ANU
MATH3015 - Mathematics of Finance
Description: This course provides an introduction to the theory of stochastic processes and its application in the mathematical finance area. The course starts with background material on markets, modelling assumptions, types of securities and traders, arbitrage and maximisation of expected utility. Basic tools needed from measure and probability, conditional expectations, independent random variables and modes of convergence are explained. Discrete and continuous time stochastic processes including Markov, Gaussian and diffusion processes are introduced. Some key material on stochastic integration, the theory of martingales, the Ito formula, martingale representations and measure transformations are described. The well-known Black-Scholes option pricing formula based on geometric Brownian motion is derived. Pricing and hedging for standard vanilla options is presented. Hedge simulations are used to illustrate the basic principles of no-arbitrage pricing and risk-neutral valuation. Pricing for some other exotic options such as barrier options are discussed. The course goes on to explore the links between financial mathematics and quantitative finance. Results which show that the transition densities for diffusion processes satisfy certain partial differential equations are presented. The course concludes with treatment of some other quantitative methods including analytic approximations, Monte Carlo techniques, and tree or lattice methods. Mathematics of Finance provides an accessible but mathematically rigorous introduction to financial mathematics and quantitative finance. The course provides a sound foundation for progress to honours and post-graduate courses in these or related areas. Note: This is an Honours Pathway Course. It continues the development of sophisticated mathematical techniques and their application begun in MATH3029 or MATH3320.
Pre-Requisites: MATH3029 Probability Modelling with Application
Description: The course introduces stochastic processes with a view towards applications in fields such as finance, insurance, risk management, and operations research. The aim is to provide mathematics students with basic knowledge of stochastic processes where practical rather than theoretical aspects are emphasized. Probability Modelling and Applications provides a sound foundation to progress to honours and post-graduate courses emphasizing the theory of mathematical finance and stochastic analysis. The course contains sufficient material for students to feel comfortable with Markov chains, Poisson processes, and Brownian motion, and the conceptual formulation of topics in continuous time finance, insurance and risk management, where these processes are applied. Also the concept of martingales, which is fundamental for understanding the modern option pricing theory of Black and Scholes, is introduced. Note: This is an HPC. It continues the development of sophisticated mathematical and probabilistic techniques and their application begun in STAT2001(HPC)
MATH3133 Environmental Mathematics
Description: In this course the major model types used to represent environmental systems are studied. Mathematical emphasis on how they are constructed will use the theory of inverse problems while the practical emphasis uses systems methodology. The focus will be on hydrological systems and their basic processes, combined with the constraints imposed by the limitations of real observational data. Case studies and project assessment will cover catchment hydrology, soil physics, subsurface hydrology and stream transport.
Pre-requisites: MATH2405 or MATH2305 Mathematical Methods 1 Honours
Description: This course provides an in depth exposition of the theory of differential equations and vector calculus. Applications will be related to problems mainly from the Physical Sciences. Topics to be covered include: Ordinary Differential Equations - Linear and non-linear first order differential equations; second order linear equations; initial and boundary value problems; Green's functions; power series solutions and special functions; systems of first and second order equations; normal modes of oscillation; nonlinear differential equations; stability of solutions; existence and uniqueness of solutions; Advanced Vector Calculus - Curves and surfaces in three dimensions; parametric representations; curvilinear coordinate systems; Surface and volume integrals; use of Jacobians; gradient, divergence and curl; identities involving vector differential operators; the Laplacian; Green's and Stokes' theorems. Note: This is an HPC, taught at a level requiring greater conceptual understanding than MATH2305.
MATH3512 Matrix Computations and Optimisation
Description: In this course, students will be introduced to important algorithms and techniques of scientific computing, focussing on the areas of linear algebra and optimisation. The course will present both theoretical and practical aspects of the algorithms. Students who have previously taken the course have come from areas such as science engineering and economics. Honours Pathway Option: Students must have completed MATH2405 or MATH2320 or Real Analysis (3rd year) to choose this option. In this option we will expand on the theoretical aspects of the underlying algorithms. Alternative assessment in the assignments and exam will be used to assess these theoretical aspects.
Pre-requisites: MATH2405 or MATH2306 Mathematical Methods 1 Honours
Description : See above
MATH3301 Number Theory and Cryptology
Description: This course is intended for students who want an introduction to elementry number theory, with an application to cryptography. Useful for mathematics engineering and computer science students. Topics chosen from: the Euclidean algorithm, congruences, prime numbers, highest common factor, prime factorisation, diophantine equations, sums of squares, Chinese remainder theorem, Euler's function, continued fractions, Pell's equation, quadratic residues, quadratic reciprocity, cryptography and RSA codes.
Pre-requisites: MATH2016; or MATH2302; or MATH2303; or MATH2301 with a mark of 60 or better;
General Pre-requisite Description: This course is designed to show some of the interdependence of mathematics and computing, and is designed for students in both computer science and mathematics. Topics to be covered include: Foundations - Relations on sets, including equivalence, partial order relations and relational databases; properties of functions, permutations, arithmetic of integers modulo n. Grammars and Automata - Phrase structure grammars, finite state automata, and the connections between the language accepted by an automaton, regular sets and regular grammars. Graph Theory - Hamiltonian circuits, vertex colouring and the chromatic polynomial of a graph, planar graphs, applications including the travelling salesperson problem and scheduling problems. Game Theory - Games of strategy as an application of graph theory, matrix games and solution of matrix games.
MATH3104 Groups and Rings Honours
Description: This course introduces the basic concepts of modern algebra such as groups and rings. The philosophy of this course is that modern algebraic notions play a fundamental role in mathematics itself and in applications to areas such as physics, computer science, economics and engineering. This course emphasizes the application of techniques.
Pre-requisites: A mark of 80 or more in MATH2305 and MATH2306 or a mark of 60 or more in MATH2405. Mathematical Honours 1
Description: This course provides an in depth exposition of the theory of differential equations and vector calculus. Applications will be related to problems mainly from the Physical Sciences. Topics to be covered include: Ordinary Differential Equations - Linear and non-linear first order differential equations; second order linear equations; initial and boundary value problems; Green's functions; power series solutions and special functions; systems of first and second order equations; normal modes of oscillation; nonlinear differential equations; stability of solutions; existence and uniqueness of solutions; Advanced Vector Calculus - Curves and surfaces in three dimensions; parametric representations; curvilinear coordinate systems; Surface and volume integrals; use of Jacobians; gradient, divergence and curl; identities involving vector differential operators; the Laplacian; Green's and Stokes' theorems. Note: This is an HPC, taught at a level requiring greater conceptual understanding than MATH2305.
ASTR3002 Galaxies and Cosmology
Description: Galaxies: classification and dynamics. Luminous matter and dark matter in galaxies. The expanding universe and cosmological models
Pre-requisites: Either one of MATH2305, MATH2405, ENGN2212, MATH2320, MATH2023 - Mathematical Honours 1
Description: see above
MATH3329 Relativity, Black Holes and Cosmology
Description: The theories of special and general relativity are presented with applications to black holes and cosmology. Topics to be covered include the following. Metrics and Riemannian tensors. The calculus of variations and Lagrangians. Spaces and space-times of special and general relativity. Photon and particle orbits. Model universes. The Schwarzschild metric and black holes. Gravitational lensing.
Pre-requisites: see above
Description: see above
MATH3325 Complex Honours Analysis
Description: This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics. Topics to be covered include: Complex differentiability, conformal mapping; complex integration, Cauchy integral theorems, Taylor series representation, isolated singularities, residue theorem and applications to real integration. Topics chosen from: argument principle, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler. Note: This is an HPC. It emphasizes mathematical rigour and proof and develops the material from an abstract viewpoint.
Pre-requisites: MATH3320 with 60+ mark. Analysis 2 Honours.
Description: This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics. Topics to be covered will include: Topological Spaces - continuity, homeomorphisms, convergence, Hausdorff spaces, compactness, connectedness, path connectedness. Measure and Integration - Lebesgue outer measure, measurable sets and integration, Lebesgue integral and basic properties, convergence theorems, connection with Riemann integration, Fubini's theorem, approximation theorems for measurable sets, Lusin's theorem, Egorov's theorem, Lp spaces as Banach spaces. Hilbert Spaces - elementary properties such as Cauchy Schwartz inequality and polarization, nearest point, orthogonal complements, linear operators, Riesz duality, adjoint operator, basic properties or unitary, self adjoint and normal operators, review and discussion of these operators in the complex and real setting, applications to L2 spaces and integral operators, projection operators, orthonormal sets, Bessel's inequality, Fourier expansion, Parseval's equality, applications to Fourier series. Calculus in Euclidean Space - Inverse and implicit function theorems. Note: This is an HPC. It emphasises mathematical rigour and proof and develops modern analysis from an abstract viewpoint.
MATH3344 Algebraic Topology Honours
Description: Algebraic topology studies properties of topological spaces and maps between them by associating algebraic invariants (fundamental groups, homology groups, cohomology groups) to each space. This course gives a solid introduction to fundamental ideas and results that are employed nowadays in most areas of mathematics, theoretical physics and computer science. This course aims to understand some fundamental ideas in algebraic topology; to apply discrete, algebraic methods to solve topological problems; to develop some intuition for how algebraic topology relates to concrete topological problems. Topics to be covered include: Fundamental group and covering spaces; Brouwer fixed point theorem and Fundamental theorem of algebra; Homology theory and cohomology theory; Jordan-Brouwer separation theorem, Lefschetz fixed theorem; some additional topics (Orientation, Poincare duality, if time permits) Note: This is an HPC. It builds upon the material of MATH3302 and MATH2322 and emphasises mathematical rigour and proof.
Pre-requisites: MATH3320 Analysis 2 Honours & MATH 2322 Algebra Honours 1
Description: See above for MATH3320. Algebra 1 is a foundational course in Mathematics, introducing some of the key concepts of modern algebra. The course leads on to other areas of algebra such as Galois Theory, Algebraic Topology and Algebraic Geometry. It also provides important tools for other areas such as theoretical computer science, physics and engineering. Topics to be covered include: Group Theory - permutation groups; abstract groups, subgroups, cyclic and dihedral groups; homomorphisms; cosets, Lagrange's Theorem, quotient groups; group actions; Sylow theory. Ring Theory - rings and fields, polynomial rings, factorisation; homomorphisms, factor rings. Linear algebra - real symmetric matrices and quadratic forms, Hermitian matrices, canonical forms. Set Theory - cardinality. Note: This is an HPC. It emphasises mathematical rigour and proof and develops modern algebra from an abstract viewpoint.
MATH3346 Data Mining Honours
Description: Topic to be covered include: Pre-requisites: entry is by invitation only. University of Sydney
AMH2 Modern Asymptotics and Peerturbation Theory
Description: Differential equations model most natural phenomena we know. Yet their so-
Pre-requisites: Available to Honours students only
AMH1 Computational Projects in applied Mathematics
Description: Lectures will be given on five projects over the semester, from which the
Pre-requisites: Available to Honours students only
AMH8 Mathematical Physiology
Description: Physiological rhythms are central for life. Prominent examples are the beating
Pre-requisites: MATH3963 Differential Equations and Biomathematics (Advanced)
Description: The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods of different nature. The theory has many applications and stimulates new developments in almost all areas of mathematics. The applications in this unit will be drawn from predator-prey systems, transmission of diseases, chemical reactions, beating of the heart and other equations and systems from mathematical biology. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, stability analysis, linearization, and hyperbolic critical points, and omega limit sets.
PMH7 Representations of the Symetric Groups
Description: Classical families of symmetric polynomials and relations between them,
Pre-Requistes: Available to Honours students only
PMH8 Partial Differential Equations
Description: The course is an introduction to the modern theory of partial differential
Pre-requistes: First Semester Honours - Functional Analysis
MSH5 Advance Time Series Analysis
Description: Course description: This is an advanced course in time series analysis and
Pre-requistes: Assumed knowledge is STAT 3011 Stochastic Processes and Time Series
Description: Section I of this course will introduce the fundamental concepts of applied stochastic processes and Markov chains used in financial mathematics, mathematical statistics, applied mathematics and physics. Section II of the course establishes some methods of modeling and analysing situations which depend on time. Fitting ARMA models for certain time series are considered from both theoretical and practical points of view. Throughout the course we will use the S-PLUS (or R) statistical packages to give analyses and graphical displays.
MSH6 Aysmptotics and Extreme Value Theory
Description: This course will cover theory of stochastic di erential equations. We will
Pre-requistes: Assumed knowledge: STAT 3911 Stochastic Processes and Time Series
Description: This is an Advanced version of STAT3011. There will be 3 lectures in common with STAT3011. In addition to STAT3011 material, theory on branching processes and birth and death processes will be covered. There will be more advanced tutorial and assessment work associated with this unit.
MSH7 Applied Probability & Stochastic DE's
Description: This course has two sections. Section 1 covers applications of extreme
Pre-requistes: Available to Honours students only
MATH3075/3975 Financial Mathematics (Normal and Advanced)
Description: This unit is an introduction to the mathematical theory of modern finance. Topics include: notion of arbitrage, pricing riskless securities, risky securities, utility theory, fundamental theorems of asset pricing, complete markets, introduction to options, binomial option pricing model, discrete random walks, Brownian motion, derivation of the Black-Scholes option pricing model, extensions and introduction to pricing exotic options, credit derivatives. A strong background in mathematical statistics and partial differential equations is an advantage, but is not essential. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. The lectures in the Normal unit are held concurrently with those of the corresponding Advanced unit. Note that students enrolled in MATH3075 and those enrolled in the advanced level unit MATH3975 attend the same lectures, but the assessment tasks for MATH3975 are more challenging than those for MATH3075.
Pre-requistes: 12 credit point of intermediate maths
MATH3078/3978 Partial Differential Equations and Waves (Normal and Advanced)
Description: This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics - the wave equation, the diffusion (heat) equation and Laplace's equation - are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of sound waves and water waves. MATH3978 As for MATH3078 PDEs & Waves but with more advanced problem solving and assessment tasks. Some additional topics may be included.
Pre-requistes: 12 credit point of intermediate maths
MATH3964 Complex Analysis with Applications(Advanced) (Not offered in 2009)
Description: This unit continues the study of functions of a complex variable and their applications introduced in the second year unit Real and Complex Analysis (MATH2962). It is aimed at highlighting certain topics from analytic function theory and the analytic theory of differential equations that have intrinsic beauty and wide applications. This part of the analysis of functions of a complex variable will form a very important background for students in applied and pure mathematics, physics, chemistry and engineering. The course will begin with a revision of properties of holomorphic functions and Cauchy's theorem with added topics not covered in the second year course. This will be followed by meromorphic functions, entire functions, harmonic functions, elliptic functions, elliptic integrals, analytic differential equations, hypergeometric functions. The rest of the course will consist of selected topics from Greens functions, complex differential forms and Riemann surfaces.
Pre-requistes: MATH2962 Real and Complex Analysis
Description: Analysis is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. Starting off with an axiomatic description of the real number system, this first course in analysis concentrates on the limiting behaviour of infinite sequences and series on the real line and the complex plane. These concepts are then applied to sequences and series of functions, looking at point-wise and uniform convergence. Particular attention is given to power series leading into the theory of analytic functions and complex analysis. Topics in complex analysis include elementary functions on the complex plane, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.
MATH3974 Fluid Dynamics (Advanced)
Description: This unit of study provides an introduction to fluid dynamics, starting with a description of the governing equations and the simplifications gained by using stream functions or potentials. It develops elementary theorems and tools, including Bernoulli's equation, the role of vorticity, the vorticity equation, Kelvin's circulation theorem, Helmholtz's theorem, and an introduction to the use of tensors. Topics covered include viscous flows, lubrication theory, boundary layers, potential theory, and complex variable methods for 2-D airfoils. The unit concludes with an introduction to hydrodynamic stability theory and the transition to turbulent flow.
Pre-requistes: MATH2961 Linear algebra & Vector Calculus & MATH2965 Introduction to Partial Differential Equations
Description: LA: This unit is an advanced version of MATH2061, with more emphasis on the underlying concepts and on mathematical rigour. Topics from linear algebra focus on the theory of vector spaces and linear transformations.
PDE: This unit of study is essentially an Advanced version of MATH2065, the emphasis being on solutions of differential equations in applied mathematics. The theory of ordinary differential equations is developed for second order linear equations, including series solutions, special functions and Laplace transforms. Some use is made of computer programs such as Mathematica. Methods for PDEs (partial differential equations) and boundary-value problems include separation of variables, Fourier series and Fourier transforms.
MATH3966 Modules & Group Representations (Advanced)
Description: This unit deals first with generalized linear algebra, in which the field of scalars is replaced by an integral domain. In particular we investigate the structure of modules, which are the analogues of vector spaces in this setting, and which are of fundamental importance in modern pure mathematics. Applications of the theory include the solution over the integers of simultaneous equations with integer coefficients and analysis of the structure of finite abelian groups. In the second half of this unit we focus on linear representations of groups. A group occurs naturally in many contexts as a symmetry group of a set or space. Representation theory provides techniques for analysing these symmetries. The component will deals with the decomposition of representation into simple constituents, the remarkable theory of characters, and orthogonality relations which these characters satisfy.
Pre-requistes: 12 credit point of intermediate maths. MATH3962 Rings, Fields and Galois Theory (Advanced)(assumed knowledge)
Description: This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize any polynomial into a product of linear factors by working over a "large enough" field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalizes the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions.
MATH3969 Measure Theory and Fourier Analysis (Advanced)
Description: Measure theory is the study of such fundamental ideas as length, area, volume, arc length and surface area. It is the basis for the integration theory used in advanced mathematics since it was developed by Henri Lebesgue in about 1900. Moreover, it is the basis for modern probability theory. The course starts by setting up measure theory and integration, establishing important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. This is then applied to Fourier Analysis, and results such as the Inversion Formula and Plancherel's Theorem are derived. Probability Theory is then discussed, with topics including independence, conditional probabilities, and the Law of Large Numbers.
Pre-requistes: 12 credit points of intermediate maths
STAT3013/3913 Statistical Inference (Normal/Advanced)
Description: Normal: In this course we will study basic topics in modern statistical inference. This will include traditional concepts of mathematical statistics: likelihood estimation, method of moments, properties of estimators, exponential families, decision-theory approach to hypothesis testing, likelihood ratio test as well as more recent approaches such as Bayes estimation, Empirical Bayes and nonparametric estimation. During the computer classes (using R software package) we will illustrate the various estimation techniques and give an introduction to computationally intensive methods like Monte Carlo, Gibbs sampling and EM-algorithm.
Advanced: This unit is essentially an Advanced version of STAT3013, with emphasis on the mathematical techniques underlying statistical inference together with proofs based on distribution theory. There will be 3 lectures per week in common with some material required only in this advanced course and some advanced material given in a separate advanced tutorial together with more advanced assessment work.
Pre-requistes: STAT2012 or STAT2912 Statistical Tests (Normal or Advanced)or STAT2003 or STAT2903 Statistical Models (Normal or Advanced)
Description: ST: This unit provides an introduction to the standard methods of statistical analysis of data: Tests of hypotheses and confidence intervals, including t-tests, analysis of variance, regression - least squares and robust methods, power of tests, non-parametric tests, non-parametric smoothing, tests for count data, goodness of fit, contingency tables. Graphical methods and diagnostic methods are used throughout with all analyses discussed in the context of computation with real data using an interactive statistical package.
SM: This unit provides an introduction to univariate techniques in data analysis and the most common statistical distributions that are used to model patterns of variability. Common discrete random models like the binomial, Poisson and geometric and continuous models including the normal and exponential will be studied. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit will have weekly computer classes where candidates will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.
STAT3014/3914 Applied Statistics (Normal & Advanced)
Description: Normal: This unit has three distinct but related components: Multivariate analysis; sampling and surveys; and generalised linear models. The first component deals with multivariate data covering simple data reduction techniques like principal components analysis and core multivariate tests including Hotelling's T^2, Mahalanobis' distance and Multivariate Analysis of Variance (MANOVA). The sampling section includes sampling without replacement, stratified sampling, ratio estimation, and cluster sampling. The final section looks at the analysis of categorical data via generalized linear models. Logistic regression and log-linear models will be looked at in some detail along with special techniques for analyzing discrete data with special structure.
Advanced: This unit is an Advanced version of STAT3014. There will be 3 lectures per week in common with STAT3014. The unit will have extra lectures focusing on multivariate distribution theory developing results for the multivariate normal, partial correlation, the Wishart distribution and Hotellling's T2. There will also be more advanced tutorial and assessment work associated with this unit.
Pre-requisites: Normal- STAT(2012 or 2912 or 2004). Advanced - STAT2012 or STAT2912 or STAT2004.
Description: Normal: See above.
Advanced: This course will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using classical linear methods, together with concepts of collection of data and design of experiments. First we will consider linear models and regression methods with diagnostics for checking appropriateness of models. We will look briefly at robust regression methods here. Then we will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course we will use the R statistical package to give analyses and graphical displays.
University of Wollongong
MATH305 Partial Differential Equations
Description: MATH305 is in a central area of mathematics, as many physical problems in the world are modelled with partial differential equations. Various types of equations and their solutions are discussed. As many equations cannot be solved in analytical form, numerical methods of solution also are considered. The aim is to develop high level mathematical ability and problem solving skills. student who successfully completes this subject should be able to:
Pre-requisites: MATH201 Multivariate and Vector Calculus and MATH202 Differential Equations, and MATH203 Linear Algebra
Descriptions: MATH201 is one of four core 200 level Mathematics subjects and is a prerequisite. for many 300 level subjects in Mathematics and Statistics. This subject extends the calculus of one variable to the calculus of more than one variable. Applications are given to maxima and minima, multiple integrals, vector calculus, line, surface and volume integrals, and to geometrical problems. A student who successfully completes this subject should be able to:
MATH202 is one of four core 200 level Mathematics subjects. This subject introduces the student to various special functions and differential equations and to techniques (both analytic and numerical) for their solution. Topics covered include exact first order equations, Gamma, Beta and Error functions, Laplace transforms, Fourier series, separation of variables for PDE's, basic numerical techniques, computer packages, and comparative accuracy of numerical techniques. A student who successfully completes this subject should be able to:
MATH203 is one of four core 200 level Mathematics subjects. The study of systems of linear equations is important not only to mathematicians but also to scientists and engineers. Study of these systems is done both theoretically and numerically with geometrical interpretations given. It aims to build on students' knowledge of matrix algebra and vector analysis.
MATH321 Numerical Analysis
Description: MATH321 is designed to extend the ideas developed in MATH202 and MATH203 as to how numerical and computational mathematics can be used to solve problems that have no analytic solution. The foci are problems in linear algebra and applications to real world problems. Specific techniques include algorithms for calculating eigenvalues and eigenvectors of a matrix. On successful completion of this subject, a student should be able to
Pre-requisites: MATH202 and MATH203
Description: see above
MATH323 Topology and Chaos
Description: MATH323 aims to develop critical understanding and problem-solving skills in the context of topology and chaos theory. It is intended to convey some of the impact of chaos theory in other areas and encourage interest of the student in phenomena such as the Koch curve. Some concepts discussed are notions of distance, dynamical systems, fractals and the Mandelbrot set. Students who successfully complete this subject should be able to:
Pre-requisites: MATH222 Continuous and Finite Mathematics
Description: MATH222 is for students who wish to continue in the mathematical analysis strand. Continuous Mathematics is concerned with the continuation of concepts introduced in first year calculus, including those of convergent sequence, continuous function and the integral of a function. Finite Mathematics is strictly independent of earlier work, but is related to first year algebra. A student who successfully completes this subject should be able to :
MATH345 Mathematics Project B
Description: The subject is a project individually chosen for the student, at a level appropriate to the 300 classification. The content may consist of (1) a placement in business or industry where substantial use is made of mathematical techniques; or (2) a project directed towards independent investigation by the student, written and/or oral presentations, and substantial interaction of the student with the supervisors of the project and other members of staff; or (3) a project directed to mastery of a mathematical package or language, with specific use of the package or language in some application or area of mathematics; or (4) a project of research collaboration with a member or members of staff, of which written and spoken presentation would be a part. Other projects which are appropriate but not primarily in one of these single categories may occur, such as a project combining features of (1) and (2).
Pre-requisites: Only available to mathematics (honours) students
MATH371 Special Topics in Inductrial and Applied Mathematics 3
Description: Entry to this subject is at the discretion of the Head of the School of Mathematics and Applied Statistics. This subject may not be offered in any particular year. MATH371 is one of a number of elective subjects available to students enrolled in the degree courses offered by the School. The aim of this subject is to provide students with specialist applied mathematical skills. Topics will be selected from the areas of interest of staff members of the School or visiting staff members.
STAT332 Multiple Regression and Time Series
Pre-requisites: STAT332 is an advanced course covering relationships between variables and the analysis of observational studies and designed experiments. Topics covered include multiple linear regression, non-linear regression, generalised linear regression, ARIMA models, forecasting of time series and Box-Jenkin's approach. A student who successfully completes this subject should be able to: (i) explain the theory and techniques of model building; (ii) apply the theory and techniques to practical problems and to use these methods for prediction purposes; (iii) undertake model building and forecasting for problems representative of those arising in industry and commerce.
Pre-requisites: STAT232 Estimation Hypothesis Testing
Description: STAT232 develops techniques of statistical inference and statistical analysis. The inference techniques are sampling distributions (such as chi-squared, t and F distributions), methods and criteria of estimation, and hypothesis testing. The analysis techniques are nonparametric testing (such as the sign, median and Wilcoxon tests), simple linear regression and one and two-way analysis of variance. A student who successfully completes this subject should be able to: (i) apply appropriate parametric and non-parametric tests and present the conclusions of that analysis; (ii) interpret and model practical problems; (iii) explain the basic concepts of sampling theory, point and interval estimation and hypothesis testing; (iv) derive the details (such as the distribution of the test statistics, their expected mean squares, and the power functions) of the tests studied and similar tests; (v) apply and interpret appropriate procedures from a statistical package such as JMP.
STAT333 Statistical Inference and Multivariate Analysis
Description: STAT333 covers inference (estimation and hypothesis testing) in both one and many dimensions. Topics covered include transformations, maximum likelihood and minimum variance unbiased estimation, the likelihood ratio, score and Wald tests, vector random variables, the multivariate Normal distribution, principal components analysis, factor analysis and discriminant analysis.
Pre-requisites: see above
Description : see above
INFO411 Data Mining and Knowledge Discovery
Description: Introduction to Data Mining and Knowledge Discovery, Data Bases and Warehouses, Data Structures, Exploratory Data Analysis Techniques, Association Rules, Artificial Neural Networks, Tree Based Methods, Clustering and Classification Methods, Regression Methods, Overfitting and Inferential Issues, Use of Data Mining packages.
After successful completion of this subject, students should be able to plan and
carry out analyses of large and complex data sets and to identify useful
relationships and important subgroups in those datasets.
Pre-requisites: 36 cp (Knowledge of mathematical and statistical notation at an introductory level.) STAT904 Statistical Consulting Description: Project management; Client liaison; Problem identification; Consulting ethics and principles; Sources of data; Choosing design and analysis procedures; Common problems in statistical consulting; Setting sample size - power calculations; Consulting case studies; Report writing. A student who successfully completes this subject should be able to: (i) conduct
efficiently a consulting session with a client; (ii) find information on
statistical methodology using the resources of the Library and the World Wide
Web ; (iii) explain the important principles behind designing and conducting an
experiment or sample survey; (iv) determine appropriate statistical procedures
to use on a wide variety of data sets; (v) apply and interpret procedures from a
statistical package Pre-requisites: Honours Students only, available to those that have done a statistics major.
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.
University of South Australia
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.
Macquarie University
Algebra
Description: This honours unit devotes approximately half of its time to ring theory and half to representation theory,
Pre-requisite: MATH337 AlgebraIIIA
Description: This unit develops the basic ideas of modern abstract algebra by concentrating on the many facets of group theory. As well as the standard material leading to the isomorphism theorems, we cover combinational aspects such as presentations of groups, the Todd-Coxeter algorithm, and subgroups of free groups via groupoids. Also studied are permutation groups, finitely generated abelian groups, soluble groups and group representations. MATH337 is especially suitable for students majoring in the theoretical aspects of physics or computing science.
Lie Groups
Description: Topology is the study of continuity. The definition of topological space was conceived in order to say what it
Pre-requisite: MATH300 Geometry and Topology
Description: Designed to widen geometric intuition and horizons by studying topics such as projective geometry, topology of surfaces, graph theory, map colouring, ruler-and-compass constructions, knot theory and isoperimetric problems. MATH300 is especially recommended for those students preparing to become teachers of high-school mathematics.
Applied Functional Analysis
Description: This unit prepares you to use differential and integral equations to attack significant problems in the physical
Pre-requisite: MATH336 Partial Differential Equations & MATH339 Real Functional Analysis
Description: Partial differential equations form one of the most fundamental links between pure and applied mathematics. Many problems that arise naturally from physics and other sciences can be described by partial differential equations. Their study gives rise to the development of many mathematical techniques, and their solutions enrich both mathematics and their areas of origin. This unit explores how partial differential equations arise as models of real physical phenomena, and develops various techniques for solving them and characterising their solutions. Especial attention is paid to three partial differential equations that have been central in the development of mathematics and the sciences -- Laplace's equation, the wave equation and the diffusion equation.
Real Functional Analysis: This unit is concerned with a review of the limiting processes of real analysis and an introduction to functional analysis. Through the discussion of such abstract notions as metric spaces, normed vector spaces and inner product spaces, we can appreciate an elegant and powerful combination of ideas from analysis and linear algebra.
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