Undergraduate Modules for Academic Year 2012–13 (last updated 15 May 2012)

MTH6100 Actuarial Mathematics

Description

This module gives an introduction to the mathematics of life assurance. You will learn to value cash flows and use life tables for making predictions and analysing mortality patterns. This leads on to the valuation of life annuities and of the benefits paid in life assurance policies. Various life assurance products will be explained and then used for illustration of the basic principles of life assurance.

Details

Organiser: Dr R J Harris

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and either of

Helpful prerequisites:

Any of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~harris/MTH6100

Syllabus

  1. Compound interest: discounting, force of interest, nominal values (APR); annuities certain: accumulated amount; schedule of principal and interest; perpetuities.
  2. Life tables (LT): LT functions; the LT as model of cohort experience or stationary distribution; survival probabilities in terms of LTs. Reference to actual populations: tables of annuitants and assured lives. Select LTs.
  3. Valuation: monetary functions; values of endowments, annuities and assurances.
  4. Calculation of premiums; policy and surrender values; paid up policies.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6100

Learning resources

Recommended:

MTH742U Advanced Combinatorics

Description

This module builds on the combinatorial ideas of the modules Combinatorics and Extremal Combinatorics and introduces some of the more advanced tools for solving combinatorial and graph theoretic problems. The topics covered will depend on the module organiser's expertise but significant emphasis will be on the techniques used as well as the results proved.

Details

Organiser: Professor T Müller

Level: 7 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~twm/MTH742U

Syllabus

This module aims to introduce students to some of the more advanced techniques used in combinatorics such as the Regularity Lemma, probabilistic techniques, the discrete Fourier transform, eigenvalue methods and generating functions with the intention that the students will be able to recognise and then apply the appropriate tools in unfamiliar situations.

Learning outcomes

Learning resources

Main texts:

MTH6103 Advanced Statistics Project

Description

The major part of this module is an individual project on some aspect of probability, statistical theory or applied statistics. There will also be classes, which will cover an introduction to project work, statistical study skills and report writing.

Details

Organiser: Professor R A Bailey

Level: 6 Credit value: 30 Semester: A and B

Overlaps:

Essential prerequisites:

and

Assessment: Written report, presentation and (possibly) oral exam

Organiser's module website: http://www.maths.qmul.ac.uk/~rab/MTH6103

Syllabus

The major part of this module is an individual project on some aspect of probability or statistical theory or applied statistics. It must be your own work in the sense that it gives an original account of the material, but it need not contain new mathematical results. The length should be the equivalent of between 3,500 and 7,000 words.

There will also be classes, which will cover the following:

  1. Introduction to project work; development of a project proposal.
  2. Statistical study skills, including use of literature, selection of appropriate methods of data analysis, selection of appropriate computer software.
  3. Report writing.

The project will be assessed primarily by a written report and, at the examiners' discretion, an oral examination, but also by a presentation of 20–30 minutes in duration, to be given towards the end of semester 6. The contribution of the presentation will be on a sliding scale that will never decrease the project mark by more than 10% or increase it by more than 20%, and provided you make a reasonable attempt at giving a presentation it will not decrease your project mark.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6103

MTH5100 Algebraic Structures I

Description

The modern axiomatic approach to mathematics is demonstrated in the study of algebraic structures. Elementary group theory includes the study of subgroups and Lagrange's theorem. Ring theory includes integral domains, ideals, homomorphisms and isomorphism theorems, polynomial rings, the Euclidean algorithm, and fields of fractions. This module has been revised for 2011–12 to focus more on ring theory, featuring familiar examples. Group theory will now mostly be covered by the follow-on module Algebraic Structures II.

Details

Organiser: Professor L H Soicher

Level: 5 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~leonard/MTH5100

Syllabus

  1. Revision of sets, functions, operations, relations, equivalence relations.
  2. Definition of group. Examples: symmetric groups, matrix groups. Cyclic groups. Subgroups, subgroup test. Cosets, Lagrange's theorem, index. Multiplication table, Cayley's Theorem.
  3. Definition of ring. Examples: matrix rings, residue class rings, division rings, fields. Gaussian integers. Integral domains, zero divisors, units, groups of units, examples. Euclidean functions, Euclidean domains, unique factorisation domains. Subrings, subring test.
  4. Homomorphisms. Image and kernel. Ideals. Construction of factor rings. Correspondence and isomorphism theorems. Generators for ideals. Principal ideal domains, maximal ideals. Polynomial rings. Construction of fields as factor rings. Finite fields. Field of fractions.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5100

Learning resources

Reading list:

MTH6104 Algebraic Structures II

Description

This is a second module in algebraic structures, covering group theory. There will be abstract thinking and proofs but also an emphasis on examples. The module includes the basics of group actions, finite p-groups, Sylow theorems and their applications, and the Jordan-Hölder theorem. The module has been revised for 2011–12 in order to make it more accessible by focussing on group theory although some of the ideas in the group theory are parallel to those first encountered in the earlier module on rings and numbers.

Details

Organiser: Dr M Fayers

Level: 6 Credit value: 15 Semester: A

Overlaps:

Helpful prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~mf/MTH6104

Syllabus

  1. Group homomorphisms, isomorphisms, automorphisms. Conjugacy and normal subgroups; examples.
  2. Construction of factor groups, 1st, 2nd and 3rd isomorphism theorems for groups; examples.
  3. Group actions; finite p-groups; Sylow theorems and applications.
  4. Jordan-Hölder theorem; finite soluble groups; examples.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

Main text:

Other text:

MTH705U Applied Statistics

Description

The semester will be divided into three four-week periods. In each a genuine application of statistics will be studied, led by a different lecturer with at most two lectures per period. The list of topics will vary from year to year and you should obtain the current list from the module organiser.

Details

Organiser: Dr H Grossmann

Level: 7 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Helpful prerequisites:

and

and

Assessment: 3 reports (about 10–15 pages each, on separate topics), 33% each

Organiser's module website: http://www.maths.qmul.ac.uk/~hg/MTH705U

Comment: Was semester B in 2011–12.

Syllabus

The semester will be divided into three four-week 'months'. In each month there is a genuine piece of applied statistics, led by a different lecturer. The lecturer will set it up with at most 2 lectures. At the end of the month the student will hand in a report of 10–15 pages. Statistical techniques and statistical computing packages from previous statistics modules will be needed. The three topics will be chosen from the following list:

  1. Design of experiments
  2. Medical statistics
  3. Time series analysis of spacecraft data
  4. Multivariate data from crop research
  5. Agricultural statistics
  6. Economic statistics
  7. Industrial statistics

See module organiser before registering.

Learning outcomes

Not yet available

MTH709U Bayesian Statistics

Description

The module aims to introduce you to the Bayesian paradigm. The module will show you some of the problems with frequentist statistical methods, show you that the Bayesian paradigm provides a unified approach to problems of statistical inference and prediction, enable you to make Bayesian inferences in a variety of problems, and illustrate the use of Bayesian methods in real-life examples.

Details

Organiser: Dr L Pettit

Level: 7 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Either of

Helpful prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~pettit/MTH709U

Syllabus

  1. The Bayesian paradigm – likelihood principle, sufficiency and the exponential family, conjugate priors, examples of prior to posterior analysis, mixtures of conjugate priors, non-informative priors, two sample problems, predictive distributions, constraints on parameters, point and interval estimation,hypothesis tests, nuisance parameters.
  2. Linear models – use of non-informative priors, normal priors, two and three stage hierarchical models, examples of one way model, exchangeability between regressions, growth curves, outliers and influential observations.
  3. Approximate methods – normal approximations to posterior distributions, Laplace's method for calculating ratios of integrals, Gibbs sampling, finding full conditionals, constrained parameter and missing data problems, graphical models. Advantages and disadvantages of Bayesian methods.
  4. Examples – appropriate examples will be discussed throughout the course. Possibilities include epidemiological data, randomised clinical trials, radiocarbon dating.

Learning outcomes

On completion of this course students should: Appreciate the differences between the Bayesian paradigm and frequentist statistical methods; Be able to calculate posterior and predictive distributions and related quantities for a range of likelihoods and priors; Be familiar with the ideas of Gibbs sampling as an example of a Markov chain Monte Carlo method and be able to set up models in this framework; Understand the arguments for and against the Bayesian paradigm.

Learning resources

Main text:

MTH4100 Calculus I

Description

This is the first of three calculus modules, whose collective aim is to provide the basic techniques and background from calculus for the pure, applied and applicable mathematics modules that follow. This module develops the concepts and techniques of differentiating and integrating, with supporting work on algebra, coordinate transformations and curve sketching.

Details

Organiser: Dr K Malik

Level: 4 Credit value: 15 Semester: A

Overlaps:

Prerequisites: A-Level Mathematics or equivalent

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~malik/MTH4100

Comment: Syllabus revised for 2012–13 to remove polar coordinates.

Syllabus

  1. Real numbers and the real line. Manipulation of algebraic equations involving the square root and modulus functions. Proving and solving algebraic inequalities. Triangle inequality. Trigonometric functions. Manipulation of trigonometric identities. Use of addition and double-angle formulas for expressing values of trigonometric functions in the surds form.
  2. Functions and their graphs. Piecewise defined functions. Odd and even functions. Periodic functions. Monotone functions. Functions defined implicitly. Composition of functions and functional inverse. Scaling (vertical and horizontal), reflecting and shifting a graph; combinations of transformations on graphs.
  3. Limits. Calculating limits using the Limit Laws and by eliminating zero denominators algebraically. The precise definition of a limit, finding deltas algebraically for given epsilons. One-sided limits and limits at infinity. Limits involving sin x/x. Limits at infinity of rational functions, horizontal and oblique asymptotes. Infinite limits and vertical asymptotes. Continuous functions. Intermediate Value Theorem for continuous functions (without proof) and its applications.
  4. Derivatives I. Tangents and derivatives. Instantaneous rate of change (derivatives at a point). Derivative as a function. Calculating derivatives from the definition. One-sided derivatives. Basic rules of differentiation. Derivatives of trigonometric functions. The chain rule and parametric equations. Second-order derivatives. Finding derivatives using table of derivatives and rules of differentiation, including repeated use of the chain rule. (Technical dexterity of finding derivatives will be checked using test assessments.)
  5. Derivatives II. Linearisation and differentials, estimating with differentials. Extreme values of functions. The Mean Value Theorem. Monotone functions and the First Derivative Test. Concavity and the Second Derivative Test. Curve sketching.
  6. Derivatives III. Indeterminate forms (0/0, ∞/∞, ∞ − ∞) of limits and l'Hopital's Rule. Limits involving (1 + 1/x)x. Logarithmic, power and exponential rates of growth. Implicit differentiation. Tangent and normal to a curve defined by f(x,y) = 0 or parametrically. Derivatives of inverse functions. Inverse trigonometric functions (graphs, derivatives). First look at hyperbolic functions (algebra of, inverse functions, derivatives, graphs).
  7. Integration I. Indefinite integral as anti-derivative. Techniques for evaluating indefinite integrals. Integration by substitution and integration by parts as the reverse processes of the chain rule and product rules.
  8. Integration II. Definite integral as limit of Riemann sums. Properties of definite integrals. Area under the graph of a non-negative function. The Fundamental Theorem of Calculus, evaluation of definite integrals. Area between curves. Area of a circle. First look at the improper integrals: area of unbounded planar regions.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4100

Learning resources

Main text:

MTH4101 Calculus II

Description

This is the second of three Calculus modules, whose collective aim is to provide the basic techniques from Calculus for the pure, applied and applicable mathematics modules that follow. This module introduces infinite series including power series and develops techniques of differential and integral calculus in the multivariate setting.

Details

Organiser: Dr R Klages

Level: 4 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~klages/MTH4101

Comment: Syllabus revised for 2012–13 to remove complex numbers and add first-order separable and linear ODEs.

Syllabus

  1. First-order separable differential equations; first-order linear differential equations by the method of integrating factor.
  2. Derivatives IV: Functions of two variables. Domain, range and level curves. Graphing a function of two variables. Limits and continuity in the xy-plane. Partial derivatives. Statement and use of ''mixed derivatives theorem'' without proof. Differentiability and continuity and related theorems without proof. Chain rule for functions of two and three variables. Implicit differentiation revisited (Implicit Function Theorem).
  3. Derivatives V. Directional derivatives and gradient vector. Tangent planes and normal lines. Estimating change in a specific direction. Linearization and differentials. Extreme values and saddle points. Lagrange multipliers.
  4. Integration III. Double integrals as volumes under surfaces and areas in the plane. Properties of double integrals. Double integrals as repeated integrals, rectangular regions, simple non-rectangular regions.
  5. Integration IV. Double integrals over non-rectangular regions, bounded and unbounded. Determining limits of integration. Reversing the order of integration. Transformations of variables – maps, domains, one-to-one maps, inverse maps.
  6. Integration V. Change of variables in double integrals. Jacobians. Transformation to polar coordinates. Other transformations. Applications to normal distribution in probability. Triple and multiple integrals. Change of variables of integration. Applications of integrals.
  7. Series I. Infinite sequences. Converging sequences. Diverging to infinity sequences. Calculating limits of sequences, including use of l'Hopital's rule. Infinite series (n-th term, partial sum, convergence, sum). Examples of converging series. Examples of diverging series. The n-th term test for divergence.
  8. Series II. Series of positive terms and the Integral Test for convergence (second look at improper integrals). The ratio test.
  9. Series III. Power series and convergence. Term-by-term differentiation and integration. Power series in the complex plane including existence of radius of convergence. Taylor and Maclaurin series and polynomials (via integration by parts). First look at the remainder term. Taylor series for common transcendental functions (exp, log, sin, cos, square root, cosh and sinh). Examples of applications of power series (power series solutions of differential equations, evaluating non-elementary integrals, evaluating indeterminate forms).

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4101

Learning resources

Main text:

MTH5102 Calculus III

Description

The module develops the elements of vector calculus and advanced topics in ordinary and partial differential equations, such as special functions, Fourier series and Laplace's equation, for application in subsequent applied mathematics modules.

Details

Organiser: Dr W J Sutherland

Level: 5 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~wjs/MTH5102

Syllabus

  1. Arc-length of plane curves: length of a parametric curve, length of a curve y = f(x) . Length of the circumference of a circle, ellipse. Area and length in polar coordinates.
  2. Vector fields, line, surface and volume integrals.
  3. Grad, div and curl operators in Cartesian coordinates. Grad, div, and curl of products etc. Vector and scalar forms of divergence and Stokes's theorems. Conservative fields: equivalence to curl-free and existence of scalar potential. Green's theorem in the plane.
  4. Orthogonal curvilinear coordinates; length of line element; grad, div and curl in curvilinear coordinates; spherical and cylindrical polar coordinates as examples.
  5. A first look at Legendre polynomials.
  6. Fourier series: full, half and arbitrary range series. Parseval's Theorem.
  7. Laplace's equation. Uniqueness under suitable boundary conditions. Separation of variables. Two-dimensional solutions in Cartesian and polar coordinates. Axisymmetric spherical harmonic solutions.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5102

Learning resources

Main text:

Other texts:

MTH6107 Chaos and Fractals

Description

The main aims are two fold: to illustrate (rigorously) how simple deterministic dynamical systems are capable of extremely complicated or chaotic behaviour; to make contact with real systems by considering a number of physically motivated examples and defining some of the tools employed to study chaotic systems in practice. Discrete and continuous dynamical systems, repellers and attractors, Cantor sets, symbolic dynamics, topological conjugacy for maps, definition of chaos. Fractals, iterated function systems, Julia sets.

Details

Organiser: Professor F Vivaldi

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~fv/MTH6107

Syllabus

  1. Continuous-time and discrete-time dynamical systems, Poincaré surface of section.
  2. Fixed points, periodic orbits and their stability, 1-dimensional diffeomorphisms and their periodic orbits. Sharkovsky's theorem.
  3. The logistic map, period-doubling scenario, Feigenbaum constants and Feigenbaum-Cvitanovic equation, tangent bifurcation and intermittency.
  4. Definition of chaos, Lyapunov exponents, Bernoulli shift, topological conjugacy, symbolic dynamics.
  5. Invariant measures and invariant densities, Perron-Frobenius operator, time and ensemble average, ergodicity.
  6. Higher-dimensional maps, Jacobian matrix and stability of periodic orbits.
  7. Examples of simple fractals, fractal dimension, Renyi dimensions.
  8. Complex dynamics, Julia sets and Mandelbrot set, iterated function systems.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6107

Learning resources

Essential:

Recommended:

MTH6108 Coding Theory

Description

The theory of error-correcting codes uses concepts from algebra, number theory and probability to ensure accurate transmission of information through noisy communication links. Basic concepts of coding theory. Decoding and encoding. Finite fields and linear codes. Hamming codes. Parity checks. Preliminary algebra on vector spaces and finite fields will be included in the module.

Details

Organiser: Dr M Fayers

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~mf/MTH6108

Syllabus

The concept of an error-correcting code is a very important one, with wide applications in communications. This module approaches the subject from a pure-mathematics perspective, to give the student a thorough grounding in construction of codes and decoding algorithms, and the main coding theory problem.

  1. Basic concepts: codes, minimum distance, equivalence of codes.
  2. The Main Coding Theory Problem. The Hamming, Singleton and Plotkin bounds.
  3. Noisy channels and nearest-neighbour decoding. Statement of Shannon's Noisy Coding Theorem.
  4. Linear codes. Generator matrices and parity-check matrices. Syndrome decoding. (A very brief review of the required linear algebra will be given.)
  5. Examples: Reed-Muller, Hamming, Golay and MDS codes.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6108

Learning resources

Main text:

Other text:

MTH6109 Combinatorics

Description

Combinatorics involves reasoning about 'discrete' structures, particularly finite sets of objects where there are links or relationships among the objects. The module is largely concerned with concepts and theory, but this is a subject that has many practical applications. Counting, recurrence relations, permutations. Steiner triple systems: construction and properties. Ramsey's theorem and applications. Transversal theory.

Details

Organiser: Professor R A Wilson

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Helpful prerequisites:

Either of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~raw/MTH6109

Syllabus

  1. Counting, binomial coefficients, recurrence relations, generating functions, partitions and permutations, finite fields, Gaussian coefficients.
  2. Steiner triple systems, necessary conditions, direct and recursive constructions, structural properties and characterisations.
  3. Ramsey's theorem, illustrations, proof and applications.
  4. Transversal theory, Latin squares, Hall's theorem, upper and lower bounds.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6109

Learning resources

Main text:

Other texts:

MTH6110 Communicating and Teaching Mathematics: the Undergraduate Ambassadors Scheme

Description

This module allows undergraduates to gain valuable transferable skills whilst exploring the teaching profession first hand by working with a teacher in a local school. The key skills gained include communication and presentation of mathematics, team-working, active listening, time management and prioritisation. The module will be supported by regular classes and assessed by a combination of written reports and an oral presentation. Registration for this module requires validation; places will be limited and interviews to assess suitability will be held during Semester A.

Details

Organiser: Dr C Agnor / Dr D Ellis

Level: 6 Credit value: 15 Semester: B

Overlaps: None

Essential prerequisites:

Restrictions: Restricted to Queen Mary Mathematical Sciences students.

Assessment: Journal of teaching activity (2,500 words) 30%; end-of-module report on ``special project'' (2,500 words) 30%; end-of-module presentation on ``special project'' (10 minutes plus 5 minutes discussion) 20%; teacher's end-of-module report 20%

Organiser's module website: http://www.maths.qmul.ac.uk/~agnor/MTH6110

Syllabus

Students will typically begin by observing the teacher's handling of the class and progress from this classroom assistant stage through small teaching tasks to at least one opportunity to undertake whole class teaching, possibly for a short part of a lesson. They will represent and promote mathematics as a potential university choice.

Students will undertake and evaluate a special project on the basis of discussion with the teacher. This may involve a specific in-class teaching problem or an extra-curricular project such as a lunchtime club or special coaching periods for higher ability pupils. The student will keep a journal of their own progress in working in the classroom environment, and they will be asked to submit a reflective written report on the special project and other relevant aspects of the school placement experience. This format is standard within the Undergraduate Ambassadors Scheme (http://www.uas.ac.uk/).

  1. Initial day of training.
  2. Competitive interview system to ensure students' suitability for the module.
  3. Student will be matched with an appropriate school and a specific teacher in the local area.
  4. Student will spend the equivalent of half a day a week in the school every week for a semester.
  5. No formal lectures.
  6. A supporting tutorial for one hour once a week for students to share experiences.
  7. Teachers will act as the main source of guidance but students will also be able to discuss their progress with the module organiser and the More Maths Grads team as needed.
  8. End of unit presentation of special project (15 minutes per student)

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6110

Learning resources

You should consult:

MTH6111 Complex Analysis

Description

This is a rigorous module in complex analysis. The first part of the course will be concerned with detailed analysis of topics already seen in Complex Variables, such as differentiation, integration, Taylor and Laurent series, conformal mappings and the residue theorem. The second part of the course will introduce more advanced topics, perhaps including Riemann surfaces and elliptic functions.

Details

Organiser: Professor C-H Chu

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~cchu/MTH6111

Syllabus

A rigorous module in complex analysis. The first part of the module will be concerned with detailed analysis of topics already seen in Complex Variables:

  1. Differentiation and integration
  2. Cauchy's theorem, Taylor and Laurent series
  3. Conformal mappings and harmonic functions
  4. The residue theorem and the calculus of residues

The second part of the module will introduce more advanced topics, e.g. some or all of

  1. Riemann surfaces
  2. Complex gamma, beta and zeta functions
  3. Elliptic functions
  4. Picard's theorem

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6111

Learning resources

Consult module organiser before buying any book specifically for this module since we shall be using a number of texts. Possibilities include

MTH6142 Complex Networks

Description

This module provides an introduction to the basic concepts and results of complex network theory. It covers methods for analyzing the structure of a network, and for modeling it. It also discusses applications to real systems, such as the Internet, social networks and the nervous system of the C. elegans.

Details

Organiser: Professor V Latora

Level: 6 Credit value: 15 Semester: B

Overlaps:

Prerequisites: Level-5 mathematical background

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~latora/MTH6142

Comment: New module in 2012–13 to replace MTH6105 Algorithmic Graph Theory.

Syllabus

  1. Basic definitions and metrics: walks, paths, cycles, connectedness, trees.
  2. Social networks and centrality measures: degree and eigenvector centrality, closeness, betweenness. The collaboration network of movie actors.
  3. Erdős-Renyi random graph models: degree distribution, giant connected component, characteristic path length.
  4. Small-world networks. Six degrees of separation and the nervous system of C.elegans. The clustering coefficient.
  5. The World Wide Web. Scale-free networks. Random graphs with a given degree sequence. The Molloy and Reed criterion.
  6. Citation networks and the linear preferential attachment. The Barabasi-Albert model and other models of growing graphs.
  7. Degree correlations. The Internet and other assortative and disassortative networks.
  8. Motif analysis and network superfamilies. Community structures: spectral bisection and hierarchical clustering methods. The modularity and Girvan-Newman algorithm.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

Main text:

A printed detailed course summary will be available from the web.

Other texts:

Newman, Networks: An Introduction, Oxford University Press 2010

MTH743U Complex Systems

Description

Complex systems can be defined as systems involving many coupled units whose collective behaviour is more than the sum of the behaviour of each unit. Examples of such systems include coupled dynamical systems, fluids, transport or biological networks, interacting particle systems, etc. The aim of this module is to introduce students with a number of mathematical tools and models used to study complex systems and to explain the mathematical meaning of key concepts of complexity science, such as self-similarity, emergence, and self-organisation. The exact topics covered will depend on the module organiser's expertise with a view to cover practical applications using analytical and numerical tools drawn from other applied modules.

Details

Organiser: Dr W Just

Level: 7 Credit value: 15 Semester: B

Overlaps: None

Prerequisites: Level-6 mathematical background

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~wj/MTH743U

Syllabus

  1. Introduction to the field of complex systems via a number of representative examples and models of these systems (e.g., coupled dynamical systems, time-delayed systems, stochastic processes, networks, time series, fractals, multifractals, particle models)
  2. Introduction to basic tools and quantities used in the study of complex systems (e.g., bifurcation diagram, symbolic dynamics, dimensions, Lyapunov exponents, complexity measures, entropies)
  3. Introduction to the concepts of emergence and self-organisation in the context of basic models of complex models
  4. Introduction to basic computational and numerical methods used to study complex systems

Learning outcomes

Learning resources

Main texts:

MTH5103 Complex Variables

Description

The integral and differential properties of functions of a complex variable. Complex differentiation, Cauchy-Riemann equations, harmonic functions. Sequences and series, Taylor's and Laurent's series, singularities and residues. Complex integration, Cauchy's theorem and consequences, Cauchy's integral formula and related theorems. The residue theorem and applications to evaluation of integrals and summation of series. Conformal transformations.

Details

Organiser: Professor B J Carr

Level: 5 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~bjc/MTH5103

Syllabus

  1. Complex numbers, functions, limits and continuity.
  2. Complex differentiation, Cauchy-Riemann equations, harmonic functions.
  3. Sequences and series, Taylor's and Laurent's series, singularities and residues.
  4. Complex integration, Cauchy's theorem and consequences, Cauchy's integral formulae and related theorems.
  5. The residue theorem and applications to evaluation of integrals and summation of series.
  6. Conformal transformations.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5103

Learning resources

Other texts:

MTH731U Computational Statistics

Description

This module introduces modern methods of statistical inference for small samples, which use computational methods of analysis, rather than asymptotic theory. Some of these methods such as permutation tests and bootstrapping, are now used regularly in modern business, finance and science.

Details

Organiser: Dr H Maruri-Aguilar

Level: 7 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Either of

Assessment: 30% coursework, 70% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~maruriag/MTH731U

Comment: Was Semester A in 2011–12.

Syllabus

The techniques developed will be applied to a range of problems arising in business, economics, industry and science. Data analysis will be carried out using the user-friendly, but comprehensive, statistics package R.

  1. Probability density functions: the empirical cdf; q − q plots; histogram estimation; kernel density estimation.
  2. Nonparametric tests: permutation tests; randomisation tests; link to standard methods; rank tests.
  3. Data splitting: the jackknife; bias estimation; cross-validation; model selection.
  4. Bootstrapping: the parametric bootstrap; the simple bootstrap; the smoothed bootstrap; the balanced bootstrap; bias estimation; bootstrap confidence intervals; the bivariate bootstrap; bootstrapping linear models.

Learning outcomes

Learning resources

Main text:

MTH5104 Convergence and Continuity

Description

This module introduces some of the mathematical theory behind Calculus. It answers questions such as: What properties of the real numbers do we rely on in Calculus? What does it mean to say that a series converges to a limit? Are there kinds of function that are guaranteed to have a maximum value? The module is a first introduction, with many examples, to the beautiful and important branch of pure mathematics known as Analysis.

Details

Organiser: Professor S R Bullett

Level: 5 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~sb/MTH5104

Syllabus

  1. Real numbers: Algebraic properties, inequalities, supremum and infimum, completeness axiom for the existence of the supremum.
  2. Sequences: Definition of limit and its use in specific examples, limit of sum, product and quotient of sequences. Bounded monotone sequences. Statement of the Bolzano-Weierstrass Theorem.
  3. Series: Convergent series, geometric series, harmonic series. Comparison and ratio tests. Absolutely convergent series. Power series. Examples, including sin(x), cos(x) and exp(x).
  4. Continuous functions: Definition of continuity and its use in specific examples, sum of continuous functions, composites of continuous functions (proofs), products/quotients of continuous functions (stated). Briefly, the Intermediate Value Theorem, application to roots of polynomials, boundedness of continuous functions on closed bounded intervals.
  5. Definition of derivative. Continuity of differentiable functions. (Cover if time permits, also covered at start of next module, Differential and Integral Analysis.)

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5104

Learning resources

Main texts:

MTH6115 Cryptography

Description

Cryptography is fundamental to commercial life; in particular, the principles of public-key cryptography were a major intellectual achievement of the last century. The module will give you a detailed understanding of the subject.

Details

Organiser: Dr B Noohi

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and either of

Assessment: 10% coursework, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~noohi/MTH6115

Syllabus

  1. History and basic concepts: substitution and other traditional ciphers; plaintext, ciphertext, key; statistical attack on ciphers.
  2. One-time pad and stream ciphers: Shannon's Theorem; one-time pad; simulating a one-time pad; stream ciphers, shift registers.
  3. Public-key cryptography: basic principles, including brief discussion of complexity issues; knapsack cipher; RSA cipher; digital signatures.

Examples of optional topics which may be included: secret sharing, quantum cryptography, the Enigma cipher

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6115

Learning resources

Reading list:

MTH6116 Design of Experiments

Description

Experiments are carried out in all areas of business, industry, science and medicine. To obtain reliable information, the experiments must be carefully planned. This module introduces the statistical side of the design of experiments from consultation to interpretation.

Details

Organiser: Dr H Maruri-Aguilar

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 20% coursework, 80% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~maruriag/MTH6116

Syllabus

Real life experiments will be discussed from several applications in business, especially market research, industry and science, including medicine.

  1. Treatment structure: factors, main effects, interaction.
  2. Completely randomised designs.
  3. Blocking.
  4. Row-column designs.
  5. Experiments on people and animals.
  6. Nested blocks, split-plot designs.
  7. General orthogonal designs.
  8. Incomplete-block designs.
  9. Factorial designs in incomplete blocks.

Seven or eight lectures will be replaced by discussion sessions, when students present their solutions to assignments. Solutions are discussed by the whole class because most questions have no single correct answer.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6116

Learning resources

Reading list:

MTH4102 Differential Equations

Description

This is an applied calculus module, which follows on from the Calculus I and Geometry I modules. The purpose of the module is to develop techniques of solving differential equations and also to show how a higher-order differential equation can be seen geometrically as a vector field. This brings in discussions of matrices, eigenvalues and eigenvectors. Some applications are given.

Details

Organiser: Professor Y Fyodorov

Level: 4 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and

Restrictions: Not available to first-year Mathematical Sciences students.

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~yan/MTH4102

Comment: Assessment was 20% two in-term tests, 80% final exam. A level-5 differential equations module will be introduced in 2013–14.

Syllabus

  1. Revision of geometrical meaning of derivative, anti-derivative. Differentiation of combined and composed functions. Verification of solution of differential equation by substitution. Particular and general solutions. The role of initial or boundary conditions. Solution of simplest ODEs by direct integration. Separation of variables for first order differential equations, implicitly defined solutions.
  2. First order linear differential equation (integrating factors), homogeneous and inhomogeneous equations.
  3. Differential forms, integral curves, exact differential equations.
  4. Interpretation of first order differential equation in terms of direction fields, the initial value problem, solution by geometric method.
  5. Linear second order differential equations with constant coefficients, homogeneous equations, superposition, characteristic equations, real roots (including degenerate equal roots case), complex roots.
  6. Inhomogeneous equations with constant coefficients, method of undetermined coefficients, variation of constants formula, forced oscillations and visualisation.
  7. Matrices, eigenvalues and eigenvectors (2-dimensional).
  8. Linear systems in two dimensions, reduction of linear second order ordinary differential equation to a linear system in two variables. Various types of solution in terms of exponential functions.
  9. Phase space for two dimensional linear systems, stable/unstable nodes/foci, planar phase space portraits, classification of equilibria. Stability and instability of autonomous linear equations, characterisation of equilibrium points in terms of stability. Nonlinear systems – finding fixed points and their linearisations.
  10. The Linearisation Theorem and examples. Linearisation breakdown by examples.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4102

Learning resources

Main text:

MTH5105 Differential and Integral Analysis

Description

This module provides a rigorous basis for differential and integral calculus, i.e. the theory behind differentiation and integration rather than their applications. The module will include some full proofs.

Details

Organiser: Professor O Jenkinson

Level: 5 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~omj/MTH5105

Syllabus

  1. Differentiable functions: Definition of differentiability. Algebra of derivatives, chain rule. Derivative of inverse function. Rolle's Theorem, Mean Value Theorem and applications. Taylor's Theorem.
  2. Integration: Darboux definition of Riemann integral, simple properties. Continuous functions are integrable (via uniform continuity). Fundamental Theorem of the calculus, integral form of Mean Value Theorem and of the remainder in Taylor's Theorem; applications to some well known series (log, arctan, binomial). Improper integrals.
  3. Sequences of functions: pointwise and uniform convergence. Weierstrass M-test. Term-by-term integration of power series.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5105

Learning resources

Main text:

Other texts:

MTH744U Dynamical Systems

Description

A dynamical system is any system which evolves over time according to some pre-determined rule. The goal of dynamical systems theory is to understand this evolution. This module develops the theory of dynamical systems systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations and chaos. Much emphasis is placed on applications.

Details

Organiser: Dr T Prellberg

Level: 7 Credit value: 15 Semester: A

Overlaps:

Prerequisites: Level-6 mathematical background

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~tp/MTH744U

Syllabus

  1. First-order differential equations (one-dimensional flows): linear and nonlinear equations, graphical solutions, bifurcations.
  2. Two-dimensional flows: phase plane, stability of fixed points, periodic solutions, and limit cycles. Introduction to bifurcation theory, local and global bifurcations. Tools for studying global behavior of flows: Lyapunov functions, Poincare-Bendixson Theorem, gradient flows.
  3. Three-dimensional flows: Lyapunov exponents, Poincare sections, strange attractors, chaos.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

Essential:

MTH5106 Dynamics of Physical Systems

Description

Major developments in mathematics have been driven by the desire to describe and explain phenomena in the natural world. This module introduces you to the mathematical and physical concepts used in modelling physical systems. In particular, the module will explore Newton's laws of motion that govern how systems of particles react to forces, particularly gravity.

Details

Organiser: Professor R K Tavakol

Level: 5 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~reza/MTH5106

Comment: Differential Equations is required only if Calculus II was taken before 2012–13.

Syllabus

Some topics may already have been met in A-level Physics or Mechanics.

  1. Review of motion in space: displacement, velocity and acceleration using vectors; equation of motion; concept of constants of motion, energy and potentials; circular motion (plane polar coordinates).
  2. Mathematical modelling skills; from statement of problem to mathematical model; testing and evaluating a mathematical model.
  3. Newton's laws of motion. Examples of different types of motion due to forces and force fields, including resistive forces, and restoring forces: springs, ice hockey and parachutists.
  4. Newtonian model of gravity; sphere theorem; projectile motion and escape speed; variable mass: footballs, rockets and black holes.
  5. Central forces (e.g. gravity and Coulomb electrostatic forces); conditions for conservative force; potentials and conservation of angular momentum; orbit theory: polar equation of motion, types of orbit, Kepler's Laws: planets, asteroids and impact hazards.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5106

Learning resources

Texts:

MTH6117 Entrepreneurship and Innovation

Description

The aim of this module is to investigate the processes required to conceive and launch new ventures, including but not limited to enterprises based in mathematics and the sciences. We examine how to cultivate an entrepreneurial mindset and discuss the routes available for turning your ideas into successful ventures. The module provides an introduction to a number of key entrepreneurial skills such as opportunity recognition, market analysis, positioning and branding, pricing, financial planning, bootstrapping, leadership and personnel management, and how to sell yourself and your ideas to investors and other stakeholders.

Details

Organiser: Dr M Wells, City University (Mike.Wells.1 AT city.ac.uk); QM contact Dr V Easson

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Restrictions: Restricted to Queen Mary Mathematical Sciences students.

Assessment: 35% report (group work), 15% presentation (group work), 50% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~fjw/MTH6117

Syllabus

The aim of this module is to increase your awareness of the commercial opportunities available to you in the area of mathematical sciences. We examine how to cultivate an entrepreneurial mind set and discuss the routes available for turning your ideas into business ventures. The module provides an introduction to a number of crucial business skills such as financial planning, business planning and how to sell yourself and your ideas.

  1. Highlight the importance of commercialisation of innovative ideas both in the university and the industrial environment.
  2. Creatively explore commercial opportunities within mathematics and science.
  3. Introduce the different routes available to take an idea to market.
  4. Develop the skills required to start a business venture.
  5. Explain the key considerations involved in intellectual property and idea protection.
  6. Introduce the key aspects of financial management required in the development of a business venture.

Assessment will include a group 5-page business plan and 10-minute presentation, and an individual 1,000 to 2,000 word essay/report.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6117

Learning resources

Main text:

Optional texts:

MTH3100 Essential Mathematical Skills

Description

A module in basic arithmetic and algebra. Passing this module is compulsory for progression to the second year for students on Mathematical Sciences study programmes.

Details

Organiser: Professor B J Carr

Level: 3 Credit value: 0 Semester: A and B

Overlaps:

Prerequisites: A-Level Mathematics or equivalent

Restrictions: Not open to Associate Students.

Assessment: 100% multiple choice exam

Organiser's module website: http://www.maths.qmul.ac.uk/~bjc/MTH3100

Comment: Essential Mathematical Skills (EMS) is described in more detail in the preamble to the study programmes. Please note that we do not accept extenuating circumstances for this module because you already have 4 first attempts and 3 resit attempts within the first academic year.

Syllabus

  1. Decompose an integer as a product of prime numbers
  2. Calculate the GCD and LCM of a pair of integers
  3. Compute quotient and remainder of integer division
  4. Simplify arithmetical expressions involving fractions
  5. Perform simple estimations
  6. Compute quotient and remainder of polynomial division
  7. Simplify polynomial and rational expressions
  8. Simplify expressions involving square roots
  9. Perform algebraic substitutions
  10. Solve linear and quadratic equations and inequalities

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH3100

Learning resources

Main text:

MTH711U Extremal Combinatorics

Description

An extremal result is one which determines the extreme value of some parameter over a class of combinatorial structures. A classical example is Mantel's theorem which answers the question: how many edges can a graph have without it containing a triangle. Another instance is the Erdos-Ko-Rado theorem which answers the question: how large can a family of r-subsets of an n-set be if any two of them have non-empty intersection. This module will cover results of this flavour on both graphs and set systems. There will be an emphasis on techniques as well as results, including the use of linear algebra, probabilistic methods and compressions.

Details

Organiser: Dr O Sisask

Level: 7 Credit value: 15 Semester: A

Overlaps:

Helpful prerequisites:

Either of

and either of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~olof/MTH711U

Syllabus

  1. Extremal Graph Theory: introduction (what is an extremal problem/result, some simple examples). Cycles (Dirac's theorem). Complete Graphs (Turan's theorem). Zarankiewicz problem (bipartite analogue of Turan). Erdos-Stone theorem.
  2. The Discrete Cube: Sperner's theorem. Shadows and isoperimetric inequalities (LYM inequality, the Kruskal-Katona theorem, Harper's theorem, edge isoperimetric inequality).
  3. Intersecting Families: Erdos-Ko-Rado theorem. Katona's t-intersecting theorem. Brief discussion of uniform t-intersecting problem (with statement but not proof of Ahlswede and Khachatrian's complete intersection theorem). Modular intersections (Frankl-Wilson theorem and some extensions and applications).
  4. Other Topics: other topics of a similar flavour chosen according to class interest and time.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

The lecture notes will be self contained. Examples of books giving background material and further reading are:

MTH745U Further Topics in Algebra

Description

This module provides exposure to advanced techniques in algebra at an MSc or MSci level. Algebra encompasses familiar objects such as integers, fields, polynomial rings and matrices and has applications throughout mathematics including to geometry, number theory and topology. The module will complement the algebra module offered in Semester A and will cover topics either in commutative or noncommutative algebra. Included will be basic definitions and theorems in either case, normally with rings or fields as a starting point.

Details

Organiser: Professor A MacIntyre

Level: 7 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~angus/MTH745U

Syllabus

The aim of this module is to expose students to advanced techniques in algebra, which complement those presented in the module Group Theory. The module is also seen as a way to prepare students to study more advanced algebra subjects at PhD level.

The topics covered will depend on the expertise of the lecturer. These could be drawn from the theory of rings and their modules, Galois theory or elements of algebraic geometry. Commutative algebra or noncommutative algebra could also be covered.

Learning outcomes

MTH6120 Further Topics in Mathematical Finance

Description

This module develops the ideas discussed in Introduction to Mathematical Finance. As in the former module, concepts from analysis, differential equations, probability and, to some extent, statistics are used to develop further the techniques and language of mathematical finance. The difference is that in this module these techniques are used at a more advanced level.

Details

Organiser: Professor C Beck / Dr M Phillips

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Helpful prerequisites:

Either of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~beck/MTH6120

Syllabus

  1. Revision of: geometric Brownian motion; interest rates and present value analysis; the arbitrage theorem; the Black-Scholes Formula; properties of the Black-Scholes option cost; arbitrage strategy.
  2. Additional results on option.
  3. Valuing by expected utility.
  4. Deterministic and probabilistic optimisation models.
  5. Exotic options.
  6. Some examples beyond geometric Brownian motion models.
  7. Autoregressive models and mean reversion.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6120

Learning resources

Main text:

MTH4103 Geometry I

Description

Properties of two- and three-dimensional space turn up almost everywhere in mathematics. For example, vectors represent points in space, equations describe shapes in space and transformations move shapes around in spaces; a fruitful idea is to classify transformations by the points and shapes that they leave fixed. Most mathematicians like to be able to 'see' in special terms why something is true, rather than simply relying on formulas. This module ties together the most useful notions from geometry – which give the meaning of the formulas – with the algebra that gives the methods of calculation. It is an introductory module assuming nothing beyond the common core of A-level Mathematics or equivalent.

Details

Organiser: Dr J N Bray

Level: 4 Credit value: 15 Semester: B

Overlaps:

Prerequisites: A-Level Mathematics or equivalent

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~jnb/MTH4103

Comment: Was semester A in 2011–12.

Syllabus

  1. Phrasebook up to R3.
  2. Vectors in 2-space and 3-space, expressed as xi + yj + zk or as row or column vectors. Addition of vectors. Length of vectors.
  3. Vector and cartesian equations of a straight line in R2 and R3.
  4. Scalar multiple and scalar product of vectors in R2 and R3. Cartesian equation of a plane in R3. Intersections of two or three planes. Solution of families of linear equations in x, y, z by reduction to echelon form.
  5. Vector products in R3. Volume of parallelepiped as given by triple scalar product and determinant.
  6. Linear transformations in R2, expressed by matrices with respect to the standard basis i, j. Examples: rotations, reflections, dilations, shears; their matrices.
  7. In R2, characteristic equation, eigenvalues and eigenvectors, trace. Application to the examples in (6) (e.g. rotations with integer trace and the crystallographic restriction).
  8. Extension of (6), (7) to R3.
  9. Addition and multiplication of 2 × 2 and 3 × 3 matrices. Their interpretation as addition and composition of linear transformations. Inversion of matrices in R2 and in R3. (Examples and exercises may include 2 × 3 and 3 × 2 matrices.)

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4103

Learning resources

Main text:

Other texts:

MTH5109 Geometry II: Knots and Surfaces

Description

The module provides a first introduction to abstract ideas of differential geometry of curves and surfaces, with some elements of knot theory and hyperbolic geometry. Building on experience with vectors in three dimensions and elementary calculus, the modules asks what one can say of mathematical interest about an arbitrary curve or surfaces. The module starts with the problem of how to tell if a curve or piece of string is knotted (other than by pulling it about, i.e. mathematically) before moving on to differential geometric (calculus based) methods of measuring properties such as 'torsion' and 'curvature' of a curve or surface. The module ends with exposure to more abstract hyperbolic surfaces and higher-dimensional spaces defined by identifications and symmetries. There will be some proofs.

Details

Organiser: Professor S Majid

Level: 5 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~majid/MTH5109

Syllabus

  1. Knots and the unsolved problem of their classification. Reidemeister moves, Jones polynomial. Examples including trefoil, figure-eight.
  2. Parametrised regular curves, their curvature and torsion. Unit speed parametrisation and arc length.
  3. Principal normal, co-normal and theorem that torsion and curvature can be prescribed up to rigid motions.
  4. Planar curves, signed curvature and the winding number theorem.
  5. Surfaces, doughnuts and pretzels (classification by number of holes). Surface patches of smooth surfaces.
  6. Orientability of a surface and unit normal. Examples of orientable and non-orientable surfaces such as Möbius band.
  7. Studying curves lying in surfaces. First fundamental form, second fundamental form, geodesic and normal curvatures.
  8. Principal, mean and Gauss curvature of a surface. Elliptic, hyperbolic and parabolic points. Euler's theorem.
  9. Geodesics. Great circles on spheres and other examples.
  10. Gauss-Bonnet theorem for closed surfaces, for simple closed curves and for curvilinear n-gons.
  11. Non-examinable discussion of higher-dimensional spaces and/or another topic.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5109

Learning resources

Essential:

Recommended:

Advanced knot theory:

MTH714U Group Theory

Description

This module provides an introduction to advanced group theory. The aim is to explore the theory of finite groups by studying important examples in detail, such as simple groups. In particular, the projective special linear groups over small fields provide a rich vein of interesting cases on which to hang the general theory.

Details

Organiser: Professor P J Cameron

Level: 7 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

Helpful prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~pjc/MTH714U

Syllabus

  1. Revision of basic group theory, isomorphism theorems, Jordan-Holder theorem, Sylow's theorems, the structure theorem for finite abelian groups.
  2. Permutation groups: transitivity, primitivity, symmetric and alternating groups. Maximal subgroups, wreath products, Iwasawa's Lemma. The outer automorphism of S6.
  3. Linear groups: finite fields, general linear groups, projective special linear groups. Projective lines and isomorphisms of some projective special linear groups with alternating groups. Simplicity of PSLn(q) .

Learning outcomes

A knowledge of those basic theorems in group theory mentioned in the syllabus, some knowledge of how those topics have developed in recent times and a feeling for how one goes about proving results in group theory.

Learning resources

The recommended text is J. L. Alperin and R. W. Bell, Groups and Representations, Springer (1995). Chapters 1 to 4 cover most (but not all) of what is in this module, plus a little extra, including the required background from Algebraic Structures II.

MTH4104 Introduction to Algebra

Description

This module is an introduction to the basic notions of algebra, such as sets, numbers, matrices, polynomials and permutations. It not only introduces the topics, but shows how they form examples of abstract mathematical structures such as groups, rings and fields, and how algebra can be developed on an axiomatic foundation. Thus, the notions of definition, theorem and proof, example and counterexample are described. The module is an introduction to later modules in algebra.

Details

Organiser: Dr K Ardakov

Level: 4 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Helpful prerequisites:

Either of

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~ardakov/MTH4104

Comment: Assessment was 20% two in-term tests, 80% final exam. Syllabus revised for 2012–13 - HOW? TO BE DONE!

Syllabus

  1. Mathematical basics: proofs, necessary and sufficient conditions, proofs and counterexamples, definitions, existence and uniqueness.
  2. Numbers: integers, rationals, real numbers, complex numbers. Induction. Irrationality of √2. Polynomials, matrices.
  3. Sets, subsets, functions, relations. One-to-one and onto functions. Equivalence relations and partitions.
  4. Division algorithm and Euclidean algorithm. Modular arithmetic. Solving polynomials; remainder and factor theorems.
  5. Rings and fields, ideals, factor rings.
  6. Groups, subgroups, cyclic groups, Lagrange's Theorem.
  7. Permutations, symmetric group, sign.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4104

Learning resources

Reading list:

MTH4105 Introduction to Mathematical Computing

Description

This module focuses on showing how to use Maple to do mathematics that you know from A-level or are learning in the first semester and introduces programming concepts that are relevant to mathematical computing. Coverage is broad but fairly superficial. The module is pragmatic and uses Maple's worksheet interface and packages where appropriate.

Details

Organiser: Professor R Whitty

Level: 4 Credit value: 15 Semester: A

Overlaps:

Prerequisites: A-Level Mathematics or equivalent

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~whitty/MTH4105

Comment: Assessment was 20% two in-term tests, 80% final exam.

Syllabus

Part I – Interactive Mathematical Computing

  1. Introduction to Maple: The Maple worksheet; online help; execution groups and text regions; basic computational number systems (integer, rational, float); simple arithmetic operations; factorial (!) and big numbers; Pi and numerical approximation using evalf; %; comma operator and expression sequences; command completion.
  2. Continuous Mathematics: Variables, assignment and automatic evaluation; indeterminates and (univariate) polynomials; simple polynomial algebra; expand, factor, simplify; sqrt, exp, log and trigonometric functions; substitution and evaluation using eval; equations and inequalities; solve and fsolve; diff; int and evalf(Int ...); limit; series and taylor.
  3. Discrete Mathematics: Integer arithmetic, divisibility and prime numbers: irem, iquo, igcd, ifactor, isprime; structured data: sequences, lists and sets; seq; nops; indexing using op and [ ]; index ranges; set operations; map; add, mul, sum.
  4. Vectors, Matrices and Multivariate Algebra: Inputting row/column vectors and matrices; Vector and Matrix; vector and matrix algebra; scalar and vector product; exact and approximate eigenvalues and eigenvectors. Multivariate expressions; solving coupled multivariate equations.
  5. Plotting and Tabulating: plotting univariate expressions; multiple plots; using the graphical user interface to read off intersections; lists of points; bivariate expressions as surfaces; 2D curves and 3D surfaces defined implicitly and parametrically; vectors; linear transformations; ellipses, ellipsoids and eigenvectors. Introduction to spreadsheets.

Part II – Mathematical Programming

  1. Boolean Logic: Boolean constants (true, false); relational operators, evalb, is; use of evalf; Boolean operators (and, or, not); truth tables (using spreadsheets); Boolean algebra; analogy with set theory.
  2. User-defined Functions: Arrow syntax; anonymous and named functions; polynomial and elementary transcendental examples; use with map; predicates (Boolean-valued functions); select and remove.
  3. Repeated Execution: do ... end do; for ... to; while; for ... in; applications such as recursive sequences and iterative approximation, e.g. Iterative method for solving univariate equations, power method for largest eigenvalue; single/double loops over vector/matrix elements.
  4. Conditional Execution: if ... then ... end if; else; elif; applications within loops (e.g. finding the maximum value in a list, vector or matrix and convergence of iterations); piecewise-defined functions; characteristic functions on sets; use with add.
  5. Procedures: proc ... end proc; variable scope; local; global; return value versus side effects; return; error; print; applications such as base conversion, simple statistics.
  6. Procedural Programming: The use of procedures for structuring programs; converting algorithms into programs; program design; debugging.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4105

Learning resources

You may find the following books useful:

MTH6121 Introduction to Mathematical Finance

Description

This module provides an introduction to the ideas of Mathematical Finance. It uses concepts from analysis, differential equations and probability to develop the techniques and language of Mathematical Finance.

Details

Organiser: Dr O Bandtlow

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and any of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~ob/MTH6121

Syllabus

  1. Pointers/revision of probability concepts: probability and events, conditional probability, random variables and expected values, covariance and correlation. Normal random variables and their properties, central limit theorem.
  2. Pricing models; geometric Brownian motion and its use in pricing models. Brownian motion.
  3. Interest rates and Present Value Analysis – including rate of return and continuously varying interest rates.
  4. Pricing contracts via arbitrage – options pricing and examples.
  5. The arbitrage theorem – proof and interpretation.
  6. The Black-Scholes Formula. Properties of the Black-Scholes option cost. Arbitrage strategy.
  7. A derivation of the Black-Scholes formula.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6121

Learning resources

Main text:

MTH5110 Introduction to Numerical Computing

Description

This module investigates the use of computer algebra, numerical techniques and computer graphics as tools for developing the understanding and the solution of a number of problems in the mathematical sciences. Topics that will be addressed will include linear algebra, the solution of algebraic equations, the generation and use of quadrature rules and the numerical solution of differential equations and, time permitting, some other aspects of computational mathematics. The computer language used is Maple.

Details

Organiser: Dr W Just

Level: 5 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~wj/MTH5110

Syllabus

This course investigates the use of computer algebra, numerical techniques and computer graphics as tools for developing the understanding and the solution of a number of problems in the mathematical sciences. The computer algebra system used for this course will be Maple.

  1. Brief revision of Maple. Tracing algorithms and debugging.
  2. Numerical and symbolic operations on matrices: obtaining and examining the properties of eigenvalues and eigenvectors.
  3. Numerical and symbolic solution of algebraic equations.
  4. Integration: overview of numerical techniques, symbolic generation of quadrature rules, comparison of numerical integration using numerical techniques and using symbolic analysis.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5110

Learning resources

Reading list:

MTH4107 Introduction to Probability

Description

This is the first course in probability, covering events and random variables. It introduces the basic notions of probability theory and develops them to the stage where one can begin to use probabilistic ideas in statistical inference and modelling, and the study of stochastic processes. The first section deals with events, the axioms of probability, conditional probability and independence. The second introduces random variables both discrete and continuous, including distributions, expectation and variance. Joint distributions are covered briefly.

Details

Organiser: Dr R Johnson

Level: 4 Credit value: 15 Semester: A

Overlaps:

Prerequisites: A-Level Mathematics or equivalent

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~jrj/MTH4107

Comment: Assessment was 20% two in-term tests, 80% final exam. Syllabus revised for 2012–13 to remove introduction to sets and functions. SHOULD PROOFS ALSO GO?

Syllabus

  1. Probability: frequentist vs modelling vs subjective. Finite sample spaces (equiprobable or not); events as subsets.
  2. Sequences: suffix notation, summation notation, change of suffix, manipulating sums.
  3. Elementary ideas of probability theory; Kolmogorov axioms; additivity of probabilities of disjoint events. Sigma notation with suffix i. Simple proofs from the axioms. Inclusion-exclusion. Propositions, logical operations, negation, and, or, converse, equivalent, ideas of proof.
  4. Sampling with and without replacement. Counting. Binomial Theorem.
  5. Independent events: definition, examples. Multiplication law. Three or more events.
  6. Conditional probability. Definition. Sampling without replacement done in stages rather than as set of outcomes. Proof by induction that P(E1 ∩ E2 ∩ ...  ∩ En) = P(E1) × P(E2 | E1) × ...  × P(En | E1 ∩ ...  ∩ En − 1 ) . Theorem of Total Probability. Use sigma notation with suffix i.
  7. Bayes' Theorem and its use to calculate 'inverse' probabilities like conditional probability of having disease D given that test for D is positive. Discrete random variables as functions from sample space to R.
  8. Probability mass function, mean. Variance. Mean and variance of a X + b.
  9. Familiarity with the following distributions (including pmf, mean, variance, what they are used to model): Bernoulli, binomial, geometric, hypergeometric, Poisson. Cumulative distribution function for discrete random variables. Informal introduction to continuous random variables. Cumulative distribution function, probability density function. Mean, variance. E(g(X)) . Median and quartiles.
  10. Familiarity with exponential and uniform distributions. Monotone 1-dimensional transformations of random variables. Proof of pdf of new random variables in continuous case.
  11. Joint distribution of two random variables in some simple discrete cases. Marginal distributions. Independent random variables. Covariance and correlation. Independence implies zero covariance. Mean and variance of a X + b Y in general. Hence re-derive mean and variance of binomial as sum of independent Bernoulli's. Conditional distribution in simple discrete cases. Conditional random variables. Theorem of Total Probability for expectation. Hence re-derive mean and variance of geometric distribution.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4107

Learning resources

Main texts:

Other texts:

MTH4106 Introduction to Statistics

Description

This first module in statistics introduces the fundamental ideas of classical statistics. It covers descriptive statistics, the estimation of population moments using data and the basic ideas of statistical inference, hypothesis testing and interval estimation. These methods will be applied to data from a range of applications, including business, economics, science and medicine. A simple statistics package will be used to perform the calculations.

Details

Organiser: Professor R A Bailey

Level: 4 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and either of

Helpful prerequisites:

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~rab/MTH4106

Comment: Assessment was 20% two in-term tests, 80% final exam.

Syllabus

  1. Data and statistics. Types of variables. Simple plots, mean and median. Five number summary, box plots. Sample variance, inter-quartile range, skewness. Effect of linear transformations on summary statistics. Scatterplots and marginal plots. Sample correlation r and proof that −1 < r < 1 . Introduction to Minitab.
  2. Revision of discrete random variables (rvs). Probability generating function and its use (factorial moments, sums of independent random variables).
  3. Ideas of statistical modelling, populations and samples. Random samples, sampling distribution of proportion. Point estimate of population proportion, unbiasedness.
  4. Hypothesis tests on proportions, basic ideas, type I and II errors. One- and two-sided alternative hypotheses.
  5. Continuous rvs. Cumulative distribution function and probability density function (pdf). Moments. Exponential and uniform distributions. Monotone transformations of rvs. Proof of pdf of new continuous rvs.
  6. Normal rv, standard and general. Use of normal tables. Simulation – how to sample from different distributions, simulation of simple functions of rvs.
  7. Introduction to joint distribution of two or more continuous rvs. Covariance, correlation and independence. Point estimation. Unbiasedness of sample mean and variance, calculation of bias.
  8. Statements of law of large numbers and central limit theorem. Simulations of theoretical results. Linear combinations of normal rvs. Normal approximation to binomial and Poisson rvs, continuity correction. Sampling distributions of sample total, mean and variance.
  9. 1-sample z test. Significance levels and p-values. Large sample applications.
  10. Confidence intervals (CIs) – general ideas, CI for normal mean and large sample applications. Confidence intervals for a Poisson mean.
  11. Conditioning on a continuous rv. Conditional expectation. Computing expectations by conditioning. Conditional variance.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4106

Learning resources

Main texts:

A book which suits YOU best to learn statistics is best (for you). You are encouraged to use it, whether it is one from the list below or another one.

You should already have a copy of:

MTH5112 Linear Algebra I

Description

This is a rigorous first module in linear algebra. The ideas introduced in Geometry I for two- and three-dimensional space will be developed and extended in a more general setting with a view to applications in subsequent pure and applied mathematics, probability and statistics modules. There will be a strong geometric emphasis in the presentation of the material and the key concepts will be illustrated by examples from various branches of mathematics. The module contains a fair number of proofs.

Details

Organiser: Professor C-H Chu

Level: 5 Credit value: 15 Semester: A

Overlaps:

Helpful prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~cchu/MTH5112

Syllabus

  1. Systems of linear equations: Elementary row operations, solution by Gaussian elimination, echelon forms; existence/uniqueness of solutions.
  2. Matrix algebra: addition and multiplication. Ax as a linear combination of the columns of A. Matrix inverse. Matrix transpose. Special types of square matrices. Linear systems in matrix notation. Elementary matrices and row operations. Reduced row echelon form for square matrices, conditions for non-singularity, matrix inverses by Gaussian elimination.
  3. Determinants: Cofactors and row/column expansions. Elementary row/column operations. Determinant of matrix transpose, of product of matrices. Matrix inverse in terms of adjoint. Cramer's rule.
  4. Vector spaces (over R and C): Definition and examples. Subspaces. Spanning sets. Linear independence. Basis and dimension of a vector space. Change of basis. Row and column spaces, rank. The null space.
  5. Linear Transformations: Definition and examples. Matrix representations of linear transformations. The law of change of matrix representation under a change of basis. The Rank-Nullity Theorem.
  6. Orthogonality in Rn: Scalar product – definition and properties. Orthogonal and orthonormal sets. Orthogonal complements. Orthogonal projections. Fundamental Subspace Theorem. Orthogonal matrices. Least-squares solutions of inconsistent systems. Gram-Schmidt process.
  7. Eigenvalues and Eigenvectors: The equation Ax = λx. The characteristic polynomial. Eigenvalues and eigenvectors of special classes of matrices. Real symmetric matrices: orthogonal diagonalisation. Similarity: distinct eigenvalues and diagonalisation.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

Main text:

MTH6140 Linear Algebra II

Description

This module is a mixture of abstract theory, with rigorous proofs, and concrete calculations with matrices. The abstract component builds on the theory of vector spaces and linear maps to construct the theory of bilinear forms (linear functions of two variables), dual spaces (which map the original space to the underlying field) and determinants. The concrete applications involve ways to reduce a matrix of some specific type (such as symmetric or skew-symmetric) to as near diagonal form as possible.

Details

Organiser: Professor T Müller

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~twm/MTH6140

Syllabus

  1. Bilinear forms over finite dimensional real and complex vector space. Sesquilinear forms over complex vector spaces. Proof of Sylvester's law of inertia. Positive definite forms over real vector spaces, Hermitian forms over complex vector spaces.
  2. Orthogonality, the Gram-Schmidt orthogonalisation process, orthogonal projections.
  3. Revision of vector spaces, subspaces, eigenspaces, linear maps, direct sum, kernel and image, spanning set, linear independence, basis, dimension, Steinitz Exchange Lemma, dimension formula for subspaces, with rigorous proofs.
  4. Properties of determinants and their connection with adjoints. The Cayley-Hamilton theorem and its proof. Eigenvalues, trace and determinant. Eigenvalues of a symmetric matrix.
  5. Linear functional, dual spaces, equality of row and column rank of a matrix.
  6. Symmetric, skew-symmetric and alternating bilinear forms over arbitrary fields. Skew-Hermitian forms over complex vector spaces.
  7. Simultaneous diagonalisation, for linear map and positive definite symmetric form, and for two symmetric forms.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6140

Learning resources

Main text:

MTH717U MSci Project

Description

You will write a report that must present the study of some mathematical topic at fourth-year undergraduate level and must be your own work in the sense that it gives an original account of the material, but it need not contain new mathematical results. The list of potential projects and supervisors is available on the School of Mathematical Sciences website. You will be accepted for a specific project only after agreement between the module organiser and the project supervisor.

Details

Organiser: Professor L H Soicher

Level: 7 Credit value: 30 Semester: A and B

Overlaps:

Essential prerequisites:

Helpful prerequisites:

Restrictions: Not open to Associate Students.

Assessment: Written report, presentation and (possibly) oral exam

Organiser's module website: http://www.maths.qmul.ac.uk/~leonard/MTH717U

Syllabus

A project in any area of mathematics, including astronomy and computing, which normally runs over both of semesters 7 and 8. The project will be assessed primarily by a written report and, at the examiners' discretion, an oral examination, but also by a presentation of about 30 minutes in duration, to be given towards the end of semester 8. The contribution of the presentation will be on a sliding scale that will never decrease the project mark by more than 10% or increase it by more than 20%, and provided you make a reasonable attempt at giving a presentation it will not decrease your project mark.

Your report must present the study of some mathematical topic at fourth-year undergraduate level and must be your own work in the sense that it gives an original account of the material, but it need not contain new mathematical results. To obtain a mark above a bare pass it is not sufficient to read several sources on the project topic and then write about this topic in your own words; you must also make some original contribution. For example, you might do one or more of the following: give a new interpretation or view of the mathematics; give new examples; do your own detailed calculations or data analysis; write, test and run your own computer program. You must make it completely clear where you are providing your own original contribution rather than paraphrasing published information. The length should be about 8000 words; if it is less than 6000 or more than 10000 words you may lose marks. You can write your report in a single semester or spread over two semesters, depending on your other module choices.

Learning outcomes

Not yet available

INE6001 Mathematical Education for Physical and Mathematical Sciences

Description

This module is for undergraduates in mathematics or in the physical sciences who are interested in studying mathematics education. 'Mathematics education' is an umbrella term that encompasses all aspects of learning and teaching mathematics in schools and in other settings.

Details

Organiser: Dr M Rodd, Institute of Education; QM contact Dr F Wright

Level: 6 Credit value: 15 Semester: A and B

Overlaps: None

Essential prerequisites:

Restrictions: Restricted to Queen Mary Mathematical Sciences students. Not open to Associate Students.

Assessment: 50% coursework essay (to be submitted towards the end of Semester B) and 50% final exam (to be sat in May)

Organiser's module website: http://www.maths.qmul.ac.uk/~fjw/INE6001

Comment: Note that this module spans both semesters. Note also that its availability for 2012–13 has not yet been confirmed and it may be withdrawn. If it is offered then the details will be similar to those for last year currently shown here, which will be updated later.

Syllabus

The aim of this module is to introduce you to central ideas of mathematical education. It should be relevant to you if you are considering going into teaching after you graduate and it will also be relevant to you as a learner of mathematics. This module is valued by Queen Mary as 15 credits at level 6 and may be taken as one of the non-MTH modules allowed in the third or final year of all Mathematical Sciences study programmes.

The module will be taught at the Institute of Education (IoE) at 20, Bedford Way, seven minutes walk from Euston Square tube station. Lectures will take place during both of Semesters A and B on Mondays from 5:00pm to 6:30pm for 19 sessions starting on 3 October 2011 and excluding reading weeks. When considering your timetable, you should allow 45 minutes travel time from Queen Mary to the Institute of Education. This means that you will not be able to attend any classes at Queen Mary after 4:00pm on Mondays. Individual tutorials will be arranged during Semester B to help with essay writing, and revision session(s) will be held late April / early May to help prepare for the exam.

There will be a preliminary meeting at 5:00pm on Monday 26 September 2011 in room 203 in the Mathematical Sciences Building, Queen Mary, which you are advised to attend before deciding whether to take this module.

You can select this module in MySIS as one of your options, but note the prerequisite that you must have a second-year mean mark of at least 50%. (We will check this!) The IoE code for this module is DDOPRO_69. You will also need to complete an intercollegiate module registration form, get it signed by the module organiser at the IoE, and return it to Dr Francis Wright. (You do not need any other signatures from Queen Mary.) This form is necessary for your results to be transferred from the IoE to Queen Mary. Forms will be available at the preliminary meeting or you can obtain one from Queen Mary Registry.

Learning outcomes

MTH6124 Mathematical Problem Solving

Description

The module is concerned with solving problems rather than building up the theory of a particular area of mathematics. The problems are wide ranging with some emphasis on problems in pure mathematics and on problems that do not require knowledge of other undergraduate modules for their solution. You will be given a selection of problems to work on and will be expected to use your own initiative and the library. However, hints are provided by staff in the timetabled sessions. Assessment is based on the solutions handed in, together with an oral examination.

Details

Organiser: Dr D Ellis

Level: 6 Credit value: 15 Semester: B

Overlaps:

Prerequisites: Level-5 mathematical background

Restrictions: Places are limited; contact the module organiser to validate registration.

Assessment: Written solutions to questions and oral exam

Organiser's module website: http://www.maths.qmul.ac.uk/~dellis/MTH6124

Comment: This is a reading module (i.e. primarily self-study).

Syllabus

The course is concerned with solving problems rather than building up the theory of a particular area of mathematics. The problems cover a wide range, with some emphasis on problems in pure mathematics and on problems which do not require knowledge of other undergraduate courses for their solution. Students are given a selection of problems to work on and are expected to use their own initiative and the library; however hints are provided by the staff at the timetabled sessions.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6124

MTH736U Mathematical Statistics

Description

This module covers the classical theory of statistical inference and the probability theory which are required for more advanced study in statistics. It is aimed at mathematicians who have done little statistics in their previous undergraduate studies. It will cover material which is in the equivalent of approximately two undergraduate modules, at a fast pace and taking a mathematical approach. Students will have considerably more responsibility for their own learning than they would have in undergraduate modules covering the same material. Topics covered include: conditional probability; hypothesis testing; distribution theory; estimation; multivariate normal distribution; Laws of Large Numbers and the Central Limit Theorem; confidence intervals; general theory of testing; matrix algebra; least squares; Gauss-Markov Theorem.

Details

Organiser: Dr B Bogacka

Level: 7 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~bb/MTH736U

Syllabus

This is a condensed statistics module intended primarily for MSc students with little statistical background; undergraduate students are strongly encouraged to take MTH6136 Statistical Theory instead.

  1. Probability Distribution Theory:
  2. Statistical Inference:

Learning outcomes

Ability to use probability to solve problems in mathematical statistics. Knowledge of the general principles of frequentist and likelihood inference. Ability to apply the general principles to specific methodological problems.

Learning resources

Reading list:

MTH4110 Mathematical Structures

Description

This module is intended to introduce students to the concerns of mathematics, namely clear and accurate exposition and convincing proofs. It will attempt to instill the habit of being "precise but not pedantic". The course covers an informal account of sets, functions, and relations, and a sketch of the number systems (natural numbers, integers, rational, real and complex numbers), outlining their construction and main properties.

Details

Organiser: Professor P J Cameron

Level: 4 Credit value: 15 Semester: A

Overlaps: None

Prerequisites: A-Level Mathematics or equivalent

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~pjc/MTH4110

Comment: New module in 2012–13.

Syllabus

  1. Definitions, theorems, proofs, counterexamples. Examples of proofs. How to construct proofs, and how to recognise false proofs.
  2. Sets, subsets, operations on subsets. Functions; injective, surjective and bijective functions. Relations, including equivalence relations. Counting finite sets (including the number of r-subsets of an n-set). Infinite sets; countable and uncountable.
  3. Natural numbers and induction. Integers and rational numbers with a sketch of the construction. Real numbers (treated as infinite decimal and binary expansions) including some completeness properties. Countability of the rationals and uncountability of the reals.
  4. Complex numbers. The complex plane with cartesian and polar coordinates; addition and multiplication. Statement of the Fundamental Theorem of Algebra.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

Reading list:

MTH5117 Mathematical Writing

Description

This module teaches the language of higher mathematics, and how to use it with precision and fluency in a variety of contexts. For raw material, it calls on the mathematics developed in the first year, which you will see from a more mature perspective. The module also develops some elements of logic that serve as the basis for an analysis of the main techniques used in mathematical proofs. You will get a lot of practice and feedback through the coursework.

Details

Organiser: Professor F Vivaldi

Level: 5 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Assessment: 20% coursework, 80% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~fv/MTH5117

Syllabus

  1. Basic words and symbols of higher mathematics.
  2. Mathematical notation: developing a coherent approach.
  3. Describing the behaviour of functions.
  4. Logical structures: the predicate algebra.
  5. Basic proof techniques.
  6. Existence statements.
  7. Natural numbers: inductive arguments.
  8. Definitions: what they are for and how to write them.
  9. Intellectual property: giving credit, respecting copyright

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5117

Learning resources

Essential:

Highly recommended:

Recommended:

MTH716U Measure Theory and Probability

Description

This is an introductory module on the Lebesgue theory of measure and integral with application to probability. You are expected to know the theory of Riemann integration. Measure in the line and plane, outer measure, measurable sets, Lebesgue measure, non-measurable sets. Sigma-algebras, measures, probability measures, measurable functions, random variables. Simple functions, Lebesgue integration, integration with respect to general measures. Expectation of random variables. Monotone and dominated convergence theorems, and applications. Absolute continuity and singularity, Radon-Nikodym theorem, probability densities. Possible further topics: product spaces, Fubini's theorem.

Details

Organiser: Professor O Jenkinson

Level: 7 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Helpful prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~omj/MTH716U

Comment: Was semester A in 2011–12.

Syllabus

This is an introductory course on the Lebesgue theory of measure and integral with application to Probability. Students are expected to know the theory of Riemann integration.

  1. Measure in the line and plane, outer measure, measurable sets, Lebesgue measure, non-measurable sets.
  2. Sigma-algebras, measures, probability measures, measurable functions, random variables.
  3. Simple functions, Lebesgue integration, integration with respect to general measures. Expectation of random variables. Monotone and dominated convergence theorems, and applications.
  4. Absolute continuity and singularity, Radon-Nikodym theorem, probability densities.
  5. Possible further topics: product spaces, Fubini's theorem.

Learning outcomes

Not yet available

MTH6126 Metric Spaces

Description

The study of metric spaces provides a link between geometry, which is fairly concrete, and topology, which is more abstract. It generalises to multidimensional spaces the concepts of continuity and other ideas studied in real analysis and explores the foundations of continuous mathematics.

Details

Organiser: Dr D Stark

Level: 6 Credit value: 15 Semester: A

Overlaps: None

Essential prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~dstark/MTH6126

Syllabus

  1. Definition of metric space; examples, including finite metric spaces, function spaces, normed vector spaces, product spaces.
  2. Convergence and continuity in metric spaces.
  3. Equivalent metrics.
  4. Open and closed sets, properties, continuity in terms of pre-images of open sets.
  5. (Sequential) compactness, properties of compact spaces, uniform continuity. Bolzano-Weierstrass Theorem.
  6. Completeness; examples, including C[0,1]. Examples of completions of metric spaces. Contraction mappings, Banach fixed-point theorem, applications, e.g., to solutions of differential equations.
  7. Further topics if time allows, such as the Heine-Borel Theorem.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6126

Learning resources

Main texts:

MTH6128 Number Theory

Description

This module introduces some of the more elementary aspects of algebraic number theory from a classical perspective. The main strands are continued fractions, binary quadratic forms and modular arithmetic. The theory of continued fractions serves as a unifying theme as well as a source of algorithms. Applications include the representation of primes as sums of squares and the solution of Pell's equation.

Details

Organiser: Professor P J Cameron

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~pjc/MTH6128

Syllabus

  1. Continued fractions: finite and infinite continued fractions, approximation by rationals, order of approximation.
  2. Continued fractions of quadratic surds: applications to the solution of Pell's equation and the sum of two squares.
  3. Binary quadratic forms: equivalence, unimodular transformations, reduced form, class number. Use of continued fractions in the indefinite case.
  4. Modular arithmetic: primitive roots, quadratic residues, Legendre symbol, quadratic reciprocity. Applications to quadratic forms.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6128

Learning resources

Main texts:

MTH6129 Oscillations, Waves and Patterns

Description

Waves and vibrations are present in almost all physical systems, from the vibrations in strings to the waves of the oceans and atmosphere. Waves and patterns are also seen in chemical and living systems. This module is an introduction to the mathematical theory of waves, dealing with the solution of differential equations describing, for example, vibrations on strings and waves in fluids. Elementary ideas about nonlinear waves, such as shock formation, are described. The material is illustrated with applications from a wide variety of different systems.

Details

Organiser: Professor J Lidsey

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~jel/MTH6129

Syllabus

  1. Oscillations: Review of restoring forces and SHM; damped oscillations, strong, weak and critical damping; forced damped oscillations, transient and steady state solutions; resonance.
  2. Coupled oscillators: normal coordinates, normal modes of vibrations, derivation of wave equation as the limit of many coupled oscillators.
  3. Waves: derivation of classical wave equation for string; D'Alembert's solution; travelling plane wave solutions; transverse vibrations on a string: harmonic waves, normal modes for string fixed at ends, solution by separation of variables; initial conditions and Fourier sine series; examples, such as vibrations and musical sounds.
  4. Waves in fluids: linear surface waves on deep and shallow water; dispersion relation, phase and group velocities; waves on inclined beds, tsunamis.
  5. Patterns: circular membranes (drums): modes of oscillation and their patterns; nonlinear waves and solitons; qualitative introduction to waves and pattern formation in other systems, e.g. biological and chemical systems.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6129

Learning resources

Other texts:

MTH5121 Probability Models

Description

This module develops some of the ideas first introduced in Introduction to Probability (Probability I). It will cover five main topics: how to compute probabilities and expectations by a process called conditioning; random walks and other discrete branching processes; continuous methods of conditioning; continuous probability models such as Poisson processes; and some very useful limit theorems. The material is important for applications in financial and actuarial mathematics, in the physical and life sciences, and for more advanced probability modules.

Details

Organiser: Professor I Goldsheid

Level: 5 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and either of

Helpful prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~ig/MTH5121

Syllabus

  1. Discrete Methods:
  2. Discrete Models:
  3. Continuous Methods
  4. Continuous Models
  5. Limit Theorems

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5121

Learning resources

Main text:

MTH6141 Random Processes

Description

This is an advanced module in probability, introducing various probability models used in physical and life sciences and economics. It serves as an introduction to stochastic modelling and stochastic processes. It covers discrete time processes including Markov chains and random walks, and continuous time processes such as Poisson processes, birth-death processes and queueing systems. It builds on previous probability modules but needs no background in statistics; some experience of linear algebra is also desirable.

Details

Organiser: Professor M Jerrum

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Either of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~mj/MTH6141

Syllabus

Discrete time processes [6 weeks]

  1. Discrete Markov chains (definition and examples). Description by transition graph and transition matrix.
  2. Markov chains with absorbing states (probability of absorption in a given state, expected time to absorption via first-step analysis).
  3. Long run behaviour of Markov chains. Equilibrium and limiting distributions (conditions for existence and calculation).
  4. Recurrence and transience.
  5. Extended examples: Random walks on a finite interval with absorbing and reflecting boundaries, Random walks on infinite intervals, Branching processes.

Continuous time processes [5 weeks]

  1. Definition of Poisson process as a description of rare events. Equivalence with description of Poisson Process from Probability Models. Arrival and interarrival times. Relation to the uniform and binomial distributions.
  2. Birth processes as a generalisation of the Poisson Process. Differential equation method. Linear birth process. Brief discussion of death process.
  3. Birth-death process. Forward and backward differential equations. Long run behaviour.
  4. Queueing systems (notation and examples). Relation of M/M/t queue to birth-death process. Brief discussion of M/G/1 and G/M/1 queues.

Learning outcomes

At the end of this module, students should be able to:

Learning resources

Main text:

MTH6132 Relativity

Description

This module is an introduction to Einstein's theories of special and general relativity. The first part of the module deals with special relativity, and is mainly about the strange dynamics that happen at speeds comparable to the speed of light. The second part develops the mathematical machinery needed to study the curvature of space-time and the subtle effects of gravity; this is the general theory of relativity. The third part deals with various consequences of the theory, and will touch upon topics like black holes and the big bang.

Details

Organiser: Dr J A Valiente-Kroon

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and

and

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~jav/MTH6132

Syllabus

  1. Special Relativity: Newtonian mechanics and Galilean relativity. Maxwell's equations and special relativity. Lorentz transformations and Minkowski space-time. Clocks and rods in relative motion.
  2. Vectors in Special Relativity: 4-vectors and the Lorentz transformation matrix 4-velocity, 4-momentum, 4-acceleration. Relativistic dynamics and collisions. Optics: redshift and aberration
  3. Tensors in special relativity: Metrics and forms. Tensors and tensor derivatives. Stress-energy tensor. Perfect fluids.
  4. Conceptual Basis of General Relativity: Problems with Newtonian gravity. Equivalence principle.
  5. Curved Space-time and General Relativity: Tensor calculus. Covariant derivatives and connections. Parallel transport and geodesics. Curvature and geodesic deviation. Einstein's field equations.
  6. Application of General Relativity: Schwarzschild solution. Tests of general relativity. Black holes and gravitational collapse.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6132

Learning resources

Main texts:

MTH5122 Statistical Methods

Description

This module develops some of the ideas first introduced in Introduction to Statistics. It begins by covering some of the essential theoretical notions required, such as covariance, correlation and independence of random variables. The majority of the material covers different types of statistical tests: how to use them and when to use them. This material is essential for applications of statistics in psychology, the life or physical sciences, business or economics. It is also required for further study of statistics.

Details

Organiser: Dr L Pettit

Level: 5 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~pettit/MTH5122

Syllabus

  1. Revision of conditional probability. Joint distributions. Computing probabilities from the joint probability density function. Uniform distribution. Marginal distributions. Expectation of a function of random variables (without proof).
  2. Covariance and correlation coefficient. Independence for two random variables (rvs). Independence in the multivariate setting. Transformation of rvs (technique and simple examples of its use). t- and F-distributions. 1-sample t test.
  3. Moment generating function (mgf) and its use. Sums of independent rvs. Gamma distribution. Chi-squared distribution. Test and CI on the variance.
  4. Goodness of fit tests, basic ideas, estimation of parameters, grouping classes. Goodness of fit for continuous rvs.
  5. F test, 2-sample t test and corresponding confidence intervals.
  6. Approximate 2-sample test when variances are unequal. Matched pairs t test, discussion about design and blocking and when to use which test.
  7. Contingency tables – chi-squared test of independence, different methods of sampling. Proof of formula for 2x2 tables, Yates' correction for 2x2 tables.
  8. Test of 2 proportions and relationship to contingency tables. Approximate test for correlation coefficient.
  9. Chebyshev's inequality. The weak law of large numbers.
  10. Central limit theorem (by way of mgf).
  11. Bivariate normal distribution (definition and basic properties). Multivariate normal distribution in matrix notation.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5122

MTH5120 Statistical Modelling I

Description

This is a first module on linear models and it concentrates on modelling the relationship between a continuous response variable and one or more continuous explanatory variables. Linear models are very widely used in almost every field of business, economics, science and industry where quantitative data are collected. They are also the basis for several more advanced statistical techniques covered in level-6 modules. This module is concerned with both the theory and applications of linear models and covers problems of estimation, inference and interpretation. Graphical methods for model checking will be discussed and various model selection techniques introduced. Computer practical sessions, in which the Minitab statistical package is used to perform the necessary computations and on which the continuous assessment is based, form an integral part of the module.

Details

Organiser: Dr B Bogacka

Level: 5 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and either of

Assessment: 10% coursework, 10% mid-term test, 80% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~bb/MTH5120

Syllabus

The techniques covered will be applied to data from various areas of business, economics, science and industry.

  1. Relationships among variables and basic concepts of statistical modelling, response and explanatory variables.
  2. The Normal-linear model: definition, matrix form, simple, multiple and polynomial regression models.
  3. Matrix algebra: trace, transpose and inverse of square matrices, manipulation of matrix equations, vector differentiation.
  4. Estimation: maximum likelihood, least squares, Gauss-Markov Theorem, properties of estimators, estimating mean responses, estimating σ2.
  5. Assessing fitted models: analysis of variance, R2, lack of fit, residuals and model checking, outliers.
  6. Model selection: transformation of the response variable, order of polynomial models, variable selection.
  7. Inference: confidence intervals for parameters and mean response, testing for parameters and mean response.
  8. Uses of linear models – prediction, control, optimisation.
  9. Problems: leverage and influence, multicollinearity.
  10. Use of Minitab to apply the theory to practical data analysis.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH5120

Learning resources

Main text:

Other texts:

MTH6134 Statistical Modelling II

Description

This is the part of linear models often called analysis of variance. It concentrates on models whose explanatory variables are qualitative. These methods are used in almost all areas of business, economics, science and industry where qualitative and quantitative data are collected.

Details

Organiser: Dr M Arephin

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~arephin/MTH6134

Syllabus

Extended use of the comprehensive statistical package GenStat is developed as it is required in the module. The methods introduced are applied to data from various applications in business, economics, science and industry.

  1. Qualitative explanatory variables – models, factors, main effects and interactions.
  2. Indicator variables – representation as linear regression models.
  3. Parameterisations and constraints – intrinsic and extrinsic aliasing.
  4. Vector spaces and least squares estimation using projections.
  5. Nested, crossed and general structures.
  6. Random effects – variance components, mixed models.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6134

Learning resources

Main text:

Other texts:

MTH6136 Statistical Theory

Description

The theory developed will be used to justify the methods introduced in Introduction to Statistics and will be used to analyse data from a variety of applications. The module will cover estimation, methods of estimation, confidence intervals, and testing.

Details

Organiser: Dr D S Coad

Level: 6 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

Either of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~coad/MTH6136

Syllabus

The theory developed will be used to justify the methods introduced in MTH4106 Introduction to Statistics and will be used to analyse data from a variety of applications.

  1. Estimation: bias, sufficiency, Cramer-Rao lower bound, minimum variance unbiased estimators.
  2. Methods of estimation: method of moments, maximum likelihood, least squares, properties of estimators obtained from these methods, asymptotic properties of MLEs.
  3. Confidence intervals: methods of obtaining CIs using pivots, likelihood CIs.
  4. Testing: power, simple and composite hypotheses, Neyman-Pearson Lemma, uniformly most powerful tests, likelihood ratio tests, Wilks' Theorem.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6136

Learning resources

Recommended:

Other texts:

MTH6138 Third-Year Project

Description

This module allows third-year undergraduates with suitable background to take a project, including one of the 30-credit MSci projects in a simplified form as a 15-credit project, although some MSci projects may not be available as third-year projects. The list of potential MSci projects and supervisors is available on the School of Mathematical Sciences website. You will be accepted onto this module only after agreement between your adviser, the module organiser and the project supervisor.

Details

Organiser: Professor L H Soicher

Level: 6 Credit value: 15 Semester: A or B

Overlaps:

Essential prerequisites:

Helpful prerequisites:

Restrictions: You will not normally be allowed to take this module together with another project module. Not open to Associate Students.

Assessment: Written report, presentation and (possibly) oral exam

Organiser's module website: http://www.maths.qmul.ac.uk/~leonard/MTH6138

Syllabus

A project in any area of mathematics, including astronomy and computing, which is offered in both of semesters 5 and 6. It may be a simplified version of an MSci project, although some MSci projects may not be available as third-year projects. The project will be assessed primarily by a written report and, at the examiners' discretion, an oral examination, but also by a presentation of about 20 minutes in duration, to be given towards the end of semester 6. The contribution of the presentation will be on a sliding scale that will never decrease the project mark by more than 10% or increase it by more than 20%, and provided you make a reasonable attempt at giving a presentation it will not decrease your project mark.

Your report must present the study of some mathematical topic at third-year undergraduate level and must be your own work in the sense that it gives an original account of the material, but it need not contain new mathematical results. It is sufficient to read several sources on the project topic and then write about this topic in your own words. The length should be about 4000 words; if it is less than 3500 or more than 5000 words you may lose marks.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6138

MTH6139 Time Series

Description

A time series is a collection of observations made sequentially, usually in time. This kind of data arises in a large number of disciplines ranging from economics and business to astrophysics and biology. This module introduces the theory, methods and applications of analysing time series data.

Details

Organiser: Dr D S Coad

Level: 6 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

and either of

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~coad/MTH6139

Comment: Assessment was 20% two in-term tests, 80% final exam.

Syllabus

The course includes time series analysis using MINITAB. The methods developed are applied to data arising in applications in economics, business, science and industry.

  1. General introduction and motivation.
  2. Trends and seasonality and their removal by moving averages. Differencing.
  3. Review of probability.
  4. Time series as a stationary stochastic process.
  5. Modelling of time series in the time domain. Development of AR(p) and MA(q) models in general and their detailed study for the case of p = q = 1 .
  6. ARMA models.
  7. Model identification using the ACF and PACF.
  8. Estimation of parameters by moments, least squares and maximum likelihood methods.
  9. Forecasting by least squares and conditional expectations.
  10. ARIMA models.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH6139

Learning resources

Recommended:

Other texts:

MTH734U Topics in Probability and Stochastic Processes

Description

This module aims to present some advanced probabilistic concepts and demonstrate their application to stochastic modelling of real-world situations. The topics covered vary from year to year but may include, for example, limit theorems, renewal theory, and continuous-time Markov processes. In addition to exposure to proofs and theoretical material, students develop practical skills through a large number of problems and worked examples.

Details

Organiser: Dr D Stark

Level: 7 Credit value: 15 Semester: A

Overlaps:

Essential prerequisites:

Either of

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~dstark/MTH734U

Comment: Was semester B in 2011–12.

Syllabus

Topics will be chosen from the following list:

  1. Borel-Cantelli lemma, Kolmogorov's inequalities, strong law of large numbers.
  2. Weak convergence of distributions. The Central Limit Theorem.
  3. Recurrent events and renewal theory.
  4. Further topics in random walks.
  5. General theory of Markov chains. Classification of states and ergodic properties.
  6. Continuous time Markov Processes.

Learning outcomes

Not yet available

Learning resources

Main texts:

MTH739U Topics in Scientific Computing

Description

This module focuses on the use of computers for solving applied mathematical problems. Its aim is to provide students with proper computational tools to solve problems they are likely to encounter while doing their MSc or MSci, and to provide them with a sound understanding of a programming language used in applied sciences. The topics covered will depend on the module organiser's expertise, with a view to emphasize applications rather than theory.

Details

Organiser: Dr A Baule

Level: 7 Credit value: 15 Semester: B

Overlaps: None

Prerequisites: Level-6 mathematical background

Assessment: 50% coursework, 50% project

Organiser's module website: http://www.maths.qmul.ac.uk/~baule/MTH739U

Syllabus

  1. Introduction to programming: Loops; If statements; procedures; input/output; recursion; program structure
  2. Numerical solution of ordinary differential equations: Euler scheme; higher-order schemes; Runge-Kutta integrator; applications
  3. Random number generation: random vs pseudo-random numbers; linear congruent systems; transformation of random variables; normal variates; applications
  4. Monte Carlo methods: direct sampling; importance sampling; Markov chains; applications
  5. Simulation of stochastic processes: Random walks; Markov chains; continuous-time limit; Brownian motion; stochastic differential equations; applications

Learning outcomes

At the end of this module, students should be able to:

Learning resources

MTH732U Topology

Description

Topology is the study of properties of shape which remain the same when pulled, pushed or squeezed by a continuous process of deformation. For example, the property of a space being connected or a surface having a hole is a topological property. In this module we start with general point set topology and formal definitions and move on to study powerful algebraic invariants such as the fundamental group. Topology allows access to many exciting areas of modern mathematics.

Details

Organiser: Professor I Goldsheid

Level: 7 Credit value: 15 Semester: B

Overlaps:

Essential prerequisites:

and

Helpful prerequisites:

Assessment: 100% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~ig/MTH732U

Syllabus

  1. Topological spaces: examples including discrete, indiscrete, metric and co-finite topologies.
  2. Continuity and convergence, homeomorphisms, topological and non-topological properties.
  3. Paths and path connectedness.
  4. Compactness in a topological space, Heine-Borel theorem, compact implies sequentially compact in metric spaces. Statement of converse.
  5. New spaces: subspaces, product spaces, identification spaces (especially of a square).
  6. Paths and path homotopies. Simply connectedness.
  7. The fundamental group, definition and elementary properties. Fundamental group of a circle. Path and homotopy lifting (proofs non-examinable).
  8. Brouwer Fixed Point Theorem in two dimensions. Borsuk-Ulam Theorem in two dimensions.
  9. Fundamental group of identifications of the square including the torus and Klein bottle.

Learning outcomes

Know the definition and properties of a metric space and understand the standard examples of metric spaces. Know the definition of a topological space in general and understand specific examples. Know the definition and uses of equivalent metrics. Know and be able to use the definition of a continuous map between topological spaces. Know the definition of the relative (subspace) topology and the definition of homeomorphism. Know the definition of closed set and related concepts. Know the definition and use of convergent sequence. Know the definition and properties of Hausdorff space. Know the definition and properties of compact space, and the statement of the Heine-Borel Theorem. Know the definition and properties of connectedness and paths. Know the definition of quotient space and quotient map, and be able to describe some examples. Know the definition of path homotopy, and be able to give a summary of the construction of the fundamental group.

Learning resources

Main texts: