Text only University | Catalogues for 2005/06 | for UGs | for PGs     # Department of Mathematical Sciences, Unit Catalogue 2005/06

### MA10001: Numbers

 Credits: 6
 Level: Certificate
 Semester: 1
 Assessment: EX100
 Requisites:
 Students must normally have A-level Mathematics, Grade A, 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 normally have A-level Mathematics, Grade A, 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: CW25EX75
 Requisites:
 Students must normally have A-level Mathematics, Grade A, 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 3n ® 3m 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 3² ® 3 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 3³. 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: Introduction to statistics & probability 1

 Credits: 6
 Level: Certificate
 Semester: 1
 Assessment: EX100
 Requisites:
 Students must normally have A-level Mathematics, Grade A, or equivalent, in order to undertake this unit. Aims & Learning Objectives: Aims: To provide a solid foundation in discrete probability theory that will facilitate further study in probability and statistics. Objectives: Students should be able to: apply the axioms and basic laws of probability using proper notation and rigorous arguments; solve a variety of problems with probability, including the use of combinations and permutations and discrete probability distributions; perform common expectation calculations; calculate marginal and conditional distributions of bivariate discrete random variables; calculate and make use of some simple probability generating functions. Content: Sample space, events as sets, unions and intersections. Axioms and laws of probability. Equally likely events. Combinations and permutations. Conditional probability. Partition Theorem. Bayes' Theorem. Independence of events. Bernoulli trials. Discrete random variables (RVs). Probability mass function (PMF). Bernoulli, Geometric, Binomial and Poisson Distributions. Poisson limit of Binomial distribution. Hypergeometric Distribution. Negative binomial distribution. Joint and marginal distributions. Conditional distributions. Independence of RVs. Distribution of a sum of discrete RVs. Expectation of discrete RVs. Means. Expectation of a function. Moments. Properties of expectation. Expectation of independent products. Variance and its properties. Standard deviation. Covariance. Variance of a sum of RVs, including independent case. Correlation. Conditional expectations. Probability generating functions (PGFs).

### MA10032: Introduction to statistics & probability 2

 Credits: 6
 Level: Certificate
 Semester: 2
 Assessment: CW25EX75
 Requisites:
 Before taking this unit you must take MA10031
 Aims & Learning Objectives: Aims: To introduce probability theory for continuous random variables. To introduce statistical modelling and parameter estimation and to discuss the role of statistical computing. Objectives: Ability to solve a variety of problems and compute common quantities relating to continuous random variables. Ability to formulate, fit and assess some statistical models. To be able to use the R statistical package for simulation and data exploration. Content: Definition of continuous random variables (RVs), cumulative distribution functions (CDFs) and probability density functions (PDFs). Some common continuous distributions including uniform, exponential and normal. Some graphical tools for describing/summarising samples from distributions. Results for continuous RVs analogous to the discrete RV case, including mean, variance, properties of expectation, joint PDFs (including dependent and independent examples), independence (including joint distribution as a product of marginals). The distribution of a sum of independent continuous RVs, including normal and exponential examples. Statement of the central limit theorem (CLT). Transformations of RVs. Discussion of the role of simulation in statistics. Use of uniform random variables to simulate (and illustrate) some common families of discrete and continuous RVs. Sampling distributions, particularly of sample means. Point estimates and estimators. Estimators as random variables. Bias and precision of estimators. Introduction to model fitting; exploratory data analysis (EDA) and model formulation. Parameter estimation via method of moments and (simple cases of) maximum likelihood. Graphical assessment of goodness of fit. Implications of model misspecification.

### MA10095: Quantitative methods

 Credits: 6
 Level: Certificate
 Semester: 1
 Assessment: EX100
 Requisites:
 Before taking this unit you must 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: Aims - To provide an elementary introduction to the Calculus, including a review of foundations in algebra and geometry. Objectives - Students should be able to demonstrate knowledge of and proficiency in: manipulation of simple algebraic expressions, the solution of simple algebraic equations and systems, trigonometric functions and identities and their use, the derivative of a function and the finding of derivatives, coordinate geometry and curve sketching including simple use of the second derivative, solution of maximization and minimization problems, the binomial theorem and its use. Content: Numbers: integers, rationals, reals. Algebra: straight lines, quadratics, functions, binomial theorem, exponential function. Trigonometry: ratios for general angles, sine and cosine rules, compound angles. Calculus: differentiation, tangents, normals, rates of change, maxima and minima.

### MA10104: Foundation mathematics 2

 Credits: 6
 Level: Certificate
 Semester: 2
 Assessment: EX50CW50
 Requisites:
 Before taking this unit you must take MA10103
 Aims & Learning Objectives: Aims - To provide an elementary introduction to integration and to statistics. Objectives - Students should know the integrals of standard functions and be able to apply basic integration techniques to evaluate definite and indefinite integrals. Students will know the definitions of basic statistical measures and be able to apply the definitions to analyse a given set of data. Content: Brief review of differentiation, standard integrals, definite integrals and their relation to areas, volumes of revolution, integration by substitution, integration by parts, first order ordinary differential equations, mean, mode, median, quartiles, histograms, standard deviation.

### MA10126: Introduction to computing with applications

 Credits: 6
 Level: Certificate
 Semester: 1
 Assessment: CW100
 Requisites:
 Students must normally have A-level Mathematics, Grade A, or equivalent, in order to undertake this unit. 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.

### MA10192: Mathematics 1

 Credits: 6
 Level: Certificate
 Semester: 1
 Assessment: EX75CW25
 Requisites:
 Aims & Learning Objectives: To provide students with a basic introduction to the mathematical skills in calculus, functions of several variables and differential equations necessary to tackle process engineering design applications. After successfully completing this unit students should be able to use concepts, ideas and results from calculus, functions of several variables and differential equations in the context of relevant applications. Content: * Calculus: Limits, differentiation, integration. Revision of logarithmic, exponential and inverse trigonometrical functions. Revision of integration including polar and parametric co-ordinates, with applications. * Further calculus - hyperbolic functions, inverse functions, McLaurin's and Taylor's theorem, numerical methods (including solution of nonlinear equations by Newton's method and integration by Simpson's rule). * Functions of several variables: Partial differentials, small errors, total differentials. * Differential equations: Solution of first order equations using separation of variables and integrating factor, linear equations with constant coefficients using trial method for particular integration.

### MA10193: Mathematics 2

 Credits: 3
 Level: Certificate
 Semester: 2
 Assessment: CW 25%, EX 75%
 Requisites:
 Before taking this unit you must take MA10192
 Aims: To provide students with a basic introduction to the mathematical skills in linear algebra, complex numbers and linear differential equations. Learning Outcomes: After successfully completing this unit students should be able to: use concepts, ideas and results from linear algebra, complex numbers, and differential equations in the context of relevant engineering applications. Skills: Mathematical methods (taught and assessed). Content: * Linear algebra: Matrix algebra, Determinants, Inverse, Numerical methods, Solution of systems of linear algebraic equation. * Complex numbers: Argand diagram, Polar coordinates, nth roots, Elementary functions of a complex variable. * Linear differential equations: Second order equations, Systems of first order equations.

### 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: EX75CW25
 Requisites:
 Before taking this unit you must take MA10002 and take MA10003 and take MA10005 and take MA10006
 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 forms an introduction to 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 separation of variables. 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 2D, in rectangular and circular domains; diffusion equation and wave equation in one space dimension; solution by separation of variables.

### 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'Hopital'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: EX75CW25
 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. Ability to use R to calculate estimates, carry out hypothesis tests and compute confidence intervals. Content: Point estimation: Maximum-likelihood estimation, including computational aspects; further properties of estimators, including mean square error, efficiency and consistency; robust methods of estimation such as the median and trimmed mean. Confidence intervals. Hypothesis testing: Size and power of tests; Neyman-Pearson lemma. One-sided and two-sided tests. Distributions related to the normal: t, chi-square and F distributions. Interference 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. Examples of all the above, including case studies in R.

### 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, introduce the principles of statistical modelling. Objectives: Ability to carry out analyses using linear Gaussian models, including regression and ANOVA. Ability to manipulate joint, marginal and conditional distributions. Ability to represent normal linear models in vector and matrix form. Content: Regression: Estimation of model parameters, tests and confidence intervals, prediction intervals, polynomial and multiple regression. One-way analysis of variance (ANOVA): One-way classification model. Main effects and interaction, parameter estimation, F- and t-tests. 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 studentised 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: 6
 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: 6
 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:
 Content: Industrial placement year

### MA20106: Study year abroad (BSc)

 Credits: 60
 Level: Intermediate
 Academic Year
 Assessment:
 Requisites:
 Content: Study year abroad (BSc)

### 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

### 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 3n 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. Picard's Theorem for u=f(t,u(t)). Sequential compactness, Bolzano-Weierstrass theorem and applications. Open and closed sets (with emphasis on 3n). 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.

### 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. Poincare-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.

### 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.

### 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 3n to 3n. 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.

### 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: 2
 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.

### 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: Classical statistical inference

 Credits: 6
 Level: Honours
 Semester: 1
 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 including criteria for the comparison of procedures. To give an in depth description of the asymptotic theory of maximum likelihood methods and hypothesis testing. Objectives: On completing the course, students should be able to: * 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. Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. 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. 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:
 Content: MMath study year abroad.

### MA30118: Management statistics

 Credits: 6
 Level: Honours
 Semester: 2
 Assessment: EX60CW40
 Requisites:
 Before taking this unit you must take MA20097 or take MA20035 or take EC20019
 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.

### 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. * Brief introduction to time dependent problems.

### 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 to explore pure mathematics from a problem-solving point of view. In addition to conventional lectures, we aim to encourage students to work on solving 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.

### MA40037: Galois theory

 Credits: 6
 Level: Masters
 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.

### MA40038: Advanced group theory

 Credits: 6
 Level: Masters
 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.

### 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, Holder, 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, and to lay the foundations for the year 4 unit in functional analysis. 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 the following: 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. Metric completion M of a metric space M. Real inner-product spaces. Hilbert spaces. Cauchy-Schwarz inequality, parallelogram identity. Examples: l2,L2[0,1]:=C[0,1]. Separability of L2. Orthogonality, Gram-Schmidt process. Bessel's inquality, Pythagoras' Theorem. Projections and subspaces. Orthogonal complements. Riesz Representation Theorem. Complete orthonormal sets in separable Hilbert spaces. Completeness of trigonometric polynomials in L2 [0,1]. Fourier Series.

### 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, Poincare recurrence theorem, Noether's Theorem.

### MA40049: Linear elasticity

 Credits: 6
 Level: Masters
 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.

### 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. Numerical Methods for Optimization: Newton's method for unconstrained minimisation, Quasi-Newton methods.

### MA40054: Representation theory of finite groups

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: EX100
 Requisites:
 Before taking this unit you must take MA40038
 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.

### 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 Holder inequalities. Examples - 3n, 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.

### MA40065: Viscous fluid mechanics

 Credits: 6
 Level: Masters
 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.

### MA40090: Multivariate analysis

 Credits: 6
 Level: Masters
 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.

### 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.

### MA40125: Markov processes & applications

 Credits: 6
 Level: Masters
 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.

### MA40146: Mathematical & statistical modelling for biological sciences

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: CW50EX50
 Requisites:
 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.

### MA40171: Numerical solution of PDEs II

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: CW25EX75
 Requisites:
 Before taking this unit you must take MA30170
 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.

### MA40188: Algebraic curves

 Credits: 6
 Level: Masters
 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.

### MA40189: Topics in Bayesian statistics

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: EX100
 Requisites:
 Before taking this unit you must take MA20035 and take MA30092
 Aims & Learning Objectives: Aims: To introduce students to the ideas and techniques that underpin the theory and practice of the Bayesian approach to statistics. Objectives: Students should be able to formulate the Bayesian treatment and analysis of many familiar statistical problems. Content: Bayesian methods provide an alternative approach to data analysis, which has the ability to incorporate prior knowledge about a parameter of interest into the statistical model. The prior knowledge takes the form of a prior (to sampling) distribution on the parameter space, which is updated to a posterior distribution via Bayes' Theorem, using the data. Summaries about the parameter are described using the posterior distribution. The Bayesian Paradigm; decision theory; utility theory; exchangeability; Representation Theorem; prior, posterior and predictive distributions; conjugate priors. Tools to undertake a Bayesian statistical analysis will also be introduced. Simulation based methods such as Markov Chain Monte Carlo and importance sampling for use when analytical methods fail.

### MA40195: Project (MMath)

 Credits: 6
 Level: Masters
 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.

### MA50046: Linear control theory

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: EX75CW25
 Requisites:
 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. Students should be able to demonstrate an in-depth understanding of the subject. 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.

### MA50059: Mathematical methods 2

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: EX75CW25
 Requisites:
 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. They should be able to demonstrate an in-depth understanding of the subject. 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.

### MA50063: Mathematical biology 2

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: EX75CW25
 Requisites:
 Aims & Learning Objectives: Aims: The aim of the course is to introduce students to applications of partial differential equations to model problems arising in biology. The course will complement Mathematical Biology I where the emphasis was on ODEs and Difference Equations. Objectives: Students should be able to derive and interpret mathematical models of problems arising in biology using PDEs. They should be able to perform a linearised stability analysis of a reaction-diffusion system and determine criteria for diffusion-driven instability. They should be able to interpret the results in terms of the original biological problem. They should be able to demonstrate an in-depth understanding of the subject. 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.

### MA50084: Linear models

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: CW25EX75
 Requisites:
 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. To facilitate an in-depth understanding of the topic.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; (d) demonstrate an in-depth understanding of the topic. 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.

### MA50085: Time series

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: CW25EX75
 Requisites:
 Aims & Learning Objectives: Aims: To introduce a variety of statistical models for time series and cover the main methods for analysing these models. To facilitate an in-depth understanding of the topic. 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; * demonstrate an in-depth understanding of the topic. 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.

### MA50087: Optimisation methods of operational research

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: EX75CW25
 Requisites:
 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. To facilitate an in-depth understanding of the topic. 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. * Demonstrate an in-depth understanding of the topic. 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.

### MA50089: Applied probability & finance

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: CW25EX75
 Requisites:
 Aims & Learning Objectives: Aims To develop and apply the theory of probability and stochastic processes to examples from finance and economics. To facilitate an in-depth understanding of the topic. 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; * demonstrate an in-depth understanding of the topic. 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.

### MA50092: Classical statistical inference

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: CW25EX75
 Requisites:
 Aims & Learning Objectives: Aims: To develop a formal basis for methods of statistical inference including criteria for the comparison of procedures. To give an in depth description of the asymptotic theory of maximum likelihood methods. To facilitate an in-depth understanding of the topic. Objectives: On completing the course, students should be able to: * 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; * demonstrate an in-depth understanding of the topic. Content: Revision of standard distributions: Bernoulli, binomial, Poisson, exponential, gamma and normal, and their interrelationships. Sufficiency and Exponential families. Point estimation: Bias and variance considerations, mean squared error. Rao-Blackwell theorem. 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. Compound null hypotheses, monotone likelihood ratio property, uniformly most powerful unbiased tests. Nuisance parameters, generalised likelihood ratio tests.

### MA50170: Numerical solution of PDEs I

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: CW25EX75
 Requisites:
 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. They should be able to demonstrate an in-depth understanding of the subject. 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.

### MA50174: Advanced numerical methods

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: CW100
 Requisites:
 Aims & Learning Objectives: Aims: To provide an introduction to the numerical solution of differential equations and how they arise in applications. To provide background linear Algebra. Objectives: The students should become familiar with the software packages Matlab and Maple, should learn computational methods for solving differential equations, should see how differential equations can be applied to a wide variety of modl problems, and should have a background knowledge of numerical linear algebra and how problems are formulated and solved using Matlab. Content: 1. 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. 2. 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, Sturn-Liouville theory, eigenvalues, and orthogonal basis expansions.

### MA50175: Topics in differential equations

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: EX75CW25
 Requisites:
 Before taking this unit you must take MA50190
 Aims & Learning Objectives: Aims: To investigate and model phenomena associated with multiple scales; to describe mathematical methods for modelling, analysis and numerical approximation of multiscale problems; to apply these techniques to problems in materials science; and to introduce students to the literature on the subject. On completing the course, students will have an advance knowledge of a modern area of the study of differential equatios and their applications. Objectives: At the end of the course, students should be familiar with various multiscale problems. They should be able to analyse selected problems, such as phase transitions and the formation of microstructures; they should be able to model multiscale problems; and they should be able to investigate the behaviour of multiscale materials analytically and numerically. Content: Review of continuum mechanics. Example of multiscale problems in materials science. Crystallographic description. Twinning and compatibility. Microstructures and their variational treatment. Averaging and relaxation. Recoverable strains. Gamma-convergence. Atomic to continuum passage. Diffusion in periodic media. Homogenisation. Two-scale methods. References: * K. Bhattacharya, Microstructure of Martensite. Why it forms and how it gives rise to the shape-memory effect, Oxford Series of Materials Modelling 2, Oxford University Press, 2003. * A. Braides, Gamma-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22, Oxford University Press, 2002. * D. Cioranescu & P. Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications 17, Oxford University Press, 1999. * U. Hornung, Homogenization and porous media, Springer Series on Interdisciplinary Applied Mathematics 6, Springer, 1997. Supplementary texts, in addition to the above and selected articles: * J. Jost & Y. Li-Jost, Xianqing, Calculus of variations, Cambridge Studies in Advanced Mathematics 64, Cambridge University Press, 1998. * S. Müller, Variational models for microstructure and phase transitions, in: Calculus of variations and geometric evolution problems (Cetraro, 1996), Springer Lecture Notes in Math. 1713, 85-210, Springer, 1999.

### 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. They should be able to demonstrate an in-depth understanding of the subject. 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. They should be able to demonstrate an in-depth understanding of the subject. 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.

### MA50181: Mathematical methods 1

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: EX75CW25
 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: EX75CW25
 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. Objectives: To provide a framework for further studies in Analysis and Topology. Topics useful in applied areas such as the Contraction Mapping Principle will be emphasized. Students will know the fundamental results listed in the syllabus and have an instinct for their utility in analysis and numerical analysis. They should be able to demonstrate an in-depth understanding of the subject. 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 3n). 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
 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.

### MA50186: Complex analysis

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: EX75CW25
 Requisites:
 Aims & 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. They should be able to demonstrate an in-depth understanding of the subject. 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.

### MA50190: Advanced mathematical methods

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: EX100
 Requisites:
 Aims: To study advanced mathematical methods and techniques relevant to solving problems which arise in applications modelled by differential equations (both ordinary and partial) and integral equations. To provide a firm basis of useful techniques for many subsequent courses in applied mathematics. Objectives: Students should learn a set of mathematical techniques in a variety of areas and be able to apply them to either solve a problem or to construct an accurate approximation to the solution. They should demonstrate an understanding of both the theory and the range of applications (including the limitations) of all the techniques studied. Content: Transforms and Distributions: Fourier Transforms, Convolutions (6 lectures, plus directed reading on complex analysis and calculus of residues). Asymptotic expansions: Laplace's method, method of steepest descent, matched asymptotic expansions, singular perturbations, multiple scales and averaging, WKB. (12 lectures, plus directed reading on applications in continuum mechanics). Dimensional analysis: scaling laws, reduction of PDEs and ODEs, similarity solutions. (6 lectures, plus directed reading on symmetry group methods). References: L. Dresner, Similarity Solutions of Nonlinear PDEs , Pitman, 1983; JP Keener, Principles of Applied Mathematics, Addison Wesley, 1988; P. Olver, Symmetry Methods for PDEs, Springer; E.J. Hinch, Perturbation Methods, CUP.

### MA50194: Advanced statistics for use in health contexts 2

 Credits: 6
 Level: Masters
 Semester: 2
 Assessment: CW100
 Requisites:
 Before taking this unit you must take PS50065 and take PS50066
 Aims: * To equip students with the skills to use and interpret advanced multivariate statistics; * To provide an appreciation of the applications of advanced multivariate analysis in health and medicine. Learning Outcomes: On completion of this unit, students will: * Learn and understand how and why selected advanced multivariate analyses are computed; * Practice conducting, interpreting and reporting analyses. Skills: Intellectual: * To learn independently; * To critically evaluate and assess research and evidence as well as a variety of other information; * To utilise problem solving skills. Professional / Practical: * Advanced information technology and computing technology (e.g. SPSS); * Independent working skills; * Advanced numeracy skills. Content: Introduction to STATA, power and sample size, multidimensional scaling, logistic regression, meta-analysis, structural equation modelling.

### MA50196: Financial derivatives

 Credits: 6
 Level: Masters
 Semester: 1
 Assessment: EX100
 Requisites:
 Aims: To provide an introduction to derivative securities including futures and options, their valuation and management. Learning Outcomes: At the end of this course, students should be able to: demonstrate an understanding of the outcomes of portfolios including a variety of financial instruments; apply principles of arbitrage, hedging and risk management; demonstrate appreciation of the derivation of the Black-Scholes pricing theory, and perform calculations based on the Black-Scholes analysis. Skills: Written communication (T/F/A), Numeracy (T/F/A), Time management and Organisational skills (F), Data acquisition, handling and analysis (F), Problem solving (T/F/A), Working independently (F), Critical/analytical skills (F), Precise thinking (T/F/A), Accuracy and attention to detail (T/F/A). Content: Introduction to derivatives, including futures and forward contracts, call and put options. Valuation of futures and forwards contracts. Time value of money, interest rates, bonds. Option pricing; put-call parity, trading strategies, arbitrage. Random asset price modelling. Introduction to stochastic calculus. The Black-Scholes analysis for pricing derivatives, including options. Other topics to be chosen from: interest rate swaps and their valuation, assets paying dividends, the binomial pricing model, risk management of options (delta, theta and gamma).  University | Catalogues for 2005/06 | for UGs | for PGs Student Records & Examinations Office, University of Bath, Bath BA2 7AY Tel: +44 (0) 1225 383019 Fax: +44 (0) 1225 386366 To request a copy of this information (Prospectus): Prospectus request To report a problem with the catalogue click here This page maintained by H.E.Buick@bath.ac.uk. Last update: 3 February, 2006 Copyright ©2005 University of Bath