# Mathematics Module Details

### Year One

Statistics and Mechanics I covers two connected and key fundamental subjects:

Mechanics

The Mechanics component can be split into three areas all concerned with rigid bodies: the simulation of motion, the detection of collisions, and the resolution of collisions. Students will be introduced to the mathematical techniques required for these models, and will be able to model the motion of rigid bodies under constant and variable acceleration.This will be done through Newton’s equations of motion for constant acceleration and numerical approximations methods such as Euler’s method for variable acceleration. Students will be able to detect collisions between pairs of convex rigid bodies using the SAT algorithm and be able to use simplified approaches for circles, axis-aligned bounding boxes, and oriented bounding boxes. Collision resolution will be explored through the conservation of linear and rotational momentum with simple coefficient simulations of elasticity and friction.

Statistics

The Statistics component in this module introduces fundamental statistical techniques for data collection, summary and presentation. Students will develop an understanding of key concepts and techniques associated with the collection, analysis and interpretation of statistical data. Extensive use will be made of a variety of industry standard software, e.g. Minitab and SAS.

Operational Research introduces fundamental techniques for linear programming and network models for decision-making. The module will introduce the formulation, solution and interpretation of linear programming models, and cover network models and project management. Extensive use will be made of a variety of software, e.g., Excel and The Management Scientist.

Linear Algebra introduces basic mathematical concepts as well as the idea of mathematical modelling. It provides a basic knowledge of linear algebra including functions, Cartesian product, matrices, systems of linear equations, vector algebra and complex numbers and complex polynomials.

Mathematical Analysis provides students with an introductory course on mathematical analysis of concepts related to real numbers. There are several methods to introduce real numbers. The method that students will learn is the one that assumes that real numbers are undefined objects satisfying certain axioms from which further properties are derived.

### Year Two

Statistics and Mechanics II covers content extending the subjects of mechanics and statistics from that covered in the first year Statistics and Mechanics 1 module:

Mechanics

The Mechanics component in this module aims to provide students with a clear understanding of applied mathematics, focused on accurate simulation and relative problem solving. Building upon a foundation of Newton’s Laws and kinematics, students will be presented with mathematical techniques to model advanced types of motion and interaction. This will be done through an expanded look at modelling the motion of rigid bodies, including momentum, moments of inertia, and impulse resolution. Circular and vertical circle motion is considered, as well as advanced projectile motion. A view on elastic strings and springs provides a look at elastic energy and considerations for impacts. This extends towards a further exploration of dynamics, considering oscillation through simple harmonic motion, including damped and forced harmonic motion.

Statistics

The Statistics component in this module presents a range of advanced statistical techniques such as ANOVA and multiple regression, which builds on knowledge of hypothesis testing and simple linear regression, and considers their applications to business decision making. Students will further develop their skills in the selection of statistical analyses and the interpretation and reporting of data.  Extensive use will be made of industry standard software, e.g. SAS.

Abstract Algebra furthers students’ knowledge of discrete mathematics and linear algebra and focuses on abstract algebra, inner product spaces and linear mapping. The module will show how theory can be applied to produce solutions to practical problems.

Numerical Methods help us estimate solutions of fundamental problems of mathematics that an analytical solution is not possible to find or is impractical to use. Numerical methods are often implemented in computer code, so in this module we will use MATLAB, in addition to finding numerical solutions by hand. Additionally, we will study the convergence and error of numerical methods, and in some cases their development.

Multivariable Analysis extends the content of the first year Mathematical Analysis module of real-valued function of one real-variable to multivariable real functions to complete a course on ‘Real Mathematical Analysis’.

### Year Three

Modelling Ordinary and Partial Differential Equations covers some of the ways in which ordinary differential equations (ODEs) are used by mathematicians, engineers and scientists to model, explain and predict the behaviour of physical and biological systems. Some problems can alternatively be formulated as variational principles through which a definite integral must be maximised or minimised. This module will draw upon a wide range of applications to demonstrate the modelling processing and solution using ordinary, and partial differential equations and variational principles.

Non-linear Dynamical Systems and Nonlinear Optimisation consists of two parts. The first part covers the principles and methods for investigation of nonlinear dynamical systems. It provides knowledge of modern nonlinear dynamical system theory and numerical methods for nonlinear analysis using MATLAB. The second part focuses on the solution of nonlinear problems, using analytical and numerical methods, both by hand and using MATLAB.

Block 3 Choose one of:

• Statistical Modelling: Analysis of Time Series Data, Categorical Data and Multivariate Data focuses on the theoretical detail and practical application of analysing time series methods in the generation of time series models, and also focuses on the theoretical detail and practical application of analysing both categorical data and multivariate data in the generation of statistical models. Taking a practical approach, advanced statistical techniques will be introduced and students will use real problems to analyse time series data, categorical data and multivariate data, thus building relevant statistical models, which are interpreted and critiqued. The statistical analyses are analysed from two complementary perspectives: the statistical theory and the implementation of the theory using industry standard software, e.g. SAS and Minitab.
• Data Mining aims to review the methods available for uncovering important information from large data sets; to discuss the techniques and when and how to use them effectively. The module uses the data mining tool SAS Enterprise Miner. SAS is a comprehensive data management software package that combines data entry and manipulation capabilities with report production, graphical display and statistical modelling.
• Fuzzy Logic and Inference Systems presents the core and fundamental concepts of fuzzy logic, from theory to application. The understanding developed will allow for a fuzzy perspective to be adopted, understood and appreciated. The ability to create specialised fuzzy inference systems will be achieved and so too will the ability to articulate on thought processes needed to create such systems. A comprehensive understanding of fuzzy logic, theory and application will also be covered. The module will also investigate the literature on fuzzy and its areas of application to further instil the applicability of a fuzzy approach and the ethical implications of modelling subjective perception-based uncertainty

The Final Year Project enables students to undertake an individual project on an approved topic of interest, that addresses significant Mathematics, Statistics or Operational Research problems relevant to the Programme of study. The Project provides an opportunity for the students to integrate many of the threads of their Programme of study and to extend their work beyond the taught elements through research, data analysis and self-learning.