# Mathematics

Mae'r cynnwys hwn ar gael yn Saesneg yn unig.

Module code MA0003 L3 Autumn Semester 10

The module covers the basic manipulative skills in mathematics which are required by students studying a scientific discipline. The module begins with a review of arithmetic skills, before illustrating how algebraic expressions can be manipulated and rearranged. This includes exploration of fractions, powers, brackets, factors etc.

The concept of a function is introduced by investigating graphical techniques and then re-enforced by applying many of the algebraic methods introduced at the start of the module. The properties and graphs of a range of functions are analysed including polynomials, exponentials and trigonometric functions, with a range of examples explored.

Prerequisites:  GCSE Mathematics or its equivalent

### Assessment

• Examination - autumn semester: 85%
• Written assessment: 15%
Module code MA0004 L3 Spring Semester 10

The module begins by introducing some basic notation from set theory and how describing ‘sets’ or ‘events’ in this way can be used to define and calculate probabilities. This leads onto further investigation of some particular probability distributions (discrete and continuous) and their applications. Ways of exploring and illustrating data is also examined, before studying some statistical techniques that can be used to draw conclusions to observations or hypotheses.

Prerequisites: GCSE mathematics or equivalent

### Assessment

• Examination - spring semester: 85%
• Written assessment: 15%
Module code MA0232 L5 Spring Semester 10

This module considers the study of pairs of differential equations. Theoretical analysis, complemented by results obtained using computer simulation, will be used to study models drawn from a variety of disciplines.

### Assessment

• Examination - spring semester: 100%
Module code MA0235 L5 Spring Semester 10

The behaviour of fluid flows is important in a very wide variety of systems.  In weather and climate change studies it is necessary to predict and understand the general motion of both the air and the ocean. In medicine, it is important to know how blood flows in the arteries and the heart, and how air flows in the lungs. For instance, mathematically based simulations of the motion of blood in the heart have recently become sophisticated enough to guide surgeons when they take interventive action to treat various heart problems. In order to design aircraft it is necessary to know how wings can create a lifting force and how so-called viscous skin-friction can increase drag forces. If some means could be found to reduce skin-friction drag forces on aircraft by even just a few per cent, then this would translate into billions of pounds of savings in fuel costs for the airline industry every year.

The fundamental Euler and Navier-Stokes equations of fluid dynamics have been known for about two hundred and fifty years and a hundred and fifty years, respectively. Yet there remain many open and interesting questions about their solutions. This is despite the fact that, using a suitably compact notation, the equations are so short that they can each be written down in two lines.  The behaviour of turbulent flows, for example, can be described by solutions of these equations. Turbulent flows are ubiquitous in the natural world, as well as in engineered systems.  But no systematic means of obtaining turbulent solutions is known. Thus turbulence is an area of work that continues to attract the attention of many thousands of researchers, both in industry and over a range of academic departments within universities.

This module aims to provide students with a first look at the equations that govern the motion of fluids. We will extract a few simple solutions of these equations and discuss how they can be interpreted. To do this we will need to introduce various fundamental notions such as: particle paths; rates of change following the fluid flow (so-called material derivatives); mass and momentum conservation equations; and vorticity, which leads to an important distinction between two possible types of flow.

Prerequisite Modules: MA1300 Mechanics I

Corequisite Modules: MA2301 Vector Calculus

### Assessment

• Examination - spring semester: 100%
Module code MA0261 L5 Spring Semester 20

Operational Research (OR) is the application of advanced analytical methods to help make better decisions.  Often this takes the form of developing a mathematical model of a system under consideration and then using the model to examine and quantify “What if?” type questions in order to improve its performance.

This double module provides an introduction to a number of topics in OR, viz Queueing Theory, Simulation, Linear Programming and Network Analysis.  These topics are orientated towards applications of mathematics in real-life situations.  This module is a prerequisite to certain Year Three modules in OR.

Prerequisite Modules: MA1500 Introduction to Probability Theory

Recommended Modules: MA1501 Statistical Inference

### Assessment

• Examination - spring semester: 90%
• Written assessment: 10%
Module code MA0317 L6 Spring Semester 10

This introductory 10 credit year-three module uses elementary and combinatorial techniques to study how properties of divisibility can be used to characterise the integers. It also touches on some special numbers, such as Bernoulli and Euler numbers, and their corresponding polynomials. At the heart of the syllabus content is the topic of multiplicative arithmetic functions.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function f(n) defined on the set of natural numbers. We say that an arithmetic function which maps the natural numbers onto the complex plane, is multiplicative if f(mn) = f(m)f(n), for all natural numbers m and n with (m, n) = 1. If f(mn) = f(m)f(n) for all integers m and n, then we say that the function f is totally multiplicative.

In addition to multiplicative functions, this module contains sections on division in congruences, solving equations modulo a prime power, quadratic reciprocity, roots of unity, Dirichlet Series, Dirichlet Characters and Mobius inversion.

Pre-requisite module: MA0111 Elementary Number Theory I
Recommended module: MA0216 Elementary Number Theory II

### Assessment

• Examination - spring semester: 100%
Module code MA0322 L6 Autumn Semester 10

Knots are closed strings in three dimensional space. The fundamental question is to decide when two given knots are the same or if a particular knot is equivalent to another or even knotted at all. Knots have been studied by mathematicians for over a century but in the last 25 years a number of new simple ideas have contributed to remarkable breakthroughs which have helped clear up a large number of outstanding problems and conjectures. These ideas have come from a number of branches of mathematics and not only have influenced knot theory itself but have revolutionised several branches of mathematics and even mathematical physics. Applications have also been found in biology in understanding how DNA strands are knotted. This course is an elementary introduction to modern knot theory as it now stands and some of the tools which are now available for understanding knots. The style and emphasis is on using and understanding the tools rather than a traditional definition-theorem-proof approach.

Prerequisite Module: MA0212 Linear Algebra

Recommended Module: MA0213 Groups

### Assessment

• Examination - autumn semester: 100%
Module code MA0332 L6 Spring Semester 10

A lecture based module which develops classical applied mathematical material introduced in Level Two modules and in Autumn Semester Level Three modules.

Prerequisite Modules: MA0235 Elementary Fluid Dynamics, MA2301 Vector Calculus

### Assessment

• Examination - spring semester: 100%
Module code MA0367 L6 Spring Semester 10

This is a lecture based module designed to acquaint students with the principles of fitting time series models to data and with use of such model in forecasting.  The goals of this module are to develop an appreciation for the richness and versatility of modern time series analysis as a tool for analyzing data. This module is aimed at the students who wish to gain a working knowledge of time series and forecasting methods as applied in economics, engineering and the natural and social sciences.

Prerequisite Modules: MA2500 Foundations of Probability and Statistics
Recommended Modules: MA3502 Regression Analysis and Experimental Design

### Assessment

• Examination - spring semester: 90%
• Written assessment: 10%
Module code MA1001 L4 Spring Semester 10

The first part of the module aims to introduce students to first-order differential equations. Calculus techniques will be deployed to find simple solutions of such differential equations. In addition, students will be expected to develop an appreciation of how the solutions can be given a geometric interpretation, even when it is not possible to use calculus techniques to obtain solutions that can be written in a simple form.

The second part of the module is concerned with the solution of second-order differential equations. Manipulative techniques will be used to determine solutions of second-order differential equations for cases where the equation takes a specific and relatively simple form. There will also be some general discussion about the circumstances under which it is possible to know that there is a solution of a differential equation, even if a simple mathematical formula for the solution cannot be obtained.

### Assessment

• Examination - spring semester: 100%
Module code MA1003 L4 Double Semester 20

In the modern world it is imperative for a mathematician to know how to program. This module will give students an introduction to general concepts of programming that should empower them through their degree and beyond.

This module will introduce Students to programming through Python. The module will also teach particularities of programming applied to mathematics.

Prerequiste:  A pass in A-level Mathematics of at least grade A.

### Assessment

• Written assessment: 70%
• Presentation: 30%
Module code MA1004 L4 Autumn Semester 10

This module gives an introduction to elementary plane Euclidean geometry. We present this material in a way which emphasises axiomatic approach, logical thinking and rigorous proofs, as well as careful use of diagrams as an aid to understanding problems and finding solutions. In the latter half of the module we also introduce basic notions of spherical geometry, emphasising the differences between it and Euclidean geometry.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

### Assessment

• Examination - autumn semester: 100%
Module code MA1005 L4 Autumn Semester 20

This module will familiarise you with the basic structures of mathematics: Sets, numbers, basic algebraic structures, and the notion of a limit. Moreover you will learn what a mathematical proof is and how to prove a mathematical statement.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

### Assessment

• Examination - autumn semester: 100%
Module code MA1006 L4 Spring Semester 20

In this module we will study rigorously real functions and their properties, focussing in particular on continuity and differentiability. We will give a mathematical definition of limits at a point, continuity, the derivative and the Riemann integral. We will show how to derive rigorously many of the computational rules already used at A-level.

Particular attention will be given to proving theorems for differentiable functions (as e.g. the Intermediate Value Theorem) and applications to maxima and minima, convexity and concavity. These tools can later be applied to qualitative study of functions and their graphs.

Later in the module we will introduce the Taylor expansion, which allows us to approximate most mathematical functions by polynomials. We will then study the general properties of the Riemann integral in detail, followed by the demonstration of the techniques of integration.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

Precursor Modules: MA1005 Foundations of Mathematics I

### Assessment

• Examination - spring semester: 100%
Module code MA1008 L4 Autumn Semester 10

Linear algebra is one of the foundation stones of mathematics.  The module introduces students to its most fundamental abstract notions: vector spaces, linear subspaces, linear maps, linear independence, and linear spans.  These are accompanied by a plethora of numerical and geometrical examples, firmly establishing the link between the abstract and the concrete.  In particular, the module culminates in the notion of a basis of a vector space, and the demonstration that it identifies every abstract (finite-dimensional) vector space with that of n-tuples of numbers.

### Assessment

• Examination - autumn semester: 100%
Module code MA1300 L4 Spring Semester 10

Classical continuum mechanics is a branch of mechanics, physics, and mathematics concerned with the behaviour of physical bodies which are either moving or at rest under the action of forces. This lecture based module focuses on basic continuum mechanics concepts and in particular on Newton's laws of dynamics, which are presented using modern mathematical tools and are applied to solve a number of mechanical problems taken from the physical world. The module is strongly recommended to all those who intend to pursue further study in applied mathematics, as well as to those interested in the roots of mathematics.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

### Assessment

• Examination - spring semester: 100%
Module code MA1500 L4 Autumn Semester 10

The module begins with the idea of a probability space, which is how we model the possible outcomes of a random experiment. Concepts such as statistical independence and conditional probability are introduced, and a number of practical problems are studied. We then turn our attention to random variables, and look at some well-known probability distributions. Following this we focus on discrete distributions, and introduce the idea of independence for random variables, and the important concept of mathematical expectation. This leads on to the study of random vectors, where we introduce covariance and correlation, conditional distributions and the law of total expectation. Finally, we show how the ideas developed for discrete distributions can be carried over to continuous distributions, and conclude with some approximation theorems.

This is a lecture-based module. Students will be required to demonstrate problem-solving skills throughout the module. No previous knowledge of probability theory is assumed.

The module is intended to prepare students for subsequent modules involving probability and statistics within the degree scheme.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

### Assessment

• Examination - autumn semester: 100%
Module code MA1501 L4 Spring Semester 10

The role of statistics in the modern world is ever increasing and applications can be found in a wide variety of areas including science, industry, government and commerce making a basic understanding of statistics an essential skill.  This is a lecture based module given at an introductory level on statistical inference to develop an understanding of the basic principles of mathematical statistics, used in situations where the full picture of a problem (population) is unknown and must be inferred from collected data (random sample).

This module will be accessible to those who have knowledge of A-level Pure Mathematics and an Introduction to Probability Theory.  It will prepare students for all modules with statistics and probability content in future years of the degree scheme.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

Precursor Module: MA1500 Introduction to Probability Theory

### Assessment

• Examination - spring semester: 100%
Module code MA1800 L4 Autumn Semester 10

This module provides a rigorous knowledge and understanding of basic macro- and micro-economic principles by combining theories and applications to contemporary economic issues. The macro-economic topics include income-expenditure model, measuring GDP, unemployment and inflation, aggregated demand/supply, expenditure multipliers and the Keynesian model, and fiscal and monetary policy. The micro-economic topics include supply/demand, elasticity, utility and indifference curves, production, general equilibrium analysis, market conditions, competition and efficiency, monopoly and oligopoly and operation of economic markets etc.

### Assessment

• Examination - autumn semester: 100%
Module code MA1801 L4 Spring Semester 10

This module provides students with concepts and knowledge of the structure and function of financial systems. It also introduces students with techniques and tools of corporate financial management at firm level. Students need to learn various fundamental topics on value of money, investment appraisal, cost of capital; capital structure and firm valuation; working capital management and capital budgeting; dividend policy. Depending on contemporary debates some further advanced topics may be addressed as well.

### Assessment

• Examination - spring semester: 100%
Module code MA2003 L5 Spring Semester 10

A lecture based module, providing an exposition of the basic theory and methods of complex analysis which are fundamental in mathematics and many of its applications.

The course shows how the concepts of differentiation and integration of real functions can be extended to complex functions. Complex functions map complex numbers to complex numbers. For a special subset of these functions it is possible to define a derivative. These differentiable complex functions have particularly nice properties. The real integral between two points x1 and x2 on the real axis is generalised to a complex integral along a path between two points z1 and z2 in the complex plane. These integrals are called contour integrals. Theorems of Cauchy show how some contour integrals of differentiable complex functions can be evaluated in a beautiful and simple way using methods known as the residue calculus. The residue calculus can be used to evaluate real integrals.

This course is essential for all mathematics students.

### Assessment

• Examination - spring semester: 100%
Module code MA2004 L5 Spring Semester 10

A lecture based module, which deals with fundamental mathematical methods which are essential to all students of mathematics or statistics. In particular the theory of certain important series and transforms is developed.

### Assessment

• Examination - spring semester: 100%
Module code MA2006 L5 Spring Semester 10

This module follows up on the introduction to mathematical analysis provided in the first-year modules MA1005 Foundations of Mathematics I and MA1006 Foundations of Mathematics II. It develops further important basic concepts of analysis, including convergence of functional sequences and series, the interchangeability (or otherwise) of limits, uniform continuity, as well as their applications to the study of functions defined through series, integrals or differential equations. These concepts form the foundation for later courses on Complex Analysis, Differential Equations and Fourier and Functional Analysis.

### Assessment

• Examination - spring semester: 100%
Module code MA2007 L5 Autumn Semester 20

Linear algebra is one of the foundation stones of mathematics. In its first half, this module introduces the students to its most fundamental abstract notions: vector spaces, linear subspaces, linear maps, linear independence, and linear spans. These are accompanied by a plethora of numerical and geometrical examples, firmly establishing the link between the abstract and the concrete. This culminates in the notion of a basis of a vector space, and the demonstration that it identifies every abstract (finite-dimensional) vector space with that of n-tuples of numbers.

We then interpret bases as coordinate isomorphisms with kn and explain how this allows us to write down and manipulate abstract linear maps as matrices. Matrix computation methods are introduced, and studied at length. The students are taught how to compute the kernel and the image of a linear map via matrices, and as an application — how to solve systems of homogeneous and non-homogeneous linear equations. Finally, we study how the matrix of a linear map changes with a change of basis and the standard form that it can be brought to.

### Assessment

• Examination - autumn semester: 100%
Module code MA2010 L5 Autumn Semester 20

This module will be dedicated to transferring all basic notions of calculus of functions of one variable to functions of several variables including limits, continuity, differentiation and integration. The second part of the module extends the calculus of several variables to the description and analysis of vector and scalar fields. There will be an emphasis on ideas and results that can be applied in many areas of mathematical modelling. But the main vector calculus theorems that will be presented are also of significance without regard to any such applications. This is because they can be viewed as being natural extensions – to cases involving more than one-dimension - of the fundamental theorem of calculus which relates the process of integration to that of anti-differentiation.

### Assessment

• Examination - autumn semester: 85%
• Written assessment: 15%
Module code MA2011 L5 Spring Semester 10

We use numbers in two different ways: the natural numbers 1, 2, 3... for counting; and the real numbers - fractions and decimals - for sizes, lengths, areas etc.

Number theory is about problems and methods where you just use the integers, meaning the extension of the natural numbers to include zero and negative numbers also:…-3, -2, -1, 0, 1, 2, 3…You can add, subtract and multiply integers, but you can't always divide exactly, so unless said otherwise, division means finding the quotient and remainder: m goes into n, q times with r left over, n = qm + r. If you fix m (the modulus) you can classify numbers according to the remainder r. This is called congruence or modulo arithmetic.

Addition and multiplication behave very differently in number theory. Given a number n, the equation x + y = n has lots of (integer) solutions, but xy = n has very few solutions, unless n = 0. If n is a prime number, the only solutions are x = 1, y = n or vice versa. The ancient Greeks knew, and proved, there are infinitely many prime numbers.

Number theory began with number puzzles. “Cattle problems" were ancient Babylonian and

Greek recreational maths puzzles - the Sudoku of their time. Now they are part of the number theory we meet in this module, with the name of Diophantine equations.

Number theory used to be the prize example of “useless" pure mathematics. Now, most of the security systems that protect our computers and bank accounts are built around disguising patterns by taking the remainders for some modulus m, and around the difficulty of factorising large numbers - i.e. solving xy = n with x; y > 1. E.g., even with a (simple) calculator it would take most humans quite a while to find that 4294967293 has just two prime factors. The numbers used in practical security are much bigger.

This part 1 introductory 10 credit second-year module contains sections on factorising integers, Euclid’s algorithm, congruence arithmetic, cattle problems, irrationality, and on continued fractions and quadratic equations.

### Assessment

• Examination - spring semester: 100%
Module code MA2013 L5 Autumn Semester 10

A group consists of a set and a binary operation which satisfy certain axioms. Many important classes of mathematical objects can be regarded as groups, some examples being certain symmetry transformations or permutations under the operation of composition, integers under the operation of addition, nonzero rational numbers under the operation of multiplication, and invertible real square matrices of fixed size under the operation of matrix multiplication.

Some of the basic definitions and concepts in group theory were introduced in the Year One module MA1005 Foundations of Mathematics I.  This Year Two module will provide a reinforcement and extension of material from Year One, together with the introduction and study of further important definitions, theorems, proofs and examples.

### Assessment

• Examination - autumn semester: 100%
Module code MA2300 L5 Autumn Semester 10

This module builds on the module Mechanics I (MA1300) by extending the study to general particle motion in 2 and 3 dimensions using vector methods. This is followed by studying systems of 2 and more particles leading to rigid bodies. Conservation principles are discussed and used. Finally, a brief introduction to Lagrangian mechanics is given.

Prerequisite Modules: MA1300 Mechanics I

### Assessment

• Examination - autumn semester: 100%
Module code MA2500 L5 Autumn Semester 20

Knowledge of probability and statistics is useful in many graduate careers. This double module gives students an understanding of the principles underlying statistical methods commonly used by professional statisticians, and is intended to prepare students for a career involving statistical analysis.

The first part of the module begins with the study of probability spaces, random variables and distributions, followed by the theory of mathematical expectation and conditional expectation. We then look at moment generating functions, which are used to prove classical limit theorems such as the law of large numbers and the central limit theorem. The second part of the module begins with a study of parameter estimation, including the notions of consistency and efficiency, and an introduction to Bayesian inference. We then look at the theory of statistical hypothesis testing, focusing in particular on the likelihood ratio test and a number of different non-parametric tests.

Prerequisite Module: MA1500 Introduction to Probability Theory

Recommended Module: MA1501 Statistical Inference

### Assessment

• Examination - autumn semester: 100%
Module code MA2501 L5 Spring Semester 10

This module will introduce the basics of programming in statistical software and will give students the opportunity to apply standard statistical methods to real data.  The module will introduce basic principles of programming like Input/Output of data, defining elements of data in programming, creating small statistical functions for the program.  It will also give the student the opportunity to see how basic statistical theory is applied in real datasets and recognize the different steps one takes in practice.

Prerequisite Modules: MA1501 Statistical Inference

Corequsite Modules: MA2500 Foundations of Probability and Statistics

### Assessment

• Class test: 60%
• Written assessment: 15%
• Written assessment: 25%
Module code MA2701 L5 Autumn Semester 10

Numerical Analysis is concerned with the development of numerical methods to solve mathematical problems in a reliable and efficient way. The ability to compute numerical solutions to mathematical problems has always been an important part of mathematics. For instance, an effective method for the evaluation of the square root of a number was discovered over 3600 years ago. Nowadays, numerical methods are under continuous research and development and are widely used in science, engineering, finance and other areas, to formulate theories, to interpret data, and to make predictions.

This module provides an introduction to computational methods for the approximation of functions on an interval of the real line. We begin with the study of methods and errors associated with the solution of systems of linear equations. We then study the interpolation of functions by polynomials of a given degree, and use these techniques for the derivation of numerical integration rules and their error analysis. In particular, we shall describe the use of orthogonal polynomials for the construction of polynomials of best approximation as well as their relevance in deriving Gauss-type numerical integration rules. We shall also turn our attention to interpolation by piecewise polynomials of low degree, known as splines. These are the building blocks for the construction of more advanced numerical methods, and require only minimal prerequisites in differential and integral calculus, differential equations, and linear algebra. Experience of numerical computation is essential for a true understanding of the applications and limitations of numerical methods, and numerous examples will be presented to explain and interpret the theoretical results.

### Assessment

• Examination - autumn semester: 100%
Module code MA2800 L5 Spring Semester 10

This module extends MA1801 Finance I (Financial Markets and Corporate Financial Management) and focuses on financial markets, investments and instruments. The key topics include investment risk/returns, portfolio analysis, capital asset pricing model (CAPM) and arbitrage pricing theory (APT), Efficient Market Hypothesis (EMH), equity dynamics and valuation, bond valuation (e.g. duration analysis, term structure determination and yield spreads), and interest rate and credit risk analysis.

Prerequisite Module: MA1801 Finance I: Financial Markets and Corporate Financial Management

### Assessment

• Examination - spring semester: 80%
• Written assessment: 20%
Module code MA2801 L5 Autumn Semester 10

This 10 credit module aims to provide students with an understanding and appreciation of recent developments in empirical finance. The module focuses on empirical modelling techniques and the interpretation of empirical results. It will cover volatility modelling with GARCH and its extensions; asset pricing and factor models; empirical analysis of risk; panel data analysis; Logit and Probit analysis for corporate finance; the econometrics for regime shifts/switching; equity dynamics and returns; empirical analysis of finance-macroeconomics nexus (interest rate, inflation, the housing market etc.).

### Assessment

• Examination - autumn semester: 70%
• Written assessment: 30%
Module code MA3000 L6 Spring Semester 10

A lecture based module  covering some advanced topics in complex analysis. The module aims to cover topics which are of particular relevance to spectral theory, differential equations and special functions.

Prerequisite Modules: MA2003 Complex Analysis

Recommended Modules: MA0221 Analysis III or MA2006 Real Analysis

### Assessment

• Examination - spring semester: 100%
Module code MA3005 L6 Autumn Semester 20

The double module introduces students to some of the techniques of modern analysis which are indispensable tools to the present-day mathematician. The expansion of functions in Fourier series (if the function is defined on a bounded interval or periodic) or Fourier integrals is a very efficient method for solving a variety of problems in pure and applied mathematics – compared to power series expansion, it works under very weak assumptions on the regularity of the function. Indeed, even discontinuous functions can reasonably be expanded in a Fourier series, an observation which led to the modern definition of the concept of a function and to the development of mathematical analysis during the 19th and 20th centuries. The desire to give a satisfactory answer to the question which functions have a Fourier expansion, and in what sense, led to the abstract notions of normed vector spaces and Hilbert spaces, which have become the foundation of modern analysis and are used in all areas of mathematics. The fundamental idea is to try and extend the framework of linear algebra (matrix theory) to the study of more complicated linear operators, such as differential operators. This requires an infinite-dimensional setting, and ideas of analysis such as convergence and continuity become important. The aim of the course is to study Fourier series and integrals, with emphasis on conditions ensuring their pointwise, uniform or mean convergence, and to give an introduction to the more general theory of functional analysis, illustrated with some further applications.

Prerequisite Modules: MA0212 Linear Algebra and either MA0221 Analysis III or MA2006 Real Analysis

### Assessment

• Examination - autumn semester: 100%
Module code MA3006 L6 Autumn Semester 20

This double module introduces the fundamentals of coding theory and data compression.

The first part is devoted to coding theory and will mainly focus on error-correcting codes, their properties and applications. No document or computer files can be guaranteed free from error.  Error-correcting codes are used to spot mistakes and suggest the most likely correction. If the rate of errors is such that several mistakes are likely in a single ‘word’ (e.g. radio transmissions), then the codes used are more combinatoric.  If errors are so rare that having two mistakes in the same ‘word’ is very unlikely (e.g. brand new computer disc), then the codes used are more algebraic. Many error-correcting codes correspond to geometrical patterns.

The second part of the module deals with the broad field of data compression. We will first study lossless compression schemes, including the fundamental algorithms of Shannon, Huffman, Lempel-Ziv and arithmetic coding. Finally, the module will give the basic principles of lossy compression, such as quantization and transform coding. For instance, we will see the role wavelets (“the mathematical microscope”) play in data compression.

### Assessment

• Examination - autumn semester: 100%
Module code MA3008 L6 Spring Semester 10

Topology is a subject of fundamental importance in many branches of modern mathematics.  Basically, it concerns properties of objects which remain unchanged under continuous deformation, which means by squeezing, stretching and twisting, but not cutting.  Apples and oranges are topologically the same, but you can’t deform an orange into a doughnut! More precisely, we can never deform in a continuous way a sphere (the surface of an orange) into a torus (the surface of a doughnut). Knots are also examples of topological objects, where a trefoil knot can never be deformed into an unknotted piece of string. It's the business of topology to describe more precisely such phenomena.

The aim of this module is to explore properties of topological spaces. To distinguish topological spaces we will consider topological invariants such as the fundamental group, which is a powerful way of using an algebraic invariant to detect topological features of spaces.

Prerequisite Modules: MA0213 Groups

### Assessment

• Examination - spring semester: 100%
Module code MA3009 L6 Autumn Semester 10

The purpose of this module is to show students to how ideas from multivariable calculus and linear algebra can be put together to form an invariant language which is quite useful in understanding the structure of very naturally occurring geometric objects in mathematics.

Recommended Module: MA0212 Linear Algebra

### Assessment

• Examination - autumn semester: 100%
Module code MA3301 L6 Spring Semester 10

This module provides an introduction to nonlinear systems and their applications in modelling. The aims of the module are:

• To introduce students to various aspects of the mathematical theory of nonlinear systems
• To illustrate the use of nonlinear systems in mathematical modelling of various phenomena, particularly those that involve physical oscillations
• To describe the qualitative changes in the behaviour of solutions of nonlinear systems that can arise when a system parameter is varied

Prerequisite Modules: MA0232 Modelling with Differential Equations

### Assessment

• Examination - spring semester: 100%
Module code MA3303 L6 Autumn Semester 20

Partial differential equations are a central modelling tool in applied mathematics and mathematical physics. They also play an important role in pure mathematics, not least as a stimulus in the development of concepts and methods of classical and modern analysis.

This double module provides an introduction to the classical analytical treatment of second-order linear partial differential equations and techniques for their numerical solution. The essential concepts and methods are introduced and developed for prototype partial differential equations representing the three classes: parabolic; elliptic; hyperbolic. Finite difference and finite element approximations to the solutions of partial differential equations are developed. The accuracy and stability of the numerical schemes are investigated. Direct and iterative methods for solving the linear systems arising from the numerical approximation of partial differential equations are described.

Recommended Modules: MA0212 Linear Algebra, MA0232 Modelling with Differential Equations, MA2301 Vector Calculus

### Assessment

• Examination - autumn semester: 85%
• Written assessment: 15%
Module code MA3304 L6 Spring Semester 20

The purpose of this double module is to consolidate students’ knowledge of and skills in modelling, analysis and applications. The module therefore is situated on the interface between Pure and Applied Mathematics and encompasses three important themes relevant to investigating physical phenomena, which will be addressed in series.

Theme 1. Asymptotic Methods

Many mathematical problems contain a small or large parameter that may be exploited to produce approximations to integrals and solutions of differential equations, for example. This theme provides an introduction to asymptotic approximations and perturbation analysis and their applications. Such techniques are important in almost every branch of applied mathematics especially those where exact analytic solutions are not available and numerical solutions are difficult to obtain.

Theme 2. Integral Equations

Many mathematical problems, particularly in applied mathematics, can be formulated in two distinct but related ways, namely as differential equations or integral equations. In the integral equation approach the boundary conditions are incorporated within the formulation of the problem and this confers a valuable advantage to the approach. The integral approach leads naturally to the solution of the problem in terms of an infinite series, known as the Neumann expansion. Integral equations have played a significant role in the history of mathematics. The Laplace and Fourier transforms are examples of integral equations. Another interesting problem is Huygens’ tautochrone problem, which is a special case of Abel’s integral equation. This course is concerned for the most part with linear integral equations. This module will introduce different types of integral equations and develop methods for their analysis and solution.

Theme 3. Calculus of Variations

What is the shortest distance between two points on a surface?  What is the shape of maximum area for a given perimeter?  These are two questions of the many that can be answered using calculus of variations.  The central problem involves an integral containing an unknown function – for example the length of a curve can be expressed as an integral along that curve.  Calculus of variations provides techniques for investigating minima of such integral functionals, which usually represent some physically or geometrically meaningful quantity. One example of great importance in modern technology is the use of minimisation in studying complex patterns observed under some conditions in shape-memory alloys. The course will consider the classical ``indirect'' approach to minimisation problems, through finding solutions of some related differential equations. However, due to some inherent (and indeed physically relevant) limitations of this method, which will become evident during the course, one has to combine it with a ``direct’’ variational technique. The power of the direct method spreads far and wide across the modern applications of mathematics. In particular, it provides a key to various techniques for finding approximate solutions to differential equations.

This module can be taken by any student who is prepared to solve some differential equations and manipulate integrals.  Although some of the problems studied are of a physical origin, these will be presented in a self-contained way and there are no applied mathematics pre-requisites.

### Assessment

• Examination - spring semester: 100%
Module code MA3502 L6 Autumn Semester 20

Regression analysis is arguably the most widely used in practice statistical tool. Fundamentals of regression analysis are thus the must for every student who will be seeking a statistics-related job. In a similar vein, the methods and principles of designing experiments are extremely important and regularly used by practitioners in a variety of disciplines. All the theoretical discussions are accompanied with solving practical problems.

Prerequisite Modules: MA1501 Statistical Inference

### Assessment

• Examination - autumn semester: 100%
Module code MA3503 L6 Spring Semester 20

Stochastic processes play a key role in analytical finance and insurance, and in financial engineering. This course presents the basic models of stochastic processes such as  Markov chains, Poisson processes and Brownian motion. It provides an application of stochastic processes in finance and insurance. These topics are oriented towards applications of stochastic models in real-life situations.

Prerequisite Modules: MA2500 Foundations of Probability and Statistics

### Assessment

• Examination - spring semester: 90%
• Written assessment: 10%
Module code MA3504 L6 Autumn Semester 10

This module is designed to provide you with the specialist skills and knowledge which are central to the conducting professional statistical work in civil service. The module introduces standard methods of drawing samples from finite populations, making inferences about population characteristics, theory in survey inference, the methodology for survey based estimation for population totals and related quantities, regression estimation for modelling relationships between variables with an emphasis on practical considerations, the principles and methods used to compensate for non-response following survey data collection, calibration methods for household surveys, the theory of index numbers as a means of making price and quantity comparisons.

Several lectures will be delivered by practitioners from the Office of National Statistics who will describe the case studies with application of official statistics.

Prerequisite Modules:  MA1501 Statistical Inference

Recommended Modules: MA2500 Foundations of Probability and Statistics

(In the academic year 2017/18 this module will only be available to students registered on the degree schemes BSc Mathematics, Operational Research and Statistics or MORS Mathematics, Operational Research and Statistics.  For the academic year 2018/19 it is also planned to make this module available to students registered on the degree scheme BSc Financial Mathematics).

### Assessment

• Examination - autumn semester: 90%
• Written assessment: 10%
Module code MA3505 L6 Autumn Semester 10

This module will introduce the basics of multivariate statistical analysis to students.  The first few weeks the module deals with classic multivariate topics.  The last few weeks the module presents modern multivariate tools for classification and clustering and dimension reduction.  Throughout the semester the students will also use the lab to learn how to apply the techniques taught in class on large datasets using statistical software.

The goal of the module is to help the students get a broad knowledge of how to handle multivariate problems.  The module is aimed to students with an OR/Stats degree who will likely encounter multivariate data in their careers as these become the norm in most real life problems.

Prerequsite Modules: MA2500 Foundations of Probability and Statistics

(In the academic year 2017/18 this module will only be available to students registered on the degree schemes BSc Mathematics, Operational Research and Statistics or MORS Mathematics, Operational Research and Statistics.  For the academic year 2018/19 it is also planned to make this module available to students registered on the degree scheme BSc Financial Mathematics).

### Assessment

• Examination - autumn semester: 75%
• Written assessment: 25%
Module code MA3602 L6 Spring Semester 10

This module examines the behaviours and properties of various algorithms that can be used for solving combinatorial operational research problems. Primarily the focus is on problems involving networks and graphs, though other sorts of problems such as packing and partitioning are also considered.

A number of problems are looked at, including graph connectivity, shortest path problems, spanning trees, maximum flow problems, matchings, routing problems, bin packing problems, and graph colouring problems. It is shown that some of these problems can be solved using efficient exact algorithms, while others require heuristic techniques and/or approximation algorithms. Hence key concepts surrounding intractability (NP-completeness) are also considered.

The module also considers a number of general purpose heuristic algorithms for intractable problems including metaheuristics. In particular, methods such as neighbourhood search, backtracking, simulated annealing, and evolutionary algorithms are considered.

Recommended Modules: MA0261 Operational Research

(In the academic year 2017/18 this module will only be available to students registered on the degree schemes BSc Mathematics, Operational Research and Statistics or MORS Mathematics, Operational Research and Statistics.  For the academic year 2018/19 it is also planned to make this module available to students registered on the degree scheme BSc Financial Mathematics).

### Assessment

• Examination - spring semester: 100%
Module code MA3603 L6 Autumn Semester 20

This double module is an introduction to the mathematical foundations of optimisation and integer programming. In these topics one seeks to minimize or maximize a real function of real or integer valued variables, subject to constraints on the variables (equations or inequalities that need to be satisfied by all solutions). The main goals of the module are to describe the main mathematical properties of the optimisation problems, to introduce algorithms for finding optimal solutions and to show how these algorithms can be applied to a selection of classical optimisation problems.

Recommended Modules: MA0261 Operational Research

### Assessment

• Examination - autumn semester: 100%
Module code MA3604 L6 Spring Semester 10

This module introduces students to the mathematical study of multiple interactive agent decision making. This is an introduction to Game Theory through notions such as Nash Equilibria and Evolutionary Game Theory. Students will learn Game Theory in an active way through role playing and student-led activities.

### Assessment

• Examination - spring semester: 75%
• Written assessment: 25%
Module code MA3606 L6 Spring Semester 10

In this module we explore techniques for modelling systems that exhibit uncertainty. Markov chains will be the principle tool, but we will also consider some important related processes such as renewal processes and diffusion processes.

Queueing theory is an important application of Markov chains, and this module will build on the material covered in MA0261. The module will also include numerous other applications, for example to problems such as insurance claims, epidemic growth, reliability, inventory management, and stock prices. Many of the exercises will involve simulation or require numerical solution methods, so it is recommended that students have some programming experience.

Prerequisite Modules: MA0261 Operational Research

(In the academic year 2017/18 this module will only be available to students registered on the degree schemes BSc Mathematics, Operational Research and Statistics or MORS Mathematics, Operational Research and Statistics.  For the academic year 2018/19 it is also planned to make this module available to students registered on the degree scheme BSc Financial Mathematics).

### Assessment

• Examination - spring semester: 100%
Module code MA3700 L6 Spring Semester 10

Recent tremendous technical advances in processing power, storage capacity, and inter-connectivity of computer technology are creating unprecedented quantities of digital data. Data mining (also known as Knowledge Discovery in Data, or KDD), the science of extracting useful knowledge from such huge data repositories, has emerged as a young and interdisciplinary field. Data mining techniques have been widely applied to problems in industry, science, engineering and government, and it is widely believed that data mining will have profound impact on our society.

This module provides an introduction to the basic ideas and methods of mathematical data mining. In this course, we will consider the following problems: classification, cluster and outlier analysis, mining time-series and sequence data, text mining and web mining, pattern analysis.

A lecture-based module open to all students with a suitable grounding. It covers the fundamental data mining ideas (clustering, support vector machine analysis, semi-supervised learning, information retrieval, collaborative filtering, harmonic analysis) and the most important algorithms (the k-means algorithm,  support vector machines,  PageRank algorithm, k-nearest neighbour classification, Naive Bayes).

### Assessment

• Examination - spring semester: 100%