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MA3TPA: Topics in Pure and Applied Mathematics

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MA3TPA: Topics in Pure and Applied Mathematics

Module code: MA3TPA

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 3 (Honours)

When you'll be taught: Semester 1

Module convenor: Professor Simon Chandler-Wilde, email: s.n.chandler-wilde@reading.ac.uk

Module co-convenor: Dr Jochen Broecker, email: j.broecker@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST ( TAKE MA2DE AND TAKE MA2RA2 ) OR ( TAKE MA2DE2NU AND TAKE MA1RA2NU ) (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA4TPA (Compulsory)

Placement information: NA

Academic year: 2024/5

Available to visiting students: No

Talis reading list: Yes

Last updated: 21 May 2024

Overview

Module aims and purpose

This module tackles two topics in pure and applied mathematics, namely i) integral equations; ii) stochastic processes. The integral equations part of the module provides a first introduction to the theory of integral equations, numerical methods for their solution, and applications, including in meteorology. The stochastic processes part of the module provides a rigorous introduction to the theory of Brownian motion and martingales, and discusses application to stochastic analysis (including stochastic integrals and stochastic differential equations).Ìý

Module learning outcomes

By the end of the module, it is expected that students will be able to:Ìý

  1. Formulate integral equations as problems in a Banach space, and apply simple approximation techniques for their solutionÌý
  2. Formulate wave-scattering problems as integral equationsÌý
  3. Carry out analyses of Brownian Motion and ²Ñ²¹°ù³Ù¾±²Ô²µ²¹±ô±ð²õÌý
  4. Carry out elementary analyses of stochastic differential equationsÌý

Module content

This module will tackle two topics in pure and applied mathematics, namely: i) integral equations, their theory and applications; ii) stochastic processes.Ìý

In the integral equations section of the module we will provide a general introduction to the key aspects of the theory, introducing functional analysis ideas as needed, following this up by a discussion of widely used approximation and numerical techniques, and a detailed examination of the application of integral equations to model real-world wave-scattering problems, arising in our research in mathematics, and in applications in meteorology. The main elements of the integral equations section are:Ìý

  • Classification of integral equationsÌý
  • Exact solution of degenerate kernel Fredholm integral equationsÌý
  • Questions of uniqueness and existence of solution (in part tackled by simple functional analysis methods): the Fredholm alternative and Neumann seriesÌý
  • Simple numerical methods for Fredholm and Volterra integral equationsÌý
  • Applications of integral equation methods to wave scatteringÌý
  • A partial analysis of the trapezium rule method for Volterra integral equations via Gronwall inequalitiesÌý

In the stochastic processes section of the module we will construct Brownian motion, a stochastic process which can be interpreted as the integral of white noise. Then we discuss stochastic integrals, that is, integrals against Brownian motion, leading to the theory of stochastic differential equations. Key aspects of the theory will be discussed, including applications of Stochastic Differential Equations as well as other, related processes called Martingales. The main elements are:Ìý

  • Construction of Brownian motionÌý
  • Properties of Brownian motion: Scaling, Law of large numbers, law of the iterated logarithm, Hölder continuity of pathsÌý
  • Construction of the Ito integralÌý
  • ²Ñ²¹°ù³Ù¾±²Ô²µ²¹±ô±ð²õÌý
  • Ito formula and stochastic differential equationsÌý

Structure

Teaching and learning methods

Lectures supported by problem sheets and tutorials as well as a small peer-to-peer reading group on measure and integration basics.Ìý

Study hours

At least 30 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.


ÌýScheduled teaching and learning activities ÌýSemester 1 ÌýSemester 2 ÌýSummer
Lectures 30
Seminars
Tutorials 14
Project Supervision
Demonstrations
Practical classes and workshops
Supervised time in studio / workshop
Scheduled revision sessions 2
Feedback meetings with staff
Fieldwork
External visits
Work-based learning


ÌýSelf-scheduled teaching and learning activities ÌýSemester 1 ÌýSemester 2 ÌýSummer
Directed viewing of video materials/screencasts
Participation in discussion boards/other discussions 5
Feedback meetings with staff 2
Other
Other (details)


ÌýPlacement and study abroad ÌýSemester 1 ÌýSemester 2 ÌýSummer
Placement
Study abroad

Please note that the hours listed above are for guidance purposes only.

ÌýIndependent study hours ÌýSemester 1 ÌýSemester 2 ÌýSummer
Independent study hours 147

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 40% to pass this module.

Summative assessment

Type of assessment Detail of assessment % contribution towards module mark Size of assessment Submission date Additional information
In-person written examination Exam 50 2 hours Semester 1, Assessment Period
Set exercise Problem sheet 15 Semester 1, Teaching Week 5 Assignment on integral equations.
Set exercise Problem sheet 35 Assignment on stochastic processes.

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

  • where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
  • the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
  • where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
  • where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

  • where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
  • where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Peer-to-peer reading group on measure and integration basics: students will produce a short report which will be discussed during two feedback sessions.Ìý

Reassessment

Type of reassessment Detail of reassessment % contribution towards module mark Size of reassessment Submission date Additional information
In-person written examination Written exam 50 2 hours During the University resit period
Set exercise Problem sheet 50

Additional costs

Item Additional information Cost
Computers and devices with a particular specification
Required textbooks
Specialist equipment or materials
Specialist clothing, footwear, or headgear
Printing and binding
Travel, accommodation, and subsistence

THE INFORMATION CONTAINED IN THIS MODULE DESCRIPTION DOES NOT FORM ANY PART OF A STUDENT'S CONTRACT.

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