Maths modules - Second year
Multivariable and Vector Analysis
Most models of real world situations depend on more than one variable and the techniques of calculus can be extended to solve problems arising in such situations. Typically these are problems whose solutions are functions of position, describing, for example, heat distribution or velocity potential, and involve the partial differentiation or multiple integration of functions of more than one variable. The theory and classification of stationary points of functions of two or more variables is developed allowing maxima and minima, including those subject to constraints, to be identified. The differential operators div, grad, curl and the Laplacian are introduced. These are used in particular in the integral theorems (the Divergence theorem and the theorems of Green and Stokes) that relate line, surface and volume integrals and are used in the mathematical formulation of physical conservation laws. This module develops fundamental ideas that are used both in applied mathematics and in the development of analysis.
To introduce the student to the fundamental structures and techniques of Linear Algebra, combining the necessary algebraic background with the methods needed for future applications
Probability and Statistics
Statistics, often regarded as distinct science rather than a branch of mathematics, is the study of data and uncertainty. Statistical techniques allow us to make conclusions, such as whether or not living near electricity pylons is dangerous, from sets of data. Statistics is also used in the design of effective experiments and in determining what data should be collected. For example, statistical techniques might be used to determine the frequency with which aircraft components should be tested for safety. Underlying these techniques is the assumption that these data are samples of a random variable that follows a probability distribution describing their behaviour. This module provides an introduction to probability and statistics. Axiomatic probability theory, including Bayes’ Theorem, is discussed briefly. Key discrete and continuous probability modules (such as the binomial, Poisson and normal distributions) are introduced. Properties of expectation and variance are discussed. The Weak Law of Large Numbers and the Central Limit Theorem are covered before basic statistical ideas, such as statistical inference and hypothesis testing are introduced. Real world data are used to illustrate the theory.
Algebra & Combinatorics 1
The first part of the module will give an introduction to several combinatorial structures, which have applications in different areas. Topics are likely to include combinatorial games, applications of counting principles to discrete probability and basic Ramsey theory. (Here Ramsey theory can be viewed as a formalization of the notion that `complete disorder is impossible’ - this surprising phenomenon will be investigated for graph colourings and arithmetic properties of the integers).
The second part of the module consists of an introduction to information theory and coding theory. The aim here is to transmit information (i) efficiently and (ii) reliably over a noisy channel. For (i), the main result will be Shannon’s noiseless coding theorem, which relates coding efficiency to the entropy of a source. For (ii), we will discuss error correcting codes, including several linear codes, such as Hamming codes. Both parts of the module are linked by the methods and ideas that are used.