# Intermediate Mathematics and Statistics Handbook

## Introduction

This handbook contains information about Intermediate Units of Study offered by the School of Mathematics and Statistics. If you enrol in an intermediate mathematics or statistics unit you will need to refer to this handbook throughout the year.

This introductory section aims to provide you with sufficient information to be able to choose the units which most suit your interests and abilities. The School offers a variety of one semester 6 credit point intermediate units in Mathematics and Statistics. There are units at both Advanced and Mainstream levels. Mathematics units are grouped into two disciplines, Applied Mathematics and Pure Mathematics.

If you are a student in the Science Faculty you are required to complete a major in a recognized subject area (such as Mathematics or Statistics). For this you must complete at least 24 senior credit points in that subject area. You can, if you wish, attempt two majors. You should keep this in mind when planning your intermediate year and consult the Senior Mathematics and Statistics Handbook and Science Faculty handbook, if necessary. The School of Mathematics and Statistics offers majors in the following subject areas:

- Mathematics, which includes a double major in Mathematics consisting of at least 48 senior credit points;
- Statistics;
- Financial Mathematics and Statistics.

Although it is possible for you to do a major in Mathematics with only 12 intermediate level credit points of Mathematics, you are strongly advised to do more than 12 credit points of Intermediate Mathematics. Completing 18 or 24 credit points of intermediate level Mathematics will provide you with a better foundation for your senior year, and open up a wider choice of senior level units to you. If your main interest is Mathematics, you could well choose to do a double major in Mathematics, and the ideal preparation for this would be 36 credit points of intermediate level Mathematics. You may, if you wish, specialize completely in Mathematics and Statistics in your intermediate and senior years. If you are in a Faculty different from Science consult the relevant Faculty handbook about your degree requirements in Mathematics and Statistics. Engineering students should also consult their department.

The School offers honours degrees in Applied Mathematics, Pure Mathematics and Mathematical Statistics. If you are considering an honours degree in Mathematics and Statistics, most of your units should be at the advanced level.

A brief description of each of the units of study, and recommended combinations of units, follows. You may, of course, choose any combination of units for which you qualify. Prerequisites for each unit are given in Chapter 2. However, you should follow the recommendations if you wish to major in, or complete an honours degree in, one of the subject areas described below.

## Pure Mathematics

Pure Mathematics units are designed to provide students with the mathematical knowledge and techniques necessary in all scientific and engineering disciplines, as well as the solid mathematical base necessary for a career in mathematics. Recent developments in pure mathematical research have led to important and exciting applications in various branches of theoretical and applied sciences, and in engineering. Some examples include the application of singularity theory and group theory to symmetry breaking and bifurcation in engineering, the application of number theory to cryptography, the application of category theory to computer science, and the application of complex analysis and algebraic geometry to physics.

Most importantly, units in Pure Mathematics provide invaluable training in logical thinking and problem solving skills which you will find invaluable no matter what career you subsequently choose. Graduates with a major in Pure Mathematics may choose a career in teaching or research, or in one of the many fields in which mathematical ability and expertise are important, such as finance, operations research, computing and management. Even if your intended career path does not seem to involve much mathematics, you will find that employers generally hold mathematical ability in very high regard.

If your interest is in Pure Mathematics, the following units are recommended.

Mainstream Level

There are three 6 credit point intermediate units recommended for students of Pure Mathematics:

- MATH2061 Linear Mathematics and Vector Calculus,
- MATH2068 Number Theory and Cryptography,
- MATH2069 Discrete Mathematics and Graph Theory.

We recommend that students wishing to study at the mainstream level and complete a Mathematics major in Pure Mathematics do MATH2061 at least one of the other two. Note that MATH2061 is also recommended for students intending to do a major in applied mathematics.

Note that all students intending to do a mathematics major are strongly advised to more than 12 credit points of intermediate mathematics. Although it is possible to proceed to a major with only 12 credit points of intermediate mathematics, taking 18 or 24 credit points will give you a better background and a wider range of options for your senior year.

Advanced Level

If your grades have been a credit average or better in Junior Mathematics units we encourage you enrol in some advanced units in your intermediate year. There are five 6 credit point advanced level intermediate units recommended for students of Pure Mathematics:

- MATH2961 Linear Mathematics and Vector Calculus (Advanced),
- MATH2962 Real and Complex Analysis (Advanced),
- MATH2968 Algebra (Advanced),
- MATH2969 Discrete Mathematics and Graph Theory (Advanced),
- MATH2988 Number Theory and Cryptography (Advanced).

MATH2961, which is the advanced level analogue of MATH2061, is considered core material for both pure mathematics and applied mathematics, and so every student intending to do a mathematics major is expected to take either MATH2061 or MATH2961. Students who are considering ultimately proceeding to honours in Pure Mathematics are advised to do either MATH2962 or MATH2968, or preferably both, since the topics introduced in these units are developed further in senior year pure mathematics units. A minimum of 12 credit points of intermediate mathematics is needed if you wish to do a mathematics major, but doing 18 or more provides a much better preparation.

Note that the lectures for MATH2969 are the same as the the lectures for the mainstream level unit MATH2069; similarly MATH2988 and MATH2068 share common lectures. Despite the similarity of the unit codes there is no close connection between the subject matter of MATH2968 and that of MATH2068, and no preclusion on enrolling in both.

## Applied Mathematics

Applied Mathematics is concerned with the development of mathematical models and techniques, both classical (such as calculus) and modern (such as discrete optimisation), with application to the solution of problems in the physical, biological and social sciences, engineering, medicine and computer science. The School has particular expertise in the general fields of astrophysics, relativity, engineering mathematics, biological sciences and earth sciences. Computer algebra and computational techniques are of considerable interest to Applied Mathematics and units are offered reflecting this.

Units in Applied Mathematics give a student a broad but rigorous training. These qualities are much in demand in the computer industry, engineering, banking and commerce, government organisations and teaching, and thus the study of Applied Mathematics can lead to a rewarding and satisfying career in a wide variety of disciplines. Applied Mathematics is complementary to units in Physics, Marine Sciences and Geophysics. For the mathematically inclined student the combination of Applied Mathematics with one or both of Pure Mathematics and Mathematical Statistics is an obvious possibility.

If your interest is in Applied Mathematics, the following units are recommended.

Mainstream Level

There are four 6 credit point intermediate units recommended for students of Applied Mathematics:

- MATH2061 Linear Mathematics and Vector Calculus,
- MATH2065 Introduction to Partial Differential Equations,
- MATH2070 Optimisation and Financial Mathematics.

Advanced Level

Serious students of Applied Mathematics are strongly encouraged to enrol in the following units:

- MATH2961 Linear Mathematics and Vector Calculus (Advanced)
- MATH2965 Introduction to Partial Differential Equations (Advanced)
- MATH2970 Optimisation and Financial Mathematics (Advanced).

This advanced level package of units will allow you to complete a Mathematics major in Applied Mathematics and proceed to an honours year in Applied Mathematics. It is not strictly necessary to take all units at the advanced level in order to do honours, but the advanced units do provide a better foundation for senior units and honours.

## Statistics

Mathematical Statistics is concerned with developing the theory necessary for the analysis and modelling of data and the testing of scientific hypotheses. Statistical ideas permeate the whole fabric of scientific investigation. Thus the units are found valuable not just for mathematicians but for students specialising in areas such as psychology and biological sciences. Statistical expertise is a valued commodity in industry, commerce and government agencies. Thus recent graduates with a major in mathematical statistics have had no difficulty in finding employment either in the private sector (e.g., insurance, banking) or the public sector (e.g., ABS, CSIRO, SRA, Telstra). There is at present an Australiawide shortage of honours graduates and postgraduates in statistics.

Modern statistics involves considerable use of highspeed computers. Students taking the intermediate Statistics units (STAT2011 or STAT2911 and STAT2012 or STAT2912) will have weekly computing sessions using the package called R.

Mainstream Level

The standard 12 credit point Statistics intermediate combination of units is as follows:

- STAT2011 Statistical Models.
- STAT2012 Statistical Tests.

Advanced Level

The standard 12 credit point Statistics intermediate combination of units is as follows:

- STAT2911 Probability and Statistical Models (Advanced),
- STAT2912 Statistical Tests (Advanced).

Science students who do not wish to major in Statistics, but want to learn further statistical techniques, without the mathematical theory, could choose STAT2012, following any Junior Statistics course. These students can then proceed to up to 12 credit points of Senior Statistics if they wish.

## Financial Mathematics and Statistics

Students in the BSc and BSc (Advanced) degrees can choose to major in Financial Mathematics and Statistics. For this you must complete the core intermediate units of study listed below.

Mainstream Level

The core 18 credit point combination of intermediate mainstream units in Financial Mathematics and Statistics is

- MATH2070 Optimisation and Financial Mathematics,
- STAT2011 Statistical Models,
- STAT2012 Statistical Tests.

Advanced Level

The core 18 credit point combination of intermediate advanced units in Financial Mathematics and Statistics is

- MATH2970 Optimisation and Financial Mathematics (Advanced),
- STAT2911 Probability and Statistical Models (Advanced),
- STAT2912 Statistical Tests (Advanced).

It is recommended that you also do the following Intermediate Mathematics units: MATH2061 or MATH2961, and MATH2065 or MATH2965.

## Progression to Senior Units

- In general, you need 12 credit points of intermediate mathematics to qualify for a mainstream level unit of senior mathematics.
- If your major interest is in pure or applied mathematics, then you are strongly encouraged to enrol in 6 units (36 credit points) of intermediate mathematics. If you are considering doing honours in mathematics, the units should include the core advanced units.
- Students intending to specialise in Applied Mathematics should choose at least 3 units (18 credit points) from the units labelled "Applied" or "Pure and Applied". The units should include MATH2061 or MATH2961, and MATH2065 or MATH2965.
- Students intending to specialise in Pure Mathematics should choose at least 3 units (18 credit points) from the units labelled "Pure" or "Pure and Applied". These should include MATH2061 or MATH2961.
- Students intending to specialise in Statistics should enrol in STAT2011 or STAT2911, and also STAT2012 or STAT2912. Intending honours students should choose the advanced versions.
- Students intending to specialise in Financial Mathematics and Statistics should enrol in MATH2070 or MATH2970, STAT2011 or STAT2911, and STAT2012 or STAT2912.
- Computer Science students should normally include MATH2069 and STAT2012 (or the advanced counterparts) among their choices, and will wish to consider MATH2068 or MATH2988 as well.
- Physics students would be well advised to choose MATH2061 or MATH2961, and MATH2065 or MATH2965.
- Most Engineering students are required to do either MATH2061 or MATH2067, and would be well advised to do MATH2065. (Check with the Engineering Faculty to see which of MATH2061 or MATH2067 is specified for your degree program.) Students who have obtained high marks in advanced level junior mathematics units are encouraged to see permission from the Engineering Faculty to replace MATH2061 by MATH2961, and to take MATH2965 rather than MATH2065.