MATH3018 Financial Time Series

Description: The components of a time series model. Additive and multiplicative models. Multiple regression analyses. Spectral decomposition. Box-Jenkins models. Forecasting techniques. Smoothing of time series. Recursive parameter estimation. Applications.

Pre-requisites: MATH2020 Statistical Foundations

Description: The course covers probability, probability distributions and densities, mathematical expectation, special probability distributions and densities, functions of random variables, sampling distribution, estimation theory, hypothesis testing theory, regression and experimental design.

MATH3010 Advanced Operations Research

Description: A selection of topics from advanced linear and nonlinear programming: large scale simplex methods, interior-point methods, or stochastic LPs. Integer programming heuristics (including heuristics for the travelling salesman problem). Networks flows and scheduling. Markov decision processes, Dynamic programming and Game Theory. Nonlinear programming algorithms. Inventory Theory and manufacturing systems. Decision theory, portfolio analysis and financial mathematics.

Pre-requisites: MATH 2014 Linear Programming & MATH 3009 Optimisation

Description: Linear Programming - Modelling in Operations Research. Linear optimisation models and their solution using GAMS. Solution to LP problems; geometry, simplex, sensitivity, duality. Solution to IP problems; geometry, branch and bound. Network models; transportation; assignment, shortest path. Case studies will be discussed with some emphasis on client - consultant interaction in practice.

Optimisation: Linear programming: convex sets, separating-hyperplane theorem, duality, interior-unit methods. Kuhn-Tucker conditions. Constrained optimisation methods. Nonlinear programming algorithms. Case studies.

MATH3017 Decision Science

Description: Fundamental concepts of decision analysis, utility, risk analysis, Bayesian statistics, game theory, Markov decision processes and optimisation. The value of sampling information and optimal sample sizes, given sampling costs, and the economics of terminal decision problems.

Pre-requisites: Basic understanding of elementary probability and matrices

MATH3022 Mathematics Clinic II

Description: Due to the individual nature of projects there is no prescribed syllabus for the Mathematics Clinic. Projects are sourced from industry. Students are required to provide the client company with a final report and presentation at the end of study period 5

Pre-requistes: Mathematics Clinic 1

MATH3026 Applied Functional Analysis

Description: Selected key theorems of modern analysis; Arzela-Ascoli, Picard, Weierstrass, Fejer, Implicit Function, Dominated Convergence, Reisz Representation, Reisz-Fischer, Projection, Hahn-Banach, Open Mapping.

Pre-requisites: MATH 3025 Differential Equations 2 & MATH 2025 Real and Complex Analysis

Description: Series solutions of differential equations, Legendre and other orthogonal polynomials, the Method of Frobenius. Bessel Functions. Self adjoint form, Sturm Liouville problems, eigenfunction expansions. Fourier series. Partial differential equations. Problems from applied physics.

MATH1005 Advanced Optimization (offered as part Honours Mathematical Studies 2)

Description: Overview of subject content: We study optimization problems and methods in finite dimensional spaces. The course consists of two parts: (I) Theory and (II) Methods.
Part (I) includes: (i) classification of optimization problems, (ii) existence and uniqueness of solutions, and (iii) optimality conditions for unconstrained and constrained optimization: differentiable case and convex case (differentiable and non-differentiable). Part (II) includes the definition and convergence analysis of: (i) Methods for unconstrained problems: Newton's method, Steepest descent method (Cauchy and Armijo variants), Quasi-Newton methods, and (ii) Methods for constrained problems: Penalty and Barrier Methods.

The course may also include some mathematical background on topology of Rn and convex analysis, as well as some classical topological results involving continuous functions.

Pre-requisites: Available to honours students only.