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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.

## Homework

The closing date is 11 April at 2 pm. Work handed up late will be assigned a mark of zero. Note what the homework says”

Answer the following using geometric arguments.

## Week 12

We finished on Monday by talking a little bit more about taking roots of complex numbers.

## Week 13 Review Week (21-25 April)

I will book a two hour slot and we will first go through the layout of the paper and then I will answer any questions that ye might have.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

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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.

## Homework

The closing date is tomorrow three weeks: that is 11 April at 2 pm. Work handed up late will be assigned a mark of zero.

## Week 11

More on complex numbers including De Moivre’s Theorem and taking roots.

## Week 12

On Monday we will talk a little bit more about taking roots of complex numbers and also do an actual example. This will wrap things up in terms of lectures.

On Wednesday we will have an additional tutorial so that we can get through some more examples. As normal there will be a tutorial on Thursday (and a review lecture after the break).

## Exercises

For the Week 12 tutorial you should look back over the exercises in general.

If you are happy with them then maybe you could have a look at some of the harder applied problems — which are definitely not examinable but will test your understanding.

If you are doing abstract algebra Q.1 is a good, tough exercise. Q. 6 (h) is another good question bringing in complex numbers.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

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.

## Homework

Will be discussed in class on Monday.

## Week 10

On Monday we showed that to multiply one complex number $z_1$ by another $z_2$ involves stretching $z_1$ by a factor of $|z_2|$ and then rotating through an angle $\text{arg }z_2$. On Tuesday we used this information to find roots of unity. We also spoke about the Conjugate Root Theorem.

## Week 11

More on complex numbers including De Moivre’s Theorem and taking roots.

## Exercises

For the Week 11 tutorial you should look at Q.54 and 61-63.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this conjecture about calculating $i^i$.

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.

## Homework

Will be discussed in class on Monday.

## Week 9

We finished our study of the Doubling Mapping and started talking about complex numbers and how they arise both algebraically and geometrically.

## Week 10

More on complex numbers.

## Exercises

For the Week 10 tutorial you should look at Q.44, 45, 52, 53, 56-60. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this conjecture about solving a complex numbers problem geometrically.

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.

## MCQ Results

As discussed previously, the MCQ results will come out the day you hand in the homework: 11 April 2014. I will send the homework next week.

## Week 8

We finished describing Chaos Theory and began proving that the Doubling Mapping is chaotic.

## Week 9

We will finish our study of the Doubling Mapping and possibly start talking about complex numbers.

## Exercises

I have emailed ye a copy of the exercises. For the Week 9 tutorial you should look at Q.46 and Q. 47. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this conjecture about the iterates of the tangent function.

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.

## MCQ Announcement

I have decided to make the MCQ open book and also ye will have 50 minutes and not 30 minutes as previously stated.

• Do bring in paper for you to tease out questions
• Do expect to do just as good without notes as with notes
• Don’t expect it take you 50 minutes to do the test. You can leave early or if you want get some study in

The Concept MCQ will take place this Wednesday 26 February Week 8 in WGB G05. It starts at 10.05 and runs until 10.55.

## Week 7

In Week 7 we finished of our study of the Tent Mapping and began talking about Chaos Theory.

## Week 8

In Week 8 we will begin our study of the Doubling Mapping.

## Exercises

I have emailed ye a copy of the exercises. For the Week 8 tutorial you should look at Q.48 and Q. 49. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## Test and Other CA

The results and solutions of the test were emailed to the class.

The Concept MCQ will take place on Wednesday 26 February Week 8 in WGB G05.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will be on complex numbers and won’t be as long as last year’s homework (MS2001 nor MS3011).

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this result about the periodic points of the tent mapping.

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.

## Week 6

We finished our study of the Logistic Family by analysing when zero was attracting/repelling and when $\displaystyle \frac{\mu-1}{\mu}$ was attracting/repelling. We summarised our results in a bifurcation diagram

The green line corresponds to attracting fixed points and the red, dashed line to repelling fixed points. The graph is of fixed points vs $\mu$.

The fact that there are no attracting fixed points for $\mu>3$ indicates that the behaviour is more complicated when the growth rate, $\mu$, gets large. We could have periodic behaviour and perhaps more strange, chaotic behaviour.

We studied therefore the case where $\mu=4$. We said that for $\mu=4$$Q_\mu$ is symmetric about $x=1/2$ and is unimodal. We showed that  $Q_4^n$ has $2^{n-1}$ branches and hence $2^n$ period-$n$ points.

## Week 7

In Week 7 we will finish of our study of the Tent Mapping and perhaps begin our study of the Doubling Mapping.

## Exercises

I have emailed ye a copy of the exercises and ye have looked at questions 38, 39, 42, 43 for the Week 6 tutorial (assuming that ye were ready for Wednesday’s Test). For the Week 7 tutorial you should look at these first and then backtrack and look at questions 30-37.

## Test and Other CA

The results of the test will be revealed in time.

The Concept MCQ will take place on Wednesday 26 February Week 8 in WGB G05. It will be a half hour test and starts at 10.25 and runs until 10.55.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about finding a formula for the iterates of $Q_1$(x).

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.

## Week 5

We began our study of the Logistic Family. We postulated the equation as a model of population growth with two assumptions and ended up with

$Q_\mu(x)=\mu x(1-x)$

where $x\in[0,1]$ can be interpreted as the proportion of a maximum population with a growth rate $\mu\in[0,4]$.  We began analysing when zero was attracting/repelling and when $\displaystyle \frac{\mu-1}{\mu}$ was attracting/repelling. We were rudely interrupted by the fire drill!

http://kkcb.com/one-of-the-best-scenes-of-the-office-is-dwights-fire-drill/

## Week 6

In Week 6 we will finish of our study of the Logistic Mapping and perhaps begin our study of the Tent Mapping.

## Exercises

I have emailed ye a copy of the exercises and ye should be able to look at these questions 38, 39, 42, 43 for the Week 6 tutorial (assuming that ye are ready for Wednesday’s Test).

## Test and Other CA

The test will take place on February 12 in Week 6.

Everything up to but not including section 3.4 in the typeset notes is examinable. We have everything for the test covered. Please find a sample test and the test I gave last year after page 31. in these typeset notes.

The following theorems from the notes are examinable: the very bottom of page 12 and start of page 13. Also the Fixed-Point Factor-Theorem (which was called such on the board. It is not in the notes but is found here. When I say examinable you should be able to

• state the theorem
• prove the theorem
• understand the theorem and the proof

Learning off the proof letter by letter won’t do you!

The Concept MCQ will take place in Week 8.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about finding a formula for the iterates of $Q_1$(x).

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.

## Week 4

On Monday we spoke about cobweb diagrams that are a graphical method of locating fixed points and determining whether they are attracting, repelling or indifferent. At this point you should look at Section 2.4 of these notes (NOT mine).

In Section 3.1, 3.2 & 3.3, the cobweb diagrams suggest that if $|f'|<1$ ($f'$ aka the slope) at the fixed point that the point is attracting and if $|f'|>1$ the fixed point is repelling.

A normal person would have proved this as per page 25 of the notes but I wanted to show you the beautiful contraction mapping principle and do it that way… we did conclude, loosely but correctly,

### Theorem The: Contraction Mapping Principle

If $f:I\rightarrow I$ is a contraction on a closed interval $I$then $f$ has a unique, attracting fixed point in $I$.

I mainly wanted to show ye this for the nice pictures! There is one here that I got off the internet but I think our blobs were better:

O.K.

In the end what I did was a hodge-podge of the (correct) contraction mapping principle we did on Monday with the proof in the notes of the Fixed Point Dynamics Theorem below which isn’t really satisfactory to me. For next year I will probably try and take this approach but as my proof had a few holes, I am just going to say that you need to know the following:

### Theorem: Fixed Point Dynamics

Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is an iterator function with a fixed point at $x_f\in \mathbb{R}$. If $f$ is differentiable in an interval $I$ containing $x_f$ then

• If $|f'(x_f)|<1$then $x_f$ is an attracting fixed point
• If $|f'(x_f)|>1$then $x_f$ is a repelling fixed point
• If $|f'(x_f)|=1$then we can make no conclusion and we call $x_f$ an indifferent or neutral fixed point

### Week 5

In Week 5 we will begin our study of the Logistic Mapping — rabbits!

## Exercises

I have emailed ye a copy of the exercises and ye should be able to look at these questions for the Week 5 tutorial.

• 19 — a good question for more understanding for test
• 23-25, 27-29 — good basic practise for the test. Do 25 and you can maybe leave the others
• 26 — more theoretical than other questions of that type. You need to understand parts (a) to (c). These are the three theorems whose proofs are examinable.

The rest of these are not going to be on the test but the exam. If you forget them now don’t forget to look at before the exam to boost understanding. The theory is the same as before

• 30-32 — not examinable on the test but have occurred in Q.2 of the exam. Week 4 theory: same as as 23-25, 27-29
• 33 — full analysis of the dynamical system required using theory of Week 4.
• 34-34 — given the orbit. Find the iterator function and analyse using theory of Week 4.
• 36-37 — Newton-Raphson method is a dynamical system. Analyse using Week 4 theory.

As there are a lot of questions it might make sense to allocate so much time and say do (A)s first, then (B)s then (C)s or whatever.

## Test and Other CA

The test will take place on February 12 in Week 6. Everything up to but not including section 3.4 in the typeset notes is examinable: we will have this covered by Februaray 3 but probably January 29. I have emailed ye a copy of a sample test.

The Concept MCQ will still take place in Week 8.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about where the OP didn’t understand why roots of $f(x)-x$ are roots of $f^2(x)-x$.

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.

## Important Tutorial Announcement

If you can’t make the tutorial on account of a clash please email me with the module code of the module the tutorial is clashing with.

## Question 13 (c) from Tutorial

Solution: We say that $y$ is eventually fixed point of $g$ if some (finite) iterate of $y$, say $g^{N}(x)$ is a fixed point.

Now suppose that $y$ is eventually fixed, at say $x_f=g^N(y)$ so that the orbit of $y$ is

$\text{orb}(y)=\{y,g(y),\dots,g^N(y)=x_f,x_f,x_f,\dots\}$.

Now by part (b) the orbit of $x_f$ under $g^{-1}$ is

$\text{orb}(x_f)=\{x_f,x_f,\dots,y\}$.

However by part (a), $x_f$ is also a fixed point for $g^{-1}$ so it follows that

$\text{orb}(x_f,)=\{x_f,x_f,\dots,x_f\}\Rightarrow x_f=y$,

that is $y$ is a fixed point of $g$ $\bullet$

## Week 3

On Monday we proved two facts about periodic orbits (on the  bottom of p.12 and the top of p.13 in the course notes)

On Wednesday we learnt how to find the period-2 points of a polynomial mapping. Finding periodic points, say period-2 points means finding points $x\in S$ such that if we apply the iterator function twice, then we get back to $x$:

$f(f(x))=f^2(x)=x$.

Solving this equation is not necessarily that easy but we proved that if $f:S\rightarrow S$, then the fixed-point factor-theorem applies: $f(x)-x$ divides into $f^2(x)-x$ and this helps immensely.

We also learnt how to find eventually fixed points.

## Week 4

In Week 4 we will study  attracting fixed points.

## Exercises

I have emailed ye a copy of the exercises and ye should be able to look at questions

• 10, 12(17), 16 – 18, 20 – 22
• 13 is hard
• 14 & 15 were done in Monday’s lecture

As there are a lot of questions it might make sense to allocate so much time and say do (A)s first, then (B)s then (C)s or whatever.

## Test Postponement and Other CA Information

To give ye adequate time to prepare, the test will take place on February 12 in Week 6. Everything up to but not including section 3.4 in the typeset notes is examinable: we will have this covered by Februaray 3 but probably January 29. I have emailed ye a copy of a sample test.

The Concept MCQ will still take place in Week 8. I have decided not to give ye a sample and I might make it a half hour test rather than an hour. The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about where the OP didn’t understand why roots of $f(x)-x$ are roots of $f^2(x)-x$.