 # Senior Mathematics and Statistics Handbook

## Applied Mathematics Units of Study

This chapter contains descriptions of units of study in the Applied Mathematics program, arranged by semester. Students who wish to take an advanced unit of study and who have not previously undertaken advanced level work in second year should speak to one of the coordinators and be prepared to devote extra time to the unit to compensate.

It should be noted that these lists are provisional only and that any unit of study may be withdrawn due to resource constraints.

## MATH3076/3976 Mathematical Computing (Advanced & Mainstream)

Prerequisite (MATH3076): 12 credit points of intermediate mathematics, and at least one of MATH1001, MATH1901, MATH1003, MATH1903 or MATH1907.

Prerequisite (MATH3976): 12 credit points of intermediate mathematics, and MATH1903 or MATH1907, or a credit in MATH1003.

Prohibitions: MATH3016, MATH3916.

Lecturer: Geoff Vasil .

Assessment: One two hour exam, assignments and quizzes (100%).

This unit of study provides an introduction to Python programming and numerical methods. Topics covered include computer arithmetic and computational errors, systems of linear equations, interpolation and approximation, solution of nonlinear equations, quadrature, initial value problems for ordinary differential equations and boundary value problems.

## MATH3063 Nonlinear ODEs with Applications (Mainstream)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2061.

Prohibitions: MATH3003, MATH3923, MATH3020, MATH3920, MATH3963.

Lecturer: Ian Lizarraga and Mary Myerscough.

Assessment: One two hour exam (75%), assignments (15%) and quizzes (10%).

This unit of study is an introduction to qualitative methods for systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. 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.

## MATH3963 Nonlinear ODEs with Applications (Advanced)

Prerequisite: 12 credit points of Intermediate Mathematics.

Assumed knowledge: MATH2961, MATH2962

Prohibitions: MATH3003, MATH3923, MATH3020, MATH3920, MATH3063.

Lecturer: Martin Wechselberger .

Assessment: One two hour exam, class tests, assignments. (Weighting to be advised.)

This course serves as an introduction to the more modern theory of differential equations and dynamical systems. The emphasis is on obtaining a qualitative understanding of properties of a system. This course intertwines the study of the theory of ODEs with applications to systems modelling various phenomena. The more theoretical part includes a study of existence and uniqueness theorems for linear and nonlinear equations, asymptotic behaviour for nonlinear ODEs and elementary bifurcation theory. The applications in this unit will be drawn from mechanical, biological and chemical models, and other equations and systems from applied mathematics.

## MATH3971/4071 Convex Analysis and Optimal Control (Advanced)

Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.

Assumed knowledge: MATH2021 and MATH2023 and STAT2011.

Lecturer: Ben Goldys .

Assessment: One two hour exam (70%), assignments (30%).

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimistion problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straight forward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.

Assumed knowledge: MATH2961 and MATH2965.

Prohibitions: MATH3914.

Lecturer: Georg Gottwald .

Assessment: One two hour exam (70%), assignments (30%).

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.

## MATH3075/3975 Financial Mathematics (Advanced & Mainstream)

Prerequisite (MATH3075): 12 credit points of intermediate mathematics.

Prerequisite (MATH3975): 12 credit points of intermediate mathematics at a credit or better average.

Prohibitions: MATH3015, MATH3933.

Lecturer: Marek Rutkowski .

Assessment: One two hour exam (80%), quizzes (20%).

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.

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.

## MATH3078/3978 Partial Differential Equations and Waves (Advanced & Mainstream)

Prerequisite (MATH3078): 12 credit points of intermediate mathematics.

Prerequisite (MATH3978): 12 credit points of intermediate mathematics at a credit or better average.

Assumed knowledge: MATH2061 (or MATH2961) and MATH2065 (or MATH2965).

Prohibitions: MATH3018, MATH3921.

Lecturer: Daniel Hauer.

Assessment: One two hour exam (70%), assignments (10%) and quizzes (20%).

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.

## MATH3977 Lagrangian and Hamiltonian Dynamics (Advanced)

Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.

Prohibitions: MATH2904, MATH2004, MATH3917.

Lecturer: Holger Dullin .

Assessment: One two hour exam (70%), assignments (10%) and quizzes (20%).

This unit provides a comprehensive treatment of dynamical systems using the mathematically sophisticated framework of Lagrange and Hamilton. This formulation of classical mechanics generalizes elegantly to modern theories of relativity and quantum mechanics.

The unit develops dynamical theory from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear very simple. The unit will also explore connections between geometry and different physical theories beyond classical mechanics.

Students will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. Hamilton-Jacobi theory will be used to elegantly solve problems ranging from geodesics (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes.

This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.