General information about level 3 courses

This course covers maximisation problems leading to game strategies, and some aspects of combinatorics. Topics include: Linear Programming and Game Theory - maximisation under linear constraints and applications to strategy in zero sum two person games; Combinatorics - the number of selections under various conditions, and the use here of recurrence relations and enumerator functions.

This course presents a variety of methods in the solution of ordinary and partial differential equations. Topics include: ODEs - classification and methods of solution of various types of first and second order ODEs, including the use of the Laplace transform; PDEs - classification and methods of solution of various types of first and second order PDEs, including the use of Fourier series.

The aim of this course is to introduce students to basic notions in analysis and set up the background for the subsequent course in complex methods. The course covers sequences, series and limits as well as the mean value and intermediate value theorems.

The aim of this course is to introduce students to complex functions and their applications. It will focus on properties of analytic functions, contour integration and conformal maps.

Many important problems in applied mathematics are modelled by systems of ordinary differential or difference equations. Even when one cannot solve these equations explicitly, it is important to know the qualitative
properties of their solutions. Often these system depend on parameters and at critical values of the fundamental nature of the solution may change. Identifying such bifurcation points is vital to an understanding of the model.
The topics covered in the course are:
Equilibrium solutions, fixed points and limit cycles of a nonlinear system. Linearisation of the system about such solutions and study of their stability.
Sketching a phase portrait for a two dimensional system.
Bifurcations in the system as parameters change.
Fixed points and periodic points for a nonlinear map and their stability.
Bifurcations of nonlinear maps.