciences, Unit Catalogue 2002/03

MA10001: Numbers

Credits: 6
Level: Certificate
Semester: 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.

MA10002: Functions, differentiation & analytic geometry

Credits: 6
Level: Certificate
Semester: 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. 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.

MA10003: Integration & differential equations

Credits: 6
Level: Certificate
Semester: 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.
Objectives: 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.

MA10004: Sets & sequences

Credits: 6
Level: Certificate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA10001

Aims & Learning Objectives:
Aims: To introduce the concepts of logic that underlie all mathematical reasoning and the notions of set theory that provide a rigorous foundation for mathematics. A real life example of all this machinery at work will be given in the form of an introduction to the analysis of sequences of real numbers.
Objectives: By the end of this course, the students will be able to: understand and work with a formal definition; determine whether straight-forward definitions of particular mappings etc. are correct; determine whether straight-forward operations are, or are not, commutative; read and understand fairly complicated statements expressing, with the use of quantifiers, convergence properties of sequences.
Content:
Logic: Definitions and Axioms. Predicates and relations. The meaning of the logical operators Ù, Ú, ˜, ®, «, ", $. Logical equivalence and logical consequence. Direct and indirect methods of proof. Proof by contradiction. Counter-examples. Analysis of statements using Semantic Tableaux. Definitions of proof and deduction. Sets and Functions: Sets. Cardinality of finite sets. Countability and uncountability. Maxima and minima of finite sets, max (A) = - min (-A) etc. Unions, intersections, and/or statements and de Morgan's laws. Functions as rules, domain, co-domain, image. Injective (1-1), surjective (onto), bijective (1-1, onto) functions. Permutations as bijections. Functions and de Morgan's laws. Inverse functions and inverse images of sets. Relations and equivalence relations. Arithmetic mod p. Sequences: Definition and numerous examples. Convergent sequences and their manipulation. Arithmetic of limits.

MA10005: Matrices & multivariate calculus

Credits: 6
Level: Certificate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA10002

Aims & Learning Objectives:
Aims: The course will provide students with an introduction to elementary matrix theory and an introduction to the calculus of functions from IRn ® IRm and to multivariate integrals.
Objectives: At the end of the course the students will have a sound grasp of elementary matrix theory and multivariate calculus and will be proficient in performing such tasks as addition and multiplication of matrices, finding the determinant and inverse of a matrix, and finding the eigenvalues and associated eigenvectors of a matrix. The students will be familiar with calculation of partial derivatives, the chain rule and its applications and the definition of differentiability for vector valued functions and will be able to calculate the Jacobian matrix and determinant of such functions. The students will have a knowledge of the integration of real-valued functions from IR² ® IR and will be proficient in calculating multivariate integrals.
Content:
Lines and planes in two and three dimension. Linear dependence and independence. Simultaneous linear equations. Elementary row operations. Gaussian elimination. Gauss-Jordan form. Rank. Matrix transformations. Addition and multiplication. Inverse of a matrix. Determinants. Cramer's Rule. Similarity of matrices. Special matrices in geometry, orthogonal and symmetric matrices. Real and complex eigenvalues, eigenvectors. Relation between algebraic and geometric operators. Geometric effect of matrices and the geometric interpretation of determinants. Areas of triangles, volumes etc. Real valued functions on IR³. Partial derivatives and gradients; geometric interpretation. Maxima and Minima of functions of two variables. Saddle points. Discriminant. Change of coordinates. Chain rule. Vector valued functions and their derivatives. The Jacobian matrix and determinant, geometrical significance. Chain rule. Multivariate integrals. Change of order of integration. Change of variables formula.

MA10006: Vectors & applications

Credits: 6
Level: Certificate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA10001 and take MA10002 and take MA10003

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.

MA10031: Statistics & probability 1

Credits: 6
Level: Certificate
Semester: 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. Use of Minitab package. 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.


MA10032: Statistics & probability 2

Credits: 6
Level: Certificate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA10031

Aims & Learning Objectives:
Aims: To introduce continuous random variables and to begin the study of statistical inference.
Objectives: Ability to compute quantities relating to continuous probability distributions, calculate confidence intervals for the mean of a normal distribution, and find the equation for the best fit straight line for a linear regression model.
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.

MA10095: Quantitative methods

Credits: 5
Level: Certificate
Semester: 2
Assessment: EX100
Requisites: While taking this unit take MN10003

Aims & Learning Objectives:
Aims: To teach the basic ideas of probability, data variability, hypothesis testing and of relationships between variables and the application of these ideas in management.
Objectives: Students should be able to formulate and solve simple problems in probability including the use of Bayes' Theorem and Decision Trees. They should recognise real-life situations where variability is likely to follow a binomial, Poisson or normal distribution and be able to carry out simple related calculations. They should be able to carry out a simple decomposition of a time series, apply correlation and regression analysis and understand the basic idea of statistical significance.
Content:
The laws of Probability, Bayes' Theorem, Decision Trees. Binomial, Poisson and normal distributions and their applications; the relationship between these distributions. Time series decomposition into trend and season al components; multiplicative and additive seasonal factors. Correlation and regression; calculation and interpretation in terms of variability explained. Idea of the sampling distribution of the sample mean; the Z test and the concept of significance level.

MA10103: Foundation mathematics 1

Credits: 6
Level: Certificate
Semester: 1
Assessment: EX50CW50
Requisites:

Aims & Learning Objectives:
Core 'A' level maths. The course follows closely the essential set book: L Bostock & S Chandler, Core Maths for A-Level, Stanley Thornes ISBN 0 7487 1779 X
Content:
Numbers: Integers, Rationals, Reals. Algebra: Straight lines, Quadratics, Functions, Binomial, Exponential Function. Trigonometry: Ratios for general angles, Sine and Cosine Rules, Compound angles. Calculus: Differentiation: Tangents, Normals, Rates of Change, Max/Min.

MA10104: Foundation mathematics 2

Credits: 6
Level: Certificate
Semester: 2
Assessment: EX50CW50
Requisites: While taking this unit take MA10103

Aims & Learning Objectives:
Core 'A' level maths. The course follows closely the essential set book: L Bostock & S Chandler, Core Maths for A-Level, Stanley Thornes ISBN 0 7487 1779 X
Content:
Integration: Areas, Volumes. Simple Standard Integrals. Statistics: Collecting data, Mean, Median, Modes, Standard Deviation.

MA10116: Mathematical techniques 1

Credits: 5
Level: Certificate
Semester: 1
Assessment: EX75CW25
Requisites:

Aims & Learning Objectives:
To provide students with a basic introduction to the mathematical skills necessary to tackle process engineering design applications.
Content:
Differentiation, integration. Revision of differentiation of logarithmic, exponential and inverse trignometrical functions. Revision of applications of integration including polar and parametric co-ordinates. Further calculus: Hyperbolic functions, Inverse functions, McLaurin's and Taylor's Theorems, Limits, Approximate methods including the solution of equations by Newton's method and integration by Simpsons rule. Partial differentials. Functions of several variables, small errors, totaldifferential Differential equations. Solution of first order equations using separation of variables and integrating factor. Linear equations with constant coefficients using trial method for particular integral. Simultaneous linear differential equations
* Further calculus: Hyperbolic functions, Inverse functions, McLaurin's and Taylor's theorem, Limits, Approximate methods, including solution of equations by Newton's method and integration by Simpson's rule
* Partial differentials: functions of several variables, Small errors, Total differential
* Differential equations: Solution of first order equations using separation of variables and integrating factor; Linear equations with constant coefficients using trial method for particular integral; Simultaneous linear differential equations.

MA10126: Introduction to computing with applications

Credits: 6
Level: Certificate
Semester: 1
Assessment: CW100
Requisites:

Aims & Learning Objectives:
Aims: To introduce computational tools of relevance to scientists working in a numerate discipline. To teach programming skills in the context of applications. To introduce presentational and expositional skills and group work.
Objectives: At the end of the course, students should be: proficient in elementary use of UNIX and EMACS; able to program a range of mathematical and statistical applications using MATLAB; able to analyse the complexity of simple algorithms; competent with working in groups; giving presentations and creating web pages.
Content:
Introduction to UNIX and EMACS. Brief introduction to HTML. Programming in MATLAB and applications to mathematical and statistical problems: Variables, operators and control, loops, iteration, recursion. Scripts and functions. Compilers and interpreters (by example). Data structures (by example). Visualisation. Graphical-user interfaces. Numerical and symbolic computation. The MATLAB Symbolic Math toolbox. Introduction to complexity analysis. Efficiency of algorithms. Applications. Report writing. Presentations. Web design. Group project.

MA20007: Analysis: Real numbers, real sequences & series

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10004 and take MA10005 and take MA10006

Aims & Learning Objectives:
Aims: To reinforce and extend the ideas and methodology (begun in the first year unit MA10004) 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. Brief discussion of convergence of sequences of functions.

MA20008: Algebra 1

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10004 and take MA10005 and take MA10006

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.

MA20009: Ordinary differential equations & control

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10001 and take MA10002 and take MA10003 and take MA10005

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.

MA20010: Vector calculus & partial differential equations

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10002 and take MA10003 and take MA10005 and take MA10006 and while taking this unit you must take MA20009

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.

MA20011: Analysis: Real-valued functions of a real variable

Credits: 6
Level: Intermediate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20007

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.

MA20012: Algebra 2

Credits: 6
Level: Intermediate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20008

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 MA20008. 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.

MA20013: Mathematical modelling & fluids

Credits: 6
Level: Intermediate
Semester: 2
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA20009 and take MA20010


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.


MA20014: Numerical analysis

Credits: 6
Level: Intermediate
Semester: 2
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA20007 and take MA20008

Aims & Learning Objectives:
Aims: To revise and develop elementary MATLAB programming techniques. 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. Composite formulae. Error terms. Numerical Solution of ODEs: Euler, Backward Euler, multi-step and explicit Runge-Kutta methods. Stability. Consistency and convergence for one step methods. Error estimation and control. Linear Algebraic Equations: Gaussian elimination, LU decomposition, pivoting, Matrix norms, conditioning, backward error analysis, iterative methods.

MA20033: Statistical inference 1

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10032

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.

MA20034: Probability & random processes

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10002 and take MA10004 and take MA10032 and while taking this unit you must take MA20007

Aims & Learning Objectives:
Aims: To introduce some fundamental topics in probability theory including conditional expectation and the three classical limit theorems of probability. To present the main properties of random walks on the integers, and Poisson processes.
Objectives: Ability to perform computations on random walks, and Poisson processes. Ability to use generating function techniques for effective calculations. Ability to work effectively with conditional expectation. Ability to apply the classical limit theorems of probability.
Content:
Revision of properties of expectation and conditional probability. Conditional expectation. Chebyshev's inequality. The Weak Law. Statement of the Strong Law of Large Numbers. Random variables on the positive integers. Probability generating functions. Random walks expected first passage times. Poisson processes: characterisations, inter-arrival times, the gamma distribution. Moment generating functions. Outline of the Central Limit Theorem.

MA20035: Statistical inference 2

Credits: 6
Level: Intermediate
Semester: 2
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA20033

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.

MA20036: Stochastic processes

Credits: 6
Level: Intermediate
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA10003 and take MA10005 and take MA20034
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, queuing 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, expected hitting times and invariant 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, expected hitting times, classification of states, renewal theorem, invariant distributions, symmetrizability and ergodic theorems. Markov processes with discrete states in continuous time: Examples, including Poisson processes, birth & death processes and various types of Markovian queues. Q-matrices, resolvents, waiting time distributions, equilibrium distributions and ergodicity.

MA20096: Statistics 1

Credits: 5
Level: Intermediate
Semester: 2
Assessment: EX60CW40
Requisites: Before taking this unit you must take MA10095

Aims & Learning Objectives:
Aims: To teach the fundamental ideas of sampling and its use in estimation and hypothesis testing. These will be related as far as possible to management applications.
Objectives: Students should be able to obtain interval estimates for population means, standard deviations and proportions and be able to carry out standard one and two sample tests. They should be able to handle real data sets using the minitab package and show appreciation of the uses and limitations of the methods learned.
Content:
Different types of sample; sampling distributions of means, standard deviations and proportions. The use and meaning of confidence limits. Hypothesis testing; types of error, significance levels and P values. One and two sample tests for means and proportions including the use of Student's t. Simple non-parametric tests and chi-squared tests. The probability of a type 2 error in the Z test and the concept of power. Quality control: Acceptance sampling, Shewhart charts and the relationship to hypothesis testing. The use of the minitab package and practical points in data analysis.

MA20097: Statistics 2

Credits: 5
Level: Intermediate
Semester: 1
Assessment: EX60CW40
Requisites: Before taking this unit you must take MA20096

Aims & Learning Objectives:
Aims: To teach the methods of analysis appropriate to simple and multiple regression models and to common types of survey and experimental design. The course will concentrate on applications in the management area.
Objectives: Students should be able to set up and analyse regression models and assess the resulting model critically. They should understand the principles involved in experimental design and be able to apply the methods of analysis of variance.
Content:
One-way analysis of variance (ANOVA): comparisons of group means. Simple and multiple regression: estimation of model parameters, tests, confidence and prediction intervals, residual and diagnostic plots. Two-way ANOVA: Two-way classification model, main effects and interactions. Experimental Design: Randomisation, blocking, factorial designs. Analysis using the minitab package.

MA20105: Industrial placement

Credits: 60
Level: Intermediate
Academic Year
Assessment:
Requisites:

Industrial Placement year

MA20106: Study year abroad (BSc)

Credits: 60
Level: Intermediate
Academic Year
Assessment:
Requisites:

BSc Study year abroad

MA20108: Statistics

Credits: 6
Level: Intermediate
Semester: 1
Assessment: EX50CW50
Requisites: Before taking this unit you must take BB10001 and take BB10002

Aims & Learning Objectives:
Aims: To understand the principles of statistics as applied to Biological problems.
Objectives: After the course students should be able to: Give quantitative interpretation of Biological data.
Content:
Topics: Random variation, frequency distributions, graphical techniques, measures of average and variability. Discrete probability models - binomial, poisson. Continuous probability model - normal distribution. Poisson and normal approximations to binomial sampling theory. Estimation, confidence intervals. Chi-squared tests for goodness of fit and contingency tables. One sample and two sample tests. Paired comparisons. Confidence interval and tests for proportions. Least squares straight line. Prediction. Correlation

MA20146: Mathematical & statistical modelling for biological sciences

Credits: 6
Level: Intermediate
Semester: 2
Assessment: CW50EX50
Requisites: Before taking this unit you must take MA20108 or take MA10032

and Pre-requisite: A-level Mathematics
Aims & Learning Objectives:
This unit aims to study, by example, practical aspects of mathematical and statistical modelling, focussing on the biological sciences. Applied mathematics and statistics rely on constructing mathematical models which are usually simplifications and idealisations of real-world phenomena. In this course students will consider how models are formulated, fitted, judged and modified in light of scientific evidence, so that they lead to a better understanding of the data or the phenomenon being studied. the approach will be case-study-based and will involve the use of computer packages. Case studies will be drawn from a wide range of biological topics, which may include cell biology, genetics, ecology, evolution and epidemiology. After taking this unit, the student should be able to
* Construct an initial mathematical model for a real-world process and assess this model critically; and
* Suggest alterations or elaborations of a proposed model in light of discrepancies between model predictions and observed data, or failures of the model to exhibit correct quantitative behaviour.
Content:
* Modelling and the scientific method. Objectives of mathematical and statistical modelling; the iterative nature of modelling; falsifiability and predictive accuracy.
* The three stages of modelling. (1) Model formulation, including the art of consultation and the use of empirical information. (2) Model fitting. (3) Model validation.
* Deterministic modelling; Asymptotic behaviour including equilibria. Dynamic behaviour. Optimum behaviour for a system.
* The interpretation of probability. Symmetry, relative frequency, and degree of belief.
* Stochastic modelling. Probalistic models for complex systems. Modelling mean response and variability. The effects of model uncertainty on statistical interference. The dangers of multiple testing and data dredging.


MA30037: Galois theory

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20008 and take MA20012

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.

MA30038: Advanced group theory

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20008 and take MA20012

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.

MA30039: Differential geometry of curves & surfaces

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20011 and take MA20012
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.

MA30041: Metric spaces

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20011

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.

MA30044: Mathematical methods 1

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20008 and take MA20009 and take MA20010 and take MA20012

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).

MA30046: Linear control theory

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20009 and take MA20011 and take MA20012

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.

MA30047: Mathematical biology 1

Credits: 6
Level: Honours
Semester: 1
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA20009 and take MA20013

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.

MA30049: Linear elasticity

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20010 and take MA30065

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.

MA30051: Numerical linear algebra

Credits: 6
Level: Honours
Semester: 1
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA20008 and take MA20012 and take MA20014

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.

MA30054: Representation theory of finite groups

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA30038

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.

MA30055: Introduction to topology

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA30041

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.

MA30056: Complex analysis

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20011

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.

MA30059: Mathematical methods 2

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA30044

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.

MA30060: Nonlinear systems & chaos

Credits: 6
Level: Honours
Semester: 2
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20009 and take MA20010 and take MA20011 and take MA20012 and take MA20013 and take MA20014

Aims & Learning Objectives:
Aims: The course is intended to be an elementary and accessible introduction to dynamical systems with examples of applications. 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.

MA30063: Mathematical biology 2

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20010 and take MA20013

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.

MA30065: Viscous fluid mechanics

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20010 and take MA20013

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.

MA30084: Linear models

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10002 and take MA10003 and take MA10005 and take MA20008 and take MA20035

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: Subset selection and stepwise regression methods with applications in polynomial regression and multiple regression. Effects of collinearity in regression variables. Uses of residuals: Probability plots, plots for additional variables, 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 dose response relationships, and logistic regression.

MA30085: Time series

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20035

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 an appropriate 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, Forecasting from ARIMA models. Stationary processes in the frequency domain: The spectral density function, the periodogram, spectral analysis. State-space models: Dynamic linear models and the Kalman filter.

MA30086: Medical statistics

Credits: 6
Level: Honours
Semester: 1
Assessment: CW25EX75
Requisites: Before taking this unit you must take MA20035

Aims & Learning Objectives:
Aims: To introduce students to the use of statistical methods in medical research, the pharmaceutical industry and the National Health Service.
Objectives: Students should be able to (a) recognize the key statistical features of a medical research problem, and, where appropriate, suggest an appropriate study design, (b) understand the ethical considerations and practical problems that govern medical experimentation, (c) summarize medical data and spot possible sources of bias, (d) analyse data collected from some types of clinical trial, as well as simple survival data and longitudinal data.
Content:
Ethical considerations in clinical trials and other types of epidemiological study design. Phases I to IV of drug development and testing. Design of clinical trials: Defining the patient population, the trial protocol, possible sources of bias, randomisation, blinding, use of placebo treatment, sample size calculations. Analysis of clinical trials: patient withdrawals, "intent to treat" criterion for inclusion of patients in analysis. Survival data: Life tables, censoring. Kaplan-Meier estimate. Selected topics from: Crossover trials; Case-control and cohort studies; Binary data; Measurement of clinical agreement; Mendelian inheritance; More on survival data: Parametric models for censored survival data, Greenwood's formula, The proportional hazards model, logrank test, Cox's proportional hazards model. Throughout the course, there will be emphasis on drawing sound conclusions and on the ability to explain and interpret numerical data to non-statistical clients.

MA30087: Optimisation methods of operational research

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA10002 and take MA10005

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 appropriate algorithms, and understand their procedures
* Understand the underlying theory of linear programming problems, especially 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. Applications of LP. Duality. Topics selected from: Sensitivity analysis and the dual simplex algorithm. Brief discussion of Karmarkar's method. 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.

MA30089: Applied probability & finance

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20036

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
* price an option on a stock modelled by a log 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. 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.

MA30090: Multivariate analysis

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20008 and take MA20035

Aims & Learning Objectives:
Aims: To develop skills in the analysis of multivariate data and study the related theory.
Objectives: Be able to carry out a preliminary analysis of multivariate data and select and apply an appropriate technique to look for structure in such data or achieve dimensionality reduction. Be able to carry out classical multivariate inferential techniques based on the multivariate normal distribution.
Content:
Introduction, Preliminary analysis of multivariate data. Revision of relevant matrix algebra. Principal components analysis: Derivation and interpretation; approximate reduction of dimensionality; scaling problems. Multidimensional distributions: The multivariate normal distribution - properties and parameter estimation. One and two-sample tests on means, Hotelling's T-squared. Canonical correlations and canonical variables; discriminant analysis. Topics selected from: Factor analysis. The multivariate linear model. Metrics and similarity coefficients; multidimensional scaling. Cluster analysis. Correspondence analysis. Classification and regression trees.

MA30091: Applied statistics

Credits: 6
Level: Honours
Semester: 2
Assessment: CW100
Requisites: Before taking this unit you must take MA30084

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. Analysis: Model-building. 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.

MA30092: Statistical inference

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20033

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.

MA30107: Study year abroad (MMath)

Credits: 60
Level: Honours
Academic Year
Assessment:
Requisites:
MMath study year abroad.

MA30118: Management statistics

Credits: 5
Level: Honours
Semester: 2
Assessment: EX60CW40
Requisites: Before taking this unit you must take MA20097 or take MA20035 or take XX20120

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.

MA30125: Markov processes & applications

Credits: 6
Level: Honours
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20036

Aims & Learning Objectives:
Aims: To study further Markov processes in both discrete and continuous time. To apply results in areas such genetics, biological processes, networks of queues, telecommunication networks, electrical networks, resource management, 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 variety of birth-death processes as explosive or non-explosive
* Find the Q-matrix of a time-reversed chain and make effective use of time reversal.
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. Series of M/M/s queues. Open and closed migration processes. Explosions. Birth-death processes. Branching processes. 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.

MA30128: Project

Credits: 6
Level: Honours
Semester: 2
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.

MA30170: Numerical solution of PDEs I

Credits: 6
Level: Honours
Semester: 2
Assessment: CW25EX75
Requisites: Before taking this unit you must take MA20010 and take MA20011 and take MA20014

Aims & Learning Objectives:
Aims: To teach numerical methods for elliptic and parabolic partial differential equations via the finite element method based on variational principles.
Objectives: At the end of the course students should be able to derive and implement the finite element method for a range of standard elliptic and parabolic partial differential equations in one and several space dimensions. They should also be able to derive and use elementary error estimates for these methods.
Content:
Introduction Variational and weak form of elliptic PDEs. Natural, essential and mixed boundary conditions. Linear and quadratic finite element approximation in one and several space dimensions. An introduction to convergence theory. System assembly and solution, isoparametric mapping, quadrature, adaptivity. Applications to PDEs arising in applications. Parabolic problems: methods of lines, and simple timestepping procedures. Stability and convergence.

MA30172: Conjecture & proof

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20008

Aims & Learning Objectives:
Aims: The aim is ro explore pure mathematics from a problem-solving point of view. In addition to convential lectures, we aim to encourage students to work on solvong problems in small groups, and to give presentations of solutions in workshops.
Objectives: At the end of the course, students should be proficient in formulating and testing conjectures, and will have a wide experience of different proof techniques.
Content:
The topics will be drawn from cardinality, combinatorial questions, the foundations of measure, proof techniques in algebra, analysis, geometry and topology.

MA30188: Algebraic curves

Credits: 6
Level: Honours
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20012

Aims & Learning Objectives:
Aims: This is an advanced pure mathematics course providing an introduction to classical algebraic geometry via plane curves. It will show some of the links with other branches of mathematics.
Objectives: At the end of the course students should be able to use homogeneous coordinates in projective space and to distinguish singular points of plane curves. They should be able to demonstrate an understanding of the difference between rational and nonrational curves, know examples of both, and be able to describe some special features of plane cubic curves.
Content:
To be chosen from: Affine and projective space. Polynomial rings and homogeneous polynomials. Ideals in the context of polynomial rings,the Nullstellensatz. Plane curves; degree; Bezout's theorem. Singular points of plane curves. Rational maps and morphisms; isomorphism and birationality. Curves of low degree (up to 3). Genus. Elliptic curves; the group law, nonrationality, the j invariant. Weierstrass p function. Quadric surfaces; curves of quadrics. Duals.
THIS UNIT IS ONLY AVAILABLE IN ACADEMIC YEARS STARTING IN AN EVEN YEAR.

MA40040: Algebraic topology

Credits: 6
Level: Masters
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20008 and take MA20012 and take MA30055

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.

MA40042: Measure theory & integration

Credits: 6
Level: Masters
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20008 and take MA20012 and take MA30041

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.

MA40043: Real & abstract analysis

Credits: 6
Level: Masters
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20011 and take MA20012 and while taking this unit you must take MA30041

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 x = f(x,t). 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 inequality, 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.

MA40045: Dynamical systems

Credits: 6
Level: Masters
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20009 and take MA20011 and take MA20012 and take MA30041 and take MA40062

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.

MA40048: Analytical & geometric theory of differential equations

Credits: 6
Level: Masters
Semester: 1
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20009 and take MA20010 and take MA20011 and take MA20012 and take MA20013 and take MA40062

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.

MA40050: Nonlinear equations & bifurcations

Credits: 6
Level: Masters
Semester: 2
Assessment: EX75CW25
Requisites: Before taking this unit you must take MA30041 and take MA30051

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: Newtons method for systems. Quasi-Newton Methods. Eigenvalue problems. Theoretical Tools: Local Convergence of Newton's Method. Implicit Function Theorem. Bifcurcation 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. Practical 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.

MA40057: Functional analysis

Credits: 6
Level: Masters
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA30041 and take MA40043

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 unit, 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], lp, 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.

MA40058: Martingale theory

Credits: 6
Level: Masters
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA10031 and take MA10032 and take MA30041 and take MA40042

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.

MA40061: Nonlinear & optimal control theory

Credits: 6
Level: Masters
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must (take MA30046 OR take MA40062) and take MA30041

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.

MA40062: Ordinary differential equations

Credits: 6
Level: Masters
Semester: 2
Assessment: EX100
Requisites: Before taking this unit you must take MA20007 and take MA20008 and take MA20009 and take MA20011 and take MA20013 and take MA30041

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.

MA40117: Project (MMath)

Credits: 6
Level: Masters
Semester: 1
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.

MA40171: Numerical solution of PDEs II

Credits: 6
Level: Masters
Semester: 1
Assessment: CW25EX75
Requisites:

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:
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.

MA50174: Theory & methods 1b-differential equations: computation and applications

Credits: 6
Level: Masters
Semester: 1
Assessment: CW100
Requisites:

Aims & Learning Objectives:
To provide an introduction to the numerical solution of differential equations and how they arise in applications. To provide backgound linear Algebra. The students should become familiar with the software packages Matlab and Maple, should learn basic computational methods for solving differential equations, should see how differential equations can be applied to a wide variety of model problems, and should have a background knowledge of numerical linear algebra and how problems are formulated and solved using Matlab.
Content:
Introduction to Maple and Matlab and their facilities: basic matrix manipulation, eigenvalue calculation, FFT analysis, special functions, solution of simultaneous linear and nonlinear equations, simple optimization. Basic graphics, data handling, use of toolboxes. Problem formulation and solution using Matlab. Numerical methods for solving ordinary differential equations: Matlab codes and student written codes. Convergence and Stability. Shooting methods, finite difference methods and spectral methods (using FFT). Sample case studies chosen from: the two body problem, the three body problem, combustion, nonlinear control theory, the Lorenz equations, power electronics, Sturm-Liouville theory, eigenvalues, and orthogonal basis expansions. Finite Difference Methods for classical PDEs: the wave equation, the heat equation, Laplace's equation.

MA50175: Theory & methods 2 - topics in differential equations

Credits: 6
Level: Masters
Semester: 2
Assessment: CW50EX50
Requisites:
Before taking this unit you must take MA50181

Aims & Learning Objectives: Aims: To describe the theory and phenomena associated with hyperbolic conservation laws, typical examples from applications areas, and their numerical approximation; and to introduce students to the literature on the subject.
Objectives: At the end of the course, students should be able to recognise the importance of conservation principles and be familiar with phenomena such as shocks and rarefaction waves; and they should be able to choose appropriate numerical methods for their approximation, analyse their behaviour, and implement them through Matlab programs.
Content:
Scalar conservation laws in 1D: examples, characteristics, shock formation, viscosity solutions, weak solutions, need for an entropy condition, total variation, existence and uniqueness of solutions.Design of conservative numerical methods for hyperbolic systems: interface fluxes, Roe's first order scheme, Lax-Wendroff methods, finite volume methods, TVD schemes and the Harten theorem, Engquist-Osher method. The Riemann problem: shocks and the Hugoniot locus, isothermal flow and the shallow water equations, the Godunov method, Euler equations of compressible fluid flow. System wave equation in 2D. Main text: R.J. LeVeque, Numerical Methods for Conservation Laws (2nd Edition), Birkhäuser, 1992. Supplementary texts: K.W. Morton & D.F. Mayers, Numerical Solution of Partial Differential Equations, CUP, 1994.R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, CUP, 2002.

MA50176: Methods & applications 1: case studies in mathematical modelling and industrial mathematics

Credits: 6
Level: Masters
Semester: 2
Assessment: CW75OT25
Requisites:

Aims & Learning Objectives:
Students will learn about the nature of the modelling process, starting with a physical problem, representing it mathematically, simplifying and solving the resulting model and interpreting their results. They will also interact directly with industrialists. Students should become familiar with real problems arising in industry. They should become able to apply modelling and computational methods to solve these problems. They should learn how to work in teams and to communicate their results.
Content:
Applications of the theory and techniques learnt in the prerequisites to solve real problems drawn from from the industrial collaborators and/or from the industrially related research work of the key staff involved. Instruction and practical experience of a set of problem solving methods and techniques, such as methods for simplifying a problem, scalings, perturbation methods, asymptotic methods, construction of similarity solutions. Comparison of mathematical models with experimental data. Development and refinement of mathematical models. Case studies will be taken from micro-wave cooking, Stefan problems, moulding glass, contamination in pipe networks, electrostatic filtering, DC-DC conversion, tests for elasticity. Students will work in teams under the pressure of project deadlines. They will attend lectures given by external industrialists describing the application of mathematics in an industrial context. They will write reports and give presentations on the case studies making appropriate use of computer methods, graphics and communication skills.

MA50177: Methods and applications 2: scientific computing

Credits: 6
Level: Masters
Semester: 2
Assessment: CW100
Requisites:

Aims & Learning Objectives:
To teach an understanding and appreciation of issues arising in the computational solution of challenging scientific and engineering problems. Students should be able to write code to solve efficiently a range of scientific problems. They should be able to analyse algorithm complexity and efficiency. They should be familiar with scientific libraries and parallel programming. They will be expected to have deep knowledge of at least one challenging application.
Content:
Units, complexity, analysis of algorithms, benchmarks. Floating point arithmetic. Programming in Fortran90: Makefiles, compiling, timing, profiling. Data structures, full and sparse matrices. Libraries: BLAS, LAPACK, NAG Library. Visualisation. Handling modules in other languages such as C, C++. Software on the Web: Netlib, GAMS. Parallel Computation: Vectorisation, SIMD, MIMD, MPI. Performance indicators. Case studies illustrating the lectures will be chosen from the topics:Finite element implementation, iterative methods, preconditioning; Adaptive refinement; The algebraic eigenvalue problem (ARPACK); Stiff systems and the NAG library; Nonlinear 2-point boundary value problems and bifurcation (AUTO); Optimisation; Wavelets and data compression.

MA50178: Numerical linear algebra

Credits: 6
Level: Masters
Semester: 1
Assessment: CW25EX75
Requisites:

Aims & Learning Objectives:
To teach an understanding of iterative methods for standard problems of linear algebra. Students should know a range of modern iterative methods for solving linear systems and for solving the algebraic eigenvalue problem. They should be able to anayse their algorithms and should have an understanding of relevant practical issues, for large scale problems.
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.

MA50179: Mathematical biology 1

Credits: 6
Level: Masters
Semester: 1
Assessment: CW25EX75
Requisites:

Aims & Learning Objectives:
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. Realistic biological systems will be considered. 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. They will be able to compare their results with biological systems in the real world.
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. Poincari-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.

MA50180: Linear Elasticity

Credits: 6
Level: Masters
Semester: 2
Assessment: EX100
Requisites:

Aims & Learning Objectives:
To provide an introduction to the mathematical modelling and analysis of the behaviour of solid elastic media in linearised regime, and of deformation and wave propogation in such media. Students should be able to understand the derivation of the governing equations of linear elasticity, to solve model problems of elastostatics and more advanced problems of elastostatics and elastodynamics (wave propogation) and to be able to give physical interpretation of the solutions.
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, Lami 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.

MA50181: Theory & methods 1a - differential equations: theory & methods

Credits: 6
Level: Masters
Semester: 1
Assessment: CW25EX75
Requisites:

Aims & Learning Objectives:
To furnish the student with a range of analytic techniques for the solution of ODEs and PDEs. 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 and by considering a variety of examples, to appreciate why these properties are important.
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).

MA50182: Metric spaces

Credits: 6
Level: Masters
Semester: 1
Assessment: CW25EX75
Requisites:

Aims & Learning Objectives:
This course is intended to develop the theory of metric spaces and the topology of 3n for students with both "pure" and "applied" interests. To provide a framework for further studies in Analysis and Topology. Topics useful in applied areas such as the Contraction Mapping Principle will be reinforced through assessed coursework. 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. Closure and interior of sets. Topological approach to continuity and compactness (with statement of Heine-Borel theorem). Equivalence of Compactness and sequential compactness in metric spaces. Connectedness and path-connectedness. Metric spaces of functions: C[0,1] is a complete metric space.

MA50183: Specialist reading course

Credits: 6
Level: Masters
Semester: 1
Assessment: CW100
Requisites:

Aims & Learning Objectives:
To acquaint the student with a range of material relating to the chosen topic through directed reading and independent learning. On completion of the course, the student should be able to demonstrate: advanced knowledge in the chosen field evidence of independent learning an ability to read critically and master an advanced topic in mathematics/statistics/probability
Content:
Defined in the individual course specification.

MA50183: Specialist reading course

Credits: 6
Level: Masters
Semester: 2
Assessment: CW100
Requisites:

Aims & Learning Objectives:
To acquaint the student with a range of material relating to the chosen topic through directed reading and independent learning. On completion of the course, the student should be able to demonstrate: advanced knowledge in the chosen field evidence of independent learning an ability to read critically and master an advanced topic in mathematics/statistics/probability
Content:
Defined in the individual course specification.

MA50185: Representation theory of finite groups

Credits: 6
Level: Masters
Semester: 2
Assessment: CW25EX75
Requisites:

Aims & Learning Objectives:
The course develops applications of linear algebra to the study of finite groups. In so doing, it will show by example, and reinforced by coursework, how one area of mathematics can enhance and enrich the study of another. 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 demonstrate an ability to apply these results to problems in group theory through assessed coursework.
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.

MA50186: Complex analysis

Credits: 6
Level: Masters
Semester: 2
Assessment: CW25EX75
Requisites:

Aims and Learning Objectives:
The aim of this course is to develop the theory of functions of a complex variable and to cover complex function theory up to Cauchy's Residue Theorem and its applications. On completion of the course, students should have mastered the essentials of the theory of functions of a complex variable. They should be capable of justifying, and have mastered the calculation of, power series, Laurent series, contour integrals and, through assessed coursework, their application.
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 Formula 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.

MA50187: Project

Credits: 30
Level: Masters
Dissertation period
Assessment: DS100
Requisites:

Aims & Learning Objectives:
To acquaint the student with a range of material relating to the chosen topic. To foster research skills and independent learning. On completion of the course, the student should be able to demonstrate:-Advanced knowledge in the chosen field Evidence of independent learning An ability to initiate mathematical/statistical research An ability to read critically and master an advanced topic in mathematics/statistics/probability to the extent of being able to expound it in a coherent, well-argued dissertation Competence in a document preparation language to the extent of being able to typeset a dissertation with substantial mathematical/statistical content
Content:
Defined in the individual project specification.

 

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