# Mathematics

Learn more about the modules study abroad students can take at the School of Mathematics.

Module code | MA0111 |
---|---|

Level | L4 |

Semester | Spring Semester |

Cerdits | 10 |

How do we determine the factors of an integer (a “whole number''), such as 120? There are far better methods than testing for divisibility by 2, 3, 4, 5, and so on, in turn. It is more efficient to invoke the factorisation of 120 as a product of primes.

A related question is that of determining the highest common factor of two numbers. A simple-minded approach might be to list the factors of both and then determine the largest number appearing in both lists. Better methods exist among those resting on prime factorisation, but a computationally more efficient way is that embodied in Euclid's Algorithm, which does not refer to primes and does not depend on factorising either number individually.

Euclid's Algorithm is also of great theoretical importance: it underlies the “Fundamental Theorem of Arithmetic'', to the general effect that if a prime divides a product then it divides one of the factors. This underlies the general proof of the (experimentally verifiable, but also provable) observation that any number is expressible as a product of primes in a unique way, apart from the variations induced by re-ordering the factors. It also provides an efficient procedure for finding the solutions *x, y* of an equation *ax + by = c* when integers *a, b*, and *c* are given.

An ancient observation due to Pythagoras (or some member of his school) is that there is no rational number (ratio of integers) *a / b* whose square is 2. One can prove the analogue for integers *n* other than 2, provided that *n *is not the square of an integer. Other simple examples of irrationalities include ratios of logarithms such as log 2 / log 3.

Many questions relating to integers are best expressed in congruence notation: we say that *a* is congruent to *b* mod *m* when *a *differs from *b* by a multiple of *m*. Thus *ax* is congruent to *c* mod *b* precisely when the equation *ax + by = c* can be satisfied. A simple instance in which congruence theory is a helpful concept is in the proposition that an integer is divisible by 9 precisely when the sum of its decimal digits is divisible by 9.

The theory of polynomials in a single variable is in many ways analogous to that relating to the integers. There is a Euclidean Algorithm for the highest common factor of two polynomials, *a*(*x*) and *b*(*x*), say. This is related to the partial fraction decomposition of 1 / *a*(*x*)*b*(*x*), though (as in the analogous problems about integers) there may in simple cases be easier ways of determining this decomposition.

It is known but not very easy to prove that a polynomial equation of degree *d* with real (or integer) coefficients has exactly *d* solutions in complex numbers, whence it has at most *d* real solutions. This last fact can be proved quite easily, by a method which also shows that a polynomial congruence of degree *d* mod *p*, where *p* is prime, has at most *d *solutions that are not congruent mod *p* to each other.

When a polynomial has integer coefficients it is in general not easy to decide whether it factorises into two such polynomials. It is however straightforward to determine all its linear factors. In this way we can quickly determine whether a given cubic polynomial factorises as a product of two polynomials with integer coefficients (it is guaranteed to factorise as a product of two polynomials with real but possibly irrational coefficients, but this is not the same question).

It is easily observed that the sequence of primes thins out as the numbers involved become larger: in the first ten numbers we find the primes 2, 3, 5, 7, but we never again find so many primes in an interval of length 10, because too many of them will be divisible by 2, 3 or 5. This raises the question whether the primes thin out so rapidly that there are only finitely many. The answer is that they do not, as was first shown by Euclid by a simple argument.

Also of interest is the occurrence of primes in sequences defined by simple expressions such as n^{4}+n^{2}+1 or 2^{n}-1. Some of the questions raised in this way remain unsolved, but a certain amount can be said by using considerations from polynomial algebra, in these instances using the polynomials x^{4}+x^{2}+1 and x^{n}-1.

The Pigeonhole Principle (or Box Principle) states that if *n* + 1 objects are each placed in one of *n* boxes then at least one box must contain more than one object. This simple observation is extraordinarily useful if applied in an appropriate way. A simple example of this occurs in a proof that any recurring decimal represents a rational number (a number that is the ratio of two integers). Another occurs in Dirichlet's Theorem on approximation of irrationals by rationals, which leads to a criterion for irrationality that is sufficient, for example, to prove that the number *e* is irrational.

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

### Assessment

- Examination - spring semester: 100%

Module code | MA0212 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 10 |

A lecture-based module, open to all students with suitable grounding. Vectors in geometry are lines with arrows (representing translations in space), added by the parallelogram law. Vectors in algebra are anything that can be modelled by lines with arrows in geometry, obeying certain rules. A vector space is all vectors that can be constructed from some given set of vectors using these rules; it has a ‘dimension’. A low-dimension space can sit inside a high-dimension space as a subspace. The process of modelling one vector space by another is performed by a ‘linear map’ ; it too obeys certain rules. Pairs of vectors can be related like forces and distances in Physics, with a dot product representing work done, or a quadratic form representing stored energy.

The aim of linear algebra is to recognise when these models are possible, and to choose the coordinate system to make everything as simple as possible.

### Assessment

- Examination - spring semester: 100%

Module code | MA0213 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 10 |

A group consists of a set together with a binary operation which satisfies certain axioms. Many important classes of mathematical objects can be regarded as groups, some examples being symmetry transformations together with the operation of composition, integers together with the operation of addition, and invertible real matrices together with the operation of matrix multiplication.

This module will provide an introduction to some of the fundamental concepts of group theory. In particular, various general definitions and theorems will be studied, and then illustrated using specific examples.

### Assessment

- Examination - autumn semester: 100%

Module code | MA0216 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 10 |

How can we describe the solutions in positive integers of x^{2 }+ y^{2 }= z^{2} ? This is “classical'' (there is a connection with Pythagoras' Theorem). Analogous questions include the solution of x^{2 }+ y^{2 }= Kz^{2}, with K = 2 or 3, for example. When K = 3 there are no solutions: this can be shown using congruence considerations and Fermat's notion of “infinite descent'', which is just mathematical induction expressed in a different way.

A different but related question is: which integers n (not necessarily squares) are representable as x^{2 }+ y^{2 }, and, if they are, then in how many ways? An illuminating way of considering this question is via the study of Gaussian Integers x + iy, where i^{2} = - 1. This involves the study of Gaussian Primes and the uniqueness of factorisation of Gaussian Integers as products of Gaussian Primes.

Fermat's “Little'' Theorem (nothing to do with “Fermat's Last Theorem'') says that a^{p} -a is divisible by p when p is prime. Historically, this result seems to have arisen from questions involving “perfect'' numbers n (whose factors sum to 2n), but the result has far greater significance elsewhere. It is needed, for example, in a characterisation of those primes p which divide numbers of the form n^{2}+1 (they are precisely those p which are not of the form 4k+3), a fact which is indispensable for a study of Gaussian Primes and Sums of Two Squares.

**Prerequisite Modules:** MA0111 Elementary Number Theory I

### Assessment

- Examination - spring semester: 100%

Module code | MA0221 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 10 |

This module follows up on the introduction to mathematical analysis provided in the first-year. 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.

**Prerequisite Modules:** MA0123 Analysis I, MA0126 Analysis II

### Assessment

- Examination - autumn semester: 100%

Module code | MA0232 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 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 - autumn semester: 100%

Module code | MA0235 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 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 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 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 third level 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 | MA0276 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 10 |

This module assumes knowledge of the basic concepts of spreadsheets and how they can be used to manipulate information. It then builds on this to cover the automation of tasks using macros and the use of Visual Basic programming within Microsoft Excel, thus enabling the construction of customised, user-friendly interfaces for a spreadsheet. A variety of Operational Research problems are used as the basis for this module, although no prior knowledge of OR is required.

Topics covered include simulation, logical programming ideas, algorithm design and debugging. This module can be taken successfully by any student who is prepared to learn the basics of computer programming, and who wishes to learn some practical problem solving skills which may be of benefit in future employment.

**Prerequisite Modules:** MA1003 Computing for Mathematics

### Assessment

- Written assessment: 40%
- Written assessment: 60%

Module code | MA0291 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 10 |

To give an appreciation of the nature and significance of Accounting in the private sector of the economy by an examination of the contribution it can make to the internal administration and external financing of a firm. This module also highlights the pivotal role of accounting as a service activity within a broad business context.

### Assessment

- Examination - spring semester: 100%

Module code | MA0322 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 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.

**Pre-requisite Module:** MA0212 Linear Algebra

### Assessment

- Examination - autumn semester: 100%

Module code | MA0332 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 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 | MA0358 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 10 |

This module assumes knowledge of the Simplex method for linear programming. The theory is extended to consider more complex problems, in particular instances where variables must be integer (integer programming), and analysis of the optimal solution and what-if analysis (post-optimal and parametric programming).

The second part of the module considers dynamic programming, firstly for solving shortest route problems and then, for machine replacement and production problems. The work is then extended to solve stochastic problems, where variables are not known exactly but relate to a probability.

This module includes techniques for formulating problems and recognising problems which can be solved easily. Ideas such as genetic algorithms and descent methods are introduced as means of solving problems which cannot easily be solved using linear or dynamic programming.

Emphasis is given throughout this module to practical and realistic problems.

**Recommended Modules:** MA0261 Operational Research

### Assessment

- Examination - autumn semester: 100%

Module code | MA0367 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 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: 85%
- Written assessment: 5%
- Class test: 10%

Module code | MA0391 |
---|---|

Level | L6 |

Semester | Double Semester |

Cerdits | 20 |

This double module provides an opportunity to undertake, with supervision, a relatively substantial piece of project work relevant to the student's scheme of study.

A wide range of projects will be offered to students. Some projects will require the student to engage in a detailed study of mathematical theories or techniques in an area of current interest. Other projects will be centred on specific problems that require the formulation of a mathematical model, its development and solution.

### Assessment

- Report: 85%
- Presentation: 15%

Module code | MA0392 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 10 |

This module provides an opportunity to undertake, with supervision, a piece of project work relevant to the student's scheme of study.

A range of projects will be offered to students. Some projects will require the student to engage in a study of mathematical theories or techniques in an area of current interest. Other projects will be centred on specific problems that require the formulation and development of a mathematical model.

### Assessment

- Report: 85%
- Presentation: 15%

Module code | MA0392 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 10 |

This module provides an opportunity to undertake, with supervision, a piece of project work relevant to the student's scheme of study.

A range of projects will be offered to students. Some projects will require the student to engage in a study of mathematical theories or techniques in an area of current interest. Other projects will be centred on specific problems that require the formulation and development of a mathematical model.

### Assessment

- Report: 85%
- Presentation: 15%

Module code | MA1001 |
---|---|

Level | L4 |

Semester | Spring Semester |

Cerdits | 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 |
---|---|

Level | L4 |

Semester | Double Semester |

Cerdits | 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 through Sage; an open source mathematics package built on Python.

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

### Assessment

- Class test: 40%
- Written assessment: 30%
- Presentation: 30%

Module code | MA1004 |
---|---|

Level | L4 |

Semester | Autumn Semester |

Cerdits | 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 | MA1006 |
---|---|

Level | L4 |

Semester | Spring Semester |

Cerdits | 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 | MA1300 |
---|---|

Level | L4 |

Semester | Spring Semester |

Cerdits | 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 |
---|---|

Level | L4 |

Semester | Autumn Semester |

Cerdits | 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 |
---|---|

Level | L4 |

Semester | Spring Semester |

Cerdits | 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 Modules: **MA1500 Introduction to Probability Theory

### Assessment

- Examination - spring semester: 100%

Module code | MA2001 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 10 |

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.

### Assessment

- Examination - autumn semester: 85%
- Written assessment: 15%

Module code | MA2002 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 10 |

A lecture based module, it provides an introduction to the manipulative parts of matrix algebra, and is essential for further work in all areas of mathematics.

### Assessment

- Examination - autumn semester: 100%

Module code | MA2003 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 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 x_{1} and x_{2} on the real axis is generalised to a complex integral along a path between two points z_{1} and z_{2 }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 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 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 | MA2005 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 10 |

Building upon a general understanding of the form and usefulness of ordinary differential equations and knowledge of elementary solution methods, this module explores the mathematical foundations of ordinary differential equation theory as well as methods for the asymptotic and qualitative study of their solutions.

It is an intriguing observation that only a very small number of types of differential equation can be solved in terms of the well-known elementary functions. Differential equations are therefore a fruitful source of new functions and thus are of great practical value in applications and remain of continuing interest. However, this also means that mere knowledge of techniques for the explicit solution of differential equations will not reach very far.

It is therefore essential to have a theoretical framework which ensures the existence of solutions of ordinary differential equations without the need to find them explicitly, and to study the uniqueness and continuous dependence of solutions on parameters of the equation. In the presence of singularities, the asymptotic behaviour of solutions is very valuable information. A further aspect of the qualitative study of ordinary differential equations is the question of stability: will nearby starting points lead to wildly different solutions (chaos), or will the solutions approach a fixed point or an attractive set of more complicated structure, e.g. a limit cycle?

The module will provide an introduction to the existence theory of ordinary differential equations and to fundamental techniques of the asymptotic and qualitative study of their solutions.

### Assessment

- Examination - spring semester: 100%

Module code | MA2300 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 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 | MA2301 |
---|---|

Level | L5 |

Semester | Spring Semester |

Cerdits | 10 |

The module extends the calculus of several variables (introduced in MA2001) 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 - spring semester: 85%
- Written assessment: 15%

Module code | MA2500 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 20 |

The first part of the module begins with the study of probability spaces, random variables and distribution functions. We then look at various transformations of random variables, which reveals deep connections between many well-known probability distributions. Following this, we develop the theory of mathematical expectation and conditional expectation. The first part of the module concludes with the study of moment generating functions and characteristic functions, which are then 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, along with an introduction to Bayesian inference. We then develop the theory of statistical hypothesis testing, focusing in particular on the likelihood ratio test. We then introduce order statistics, and proceed to study a number of different non-parametric tests and their various applications. The second part of the module concludes with a brief look at linear models, including least squares and logistic regression.

Knowledge of probability and statistics is useful in many graduate careers. This 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.

**Prerequisite Modules: **MA1500 Introduction to Probability Theory, MA1501 Statistical Inference

### Assessment

- Examination - autumn semester: 100%

Module code | MA2700 |
---|---|

Level | L5 |

Semester | Autumn Semester |

Cerdits | 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.

**Recommended Modules: **MA0123 Analysis I, MA0126 Analysis II

### Assessment

- Examination - autumn semester: 100%

Module code | MA3000 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 10 |

A lecture based module showing some of the ways in which Complex Function Theory developed in the nineteenth century.

The module will show how contour integration may be used to sum certain types of infinite series and to evaluate real integrals involving logarithms or non-integer powers, how Riemann surfaces may be used to represent multivalued functions and how complex functions can be used to map one region of the complex plane to another.

The final topic will be elliptic functions which may be seen as a generalisation of real periodic functions to complex functions which may have two periods.

**Prerequisite Modules:** MA2003 Complex Analysis

**Recommended Modules:** MA0221 Analysis III

### Assessment

- Examination - spring semester: 100%

Module code | MA3003 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 10 |

A group consists of a set together with a binary operation which satisfies certain axioms, and a ring or field consists of a set together with two binary operations which satisfy certain axioms. Many important classes of mathematical objects can be regarded as groups, rings or fields. For example, permutations or symmetry transformations together with the operation of composition form groups, the integers together with the operations of addition and multiplication form a ring, and the rational, real or complex numbers together with the operations of addition and multiplication form fields.

In this module, various general definitions and theorems in group, ring and field theory will be studied, and then illustrated using specific examples. Students will thereby be exposed to some of the fundamental structures and concepts of abstract algebra.

**Recommended Modules: **MA0212 Linear Algebra

### Assessment

- Examination - autumn semester: 100%

Module code | MA3004 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 10 |

Combinatorics is the branch of discrete mathematics concerned with the theory of arranging objects according to specified rules. The objects can be material (such as people in a group or cards from a pack) or abstract (such as numbers, symbols, steps in a process or choices in a procedure). A frequent aim of combinatorics, when applied to particular cases, is to determine the number of arrangements, but without actually listing them. Accordingly, combinatorics is sometimes regarded as being the study of counting or enumeration. However, some other questions which can be addressed by combinatorics are whether certain arrangements are possible at all, and, if so, what an optimal way of obtaining them might be. Also, in many cases the arrangements will depend on variables, and an aim is often then to study the number of arrangements as a function of these variables and, if possible, obtain an explicit formula for that counting function.

In this module, various general principles and methods of combinatorics will be studied, and then applied to several important enumeration problems, including some in graph theory.

### Assessment

- Examination - spring semester: 100%

Module code | MA3005 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 20 |

The course 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 19^{th} and 20^{th} 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, MA0221 Analysis III

### Assessment

- Examination - spring semester: 100%

Module code | MA3006 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 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 | MA3301 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 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 - autumn semester: 100%

Module code | MA3303 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 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 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.

**Prerequisite Modules:** MA0122 Analysis I, MA0126 Analysis II

**Recommended Modules:** MA0212 Linear Algebra, MA0221 Analysis III, MA0232 Modelling with Differential Equations

### Assessment

- Examination - spring semester: 85%
- Written assessment: 15%

Module code | MA3304 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 20 |

The purpose of the 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 - autumn semester: 100%

Module code | MA3501 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 10 |

Is the arithmetic average the best way of estimating the mean of a probability distribution? Is Student's t-test the best way of testing null hypotheses about the mean? The answers to these questions are assumed to be yes in elementary statistics, this module shows that there is a firm mathematical basis for this assumption.

The first part of the module is a study of methods of estimation of parameters of probability distributions.Brief comparisons are made of maximum likelihood estimation, the method of moments approach and Bayesian inference. The properties that are desirable in estimators are identified, and by using a series of results it is shown that under fairly general conditions maximum likelihood estimators have optimal properties. It is even possible, for some statistical estimation problems, to identify estimators that are unbiased (correct on average, in the long run) and have a smaller theoretical variance than any other unbiased estimator.

The second part of the module covers the testing of hypotheses in statistics. It is shown how optimal statistical tests can be devised and a link is made with maximum likelihood.

The module presents a coherent view of estimation and hypothesis testing in a firm theoretical framework.

**Prerequisite Modules:** MA2500 Foundations of Probability and Statistics

### Assessment

- Examination - autumn semester: 100%

Module code | MA3502 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 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: **MA0152 Statistical Inference I

### Assessment

- Class test: 15%
- Examination - spring semester: 85%

Module code | MA3503 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 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

- Class test: 10%
- Written assessment: 5%
- Examination - autumn semester: 85%

Module code | MA3700 |
---|---|

Level | L6 |

Semester | Spring Semester |

Cerdits | 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: 85%
- Class test: 15%

Module code | MA3900 |
---|---|

Level | L6 |

Semester | Autumn Semester |

Cerdits | 20 |

Mi fydd y modiwl yma yn darparu cyflwyniad i fyfyrwyr israddedig i addysgu mathemateg mewn ysgol uwchradd trwy gyfrwng y Gymraeg. Mi fydd myfyrwyr yn datblygu eu dealltwriaeth o addysg mathemateg ac o strwythur y cwricwlwm mathemateg ar lefel cyflwyniadol (israddedig).

### Assessment

- Presentation: 10%
- Written assessment: 10%
- Written assessment: 0%
- Written assessment: 15%
- Portfolio: 30%
- Written assessment: 30%
- Written assessment: 5%