La Trobe University
MAT3AMP Applied Mathematics Projects
Description: This unit introduces the student to mathematical modelling using some of the important computer-based tools available to the professional applied mathematician. Models in various areas of applied mathematics, such as heat and mass transport, financial mathematics, biomathematics, statistical mechanics and dynamic systems, are considered. The student will complete projects in these topics through integrated usage of Fortran programming for numerical analysis, Maple programming for symbolic computation and graphics, advanced spreadsheet use for data manipulation and a text processing package for mathematical document preparation.
Pre-requisite: (MAT22AM and MAT22APD and MAT31NA) or (MAT22AM and MAT22APD and CSE11OOJ and CSE12IPJ) = Mechanics and Partial Differential Equations
Description:
Mechanics - This unit deals with the kinematics and dynamics of a particle and of systems of particles. Particle dynamics and conservation laws, rigid rotating bodies and the two body problem (central forces) are the main topics. All the dynamical problems we look at are based on Newton's second law. The mathematical concepts from MAT21AVC and the differential equations from MAT12CLA are the main tools used.
Partial Differential Equations: This unit concentrates on three fundamental partial differential equations in applied mathematics, the wave equation (used, for example, to describe sound waves and vibrations of a stretched string), the heat equation (which describes heat flow in a conductor) and Laplace's equation (which is used in electrostatics, for example). A technique is used which reduces a partial differential equation to several ordinary differential equations. Along the way we extend some of the ideas developed in Mathematics 12CLA (to be able to handle boundary value problems and second-order ordinary differential equations with variable coefficients) and Mathematics 21LA (finding the Fourier expansion of a function).
MAT3DQ Dynamics of Quantum Mechanics
Description: The first component of this unit, dynamics, is concerned with the Hamiltonian description of classical mechanics (in contrast with MAT22AM, which looks at the Newtonian description). The approach due to Hamilton allows the dynamics to be derived from a scalar function (the Hamiltonian) and reveals more of the structure and underlying principles which govern the dynamics. Topics include conservation laws and canonical transformations. The second component is quantum mechanics and we use the Hamiltonian treatment of the classical central force problem of gravity (the Kepler problem) and electrostatics (the Coulomb problem) to bridge the gap between classical and quantum mechanics. Topics include energy eigenvalue problems in one, two and three dimensions and the hydrogen atom is treated as the quantisation of the classical Coulomb problem.
Pre-requisites: MAT2MEC or MAT2AM = Mechanics
Description: This unit deals with the kinematics and dynamics of a particle and of systems of particles. Particle dynamics and conservation laws, rigid rotating bodies and the two body problem (central forces) are the main topics. All the dynamical problems we look at are based on Newton's second law. The mathematical concepts from MAT21AVC and the differential equations from MAT12CLA are the main tools used.
MAT3DS Discrete Algebraic Structures
Description: This unit is a continuation and expansion of MAT22PDM. Further applications of finite groups to counting problems will be given. Finite fields and their applications will be discussed. The applications of ring theory to the classification of cyclic codes will be presented. Approximately half the unit will be devoted to ordered sets, lattices and Boolean algebras. Applications of lattices to concept analysis and applications of ordered sets to computer science will be discussed.
Pre-requisites: MAT22PDM or MAT2AAL = DISCRETE MATHEMATICAL STRUCTURES
Description: The study of discrete mathematical structures, such as groups and fields, underpins a large amount of modern computer science. Group theory, the principal mathematical tool for analysing symmetry, is genuinely 20th century mathematics and has widespread applications in all areas of science. Field theory has important applications to coding theory, which is used to preserve the security of computer networks. This unit develops the basics of both group theory and field theory, including applications to codes.
MAT3LPG Linear Programming of Game Theory
Description: Linear Programing and Game Theory are relatively new branches of mathematics. Linear Programming involves maximising and minimising a linear function subject to inequality and equality constraints. Such problems have many economic and industrial applications. Game Theory deals with decision making in a competitive environment. This unit studies the simplex technique for solving linear programming problems and gives an introduction to game theory and its applications.
Pre-requisites: MAT21LA or MAT21ELA or MAT2LAL = Linear Algebra
Description: Linear algebra is one of the cornerstones of modern mathematics, both pure and applied. Simple geometrical ideas, such as lines, planes, rules for vector addition and dot products arise in many places, including calculus, mechanics, differential equations and numerical analysis. This unit is an introduction to the mathematics which allows these geometrical ideas to be applied in non-geometrical contexts.
MAT3MFM Mathematics of Fluid Mechanic
Description: This subject is delivered in fully on-line
mode. Each fortnight lecture notes will be posted on-line. There will
be worked problems each fortnight with answers available in a separate
format. A bulletin board discussion will be provoked twice weekly. All
emails will be guaranteed a response within 48 hours of receipt. The
examination will be supplied on-line with defined start and end times.
Pre-requisites: (MAT21AVC or MAT2AVC) and (MAT31CZ or MAT3CZE) = Vector Calculus
Description: Many quantities in the physical world can be represented by smoothly varying functions of position. This unit develops, with a computational flavour, the differential and integral calculus of scalar and vector fields. These ideas are used to formulate important physical concepts, such as rates of change in a given direction, flux of a vector field through a surface, mechanical work and the local rate of expansion and rate of rotation of a fluid. In another part of the unit, Laplace transforms are introduced as a technique for solving constant coefficient ordinary differential equations with discontinuous forcing terms.
STA3AP Applied Probability for Computer Systems Engineers
Description: This unit is designed for students taking
one of the Computer Systems Engineering degrees, but is also available
to any student who has completed either STA2MDA or STA2MD.
Pre-requisites: MAT1EN and MAT1FEN (from 2009 MAT1CPE and MAT1CLA) or STA2MDA or STA2MD= Models for Data Analysis
Description: The analysis of scientific, engineering and economic data makes extensive use of probability models. This unit describes the most basic of these models and their properties. Applications of these models are illustrated with examples from digital communication systems, expert systems, financial risk assessment and bioinformatics. Specific topics covered in this unit include a wide range of discrete and continuous univariate distributions; joint distributions; mean and variance of linear combinations of random variables; Chebyshev's inequality; moment generating functions and the law of large numbers.
STA3AS Applied Statistics
Description: This unit provides advanced-level introductions to the topics of sample surveys, multivariate analysis and time series analysis. These topics are very important in applied statistics. The unit also includes an introduction to statistical consulting.
Pre-requisites: STA2MDA or STA2MD - Models for Data Analysis
Description: As above
STA3LM Analysis based on Linear Models
Description: Linear models are the most commonly used class of models in applied statistics. They are used to relate a response variable to one or more explanatory variables to both determine the form of this relationship and make predictions. The methods are widely used in many areas of application including agricultural science, biological science, economics, engineering, health science, medical science and psychological science. Topics covered in this unit include the simple linear regression model, derivation and properties of the ordinary least squares estimators, inference, diagnosis and prediction in the simple linear regression model, the multiple linear regression model, inference in the multiple linear regression model, the use of dummy variables and general regression models when the classical regression assumptions are violated. This unit has a combined flavour of both theoretical derivations and practical application through the use of a software package.
Pre-requisites: one of STA2MAS Modern Applied Statistics, STA2BS Biostatistics, STA2MS Medical Statistics, STA2MDA Models for data analysis, STA22LM Linear Models, STA22BS Biostatistics , STA2AS Modern Applied Statistic, STA2MD Models for Data Analysis , STA2LM Linear Models
Description: Any second year stats course
STA4RA Regression Analysis
Description: The main objective of this unit is to provide an introduction to the theory of regression analysis. The topics for this unit include; multiple linear regression; classical estimation and testing; residual analysis; diagnostics; variable selection and robust regression.
Pre-requisites: only available to 4th year honours students. |