## 3000-level modules

#### MT3501  Linear Mathematics 2

Credits 15.0 1 2018/9 12.00 noon Mon (even weeks), Tue and Thu This module continues the study of vector spaces and linear transformations begun in MT2501. It aims to show the importance of linearity in many areas of mathematics ranging from linear algebra through to geometric applications to linear operators and special functions. The main topics covered include: diagonalisation and the minimum polynomial; Jordan normal form; inner product spaces; orthonormal sets and the Gram-Schmidt process; adjoint and self-adjoint operators. MT2001 or MT2501 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). 2-hour Written Examination = 90%, Coursework = 10% Dr J D Mitchell Dr J D Mitchell

#### Continuous assessment

• Assessed tutorial-style questions: 10% of final mark.

#### Syllabus

• Vector spaces: subspaces, spanning sets, linear independent sets, bases.
• Linear transformations: rank, nullity, general form of a linear transformation, matrix of a linear transformation, change of basis.
• Direct sums, projection maps.
• Diagonalisation of linear transformations: eigenvectors and eigenvalues, eigenspaces, characteristic polynomial, minimum polynomial, characterisations of diagonalisable transformations.
• Jordan normal form: method to determine the Jordan normal form.
• Inner product spaces: orthogonality, associated inequalities, some examples of infinite-dimensional inner product spaces, orthonormal bases, Gram-Schmidt process, orthogonal complements, applications.

#### Assumed knowledge

• Familiarity with solving systems of linear equations.
• Matrices, their basic properties, determinants and the method of finding the inverse of a matrix (provided it has non-zero determinant).
• Students will have met the definition of a vector space, basis, linear transformation, and their properties.  These will be revised quite rapidly (and more properties discussed) at the start of the course.

• T.S. Blyth & E.F. Robertson, Basic Linear Algebra, Second Edition, Springer Undergraduate Mathematics Series, Springer-Verlag, 2002.
• T.S. Blyth & E.F. Robertson, Further Linear Algebra, Springer Undergraduate Mathematics Series, Springer-Verlag, 2002.
• R. Kaye & R. Wilson, Linear Algebra, Oxford Science Publications, OUP, 1998.

#### MT3502  Real Analysis

Credits 15.0 1 2018/9 11.00 am Mon (even weeks), Tue & Thu This module continues the study of analysis begun in the 2000-level module MT2502 Analysis. It considers further important topics in the study of real analysis including: integration theory, the analytic properties of power series and the convergence of functions. Emphasis will be placed on rigourous development of the material, giving precise definitions of the concepts involved and exploring the proofs of important theorems. The language of metric spaces will be introduced to give a framework in which to discuss these concepts. MT2502 2.5-hours of lectures and 1 tutorial. 2-hour Written Examination = 90%, Class Test = 10% Dr J M Fraser Dr J M Fraser

#### Continuous assessment

• 50-minute class test: 10% of final mark.

#### Syllabus

• Countable and uncountable sets, including standard examples, basic properties, methods for showing sets are countable or uncountable.
• Review of convergence of sequences and continuity of real functions; uniform continuity.
• Riemann integration, definition in terms of lower and upper sums, basic properties, integrability of continuous and monotonic functions; integral of the uniform limit of a sequence of functions; Fundemental Theorem of Calculus.
• Power series, radius of convergence, differentiation and integration of power series.
• Introduction to convergence and continuity in normed and metric spaces, examples, including uniform convergence and L1 convergence.

• John M. Howie, Real Analysis, Springer, 2016.
• Robert G. Bartle & Donald R. Sherbert, Introduction to Real Analysis, 4th Edition, Wiley, 2011.
• Kenneth Ross, Elementary Analysis, 2nd edition, Springer, 2014.
• David Brannan, A First Course in Mathematical Analysis, CUP, 2006.
• DJH Garling, A Course in Mathematical Analysis, Vol.1, CUP, 2014. (More advanced)

#### MT3503  Complex Analysis

Credits 15.0 1 2018/9 12.00 noon Mon (odd weeks), Wed and Fri This module aims to introduce students to analytic function theory and applications. The topics covered include: analytic functions; Cauchy-Riemann equations; harmonic functions; multivalued functions and the cut plane; singularities; Cauchy's theorem; Laurent series; evaluation of contour integrals; fundamental theorem of algebra; Argument Principle; Rouche's Theorem. MT2502 or MT2503 or MT2001 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). 2-hour Written Examination = 90%, Coursework = 10% Dr M Quick Dr M Quick

#### Syllabus

• Review of complex numbers
• Holomorphic functions
• Contour integrals and Cauchy's Theorem
• Consequences of Cauchy's Theorem, including Liouville's Theorem, the Fundamental Theorem of Algebra, and Taylor's Theorem
• Harmonic functions
• Singularities, poles and residues: Laurent's Theorem, classification of isolated singularities, and Cauchy's Residue Theorem
• Application of contour integration: calculation of various integrals and infinite sums
• Complex logarithms and related multifunctions: branch cuts
• Counting zeros and poles: Rouché's Theorem and the Argument Principle

• John M. Howie, Complex Analysis, Springer Undergraduate Mathematics Series, Springer, 2003
• H. A. Priestly, Introduction to Complex Analysis, Second Edition, OUP, 2003

#### MT3504  Differential Equations

Credits 15.0 1 2018/9 9.00 am Mon (odd weeks), Wed and Fri The object of this module is to provide a broad introduction to analytical methods for solving ordinary and partial differential equations and to develop students' understanding and technical skills in this area. This module is a prerequisite for several other Honours options. The syllabus includes: existence and uniqueness of solutions to initial-value problems; non-linear ODE's; phase-plane analysis; Green's functions for ODE's; Sturm-Liouville problems; first order PDE's; method of characteristics; classification of second order linear PDE's; method of separation of variables; characteristics and reduction to canonical form. MT2001 or MT2503 2.5 lectures (weeks 1 - 10) and 1 examples class (week 2 - 11). Written Examination = 100% (2-hour final exam = 90%, class test = 10%) Prof A W Hood Prof A W Hood, Prof D G Dritschel

#### Syllabus

• Existence and uniqueness of solutions to initial-value problems.
• Non-linear ordinary differential equations.
• Phase-plane analysis.
• Green's functions for ordinary differential equations.
• Sturm-Liouville problems.
• First-order partial differential equations; methods of characteristics.
• Classification of second-order partial differential equations; method of separation of variables.
• Characteristics and reduction to canonical form.

• W.E. Boyce & R.C. DiPrima, Elementary differential equations and boundary value problems, Wiley.
• Peter V. O'Neil, Beginning Partial Differential Equations, John Wiley, 1999.

#### MT3505  Algebra: Rings and Fields

Credits 15.0 2 2018/9 11.00 am Mon (odd weeks), Wed & Fri This module continues the study of algebra begun in the 2000-level module MT2505 Abstract Algebra. It places emphasis on the concept of a ring and their properties, which give insight into concepts of factorisation and divisibility. Important examples such as polynomial rings will be used to motivate and illustrate the theory developed. MT2505 MT4517 2.5 hours of lectures and 1 tutorial. 2-hour Written Examination = 90%, Coursework = 10% Dr S Huczynska Dr S Huczynska

#### Continuous assessment

Short piece of work examining some of the topics developed in the module: 10% of final mark.

#### Syllabus

• Rings: definitions, examples (integers, modulo arithmetic, polynomial rings, etc.), definition of a field and its characteristic.
• Subrings, the prime subfield of a field, ideals, homomorphisms, quotient rings, the Isomorphism Theorems.
• Integral domains, field of fractions.
• Euclidean domains, polynomial rings (over fields) as Euclidean domains, Euclidean algorithm, greatest common divisors.
• Prime ideals, maximal ideals, their links to the quotient rings.
• The Chinese Remainder Theorem.  Applications of rings to number theory.
• Prime ideals and maximal ideals in Euclidean domains, and in particular in polynomial rings.
• Principal ideal domains, examples.
• Unique factorisation domains, theorem that if R is a UFD, then R[X] is a UFD.

• R.B.J.T. Allenby, Rings, Fields and Groups, 2nd ed., Edward Arnold, 1991.
• T.S. Blyth & E.F. Robertson, Essential Student Algebra, Vol.3: Abstract Algebra, Chapman & Hall, 1986.
• T.S. Blyth & E.F. Robertson, Algebra Through Practice, Book 3: Groups, Rings and Fields, CUP, 1984.
• D.A.R. Wallace, Groups, Rings and Fields, Springer, 1998.

#### MT3506  Techniques of Applied Mathematics

Credits 15.0 2 2018/9 12.00 noon Mon (odd weeks), Wed & Fri Differential equations are of fundamental significance in applied mathematics. This module will cover important and common techniques used to solve the partial differential equations that arise in typical applications. The module will be useful to students who wish to specialise in Applied Mathematics in their degree programme. MT2506 and MT3504 MT3081 2.5 hours of lectures and 1 tutorial. 2-hour Written Examination = 90%, Coursework = 10% Dr R K Scott Dr R K Scott

#### Continuous assessment

Short project involving application of methods developed in the module: 10% of final mark

#### Syllabus

• Modelling and interpretation (generating ordinary and partial differential equations).
• Ordinary differential equations resulting from separation of variables of partial differential equations (Laplacian operator in cylindrical and spherical coordinates).
• Frobenius methods for regular singular points.
• Special functions, including Bessel functions, Legendre (and associated) functions and Airy functions, Hermite, Laguerre, Heaviside and Delta functions.
• Green's function solutions for partial differential equations; examples of applications (e.g., Poisson's Equation for self-gravitation or electrostatics).
• Vector calculus revision and application to physical problems: e.g., solutions to grad p = F (where curl F = 0), curl B = j (Biot-Savart law), div E = rho_c, B = curl A (using Stokes' Theorem).
• Application to conservation laws (e.g., mass continuity as physical problem).

• William E. Boyce & Richard C. DiPrima, Elementary differential equations and boundary value problems, Wiley, 2013.
• Peter V. O’Neil, Beginning partial differential equations, Wiley-Interscience, 2008.
• David Griffiths, Introduction to Electrodynamics, Pearson, 2013.

#### MT3507  Mathematical Statistics

Credits 15.0 1 2018/9 11.00 am Mon (odd weeks), Wed & Fri Together with MT3508, this module provides a bridge between second year and Honours modules in statistics. It will provide students with a solid theoretical foundation on which much of more advanced statistical theory and methods are built. This includes probability generating functions and moment generating functions, as well as widely used discrete distributions (binomial, Poisson, negative binomial and multinomial) and continuous distributions (gamma, exponential, chi-squared, beta, t-distribution, F-distribution, and multivariate normal). It will also provide a foundation in methods of statistical inference (maximum likelihood and Bayesian) and model selection methods based on information theory (AIC and BIC). MT2508 MT3606 2.5 hours of lectures and 1 tutorial. 2-hour Written Examination = 90%, Class Test = 10% Prof S T Buckland Prof S T Buckland

#### Continuous assessment

50-minute class test: 10% of final mark

#### Syllabus

• Discrete data and distributions: Recap of probability generating functions; Binomial data: normal approximation, confidence intervals, dispersion test, testing equality of two binomial proportions.
• Poisson data: point estimation, confidence intervals, dispersion test, comparison of two Poisson counts.
• Further standard discrete distributions: negative binomial, multinomial.
• Continuous distributions: Recap of moment generating functions; Distribution of a function of a single random variable, function of several random variables.
• Some standard continuous distributions: gamma (including exponential and chi-squared), beta, t, F.
• Multivariate normal distribution.
• Likelihood-based methods: The likelihood function; Maximum likelihood vs Bayesian methods.
• Maximum likelihood estimators: properties, variance and interval estimation; Sufficient statistics.
• Bayes’ Theorem, prior and posterior distribution, conjugate priors, credible intervals; Information criteria: AIC and BIC.
• General (normal) linear model: The normal equations; Hypothesis testing.

• M.H. DeGroot & M.J. Schervish, Probability and Statistics, 4th edn
• G.M. Clarke & D. Cooke, A Basic Course in Statistics, 5th edn
• M. Fisz, Probability Theory & Mathematical Statistics
• G. Casella & R.L. Berger, Statistical Inference, 2nd edn
• J.G. Kalbfleisch, Probability and Statistical Inference, volume 2

#### MT3508  Applied Statistics

Credits 15.0 2 2018/9 12.00 noon Mon (even weeks), Tue & Thu Together with MT3507, this module provides a bridge between second year and Honours modules in statistics. It deals with the application of statistical methods to test hypotheses and draw inferences from data. This includes a number of nonparametric methods and statistical tests (permutation and randomization tests, goodness-of-fit tests and tests of independence). Inference methods include model fitting by least squares and maximum likelihood, and variance estimation by means of the information matrix and by bootstrap. Applications include multiple regression, analysis of variance, the general (normal) linear model and an introduction to generalized linear models and generalized additive models. MT2508 MT3606 2.5 hours of lectures and 1 tutorial. 2-hour Written Examination = 90%, Coursework (Project) = 10% Dr V.M. Popov Dr V.M. Popov

#### Continous assessment

Computer-based project using the package R: 10% of final mark

#### Syllabus

• Nonparametric methods and goodness-of-fit: Types of data; Recap of permutation and randomization tests; Sign test.
• Wilcoxon signed ranks test; Mann–Whitney test.
• Runs test; Goodness-of-fit tests: chi-squared and Kolmogorov–Smirnov test.
• Chi-squared tests of homogeneity and independence.
• Model fitting and quantifying precision: Least squares.
• Maximum likelihood; Estimating variance using the information matrix.
• Nonparametric bootstrap; Parametric bootstrap.
• Statistical modelling: Multiple regression; Analysis of variance.
• Factorial experiments; The general (i.e., normal) linear model.
• Brief summary of GLMs and GAMs, and how to fit them.

• S. Siegel & N.J. Castellan Jr., Nonparametric Statistics for the Behavioral Sciences, 2nd edn
• W.J. Conover, Practical Nonparametric Statistics, 3rd edn
• Richard D. De Veaux, Paul F. Velleman & David E. Bock, Stats: Data and Models, Pearson/Addison Wesley, 2005.
• Bryan F.J. Manly, Randomization, Bootstrap and Monte Carlo Methods in Biology, Chapman & Hall, 2007

#### MT3802  Numerical Analysis

Credits 15.0 1 2018/9 10.00 am Mon (odd weeks), Wed and Fri The module will introduce students to some topics in numerical analysis, which may include methods of approximation, iterative methods for solving systems of linear equations, numerical techniques for differential equations. MT2001 or MT2501 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). 2-hour Written Examination = 70%, Coursework = 30% Dr A P Naughton Dr A P Naughton

#### Syllabus

• Norms (ways to measure errors).
• Iterative methods to solve linear systems of equations.
• Approximations to functions.
• Best approximations.

#### Assumed knowledge

It will be assumed that students have a good knowledge of basic matrix methods (inversion, multiplication, etc.).

• G.M. Phillips & P.J. Taylor, Theory and Applications of Numerical Analysis.

#### MT3832  Mathematical Programming

Credits 15.0 2 2019/20 12.00 noon Mon (odd weeks), Wed and Fri The aim of this module is to introduce students to the formulation and solution of various linear programming problems. The subject matter will be illustrated by applying the methods of solution to real examples. The syllabus includes: formulation of linear problems; solution graphically and by simplex algorithm; sensitivity analysis; duality; transportation and transshipment; the assignment problem. MT2001 or MT2501 or (MT1002 and MN2002) 2.5 lectures (weeks 1 - 10) and 0.5 tutorial (weeks 2 - 11). 2-hour Written Examination = 100% TBC TBC

MT3832 runs in alternate years. It is expected to be withdrawn after 2019/20.

#### MT3852  Automata, Languages and Complexity

Credits 15.0 2 2018/9 3pm Mon (even weeks), Tue, Thu. This module begins with finite state machines, context-free grammars and big-O notation. Turing machines, non-determinism and pushdown automata are introduced, followed by studies on decidability, simulation and the Halting problem. The complexity classes P, NP, co-NP, NP-hard, etc., are described via analysis of SAT and graph isomorphism. Strengths and limitations of the abstract approach to complexity are discussed, followed by an introduction to practical complexity. MT2504 or ((CS2001 or CS2101) and CS2002) CS3052 2 hours of lectures (x 11 weeks), .5-hour tutorial (x 10 weeks) 2-hour Written Examination = 60%, Coursework = 40% Prof C M Roney-Dougal Prof C M Roney-Dougal, Dr S Sarkar, Prof I Gent, Prof S Linton

MT3852 runs in alternate years.  The timetable may change for future years.