**I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.**

## Test

I am afraid I am a little on the busy side and ye might be waiting until around March 8 for your results. Just to clarify, I had intended for ye to prove rather than show in question 1; however I wrote show. Therefore if you used Theorem 1 or 2 to answer you will have to get full marks. However at the same time some people understood show as prove. By and large these people are looking at almost 100% but if they drop marks elsewhere, I will give a bonus mark to students who proved in question 1 rather than showed.

## Homework

Please find the Homework. Before you open it don’t be too alarmed: you only have to do ONE of the SIX options. All of the options are about dynamical systems & complex numbers in different areas of math:

- Discrete Mathematics, Number Theory & Abstract Algebra
- Probability
- Differential Calculus
- Integral Calculus
- Linear Algebra
- Complex Numbers

Therefore, if you are good at differential calculus, for example, you should have a look at option 3.

All of these questions are unseen to you and — with the exception of Q.6 — all require some knowledge of modules you are doing now or have done before. Although we have been concentrating on real-valued functions on the set of real numbers (i.e. , etc.), a lot of the theory carries over into more general sets and functions, and this is the main learning outcome of this homework.

I am not going to pretend that this is an easy assignment, but I will say that clear and logical thinking will reveal that the solutions and answers aren’t ridiculously difficult: a keen understanding of the principles of dynamical systems and a good ability in one of the options should see you through.

The final date for submission is 12 April 2013 and you can hand up early if you want. You will be submitting to the big box at the School of Mathematical Science. If I were you I would aim to get it done and dusted early as this is creeping into your study time and is very close to the summer examinations.

Note that you are will be free to collaborate with each other and use references but this must be indicated on your hand-up in a declaration. Evidence of copying or plagiarism (although this is unlikely as these are original problems by and large) will result in divided marks or no marks respectively. You will not receive diminished marks for declared collaboration or referencing although I demand originality of presentation. If you have a problem interpreting any question feel free to approach me, comment on the webpage or email.

Ensure to put your name, student number, module code (MS 3011), and your declaration on your homework.

## Tutorial Venue

I have applied to get this changed to WGB G03… I didn’t get everything. The timetable is

- This week coming 28 February LL2
- 7 March LL2
- 14 March WGB G03
- 21 March WGB G03
- 28 March WGB G03

## Week 6

We finished describing what a chaotic dynamical system is and began our study of the tent mapping

## What next?

We won’t be long finishing off our work on the tent mapping and then we will commence our third special study: of the doubling mapping.

## Tutorials

Exercises for Thursday 28 February are to look at the following. Not a whole pile of new stuff covered so some revision.

Autumn 2009 Q. 2(b), 4

Summer 2008, Q. 1(a), 2(a), (b), 3, 6

## 6 comments

Comments feed for this article

March 25, 2013 at 12:03 pm

Student 32Dear JP,

In relation to the project, number one of the projects, for the first part must we formulate our function?

I understand how to get an eventually periodic point from a infinite set but how do i get one from a finite set?

March 25, 2013 at 12:14 pm

J.P. McCarthyFor (a) (i) you must prove the statement for ALL points in any finite set and functions on that finite set. So no, you can’t pick a distinguished point, set or function.

You must do this abstractly by starting with: let be an element of a finite set and …

The idea is that because the ‘universe’ (of states) is so small (finite), complicated behaviour can not occur.

The corresponding statement for not-finite or infinite sets is the following:

“Suppose that is a not-necessarily finite set and . Then all of the points of ” are eventually periodic.

However this is NOT correct. Two counterexamples:

1. In the dynamical system given by the doubling mapping, the point that we constructed with a dense orbit,

is NOT eventually periodic.

2. In the dynamical system given by the squaring function on the complex numbers, . We might see on Wednesday that the point

is NOT eventually periodic when — i.e. when is irrational.

The statement is correct when is finite.

Regards,

J.P.

April 4, 2013 at 1:29 pm

Student 33Hi J.P,

I am doing the complex numbers question for the homework and I am not sure whether im answering the questions the way you want.

For 2010, question 3 b, I started by subtracting from to find and using this with the formula i was able to find a value for . I showed and the argument of on a graph.

I was just wondering is this enough information for the answers you are looking.

Thanks for your help.

April 4, 2013 at 1:30 pm

J.P. McCarthyThat sounds OK for question 3 (b) (i).

What about part (ii)

Regards,

J.P.

April 10, 2013 at 10:40 am

Student 34J.P,

We are doing the complex numbers questions for the project. For the 2012 Project Maths paper( question three) if we draw a geometric sketch for part (a) does this count as having answered part(b) which asks for a sketch on the argand diagram?

April 10, 2013 at 10:41 am

J.P. McCarthyYes.

Regards,

J.P.