Mathematics - MMath

This module serves as an introduction to algebraic methods which are central in modern mathematics and that have found applications in many other sciences, but also in our everyday life. In this course students will also gain an appreciation of the concept of proof in mathematics.

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This module introduces mathematical modelling and Newtonian mechanics. Tutorials and Maple worksheets will be used to support taught material.

The modelling cycle: General description with examples; Newton's law of cooling; population growth (Malthusian and logistic models); simple reaction kinetics (unimolecular and bimolecular reactions); dimensional consistency

Motion of a body: frames of reference; a particle's position vector and its time derivatives (velocity and acceleration) in Cartesian coordinates; mass, momentum and centre of mass; Newton's laws of motion; linear springs; gravitational acceleration and the pendulum; projectile motion

Orbital motion: Newton's law of gravitation; position, velocity and acceleration in plane polar coordinates; planetary motion and Kepler's laws.

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This module considers the abstract theory of linear spaces together with applications to matrix algebra and other areas of Mathematics (and its applications). Since linear spaces are of fundamental importance in almost every area of mathematics, the ideas and techniques discussed in this module lie at the heart of mathematics. Topics covered will include vector spaces, linear transformations, eigenvalues and eigenvectors, diagonalisation, orthogonality and applications.

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This module introduces widely-used mathematical methods for functions of a single variable. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage.

Basic notation for sets and number systems including complex numbers (a+ib representation only). Standard functions: trig functions, polynomials, rational functions, exponentials and logarithms.

Single variable calculus: Differentiation, including product and chain rules; Fundamental Theorem of Calculus (statement only), elementary integrals, change of variables, integration by parts, differentiation of integrals with variable limits.

Curve sketching: graphs of elementary functions, maxima, minima and points of inflection, asymptotes.

Algebra of matrices and vectors; addition, multiplication, transposes, inner-products.

Row reduced echelon form, solving linear systems (homogeneous and inhomogeneous).

Inverse of a matrix.

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This module introduces widely-used mathematical methods for vectors and functions of two or more variables. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Vectors: Cartesian coordinates; vector algebra; scalar, vector and triple products (and geometric interpretation); straight lines and planes expressed as vector equations; parametrized curves; differentiation of vector-valued functions of a scalar variable; tangent vectors; vector fields (with everyday examples)

Partial differentiation: Functions of two variables; partial differentiation (including the chain rule and change of variables); maxima, minima and saddle points; Lagrange multipliers

Integration in two dimensions: Double integrals in Cartesian coordinates; plane polar coordinates; change of variables for double integrals; line integrals; Green's theorem (statement – justification on rectangular domains only).

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Introduction to Probability. Concepts of events and sample space. Set theoretic description of probability, axioms of probability, interpretations of probability (objective and subjective probability).

Theory for unstructured sample spaces. Addition law for mutually exclusive events. Conditional probability. Independence. Law of total probability. Bayes' theorem. Permutations and combinations. Inclusion-Exclusion formula.

Discrete random variables. Concept of random variable (r.v.) and their distribution. Discrete r.v.: Probability function (p.f.). (Cumulative) distribution function (c.d.f.). Mean and variance of a discrete r.v. Examples: Binomial, Poisson, Geometric.

Continuous random variables. Probability density function; mean and variance; exponential, uniform and normal distributions; normal approximations: standardisation of the normal and use of tables. Transformation of a single r.v.

Joint distributions. Discrete r.v.'s; independent random variables; expectation and its application.

Generating functions. Idea of generating functions. Probability generating functions (pgfs) and moment generating functions (mgfs). Finding moments from pgfs and mgfs. Sums of independent random variables.

Laws of Large Numbers. Weak law of large numbers. Central Limit Theorem.

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Real Numbers: Rational and real numbers, absolute value and metric structure on the real numbers, induction, infimum and supremum.

Limits of Sequences: Sequences, definition of convergence, epsilon terminology, uniqueness, algebra of limits, comparison principles, standard limits, subsequences and non-existence of limits, convergence to infinity.

Completeness Properties: Cantor's Intersection Theorem, limit points, Bolzano-Weierstrass theorem, Cauchy sequences.

Continuity of Functions: Functions and basic definitions, limits of functions, continuity and epsilon terminology, sequential continuity, Intermediate Value Theorem.

Differentiation: Definition of the derivative, product rule, quotient rule and chain rule, derivatives and local properties, Mean Value Theorem, L'Hospital's Rule.

Taylor Approximation: Taylor's Theorem, remainder term, Taylor series, standard examples, limits using Taylor series.

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Introduction to R and investigating data sets. Basic use of R (Input and manipulation of data). Graphical representations of data. Numerical summaries of data.

Sampling and sampling distributions. ?² distribution. t-distribution. F-distribution. Definition of sampling distribution. Standard error. Sampling distribution of sample mean (for arbitrary distributions) and sample variance (for normal distribution) .

Point estimation. Principles. Unbiased estimators. Bias, Likelihood estimation for samples of discrete r.v.s

Interval estimation. Concept. One-sided/two-sided confidence intervals. Examples for population mean, population variance (with normal data) and proportion.

Hypothesis testing. Concept. Type I and II errors, size, p-values and power function. One-sample test, two sample test and paired sample test. Examples for population mean and population variance for normal data. Testing hypotheses for a proportion with large n. Link between hypothesis test and confidence interval. Goodness-of-fit testing.

Association between variables. Product moment and rank correlation coefficients. Two-way contingency tables. ?² test of independence.

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The concept of symmetry is one of the most fruitful ideas through which mankind has tried to understand order and beauty in nature and art. This module first develops the concept of symmetry in geometry. It subsequently discusses links with the fundamental notion of a group in algebra. Outline syllabus includes: Groups from geometry; Permutations; Basic group theory; Action of groups and applications to (i) isometries of regular polyhedra; (ii) counting colouring problems; Matrix groups.

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In this module we will study linear partial differential equations, we will explore their properties and discuss the physical interpretation of certain equations and their solutions. We will learn how to solve first order equations using the method of characteristics and second order equations using the method of separation of variables.

Introduction to linear PDEs: Review of partial differentiation; first-order linear PDEs, the heat equation, Laplace's equation and the wave equation, with simple models that lead to these equations; the superposition principle; initial and boundary conditions

Separation of variables and series solutions: The method of separation of variables; simple separable solutions of the heat equation and Laplace’s equation; Fourier series; orthogonality of the Fourier basis; examples and interpretation of solutions

Solution by characteristics: the method of characteristics for first-order linear PDEs; examples and interpretation of solutions; characteristics of the wave equation; d’Alembert’s solution, with examples; domains of influence and dependence; causality.

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This module builds on the Stage 1 Real Analysis 1 module. We will extend our knowledge of functions of one real variable, look at series, and study functions of several real variables and their derivatives.

The outline syllabus includes: Continuity and uniform continuity of functions of one variable, series and power series, the Riemann integral, limits and continuity for functions of several variables, differentiation of functions of several variables, extrema, the Inverse and Implicit Function Theorems.

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Constructing suitable models for data is a key part of statistics. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. Even if the temperature and pressure are fixed, there will be variation in the yield which motivates the use of a statistical model which includes a random component. In this module, students study linear regression models (including estimation from data and drawing of conclusions), the use of likelihood to estimate models and its application in simple stochastic models. Both theoretical and practical aspects are covered, including the use of R.

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The main aim of this module is to give an introduction to the basics of differential geometry, keeping in mind the recent applications in mathematical physics and the analysis of pattern recognition. Outline syllabus includes: Curves and parameterization; Curvature of curves; Surfaces in Euclidean space; The first fundamental form; Curvature of surfaces; Geodesics.

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This module will present a new perspective on Newton's familiar laws of motion. First we introduce variational calculus with applications such as finding the paths of shortest distance. This will lead us to the principle of least action from which we can derive Newton's law for conservative forces. We will also learn how symmetries lead to constants of motion. We then derive Hamilton's equations and discuss their underlying structures. The formalisms we introduce in this module form the basis for all of fundamental modern physics, from electromagnetism and general relativity, to the standard model of particle physics and string theory.

Indicative syllabus:

Review of Newton mechanics: Newton's law; harmonic and anharmonic oscillators (closed and unbound orbits, turning points); Kepler problem: energy and angular momentum conservation

Lagrangian Mechanics: Introdution to variational calculus with simple applications (shortest path - geodesic, soap film, brachistochrone problem); principle of least action: Euler-Lagrange equations (Newtonian mechanics with conservative forces); constraints and generalised coordinates (particle on a hoop, double pendulum, normal modes); Noether's theorem (energy and angular momentum conservation)

Hamiltonian Dynamics: Hamilton's equations; Legendre transform; Hamiltonian phase space (harmonic oscillator, anharmonic oscillators and the mathematical pendulum); Liouville's theorem; Poisson brackets.

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Probability: Joint distributions of two or more discrete or continuous random variables. Marginal and conditional distributions. Independence. Properties of expectation, variance, covariance and correlation. Poisson process and its application. Sums of random variables with a random number of terms.

Transformations of random variables: Various methods for obtaining the distribution of a function of a random variable —method of distribution functions, method of transformations, method of generating functions. Method of transformations for several variables. Convolutions. Approximate method for transformations.

Sampling distributions: Sampling distributions related to the Normal distribution — distribution of sample mean and sample variance; independence of sample mean and variance; the t distribution in one- and two-sample problems.

Statistical inference: Basic ideas of inference — point and interval estimation, hypothesis testing.

Point estimation: Methods of comparing estimators — bias, variance, mean square error, consistency, efficiency. Method of moments estimation. The likelihood and log-likelihood functions. Maximum likelihood estimation.

Hypothesis testing: Basic ideas of hypothesis testing — null and alternative hypotheses; simple and composite hypotheses; one and two-sided alternatives; critical regions; types of error; size and power. Neyman-Pearson lemma. Simple null hypothesis versus composite alternative. Power functions. Locally and uniformly most powerful tests.

Composite null hypotheses. The maximum likelihood ratio test.

Interval estimation: Confidence limits and intervals. Intervals related to sampling from the Normal distribution. The method of pivotal functions. Confidence intervals based on the large sample distribution of the maximum likelihood estimator – Fisher information, Cramer-Rao lower bound. Relationship with hypothesis tests. Likelihood-based intervals.

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This module is an introduction to the methods, tools and ideas of numerical computation. In mathematics, one often encounters standard problems for which there are no easily obtainable explicit solutions, given by a closed formula. Examples might be the task of determining the value of a particular integral, finding the roots of a certain non-linear equation or approximating the solution of a given differential equation. Different methods are presented for solving such problems on a modern computer, together with their applicability and error analysis. A significant part of the module is devoted to programming these methods and running them in MATLAB.

Introduction: Importance of numerical methods; short description of flops, round-off error, conditioning

Solution of linear and non-linear equations: bisection, Newton-Raphson, fixed point iteration

Interpolation and polynomial approximation: Taylor polynomials, Lagrange interpolation, divided differences, splines

Numerical integration: Newton-Cotes rules, Gaussian rules

Numerical differentiation: finite differences

Introduction to initial value problems for ODEs: Euler methods, trapezoidal method, Runge-Kutta methods.

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This module introduces the basic ideas to solve certain ordinary differential equations, like first order scalar equations, second order linear equations and systems of linear equations. It mainly considers their qualitative and analytical aspects. Outline syllabus includes: First-order scalar ODEs; Second-order scalar linear ODEs; Existence and Uniqueness of Solutions; Autonomous systems of two linear first-order ODEs.

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Can we square a circle? Can we trisect an angle? These two questions were studied by the Ancient Greeks and were only solved in the 19th century using algebraic structures such as rings, fields and polynomials. In this module, we introduce these ideas and concepts and show how they generalise well-known objects such as integers, rational numbers, prime numbers, etc. The theory is then applied to solve problems in Geometry and Number Theory. This part of algebra has many applications in electronic communication, in particular in coding theory and cryptography.

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The security of our phone calls, bank transfers, etc. all rely on one area of Mathematics: Number Theory. This module is an elementary introduction to this wide area and focuses on solving Diophantine equations. In particular, we discuss (without proof) Fermat's Last Theorem, arguably one of the most spectacular mathematical achievements of the twentieth century. Outline syllabus includes: Modular Arithmetic; Prime Numbers; Introduction to Cryptography; Quadratic Residues; Diophantine Equations.

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Discrete mathematics has found new applications in the encoding of information. Online banking requires the encoding of information to protect it from eavesdroppers. Digital television signals are subject to distortion by noise, so information must be encoded in a way that allows for the correction of this noise contamination. Different methods are used to encode information in these scenarios, but they are each based on results in abstract algebra. This module will provide a self-contained introduction to this general area of mathematics.

Syllabus: Modular arithmetic, polynomials and finite fields. Applications to

• orthogonal Latin squares,

• cryptography, including introduction to classical ciphers and public key ciphers such as RSA,

• "coin-tossing over a telephone",

• linear feedback shift registers and m-sequences,

• cyclic codes including Hamming,

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This module provides an introduction to the study of orthogonal polynomials and special functions. They are essentially useful mathematical functions with remarkable properties and applications in mathematical physics and other branches of mathematics. Closely related to many branches of analysis, orthogonal polynomials and special functions are related to important problems in approximation theory of functions, the theory of differential, difference and integral equations, whilst having important applications to recent problems in quantum mechanics, mathematical statistics, combinatorics and number theory. The emphasis will be on developing an understanding of the structural, analytical and geometrical properties of orthogonal polynomials and special functions. The module will utilise physical, combinatorial and number theory problems to illustrate the theory and give an insight into a plank of applications, whilst including some recent developments in this field. The development will bring aspects of mathematics as well as computation through the use of MAPLE. The topics covered will include: The hypergeometric functions, the parabolic cylinder functions, the confluent hypergeometric functions (Kummer and Whittaker) explored from their series expansions, analytical and geometrical properties, functional and differential equations; sequences of orthogonal polynomials and their weight functions; study of the classical polynomials and their applications as well as other hypergeometric type polynomials.

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This module provides a rigorous foundation for the solution of systems of polynomial equations in many variables. In the 1890s, David Hilbert proved four ground-breaking theorems that prepared the way for Emmy Nöther's famous foundational work in the 1920s on ring theory and ideals in abstract algebra. This module will echo that historical progress, developing Hilbert's theorems and the essential canon of ring theory in the context of polynomial rings. It will take a modern perspective on the subject, using the Gröbner bases developed in the 1960s together with ideas of computer algebra pioneered in the 1980s. The syllabus will include

• Multivariate polynomials, monomial orders, division algorithm, Gröbner bases;

• Hilbert's Nullstellensatz and its meaning and consequences for solving polynomials in several variables;

• Elimination theory and applications;

• Linear equations over systems of polynomials, syzygies.

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Most differential equations which arise from physical systems cannot be solved explicitly in closed form, and thus numerical solutions are an invaluable way to obtain information about the underlying physical system. The first half of the module is concerned with ordinary differential equations. Several different numerical methods are introduced and error growth is studied. Both initial value and boundary value problems are investigated. The second half of the module deals with the numerical solution of partial differential equations. The syllabus includes: initial value problems for ordinary differential equations; Taylor methods; Runge-Kutta methods; multistep methods; error bounds and stability; boundary value problems for ordinary differential equations; finite difference schemes; difference schemes for partial differential equations; iterative methods; stability analysis.

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Linear PDEs. Dispersion relations. Review of d'Alembert’s solutions of the wave equation. Review of Fourier transforms for solving linear diffusion equations.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).

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Groups arise naturally in many areas of mathematics as well as in chemistry and physics. A concrete way to approach groups is by representing them as a group of matrices, in which explicit computations are easy. This approach has been very fruitful in developing our understanding of groups over the last century. It also helps students to understand aspects of their mathematical education in a broader context, in particular concepts from earlier modules (From Geometry to Algebra/Groups and Symmetries and Linear Algebra) have been amalgamated into more general and powerful tools.

This module will provide a rigorous introduction to the main ideas and notions of groups and representations. It will also have a strong computational strand: a large part of the module will be devoted to explicit computations of representations and character tables (a table of complex numbers associated to any finite group).

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Quantum mechanics provides an accurate description of nature on a subatomic scale, where the standard rules of classical mechanics fail. It is an essential component of modern technology and has a wide range of fascinating applications. This module introduces some of the key concepts of quantum mechanics from a mathematical point of view.

The joint level 6/level 7 curriculum will consist of the following:

• The necessity for quantum mechanics. The wavefunction and Born's probabilistic interpretation.

• Solutions of the time-dependent and time-independent Schrödinger equation for a selection of simple potentials in one dimension.

• Reflection and transmission of particles incident onto a potential barrier. Probability flux. Tunnelling of particles.

• Wavefunctions and states, Hermitian operators, outcomes and collapse of the wavefunction.

• Heisenberg's uncertainty principle.

Additional topics may include applications of quantum theory to physical systems, quantum computing or recent developments in the quantum world.

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Matrix theory: Hermitian and symmetric matrices, spaces of these matrices and the associated inner product, diagonalization, orthonormal basis of eigenvectors, spectral properties, positive definite matrices and their roots

Hilbert space theory: inner product spaces and Hilbert spaces, L^2 and l^2 spaces, orthogonality, bases, Gram-Schmidt procedure, dual space, Riesz representation theorem

Linear operators: the space of bounded linear operators with the operator norm, inverse and adjoint operators, Hermitian operators, infinite matrices, spectrum, compact operators, Hilbert-Schmidt operators, the spectral theorem for compact Hermitian operators.

Additional topics, may include:

- the Rayleigh quotient and variational characterisations of eigenvalues,

- the functional calculus,

- applications to Sturm-Liouville systems.

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This is a practical module to develop the skills required by a professional statistician (report writing, consultancy, presentation, wider appreciation of assumptions underlying methods, selection and application of analysis method, researching methods).

Software: R, SPSS and Excel (where appropriate/possible). Report writing in Word. PowerPoint for presentations.

• Presentation of data

• Report writing and presentation skills

• Hypothesis testing: formulating questions, converting to hypotheses, parametric and non-parametric methods and their assumptions, selection of appropriate method,

application and reporting. Use of resources to explore and apply additional tests. Parametric and non-parametric tests include, but are not limited to, t-tests, likelihood

ratio tests, score tests, Wald test, chi-squared tests, Mann Whitney U-test, Wilcoxon signed rank test, McNemar's test.

• Linear and Generalised Linear Models: simple linear and multiple regression, ANOVA and ANCOVA, understanding the limitations of linear regression, generalised linear

models, selecting the appropriate distribution for the data set, understanding the difference between fixed and random effects, fitting models with random effects, model

selection.

• Consultancy skills: group work exercise(s)

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Revision of complex numbers, the complex plane, de Moivre's and Euler's theorems, roots of unity, triangle inequality

Sequences and limits: Convergence of a sequence in the complex plane. Absolute convergence of complex series. Criteria for convergence. Power series, radius of convergence

Complex functions: Domains, continuity, complex differentiation. Differentiation of power series. Complex exponential and logarithm, trigonometric, hyperbolic functions. Cauchy-Riemann equations

Complex Integration: Jordan curves, winding numbers. Cauchy's Theorem. Analytic functions. Liouville's Theorem, Maximum Modulus Theorem

Singularities of functions: poles, classification of singularities. Residues. Laurent expansions. Applications of Cauchy's theorem. The residue theorem. Evaluation of real integrals.

Possible additional topics may include Rouche’s Theorem, other proofs of the Fundamental Theorem of Algebra, conformal mappings, Mobius mappings, elementary Riemann surfaces, and harmonic functions.

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In this module we study the fundamental concepts and results in game theory. We start by analysing combinatorial games, and discuss game trees, winning strategies, and the classification of positions in so called impartial combinatorial games. We then move on to discuss two-player zero-sum games and introduce security levels, pure and mixed strategies, and prove the famous von Neumann Minimax Theorem. We will see how to solve zero-sum two player games using domination and discuss a general method based on linear programming. Subsequently we analyse arbitrary sum two-player games and discuss utility, best responses, Nash equilibria, and the Nash Equilibrium Theorem. The final part of the module is devoted to multi-player games and cooperation; we analyse coalitions, the core of the game, and the Shapley value.

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Integrable systems are special dynamical systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems and nonlinear wave equations to difference equations. This module covers the origins of the subject as well as modern topics like integrable maps and lattice equations.

* Liouville integrability in classical mechanics. Hamiltonian mechanics. Canonical symplectic form and Poisson brackets. Liouville's theorem (statement and examples). Lax pairs for finite-dimensional systems.

* Soliton equations. History and physical origins (e.g. Korteweg-de Vries and/or sine-Gordon). Conservation laws. Hamiltonian formalism. Lax pairs.

* Construction of solitons. Introduction to inverse scattering. Darboux-Bäcklund transformations. Hirota's method.

* Discrete integrability. Symplectic maps. Liouville's theorem (discrete version). Integrable lattice equations. Discrete Lax pairs with examples.

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Sampling: Simple random sampling. Sampling for proportions and percentages. Estimation of sample size. Stratified sampling. Systematic sampling. Ratio and regression estimates. Cluster sampling. Multi-stage sampling and design effect. Questionnaire design. Response bias and non-response.

General principles of experimental design: blocking, randomization, replication. One-way ANOVA. Two-way ANOVA. Orthogonal and non-orthogonal designs. Factorial designs: confounding, fractional replication. Analysis of covariance.

Design of clinical trials: blinding, placebos, eligibility, ethics, data monitoring and interim analysis. Good clinical practice, the statistical analysis plan, the protocol. Equivalence and noninferiority. Sample size. Phase I, II, III and IV trials. Parallel group trials. Multicentre trials.

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Multivariate normal distribution, Inference from multivariate normal samples, principal component analysis, mixture models, factor analysis, clustering methods, discrimination and classification, graphical models, the use of appropriate software.

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• Introduction to Matrix Lie Groups: Basic examples. Matrix groups GL(n), SL(n), SO(n), Sp(n).

• Representations of SU(2): Tensor product of representations, Clebsch-Gordan series for SU(2).

• The Lie algebra of a Lie group. The exponential map.

• Introduction to Lie algebras: The Lie algebras gl(n), sl(n), so(n), sp(n). Nilpotent, solvable and semi-simple Lie algebras. The adjoint action of a group on its Lie algebra, and of a Lie algebra on itself. Derivations.

• Representations of sl(2).

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This module provides an overview of analytical careers in finance and explores the mathematical techniques used by actuaries, accountants and financial analysts. Students will learn about different types of financials assets, such as shares, bonds and derivatives and how to work out how much they are worth. They will also look at different types of debt and learn how mortgages and other loans are calculated. Developing these themes, the module will explain how to use maths to make financial decisions, such as how much an investor should pay for a financial asset or how a company can decide which projects to invest in or how much money to borrow. Risk management is a vital part of most mathematical careers in finance so the module will also cover different mathematical techniques for measuring and mitigating financial risk. Extension topics may include complex derivatives, economic theories of finance and the dangers of misusing mathematics. The module provides an opportunity to apply complex mathematical techniques to important real-world questions and is excellent preparation for those considering a financial career.

Introduction to financial mathematics: Key uses of mathematics in finance; key practitioners of financial mathematics.

Financial valuation and cash flow analysis: Discounting, Interest rates and time requirements, Future and Present value. Project Evaluation.

Characteristics and valuation of different financial securities: Debt capital, bonds and stocks, valuation of bonds and stocks.

Loans and interest rates: term structure of interest rates, spot and forward rates, types of loan, APR, loan schedules.

Capital structure and the cost of capital: Gearing, WACC, understanding betas.

Additional topics that may be covered: arbitrage and forward contracts, efficient markets hypothesis, pricing and valuing forward contracts, option pricing and the Black Scholes model, credit derivatives and systemic risks, limitations of mathematical modelling.

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Introduction: Principles and examples of stochastic modelling, types of stochastic process, Markov property and Markov processes, short-term and long-run properties. Applications in various research areas.

Random walks: The simple random walk. Walk with two absorbing barriers. First–step decomposition technique. Probabilities of absorption. Duration of walk. Application of results to other simple random walks. General random walks. Applications.

Discrete time Markov chains: n–step transition probabilities. Chapman-Kolmogorov equations. Classification of states. Equilibrium and stationary distribution. Mean recurrence times. Simple estimation of transition probabilities. Time inhomogeneous chains. Elementary renewal theory. Simulations. Applications.

Continuous time Markov chains: Transition probability functions. Generator matrix. Kolmogorov forward and backward equations. Poisson process. Birth and death processes. Time inhomogeneous chains. Renewal processes. Applications.

Queues and branching processes: Properties of queues - arrivals, service time, length of the queue, waiting times, busy periods. The single-server queue and its stationary behaviour. Queues with several servers. Branching processes. Applications.

In addition, level 7 students will study more complex queuing systems and continuous-time branching processes.

This module will cover a number of syllabus items set out in Subject CS2 published by the Institute and Faculty of Actuaries. This is a dynamic syllabus, changing regularly to reflect current practice.

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Stationary Time Series: Stationarity, autocovariance and autocorrelation functions, partial autocorrelation functions, ARMA processes.

ARIMA Model Building and Testing: estimation, Box-Jenkins, criteria for choosing between models, diagnostic tests for residuals of a time series after estimation.

Forecasting: Holt-Winters, Box-Jenkins, prediction bounds.

Testing for Trends and Unit Roots: Dickey-Fuller, ADF, structural change, trend-stationarity vs difference stationarity.

Seasonality and Volatility: ARCH, GARCH, ML estimation.

Multiequation Time Series Models: transfer function models, vector autoregressive moving average (VARM(p,q)) models, impulse responses.

Spectral Analysis: spectral distribution and density functions, linear filters, estimation in the frequency domain, periodogram.

Simulation: generation of pseudo-random numbers, random variate generation by the inverse transform, acceptance rejection. Normal random variate generation: design issues and sensitivity analysis.

This module will cover a number of syllabus items set out in Subject CS2 published by the Institute and Faculty of Actuaries. This is a dynamic syllabus, changing regularly to reflect current practice.

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There is no specific mathematical syllabus for this module; students will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their coursework assessments (different topics for levels 6 and 7). The coursework is supported by a series of workshops covering various forms of written and oral communication. These may include critically evaluating the following: a research article in mathematics, statistics or finance; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

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There is no specific mathematical syllabus for this module. Students will study a topic in mathematics or statistics, either individually or within a small group, and produce an individual or group project on the topic as well as individual coursework assignments. Projects will be chosen from published lists of individual and of group projects. The coursework and project-work are supported by a series of workshops covering various forms of written and oral communication and by supervision from an academic member of staff.

The workshops may include critically evaluating the following: a research article in mathematics or statistics; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

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Statistics methods contribute significantly to areas such as biology, ecology, sociology and economics. The real data collected does not always follow standard statistical models. This module looks at modern statistical models and methods that can be utilised for such data, making use of R programs to execute these methods.

Indicative module content: Motivating examples; model fitting through maximum likelihood for specific examples; function optimization methods; profile likelihood; score tests; Wald tests; confidence interval construction; latent variable models; EM algorithm; mixture models; simulation methods; importance sampling; kernel density estimation; Monte Carlo inference; bootstrap; permutation tests; R programs.

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The module offers students the opportunity to work independently, under limited supervision, on an area of mathematics of their choice. There is no specific mathematical syllabus for this module. The topic of the dissertation will depend on the mutual interests of the student and the student's chosen supervisor. The coursework will consist of writing a dissertation plan, an oral presentation of material from the dissertation to examiners and an interview of the student by the examiners.

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• Introduction to Matrix Lie Groups: Basic examples. Matrix groups GL(n), SL(n), SO(n), Sp(n).

• Representations of SU(2): Tensor product of representations, Clebsch-Gordan series for SU(2).

• The Lie algebra of a Lie group. The exponential map.

• Introduction to Lie algebras: The Lie algebras gl(n), sl(n), so(n), sp(n). Nilpotent, solvable and semi-simple Lie algebras. The adjoint action of a group on its Lie algebra,

and of a Lie algebra on itself. Derivations.

• Representations of sl(2).

Find out more about MAST5610

Linear PDEs. Dispersion relations. Review of d'Alembert's solutions of the wave equation. Review of Fourier transforms for solving linear diffusion equations.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).

At level 7, topics will be studied and assessed to greater depth.

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Groups arise naturally in many areas of mathematics as well as in chemistry and physics. A concrete way to approach groups is by representing them as a group of matrices, in which explicit computations are easy. This approach has been very fruitful in developing our understanding of groups over the last century. It also helps students to understand aspects of their mathematical education in a broader context, in particular concepts from earlier modules (From Geometry to Algebra/Groups and Symmetries and Linear Algebra) have been amalgamated into more general and powerful tools.

This module will provide a rigorous introduction to the main ideas and notions of groups and representations. It will also have a strong computational strand: a large part of the module will be devoted to explicit computations of representations and character tables (a table of complex numbers associated to any finite group).

At level 7, topics will be studied and assessed to greater depth.

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Quantum mechanics provides an accurate description of nature on a subatomic scale, where the standard rules of classical mechanics fail. It is an essential component of modern technology and has a wide range of fascinating applications. This module introduces some of the key concepts of quantum mechanics from a mathematical point of view.

Indicative syllabus for the joint level 6/level 7 curriculum::

• The necessity for quantum mechanics. The wavefunction and Born's probabilistic interpretation.

• Solutions of the time-dependent and time-independent Schrödinger equation for a selection of simple potentials in one dimension.

• Reflection and transmission of particles incident onto a potential barrier. Probability flux. Tunnelling of particles.

• Wavefunctions and states, Hermitian operators, outcomes and collapse of the wavefunction.

• Heisenberg's uncertainty principle.

Additional topics may include applications of quantum theory to physical systems, quantum computing or recent developments in the quantum world.

At level 7, topics will be studied and assessed to greater depth.

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Matrix theory: Hermitian and symmetric matrices, spaces of these matrices and the associated inner product, diagonalization, orthonormal basis of eigenvectors, spectral properties, positive definite matrices and their roots

Hilbert space theory: inner product spaces and Hilbert spaces, L^2 and l^2 spaces, orthogonality, bases, Gram-Schmidt procedure, dual space, Riesz representation theorem

Linear operators: the space of bounded linear operators with the operator norm, inverse and adjoint operators, Hermitian operators, infinite matrices, spectrum, compact operators, Hilbert-Schmidt operators, the spectral theorem for compact Hermitian operators.

Additional topics, may include:

- the Rayleigh quotient and variational characterisations of eigenvalues,

- the functional calculus,

- applications to Sturm-Liouville systems.

At level 7, topics will be studied and assessed to greater depth.

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Discrete mathematics has found new applications in the encoding of information. Online banking requires the encoding of information to protect it from eavesdroppers. Digital television signals are subject to distortion by noise, so information must be encoded in a way that allows for the correction of this noise contamination. Different methods are used to encode information in these scenarios, but they are each based on results in abstract algebra. This module will provide a self-contained introduction to this general area of mathematics.

Syllabus: Modular arithmetic, polynomials and finite fields. Applications to

• orthogonal Latin squares,

• cryptography, including introduction to classical ciphers and public key ciphers such as RSA,

• "coin-tossing over a telephone",

• linear feedback shift registers and m-sequences,

• cyclic codes including Hamming,

At level 7, topics will be studied and assessed to greater depth.

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Integrable systems are special dynamical systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems and nonlinear wave equations to difference equations. This module covers the origins of the subject as well as modern topics like integrable maps and lattice equations.

- Liouville integrability in classical mechanics. Hamiltonian mechanics. Canonical symplectic form and Poisson brackets. Liouville's theorem (statement and examples). Lax pairs for finite-dimensional systems.

- Soliton equations. History and physical origins (e.g. Korteweg-de Vries and/or sine-Gordon). Conservation laws. Hamiltonian formalism. Lax pairs.

- Construction of solitons. Introduction to inverse scattering. Darboux-Bäcklund transformations. Hirota's method.

- Discrete integrability. Symplectic maps. Liouville's theorem (discrete version). Integrable lattice equations. Discrete Lax pairs with examples.

At level 7, topics will be studied and assessed to greater depth.

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This module provides an introduction to the study of orthogonal polynomials and special functions. They are essentially useful mathematical functions with remarkable properties and applications in mathematical physics and other branches of mathematics. Closely related to many branches of analysis, orthogonal polynomials and special functions are related to important problems in approximation theory of functions, the theory of differential, difference and integral equations, whilst having important applications to recent problems in quantum mechanics, mathematical statistics, combinatorics and number theory. The emphasis will be on developing an understanding of the structural, analytical and geometrical properties of orthogonal polynomials and special functions. The module will utilise physical, combinatorial and number theory problems to illustrate the theory and give an insight into a plank of applications, whilst including some recent developments in this field. The development will bring aspects of mathematics as well as computation through the use of MAPLE. The topics covered will include: The hypergeometric functions, the parabolic cylinder functions, the confluent hypergeometric functions (Kummer and Whittaker) explored from their series expansions, analytical and geometrical properties, functional and differential equations; sequences of orthogonal polynomials and their weight functions; study of the classical polynomials and their applications as well as other hypergeometric type polynomials.

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This module provides a rigorous foundation for the solution of systems of polynomial equations in many variables. In the 1890s, David Hilbert proved four ground-breaking theorems that prepared the way for Emmy Nöther's famous foundational work in the 1920s on ring theory and ideals in abstract algebra. This module will echo that historical progress, developing Hilbert's theorems and the essential canon of ring theory in the context of polynomial rings. It will take a modern perspective on the subject, using the Gröbner bases developed in the 1960s together with ideas of computer algebra pioneered in the 1980s.

Indicative syllabus:

• Multivariate polynomials, monomial orders, division algorithm, Gröbner bases;

• Hilbert's Nullstellensatz and its meaning and consequences for solving polynomials in several variables;

• Elimination theory and applications;

• Linear equations over systems of polynomials, syzygies.

Level 7 students will cover additional topics such as polynomial maps between varieties.

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There is no specific mathematical syllabus for this module; students will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their coursework assessments (different topics for levels 6 and 7). The coursework is supported by a series of workshops covering various forms of written and oral communication. These may include critically evaluating the following: a research article in mathematics, statistics or finance; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

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