Mathematics

This module extends some of the fundamental techniques of operational research from Level 4, and introduces some new operational research techniques to enable you to solve less well defined or more complex problems than those studied in Level 4.  The methods studied in this module are all designed to enable the solution of particular classes of business problem which are widely used in commerce and industry and many students report that they have found this module the most useful when undertaking their Industrial Placement year. Examples of techniques that may be explored in further depth are linear programming (to include sensitivity analysis for example) and network analysis (to include the process of crashing, or the Program Evaluation and Review Technique, for example).  Other potential topics are dynamic programming, game theory, inventory control, decision analysis, transportation and assignment algorithms.  You will have the opportunity for early feedback on your progress in this module as it has been designed with an early, small, assessment to help you guage your levels mastery of the tools and techniques you will need.

This course extends the algebra and calculus studied in year 1. It is designed into two streams. The first stream consists of 24 hours of lectures over 12 consecutive weeks covering Group Theory. Group Theory is the mathematical study of symmetry. This sounds a bit vague but it’s this vague definition that makes group theory such a broad and deep subject. It has its origins in the work of Galois, Lagrange and Cauchy in the 1800s. The second stream consists of 24 hours of lectures over 12 consecutive weeks on Differential Equations. The laws of physics are generally written down as differential equations. Therefore, all of science and engineering use differential equations to some degree. Understanding differential equations is essential to understanding almost any mathematical model.

This course extends the analysis and calculus studied in year 1. It is designed into two streams. The first stream consists of 24 hours of lectures over 12 consecutive weeks covering Real Analysis. In real analysis we extend and reinforce the calculus and analysis studied in year 1. The second stream consists of 24 hours of lectures over 12 consecutive weeks on Complex Analysis. Complex analysis involves the analysis of functions of a complex variable. It is a rich and deep subject with applications in fluid mechanics and engineering as well as within real analysis. Real Analysis - The Real Numbers, boundedness, supremum, infimum, completeness of the reals, the Archimedean property, the density of the rationals, defining the real numbers. - Sequences, partial sums, convergence, divergence, limit of a sequence, squeeze theorem, convergence criteria, subsequences. - Series, convergence, divergence, telescoping series, geometric series, harmonic series etc. Tests for convergence, comparison test, ratio test. - Continuity & amp; Differentiability, epsilon-delta limits, continuity, intermediate value theorem, differentiability, piecewise continuous functions, mean value theorem, Rolle& rsquo;s theorem. Complex Analysis -Review of Complex numbers: definition as an ordered pair, geometry in the complex plane, polar form, Euler& rsquo;s formula, roots of unity. -Set Theory in the Complex Plane: open/closed discs, boundary points, closure, annulus, bounded/unbounded, the Riemann Sphere, Stereographic Projection. -Complex Functions: limits, continuity, differentiability, holomorphicity, Cauchy-Riemann theorem, Laplace equation, Harmonic conjugates. -Complex Integration: curves, contours and simply connected domains, integration along curves and contours (path integral), ML-Inequality, fundamental theorem of contour integration, Cauchy-Goursat and deformation theorem, Cauchy& rsquo;s Integral Formula. -Series: Taylor and Laurent series, Laurent& rsquo;s theorem, poles, removable and essential singularities. -Cauchy& #39;s Residue Theorem: residue theory, the residue theorem, evaluation of real integrals (infinite and trigonometric).

This course extends the analysis and calculus studied in year 1. It introduces you to the hugely useful subject area of mathematical modelling. You will be introduced to the main concepts and tools within this subject as well as being introduced to the software package Matlab. Matlab is the industry standard package for mathematical modelling and is used in disciplines from finance to engineering to solve applied problems. Each week you will have 3 hours of lecture time and 1 hour of laboratory sessions using Matlab.

This course extends the probability and statistics studied in year 1. It will introduce you to the theory and practice of Linear Statistical Modelling and Survey Design. Linear statistical modelling statistical modelling where a response variable depends on one or more explanatory variables. Examples include, how well a patient responds to treatment, given their age and disease severity, or how different strains of corn compare when grown in various conditions. Survey design is a critical topic when determining the quality of survey based research. The potential for poor survey design is enormous whether intentional or not. Examples of poor survey design can be found in many real world spheres including political polling and readership surveys.

This module is designed for you to prepare for your project. There are two parts in this module: • (Part 1) Project Management principles, methodologies, tools and techniques that may be used in any industry together with organisational and human resource factors are also included. In the Social and Professional area it offers the opportunity to synthesize the wider human and social dimensions of the course in terms of social, ethical, legal, and cultural issues in order to develop your awareness of concerns which will be relevant to your chosen pathway and future.. • (Part 2) research methodology: this part forms part of the project The overall aims are: • To prepare for the level 6 project • To introduce new technologies and research trends • To provide students with the skills to critically evaluate the benefits, drawbacks and pitfalls of leading edge techniques in their chosen field. • To give the student analytical skills to determine the effect of applying new techniques to existing practices • To give the student a good grounding in new research and commercial developments that can then be developed beyond their course.

This module will help you to develop knowledge and understanding of major techniques in calculus which will either extend topics you have covered previously or will be completely new to you. For example, you will learn further methods to solve differential equations (such as Fourier Transforms, Laplace Transforms, Reduction of Order, Variation of Parameters, Power Series, the Frobenius method…etc). You will also extend your knowledge of functions of multiple variables and/or integral calculus by studying a selection of topics which might include: multiple integration, absolute extrema, Lagrange Multipliers, surface integrals or complex integrals.

Operational research is the application of mathematics to business and management problems and is a skill sought after by employers. Many mathematicians in industry use some of the OR techniques covered in this module in their daily working lives. This module extends the material taught in 5MM021 Further Techniques in Operational Research. You will apply familiar and new OR techniques to practical problems, and you will use OR specific software to help with the formulation of solutions. This module introduces practical applications of operational research techniques previously studied in first and second year but also introduces some new methods that are in common use in industry today.  This module extends some of the topics into new areas, such as integer programming, goal programming and data envelopment analysis.  It also introduces new techniques, which may include queuing theory, simulation or soft systems analysis.  Appropriate operational research software will be used to help solve some of the problems that arise. This is a practical module, and you will solve real world, extensive problems in small groups, where you will apply some of the professional and teamworking skills you have learned elsewhere in your course.

This module extends the algebra you have learned in previous modules in the shape of the two applied areas of Coding Theory and Cryptography. Coding Theory is concerned with error detection and correction in transmitted data. It utilises tools from linear algebra and field theory to do this. Cryptography is concerned with securing digital communications. You may already be familiar with Diffie-Hellmann Key Exchange and the RSA algortihm from first year. Here we build on this material from an applied standpoint.

Financial mathematics is a vast topic, and here you will be introduced to applying probaility, statistics and mathematics to finance related problems. Taking uncertainty into account in your calculations is one of the key aspects that we will focus on in particular studying some of the mathematical and statistical techniques used by life and general insurance companies. Among the topics that may be included are valuations of financial and insurance products, different ways of dealing with uncertainty and risk, premium setting, mortality, morbidity and hazard rate analysis. In addition there will be analysis of both financial and non financial data using a variety of time series analysis techniques such as using Box Jenkins and Holt Winters methods in forecasting. There is also an opportunity to undertake data analysis to underpin data informed decision making; the data will typically be from subjects that either directly or indirectly impact the finanical sector such as investment prices, economic forecasts and political trends. This is a very practical module and the emphasis is on the work you will do when preparing your report(s) and presentation(s). You will be collecting and analysing data, writing technical reports and presenting your findings to a wider audience, with plenty of opportunities for formative feedback from module tutors to help you maxiimise your potential.

Multivariate statistics is a branch of statistics involving the observation and analysis of more than one outcome variable at the same time. There are various forms of multivariate analysis and in this module you will be learn about several. Cybermetrics aims to measure aspects of the World Wide Web to gain information about hyperlinks between pages. It encompasses many techniques including network visualisation and content analysis. In this module you will learn about these and more.

This module introduces some of the basic concepts of partial differential equations (PDEs) in the context of Fluid Dynamics. It is shown how PDEs may be used to model situations in a wide variety of situations including biology, finance and applied mathematics. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced and the appropriate boundary conditions for each of these are considered. Computational methods for solving PDEs will be studied and applied with the use of mathematical software (MATLAB). A solid background in the mathematical description of fluid dynamics will also be provided. This includes the derivation of the conservation laws (mass, momentum, energy) that describe the dynamics of fluids and their application to a remarkable range of phenomena including water waves, sound propagation, atmospheric dynamics and aerodynamics. The focus will be on deriving approximate expressions using (usually) known mathematical techniques that yield analytic (as opposed to computational) solutions and then using the computational methods studied to find numerical solutions. There will be weekly MATLAB workshops to apply the theory studied in lectures.

This module investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize any polynomial into a product of linear factors by working over a "large enough" field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalizes the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions.