# Level 2 Modules

**MTH2011 Linear Algebra (Sem 1)**

*Pre-requisites:* MTH1011 Introduction to Analysis and Algebra and MTH1021 Mathematical Methods 1 or MTH1002 Numbers, Vectors and Matrices [for 2021/22]

*Lecturers:* Dr D Barnes

**Introduction**

**Content**

- Recap and extend to fields such as C, the notions of abstract vector spaces and subspaces, linear independence, basis, dimension.

- Linear transformations, image, kernel and dimension formula.

- Matrix representation of linear maps, eigenvalues and eigenvectors of matrices.

- Matrix inversion, definition and computation of determinants, relation to area/volume.

- Change of basis, diagonalization, similarity transformations.

- Inner product spaces, orthogonality, Cauchy-Schwarz inequality.

- Special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties.

- Basic computer application of linear algebra techniques.

Additional topics and applications, such as: Schur decomposition, orthogonal direct sums and geometry of orthogonal complements, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form.

**Assessment**

Assignment 30%

**MTH2012 Analysis (Sem 1)**

*Pre-requisite: *MTH1011 Introduction to Algebra and Analysis and MTH1021 Mathematical Methods 1, or MTH1001 Analysis and Calculus and MTH1002 Numbers, Vectors and Matrices [for 2021/22]

*Lecturers:* Dr A Zhigun and Dr B McMaster

**Introduction**

**Content**

Cauchy sequences, especially their characterisation of convergence. Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omitted). Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test. Mean value theorems including that of Cauchy, proof of l'Hôpital's rule, Taylor's theorem with remainder. Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus.

**Assessment**

Assignment 10%

Class Test 15%

Exam 60%

Project 1, 2, 3 5%

**MTH2013 Metric Spaces (Sem 2)**

*Pre-requisite: *MTH2011 Linear Algebra, MTH2012 Analysis

*Lecturers:* Dr Y-F Lin and Dr G Kiss

**Introduction**

**Content**

- definition and examples of metric spaces (including function spaces)

- open sets, closed sets, closure points, sequential convergence, density, separability

- continuous mappings between metric spaces

- completeness

**Assessment**

Assignment 30%

Exam 70%

**MTH2014 Group Theory (Sem 2)**

*Pre-requisite: *MTH1011 Introduction to Algebra and Analysis, MTH2021 Linear Algebra, or MTH1001 Analysis and Calculus, MTH1002 Numbers, Vectors and Matrices and MTH2021 Linear Algebra [for 2021/22]

*Lecturer:* Dr A Blanco

**Introduction**

**Content**

- definition and examples of groups and their properties

- countability of a group and index

- Lagrange’s theorem

- normal subgroups and quotient groups

- group homomorphisms and isomorphism theorems

- structure of finite abelian groups

- Cayley’s theorem

- Sylow’s theorem

- composition series and solvable groups

**Assessment**

Assignment 30%

Exam 70%

**MTH2021 - Mathematical Methods 2 (Sem 2)**

*Pre-requisite: *MTH1011 Introduction to Algebra and Analysis, MTH1021 Mathematical Methods 1, or MTH1001 Analysis and Calculus [for 2021/22]

*Lecturers*: Dr T Todorov, Dr G Gribakin

**Introduction**

**Content**

Functions of a complex variable: limit in the complex plane, continuity, complex differentiability, analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, residues, Cauchy residue theorem, evaluation of integrals using the residue theorem.

Series solutions to differential equations: Frobenius method.

Fourier series and Fourier transform. Basis set expansion.

Introduction to partial differential equations. Separation of variables. Wave equation, diffusion equation and Laplace’s equation.

**Assessment**

Assignment 15%

Exam 60%

Project 25%

**MTH2031 Classical Mechanics (Sem 1)**

*Pre-Requisites: *MTH1021 Mathematical Methods 1, or MTH1001 Analysis and Calculus and MTH1002 Numbers, Vectors and Matrices [for 2021/22]

*Lecturers: *Dr C Ramsbottom

**Introduction**

**Content**

Introduction to calculus of variations.

Recap of Newtonian mechanics.

Generalised coordinates. Lagrangian. Least action principle. Conservation laws (energy, momentum, angular momentum), symmetries and Noether’s theorem. Examples of integrable systems. D’Alembert’s principle. Motion in a central field. Scattering. Small oscillations and normal modes. Rigid body motion.

Legendre transformation. Canonical momentum. Hamiltonian. Hamilton’s equations. Liouville’s theorem. Canonical transformations. Poisson brackets.

**Assessment**

Exam 80%

Project 20%

** **

**SOR2002 Statistical Inference (Sem 2)**

*Pre-requisite:* SOR1020 & SOR1021

*Lecturer*: Dr L McFetridge

**Introduction**

The course builds upon the probability theory in SOR1001 and the statistical inference methods in SOR1002 to provide a second-level account of the principles and most important methods of estimation and hypothesis testing.

An important element of this module will be a weekly practical data analysis lab using the SAS software package. SAS is probably the leading statistical package used in industry. The lab sessions, approximately 2 hours long, will give students an opportunity to put the theory of lectures into practice. During the lab sessions students will be guided through practical tutorials on reading, accessing and editing data in SAS, descriptive statistical analyses, cross tabulations, graphs, charts, frequency tables, χ^{2} tests, *t*-tests and analysis of variance tests. Some of the lab time will also be allocated for project work.

The practical work in the lab sessions culminates in a group project. During the semester the student will work as a team member on a specific statistical investigation using the methods described in lectures and performed in the SAS lab sessions. This provides a valuable insight into the applications of statistics in industry allowing students to gain the necessary skills required for such a working environment. Near completion of the group project, each group will be required to give an oral presentation (approximately 5 mins per student) on the results of their analysis. This provides an opportunity for the group to receive feedback on their progress and helps consolidate their findings before submitting a final written report on their investigation.

**Content**

- Statistical Investigations: Understanding the problem. Collecting the data. Initial data analysis. Definitive analysis: modelling. Conclusions.
- Initial Data Analysis: Data structure. Processing data: coding, input, screening, editing and modifying. Data quality.
- Preliminary analysis: measures of location, dispersion; tables; graphs.
- Sample Diagnostics: Testing for independence - non-parametric tests, serial correlations. Testing for normality - skewness and kurtosis, goodness-of-fit tests, probability plotting. Identification of outliers. Transformations.
- Point Estimation of Parameters: Definitions of estimate, estimator, sampling distribution. Unbiasedness. Relative efficiency. Bias. Mean squared error.
- Sufficiency: Fisher-Neyman factorization theorem. Regular exponential class of distributions.
- Maximum Likelihood: Likelihood function. Calculation of MLE. Log relative likelihood function. Asymptotic properties of MLE. Applications.
- Least Squares Estimation and Linear Regression: Standard linear model: matrix notation. Properties of LSE. Weighted least squares. Fitting a straight line. Multiple regression. Goodness-of-fit: residuals. Hypothesis tests and confidence intervals using the t-distribution.
- Experimental Design and Comparative Studies: Principles of design - experimental unit, treatment, replication, randomization; factorial design. Analysis of variance. Completely randomized design. Randomized block design. Dichotomous treatment/risk and outcome studies. Sampling schemes - cross-sectional; longitudinal-cohort, case-control study.
- Measures of association: rates, relative risk, odds ratio.
- Significance Tests and Hypothesis Testing: Neyman-Pearson approach - critical region, Type I and Type II errors, significance level, power function. Best critical region. Generalized likelihood ratio test.
- Computer intensive methods: randomization tests; Monte Carlo sampling.
- Confidence Intervals: Construction - pivotal quantity; MLE procedure. Confidence region. Prediction interval.
- Bayesian Methods: Prior and posterior distributions. Conjugate families. Point estimates, confidence regions, hypothesis testing. Prediction. Improper and non-informative priors.

**Assessment**

Exam 70%

Report 20%

Presentation 10%

**SOR2003 Methods of Operational Research (Sem 1)**

*Pre-requisite:* SOR1020 & SOR1021

*Lecturer*: Dr S Moutari

**Introduction**

This course applies mathematical analysis to a series of problems which occur in business and industry. The analysis can be more far reaching if we use a deterministic model but a degree of uncertainty (e.g. about future events) is often an important feature of the situation and a stochastic model has to be used. The statistical knowledge assumed is that contained in SOR1001. Although novel ways of setting out the work may be used in some topics, the mathematical techniques required on this course are no more advanced than simple calculus and algebra and most practical problems require only arithmetic and the use of tables.

The aim of the course is to teach a range of simple techniques illustrating the application of mathematics and probability theory to the problems of business and industry. Apart from the first two chapters each chapter is a distinct and separate topic. Some topics (e.g., Forecasting) involve lengthy calculation and students are taught how to use a spreadsheet for the computation. Students who do not have access to a spreadsheet on a personal computer can use the Open Access Areas. Specific instructions on the use of the Excel spreadsheet is given on the course and there is a practical session in an Open Access Area.

Homeworks are an essential part of the learning process, but there is no continuous assessment.

Emphasis is placed on choosing the correct model for the circumstances and on presenting answers in a form intelligible to management. If a question is posed in words then the final answer should be in words and not left in algebra or in a table. The practical problems associated with obtaining data are discussed. The answer should be to a number of significant figures consistent with the accuracy of the original data, or rounded to an integer if that is appropriate.

**Content**

- Deterministic inventory including quantity discounts, common cycle production, constrained inventory and the use of Lagrange multipliers.
- Stochastic inventory models including service levels.
- Simple and adaptive forecasting.
- The use of spreadsheets and their application to forecasting and equipment replacement.
- The replacement of deteriorating equipment and the replacement of equipment liable to sudden failure.
- Acceptance sampling by attribute and variable.
- Network planning including PERT, speeding up, the use of LP, Gantt charts and resource smoothing.
- Decision analysis including utility curves, decision trees and Bayesian statistics.

**Assessment**

Exam 70%

Report 20%

Presentation 10%