# Level 2 Modules

**MTH2010 Employability for Mathematics (Sem 1)**

*Pre-requisites:* None

*Lecturer:* Dr D Dundas

**Introduction**

This is a 0 CAT point module that is compulsory for students planning to take a placement year. The module consists of 6 lectures and 2 workshops and is assessed 100% by attendance.

**Content**

Introduction to placement for mathematics and physics students, CV building, international options, interview skills, assessment centres, placement approval, health and safety and wellbeing. Workshops on CV building and interview skills. The module is delivered in-house with the support of the QUB Careers Service and external experts.

**Compulsory Element**

Attend at least five of the six lectures and attend all two workshops.

**Assessment**

Attend at least five of the six lectures and attend and participate in the two workshops.

**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]

*Lecturer:* Dr D Barnes

**Introduction**

The techniques of linear algebra are core to most areas of mathematics, statistics, data analysis, physics and computer science. This module will build upon the level 1 linear algebra to develop the theory, methods and algorithms needed throughout the degree programme and beyond.

**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**

Continuous assessment 30%

Written examination 70%

**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]

*Lecturer:* Dr A Zhigun

**Introduction**

This module extends and develops several core ideas from Level 1 analysis and seeks to deepen your understanding of them.

Its primary aim is to equip you with some key techniques of mathematical analysis, so that you can apply them both within mathematics itself and beyond in various other disciplines.

**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**

Exam 60%

Class Test 15%

Projects 15%

Assignments 10%

**MTH2013 Metric Spaces (Sem 2)**

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

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

**Introduction**

Analysis in Semester 1 is the study of convergence and continuity, it is fundamentally linked to the structure of the real numbers. The aim of this module is to move gradually away from real numbers to the more general setting of metric spaces. A metric space is a set with a notion of distance, called a metric. The most familiar example is the real line, with the distance from x to y given by |x- y|. This notion allows us to define convergence and continuity in a much more abstract setting, and induces topological properties, like open sets and closed sets which lead to the study of more abstract topological spaces.

**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**

Continuous assessment 30%

Written examination 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**

Group Theory is the branch of Algebra concerned with the study of groups. It is quite different from Linear Algebra and its origins can be traced back to the work of Lagrange on permutation groups. Groups usually arise in connection with invariance properties of the objects under study, e.g., the collection of all geometrical transformations of the Euclidean plane that leave invariant certain figure on it, form a group. Applications of group theory can be found in almost every area of mathematics, and also in chemistry and physics. This module will introduce you to some of the main concepts and techniques of the subject.

**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**

Continuous assessment 30%

Written examination 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**

This module covers complex analysis, partial differential equations and some of the interconnections between them. Complex analysis is the calculus of functions of a complex variable. It has elegant and useful applications in other areas of mathematics, in the physical sciences and in engineering. One of its most powerful uses, which we will study in the module, is in the evaluation of large classes of real integrals that are very hard or even impossible to do in any other way. Partial differential equations are differential equations for functions of more than one variable, e.g., position and time. Most areas of physics hinge on partial differential equations. They also have key applications in other subjects such as financial mathematics and mathematical biology.

**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]

*Lecturer: *Dr C Ramsbottom

This module is in transition for the 2021-2022 academic year. For this year only the module specifications, learning outcomes and assessment procedures are:

**Introduction**

Mechanics has always been a source of inspiration for mathematics and mathematicians. Calculus, as well as the famous Newton's laws, were “invented” by Newton largely because he wanted to solve a single but very important mechanical problem, the problem of planetary motion. Mechanics was at the beginning of such branches of mathematics as theory of functions, calculus of variations, differential equations and more recently, theory of chaos. Mechanics and its mathematical methods are important in optics, electromagnetic theory, statistical mechanics, quantum mechanics, theory of relativity, quantum field theory and many ‘non-physical’ applications such as theory of optimisation and control.

For students, besides giving a comprehensive picture of mechanical phenomena and teaching how to solve a wide variety of problems, Classical Mechanics offers a unique opportunity to see the mathematical methods they have learned at work and to practice their mathematical skills. Vectors, partial derivatives, single and multiple integrals, differential equations, stationary points, complex numbers, as well as matrices and determinants are all among the tools used in the course.

**Content**

*Introduction:*Basic revision of Newtonian mechanics: conservation of mechanical energy and angular momentum. Revision of key concepts in linear algebra (matrices): eigenvalues and eigenvectors. Revision of basic multivariable calculus (partial differentiation and multiple integration).*Motion of a single particle in a central potential:*Attractive forces: planetary motion and transfer orbits. Repulsive forces: Rutherford scattering.*Rotating frames of reference:*Angular velocity, effect of the Earth's rotation; apparent gravity, cyclones and anticyclones, Foucault's pendulum, Larmor precession.*Conservation laws for a system of particles:*Internal and external forces, conservation of energy, centre of mass as the origin.*Rigid body motion:*Moments and products of inertia, parallel axis theorem, Euler's equations of motion, Euler angles and rotation matrices.*Lagrange's equations:*Constraints, equations for holonomic constraints; examples.*Lagrange's equations for impulses:*Applications to systems of rods.*Small oscillations:*Normal modes of oscillation.

**Assessment**

Exam 60%

Class Test 20%

Project 15%

Assignments 5%

From 2022-2023 onwards the module specifications and assessment procedures will be:

**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.
- Small oscillations and normal modes. Rigid body motion.
- Legendre transformation. Canonical momentum. Hamiltonian. Hamilton’s equations. Liouville’s theorem. Canonical transformations. Poisson's brackets.

**Assessment**

Exam 80%

Project 20%

** **

**SOR2002 Statistical Inference (Sem 2)**

*Pre-requisites: *SOR1020 Introduction to Probability and Statistics

*Lecturer*: Dr L McFetridge

**Introduction**

Data is everywhere, being collected in all application areas and creating a huge growth in demand for qualified statisticians and data scientists. As the Economist noted, “the world’s most valuable resource is no longer oil, but data”. This module will lay the foundations for harnessing this important resource. It is therefore a key SOR module that provides the skills for extracting information from data and thus is a pre-requisite for most SOR modules in Levels 3 and 4.

There is a need for data enabled decision making. This module provides students with the skills to be able to analyse and interpret real world data, focusing first on initial data analysis and leading on to the use of linear regression to analyse quantitative information. Due to the practical nature of statistics, reference will be given throughout the module as to how the theory and computer elements that are taught relate to real world problems. 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.

To put these skills into practice, students will undertake a group project to analyse real world data, replicating a typical project undertaken in industry. The module covers data analysis, sample diagnostics, estimation, linear regression, experimental design, hypothesis testing, and Bayesian methods.

**Content**

*Statistical Investigations*: Understanding the main stages undertaken within a statistical analysis.*Initial Data Analysis & Sample Diagnostics*: This is the first step in analysing any real-world data. It focuses on gaining a better understanding of the data and therefore which statistical method would be most appropriate to use in its analysis. Testing for independence, serial correlations, normality, etc. will be explored.*Linear Regression*: Following initial data analysis, statisticians utilise statistical models to extract information from complex data. One of the fundamental statistical models is a linear model. This model will be introduced, illustrated using real world data and validated using goodness-of-fit measures.*Experimental Design and Comparative Studies*: Statisticians are often asked to design the collection of data for studies, such as clinical trials. The principles of experimental design and advantages/disadvantages of different sampling schemes will be discussed.*Estimation of parameters*: This will explore the concepts of bias and efficiency in the estimation of unknown parameters. Relative efficiency, sufficiency and mean square error will be introduced. The Fisher-Neyman factorization theorem will be explored.*Maximum Likelihood*: This is one of the most common methods of estimation. Construction of the likelihood function, calculation of a corresponding estimate and its properties will be discussed.*Significance Tests and Hypothesis Testing*: Hypothesis tests are utilised extensively in the analysis of data. The origins of some commonly used equations will be demonstrated utilising the Neyman-Pearson lemma. The use of computer intensive methods will be discussed, including randomization tests and Monte Carlo sampling.*Bayesian Methods*: The demand for Bayesian estimation is rapidly increasing in recent years. Such methods can incorporate prior beliefs about the situation under analysis, making the process extremely appealing to analysts. Conjugate families, prediction and the use of improper and non-informative priors will be discussed.

**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 and stochastic inventory models; simple and adaptive forecasting; theory of replacement of equipment; quality control, acceptance sampling by attribute and variable; network planning including the use of PERT, LP, Gantt charts and resource smoothing; decision theory, including utility curves, decision trees and Bayesian stastistics; simple heuristics.

**Assessment**

Exam 70%

Report 20%

Presentation 10%