| | La Trobe University| Unit Code: | MAT3DQ |
| Unit Short Title: | DYNAMICS AND QUANTUM MECHANICS |
| Credit Points: | 15 |
| Unit Description: | The
first component of this unit, dynamics, is concerned with the
Hamiltonian description of classical mechanics (in contrast with
MAT2MEC, 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. |
| Prerequisites: | MAT2MEC or (MAT2AM and MAT2APD) |
| Corequisites: | None |
| Incompatibles: | None |
| Class Requirements: | two 1-hour lectures and one 1-hour tutorial per week |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Geoff Prince
| Unit Code: | MAT3DS |
| Unit Short Title: | DISCRETE ALGEBRAIC STRUCTURES |
| Credit Points: | 15 |
| Special Conditions: | Assessment
and class requirements depend upon number of students enrolled. If more
than 16 students enrolled, assessment will be one 3-hour examination
(80%), four assignments (20%) and class requirements: three 1-hour
lectures per week. |
| Unit Description: | This
unit is a continuation and expansion of MAT2PDM/MAT2ALL. 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. |
| Prerequisites: | MAT2PDM or MAT2AAL |
| Class Requirements: | three 1-hour problem sessions per week |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Brian Davey | Unit Code: | MAT3LPG |
| Unit Short Title: | LINEAR PROGRAMMING AND GAME THEORY |
| Credit Points: | 15 |
| Unit 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. |
| Prerequisites: | MAT21LA or MAT21ELA or MAT2LAL |
| Incompatibles: | MAT3ALP |
| Class Requirements: | 26 lectures and 13 tutorials |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Grant Cairns | Unit Code: | MAT3MFM |
| Unit Short Title: | MATHEMATICAL FLUID MECHANICS |
| Credit Points: | 15 |
| Special Conditions: | 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. |
| Unit Description: | An
introduction to incompressible fluid flow, with emphasis on the
structure of basic approximations in the theory of fluids and solutions
of problems using the approximations. The unit is fully online. |
| Prerequisites: | (MAT21AVC or MAT2AVC) and (MAT3CZ or MAT3CZE) |
| Class Requirements: | online work equivalent to two 1-hour lectures and one 4-hour problem-solving session per week. |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Prof. Ed Smith
| Unit Code: | MAT4AA |
| Unit Short Title: | ASYMPTOTIC ANALYSIS |
| Credit Points: | 15.0 |
| Special Conditions: | This unit is offered subject to sufficient enrolments. |
| Unit Description: | This
unit examines how we can describe a function as its argument becomes
large. We first define the language of asymptotics and then consider
various techniques for obtaining asymptotic expansions. This unit also
introduces or expands your knowledge of special functions, such as the
Bessel functions and the Airy functions. |
| Prerequisites: | MAT31CZ or MAT3CZ; and requires co-ordinators approval |
| Incompatibles: | MAT41AA, MAT42AA |
| Class Requirements: | two 1-hour seminars per week requiring extensive preparation for class presentations |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Prof. Ed Smith | Unit Code: | MAT4AMP |
| Unit Short Title: | APPLIED MATHEMATICS PROJECT |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | This
unit introduces students to mathematical modelling using some of the
computer-based tools available to the professional applied
mathematician. Models in various areas of application, such as heat and
mass transport, financial mathematics, biomathematics, statistical
mechanics and dynamic systems are considered. Students will complete
projects in these topics through integrated use of Fortran programming
for numerical analysis, Maple programming for symbolical computation
and graphics, advanced spreadsheet use for data manipulations and a
text processing package for mathematical document preparation. This
unit is an honours version of the existing subject, MAT3AMP. A higher
level of understanding will be expected. |
| Prerequisites: | (MAT2AM or MAT2MEC) and (MAT3NA or MAT3SC) |
| Incompatibles: | MAT32AMP, MAT3AMP |
| Class Requirements: | One 1-hour lecture and two hours of computer laboratory sessions per week |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Geoff Prince
| Unit Code: | MAT4CI |
| Unit Short Title: | COMPUTABILITY AND INTRACTABILITY |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | When
does a problem have an effective algorithmic solution? What does it
mean for an algorithm to be effective? In this unit we attempt to give
rigorous meaning to questions of this type and investigate some
possible answers. Abstract computing machines and their role in the
definitions of various notions of computational complexity will be
discussed. Classes of problems such as P, NP will be defined and a
number of well known problems in graph theory, algebra and applied
discrete mathematics will be classified according to their
computational complexity. The second half of the unit covers
undecidability for decision problems: problems for which no algorithmic
solution is possible. This property is found amongst problems from
computing, abstract algebra, combinatorics, matrices and the theory of
tilings. |
| Prerequisites: | MAT1DM and MAT2AAL and MAT1CLA, or any third year mathematics unit and requires co-ordinators approval |
| Class Requirements: | Two 1-hour seminars per week requiring extensive preparation for class presentations |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Marcel Jackson | Unit Code: | MAT4DS |
| Unit Short Title: | CHAOS AND ORDER IN DYNAMICAL SYSTEMS |
| Credit Points: | 15 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | What
is chaos? How does it arise in dynamical systems? What other dynamical
phenomena exist, or to put it slightly differently: what are the
different kinds of dynamical systems? If one has a differential
equation that exhibits chaos, how should one solve it? These are some
of the ingredients of this unit. The exact mix of ingredients is
adjusted from year to year, depending on students' interest and
background. |
| Prerequisites: | At least 30 credit points of second or third year mathematics units and requires co-ordinators approval |
| Class Requirements: | Two 1-hour seminars per week requiring extensive preparation for class presentations |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Prof. Reinout Quispel | Unit Code: | MAT4DT |
| Unit Short Title: | DUALITY THEORY |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | The
unit will begin with a primer on category theory, general algebra and
topology. The aim will be to emphasize the algebra and use, but
down-play, the category theory and topology. We shall cover the general
theory of dualities (between classes of algebras and classes of
topological relational structures). The theory will be applied to prove
the classical dualities for Abelian groups and Boolean algebras, the
neo-classical duality for distributive lattices and a host of other
less familiar dualities as well. Applications of duality theory will
also be presented. |
| Prerequisites: | MAT3DS and MAT3TA and requires co-ordinators approval |
| Class Requirements: | Three 1-hour seminars per week requiring extensive preparation for class presentations |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Brian Davey | Unit Code: | MAT4GA |
| Unit Short Title: | GENERAL ALGEBRA |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | General
algebra, otherwise known as universal algebra, provides a theory within
which to study the common features of all algebraic systems such as
vector spaces, groups, rings, lattices and semigroups. The unit will
present all of the basic results in the theory as well as providing an
intoduction to important recent developments. The close relationship
between general algebra and lattice theory will be emphasised
throughout. |
| Prerequisites: | MAT3DS and requires co-ordinators approval |
| Class Requirements: | Three 1-hour seminars per week requiring extensive preparation for class presentations |
| Work Experience Indicator: | Not undertaking work experience in industry |
| Available to 'Study Abroad' students: | Y |
| Recommended Prior Studies: | MAT3TA |
Coordinator: Dr Brian Davey | Unit Code: | MAT4GG |
| Unit Short Title: | GROUP ACTIONS |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | This
unit studies the foundations of the theory of group actions. In doing
so, it touches on a selection of topics which display interconnections
between geometry, group theory, topology and calculus. |
| Prerequisites: | MAT3TA and MAT3DS and requires co-ordinators approval |
| Class Requirements: | Two 1-hour seminars per week requiring extensive preparation for class presentations. |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Grant Cairns | Unit Code: | MAT4GM |
| Unit Short Title: | GEOMETRIC METHODS FOR DIFFERENTIAL EQUATIONS |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | This
unit aims to show how geometric symmetry of the solutions of
differential equations can be used to find those solutions using
integrating factors. These integrating factors exist for all ordinary
differential equations, not just the ones you learnt about in first
year Mathematics. The mathematics developed includes: calculus on
manifolds, one-parameter Lie groups and flows, exterior calculus and of
course lots of differential equation theory. Computer algebra is used,
though no prior knowledge is required. |
| Prerequisites: | MAT3AC and requires co-ordinators approval |
| Class Requirements: | Two 1-hour lectures per week |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Dr Geoff Prince | Unit Code: | MAT4MFM |
| Unit Short Title: | MATHEMATICAL FLUID MECHANICS |
| Credit Points: | 15.0 |
| Special Conditions: | Offered subject to sufficient enrolments. |
| Unit Description: | An
introduction to incompressible fluid flow, with emphasis on the
structure of basic approximation in the theory of fluids. Solution of
problems using the approximations. This unit is fully online. This unit
is a substantial extension of the third year subject, MAT3MFM. A higher
level of understanding will be expected. |
| Prerequisites: | MAT2AVC and MAT3CZ and requires co-ordinators approval |
| Incompatibles: | MAT40HON, MAT32MFM |
| Class Requirements: | Online work equivalent to two 1-hour lectures and one 4-hour problem solving session per week |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Prof Edgar Smith | Unit Code: | MAT4NT |
| Unit Short Title: | NUMBER THEORY |
| Credit Points: | 15 |
| Special Conditions: | This unit will be offered subject to sufficient enrolments. |
| Unit Description: | We
will commence this unit with an introduction to number theory. In the
later parts of the unit we will treat topics such as congruences,
residues, continued fractions, diophantine equations, transcendental
numbers, and/or primality and factoring. Depending on student interest
and time constraints, we may also touch on tantalizing connections to
cryptography and/or one or more of the recent analytic and algorithmic
advances in the areas of Mersenne primes, the Riemann hypothesis, and
primality proving. This unit will be accessible and interesting to a
varied audience, including students with interests in applied
mathematics, pure mathematics, or computer science. |
| Prerequisites: | MAT1DM
and MAT2AAL and MAT1CLA, or 30 credit points of second year units, or
any third year mathematics unit and requires co-ordinators approval |
| Class Requirements: | two 1-hour lectures per week requiring extensive preparation |
| Work Experience Indicator: | Not undertaking work experience in industry |
Coordinator: Prof. Reinout Quispel
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