Mathematics MMath (Hons)

The module introduces the concepts and terminology of financial mathematics and modelling in finance. You will learn about
the properties of interest rates and the key tools of compound interest functions for modelling a range of annuity schemes. The module develops models for life insurance and endowment schemes and enables the students to analyse the behaviour of share prices. The generalised cash-flow model is introduced to describe financial transactions. The student learns how to develop simple models of financial instruments such as bonds and shares.


Outline Syllabus

Interest: Simple and compound interest. Effective and nominal interest rates. Force of interest. Interest paid monthly. Present values. Cash flows and equations of value.
Annuities: Annuities with annual payments, and payments more regularly. Payments in arrear and in advance. Deferred and varying annuities, annuities payable continuously. Loans, loan structure and equal payments.
Discounted cash flow: Generalised cash flow model. Project appraisal at fixed interest rates. Comparison of two investment projects. Different interest rates for lending and borrowing. Payback periods. Measurement of investment performance.
Investments: Types of investments. Valuation of fixed interest securities and uncertain income securities. Real rates of interest. Effects of inflation. Capital gains tax.
Arbitrage in financial mathematics: Forward contracts. Calculating delivery price and delivery value of forward contracts using arbitrage-free pricing methods. Discrete and continuous time rates.
Life Insurance: Term insurance and whole life insurance. Curtate future lifetime. Life tables, expectation of life. Annual and monthly premium. Endowments. Payment at death.
Stochastic Interest Rates: Varying interest rates. Independent rates of return. Expected values. Application of the lognormal distribution. Brownian motion.

Mathematical Cryptology concerns the creation and analysis of secret messages using mathematical techniques. You will learn about both classical and contemporary cryptology from the time of Julius Caesar until the present day. Mathematical techniques have been at the heart of many of these approaches and, on this module, you will be able to see, for example, how modular algebra can be a powerful cryptographic tool. Large prime numbers are another useful tool at the heart of modern cryptology and you will learn how to formulate an efficient approach to determining whether a large number is prime or composite.

By the end of the module, you should have developed an awareness of different approaches to deciphering various forms of ciphertext and should have an ability to appraise which cryptographical techniques are robust.

Outline Syllabus
Classical Cryptology: Encryption and decryption using direct standard alphabets and alphabets created using classical techniques from the shift cipher to polygraphic ciphers.

Contemporary Cryptology: Encryption and decryption using techniques based on Boolean functions and exploring the mathematical theorems and approaches at the heart of modern cryptology practices.

This module is designed to introduce fundamental concepts in the mathematical area of Fluid Dynamics. You will analyse the equations of continuity and momentum, and will investigate key concepts in this area. We will introduce the Navier-Stokes equations, and case studies will be used to visualise and analyse real-world problems (using appropriate software) as appropriate to delivery of the module. Initially, we will use the inviscid approximation and then utilise analytical and computational techniques to investigate flows. The second half of the module is a specialist course in laminar incompressible viscous flows, encompassing background mathematical theory allied to a case study approach, with solution to problems by both analytical and computational means.

Assessment of the module is by one individual assignment (30%) and one formal examination (70%).

The module is designed to provide you with a useful preparation for employment in an applied mathematical environment, physics environment or engineering environment.

Outline Syllabus
• Introduction of fluid dynamics, Navier-Stokes equations, equations of continuity and momentum for inviscid flow, unsteady one-dimensional flow along a pipe, irrotational flow, Bernouilli's equation, stream function formulation, flow past a cylinder, velocity potential.

• Low Reynolds Number Flow including: (i) lubrication theory, slider bearing, cylinder-plane, journal bearing, Reynolds equation, short bearing approximation; (ii) Flow in a corner, stream function formulation, solution of the biharmonic equation by separation of variables.

• High Reynolds Number Flow including boundary layer equations, skin friction, displacement and momentum thickness, similarity solutions, momentum integral equation, approximate solutions.

The module aims to present an introduction to Dynamical Systems and associated transferable skills, providing the students with tools and techniques needed to understand the dynamics of those systems. You will analyse non-linear ordinary differential equations and maps, focusing on autonomous systems, and will learn analytical and computational methods to solve them. This module offers the additional opportunity of research-orientated learning through a hands-on approach to selected research-based problems.

Topics may include (note this is indicative rather than prescriptive):
1. Autonomous linear systems, fixed points and their classification.
2. 1-dimensional non-linear systems: critical points; local linear approximations; qualitative analysis; linear stability analysis; bifurcations.
3. Multi-dimensional non-linear systems: linearisation about critical points, limit cycles, bifurcations.
4. Discrete systems: maps (such as tent map, logistic map, Henon map, standard map).
5. Numerical schemes for ordinary differential equations, such as the embedded Runge-Kutta method.
6. Numerical applications and programming: generation of the orbit of a map, Lorenz map for a dynamical system, orbit diagrams, cobwebs, simple fractals.
7. Elements of Chaos theory: Lyapunov exponents, sensitive dependence on initial conditions, strange attractors, Hausdorff dimension, self-similarity, fractals.

This module covers the three important areas of experimental design, multivariate techniques and regression. Experimental design will be developed using analysis of variance techniques to compare treatments meaningfully using replication, factorial experiments and balanced incomplete block designs. You will then move on to multivariate techniques including multivariate inference, data reduction using principal component analysis and classification with linear discriminant analysis. You will also learn how to extend regression models to the case where there are several explanatory variables including indicator variables. The models will subsequently be scrutinised using variable selection criteria and regression diagnostics to improve the model. Curvilinear and non-linear regression models cover the important aspect where different types of curves are appropriate for the data. The generalised linear model will be introduced, and the specific case of a count response variable is developed.

Outline Syllabus
Experimental Design: design and analysis of 2n factorial experiments with replication, a full replicate and balanced
incomplete block designs.
Multivariate techniques: the multivariate normal distribution and its properties. Hotellings T2 test for one, two and paired
samples. Manova, linear discriminant analysis and principal component analysis.
Multiple linear regression: least squares estimation of the parameters of the model and their properties. The analysis of variance
and the extra sum of squares method. Variable selection techniques and regression diagnostics.
Non-linear and generalised linear models: Non-linear regression models, estimation of parameters and testing the model. Analysis of deviance and the Poisson regression model.

You will learn about a range of appropriate statistical techniques that are used to analyse medical data. You will be introduced to the design and analysis of clinical trials and learn how to design the statistics of clinical trials for a variety of scenarios. These trials are the scientific tests that all medical advances need to go through to assess whether they have merit. You will learn techniques that can be used to handle various types of medical data found in epidemiology and learn when to apply them. You will investigate some of the statistical models used in survival data analysis for the analysis of time to failure data such as transplant data.

By the end of the module, you should have developed an ability to design clinical trials that are scientifically sound and be able to select and apply the appropriate statistical techniques to analyse medical data in a variety of forms.

Outline Syllabus

Design and analysis of Clinical Trials including the four main phases, estimation of sample size and power of a test. Parallel group and cross-over trials.

Categorical data analysis using contingency tables, McNemar's test, Fishers Exact test and test for trend.
Epidemiology: Prospective, retrospective and cross-sectional studies. Analysis of trials including dichotomous response and dichotomous risk factors. Study bias and reliability of a trial. Observer bias and diagnostic tests
Mortality statistics. Survival data analysis

Analysis of covariance, logistic regression

This module is designed specifically to enhance your graduate skills that are essential to your future career and/or postgraduate study. This is achieved by an individual, research-based project work in an area appropriate to your degree.

You will develop the ability to undertake independent research in an area of interest, requiring a survey of current literature, synthesis of ideas, find solutions where required and drawing a coherent appraisal of conclusions. In this process you will learn how to defining clearly a mathematical and/or statistical problem to be investigated/solved, research and appraise current thinking as regards the subject, select methodologies, include appropriate mathematical exemplars to justify your argument and present a well-integrated set of conclusions.

You will also develop the ability to critically appraise both your own work and the work of others in the field.

You will be research-tutored through the module, and you will be assessed by a written project proposal, a poster presentation and a final written report.

In this module you will learn basics concepts of continuum mechanics i.e. kinematics (material derivatives, kinematic boundary conditions), kinetics (stresses, traction), balance laws (mass, momentum, energy) and constitutive relationships (stress strain relationships for fluids).

You will learn the effects of rotation on the fluid flow and how these affect the large-scale flow of the ocean and the atmosphere. The focus is on understanding how the large-scale flow regime of the ocean, atmosphere, glaciers and the mantle can be described mathematically.

Your learning will be set within the context of global environmental changes. The module will provide you with the tool required to understand the physical principles of global circulations models used to describe the movement of the atmosphere, oceans, earth’s mantle and large ice sheets.


The course consists of lectures and exercises using a state-of-the art numerical ice sheet model. Assessment of the module is by one individual numerical modelling assignment (30%) and one formal examination (70%).

On completion of the module you will have developed an improved understanding of the earth system and the principles of climate change.
The module is designed to provide you with a useful preparation for employment in earth sciences with aim at pursuing graduate studies in environmental modelling.




Outline Syllabus

• Introduction and review of continuum mechanics, balance laws (mass, conservation, energy), kinematics of deforming bodies, constitutive laws (non-Newtonian rheology).

• Rotating shallow-water models. Large-scale flow approximations used in earth sciences. Large-scale features of ocean circulation and atmospheric circulation. Ocean gyres, boundary currents, eddy transport. Variation in flow with depth/height. Geostrophic balance. Ekman spiral.

• Hydrostasy in the ocean and atmosphere.

• Mechanics of glaciers and ice sheets. Commonly used flow approximations in glaciology. Ice-sheet instabilities. Grounding-line dynamics and instabilities.

• Scaling of flow equations and linearization. Systematic reduction of equations.

Academic skills when studying away from your home country can differ due to cultural and language differences in teaching and assessment practices. This module is designed to support your transition in the use and practice of technical language and subject specific skills around assessments and teaching provision in your chosen subject. The overall aim of this module is to develop your abilities to read and study effectively for academic purposes; to develop your skills in analysing and using source material in seminars and academic writing and to develop your use and application of language and communications skills to a higher level.

The topics you will cover on the module include:

• Understanding assignment briefs and exam questions.
• Developing academic writing skills, including citation, paraphrasing, and summarising.
• Practising ‘critical reading’ and ‘critical writing’
• Planning and structuring academic assignments (e.g. essays, reports and presentations).
• Avoiding academic misconduct and gaining credit by using academic sources and referencing effectively.
• Listening skills for lectures.
• Speaking in seminar presentations.
• Presenting your ideas
• Giving discipline-related academic presentations, experiencing peer observation, and receiving formative feedback.
• Speed reading techniques.
• Developing self-reflection skills.

Data Science concerns extracting information from data – in other words giving a voice to the data. Different analysts may have different purposes when analysing data – the intention may be to describe the information in the data, explain the relationships between parts of the data or use a subset of the data to predict the outcome of a variable of interest. For example, that variable could be whether a customer with a particular profile may buy an item of interest. Most companies collect data on their customers and are interested in how this data can be used to improve customer experience as well as profits. Depending on the intention, the approach taken by the analyst will differ and this module will cover the main tools for classification, clustering, association mining and outlier detection allowing you to analyse data with confidence.

By the end of the module, you should have developed an awareness of different approaches to analysing various forms of data and should have an ability to appraise which analytical techniques are appropriate. You will be able to perform the analysis and interpret the results correctly.

Outline Syllabus

Classification techniques that may include decision trees, support vector machines, linear discriminant techniques and logistic regression.

Clustering techniques including k-means clustering, apriori association mining, naïve Bayes and dimensionality reduction.

‘Optimisation and Decision Analytics’ is designed to introduce students to the problem solving methods that apply to specific business problems. You will learn about the two main areas of operational research - linear programming and simulation. These topics were created to analyse and solve everyday business problems and we will explore both the theoretical and practical approaches to solving such problems using software such as Excel and R.

Outline syllabus
Simulation Models: techniques for generating both uniform and non-uniform pseudo random numbers, bootstrapping, tests of randomness and stochastic integration.
Linear Programming: Formulation of a problem for two or more variables; graphical solution; sensitivity analysis; simplex algorithm and the Big M method. Duality and its interpretation. Integer programming using the Cutting Plane and Branch and Bound Algorithms. Kuhn-Tucker conditions and quadratic programming. Use of Excel to solve these linear programming problems.

With this module you will learn advanced methods and technical skills to find exact and approximate solutions to complex problems inspired by the real world. Examples of applications include traffic flow, waves in the ocean, optical telecommunications systems, models for climate and biological systems and magnetohydrodynamics.

Outline syllabus will develop the following three areas:

Exact methods

1) Integral transforms:
- Laplace transform,
- Fourier transform

2) Applications of integral transforms to linear differential equations.

3) Theory of quasilinear partial differential equations

4) Method of characteristics.

5) Conservation laws and shock waves.

6) Applications of exact methods to

- traffic flows
- water waves
- magnetohydrodynamics.



Asymptotic methods:

1) Asymptotic methods for algebraic equations
2) Regular and singular perturbation methods for ordinary differential equations;
3) Asymptotic methods for evaluations of integrals


Applications

1) Boundary layers
2) Linear and Nonlinear dispersive waves
3) Solitons.