MATHS 2203 - Advanced Mathematical Perspectives II
North Terrace Campus - Semester 2 - 2015
General Course Information
Course Code MATHS 2203 Course Advanced Mathematical Perspectives II Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Undergraduate Location/s North Terrace Campus Units 3 Contact Up to 3.5 contact hours per week Available for Study Abroad and Exchange Y Prerequisites MATHS 1012 and MATHS 1015 Restrictions Available to BmaSc(Adv) students only Course Description The aim of this course is to foster a broad appreciation of the mathematical sciences with an exposure to the areas of major research strength within the School. It will be taught in four three week blocks covering Mechanics, Operations research, Pure Mathematics and Statistics. Students will be required to participate proactively in the course by possible involvement in open ended problems, independent reading and mini projects.
Course Coordinator: Professor Michael Murray
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning Outcomes
- Be able develop rigorous mathematical arguments.
- Be able to understand and apply basic combinatorial arguments.
- Understand the connections between probability, statistical data and evidence.
- Be able to apply probabilistic reasoning in real contexts involving data and evidence.
- Appreciate the usefulness of both stochastic and deterministic modelling approaches.
- Be able to analyse simple difference equation and Markov chain models.
- Be able to develop simple mathematical models.
- Be able to write project reports.
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s) Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1, 3, 4, 7 An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 1, 2, 4, 7 Skills of a high order in interpersonal understanding, teamwork and communication. 8 A proficiency in the appropriate use of contemporary technologies. 8 A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
Recommended ResourcesCourse materials will be provided by the lecturers.
Learning & Teaching Activities
Learning & Teaching ModesThe course is taught in four blocks of three weeks. In each block there are 9 lectures and 1 tutorial.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Lecture 36 108 Tutorial 4 16 Project 3 26 Assignment 2 6 Total 156
Learning Activities SummaryDeterministic and stochastic models
Week 1: Introduction to mathematical modelling - the process and its objectives. Thinking about simple problems, and turning words into equations. Linear difference equations - definitions and methods of solution. More complicated problems, and getting to grips with data.
Week 2: Reflecting on the model. How good a fit is our model to the data? What is does it fail to take into account? Analytical methods for understanding the behaviour of difference equation models: cob-webbing, stability of a fixed point. A more detailed look at the discrete logistic equation - fixed points, periodic cycles, chaos, the idea of a bifurcation.
Week 3: Introduction to discrete-time Markov chains. Development of a simple model (a stochastic logistic model). Transition probability matrices. Analytic evaluation of mean and variance. Ability to explain variability in data.
Week 4: Introduction to the concepts of stationary and quasi-stationary behaviours. How can we find the parameters in the model? Evaluation of likelihood, and maximum likelihood estimation of parameters. Estimating parameters from data.
Week 5: Applied Probability Extended. Expected hitting times of Discrete Time Markov Chain; applied to expected time to extinction of a population. The stochastic logistic model is equivalent to S-I-S epidemic model. Extension to modelling S-I-R epidemics. Models of interacting populations - systems of difference equations. Host-parasitoid interactions: the Nicholson-Bailey model.
Week 6: Analytical tools for systems of difference equations - stability of fixed points, the Jury conditions. Analysis of the Nicholson Bailey model. Further examples of interacting population models, e.g. spread of diseases - interactions between susceptible and infected. Summary and possible case study.
Week 7: Axioms of probability, the long-run interpretation of probability and laws of large numbers, conditional probability and Bayes' Theorem, subjective probabilities.
Week 8: Probability and data, assumptions, independence, rare events, understanding coincidences in context.
Week 9: Paradoxes and common fallacies, examples of probabilistic evidence in legal and forensic contexts.
Week 10: Review from Multivariable and Complex Calculus. Examples of minimal surfaces. Second fundamental form and curvature of a surface.
Week 11: Minimal surfaces, definition and examples. Area minimising and minimal surfaces. Review complex analysis. Isothermal parameters.
Week 12: Weiestrass-Enneper representation. Examples. Test for non-minimising minimal surfaces. Examples.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Component Weighting Topic Objective assessed Assignment 1 10% Deterministic models 5, 6, 7, 8 Assignment 2 10% Stochastic models 5, 6, 7, 8 Project 1 30% Deterministic and stochastic models 5, 6, 7, 8 Project 2 25% Statistics 3, 4, 8 Project 3 25% Geometry 1, 2, 8
Assessment item Distributed Due Assignment 1 Week 3 Week 4 Assignment 2 Week 4 Week 5 Project 1 Week 4 Week 6 Project 2 Week 9 Week 11 Project 3 Week 10 Week 14
No information currently available.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description FNS Fail No Submission F 1-49 Fail P 50-64 Pass C 65-74 Credit D 75-84 Distinction HD 85-100 High Distinction CN Continuing NFE No Formal Examination RP Result Pending
Further details of the grades/results can be obtained from Examinations.
Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.
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