Level 1 modules
AMA1001 Vector Algebra and Dynamics (1st semester)
Pre-requisite: This course is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This module introduces two essential techniques used in applied mathematics: namely, differential equations and vectors. These techniques find many applications. One of the oldest applications is to dynamics. We will study the motion of a particle subject to various forces. The module introduces the concepts of linear and angular momentum, kinetic and potential energy and conservation laws.Contents
Mathematical introduction: numbers, functions, trigonometric functions, derivatives and differentials, Maclaurin series, integration, complex numbers, Euler's formula.
Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.
Vectors: definitions and notation, operations on vectors, application to geometry (ratio theorem, centroid), basis and components, scalar and vector products, triple products, equations of a plane and straight line.
Newtonian mechanics: position, velocity and acceleration (tangential and normal), motion under gravity, Newton's laws, motion under gravity with resistance, oscillatory motion, forced oscillations, damped oscillations.
Conservation laws: momentum, work, kinetic energy, conservative forces, potential energy, collisions, centre of mass, restitution.
Planetary motion: central forces, angular momentum, plane polar coordinates, orbital and escape velocities, Kepler's problem and Kepler's laws.Assessment
During the course there will be a one-hour test which will count as 10% of the total final mark. The remaining 90% comes from a 3-hour written examination containing 8 questions, of which the best 6 are counted.AMA1002 Waves and Vector Fields (2nd semester)
Co-requisite: AMA1001
Introduction
The aim of this module is to develop skills in the calculus of several variables and to apply this theory to solve differential equations such as the diffusion and wave equation, as well as investigate the properties and applications of scalar and vector fields. Module AMA1001 introduces calculus for a function of a single variable. Essentially, this theory teaches us how to solve one-dimensional problems. However, the world is a three-dimensional environment and many of the interesting and useful applications of mathematics are formulated in two, three and four-dimensional calculus. In this module, we consider functions of several variables. We learn how to define integrals and derivatives in 2D and 3D and apply this knowledge to solve problems of practical interest in acoustics, biology, engineering, physics and chemistry.Contents
Functions, integrals, series and derivatives: Definition of a function and a limit. Continuous and discontinuous functions. Definition and calculation of derivatives. Function of a function, and the chain rule for differentiation. Rolle's theorem. Mean-value theorem. Integration rules. Riemann definition of the integral. Integration by parts and change of variable of integration. Convergence of infinite series - D'Alembert ratio test. Taylor and Maclaurin series. L'Hôpital's rule.
Functions of several variables: Partial derivatives and multiple integrals. The chain rule for differentiation. Taylor series in two independent variables. Stationary points in 2D and 3D. Functions of space and time.
Diffusion and Wave equations: Partial differential equations: diffusion and wave equations in one space and one time dimension. Separation of variables method.
Periodic functions and Fourier series. Applications to random processes, heat transfer and wave motion. Mathematics of music.
Functions in 2D and 3D space: Vector algebra. Vector equations of lines and planes. Cartesian and plane polar coordinates. Jacobians. Double and triple integrals. Cylindrical and spherical coordinates, volumes of 3D shapes. Curves in 3D and tangent vectors. Equation of normal vectors and tangent planes to surfaces.
Vector fields: Scalar and vector fields. Partial differential operators. Div, grad and curl. Directional derivatives, normal derivative. Tangent line integrals, circulation, and conservative fields. Surfaces in 3D, surface integrals. Gauss's divergence theorem. Green's theorem and Stokes' theorem.Assessment
There is one 1-hour Class Test worth 10% of the final mark and a Computational Mathematics exercise worth 10% of the final mark. The remaining 80% comes from a 3-hour written examination containing 8 questions, of which 6 must be attempted.SOR1001 Introduction to Probability and Operational Research (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 1 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Probability is an important branch of statistics that deals with chance and how to predict whether a result is likely or unlikely. The applications of probability are diverse, occurring in industry, mathematics, science, technology, medicine, etc. Probability models can be described in terms of random variables (which are functions of the outcomes of a random experiment), usually described as being discrete or continuous. The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. An example of a discrete random variable is the outcome of throwing a dice. Alternatively, there are other situations which are better described by a continuous random variable such as the measure of height. The probability density function of a continuous random variable is a function which can be integrated to obtain the probability that the random variable takes a value in a given interval.Operational Research (OR) helps to improve operations in business and governments through the use of scientific methods and the development of specialised techniques. OR provides systematic and general approaches to problem solving and decision making, regardless of the nature of the system, product, or service. The approaches used in OR are based on one or more of mathematical methods, simulation, and qualitative or logical reasoning. OR originated in Great Britain during World War II to bring mathematical or quantitative approaches to bear on military operations. Since then OR has evolved to be applicable to the management of all aspects of a system, product, or service, and hence is often referred to as Systems Science or Management Science. It has now become recognised as an important input to decision-making in a wide variety of applications in business, industry, and government.
The course provides an introduction to probability theory and operational research, the foundation of further courses in Statistics and OR. No prior knowledge of Probability, Statistics or OR is assumed.
Contents
Probability: Probability theory, probability axioms, conditional probability. Random variables, discrete and continuous probability distributions. Properties and applications of standard discrete distributions: geometric, binomial, negative binomial, Poisson, hypergeometric; and standard continuous distributions: uniform, exponential, normal (Gaussian). Bivariate distributions.
Operational Research: Elementary linear programming: simplex method. Decision making under uncertainty. Random sampling and simulation.
Assessment
During the course there will be a one hour test which will count as 10% of the total final mark. The remaining 90% comes from a 3-hour written examination with two sections, A and B. Section A, accounting for 60 % of the examination mark, is compulsory and comprises a number of short questions of variable length designed to test the student's basic understanding of the module. Section B, accounting for 40% of the examination mark, comprises 6 longer questions of equal length, with 4 questions to be attempted.SOR1002 Statistical Methods (2nd semester)
Co-requisite: SOR1001
Introduction
When we wish to know something about a particular population it is usually impractical, especially in large populations, to collect data from every unit of that population. It is more efficient to collect data from a sample of the population under study and, from that sample, make estimates of the population parameters. The method of using samples to estimate population parameters is known as statistical inference. The course provides an elementary introduction to statistical inference; the body of principles and methods underlying the statistical analysis of data. The methods draw on the probability distributions previously discussed in SOR1001.Contents
Data Collection and Preliminary Analysis: Statistical models. Sampling. Initial data analysis: discrete and continuous variables. Descriptive statistics and graphical/diagrammatic representations. Descriptive statistics for ungrouped data; box-plot. Frequency table for grouped discrete data: relative frequency, cumulative frequency, bar diagram; sample mean, variance, standard deviation and percentiles. Frequency table for grouped continuous data: stem-and-leaf plot, histogram, cumulative percentage frequency plot; sample mean, variance, standard deviation and percentiles. Linear transformations. Bivariate data; scatter diagram, sample correlation coefficient.
Estimation: Properties of an estimator, point estimates, unbiasedness, efficiency and the three methods of estimation. Methods of moments, maximum likelihood and least squares. Regression.
Hypothesis Testing: Parametric methods: tests based on the Normal distribution, t-distribution, F-distribution and χ2 distribution. Non-parametric methods: tests on contingency tables, testing for association, homogeneity, Sign test, Wilcoxon signed-rank test, Mann-Whitney test and the run test.
Statistical Quality Control: Shewart control charts; upper and lower control and warning limits. Analysis of patterns. Control charts for attributes and for variables.Assessment
During the course there will be a one hour test which will count as 10% of the total final mark. The remaining 90% comes from a 3-hour written examination with two sections, A and B. Section A, accounting for 60% of the examination mark, is compulsory and comprises a number of short questions of variable length designed to test the student's basic understanding of the module. Section B, accounting for 40% of the examination mark, comprises 6 longer questions of equal length, with 4 questions to be attempted.
Level 2 modules
AMA2001 Classical Mechanics (2nd semester)
Pre-requisite: AMA1001 and AMA1002
Introduction
Mechanics has always been a source of inspiration for mathematics and mathematicians. Calculus, as well as the famous Newton's laws, were “invented” by Newton largely because he wanted to solve a single but very important mechanical problem, the problem of planetary motion. Mechanics was at the beginning of such branches of mathematics as theory of functions, calculus of variations, differential equations and more recently, theory of chaos. Mechanics and its mathematical methods are important in optics, electromagnetic theory, statistical mechanics, quantum mechanics, theory of relativity, quantum field theory and many ‘non-physical’ applications such as theory of optimisation and control.For students, besides giving a comprehensive picture of mechanical phenomena and teaching how to solve a wide variety of problems, Classical Mechanics offers a unique opportunity to see the mathematical methods they have learned at work and to practice their mathematical skills. Vectors, partial derivatives, single and multiple integrals, differential equations, stationary points, complex numbers, as well as matrices and determinants are all among the tools used in the course.
For those who have studied mechanics before this course offers a completely new look on familiar facts, and shows a more universal approach to solving mechanical problems. For those who have not had much experience with mechanics the course will provide a comprehensive introduction into the subject, as it does not require any previous physics knowledge. Finally, for those interested in theoretical physics, the course will serve as a first step into this wide and extremely interesting area.
Contents
Introduction: Basic revision of Newtonian mechanics: conservation of mechanical energy and angular momentum. Revision of key concepts in linear algebra (matrices): eigenvalues and eigenvectors. Revision of basic multivariable calculus (partial differentiation and multiple integration).
Motion of a single particle in a central potential: Attractive forces: planetary motion and transfer orbits. Repulsive forces: Rutherford scattering.
Rotating frames of reference: Angular velocity, effect of the Earth's rotation; apparent gravity, cyclones and anticyclones, Foucault's pendulum, Larmor precession.
Conservation laws for a system of particles: internal and external forces, conservation of energy, centre of mass as the origin.
Rigid body motion: Moments and products of inertia, parallel axis theorem, Euler's equations of motion, Euler angles and rotation matrices.
Lagrange's equations: Constraints, equations for holonomic constraints; examples.
Motion of a top: Precession and nutation; stability of a sleeping top; gyroscopes and the gyrocompass.
Lagrange's equations for impulses: Applications to systems of rods.
Small oscillations: Normal modes of oscillation.
Lagrange's equations with non-holonomic constraints: Constrained optimisation; Lagrange multipliers.Assessment
One three-hour written examination with 7 questions: all questions carry equal marks; the best 5 answers are credited.AMA2003 Methods of Applied Mathematics (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 2 of either an MSci or a BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This is a core Level 2 Applied Mathematics module. It provides an introduction to important techniques of Applied Mathematics. Students completing the module will be able to understand and apply these methods to solve problems. These skills are required for further studies in science and mathematics. Often physical problems are framed in terms of linear equations, requiring their numerical solution with modern computers. Students will have the basic knowledge of linear algebra necessary to study such highly-developed numerical and computational methods. Functions of a complex variable and associated calculus lead to a range of techniques that enable analytical solutions to physical problems. These techniques link up also with other modules later on. One of the most powerful and elegant applications of complex analysis is in the calculation of integrals, and the module places an emphasis on that.Contents
PART A, Linear Algebra: Matrix algebra; Gaussian elimination; LU-factorization; matrix inversion; vector spaces; solution of m equations in n unknowns; linear independence, null spaces, row and column spaces; inner products of vectors; fundamental theorem of linear algebra; projections and the least-squares approximation; determinants; complex matrices; eigenvalues and eigenvectors; matrix diagonalisation, positive definite matrices.
PART B, Complex Functions: Algebra; analytic functions; Cauchy-Riemann equations; Cauchy's theorem; infinite series; Taylor's theorem; Laurent expansions; theory of residues; evaluation of integrals using the residue theorem.Assessment
One 3-hour written examination. The paper has 8 questions divided into two sections: Section A - linear algebra, Section B - complex functions. In each section 3 questions must be answered from a choice of 4.AMA2004 Numerical Analysis (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 2 of either an MSci or a BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Numerical Analysis is concerned with devising methods for finding approximate, numerical solutions to mathematically expressed problems. The methods are analysed for their accuracy, efficiency and robustness. As a simple example, consider solving the equation f(x) = 0, where f(x) is a specified function. If f(x) is of even moderate complexity, we will not be able to solve this equation analytically. In Numerical Analysis we develop different procedures, or algorithms, to solve this problem. Finding the most suitable requires an appreciation of the methods. For example, some will guarantee convergence to a solution, but may require much effort, while other methods may converge quickly with less effort, but may also diverge, depending on the function and initial guess. Faced with such differing behaviour of the methods, what is the ‘best’ strategy to adopt? In AMA2004 we cover the basic introductory material of Numerical Analysis. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations.An important element of this module are the practical classes, in which algorithms developed during the lectures are implemented using MATLAB, which is an interactive and programmable software package. The practical work culminates in a numerical project, in which a MATLAB program is written to implement a particular algorithm, or to investigate the behaviour of a method. Although prior familiarity with computers is helpful, it is not essential.
A third element of the module is the oral presentation. You are required to give a 5-minute presentation to a small group of students and staff.
Contents
Introduction and basic properties of errors: Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.
Solution of equations in one variable: Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's δ2 process; Roots of polynomials.
Solution of linear equations: Gaussian elimination; Pivoting strategies; Calculating the inverse; LU decomposition; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.
Interpolation and polynomial approximation: Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.
Approximation theory: Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.
Numerical quadrature: Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature.
Numerical solution of ordinary differential equations: Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.Assessment
The module is assessed by: an examination of 3 hours duration; a project undertaken during the semester; and an oral presentation given during the semester. The examination is worth 60% of the final mark, the project is worth 30%, and the presentation is worth 10%. The project is assessed through a report to be handed in at the end of week 12. The project and the presentation are compulsory, and a score of at least 40% is required for both the project report (i.e., 12 out of 30 marks) and the presentation (i.e., 4 out of 10 marks) to pass the module. In the written examination 5 questions must be attempted from a choice of 7.AMA2005 Fluid Mechanics (2nd semester)
Pre-requisite: AMA1002
Introduction
This course provides an elementary introduction to the mechanics of non-viscous and viscous fluids. The course uses the methods of vector field theory that have been introduced at level 1 in course AMA1002, and it is thus a natural follow-on from AMA1002. It also provides a useful link to several level-3 courses, such as AMA3001 Electromagnetic Theory and AMA3003 Tensor Field Theory.The course starts with a revision of the main aspects of vector field theory that are needed later in the course. This requires several extensions of the material covered in AMA1002.
The main body of the course is concerned with fluid mechanics. The fundamental equations are derived. The theory is then applied to situations in which the equations simplify to the extent that analytic solutions are possible. This course does not consider computational solutions. It is hoped that, in addition to some reasonably challenging mathematics, the course contains enough simple treatments of every-day phenomena that all level-2 students will find manageable, stimulating and rewarding.
Contents
I. Review of Vector Field Theory (AMA1002)
Vector algebra: scalar product; vector product; triple products. Co-ordinates: Cartesian; general orthogonal curvilinear coordinates; spherical polar; cylindrical polar. Scalar Fields: gradient; time-dependent fields. Vector fields: flux and divergence; Gauss divergence theorem; line-integrals and curl; Stokes theorem. Miscellaneous topics: ∇ notation; ∇2; identities; Green's Theorem.II. Fluid Mechanics
1) General: Basic definitions and properties: materials; solids, plastics and fluids; density; pressure; equation of state; body forces; surface forces; viscosity. Velocity: pathlines; streamlines; boundary conditions. Continuity equation; incompressible fluids; streamtubes. Euler's Momentum Equation.
2) Applications: Hydrostatics; Sound waves.
3) Vorticity and circulation: Definitions; Vortex lines and tubes; Vorticity equation; Kelvin's Circulation Theorem.
4) Bernoulli's equation: Derivation and conditions; Simple examples; Class demonstrations; Open channel flows.
5) Two-dimensional flow of incompressible Fluid: Stream function: properties; flux; vorticity; solid boundaries. Some model flows. Steady flow of an inviscid liquid past a cylinder.
6) Irrotational flow in two dimensions: Review of complex variables. Complex potential: definition; some model flows; Shifts and rotations; examples. Image theorems for walls and circles. Conformal mapping: general theory; particular mappings; the Joukowski transformation; the cambered aerofoil.
7) Irrotational flow in three dimensions: Method of separation of variables. Spherical polar solution; Cylindrical polar solution; Cartesian solution. Gravity surface-waves.
8) Viscous fluids: The stress tensor; surface forces revisited; deformations and the strain tensor; Newtonian fluids. Navier-Stokes equation: general derivation; incompressible fluids; boundary conditions. Examples. Laminar and turbulent flow; Reynolds and Froude numbers; dimensional analysis.Assessment
One 3-hour examination with 5 questions to be attempted from a choice of 7.SOR2002 Statistical Inference (1st semester)
Pre-requisite: SOR1002
Introduction
The course builds upon the probability theory in SOR1001 and the statistical inference methods in SOR1002 to provide a second-level account of the principles and most important methods of estimation and hypothesis testing.An important element of this module will be a weekly practical data analysis lab using the SAS software package. SAS is probably the leading statistical package used in industry. The lab sessions, approximately 2 hours long, will give students an opportunity to put the theory of lectures into practice. During the lab sessions students will be guided through practical tutorials on reading, accessing and editing data in SAS, descriptive statistical analyses, cross tabulations, graphs, charts, frequency tables, χ2 tests, t-tests and analysis of variance tests. Some of the lab time will also be allocated for project work.
The practical work in the lab sessions culminates in a group project. During the semester the student will work as a team member on a specific statistical investigation using the methods described in lectures and performed in the SAS lab sessions. This provides a valuable insight into the applications of statistics in industry allowing students to gain the necessary skills required for such a working environment. Near completion of the group project, each group will be required to give an oral presentation (approximately 5 mins per student) on the results of their analysis. This provides an opportunity for the group to receive feedback on their progress and helps consolidate their findings before submitting a final written report on their investigation.
Contents
Statistical Investigations: Understanding the problem. Collecting the data. Initial data analysis. Definitive analysis: modelling. Conclusions.
Initial Data Analysis: Data structure. Processing data: coding, input, screening, editing and modifying. Data quality.
Preliminary analysis: measures of location, dispersion; tables; graphs.
Sample Diagnostics: Testing for independence - non-parametric tests, serial correlations. Testing for normality - skewness and kurtosis, goodness-of-fit tests, probability plotting. Identification of outliers. Transformations.
Point Estimation of Parameters: Definitions of estimate, estimator, sampling distribution. Unbiasedness. Relative efficiency. Bias. Mean squared error.
Sufficiency: Fisher-Neyman factorization theorem. Regular exponential class of distributions.
Maximum Likelihood: Likelihood function. Calculation of MLE. Log relative likelihood function. Asymptotic properties of MLE. Applications.
Least Squares Estimation and Linear Regression: Standard linear model: matrix notation. Properties of LSE. Weighted least squares. Fitting a straight line. Multiple regression. Goodness-of-fit: residuals. Hypothesis tests and confidence intervals using the t-distribution.
Experimental Design and Comparative Studies: Principles of design - experimental unit, treatment, replication, randomization; factorial design. Analysis of variance. Completely randomized design. Randomized block design. Dichotomous treatment/risk and outcome studies. Sampling schemes - cross-sectional; longitudinal-cohort, case-control study.
Measures of association: rates, relative risk, odds ratio.
Significance Tests and Hypothesis Testing: Neyman-Pearson approach - critical region, Type I and Type II errors, significance level, power function. Best critical region. Generalized likelihood ratio test.
Computer intensive methods: randomization tests; Monte Carlo sampling.
Confidence Intervals: Construction - pivotal quantity; MLE procedure. Confidence region. Prediction interval.
Bayesian Methods: Prior and posterior distributions. Conjugate families. Point estimates, confidence regions, hypothesis testing. Prediction. Improper and non-informative priors.
Assessment
One three hour written examination with 5 questions to be attempted out of a choice of 7. The examination is worth 70% of the final mark, the project report is worth 20%, and the oral presentation is worth 10%. All aspects of the module assessment are compulsory: each student must sit the examination, submit a written report and give an oral presentation.SOR2003 Methods of Operational Research (2nd semester)
Pre-requisite: SOR1001
Introduction
This course applies mathematical analysis to a series of problems which occur in business and industry. The analysis can be more far reaching if we use a deterministic model but a degree of uncertainty (e.g. about future events) is often an important feature of the situation and a stochastic model has to be used. The statistical knowledge assumed is that contained in SOR1001. Although novel ways of setting out the work may be used in some topics, the mathematical techniques required on this course are no more advanced than simple calculus and algebra and most practical problems require only arithmetic and the use of tables.The aim of the course is to teach a range of simple techniques illustrating the application of mathematics and probability theory to the problems of business and industry. Apart from the first two chapters each chapter is a distinct and separate topic. Some topics (e.g., Forecasting) involve lengthy calculation and students are taught how to use a spreadsheet for the computation. Students who do not have access to a spreadsheet on a personal computer can use the Open Access Areas. Specific instructions on the use of the Excel spreadsheet is given on the course and there is a practical session in an Open Access Area.
Homeworks are an essential part of the learning process, but there is no continuous assessment.
Emphasis is placed on choosing the correct model for the circumstances and on presenting answers in a form intelligible to management. If a question is posed in words then the final answer should be in words and not left in algebra or in a table. The practical problems associated with obtaining data are discussed. The answer should be to a number of significant figures consistent with the accuracy of the original data, or rounded to an integer if that is appropriate.
Contents
Deterministic inventory including quantity discounts, common cycle production, constrained inventory and the use of Lagrange multipliers. Stochastic inventory models including service levels. Simple and adaptive forecasting. The use of spreadsheets and their application to forecasting and equipment replacement. The replacement of deteriorating equipment and the replacement of equipment liable to sudden failure. Acceptance sampling by attribute and variable. Network planning including PERT, speeding up, the use of LP, Gantt charts and resource smoothing. Decision analysis including utility curves, decision trees and Bayesian statistics.Assessment
One 3-hour written examination in which 5 questions are to be attempted out of a choice of 7.SOR2004 Linear Models (2nd semester)
Co-requisite: SOR2002. Make sure you are enrolled for this module in the 1st semester. This co-requisite is frequently overlooked!
Introduction
The aim of this module is to cover linear models encompassing multiple linear regression and analysis of variance (ANOVA). These models are the workhorses of statistical data analysis and are found in virtually all branches of the sciences as well as in the industrial and financial sectors.Multiple linear regression is concerned with modelling a measured response as a function of explanatory variables. For example, a pharmaceutical company might use a a regression model to relate the effectiveness of a new cancer drug to the patients age, gender, weight, diet, tumour size, etc. ANOVA is concerned with the analysis of data from designed experiments. A materials manufacturer for example, may wish to analyse the results from an experiment to compare the heat resisting properties of four different polymers.
Regression and ANOVA will be initially developed using a classic least squares approach and later the correspondence between least squares and the method of maximum likelihood will be examined. After a thorough development of linear models the groundwork will have been laid to allow an extension to the broader class of Genealized Linear Models (GLM). These permit regression models to be applied to situations where the recorded response is not normally distributed. One famous example of the use of GLM was the analysis of O-ring failures on the space shuttle Challenger.
An important element of this module will be a weekly practical data analysis class using the SAS software package. SAS is probably the leading statistical package used in industry. These classes, lasting up to three hours, will introduce the student to elementary data entry in SAS, elementary matrix manipulation using the SAS Interactive Matrix Language (IML) and analysis of data using linear and generalized linear models. Each week the student will complete a data analysis task using SAS and is required to submit a report the following week.
Contents
Multiple linear regression: ordinary least squares, model selection and diagnostics, weighted least squares.
Analysis of variance: Non-singular and singular cases; extra sum of squares principle, analysis of residuals, generalized inverse solution, estimable functions, testable hypotheses.
Experimental designs: completely randomized, randomized block, factorial, contrasts, analysis of covariance.
Generalised linear model: maximum likelihood and least squares, exponential family, Poisson and logistic models, model selection for GLM.Assessment
One three hour written examination with 5 questions to be attempted out of a choice of 7, comprising 80%. Ten weekly assignments worth a total of 20%.
Level 3 modules
AMA3001 Electromagnetic Theory (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The ideas of electromagnetic theory are developed at a comparatively sophisticated mathematical level making extensive use of the methods of vector calculus. The module develops the subject from the simplest situation - charges at rest (electrostatics), through charges in uniform motion (magnetostatics) to charges in non-uniform motion (electrodynamics). The full Maxwell equations appropriate to the latter general situation are built up from simpler forms, with their origins in experimental findings.Students completing this module will understand the mathematical treatment of electromagnetism and through solving the problems, will have experience in mathematical modelling. They will be fluent in the use of vector calculus. The methods used here find application in more advanced studies of theoretical physics.
Contents
Electrostatics: electric field; Gauss's law; Poisson's equation; Laplace's equation; polarisation; electric displacement; boundary conditions in electrostatics; methods for solving Laplace's equation; electrostatic energy density.
Steady currents: electric current; equation of continuity.
Magnetostatics: magnetic scalar and vector potentials; magnetic dipole; magnetic field; magnetisation; boundary conditions in magnetostatics; potential problems.
Electromagnetic induction: electromotive force; magnetic energy; energy density of the magnetic field.
Maxwell's equations: electromagnetic energy; electromagnetic potentials; the wave equation; plane waves; electromagnetic radiation; wave-guides.
Assessment
One 3 hour written examination with 4 questions to be attempted out of a choice of 6.AMA3002 Quantum Theory (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Though the development of quantum theory is one of the greatest intellectual achievements this introductory course does not require advanced mathematical skills or advanced knowledge of physics.After a brief historical survey, the Schrödinger equation is derived and the Postulates of quantum theory are stated. To illustrate them some exactly solvable simple problems are treated, for example a free particle, a particle in a rectangular box with rigid walls, a simple harmonic oscillator, a particle in a central potential field and the hydrogen atom. Heisenberg's uncertainty principle (which sets limits on what may be simultaneously known about, for instance, the x position coordinate and the px component of momentum of a particle) is proved. It is shown that Newton's Laws of Motion may be deduced from the postulates of quantum theory. Most stationary problems cannot be treated exactly because it is not possible to separate the variables. Approximate methods are therefore introduced and applied to two-electron systems and the effect of electric and magnetic fields on the energy levels of atomic systems. The electron is not a structureless charged particle: it possesses spin. A mathematical description of this is given. The Pauli exclusion principle is introduced and with its aid the periodic table of the elements is discussed. A brief introduction to quantum mechanical scattering theory is given.
The course is largely self contained and no pre-requisite from Level 2 is required though advantage would be gained by attending courses AMA2001 on classical mechanics, and AMA2003 on the methods of applied mathematics.
Contents
Introduction and experimental background: Introductory comments; experimental background - black-body radiation, photoelectric effect, Bohr atom, wave nature of electrons.
The Schrödinger Equation: derivations of the wave equation; probability current density; expectation values; Ehrenfest's Theorem; energy eigenfunctions; one-dimensional square well potential with rigid walls.
Principles of Quantum Mechanics: postulates of quantum mechanics; commutability and compatibility; Heisenberg's Uncertainty Principle; momentum eigenfunctions; Dirac δ-function; δ-function normalisation and closure; momentum space and momentum representation.
Exactly solvable bound state problems: one-dimensional square well potential; the linear harmonic oscillator; spherically symmetric potential in three dimensions; angular momentum; three-dimensional square well potential; the hydrogen atom.
Approximate methods for the solution of bound state problems: Variational Method - one and two-electron systems; stationary perturbation theory - first- and second-order perturbation theory; semi-classical approximation - asymptotic connection formulae, Bohr-Sommerfeld quantization.
Spin Angular Momentum: angular momentum shift operators; spin operators and eigenvectors; Pauli exclusion principle and the periodic table.
Scattering Theory: one-dimensional square potential barrier; scattering in three dimensions; partial wave analysis of the scattering amplitude, scattering phase shift, Optical Theorem.
Assessment
One 3 hour written examination with 4 questions to be attempted out of a choice of 6. Best 4 questions are counted.AMA3003 Tensor Field Theory (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The notion of scalar and vector quantities is essential in the application of mathematics to physical phenomena. For example it plays a pivotal role in dynamics: Newton's 2nd law relates the vector force to the rate of change of the vector momentum. A vector quantity such as a particle's position is often given as 3 components referred to a rectangular set of axes, the familiar Cartesian components. However the physics does not depend on the choice of reference axes. Thus the equations that describe the motion are independent of coordinate system. The vector components in one coordinate system will transform in a prescribed way to another coordinate system so as to ensure that this independence is preserved. The key idea is that of invariance with respect to coordinate transformations. A tensor is a set of quantities that transform in a prescribed manner when a coordinate transformation is made. It is a generalisation of the notion of a vector and a vector is in fact a tensor of rank 1. A scalar is a tensor of rank 0. Tensors are the appropriate objects for describing many physical phenomena, such as solid and fluid mechanics, elasticity, special and general relativity.General tensors allow coordinate transformations of any type. Cartesian tensors axe restricted to orthogonal transformations. The possible transformations are then rotations and/or reflections so that the reference axes remain orthogonal. While the theory of Cartesian tensors is simpler than that of general tensors, it still has wide application in physics, including special relativity. However general tensors are required for a study of general relativity.
In this module you will study the algebra and calculus of Cartesian and general tensors with applications in special and general relativity. The pure mathematician will delight in the tensors and the differential geometry while the theoretical physicist can wallow in the relativity. There is something here for everyone. You should be confident with vectors.
Contents
Cartesian tensors (6 lectures): Notation, Kronecker delta, permutation symbol, determinants, Euclidean space, orthogonal transformations, definition of a Cartesian tensor, tensor algebra, addition and subtraction, outer product, inner product, contraction, symmetry and skew-symmetry, quotient rule, tensor equations, tensor densities, isotropic tensors, tensor fields and calculus, vectors in E3, vector identities.
Special relativity (9 lectures): Newtonian mechanics, Galilean transformation, Lorentz transformation, Minkowski space-time, Minkowski diagram, causality Lorentz-Fitzgerald contraction, time dilation. length paradox, relativistic mechanics, 4-velocity, transformation of 4-velocity, 4-acceleration, transformation of 4-acceleration, 4-momentum and 4-force, mass-energy relationship, energy-momentum relationship, transformation of 4-momentum, conservation of 4-momentum, potential energy, electromagnetism, Maxwell's equations, Lorentz force, scalar and vector potentials, Lorentz gauge and wave-equations, tensor formulation of electromagnetism, electromagnetic tensor, dual electromagnetic tensor density, invariants, transformation of fields.
General tensors (9 lectures): Notation, Riemann space, contravariant vectors, covariant vectors, tensors, Cartesian tensors, tensor algebra, raising and lowering indices, relative tensors, tensor calculus, covariant derivative, constant vector fields, geodesics, covariant derivative of tensors, geodesic coordinates, Bianchi identities, metric affinity, 3-dimensional vectors, symmetries of curvature tensor, Ricci tensor, divergence and Laplacian, Einstein tensor.
General relativity (6 lectures): General relativity, special relativity, Newtonian gravitation, field equations, cosmology, Robertson-Walker metric, Friedmann equation, models, gravitational red-shift, Schwarzschild solution, time-like geodesics, orbit equation, advance in the perihelion of planets, null geodesics, deflection of a light ray near the sun, black-holes, Eddington form, radial motion.
Assessment
One 3 hour examination with 4 questions to be attempted out of a choice of 6.AMA3004 Advanced Numerical Analysis (1st semester)
Pre-requisite: AMA2004
Introduction
The aim of this course is to develop further the ideas introduced in the level 2 course on Numerical Analysis. In particular, it aims to give further insight into modern approximation techniques; to explain how, why and when they work; and to provide a firm basis for future study in numerical analysis and scientific computing.The course covers three areas: approximation theory, eigenvalue and eigenvector approximation and numerical integration. In the first of these we attempt to approximate a continuous, real-valued function by a polynomial (e.g. Chebyshev polynomials) or by a rational function in order to provide computationally economical ways of evaluating the particular function. We also discuss polynomial interpolation techniques such as the Lagrange and Newton forms of interpolation and cubic splines - a technique widely used in computer aided design.
The solution to many physical problems requires the calculation of eigenvalues and eigenvectors of a matrix. Theoretically, the eigenvalues of a matrix A, can be obtained by finding the roots of the characteristic polynomial p(λ) = det(A−λI). In practice, p(λ) may be hard to obtain and its roots even harder to compute directly. Various elegant and powerful approximation techniques have therefore been developed to find the eigenvalues and eigenvectors of a matrix.
Numerical integration, or quadrature, is a hugely enjoyable subject. Like many other areas in numerical analysis, the development of good approximate integration techniques requires a wonderful mixture of mathematical rigour and intuition. One needs numerical integration when faced with definite integrals which have no analytical solution. For example, while the sine function is one of the most common mathematical functions, calculation of its length gives rise to an elliptic integral of the second kind, which cannot be evaluated analytically. In such cases approximation techniques provide the only way forward.
Contents
Approximation theory: Norms, polynomial minimax approximation, Chebyshev polynomials, summation of Chebyshev series, Chebyshev expansions, rational Padé approximation, rational minimax approximation, numerical calculation of the rational minimax approximation, piecewise polynomial approximation, polynomial interpolation, divided differences and the Lagrange and Newton forms of interpolation, cubic splines.
Eigenvalues/eigenvector analysis: Definitions, vector and matrix norms, matrix algorithms, rounding errors and error analysis, Givens' and Householder zeroising transformations, power methods and inverse iteration, methods for symmetric matrices, Jacobi's method, tri-diagonalisation using Householder transformations, general matrices and the QR method.
Numerical integration techniques: Review of basic techniques; interpolatory quadrature; Gaussian quadrature - types, properties and ways of constructing the nodes and weights, error estimation; automatic adaptive quadrature; integration over infinite intervals.Practical assignments: you will be asked to write and run Matlab programs for some of the algorithms discussed in lectures, and you will be required to maintain a portfolio demonstrating your efforts in carrying out the work.
Assessment
One 3-hour written examination contributes 70% of the final mark for the module, with the additional 30% coming from the practical assignment portfolio (see above). The examination paper consists of two sections with three questions in each, and you will be asked to answer two questions from each section.AMA3006 Partial Differential Equations (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This module develops further some ideas and methods initially introduced in AMA1002. It concerns those problems in Applied Mathematics which can be formulated as differential equations involving functions of more than one variable (e.g., position and time, or several coordinates). These are partial differential equations (PDE). Besides AMA1002, students deal with examples of such equations in AMA2005, AMA3001, AMA3002, AMA3007, and SOR3012. This course studies properties of these equations, and develops methods for solving them.Contents
Introduction: Terminology: order, linear and nonlinear equations, initial and boundary conditions. Derivation of the wave equation (in one dimension, 1D) and heat equations. Further examples: Laplace's equation, Poisson's equation, Schrödinger equation. Method of Separation of Variables: for the wave equation and heat equation in 1D, and for the circular membrane. Bessel functions. Eigenvibrations (modes).
Fourier Method: Fourier expansion of a function. Piecewise continuous functions. Convergence of Fourier series (FS). FS for even and odd functions. Half-range FS. FS near discontinuities - the Gibbs phenomenon. Application of FS to solving PDE. Laplace's equation for a disk: Poisson's integral.
Integral Transform Methods: Fourier transform (FT) as a limit of the complex Fourier series for the infinite interval. Notion of the Dirac delta function. FT of even and odd functions, sine and cosine FT, and FT of partial derivatives. Application to PDE (heat, wave, Laplace). Laplace transform (LT). LT of some common functions, convolution and shift theorems. Applications of LT to ordinary and partial differential equations.
Orthogonal expansions: Sturm-Liouville Theory. Inner product and norm, orthogonal systems of functions. Gram-Schmidt process. Expansion of functions in orthogonal systems. Convergence in the mean and completeness. Self-adjoint differential operators and Green's formula. Singular, periodic and homogeneous boundary conditions. Sturm-Liouville theory: properties of the eigenfunctions and eigenvalues; degeneracy. Generalised Fourier series. Fourier-Bessel expansion.
Green's Functions: Dirac delta function. Green's function of the Sturm-Liouville equation. Green's functions in several dimensions (Dirichlet problem for the Laplace equation; heat and wave equations with source terms).
Normal Forms of 2nd-order PDE in Two Variables: Linear and quasi-linear PDE. Hyperbolic, parabolic and elliptic types. Reduction to the normal form, and use of this method for solving PDE.Assessment
One three-hour examination with the mark based on the 4 best questions out of total of 6.AMA3007 Financial Mathematics (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Mathematical skills are highly sought after in the financial services industries, and this employment sector remains a favoured destination for graduates. Around 40% of Mathematics graduates entering employment across the UK (see www.prospects.ac.uk for recent data) go into financial services, which includes, accountancy, retail and investment banking, mergers and acquisitions, insurance and actuarial work, capital market trading, and hedge fund employment, and so on.At the low end of this sector, retail banking for example, a degree in mathematics is certainly not essential. This work is mainly concerned with simple arithmetic operations. However, at the high end of financial services, in a hedge fund for example, employers expect to see PhD-level qualifications in mathematics from their applicants along with excellent software skills. These mathematicians are involved in the business of derivative pricing and trading and earn salaries well over 100k. Derivatives are financial products (instruments as they are called in the trade) derived from assets that have an unpredictable price. The total outstanding notional value of derivatives contracts today has grown beyond a quadrillion dollars (that's 1015 to you and me). It is a perilous and lucrative business!
Derivatives were originally devised to avoid risk by providing an insurance on a risky asset. Nowadays, they are an essential part of risk taking in capital markets. Indeed the speculation in buying and selling these instruments, specifically credit derivatives, precipitated the current credit crunch. Of course, this trade relies upon knowing the fair price of a derivative. Pioneering work by Black, Merton and Scholes, showed that, under certain assumptions for the unpredictability of the asset, the price of the derivative obeys a partial-differential equation. The construction of such equations and their solution is where mathematicians come in!
The objective of the course is to provide an introduction to the mathematical techniques which can be applied to pricing problems for financial derivatives. Specifically, our focus is on stochastic calculus and the theory and practice of pricing simple derivatives such as contracts and options.
We are grateful to First Derivatives plc for their support of this course and the provision of prizes for the best examination performance.
Contents
Introduction to financial derivatives: forwards, futures and options. Future markets and prices. Option markets. Binomial models and the risk-free portfolio. Stochastic calculus and random walks. Black-Scholes equation. Pricing European options. Various option pricing models. Interest-rate derivatives. Credit derivatives. Swaps.Assessment
One 3-hour written examination with 4 questions to be attempted out of a choice of 6.AMA3010 Dynamical Systems (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
A picture is sometimes worth a thousand words.Dynamical systems theory gives a graphical, qualitative understanding of differential equations, which arise ubiquitously in nature when properties such as position, density, or temperature change in time. Frequently these equations are nonlinear - the sum of two solutions is not again a solution - and are impossible to solve in closed form. Also, many of the quantitative details of the solutions are not of interest to us. To gain qualitative insight the graphical approach is then useful.
Dynamical systems is a new area of mathematics and has many applications: e.g., weather forecasting, how Google works, the solar system's future. This course teaches you how to draw the phase portrait of a differential equation and how to use the phase portrait to understand the long- and short-term behaviour of the solutions. Bifurcations, the butterfly effect, chaos and fractals are also introduced.
As an example, take the damped pendulum ml d2θ/dt2 + B dθ/dt + mg sinθ = 0, where a mass m at the end of a light rod of length l rotates under the influence of gravity g and friction B. The sinθ term makes this difficult to solve: routinely sinθ ≈ θ approximation is made to give a linear differential equation. To deal with the exact non-linear equation you will learn how this equation can be represented by the graph below: in particular why we use two dimensions for the graph and not one or three, what the curved lines mean and why the points towards which the lines swirl matter.
Contents
Flows; Fixed points and stability; Phase-space solutions; Vector fields; Linear flows; One-, Two- and Three-dimensional nonlinear flows; Linearisation about fixed points; Models of competition and predation; Limit cycles; Poincare-Bendixson theorem; Bifurcations; Lorenz equations; Chaos on a strange attractor; Iterative maps; Google; The logistic map; The Mandelbrot set; Fractals; Fractional dimensions.Assessment
One 3-hour written examination with 4 questions to be attempted out of a choice of 6.AMA3013 Calculus of Variations and Hamiltonian Mechanics (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
The aim of this course is to develop the essential ideas and mathematics of the Calculus of Variations, with applications to various problems.In its simplest form the Calculus of Variations is concerned with finding a function which makes a given integral a maximum or minimum, or, more generally, ‘stationary’. A famous historical example (solved by the Bernoullis, Newton and others) is the following. A particle, which is initially at rest, slides under gravity along a smooth wire connecting two points: find the shape of the wire, which minimizes the time taken for the particle to travel between the two points. This corresponds to finding the function which minimizes the time integral. More generally, ‘variational’ problems may involve integrals containing more than one function or higher derivatives of the functions than the first, or involve constraints on the functions, which may be given in integral form or point-wise. In addition, the variational integral may be multi-dimensional.
In the formulation of physical laws variational methods have a much deeper significance. This was first suggested by the elegant work of Lagrange and the Irish mathematician Hamilton who looked at the basic mathematical structure of classical mechanics. The formalism that they developed is not only essential for a full understanding of quantum mechanics and statistical mechanics but turns out to have a much wider application in that it can be extended to systems that are not normally considered in dynamics, e.g., the Electromagnetic Field. In this sense variational methods provide a ‘unifying’ principle of physics.
The first part of this course concentrates on the basic mathematics of the Calculus of Variations. The second part deals with Lagrangian and Hamiltonian mechanics and their variational basis. While the course is more or less self-contained, it does requires a knowledge of basic first-year calculus, including functions of several variables and a smattering of ordinary differential equations. Newton's laws of motion in vector form and the elementary ideas of kinetic and potential energy are also needed.
Contents
Part I: The Calculus of Variations (approximately 15 lectures): Motivation: the Brachistochrone and Isoperimetric problems. Functional; extremum, stationary point. Function classes. Weak and strong variations. The simplest variational problem: necessary condition for an extremum; Euler-Lagrange lemma; Euler's equation. Several unknown functions. Fermat's principle. Geodesics. Functionals depending upon higher order derivatives. Variational problems with subsidiary conditions: Lagrange multipliers; finite subsidiary conditions. Variable end-point theorem: broken extremals; Weierstrass-Erdmann corner condition. Second variation of a functional: Legendre's necessary condition for a minimum. Direct methods: the Ritz method; the method of finite differences; the Sturm-Liouville problem.
Part II: Analytical Mechanics (approximately 15 lectures): Constraints and generalized coordinates; holonomic constraints. Virtual displacement; D'Alembert's principle; Lagrange's equations. Action integral. Hamilton's principle. Generalized momentum; cyclic coordinates; conservation laws. The Hamiltonian; Hamilton's equations; derivation of Hamilton's equations from a variational principle. Principle of least action; Jacobi's form of the principle of least action. Canonical transformations: generating function; symplectic matrices and canonical transformations; Hamilton's equations in symplectic form. Poisson brackets: Jacobi's identity; canonical transformations of Poisson brackets; Hamilton's equations in Poisson bracket form; Poisson's theorem. The Hamilton-Jacobi equation; Hamilton's characteristic function. Action-angle variables. Phase-space diagram.Assessment
One three-hour exam. Two questions to be answered from Section A and two questions to be answered from section B. There are three questions in each section.AMA3014 Mathematical Modelling in Biology and Medicine (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Mathematics has always benefitted from its involvement with developing sciences. Biomedical science is clearly the premier science of the forseeable future. With the example of how mathematicians have have benefited from and influenced physics, it is clear that if mathematicians become involved in the biosciences they will be part of a field of important and exciting scientific discoveries.In this module no previous knowledge of biology is assumed. With each topic a brief description of the biological background will be provided sufficient to understand the models being studied. The emphasis throughout the course is on the practical application of mathematical models in helping unravel the underlying mechanisms involved in the biological processes. The mathematics in the module is dictated by the biology and not vice-versa.
By the end of the course the student should have a good appreciation of the art of modelling which relies on: (i) a sound understanding and appreciation of the biological problem; (ii) a realistic mathematical representation of the important biological phenomena; (iii) finding useful solutions, preferably quantitative; and finally what is crucially important (iv) a biological interpretation of the mathematical results in terms of predictions and insights.
Contents
The module will cover models used in population biology and ecology. These are listed below.
1. Continuous population models for single species.
2. Discrete population models for single species.
3. Models for interacting populations.
4. Biochemical genetics.
5. Biological motion including reaction diffusion models and chemotaxis.
6. Biological waves for single species models.Assessment
One three hour exam. Two questions to be answered from Section A and two questions to be answered from section B. There are three questions in each section.SOR3001 Linear and Dynamic Programming (1st semester)
Pre-requisite: AMA1001 or PMA1012
Introduction
Operational Research is the application of quantitative analysis to problems outside of the physical sciences and in particular to the problems of business, industry and administration. However, all the techniques can be used outside of Operational Research and emphasis is given to formulation of problems and recognition of the appropriate technique. Blackboard examples are given for each technique and the homework problems are drawn from a wide field. Most techniques are in the form of algorithms and for hand calculations the problems have to be fairly simple, but indications are given how computers could be applied to bigger problems and computer outputs are examined.The aim of this course is to develop competence in two of the most important mathematical techniques used in Operational Research, in formulating problems and expressing answers clearly.
This course covers the two main general-purpose mathematical techniques used in Operational Research together with some specialized applications. These are all optimisation techniques, although they also give insight into the problems and the interpretation of answers is stressed. Calculus cannot be used because either (i) the objective function and the constraints cannot be expressed simply, or (ii) the constraints dominate the problem and the optimal solution will be at the edge of the feasible region, or (iii) the variables are not continuous (e.g., they are integer).
Some students may have met some of the techniques in courses at other levels, but no previous knowledge is assumed and more mathematical rigour and understanding is required than at those levels. However, the mathematical knowledge assumed does not extend beyond that in Level 1. The main knowledge assumed is elementary linear algebra (bases, linear dependence of vectors, matrix notation, partitioning of matrices). No knowledge of economics is required but the economic interpretation of some results is explored. Apart from a passing reference to stochastic Dynamic Programming this course is purely deterministic and no knowledge of statistics is required.
Contents
The scope of Operational Research, formulating a problem from a verbal description.
Dynamic programming: formulation, principle of optimality, value iteration, applications including equipment replacement, allocation, production planning and optimal routes. Special algorithms for production planning and optimal routes.
Linear programming: formulation, theory, Primal Simplex Method, interpretation of the final tableau, Revised Simplex Method, duality theory including economic interpretation, Dual Simplex Method, Post-optimal Analysis, Transportation and Assignment problems. Network flows: maximal flow problem, minimal cost flows, the out-of-kilter algorithm. A wide variety of practical problems and applications is discussed.Assessment
One 3-hour written examination in which 4 questions are to be attempted out of a choice of 6.SOR3007 Survival Analysis for Medicine (2nd semester - half module)
Pre-requisite: SOR2004
Introduction
Survival analysis is an important tool for research in medicine and epidemiology. It is that part of statistics that deals with time-to-event data. For example, in a clinical study the data might consist of the post-treatment survival times of patients with hypernephroma (i.e. a malignant tumour of the kidney). Survival analysis might address questions such as: How does the patient's survival time depend on her age at treatment? What is the affect of kidney removal on the survival times of patient compared with others who are treated just with chemotherapy? Is the size of the tumour an equally good predictor of survival for patients under 60 years of age as for the over 60s?While survival analysis is used in many other settings (e.g. actuarial science, engineering), this module will focus on applications of survival analysis in the medical field.
The module introduces the student to the special features of survival data such as censoring (e.g. where a patient is lost to follow up but is known to have survived to a particular time) and positive skew in the distribution of survival times. Fundamental concepts of survival analysis will be introduced including the survivor function, the hazard function and the hazard ratio. The course will build from some elementary nonparametric techniques such as the Kaplan-Meier estimate of the survival curve to the Cox proportional hazards model - one of the most flexible and widely used tools for the analysis of survival data.
An important element of this module will be two hours a week of survival data analysis classes using the SAS software package. SAS is probably the leading statistical package used in industry. Some previous experience of SAS is assumed.
Contents
Introduction to survival data: Features of survival data, distribution of survival times, survivor function, hazard function, cumulative hazard function.
Some nonparametric procedures: Estimating the survivor function - life-table, Kaplan-Meier, Nelson-Aalen, confidence intervals. Estimating the hazard function, estimating median and percentile survival and confidence intervals. Comparing two groups of survival data, the log-rank and Wilcoxon tests. Comparison of k-groups of survival data.
The Cox proportional hazards model: The Cox proportional hazard model (Cox model), baseline hazard function, hazard ratio, including variates and factors, maximum likelihood for the Cox model. Treatment of ties in the Cox model. Confidence intervals for the Cox model regression parameters and hypothesis testing. Estimating the baseline hazard. Model building, Wald tests, likelihood ratio tests and nested models.Assessment
Assessment is based on one written examination paper (contributing a maximum of 85% to the overall module mark) and coursework (contributing a maximum of 15% to the overall module mark). The 2-hour written examination paper comprises two sections, A and B. Section A, accounting for 60% of the examination mark, is compulsory. The material examined in Section A will require in-depth knowledge of one of the techniques from a range indicated at the start of the module. Section B, accounting for 40% of the examination mark, comprises 2 questions of equal marks, of which 1 question is to be attempted. Coursework is linked with the computing sessions, assessing students' use of statistical software to produce and interpret results.SOR3008 Statistical Data Mining (2nd semester - half module)
Pre-requisite: SOR1002
Introduction
In the 1990's there was an explosive growth in both the generation and collection of data due mainly to the advancement of computing technology in processing and storage of data and the ease of scientific data collection. As a result, overwhelming mountains of data are being generated and stored. For example, in the business world large supermarket chains such as Wal-Mart and Sainsbury's collect data amounting to millions of transactions per day. In the US all health-care transactions are stored on computers yielding terabyte databases which are constantly being analysed by insurance companies. There are huge scientific databases as well. Examples include the human genome database project and NASA's Earth Observatory System. This has brought about a need for vital techniques for the modelling and analysis of these large quantities of data: data mining.Data Mining is the process of selection, exploration, and modelling of large quantities of data to discover previously unknown regularities or relations with the aim of obtaining clear and useful results for the owner of the database. The application of data mining includes many different areas, such as market research (customer preferences), medicine, epidemiology, risk analysis, fraud detection and more recently within bioinformatics for modelling DNA.
This module will focus on data mining techniques which have evolved from and are strongly based on statistical theory.
Contents
Introduction to Data Mining. Exploratory data analysis. Cluster analysis. Classification: Decision tree analysis, Probabilistic Modelling, Bayesian Networks. Association Rule Mining. Genetic Algorithms. Neural nets.Assessment
Assessment is based on one written examination paper (contributing a maximum of 85% to the overall module mark) and coursework (contributing a maximum of 15% to the overall module mark). The 2-hour written examination paper comprises two sections, A and B. Section A, accounting for 60% of the examination mark, is compulsory. The material examined in Section A will require in-depth knowledge of one of the techniques from a range indicated at the start of the module. Section B, accounting for 40% of the examination mark, comprises 2 questions of equal marks, of which 1 question is to be attempted. Coursework is linked with the computing sessions, assessing students' use of statistical software to produce and interpret results.SOR3012 Stochastic Processes and Risk (1st semester)
Pre-requisite: SOR1001
Introduction
Uncertainty or risk is a natural part of life. Continually faced with choices, we are required to make decisions based on the possible consequences or outcomes. We often use the phrase ‘a calculated risk’ to describe how we analyse a problem and come up with our decision. Now the question is: how do we calculate risk - mathematically.Statistical analysis of a random process helps determine whether there are any underlying patterns. If there are, we can use this to our advantage in predicting the future - or at least make an educated guess! Not surprisingly, this topic is primarily recognised for its career opportunities where ‘risk’ assessment and planning matter: for example the provision of power stations, telecoms, transport planning, the spread of infectious diseases (swine flu, for example). A major application of this kind of mathematics is the financial services sector: insurance and investment, credit risk, capital market trading etc. Many students that have taken this course are currently employed as highly-paid actuaries.
Aside from the educational and commercial value of this subject, it leads into very interesting and fun topics such as measure theory, martingales and potential theory, noise, fractals, etc.
Contents
Introduction to stochastic processes and some applications. Counting and combinatorics. Set algebra. Inclusion-exclusion theorem. Mutually exclusive events. De Morgan Laws. Philosophy and axioms of probability. Events and probability spaces. σ-fields.
Simple random walk - gambler's ruin. Moments and generating functions. Binomial and Poisson distributions. Laws of large numbers. Central-limit theorem. Correlation and covariance.
Discrete Markov chains. Chapman-Kolmogorov relation. Transient and absorbing states. Ergodic theorem. Martingales. Hitting times. Monte-Carlo Markov chains.
Continuous Markov processes. Poisson processes. Queueing theory. Random walks in continuous time. Chronic and infectious diseases.Assessment
One 3-hour written examination with a choice of 4 questions from 6.
Level 4 modules
AMA4001 Theoretical Atomic Structure and Collisions (1st semester)
Pre-requisite: AMA3002 or, subject to the approval of HoT (AMA), PHY3011
Introduction
The aim of this course is to continue the development of students' understanding of the mathematical foundations of quantum theory, and to develop students' understanding of how quantum theory used in current theoretical research in atomic physics.Theoretical atomic physics is of great interest, since atoms serve as an important test bed for new experimental techniques. In many cases, atoms are the first system in which new physical effects are observed, and it is up to theoreticians to provide explanations for these new effects. The agreement between theory and experiment in atomic physics is unparallelled, which means that theoretical calculations require a high degree of precision.
The module consists of two sections: The first section discusses theoretical atomic structure, and the second section discusses collisions or scattering processes.
The theoretical atomic structure section starts with a basic review of quantum mechanics. At the heart of quantum mechanics lies the mathematical concept of operators. The first part of the module will investigate basic operators and their mathematical properties in detail. Central amongst these properties is the commutator.
Special emphasis is placed on the angular-momentum operators. Atomic systems are spherically symmetric, and we will investigate the connection between spherical symmetry and angular-momentum operators. The influence of spherical symmetry on atomic systems is then further studied by exploring how we can simplify numerical calculations for atoms by obtaining analytic expressions for the angular part of matrix elements: matrix elements involving the spherical harmonics only (i.e., the angular part of the full wavefunction) can be evaluated analytically in terms of Clebsch-Gordan coefficients.
Following this section on the theory of spherical symmetry, we finish the atomic structure section with a look at theoretical methods that can be used to obtain energies and wavefunctions for atomic systems, such as Hartree-Fock theory and configuration-interaction. We will also study the effects of the Pauli principle on two-electron systems, and demonstrate how this principle leads to the classically forbidden exchange integrals.
The second part of the module focuses on collisions or scattering processes. This aspect of quantum theory is extremely important, since atomic systems are extremely tiny: we can only study atomic systems in detail through their interaction with colliding particles, such as photons, electrons or other atoms. In this section, we will introduce students to a variety of methods that can be applied to the theoretical study of collisions between electrons and atoms (or, more generally, systems with a spherical symmetry).
We will first develop the basic principles of scattering calculations by studying basic one-dimensional scattering processes. These processes allow us to introduce a fundamental scattering quantity, the S-matrix, and to investigate the properties of the S-matrix. Using one-dimensional scattering, the differences between the quantum world and the classical world will be demonstrated through the quantum mechanical phenomenon of tunneling.
We then continue by investigating scattering processes in three dimensions. We will introduce the mathematical description of spherical waves using spherical Bessel functions. We will then link the phase shift of the outgoing spherical waves to the S-matrix. We will end this section by studying how resonances affect the scattering cross section, and how interference between the resonance and the background leads to so-called Fano resonance profiles.
Scattering processes can also be investigated using approximate techniques. We will investigate two such techniques: the Born approximation for particles that collide with a large energy, and the semi-classical approximation for collisions involving heavy particles. We will end the collision section by introducing R-matrix theory, a technique used extensively at Queen's in theoretical atomic physics research.
Contents
Theoretical atomic structure
Review of quantum mechanics: wave functions, operators and orbitals, expectation values, parity, the Dirac Hamiltonian, two-electron systems.
Angular momentum operators: relation to rotations, step-up and step-down operators. Coupling of angular momenta: Clebsch-Gordan coefficients.
Evaluation of angular matrix elements: Clebsch-Gordan series, irreducible tensor operators, Wigner-Eckart theorem, spherical harmonic addition theorem. Wavefunctions for two-electron systems: 1s2s states of helium, equivalent electrons, direct and exchange integrals, calculations of electron repulsion.
Multi-electron wave functions: Hartree-Fock theory, configuration interaction, Hylleraas-Undheim theorem, Rayleigh-Ritz variational principle.
Collisions
The classical scattering problem. Quantum scattering in one dimension: Scattering off a square barrier and tunnelling, The S-matrix and its symmetries. Quantum scattering in three dimensions: spherical waves, Bessel functions, scattering amplitude and phase shifts, the optical theorem, resonances. Perturbation theory for scattering: Green's function, Born series and the first Born approximation. Semiclassical approach to scattering. R-matrix theory.Assessment
One 3-hour written examination with 3 questions in the theoretical atomic structure section and 3 questions in the collisions section. Two questions must be attempted from each section.AMA4003 Advanced Mathematical Methods (1st semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 4 of an MSci Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
This course falls into two equal parts. The first part is devoted to a study of partial differential equations of the first order. Its orientation is very much geometrical. Here the solution of a partial differential equation is envisaged as a surface in the space of the dependent and independent variables. It is first shown how solution surfaces of the quasilinear Lagrange equation may be generated from families of curves. Knowledge of how to solve this equation enables us to find, by methods due to Charpit and Jacobi, so-called ‘complete integrals’ of the general first-order partial differential equation. A complete integral may be viewed as a family of surfaces. Envelopes of this family, and of sub-families of this family, also solve the differential equation and give solutions known as ‘singular integrals’ and ‘general integrals’.The second half of the course deals with linear integral equations. Often many problems can be formulated in terms of an integral equation rather than the more familiar differential equation. This can have some advantages, for example, usually the boundary conditions are automatically included in the integral equation formulation and do not have to be imposed afterwards as is usual with differential equations. In addition, linear integral equation theory represents a generalization of matrix theory in which the discrete indices of a matrix element become continuous variables.
There are many analogies between matrix theory and linear integral equation theory. For example, a linear matrix equation is solved by constructing the inverse matrix. Similarly, a solution of a linear integral equation is obtained if we can find the resolvent kernel. In matrix theory Hermitian matrices, with their eigenvalues and eigenvectors, play an important role. Analogously, Hermitian kernels, with their characteristic values and characteristic functions, play an important part in the Hilbert-Schmidt theory of linear integral equations.
While the course is more or less self-contained, some basic knowledge of other areas of mathematics is required, in particular, competence at solving ordinary differential equations, a clear understanding of partial differentiation, matrix theory, and, for the section on integral transform methods for solving integral equations, some knowledge of complex contour integration (complex variable theory).
Contents
Partial Differential Equations of the First Order: Integral surface. Curves parallel to the vector (P,Q,R). Families of curves. Surfaces generated by curves. Envelope of a family of curves. Surfaces: normals, envelopes. Lagrange's equation: a solution; the most general solution; integral surface passing through a given curve; characteristic curves; generalization to n independent variables.
The general first-order equation: classification of integrals, complete integral, singular integral, general integral; integral surface passing through a given curve; Charpit's method for finding a complete integral; Cauchy's method of characteristics; characteristic strips; special types of first-order equations, equations only involving the derivatives, equations not involving the independent variables, separable equations, Clairaut equations; Jacobi's method for finding a complete integral, generalization to n independent variables.
Integral Equations: Fredholm and Volterra equations of the first and second kinds; kernel. Operator notation and scalar product. Solving integral equations : general solution; characteristic values and characteristic functions; resolvent kernel; separable kernel; conversion to differential equation. Connection with matrix theory. Square integrable functions and kernels. Singular kernel. Singular integral equation.
Use of Integral Transforms to Solve Integral Equations: Convolution kernels. Laplace and Fourier Transforms; convolution theorems. Application to Fredholm and Volterra equations with convolution kernels. Abel's integral equation.
Solution of Integral Equations by the Method of Successive Approximations: Neumann series; iterated kernels; resolvent kernel; convergence properties of Neumann series.
Degenerate Kernels: Fredholm Formulae for Continuous Kernels; Fredholm determinant; recurrence relations; characteristic values. Hilbert-Schmidt Theory: Hermitian kernel - characteristic values and functions; orthonormality. Hilbert-Schmidt theorem. Solution of Fredholm equation of the second kind.Assessment
One 3 hour written examination, divided into 2 sections containing 3 questions each and corresponding to the first (partial differential equations) and second (integral equations) halves of the course, with the requirement of answering 2 questions out of each section.AMA4004 Statistical Mechanics (2nd semester)
Pre-requisite: AMA3002 or PHY3011
Introduction
Statistical mechanics is a formalism that aims at explaining the physical properties of matter at the macroscopic level in terms of the dynamical behavior of its microscopic constituents. The scope of the formalism is almost unlimited as it is applicable to matter in any state of aggregation, ranging from gases, liquids and solids to matter in equilibrium with radiation and biological specimens. The aim of this course is to introduce the basic principles and methods of statistical mechanics and to apply them to a number of model systems in order to illustrate their use and potential in a systematic manner.We start by introducing the fundamentals of thermodynamics and then proceed to develop the concepts and techniques needed t o evaluate probability distributions and partition functions. This later quantity establishes the link between the microscopic description of a system, based on quantum states or positions in phase space, and the macroscopic characterization provided by a small set of independent thermodynamic variables.
Among the applications we consider ideal gases made up of classical and quantum particles, vibrations in solids and electro magnetic radiation. We discuss in detail Bose-Einstein condensation and the behavior of electrons in metals. We then move to interacting classical systems, where we consider the treatment of real gases and liquids and introduce the concept of phase transition. The Ising model and some of its variants are solved in the mean-field approximation. Time allowing we provide a general introduction to Monte Carlo methods, a set of powerful numerical techniques used to investigate many-body interacting systems.
Contents
Applications of Boltzmann statistics to ideal systems: Factorisation approximation. Mono-atomic gases. Gases with internal degrees of freedom: vibrations and rotations.
Quantum statistical mechanics: Indistinguishable quantum particles and symmetry requirements. Fermi-Dirac and Bose-Einstein statistics: derivation of partition functions. Recovering the classical limit. Statistical mechanics of quasi-particles: phonons and photons. Ideal Fermi gas: low and high-density limits. Electrons in metals. Ideal Bose gas: low and high-density limits. Bose-Einstein condensation.
Statistical mechanics of interacting systems: Interaction potentials. Perturbation theory using a control parameter. Cluster and virial expansions. The van der Waals equation of state and the liquid-vapor phase transition. Models of Ising and Heisenberg. Mean field theories. Introduction to Monte Carlo methods in statistical physics.Assessment
One 3-hour written examination with 4 questions to be attempted out of a choice of 6.AMA4005 Project (full year)
Pre-requisite: This two-semester-long double module is only available to students on the Mathematics/Applied Mathematics and Physics/Theoretical Physics MSci pathways.
Introduction
The project involves a substantial investigation of a research problem incorporating literature survey, development of appropriate theoretical models and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed. Each student will work under individual supervision of a member of staff.Contents
The mathematical contents of the project will depend on the nature of the research problem.Assessment
80% dissertation, 20% oral presentation.AMA4006 Practical Methods for Partial Differential Equations (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 4 of an MSci Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Many problems in applied mathematics reduce to solving either ordinary or, more usually, partial differential equations subject to certain boundary conditions. In many cases of practical interest exact analytical solutions are not available.In non-relativistic quantum mechanics for example the problems involve finding numerical solutions to the Schrödinger equation. In fluid mechanics solutions of the Navier-Stokes equation are needed in the context for example of weather systems, or of hypersonic flow around a space shuttle, or the flow of blood through arteries. In financial mathematics solutions of the Black-Scholes equation that describes the pricing of options are required. A familiar partial differential equation arising in many problems in physics and engineering is the diffusion equation, which describes how heat flows from hot to cold regions and many other processes. Another familiar example is the wave equation that governs phenomena all around us: electromagnetic waves, the vibrations of guitar strings, and the propagation of the sound they produce.
In order to solve these or other problems we need to understand first the conditions that give rise to a unique solution. Since analytical solutions are only rarely available we examine techniques that have been developed to acquire a numerical approximation to the solution of a particular problem. The third feature of the course is to actually find numerical solutions to a set of representative model problems, by writing computer programs to implement the requisite methods.
Contents
1. Introduction. Examples of partial differential equations in mathematical physics.
2. First-order linear and quasi-linear equations. Method of characteristics.
3. Second-order linear equations and their classification. Canonical forms.
4. The method of separation of variables. Applications.
5. Introduction to numerical methods. Explicit and implicit finite-difference schemes. Applications.
6. Practical assignment. Numerical solutions to model problems.
7. Sturm-Liouville problem. Eigenfunctions and eigenvalues. Eigenfunction expansion.
8. Elliptic equations.
9. Dirac delta function. Green's functions and integral representation.
Assessment
The assessment consists of two parts. The written examination accounts for 75% credit. Three questions may be attempted from a choice of five. There is also a practical assignment, in which candidates are asked to solve numerically a set of model problems by implementing the requisite methods on the computer, and to submit a short written report of their solutions and associated codes. The emphasis of the written account should be on the practical aspects of assessing the accuracy of the solution and understanding its properties. The practical part accounts for 25% credit.AMA4009 Information Theory (2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended primarily for students at stage 4 of the MSci Mathematics/MSOR/M&CS pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.
Introduction
Information is a message received and understood. It was Claude Shannon who, in 1948, came up with an idea to quantify the amount of information with a well-defined means of mathematics. Shannon's creation of the subject of information theory was one of the great intellectual achievements of the 20th century and it is an important basis of the development of information technology in the 21st century. Theory of information, which may come in various forms including a sound pattern, a page of a book and a TV message, is one of the important applications of mathematics for the modern society. In fact, information theory is not merely an application of mathematics. Information theory has had an important and significant influence on mathematics and has given profound contribution to pure mathematics.Information transfer occurs through “codes”. These codes can take various forms, ranging from the English language to the Morse code to the ASCII code for computer data storage. In this module, we will start by defining what we mathematically mean with a code. To measure the amount of information stored within a message expressed in the code, we will then introduce the key quantity of information theory, entropy. We will also introduce the basics of coding theory: if a message is received with erorrs, how can we retrieve the original information content.
Some of the basic concepts of probability theory are discussed. When the information is transmitted from one point to another, we quantify how much information is transmitted through a noisy communication channel. We discuss how the information content of a random variable can be measured. How to infer the original message from the distorted data received is considered.
Contents
Introduction to information theory. Random variables and some concepts of probabilities. Basic number theory and factoring. Secret coding. Uniquely decipherable and instantaneous codes. Code extensions. Huffman coding. Entropy, conditional entropy, joint entropy and mutual information. Noisy channel. Ideal observer. The fundamental theorem of information theory. Basic coding theory. Linear codes. Error correction.Assessment
One 3-hour written examination with 4 questions to be attempted out of a choice of 6.AMA4020 Investigations (2nd semester)
Pre-requisite: Only available to students on the MSci pathways. This module is taken in stage/year 3.
Introduction
Problems in real life come in various shapes and sizes, usually with no answer “at the back of the book”. The purpose of this module is to equip the MSci students with problem-solving and report-writing skills that they will need for the year-long Project (AMA4005 or SOR4001) that they do in their final year. The problems that the students are offered to study will often be open-ended, and in many cases will require the student to set the problem in mathematical terms before attempting its solution.Contents
Students conduct a short practice investigation, followed by two short investigations (one in pairs - two weeks, and one solo - three weeks) in a range of problems in Applied Mathematics and Theoretical Physics. This is followed by a long investigation (five weeks), which is a literature study of a Mathematical or Theoretical Physics topic not covered in the offered (or chosen) lecture modules. The two short and the long investigation are typed up in reports and submitted for assessment. [The practice report is also typed up.] Feedback on reports is given to assist students with improving their writing skills. A short introduction to LaTeX, possibly the best tool for typesetting mathematical papers and reports, will be provided near the beginning of the solo investigation.Assessment
The final mark will be based on the two short investigation reports (pairs and solo), each contributing 25%, and long investigation report (50%).SOR4001 Project (full year)
Pre-requisite: This two-semester-long double module is only available to students on the Mathematics and Statistics & Operational Research MSci pathway.
Introduction
A substantial investigation of a statistical research problem incorporating literature survey, use of relevant statistical packages and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed. Each student will work under individual supervision of a member of staff.Contents
The mathematical contents of the project will depend on the nature of the research problem.Assessment
80% dissertation, 20% oral presentation.
