Mathematics

Mathematics including a Foundation Year - BSc (Hons)

UCAS code G108

2019

Mathematics is important to the modern world. All quantitative science, including both physical and social sciences, is based on it. It provides the theoretical framework for physical science, statistics and data analysis as well as computer science. Our programmes reflect this diversity and the excitement generated by new discoveries within mathematics.

Overview

This programme is for you if you do not meet the entry requirements for direct entry to a Mathematics degree, as you spend a foundation year developing your mathematical knowledge and skills. You then have the option to continue to one of the following programmes: Mathematics - BSc (Hons), Mathematics and Statistics - BSc (Hons), Mathematics and Accounting and Finance - BA (Hons), Financial Mathematics - BSc (Hons). 

You also have the option of spending a year working in industry between Stages 2 and 3. We can offer help and advice in finding a placement.

Independent rankings

Mathematics at Kent was ranked 19th for course satisfaction in The Guardian University Guide 2017.

For graduate prospects, Mathematics was ranked 19th in The Complete University Guide 2017. Of Mathematics and Statistics students who graduated from Kent in 2015, 92% were in work or further study within six months (DLHE).

Teaching Excellence Framework

Based on the evidence available, the TEF Panel judged that the University of Kent delivers consistently outstanding teaching, learning and outcomes for its students. It is of the highest quality found in the UK.

Please see the University of Kent's Statement of Findings for more information.

TEF Gold logo

Course structure

The following modules are indicative of those offered on this programme. This listing is based on the current curriculum and may change year to year in response to new curriculum developments and innovation.  

On most programmes, you study a combination of compulsory and optional modules. You may also be able to take ‘wild’ modules from other programmes so you can customise your programme and explore other subjects that interest you.

Foundation year

If your qualifications are not sufficient, for whatever reason, for direct entry onto a degree programme, you can apply for this programme. It covers the mathematical skills you need to enter Stage 1 of the degree. 


Modules may include Credits

This module will focus on the topics which are fundamental across mathematics and the sciences. We will learn about the properties of many functions such as straight lines, quadratics, circles, exponentials, logarithms and the trigonometric functions. The focus of this module is on applied problem solving in many real-life situations, as well as some coverage of the rigorous theory behind many of these ideas. The material is delivered through lectures and examples classes, so that students have many different ways to learn. Many harder, extra-curricular examples are provided for keen students.

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15

Statistical techniques are a fundamental tool in being able to measure, analyse and communicate information about sets of data. Even the most basic methods can be indispensable in applied sciences and other quantitative areas and in this module we demonstrate why this is the case. In addition to learning about the common methods used in Statistics and how these can be applied meaningfully to other disciplines, we cover theoretical properties of Statistics.

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15

a) Functions: Functions, inverse functions and composite functions. Domain and range.

Elementary functions including the exponential function, the logarithm and natural logarithm functions and ax for positive real numbers a. Basic introduction to limits and continuity of a function, without epsilon-delta proofs.

b) The derivative: The derivative as the gradient of the tangent to the graph; interpretation of the derivative as a rate of change. The formal definition of the derivative and the calculation of simple examples from first principles. Elementary properties of the derivative, including the product rule, quotient rule and the chain rule; differentiation of inverse functions; calculating derivatives of familiar functions, including trigonometric, exponential and logarithmic functions. Applications of the derivative, including optimisation, gradients, tangents and normal. Parametric and implicit differentiation of simple functions. Taylor series.

c) Graphs: Curve sketching including maxima, minima, stationary points, points of inflection, vertical and horizontal asymptotes and simple transformations on graphs of functions. Additional material may include parametric curves and use of Maple to plot functions.

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15

a) Vectors: Vectors in two and three dimensions. Magnitude and direction. Algebraic operations involving vectors and their geometrical interpretations including the scalar product between two vectors. Use vectors to solve simple problems in pure mathematics and applications.

b) Kinematics: Fundamental and derived quantities and units in the S.I. system. Position, displacement, distance travelled, velocity, speed, acceleration. Constant acceleration for motion in one and two dimensions. Motion under gravity in a vertical plane. Projectiles. Use of calculus for motion in a straight line.

c) Forces and Newton's Laws: Newton's laws of motion applied to simple models of single and coupled bodies.

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15

This module introduces the ideas of integration and numerical methods.

a) Integration: Integration as a limit of a sum and graphical principles of integration, derivatives, anti-derivatives and the Fundamental Theorem of Calculus (without proof), definite and indefinite integrals, integration of simple functions.

b) Methods of integration: integration by parts, integration by change of variables and by substitution, integration by partial fractions.

c) Solving first order differential equations: separable and linear first order differential equations. Construction of differential equations in context, applications of differential equations and interpretation of solutions of differential equations.

d) Maple: differentiation and integration, curve sketching, polygon plots, summations.

e) Numerical integration: mid-ordinate rule, trapezium rule, Simpson's rule, use of Maple in estimating definite integrals.

Additional material may include root finding using iterative methods, parametric integration, surfaces and volumes of revolution.

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15

Students will be introduced to key mathematical skills, necessary in studying for a mathematics degree: use of the University Library and other sources to support their learning, present an argument in oral or written form, learn about staff in the School and beyond, etc. In particular, students will study various techniques of proof (by deduction, by exhaustion, by contradiction, etc.). These techniques will be illustrated through examples chosen from various areas of mathematics (and in particular co-requisite modules).

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Lecture Syllabus

INTRODUCTION TO MATLAB – 20 Lectures

An introduction to the use of computers and the process of programming them

Introduction to the MATLAB programming environment

MATLAB basics: Variables and Arrays, Displaying Output Data, Data Files, Operations

Built-in MATLAB Functions

Branching statements and Loops

An introduction to problem solving techniques and the Program development cycle

Program design tools: Flowcharts and Pseudocode

User-defined functions

Introduction to Plotting: Two-Dimensional, Three-Dimensional, Multiple Plots and Animation

Additional data types: Cell arrays, Structures and Graphics handles.

Coursework

22 hours terminal based exercises integrated with the lectures. This will take the form of 11, 2-hour exercises during the year of which 6 will be assessed.

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15

Through this module, students will develop the transferable linguistic and academic skills necessary to successfully complete other modules on their programme and acquire the specific language skills that they will require when entering SMSAS and SPS Stage 1 programmes. The programme of study focuses on writing and speaking skills, enhancing academic language through classroom, homework and assessed activities. Writing skills will be used to write a technical report, interpret data and describe processes. Spoken skills will be used in presentations and seminars.

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15

Since Plato's Dialogues, it has been part of philosophical enquiry to consider philosophical questions using logic and common sense alone. This module aims to train students to continue in that tradition. In the first part students will be introduced to basic themes in introductory formal logic and critical thinking. In the second part students will be presented with a problem each week in the form of a short argument, question, or philosophical puzzle and will be asked to think about it without consulting the literature. The problem, and students’ responses to it, will then form the basis of a structured discussion. By the end of the module, students (a) will have acquired a basic logical vocabulary and techniques for the evaluation of arguments; (b) will have practised applying these techniques to short passages of philosophical argument; and (c) will have acquired the ability to look at new claims or problems and to apply their newly acquired argumentative and critical skills in order to generate philosophical discussions of them.

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Stage 1

Modules may include Credits

Increasingly data are collected to inform future decisions, varying from which websites people access on a regular basis to how patients respond to new drugs, to how the stock market responds to global events, or to how animals move around their local habitat. Therefore, most professionals will need to extract useful information from data and to manage and present data in their working lives.  This module explores some of the basic concepts of statistics, from data summarisation to the main methods of statistical inference. The techniques that are discussed can be used in their own right for simple statistical analyses, but serve as an important foundation for later, more advanced, modules. The statistical computing package R is used throughout the module for data analysis. The syllabus includes: an introduction to R and investigating data sets, sampling and sampling distributions, point and interval estimation, hypothesis testing, association between variables.

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15

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

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15

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

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

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

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

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15

This module is a sequel to Algebraic Methods. It considers the abstract theory of linear spaces together with applications to matrix algebra and other areas of Mathematics (and its applications). Since linear spaces are of fundamental importance in almost every area of mathematics, the ideas and techniques discussed in this module lie at the heart of mathematics.

Topics covered will include:

1 Vector Spaces: definition, examples, linearly independent and spanning sets, bases, dimension, subspaces.

2 Linear transformations: definition, examples, matrix of a linear transformation, change of basis, similar matrices.

3 Determinant of a linear transformation.

4 Eigenvalues/eigenvectors and diagonalisation: characteristic polynomial, invariant subspaces and upper triangular forms. Cayley-Hamilton Theorem.

5 Bilinear forms: inner products, norms, Cauchy-Schwarz inequality.

6 Orthonormal systems, the Gram-Schmidt process.

7 Symmetric Matrices. Every real symmetric matrix is diagonalisable.

8 Quadratic forms: Sylvester's Law of Inertia; signature of a quadratic form; application to conics (and quadrics if time permits).

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15

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

Complex numbers: Complex arithmetic, the complex conjugate, the Argand diagram, de Moivre's Theorem, modulus-argument form; elementary functions

Polynomials: Fundamental Theorem of Algebra (statement only), roots, factorization, rational functions, partial fractions

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

Scalar ordinary differential equations (ODEs): definition; methods for first-order ODEs; principle of superposition for linear ODEs; particular integrals; second-order linear ODEs with constant coefficients; initial-value problems

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

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15

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

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

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

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

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15

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

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

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

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

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

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

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

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Topics covered will include:

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

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

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

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

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

Taylor Approximation: Taylor's Theorem, remainder term, Taylor series, standard examples, O and o notation, limits using Taylor series.

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15

Stage 2

Modules may include Credits

This module builds on the Stage 1 Real Analysis 1 module. You will extend your knowledge of functions of one real variable, look at series, and study functions of several real variables and their derivatives. Outline syllabus includes: Continuity and uniform continuity of functions of one variable; Sequences of functions; Series; The Riemann integral; Functions of several variables; Differentiation of functions of several variables; Extrema; Inverse function and Implicit function theorems.

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15

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

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15

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

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15

This module is a pre-requisite for many of the other statistics modules at Stages 2, 3 and 4, but it can equally well be studied as a module in its own right, extending the ideas of probability and statistics met at Stage 1 and providing practice with the mathematical skills learned in MA348 and MA349. It starts by revising the idea of a probability distribution for one or more random variables and looks at different methods to derive the distribution of a function of random variables. These techniques are then used to prove some of the results underpinning the hypothesis test and confidence interval calculations met at Stage 1, such as for the t-test or the F-test. With these tools to hand, the module moves on to look at how to fit models (probability distributions) to sets of data. A standard technique, the method of maximum likelihood, is used to fit the model to the data to obtain point estimates of the model parameters and to construct hypothesis tests and confidence intervals for these parameters. Outline Syllabus includes: Joint, marginal and conditional distributions of discrete and continuous random variables; Transformations of random variables; Sampling distributions; Point and interval estimation; Properties of estimators; Maximum likelihood; Hypothesis testing; Neyman-Pearson lemma; Maximum likelihood ratio test.

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15

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

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15

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

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15

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

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15

Constructing suitable models for data is a key part of statistics. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. Even if the temperature and pressure are fixed, there will be variation in the yield which motivates the use of a statistical model which includes a random component. In this module, we study how suitable models can be constructed, how to fit them to data and how suitable conclusions can be drawn. Both theoretical and practical aspects are covered, including the use of R.

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15

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

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15

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

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15

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

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15

Year in industry

The year in industry included in this programme provides you with the opportunity to gain valuable work experience and greatly enhances your cv. We can help you to find a placement and support you while you are there.

Stage 3

Modules may include Credits

This module is an introduction to point-set topology, a topic that is relevant to many other areas of mathematics. In it, we will be looking at the concept of topological spaces and related constructions. In an Euclidean space, an "open set" is defined as a (possibly infinite) union of open "epsilon-balls". A topological space generalises the notion of "open set" axiomatically, leading to some interesting and sometimes surprising geometric consequences. For example, we will encounter spaces where every sequence of points converges to every point in the space, see why for topologists a doughnut is the same as a coffee cup, and have a look at famous objects such as the Moebius strip or the Klein bottle. 

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15

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

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15

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

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

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

• Elimination theory and applications;

• Linear equations over systems of polynomials, syzygies.

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15

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

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

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15

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

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15

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

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

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

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

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

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

• Heisenberg's uncertainty principle.

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

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15

A stochastic process is a process developing in time according to probability rules, for example, models for reserves in insurance companies, queue formation, the behaviour of a population of bacteria, and the persistence (or otherwise) of an unusual surname through successive generations.The syllabus will include coverage of a wide variety of stochastic processes and their applications: Markov chains; processes in continuous-time such as the Poisson process, the birth and death process and queues.

Marks on this module can count towards exemption from the professional examination CT4 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

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A time series is a collection of observations made sequentially in time. Examples occur in a variety of fields, ranging from economics to engineering, and methods of analysing time series constitute an important area of statistics. This module focuses initially on various time series models, including some recent developments, and provides modern statistical tools for their analysis. The second part of the module covers extensively simulation methods. These methods are becoming increasingly important tools as simulation models can be easily designed and run on modern PCs. Various practical examples are considered to help students tackle the analysis of real data.The syllabus includes: Difference equations, Stationary Time Series: ARMA process. Nonstationary Processes: ARIMA Model Building and Testing: Estimation, Box Jenkins, Criteria for choosing between models, Diagnostic tests.Forecasting: Box-Jenkins, Prediction bounds. Testing for Trends and Unit Roots: Dickey-Fuller, ADF, Structural change, Trend-stationarity vs difference stationarity. Seasonality and Volatility: ARCH, GARCH, ML estimation. Multiequation Time Series Models: Spectral Analysis. Generation of pseudo – random numbers, simulation methods: inverse transform and acceptance-rejection, design issues and sensitivity analysis.

Marks on this module can count towards exemption from the professional examination CT6 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

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15

The aim of this module is to equip students with the skills needed to communicate mathematics effectively to the world. This module is supported by a series of workshops covering various forms of written and oral communication. Each student will choose a topic in mathematics, statistics or financial mathematics from a published list on which to base their three coursework assessments which include a scientific writing assessment and an oral presentation.

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15

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

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

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

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

• Presentation of data

• Report writing and presentation skills

• Hypothesis testing: formulating questions, converting to hypotheses, parametric and non-parametric methods and their assumptions, selection of appropriate method, application and reporting. Use of resources to explore and apply additional tests. Parametric and non-parametric tests include, but are not limited to, t-tests, likelihood ratio tests, score tests, Wald test, chi-squared tests, Mann Whitney U-test, Wilcoxon signed rank test, McNemar's test.

• Linear and Generalised Linear Models: simple linear and multiple regression, ANOVA and ANCOVA, understanding the limitations of linear regression, generalised linear models, selecting the appropriate distribution for the data set, understanding the difference between fixed and random effects, fitting models with random effects, model selection.

• Consultancy skills: group work exercise(s)

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

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

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

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

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

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

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15

Combinatorial games, game trees, strategy, classification of positions. Two-player zero-sum games, security levels, pure and mixed strategies, von Neumann's minimax theorem. Solving zero-sum two player games using linear programming. Arbitrary sum games, utility, and matrix games. Nash equilibrium, Nash equilibrium theorem, applications, and cooperation. Multi-player games, coalitions, and the Shapley value.

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15

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

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

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

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

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

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15

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

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

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

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15

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

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15

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

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

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

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

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

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

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

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Over a century ago, the Norwegian mathematician Sophus Lie made a simple but profound observation: each well-known method for solving a class of ordinary differential equations (ODEs) uses a change of variables that exploits symmetries of the class. Lie went on to develop this idea into a systematic method for attacking the problem of solving unknown differential equations. Essentially, one can use mathematical tools to force a given differential equation to reveal whether or not it has certain symmetries – provided it has, they can be used to simplify or solve the equation. This module is designed to enable students to understand the mathematics behind Lie's methods and to become proficient in using these powerful tools.

The following topics are covered.

Introduction: Symmetries of geometrical objects, symmetries of some first-order ODEs, solution via symmetries.

Lie symmetries of first-order ODEs: The infinitesimal generator, canonical coordinates, invariant points, Lie symmetries and standard solution methods.

How to find Lie symmetries: The linearized symmetry condition, solution of overdetermined systems, the Lie algebra of point symmetry generators.

Solution of higher-order ODEs: Solvability, differential invariants, reduction of order, invariant solutions.

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Linear PDEs. Dispersion relations. Review of d'Alembert's solutions of the wave equation.

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

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

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

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

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

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

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

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The origins of Bayesian inference lie in Bayes' Theorem for density functions; the likelihood function and the prior distribution combine to provide a posterior distribution which reflects beliefs about an unknown parameter based on the data and prior beliefs. Statistical inference is determined solely by the posterior distribution. So, for example, an estimate of the parameter could be the mean value of the posterior distribution. This module will provide a full description of Bayesian analysis and cover popular models, such as the normal distribution. Initially, the flavour will be one of describing the Bayesian counterparts to well known classical procedures such as hypothesis testing and confidence intervals. Outline Syllabus includes: Bayes Theorem for density functions; Exchangeability; Choice of priors; Conjugate models; Predictive distribution; Bayes estimates; Sampling density functions; Gibbs samplers; OpenBUGS; Bayesian hierarchical models; Applications of hierarchical models; Bayesian model choice.

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

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

• orthogonal Latin squares,

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

• "coin-tossing over a telephone",

• linear feedback shift registers and m-sequences,

• cyclic codes including Hamming,

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Teaching and assessment

Teaching amounts to approximately 16 hours of lectures and classes per week. Modules that involve programming or working with computer software packages usually include practical sessions.

The majority of Stage 1 modules are assessed by end-of-year examinations. Many Stage 2 and 3 modules include coursework which normally counts for 20% of the final assessment. Both Stage 2 and 3 marks count towards your final degree result.

Programme aims

The programme aims to:

  • equip students with the technical appreciation, skills and knowledge appropriate to a degree in Mathematics
  • develop students’ facilities of rigorous reasoning and precise expression
  • develop students’ abilities to formulate and solve mathematical problems
  • encourage an appreciation of recent developments in mathematics and of the links between the theory of mathematics and its practical application
  • provide students with a logical, mathematical approach to solving problems
  • provide students with an enhanced capacity for independent thought and work
  • ensure students are competent in the use of information technology and are familiar with computers and the relevant software
  • provide students with opportunities to study advanced topics in mathematics, engage in research at some level, and develop communication and personal skills.

Learning outcomes

Knowledge and understanding

You gain knowledge and understanding of:

  • the core principles of calculus, algebra, mathematical methods, discrete mathematics, analysis and linear algebra
  • statistics in the areas of probability and inference
  • information technology as relevant to mathematicians
  • methods and techniques of mathematics
  • the role of logical mathematical argument and deductive reasoning.

Intellectual skills

You develop your intellectual skills in the following areas:

  • the ability to demonstrate a reasonable understanding of mathematics
  • the calculation and manipulation of the material written within the programme
  • the ability to apply a range of concepts and principles in various contexts
  • the ability to use logical argument
  • the ability to solve mathematical problems by various methods
  • the relevant computer skills
  • the ability to work independently.

Subject-specific skills

You gain subject-specific skills in the following areas:

  • the ability to demonstrate knowledge of key mathematical concepts and topics, both explicitly and by applying them to the solution of problems
  • the ability to comprehend problems, abstract the essentials of problems and formulate them mathematically and in symbolic form so as to facilitate their analysis and solution
  • the use of computational and more general IT facilities as an aid to mathematical processes
  • the presentation of mathematical arguments and conclusions with clarity and accuracy.

Transferable skills

You gain transferable skills in the following areas:

  • problem-solving skills, relating to qualitative and quantitative information
  • communication skills
  • numeracy and computational skills
  • information-retrieval skills, in relation to primary and secondary information sources, including through online computer searches
  • information technology skills such as wordprocessing, spreadsheet use and internet communication
  • time-management and organisational skills, as shown by the ability to plan and implement effective modes of working
  • study skills needed for continuing professional development.

Careers

Those students who choose to take the year in industry option find the practical experience they gain gives them a real advantage in the graduate job market. Through your studies, you also acquire many transferable skills including the ability to deal with challenging ideas, to think critically, to write well and to present your ideas clearly, all of which are considered essential by graduate employers.

Recent graduates have gone into careers in medical statistics, the pharmaceutical industry, the aerospace industry, software development, teaching, actuarial work, Civil Service statistics, chartered accountancy, the oil industry and postgraduate research.

Entry requirements

Home/EU students

The University will consider applications from students offering a wide range of qualifications. Typical requirements are listed below. Students offering alternative qualifications should contact us for further advice. 

It is not possible to offer places to all students who meet this typical offer/minimum requirement.

New GCSE grades

If you’ve taken exams under the new GCSE grading system, please see our conversion table to convert your GCSE grades.

Qualification Typical offer/minimum requirement
A level

Applications are individually considered. Please contact an Admissions Officer

Access to HE Diploma

The University will not necessarily make conditional offers to all Access candidates but will continue to assess them on an individual basis. 

If we make you an offer, you will need to obtain/pass the overall Access to Higher Education Diploma and may also be required to obtain a proportion of the total level 3 credits and/or credits in particular subjects at merit grade or above.

BTEC Level 3 Extended Diploma (formerly BTEC National Diploma)

The University will consider applicants holding BTEC National Diploma and Extended National Diploma Qualifications (QCF; NQF; OCR) on a case-by-case basis. Please contact us for further advice on your individual circumstances.

International Baccalaureate

Applications are individually considered please contact an Admissions Officer.

International students

The University welcomes applications from international students. Our international recruitment team can guide you on entry requirements. See our International Student website for further information about entry requirements for your country.

If you need to increase your level of qualification ready for undergraduate study, we offer a number of International Foundation Programmes.

Meet our staff in your country

For more advice about applying to Kent, you can meet our staff at a range of international events.

English Language Requirements

Please see our English language entry requirements web page.

Please note that if you are required to meet an English language condition, we offer a number of 'pre-sessional' courses in English for Academic Purposes. You attend these courses before starting your degree programme. 

General entry requirements

Please also see our general entry requirements.

Fees

The 2019/20 tuition fees have not yet been set. As a guide only, the 2018/19 annual tuition fees for this programme are:

UK/EU Overseas
Full-time £9250 £15200

For students continuing on this programme, fees will increase year on year by no more than RPI + 3% in each academic year of study except where regulated.* 

Your fee status

The University will assess your fee status as part of the application process. If you are uncertain about your fee status you may wish to seek advice from UKCISA before applying.

Fees for Year in Industry

For 2018/19 entrants, the standard year in industry fee for home, EU and international students is £1,385

Fees for Year Abroad

UK, EU and international students on an approved year abroad for the full 2018/19 academic year pay £1,385 for that year. 

Students studying abroad for less than one academic year will pay full fees according to their fee status. 

General additional costs

Find out more about accommodation and living costs, plus general additional costs that you may pay when studying at Kent.

Funding

University funding

Kent offers generous financial support schemes to assist eligible undergraduate students during their studies. See our funding page for more details. 

Government funding

You may be eligible for government finance to help pay for the costs of studying. See the Government's student finance website.

Scholarships

General scholarships

Scholarships are available for excellence in academic performance, sport and music and are awarded on merit. For further information on the range of awards available and to make an application see our scholarships website.

The Kent Scholarship for Academic Excellence

At Kent we recognise, encourage and reward excellence. We have created the Kent Scholarship for Academic Excellence. 

For 2018/19 entry, the scholarship will be awarded to any applicant who achieves a minimum of AAA over three A levels, or the equivalent qualifications (including BTEC and IB) as specified on our scholarships pages

The scholarship is also extended to those who achieve AAB at A level (or specified equivalents) where one of the subjects is either Mathematics or a Modern Foreign Language. Please review the eligibility criteria.

The Key Information Set (KIS) data is compiled by UNISTATS and draws from a variety of sources which includes the National Student Survey and the Higher Education Statistical Agency. The data for assessment and contact hours is compiled from the most populous modules (to the total of 120 credits for an academic session) for this particular degree programme. 

Depending on module selection, there may be some variation between the KIS data and an individual's experience. For further information on how the KIS data is compiled please see the UNISTATS website.

If you have any queries about a particular programme, please contact information@kent.ac.uk.