 # Intermediate Mathematics and Statistics Handbook

## Units of Study

Each unit of study has a web page, accessed by following the links on the main Intermediate Mathematics and Statistics web page.

In this chapter, Mathematics units are listed, by semester, in numerical order; then Statistics and Data Science units are listed, by semester, in numerical order.

Units are designated Mainstream or Advanced. Entry to an Advanced level unit normally requires a Credit or better in a Mainstream level prerequisite, or a Pass or better in an Advanced level prerequisite.

Mathematics units are also labelled Applied, or Pure, or both. Although there is no clear distinction between applied mathematics and pure mathematics at the intermediate level, this labelling gives a rough guide as to which senior level units the intermediate level units are most closely allied with.

The unit code for an intermediate unit of study in the School consists of MATH or STAT or DATA followed by four digits: for example MATH2088 or STAT2011. The first digit is 1 for junior level units, 2 for intermediate level units, 3 for senior level units. The second digit indicates whether the unit is mainstream (0 or 1) or advanced (9). In most cases two units which share the same last two digits are mutually exclusive: for example, MATH2021 may not be counted with MATH2921.

Text and reference books are yet to be advised. Except for The Little Blue Book it is suggested that you do not purchase any books until recommendations are made by lecturers.

The Little Blue Book is a compact reference book: it contains definitions, formulas and important results from Junior Mathematics which are used in Intermediate Units. It is recommended that all students have access to this book: it is available from the Coop Bookshop.

## MATH2021 Vector Calculus and Differential Equations

(6 credit points, Mainstream, Pure and Applied)

Prerequisites: MATH1021 or MATH1921 or MATH1931 or MATH1001 or MATH1901 or MATH1906,
and MATH1002 or MATH1902,
and MATH1023 or MATH1923 or MATH1933 or MATH1003 or MATH1903 or MATH1907.

Prohibitions: May not be counted with MATH2921 or MATH2065 or MATH2965 or MATH2961 or MATH2067. Additionally, MATH2021 cannot be counted with both MATH2061 and MATH2022.

Lecturer(s): Zhou Zhang.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

This unit opens with topics from vector calculus, including vector-valued functions (parametrized curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals, polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, through cylinders, spheres and other parametrized surfaces), Gauss' and Stokes' theorems.

The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

## MATH2022 Linear and Abstract Algebra

(6 credit points, Mainstream, Pure and Applied)

Prerequisites: MATH1002 or MATH1902.

Prohibitions: May not be counted with MATH2922 or MATH2968 or MATH2961. Additionally, MATH2022 cannot be counted with both MATH2061 and MATH2021.

Lecturer(s): David Easdown.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

Linear and abstract algebra form one of the cornerstones of mathematics and is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit investigates and explores properties of linear functions, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract group theory.

## MATH2061 Linear Mathematics and Vector Calculus

(6 credit points, Mainstream, Pure and Applied)

Prerequisites: One of MATH1011 or MATH1001 or MATH1901 or MATH1906 or MATH1021 or MATH1921 or MATH1931,
and one of MATH1014 or MATH1002 or MATH1902,
and one of MATH1003 or MATH1903 or MATH1907 or MATH1021 or MATH1923 or MATH1933.

Prohibitions: May not be counted with MATH2067 or MATH2961. Additionally, only one of MATH2021 and MATH2022 may be counted with MATH2061.

Lecturer(s): Laurentiu Paunescu (Linear Mathematics), Leo Tzou (Vector Calculus).

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

This unit starts with an investigation of linearity: linear functions, general principles relating to the solution sets of homogeneous and inhomogeneous linear equations (including differential equations), linear independence and the dimension of a linear space. The study of eigenvalues and eigenvectors, begun in junior level linear algebra, is extended and developed. Linear operators on two dimensional real space are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors.

The unit then moves on to topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; pathindependent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals; polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, though cylinders, spheres and parametrised surfaces), Gauss' Divergence Theorem and Stokes' Theorem.

## MATH2069/2969 Discrete Mathematics and Graph Theory

Prerequisites (MATH2069): 6 credit points of Junior Mathematics.

Prerequisites (MATH2969): 9 credit points of Junior Mathematics at the advanced level or at the mainstream level with credit.

Prohibitions: MATH2069 and MATH2969 may not both be counted.

Lecturer(s): Andrew Mathas .

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

We introduce students to several related areas of discrete mathematics, which serve their interests for further study in pure and applied mathematics, computer science and engineering. Topics to be covered in the first part of the unit include recursion and induction, generating functions and recurrences, combinatorics, asymptotics and analysis of algorithms. Topics covered in the second part of the unit include Eulerian and Hamiltonian graphs, the theory of trees (used in the study of data structures), planar graphs, the study of chromatic polynomials (important in scheduling problems), maximal flows in networks, matching theory.

## MATH2921 Vector Calculus and Differential Equations (advanced)

(6 credit points, Advanced, Pure and Applied)

Prerequisites: MATH1921 or MATH1931 or MATH1001 or MATH1901 or MATH1906 or a mark of 65 or above in MATH1001 or MATH1021,
and MATH1902 or a mark of 65 or above in MATH1002,
and MATH1923 or MATH1933 or MATH1903 or MATH1907 or a mark of 65 or above in MATH1003 or MATH1023.

Prohibitions: May not be counted with MATH2021 or MATH2065 or MATH2965 or MATH2061 or MATH2961 or MATH2067

Classes: 3 lectures and 1 tutorial and 1 practice class per week.

Assessment: quizzes, assignments and a final exam.

This is the advanced version of MATH2021, with more emphasis on the underlying concepts and mathematical rigour. The vector calculus component of the course will include: parametrised curves and surfaces, vector fields, div, grad and curl, gradient fields and potential functions, lagrange multipliers line integrals, arc length, work, path-independent integrals, and conservative fields, flux across a curve, double and triple integrals, change of variable formulas, polar, cylindrical and spherical coordinates, areas, volumes and mass, flux integrals, and Green's Gauss' and Stokes' theorems. The Differential Equations half of the course will focus on ordinary and partial differential equations (ODEs and PDEs) with applications with more complexity and depth. The main topics are: second order ODEs (including inhomogeneous equations), series solutions near a regular point, higher order ODEs and systems of first order equations, matrix equations and solutions, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, elementary Sturm-Liouville theory, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a more thorough grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

## MATH2922 Linear and Abstract Algebra (advanced)

(6 credit points, Advanced, Pure and Applied)

Prerequisites: MATH1902 or a mark of 65 or above in MATH1002.

Prohibitions: May not be counted with MATH2022 or MATH2968 or MATH2061 or MATH2961.

Classes: 3 lectures and 1 tutorial and 1 practice class per week.

Assessment: quizzes, assignments and a final exam.

Linear and abstract algebra form one of the cornerstones of mathematics and is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit is an advanced version of MATH2022, with more emphasis on the underlying concepts and on mathematical rigour. This unit investigates and explores properties of vector spaces, matrices and linear transformations, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract group theory. The unit culminates in studying inner spaces, quadratic forms and normal forms of matrices together with their applications to problems both in mathematics and in the sciences and engineering.

## MATH2023 Analysis

(6 credit points, Mainstream, Pure and Applied)

Prerequisites: MATH1021 or or MATH1921 or MATH1931 or MATH1001 or MATH1901 or MATH1906,
and MATH1023 or or MATH1923 or MATH1933 or MATH1003 or MATH1903 or MATH1907,
and MATH1002 or MATH1902.

Prohibitions: May not both be counted with MATH2923 or MATH3068 or MATH2962.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam (70%), mid-semester test (20%), assignments (10%).

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This unit introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students of mathematics, science and engineering and in particular to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit starts with the foundations of calculus and the real numbers system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

## MATH2088/2988 Number Theory and Cryptography

(6 credit points, Mainstream/Avanced, Pure)

Prerequisites (MATH2088): 6 credit points of Junior Mathematics.

Prerequisites (MATH2988): 9 credit points of Junior Mathematics at the advanced level or at the mainstream level with credit.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam (60%), quiz (20%), assignments (20%).

Cryptography is the branch of mathematics that provides the techniques for confidential exchange of information sent via possibly insecure channels. This unit introduces the tools from elementary number theory that are needed to understand the mathematics underlying the most commonly used modern public key cryptosystems. Topics include the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, Möbius Inversion, the RSA Cryptosystem, the Elgamal Cryptosystem and the Diffie-Hellman Protocol. Issues of computational complexity are also discussed.

## MATH2070/2970 Optimisation and Financial Mathematics

(6 credit points, Mainstream/Avanced, Applied)

Prerequisites (MATH2070): MATH1011 or MATH1001 or MATH1901 or MATH1906, and MATH1014 or MATH1002 or MATH1902.

Prerequisites (MATH2970): MATH1901 or MATH1906 or credit in MATH1001, and MATH1902 or credit MATH1002.

Assumed knowledge: MATH1003 for MATH2070, MATH1903 (or MATH1907 or Credit in MATH1003) for MATH2970.

Lecturer(s): Chunxi Jiao and Desmond Ng.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam (70%), assignments (10%), quizzes (10%), project (10%).

Problems in industry and commerce often involve maximising profits or minimising costs subject to constraints arising from resource limitations. The first part of this unit looks at the important class of linear programming problems and their solution using the simplex algorithm, and the minimisation of functions of several variables with constraints, including Lagrange multipliers, Kuhn-Tucker theory and quadratic programming.

The second part of the unit deals with utility theory and modern portfolio theory. Topics covered include: pricing under the principles of expected return and expected utility; mean-variance Markowitz portfolio theory, the Capital Asset Pricing Model, log-optimal portfolios and the Kelly criterion; dynamical programming. Some understanding of probability theory including distributions and expectations is required in this part. Theory developed in lectures will be complemented by computer laboratory sessions using MATLAB. Minimal computing experience will be required.

## MATH2923 Analysis

(6 credit points, Advanced, Pure and Applied)

Prerequisite: MATH1921 or MATH1931 or MATH1901 or MATH1906 or a mark of 65 or above in MATH1021 or MATH1001,
and MATH1902 or a mark of 65 or above in MATH1002,
and MATH1923 or MATH1933 or MATH1903 or MATH1907 or a mark of 65 or above in MATH1023 or MATH1003.

Prohibitions: May not both be counted with MATH2023 or MATH3068 or MATH2962.

Lecturer(s): Laurentiu Paunescu.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: quizzes, an assignment, and a final exam .

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This advanced unit introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students with more mathematical maturity who study mathematics, science, or engineering. The unit starts with the foundations of calculus and the system of real numbers, with more emphasis on the topology. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

## STAT2011 Statistical Models

(6 credit points, Mainstream)

Prerequisites: MATH1001 or MATH1011 or MATH1901 or MATH1906, and MATH1005 or MATH1015 or STAT1021 or MATH1905 or ECMT1010.

Prohibition: May not be counted with STAT2911.

Lecturer(s): Shelton Peiris and Rachel Wang .

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam, assignments, quizzes, computer practical reports, and a one-hour computer practical class assessment task.

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 variable 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.

## STAT2911 Probability and Statistical Models

Prerequisites: MATH1903 or MATH1907 or credit in MATH1003, and and MATH1905 or credit in MATH1005 or MATH1015 or ECMT1010.

Prohibition: May not be counted with STAT2011.

Lecturer(s): Uri Keich .

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam, assignments, quizzes, computer practical reports, and a one-hour computer practical class assessment task.

This unit is essentially an advanced version of STAT2011 with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common random variables including the Poisson, normal, beta and gamma families are introduced. Probability generating functions and convolution methods are used to understand the behaviour of sums of random variables. 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 students will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

## DATA2001 Data Science: Big Data and Data Diversity

(6 credit points)

Prerequisites: DATA1002 or INFO1110 or INFO1903 or INFO1103.

Lecturer(s): TBA.

Classes: Lectures, Laboratories, Project Work — own time.

Assessment: through semester assessment (50%), final exam (50%).

This unit focuses on methods and techniques to efficiently explore and analyse large data collections. Where are hot spots of pedestrian accidents across a city? What are the most popular travel locations according to user postings on a travel website? The ability to combine and analyse data from various sources and from databases is essential for informed decision making in both research and industry. Students will learn how to ingest, combine and summarize data from a variety of data models which are typically encountered in data science projects, such as relational, semi-structured, time series, geospatial, image, text. As well as reinforcing their programming skills through experience with relevant Python libraries, this course will also introduce students to the concept of declarative data processing with SQL, and to analyse data in relational databases. Students will be given data sets from, for example, social media, transport, health and social sciences, and be taught basic explorative data analysis and mining techniques in the context of small use cases. The unit will further give students an understanding of the challenges involved with analysing large data volumes, such as the idea to partition and distribute data and computation among multiple computers for processing of 'Big Data'.

Assessment: One 2 hour exam, assignments, quizzes, computer practical reports, and a one-hour computer practical class assessment task.

Prerequisites: MATH1905 or credit in MATH1005 or MATH1015 or ECMT1010.

Prohibition: May not be counted with STAT2012.

Lecturer(s): Neville Weber .

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam (65%), assignments (10%), quizzes (5%), computer practical reports (10%), and a one-hour computer practical class assessment task (10%).

The 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, nonparametric tests, nonparametric smoothing, tests for count data goodness of fit, contingency tables. Graphical methods and diagnostics are used throughout with all analyses discussed in the context of computation with real data using an interactive statistical package.

There is emphasis both on the methods and their mathematical derivations.

## DATA2002 Data Analytics: Learning from Data

(6 credit points)

Assumed knowledge: basic linear algebra and some coding, or QBUS1040.

Prohibitions: STAT2012 or STAT2912.

Lecturer(s): Garth Tarr.

Classes: 3 lectures and 2 computer tutorials per week.

Assessment: written assignment, presentation, exams.

Technological advances in science, business, engineering has given rise to a proliferation of data from all aspects of our life. Understanding the information presented in these data is critical as it enables informed decision making into many areas including market intelligence and science. DATA2002 is an intermediate level unit in statistics and data sciences, focusing on learning data analytic skills for a wide range of problems and data. How should the Australian government measure and report employment and unemployment? Can we tell the difference between decaffeinated and regular coffee ? In this course, you will learn how to ingest, combine and summarize data from a variety of data models which are typically encountered in data science projects as well as reinforcing their programming skills through experience with statistical programming language. You will also be exposed to the concept of statistical machine learning and develop the skill to analyse various types of data in order to answer a scientific question. From this unit, you will develop knowledge and skills that will enable you to embrace data analytic challenges stemming from everyday problems.

## DATA2902 Data Analytics: Learning from Data (Advanced)

(6 credit points)

Prohibitions: may not be counted with any of STAT2012, STAT2912, DATA2002.

Lecturer(s): Garth Tarr.

Classes: 3 lectures and 2 computer tutorials per week.

Assessment: Computer practicals in-class (10%), Online quizzes (15%), Project group work assignment (5%), Project group work presentation (10%), Final exam (60%).

This is an advanced level version of DATA2002.