Studying Mathematics at Royal Holloway Assessment & Feedback: Formative assignments in the form of 11 problem sheets. ... Sets: intersection, union, complement, Venn diagrams, De Morgan’s

The Faculty of Science

Studying Mathematics at Royal Holloway

COURSE OPTIONS FOR VISITING STUDENTS 2011-12

ABOUT THE DEPARTMENT The Mathematics Department at Royal Holloway is a lively and friendly place with an international reputation for the quality of its teaching and research. Academic staff are active in pioneering research which is making an impressive impact on the world stage. This strong research culture influences our curriculum, helping students to keep in touch with the latest developments in the field. Mathematics courses at Royal Holloway cover a diverse spectrum from abstract pure mathematics to applications in information security, theoretical physics, finance and statistics.

ENTRY REQUIREMENTS The below listed courses are open to all Study Abroad, International Exchange and Erasmus students, provided sufficient previous experience and knowledge as stated in the individual course prerequisites can be evidenced.

Click on any course headings or codes on the pages below to access further information. Term 1 = Autumn Term Term 2 = Spring Term

The information contained in the course outlines on the following pages is correct at the time of publication but may be subject to change as part of our policy of continuous improvement and development.

Level One Courses:

MT1100 From Euclid to Mandelbrot ½ unit Term 1

This course is available to the following students:

ERASMUS students

INTERNATIONAL EXCHANGE students.

STUDY ABROAD students The course lasts 1 term and cannot be started in January

MT1710 Calculus ½ unit Term 1


ERASMUS students



MT1810 Number Systems ½ unit Term 1


ERASMUS students



MT1940 Numbers and Functions ½ unit Term 1


ERASMUS students



MT1210 Introduction to Applied Mathematics ½ unit Term 2


ERASMUS students


STUDY ABROAD students The course lasts 1 term and cannot be started in September

MT1300 Principles of Statistics ½ unit Term 2


ERASMUS students



MT1510 Computational Mathematics I ½ unit Term 2


ERASMUS students



MT1720 Functions of Several Variables ½ unit Term 2


ERASMUS students



MT1820 Matrix Alegbra ½ unit Term 2


ERASMUS students



Level Two Courses:

MT2220 Vector Analysis and Fluids ½ unit Term 1


ERASMUS students



MT2300 Statistical Methods ½ unit Term 1


ERASMUS students



MT2680 Groups and Graphs ½ unit Term 1


ERASMUS students



MT2800 Linear Algebra and Project ½ unit Term 1


ERASMUS students



MT2940 Real Analysis ½ unit Term 1


ERASMUS students



MT2320 Probability ½ unit Term 2


ERASMUS students



MT2620 Mathematical Programming ½ unit Term 2


ERASMUS students



MT2720 Ordinary Differential Equations and Fourier Analysis

½ unit Term 2


ERASMUS students



MT2830 Primes and Factorisation ½ unit Term 2


ERASMUS students



MT2900 Complex Variable ½ unit Term 2


ERASMUS students



Course Code: MT1100 Course Value: 0.5 Availability: Term 1

Course Title: From Euclid to Mandelbrot

Prerequisites: Mathematics A-level or equivalent

Aims: This course aims to show how mathematics has been used to describe space over the last 2500 years and use this to motivate the study of various aspects of the subject.

Learning Outcomes:

On completion of the course, students should be able to • Appreciate what can be done with ruler and compass constructions; • Sketch simple curves using plane polar coordinates; • Sketch and classify conics, and find their foci and directrices; • Understand the concepts of self-similarity and fractal dimension; • Use simple arguments to distinguish between countable and uncountable sets; • Analyse the logistic map and similar iterated maps; • Explain the period-doubling route to chaos.

Course Content:

Geometry. Ruler and compass constructions (up to the regular pentagon). Platonic solids. Euler’s formula. Plane polar coordinates. Conics: Cartesian and polar forms, focus and directrix. Fractals. Self-similarity, fractal dimension, Koch snowflake, Cantor dust, Sierpinski gasket. Countability. Countability of rationals, uncountability of reals and of theCantor set. Iteration. Iterative maps, cobwebbing, fixed points, limit cycles, stability, logistic equation, period doubling. Chaos. The Mandelbrot set. Bilinear transformations. Fibonacci numbers, the golden mean.

Teaching & Learning Methods:

33 hours of lectures and examples classes, 6 hours tutorials. 111 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Elementary Geometry – J Roe (Oxford UP 1993) Library reference 516.ROE Fractals, Chaos, Power Laws – M Schroeder (Freeman 1991) Library reference 513.15 SCH

Formative Assessment & Feedback:

Formative assignments in the form of 11 problem sheets. The students will receive feedback as written comments on their attempts.

Summative Assessment:

Exam (100%) Four questions out of five in a two-hour paper.


Course Title: Calculus

Prerequisites: A-level Mathematics or equivalent

Aims:

This course aims to develop the students’ confidence and skill in dealing with mathematical expressions, to extend their understanding of calculus, and to introduce some topics which they may not have met at A-level. It also aims to ease the transition to university work and to encourage the student to develop good study skills. Mathematica is to be used as a calculating and graphical aid.

Learning Outcomes:

On completion of the course the student should be able to: • factorize polynomials and separate rational functions into partial fractions; • sketch the graphs of polynomials, rational functions and other elementary functions, identifying turning points and asymptotes where appropriate; • differentiate commonly occurring functions, and find indefinite and definite integrals of a wide variety of functions, using substitution or integration by parts; • recognize the standard forms of first-order differential equations, reduce other equations to these forms, and solve them; • solve certain second and higher order differential equations; • demonstrate that he or she can use Mathematica as an aid in the solution of problems or to illustrate the ideas met in the course.

Course Content:

Polynomials and rational functions: asymptotes, sketching, differentiation. Transcendental functions: ex , ln x , trigonometric and hyperbolic functions (differentiation, zeros, turning points, sketching, symmetry, periodicity). Calculus: chain rule, integration by parts, substitution, use of trigonometric formulae, partial fractions. First-order differential equations: separable equations, linear equations. Second-order differential equations: constant coefficients, complementary function and particular integral. Use of the Mathematica package: including polynomials, integrals and derivatives, plots, and general applications to many of the above topics.


33 hours of lectures and examples classes, 11 hours of problem workshops, 3 hours of Mathematica training, 6 hours tutorials, 97 hours of private study, including work on problem sheets (including Mathematica exercises) and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Calculus (5th edition) – J Stewart (Brooks-Cole 2003). Library Ref. 515 STE Elementary Differential Equations and Boundary Value Problems − W E Boyce & R C di Prima (Wiley). Library Ref. 515.41 BOY




Exam (90%) Four questions out of five in a two-hour paper Coursework (10%) One 45 minute test in November: 0%; one 45 minute test in January


Course Title: Number Systems


Aims: This course aims to introduce fundamental algebraic structures used in subsequent courses and the notion of formal proofs, and to illustrate these concepts with examples.

Learning Outcomes:

On completion of the course, students should be able to: • apply Euclid’s algorithm to find the greatest common divisor of two integers; • use mathematical induction in a careful and logical way to prove simple results; • perform arithmetic operations on complex numbers, using x + iy and reiq forms, locate points on the Argand diagram, and extract roots of complex numbers; • prove De Morgan’s laws and the distributive laws of set theory, and use the principle of inclusion/exclusion in simple cases; • determine whether a given mapping is bijective and if so find its inverse; • establish whether a given relation on a set is an equivalence relation and find the corresponding equivalence classes; • compile truth tables to determine whether two statements are logically equivalent; • define a ring, integral domain and field, establish some of their simple properties.

Course Content:

The integers: division with quotient and remainder, binary numbers, the Euclidean algorithm, greatest common divisors, gcd(m,n) = sm + tn, primes, statement of the fundamental theorem of arithmetic, the principle of mathematical induction. Complex numbers: Cartesian addition and multiplication, the complex conjugate, rules of manipulation (the field axioms), inversion, the Argand diagram, modulus and argument, extraction of nth roots (quadratic equations and roots of unity (cyclic groups)), e i iq = cosq + sinq, ez and log z . Sets: intersection, union, complement, Venn diagrams, De Morgan’s laws. Mappings: composition, associative law, injections, surjections, bijections and inverses. Equivalence relations and partitions. Propositional logic, truth tables. Rings and fields: the ring Zn of integers modulo n, the field Zp . The ring F[x] of polynomials over a field, analogy with Z (division law, monic polynomials, gcds), zeros, remainder and factor theorems, a polynomial of degree n over F has at most n zeros. The ring of 2 × 2 matrices over a field.


33 hours of lectures and examples classes, 11 hours of problem workshops, 6 hours tutorials. 100 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

A Concise Introduction to Pure Mathematics – M Liebeck (Chapman and Hall/CRC Mathematics 2000) Library Ref. 510 LIE Discrete Mathematics (2nd edition)− N L Biggs (Oxford UP 2002). Library Ref. 510 BIG




Exam (90%) Four questions out of five in a two-hour paper Coursework (10%) One 45 minute test in November: 0%; one 45 minute test in January


Course Title: Numbers and Functions


Aims:

• To kindle an interest in analysis, and to provide a taste of what the subject is about; • To give a user-friendly introduction to key ideas of analysis, illustrated with copious examples; • To provide a structure that enables students to gain confidence in handling concepts in analysis.

Learning Outcomes:

On completion of the course, the student should be able to: • appreciate the significance of the completeness property distinguishing R from Q; • find sups and infs of elementary sets; • find out whether a given (simply defined) function is continuous at a point, and apply properties of continuous functions in examples; • determine whether a given (simple) sequence tends to a limit, using monotonicity when appropriate; • find limits of sequences defined recursively.

Course Content:

The real number system: The natural numbers N, the binomial theorem using induction. The integers Z and the rational field Q. Order properties, manipulation of inequalities, the triangular inequality. Irrationality of 2 . Decimal representation of real numbers. Null sequences. Subsets, maximum, upper bound, least upper bound. Every non-empty subset of the reals which is bounded above has a least upper bound. Bernoulli’s inequality with applications. Continuity: Discussion of continuity, continuity of f at a point. A continuous function is specified by its values at rational points. Discussion of f x x ( ) = 2 . Continuity on an interval. A continuous function on a closed interval. Discussion of the intermediate value theorem . Sequences: Limits of sequences, limits of sequences defined recursively. Monotonic sequence theorem. Connection with continuity.


33 hours of lectures and examples classes, 6 hours tutorials. 111 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Numbers and Functions (steps into analysis) – R P Burn (Cambridge 2000). Library Ref. 515.23 BUR Yet Another Introduction to Analysis – Victor Bryant (Cambridge 1990). Library Ref. 515 BRY




Exam (90%) Four questions out of five in a two-hour paper Coursework (10%) One written paper of 45 minutes during January


Course Title: Introduction to Applied Mathematics

Prerequisites: MT1710

Aims:

The aim of this course is to provide an introduction to some key ideas and methods of classical mechanics, chaos theory and special relativity. The course covers Newton’s equations of motion for a single particle, shows how these equations can give rise to chaotic behaviour, and shows how they need to be modified for velocities close to the speed of light.

Learning Outcomes:

On completion of the course, the student should be able to: • solve Newton’s equations of motion for a variety of problems, including the damped, driven harmonic oscillator; • use Mathematica where a closed solution cannot be found; • use the conservation laws for energy and momentum; • work with co-ordinate systems that accelerate or rotate; • explain how chaos arises for the forced pendulum; • state Einstein’s principle of relativity and explain how it leads to special relativity; • use the Lorentz transformation and draw Minkowski diagrams.

Course Content:

Classical Dynamics: Dimensional analysis, units, forces. Newton's laws, One dimensional motion; Conservation of energy and momentum. Stable and unstable equilibrium points. Simple Harmonic motion, damped and harmonic forced motion. Three dimensional motion. Projectile in the presence of friction. Circular motion. Angular momentum. Numerical solution of Newton's equation, application to planetary motion, Coriolis force. Chaos: The damped forced pendulum. Limit cycles, attractors. Period doubling. Chaotic motion of three gravitating bodies. Special Relativity: Galilean invariance. Inertial systems. Einstein's principle of relativity. The Lorentz transformation. Length contraction, time dilation, the twin paradox. The geometry of space-time. Energy-mass equivalence E = mc2.


33 hours of lectures and examples classes, 8 hours of workshops. 109 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Mechanics − P Smith & R C Smith (Wiley) 2nd edition. Library Ref. 531 SMI




Exam (100%) Four questions out of five in a two-hour paper


Course Title: Principles of Statistics


Aims:

This course introduces the notion of probability and the basic theory and methods of statistics, aiming to give an understanding of random variables and their distributions, data sets and their initial analysis, estimation and inference concerning means and variances. The overall aim of the course is to show students how to analyse a variety of different sorts of data sets in a scientific way.

Learning Outcomes:

At the end of the course the students should be able to: • calculate probabilities of events that arise from the standard distributions; • examine data critically, calculate summary statistics and display main features graphically; • calculate estimates of means and variances, deriving the corresponding sampling distributions; • derive confidence intervals for means and differences of means; • carry out t tests for means and differences of means; • analyse two-factor contingency tables using c2; • specify null/alternative hypotheses and calculate the corresponding acceptance/rejection regions. The student should be familiar with the notions of types of error, power and significance level. The student will have had a good experience of MINITAB, and should be proficient in its use for the applied parts of the course.

Course Content:

Descriptive Statistics: Organizing data; histogram dotplot, boxplot and stem-and-leaf; descriptive measures; plots of bivariate data; empirical distribution function. Probability: Elementary notion of probability in terms of distribution of random variables as models for experiments. The Binomial, Poisson, Discrete and Geometric distributions; the normal distribution, 2 and t distributions; the Exponential distribution. Expectation, variance and covariance. Moment generating function methods. Statistics: Simple random sampling, estimation (point and interval); maximum likelihood estimation; tests of hypotheses, null and alternative hypotheses, error types and power, sample size/power relation, large and small samples. One sample, two sample and paired comparison tests, 2and contingency tests.


33 hours of lectures and examples classes, 11 hours practical work, 6 hours tutorials. 100 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

A Basic Course in Statistics (5th edition) – G M Clarke and D Cooke (Arnold 2004). Library Ref. 518.3 CLA




Exam (80%) Four questions out of five in a two-hour paper Coursework (20%) One MINITAB project


Course Title: Computational Mathematics I

Prerequisites: MT1710 and MT1810

Aims: This course teaches how to program in C++ and how to solve a range of mathematical problems using a computer. Programming techniques and numerical algorithms are developed in parallel and applied to practical examples.

Learning Outcomes:

On completion of the course, the student should be able to • edit, compile, run and debug C++ programs; • handle input and output; • use standard constructs like loops, arrays and functions; • analyse numerical errors and the precision of results; • understand a range of simple mathematical algorithms

Course Content:

C++ Programming: Editing, compiling and running a program; input and output; common errors and debugging, types of variables, branching and loops; arrays and pointers; functions; structures. Theory and applications: Numerical error and precision. Applications from numerical analysis, such as numerical integration, numerical derivatives, the Horner scheme to evaluate a polynomial, evaluation of series, finding zeros of a function, finding maxima of a function. Applications from algebra, such as modular arithmetic, the Euclidean algorithm, random number generation. Applications from combinatorics, such as recurrence relations, using a computer to solve simple enumeration problems, counting integer points in a polygon. Searching and sorting.


22 hours of lectures and examples classes, 22 hours practical classes. 106 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

An Introduction to C++ and numerical methods – J M Ortega & A S Grimshaw (Oxford UP 1999) Library reference 001.6424 ORT


Formative assignments in the form of 8 problem sheets. The students will receive feedback as written comments on their attempts. They will also get feedback on the mini-projects.


Exam (75%) Four questions out of five in a two-hour paper Coursework (8.33%) Three mini-projects in C++


Course Title: Functions of Several Variables


Aims:

This course aims to introduce students to the calculus of functions of more than one variable, and show how it may be used in such areas as geometry and optimization, and to demonstrate how simple functions may be represented as a power series under certain conditions.

Learning Outcomes:

On completion of the course, students should be able to • manipulate partial derivatives; • use partial derivatives to determine the nature of stationary points and to analyse certain properties of surfaces; • construct and manipulate line integrals; • evaluate double integrals, including the use of change of order of integration and change of coordinates; • expand functions such as ex trigonometrical and hyperbolic functions, ln(1+ x) , arctan x and simple variants as power series; • generate Taylor and Maclaurin series, including the remainder terms.

Course Content:

Partial differentiation: partial derivatives (using Mathematica to check), exact first order differential equations; chain rule for differentiation; stationary points; use of Mathematica for visualization; geometry: gradient, directional derivative, normals, tangents. Applications of calculus: intuitive notions of continuity and differentiability, intermediate value theorem, Rolle’s theorem and mean value theorem, all stated without proof but illustrated by examples; l’Hôpital’s rule. Series: idea of a power series, Taylor and Maclaurin series; binomial, geometric, exponential, sin and cos, ln(1+ x) , arctan x . Remainder terms (one type only). Integration in more than one dimension: curves in three dimensions: parametric equations, distance along a curve; line integrals; line integral of a gradient; double integrals; use of Mathematica; change of order; change of variables, Jacobian; plane polar coordinates.


33 hours of lectures and examples classes, 11 hours of problem workshops. 106 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Calculus (5th edition) – J Stewart (Brooks-Cole 2003). Library Ref. 515 STE






Course Title: Matrix Algebra


Aims:

This course aims to give students a working knowledge of basic linear algebra, with an emphasis on an approach via matrices and vectors. The course introduces some of the basic theoretical and computational techniques of matrix theory, and illustrates them by examples.

Learning Outcomes:

On completion of the course, students should be able to: • appreciate the power of vector methods, use vector methods to describe threedimensional space, and apply scalar and vector products of two vectors and triple products appropriately; • understand the notions of field, vector space and subspace; • calculate the determinant of an n × n matrix; calculate the inverse of a non-singular matrix; • appreciate the significance of the characteristic polynomial of a matrix, compute the eigenvalues and eigenspaces of a matrix, and diagonalize it when possible; • understand the notions of linear independence and dimension; • reduce a matrix to row-reduced echelon form.

Course Content:

Vectors in R3: vectors as directed line segments; addition, scalar multiplication, parallelogram law. Dot product, length, distance, perpendicular vectors, angle between

vectors, u.v u v cos. Lines and planes in R3. Cross product, area u × v and volume (u × v).w . 2 × 2 matrices over a field: determinants, inverses, rotation and reflection matrices. Eigenvalues and eigenvectors, the characteristic polynomial and trace. Diagonalization

P AP �1with applications. 3 × 3 matrices: permutations, compositions, parity. Definition of the determinant of an n × n matrix, cofactors, row and column expansion, the adjugate matrix, inverse of a non-singular matrix. Examples of characteristic polynomial and diagonalization of 3 × 3 matrices A. Statement of the properties of determinants, including AB = A B. Vector spaces: axioms, linear independence, span, dimension. Subspaces. Row-reduction: elementary row operations, echelon and row-reduced echelon form, rank of a matrix. Applications: solution of systems of linear equations, deriving a basis from a spanning set, computing the inverse of a matrix.


33 hours of lectures and examples classes, 11 hours of problem workshops. 106 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Linear Algebra (Schaum Series ) − S Lipschutz (McGraw-Hill). Library Ref. 510.76 LIP Undergraduate Algebra − C W Norman (Oxford 1986). Library Ref. 512.11 NOR Linear Algebra: a Modern Introduction – D Poole (Brooks-Cole 2005). Library Ref. 512.3 POO






Course Title: Vector Analysis and Fluids


Aims:

to study the integration and differentiation of vectors and scalars defined at points in space, introducing the concepts of scalar and vector fields; • to familiarize the student with the use of general orthogonal curvilinear coordinates and the evaluation of differential operators; • to introduce integral theorems and demonstrate their usefulness; • to show how simple partial differential equations may be solved by the technique of separation of variables; • to show how the acquired concepts can be applied in the field of dynamics of inviscid fluids.

Learning Outcomes:

On completion of the course students should be able to: • identify scalar and vector fields; • calculate the gradient of a scalar field and the divergence and curl of a vector field; • use general orthogonal curvilinear co-ordinates and, in particular, cylindrical and spherical polar co-ordinates; • use the divergence theorem and Stokes’ theorem; • recognize when and how variables separate in a partial differential equation; • apply the equations of continuity and motion for an inviscid fluid and use Bernoulli’s equation; • use velocity potential and apply it to examples of irrotational flow.

Course Content:

Vector analysis: scalar and vector fields. Field lines for a vector field. Gradient of a scalar field, divergence and curl of a vector field. The del-operator. Cylindrical and spherical polar coordinates. General orthogonal curvilinear coordinates. Surface and volume integrals. The divergence theorem and Stokes’ theorem. Green’s theorem. Partial differential equations: Laplace’s equation,, the diffusion equation and the wave equation in Cartesian coordinates. Separation of variables, used in plane polar and spherical coordinates. Dynamics of inviscid fluids: equation of continuity. Velocity and acceleration. Equation of motion. Bernoulli’s equation. Irrotational flow and velocity potential. Examples of potential flow of incompressible fluids.


33 hours of lectures and examples classes. 117 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Calculus (5th edition) – J Stewart (Brooks-Cole 2003). Library Ref. 515 STE Vector Analysis and Cartesian Tensors − (Third Edition) D E Bourne and P C Kendall (Chapman and Hall 1992). Library Ref. 515.34 BOU






Course Title: Statistical Methods


Aims: To study important aspects of statistical modelling in an integrated way and develop some expertise both in the theory and applications of linear models.

Learning Outcomes:

On completion of the course, students should be able to • demonstrate familiarity with the main methods based on linear models; • apply these methods to analyse data and interpret the results from such analysis; • understand and apply non-parametric methods; • use MINITAB effectively in the analysis of relevant data.

Course Content:

Principles of statistical modelling and terminology: Systematic and random components, types of variables. Simple and multiple linear regression: Matrix notation, fitting the model, inferences about individual regression parameters, prediction, assessing the regression. Some special cases: Polynomial models, models that incorporate factors. Model building: Testing significance of specified subsets of variables, examining all subsets. Model validation and comparison of regressions: Examination of residuals, influential observations, some possible problems and remedial actions, dummy variables. Qualitative explanatory variables - analysis of variance: One-way and two-way ANOVA, point estimation, linear contrasts, a general approach via multiple regression. Some non-parametric methods: The sign test, the Wilcoxon test, the Kolmogorov-Smirnov goodness-of-fit test.


33 hours of lectures and examples classes, 11 hours of practical classes on MINITAB. 106 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Introduction to Statistical Modelling − W J Krzanowski (Arnold) Library Ref. 518.3 KRZ Applied Regression Analysis and Other Multivariable Methods − D G Kleinbaum, L L Kupper and K E Muller (Duxbury Press 1998) Library Ref. 518.3 KLE A Second Course in Statistics: Regression Analysis – W Mendenhall and T Sincich (Prentice Hall 2003) Library Ref. 518.3 MEN John E Freund’s Mathematical Statistics – I Miller and M Miller (Prentice Hall 1999) Library Ref. 518.3 FRE




Exam (80%) Four questions out of five in a two-hour paper Coursework (20%) Miniproject


Course Title: Groups and Graphs

Prerequisites: MT1810, MT1820

Aims:

One of the most significant developments in 20th century mathematics was the inux of algebra into all parts of mathematics, including classical areas such as geometry and number theory. The course introduces and gently develops the elementary theory of two fundamental kinds of algebraic objects: groups and graphs. Groups, which measure `symmetry', and graphs, which provide discrete models of `connectivity', are ubiquitous in modern mathematics and play an important role in cognate areas of computer science and physics. The course places an emphasis on concrete examples. Key aims of the course include:

to provide an introduction to groups and graphs;

to develop the skill of connecting concrete examples with general mathematical concepts, and to practise algebraic reasoning.

Learning Outcomes:

On completion of the course, students should be able to:

understand and apply the fundamental concepts of group theory;

know basic examples of groups, subgroups and factor groups;

recognise multiplicative and additive groups when they occur as automorphism

groups or as part of more complex structures such as rings or vector spaces;

be able to recognise and to construct homomorphisms and isomorphisms;

know and apply Lagrange's Theorem;

understand and apply the fundamental concepts of graph theory.

Course Content:

Groups: Definition of a group; basic examples of groups like cyclic groups, permutation groups, matrix groups; basic properties of groups and of group elements, e.g. order and (non-)commutativity; group homomorphisms and isomorphisms; conjugation; subgroups, cosets and Lagrange's Theorem; normal subgroups and factor groups; generators and relations. Graphs: Definition of a graph; basic examples; elementary properties of graphs, e.g. connectivity; early highlights such as the Handshaking Lemma; graph isomorphisms; special graph classes, e.g. complete graphs, paths, cycles, planar graphs, trees; characterisations of trees. Connections between groups and graphs: Cayley graphs, automorphisms of graphs.


33 hours of lectures and examples classes. 117 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course lecturer if the student wishes.

Key Bibliography:

Groups and Symmetry - M.A. Armstrong A first course in Discrete Mathematics - I. Anderson


Formative assessment in form of 8 problem sheets. The student will receive feedback as written comments on their attempts.




Course Title: Linear Algebra and Project


Aims: • To develop the matrix theory covered in MT1820. • To learn how to put together different aspects of mathematics via a project. • To improve spoken and written communication.

Learning Outcomes:

On completion of the course, the student should be able to: • understand rank and nullity of a linear map and the connection between them; • calculate orthonormal bases; • diagonalise real symmetric matrices; • check positivity of a real symmetric matrix in various ways; • apply the least squares method; • work in a team to prepare a (mathematical) report and a presentation.

Course Content:

Linear algebra: Laplace expansion of determinants, minors and cofactors. Bases and change of basis. Linear maps and their matrix representations. Range and kernel. Row rank, column rank, rank and nullity. Sylvester’s law of nullity. Diagonalisation of a diagonalisable matrix. Algebraic and geometric multiplicities. Introduction to Jordan normal form. Abstract definition of an inner product. Orthonormal bases. Gram-Schmidt orthogonalisation procedure. Symmetric and positive definite matrices. Quadratic and positive definite forms. Bringing a quadratic form into canonical form. Sylvester’s law of inertia. Testing positivity of a real symmetric matrix. Change between orthonormal bases. Orthogonal matrices. Spectral theorem. Singular value decomposition of real matrices. Least squares approximation. Moore-Penrose inverse. Project: Do’s and dont’s of giving an effective presentation. Preparation of a small project on a mathematical topic chosen from an approved list, together with three or four other students. Writing of an agreed project on the project, and a joint oral presentation on the project.


28 hours of lectures and examples classes, 10 hours of problem workshops, 12 hours of group study and preparation of report and oral presentation 100 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

Linear Algebra: a Modern Introduction – D Poole (Brooks-Cole) 512.3 POO


Formative assignments in the form of 8 problem sheets. The students will receive feedback as written comments on their attempts. For the project, the students in each group will receive 30 minutes of guidance from a staff member, who may comment on one draft.


Exam (80%) Four questions out of five in a two-hour paper Coursework Group project (see above): oral presentation in which each student must participate (max 20 minutes total) 10%; written report 10%. The written report must be word processed and no more than 3000 words in length. Two copies will be required. It should be submitted jointly, with all group members signing a form to indicate that it was their agreed work. In addition, each student must submit a signed short statement (no more than a page of A4) indicating in general terms their own contribution to the project.


Course Title: Real Analysis


Aims:

• To explain the rigorous definition of limit of a function of a positive integer variable; • To discuss convergence of series, including power series; • To discuss the concepts of continuity and differentiability of functions of a real variable x; • To show how the Riemann integral is constructed.

Learning Outcomes:

On completion of the course, students should be able to: • quote the Weierstrass definition of a limit and verify it in simple cases; • use standard tests to investigate the convergence of commonly occurring series; • specify the power series of standard functions; • understand the Intermediate Value Theorem and the Mean Value Theorems; • understand the constructive approach of the Riemann integral.

Course Content:

Sequences and series: Sequences which tend to a limit. Absolute convergence of series; use of comparison and ratio tests for absolute convergence; absolute convergence implies convergence. Conditional convergence, alternating series test. Differentiation: The intermediate value theorem. Differentiability at a point – definition and geometric interpretation, with examples. Differentiability implies continuity. Derivative of a sum, product, quotient, and the chain rule (with application to inverse functions). Differentiability on an open interval; Rolle’s theorem, Mean Value Theorem, Cauchy’s Mean Value Theorem, with applications including l’Hôpital’s rule. Taylor’s theorem with (one) remainder. Power series: Existence of radius of convergence, and use of ratio test to find it. Power series can be differentiated term-by-term within the circle of convergence. Formal definition and properties of exp, sin, cos, etc., and (using the inverse function) of log, sin-1 etc. Periodicity of sin and cos. Riemann integral: Upper and lower sums, leading to definition and properties of Riemann integral. Fundamental theorem of calculus. Integral test for convergence of series.



Key Bibliography:

Numbers and Functions (steps into analysis) – R P Burn (Cambridge 2000). Library Ref. 515.23 BUR Yet Another Introduction to Analysis – Victor Bryant (Cambridge 1990). Library Ref. 515 BRY






Course Title: Probability


Aims:

To introduce the formalism of the mathematical theory of probability and thereby to lay a firm foundation for applications of probability in virtually all areas of science, including statistics, economics, the mathematics of financial markets, operational research, information theory, number theory, quantum theory and statistical physics.

Learning Outcomes:

On completion of the course, the student should be able to • demonstrate an understanding of the basic principles of the mathematical theory of probability; • use the fundamental laws of probability to solve a range of problems; • prove simple theorems involving both discrete and continuous random variables; • explain the weak law of large numbers and the central limit theorem.

Course Content:

Elements of probability: Sets and events. Axioms of probability. Independent events. Conditional probability. Bayes’ theorem. Discrete random variables: Probability distribution function and cumulative distribution function. Joint distribution, marginal distribution, independence. Distribution of a function of a random variable. Expectation, variance, covariance. Binomial and Poisson distributions, with application to simple combinatorial problems. Chebychev’s inequality and the weak law of large numbers. Further topics, such as probability generating functions, the hypergeometric and negative binomial distributions. Continuous random variables: Expectation and variance. Normal and exponential distributions. Joint and marginal densities, independence. Transformations of random variables. Normal approximation to the binomial distribution. Statement of the central limit theorem. Further topics such as moment generating functions and the gamma distribution. Further topics: Possible further topics include applications in the fields of geometric probability and simple random walks, such as Buffon’s needle problem, simple problems related to the geometry of points chosen at random in the interior of squares and circles, the gambler's ruin problem, the reflection principle, the ballot theorem, first passage times.



Key Bibliography:

TBC






Course Title: Mathematical Programming


Aims:

• To show how a wide range of problems may be formulated as linear programming (LP) problems; • To show how two-variable problems may be solved using graphical methods; • To show how the general LP problem can be solved using the simplex algorithm; • To show how duality in LP fits into a more general framework using Lagrange multipliers; • To show how the transportation and allocation problems may be solved using special algorithms; • To show how the integer constraint may be taken into account.

Learning Outcomes:

On completion of the course the students should be able to: • formulate problems as LP problems; • solve two-variable problems by graphical means; • use the simplex algorithm to solve small LP problems; • use the stepping-stone method to solve transportation problems; • use the Hungarian algorithm to solve assignment problems; • solve integer programming problems using the branch and bound method.

Course Content:

Linear programmes: Examples of formulation of problems as LPs. Examples with solutions by diagram. Unbounded and infeasible LPs. General LPs: feasible and basic feasible solutions. Simplex method: How you solve with the simplex method, with the emphasis on

elementary row operations. The formula for the j. Getting started: the two-phase method, infeasibility. Proof of termination, cycling. Brief mention of complexity. Duality: Its meaning; dual of the dual is the primal; the dual solution; the duality theorem; the simplex tableau displays the dual solution. Complementary slackness and testing for optimality. Dual simplex method. Duality in a wider context: graphical description of Lagrange multipliers in a constrained optimization; the dual variables as Lagrange multipliers. Sensitivity analysis: Examples of different types. Illustrations using a package. Transportation problem: Loops and basic feasible solutions; setting out the algorithm; formulation of problems as transportation problems. Assignment problems: Why the transportation method is inefficient. Hungarian algorithm, proof of convergence. Integer programming: Formulation of problems using binary variables; capital budget problems, fixed charge problems. Solution methods: branch and bound.



Key Bibliography:

Elementary Linear Programming with Applications – B Kolman and R E Beck (Academic Press). Library Ref. 519.41 KOL






Course Title: Ordinary Differential Equations and Fourier Analysis

Prerequisites: MT1710, MT1720 and MT1820

Aims:

This course aims to introduce the concepts of eigenvalues and eigenfunctions in the familiar situation of the trigonometric differential equation and to show how these yield Fourier series expansions for a general function. These Fourier series can be generalized to (a) generate more general eigenfunction expansions for a given function and (b) develop the Fourier transform, which is used in a variety of applications; its properties are investigated. The final step is to introduce a technique for solving differential equations where the coefficients are no longer constant.

Learning Outcomes:

On completion of the course, students should be able to: • locate eigenvalues both analytically and graphically; • determine the Fourier series for a periodic function, including odd and even functions, and recognize the function represented by a given Fourier series; • understand the role of eigenfunctions in building up a general function; • orthogonalize a set of polynomials over a specified interval; • manipulate the Dirac delta-function; • manipulate and apply the Fourier transform; • complete a solution-in series in straightforward cases.

Course Content:

Introduction to Sturm-Liouville theory: eigenvalues and eigenfunctions; self-adjoint operators, orthogonal functions and their properties, orthogonalization, completeness of eigenfunctions. Laguerre polynomials, Legendre polynomials. Fourier series: Fourier-Euler formulae and statement of Fourier Theorem on [−p ,p ], Fourier sine and cosine formulae, extension to general analysis. The Fourier transform: Fourier transform of derivatives, statement of Inversion Theorem, Dirac delta-function, Convolution Theorem, Parseval Theorem. Ordinary differential equations: The Cauchy-Euler equation. Solution in series for a second-order linear differential equation, for two out of the four cases that can arise.



Key Bibliography:

Elementary Differential Equations and Boundary Value Problems − W E Boyce & R C di Prima (Wiley). Library Ref. 515.41 BOY






Course Title: Primes and Factorisation


Aims:

One of the most significant developments in 20th century mathematics was the influx of algebra into all parts of mathematics, including classical areas such as number theory. The course introduces and develops the elementary theory of rings. It shows how this theory can be applied to study the problem of factorising integers into primes and shows how this situation can be generalised in a natural way. The course places an emphasis on concrete examples, such as rings of integers and polynomial rings; it aims to develop the skill of connecting these concrete examples with general mathematical concepts, and to practice algebraic reasoning.

Learning Outcomes:

On completion of the course, students should be able to:

understand and apply the fundamental concepts of ring theory;

know basic examples of rings;

investigate the structure and detect properties of explicit rings;

recognise and construct ring homomorphisms and quotients;

know and apply key theorems such as Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

Course Content:

Commutative ring theory: Rings and subrings; homomorphisms and ideals; factor rings; the first isomorphism theorem for rings and the correspondence theorem; prime ideals, maximal ideals and their use in constructing fields; Kronecker's theorem that every field can be extended to include a root of a given irreducible polynomial. Factorisation, applications and further topics: Integers and polynomial rings; zerodivisors and group of units; irreducibles and factorisation; unique factorisation domains; principal ideal domains; the Gaussian integers as an example of a Euclidean ring; quotients of integers and polynomial rings, including their groups of units, via the Chinese Remainder Theorem; if time permits, suitable further topics, for instance, simple primality tests, applications to algebraic geometry, or the Hilbert basis theorem.



Key Bibliography:

Introduction to Algebra - P.J.Cameron (Oxford Univ Press) 512.11 CAM A First Course in Abstract Algebra with Applications - J.J.Rotman (Pearson Prentice Hall) 512.02 ROT


Formative assessment in form of 8 problem sheets. The student will receive feedback as written comments on their attempts.


Exam (100%) Four questions out of five in a two hour paper


Course Title: Complex Variable

Prerequisites: MT1710, MT1720 and MT1810

Aims: This course is designed to provide an outline of the basic complex variable theory with some proofs. Applications are exhibited as used in other areas of mathematics. The object is to equip students to be able to use complex analysis to solve specific problems.

Learning Outcomes:

On completion of the course, the students should be able to: • use the definitions of continuity and differentiability of a complex valued function at a point, establish the necessity of the Cauchy-Riemann equations and apply this result; • use a power series to define the complex exponential function and hence define the trigonometric and hyperbolic functions and the complex logarithm, and establish their properties; • use the parametric definition of a contour integral in specific straightforward examples; • state and use Cauchy’s Theorem, and apply Cauchy’s Integral Formulae to evaluate integrals; • obtain Taylor series of rational and other functions of standard type; • determine zeros and poles of given functions, and the residue at a simple pole and at higher order poles; • state Cauchy’s Residue Theorem and apply it to evaluate real integrals (using Jordan's lemma when relevant) and to sum certain series, and state and use Rouché’s Theorem.

Course Content:

Special functions: Power series and radius of convergence. Discussion of the exponential, trigonometric and hyperbolic functions for both real and complex variable. Definition of log z and a z . Topology: An open (pathwise) connected set of the plane. Functions of a complex variable: Continuity and differentiability of functions defined on an open set. The Cauchy-Riemann equations and Laplace's equation. Contour integrals along piecewise smooth curves C, defined by ∫ f (z)dz = ∫ f (z(t))z'(t)dt. Cauchy's theorem and Cauchy's integral formulae. Taylor series with examples, removable singularities, zeros and poles. Residue theorem and applications: calculation of simple integrals, including use of Jordan's lemma, and summation of infinite series. Principle of the argument. Rouché's theorem and the location of zeros of polynomials.


33 hours of lectures and examples classes, 117 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the courseleader if the student wishes.

Key Bibliography:

Complex Analysis – J M Howie (Springer 2003). Library Ref. 515.24 HOW Theory and Problems of Complex Variables − M R Spiegel (Schaum 2007). Library Ref. 510.76 SPI Advanced Engineering Mathematics 8th ed. – E Kreysig (Wiley 2005). Library Ref. 510.245 KRE





Documents

Studying Mathematics at Royal Holloway Assessment & Feedback: Formative assignments in the form of 11 problem sheets. ... Sets: intersection, union, complement, Venn diagrams, De Morgan’s