# 35004 Mathematical Analysis and Applications

**Warning:** The information on this page is indicative. The subject outline for a
particular session, location and mode of offering is the authoritative source
of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.

Subject handbook information prior to 2023 is available in the Archives.

*UTS: Science: Mathematical and Physical Sciences*

*Credit points:*8 cp

*Result type*: Grade and marks

*There are course requisites for this subject. See access conditions.*

### Description

This subject introduces some of the most important and powerful mathematical tools developed over the last one hundred years. Topics include measure spaces; Lebesgue measure; borel sets and sigma algebra; Lebesgue integrals; product measures; probability as a measure; metric spaces; normed linear spaces; Banach spaces; Hilbert spaces; Lp spaces; applications to problems in probability and Fourier series. These are essential for the modern theory of probability and stochastic processes that underpin the pricing of derivative securities traded in international financial markets as well as the mathematical foundations of quantum physics.

### Subject learning objectives (SLOs)

Upon successful completion of this subject students should be able to:

1. | Elaboratively build one's mathematical language and arguments to explain definitions and theorems relevant to the subject, presenting the information with the highest standards of rigour and professionalism |
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2. | Demonstrate how language is used when referencing definitions and theorems at a conceptual level to one's peers |

3. | Analyse and synthesise the methods of proof of the theorems and be able to prove these theorems. Validate and test ideas in the theory of the integral, Fourier analysis, functional analysis and distribution theory. |

4. | Hypothesise and prove or disprove results the students have not seen before, using methods of advanced mathematical analysis. |

5. | Apply results from the subject to problems arising in related disciplines including probability theory and engineering. |

### Course intended learning outcomes (CILOs)

This subject also contributes specifically to the development of following course intended learning outcomes:

- Apply: Develop and distinguish between logical, clearly-presented, and justified arguments incorporating deductive reasoning to solve complex problems. (1.1)
- Analyse: Examine and combine the principles and concepts of a broad understanding in a range of fundamental areas in the mathematical sciences (calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management). (1.2)
- Synthesise: Integrate extensive knowledge of at least one sub-discipline of the Mathematical Sciences. (1.3)
- Apply: Formulate and model practical and abstract problems that are complex in nature using advanced quantitative principles, concepts, techniques, and technology. (2.1)
- Analyse: Devise solutions to problems based on a selection of approaches (e.g. analytic vs numerical/experimental, different statistical tests, different heuristic algorithms) and various sources of data and knowledge. (2.2)
- Apply: Ability to work effectively and responsibly in an individual or team context. (3.1)
- Apply: Succinct and accurate presentation of information, reasoning, and conclusions in a variety of modes to diverse expert and non-expert audiences. (5.1)

### Contribution to the development of graduate attributes

This subject is expected to contribute to the following honours level graduate profile attributes:

**1 . Disciplinary Knowledge**

The subject introduces the students to the advanced mathematical tools and ideas that are essential to modern mathematics. The material they learn lies at the very foundations of modern mathematics and an enormous range of applications. For example, modern signal processing, which is essential to mobile phone technology, is based on Fourier analysis, which the students learn in this subject. This is assessed in all assessment tasks.

**2. Research, Inquiry and Critical Thinking**

In the subject students learn how new results and areas of research arise from actual problems. For example the modern theory of integration, which is covered in the subject, developed because the existing theory proved inadequate for the solution of problems in Fourier analysis. The student will be asked to prove results that they have not seen before using the methods developed in the subject. This is developed and assessed across the whole subject in all assessments.

**3. Professional, Ethical and Social Responsibility**

The ability to acquire, develop, employ and integrate a range of technical, practical and professional skills, in appropriate and ethical ways within a professional context, autonomously and collaboratively and across a range of disciplinary and professional areas. Students will be expected to take an unfamiliar problem, analyse it, formulate a solution and present this in a clear and professional manner. This attribute is developed throughout the subject, particuarly in the assignments, which involve advanced problem solving skills.

**4. Reflection, Innovation, Creativity**

An ability to think and work creatively, including the capacity for self-starting, and the ability to apply mathematical skills to unfamiliar problems. This attribute is developed and assessed throughout the subject, particularly in the assignments.

**5. Communication **

Students learn to communicate mathematical ideas and results for an audience of peers and the wider mathematical community. This attribute is developed and assessed throughout the subject, particularly in the assignments.

### Teaching and learning strategies

*The material online will be discussed actively with students strongly encouraged to attend and participate, as this will benefit their performance in the subject. Online attendance will be possible if students are not able to attend for some reason.*

Canvas is used extensively. A complete set of class notes will be posted, as well as class problems with solutions and assignments. In addition, informal notes will be posted to stimulate class discussion.

Students are strongly encouraged to prepare for each class by doing the problem sets and reading material from the notes.

Students are able to collaborate with their peers on the solution of the weekly problem sets. Students may be asked to present solutions to problems to the class.

Students are given regular written and verbal feedback in class. The most common errors being made by the students are addressed in class with explanations and advice on how to avoid them.

There will be three assessment tasks, two assignments due in weeks 5 and 10 and a final assignment in week 12. Feedback is given on assignments when they are returned to the students.

The assessment tasks are based upon the application of the theory learned in class and through other resources, such as the online notes, which provide greater depth than can be covered in a two hour lecture. Reading recommended texts and doing exercises from the texts plays an essential part in the study of this subject.

In general, students should expect to spend 6 hours per week on this subject outside class time.

### Content (topics)

**Module 1 Convergence in Analysis **

This module starts from a study of real numbers, and the study of convergent sequences. Then we move on to completeness and compactness in metric spaces. We finish with a discussion of the Riemann integral.

**Module 2 Integration and Probability**

This module will teach you about the integration theory professionals use: the Lebesgue integral. We first study how this works on the real numbers, and then how it can be extended to general abstract measure spaces and probability.

**Module 3 Functional Analysis**

We introduce the idea of normed linear spaces, Banach spaces and Hilbert spaces. We then study the Fourier transform on L^p spaces, and introduce the theory of Distributions. Finally, we use this theory to find Fundamental solutions of Partial Differential Equations.

### Assessment

#### Assessment task 1: Assignment 1

Intent: | This assessment contributes to the following Graduate attributes 1. Disciplinary knowledge 2. Research, Inquiry and Critical Thinking 3. Professional, Ethical and Social Responsibility 5. Communication |
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Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 1.2, 1.3, 2.1, 3.1 and 5.1 |

Type: | Exercises |

Groupwork: | Group, individually assessed |

Weight: | 25% |

Criteria: | Accuracy of proofs and calculations, Clarity of answers, Correctness of results, insight and creativity, communication of ideas |

#### Assessment task 2: Assignment 2

Intent: | This assessment contributes to the following Graduate attributes 1. Disciplinary knowledge 2. Research, Inquiry and Critical Thinking 3. Professional, Ethical and Social Responsibility 5. Communication |
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Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 1.2, 1.3, 2.1, 3.1 and 5.1 |

Type: | Exercises |

Groupwork: | Group, individually assessed |

Weight: | 25% |

Criteria: | Accuracy of proofs and calculations, Clarity of answers, Correctness of results, insight and creativity, communication of ideas |

#### Assessment task 3: Assignment 3

Intent: | This assessment contributes to the following Graduate attributes 1. Disciplinary knowledge 2. Research, Inquiry and Critical Thinking 5. Communication |
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Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 1.2, 1.3, 2.1, 2.2 and 5.1 |

Type: | Exercises |

Groupwork: | Group, individually assessed |

Weight: | 50% |

Criteria: | Accuracy of proofs and calculations, Clarity of answers, Correctness of results, insight and creativity, communication of ideas |

### Minimum requirements

The final grade will be F = A1+A2+A3, where A1 and A2 are the marks for the first two assignments out of 25 and A3 is the mark for assignment three out of 50.

### Recommended texts

The following textbooks are useful, but not required.

- Capinski, M. and Kopp, E. Measure, Integral and Probability, 2nd Edition., Springer 2005.
- Robdera, M. A. A Concise Approach to Mathematical Analysis, Springer 2003.
- Cohen, G. l. A Course in Modern Analysis and its Applications. Cambridge University Press, 2003.

### Other resources

A complete set of written lecture notes covering the subject will be provided on UTSCanvas.