FInal Year Mathematics- Book recommendations?

Description

This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&©s functions.
Aims

The aim of this module is to provide an introduction to analytical methods for solving partial differential equations.

Intended Learning Outcomes

Classify partial differential equations (PDEs) into linear, quasi-linear and non-linear types (1, 2).
Solve first order linear and quasi-linear PDEs by the method of characteristics (1, 2).
Classify linear second order PDEs into hyperbolic, parabolic and elliptic types, reduce to canonical form and solve by the method of characteristics (1, 2).
Solve linear second order PDEs by separation of variables, with applications to the wave, diffusion and Laplace&©s equations (1, 2).
Demonstrate knowledge of Sturm-Liouville theory and the application of generalised Fourier series to solve boundary value problems (1, 2).
Solve inhomogeneous linear second order PDEs using the method of eigenfunction expansion (1, 2).
Solve boundary value problems using Fourier transform techniques (1, 2)
Solve linear second order PDEs using Green&©s functions (1, 2).

Description

Number Theory studies the properties of the natural numbers and the integers. It is one of the oldest and most beautiful areas of Pure Mathematics, first studied by the Ancient Greeks, and yet has a surprising number of modern applications. It is a topic which is famous for a large number of results which are extremely simple to state, but turn out to be difficult to prove. Indeed, many of these problems remain unsolved, and so Number Theory is one of the most active areas of modern research. This module will build upon the concepts and techniques introduced in the second year module "Analysis and Abstract Algebra", and consider how these can be extended and applied to both ancient and modern problems, from finding whole number solutions to polynomial equations to primality testing and cryptography.

Aims

This module will provide students with a sound foundation in Number Theory from a modern perspective.
It will cover one of the oldest and most beautiful areas of Pure Mathematics, from basic ideas to modern applications. One of the objectives is to give a self-contained module with modern answers to ancient problems and modern applications of classical ideas.


Intended Learning Outcomes

State clearly the key definitions and theorems of elementary number theory, including the Fundamental Theorem of Arithmetic, the Chinese Remainder Theorem, the definition of a multiplicative function, Euler's Theorem, Wilson's Theorem, the definition of a quadratic residue, the Legendre symbol and Quadratic Reciprocity. will be achieved by assessments: 2
Apply modern number theoretic algorithms to problems of primality testing and cryptography. will be achieved by assessments: 1,2
Apply appropriate techniques to solve Diophantine equations, or to show insolubility. will be achieved by assessments: 1,2
Use the Theorems of Fermat and Lagrange to identify representations of integers as sums of 2 and 4 squares, and calculate the number of such representations in certain cases. will be achieved by assessments: 1,2

Description

This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible.

We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).

Aims

This module aims to develop students&© ability to view mathematics as an interdisciplinary subject and to provide some applications in biology.





Intended Learning Outcomes

Demonstrate ability to analyse qualitative aspects of ODEs in a biological modelling context. will be achieved by assessments: 1,2
Apply appropriate techniques to solve a given model of a biological problem. will be achieved by assessments: 1,2
Select appropriate approaches/methods and tools to generate mathematical models of aspects of biology. will be achieved by assessments: 1,2
Be able to formulate and critically evaluate epidemiological models. will be achieved by assessments: 1,2
Critically evaluate the merits and weaknesses of biological models. will be achieved by assessments: 1,2

Description

This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with particular emphasis placed on cancer studies. No prior knowledge of medicine or biology is required. The module commences with a revision of hypothesis testing procedures. This is followed by three main topics: clinical trials, survival analysis and epidemiology. Clinical trials are immensely important for evaluating the relative effectiveness of different treatments, and their design and analysis are considered in-depth. Survival analysis looks at the features and analysis of data from studies of patients with a potentially fatal disease. Epidemiology explores the distribution of disease in a population and discusses studies for assessing whether there is a possible association between a factor (such as, smoking, eating beef, using a mobile phone) and the subsequent development of a disease.

Aims

The aim of this module is to study the application of specialised statistical techniques to health related research.

Intended Learning Outcomes

Demonstrate knowledge of the concepts of hypothesis testing (1, 2).
Demonstrate knowledge of the theory and application of clinical trials (1, 2).
Demonstrate knowledge of survival analysis, including functions of survival time, comparison of survival distributions, and multivariate techniques (1, 2).
Demonstrate knowledge of the science of epidemiology and its applications (1, 2).

(Note Stochastic Processes is a pre-requisite.)

Probability models are widely used in natural, social and financial sciences
for the purposes of understanding, predicting and controlling random
phenomena. The module builds on the earlier study of probability and
stochastic processes. It aims to (a) provide students with knowledge of
some advanced concepts in probability theory, in particular: conditional
expectation, convergence of random variables, martingale sequences, and
brownian motion; (b) apply probabilistic techniques to a wide range of
stochastic modules and processes.

(Note Linear and Metric Spaces is a pre-requisite.)

This module will illustrate how abstract ideas from group theory, number
theory and linear spaces can be bought together to solve problems
concerning reliable, efficient and secure communication of information.
Coding theory studies methods and algorithms for reliably and efficiently
transmitting information in a situation where there is a risk of ‘noise’
disrupting the communication. Cryptography adds the requirement that
this communication should be secure. Both of these areas depend on the
study of finite fields and linear spaces over them. In addition to studying
the theory, there will be an opportunities to implement the techniques
and algorithms via group activities.