Things are definitely getting more difficult now folks. I am happy with the work a lot of ye are putting in but from now on everyone will have to work hard if they want to succeed. The good news is that Easter might provide extra time to put into Chapter 3.

## Week 8

### Lectures

Two sections of Chapter 3:

If you are interested in a very “mathsy” approach to curves you can look at this. I have live video of some of the above material here.

### Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises.

When you are finished with the material for Test 2 (Weeks 4-6) you can try:

• p. 120, Q. 1-3
• p. 128, Q. 1-5 [Note these exercises are interleaved – there are questions here from earlier sections in Chapter 2]

Additional Exercises: p. 121, Q. 4-7, p. 130, Q.5-7

Submit work for Canvas feedback by Sunday 21 March for video feedback after Monday 22 March.

## Week 9

We will look at partial differentiation and its applications to error analysis.

Looking further ahead, a good revision of integration/antidifferentiation may be found here.

## Assessment Schedule

Week 5 – 25% Vectors Test

Week 8 – 25% Matrices Test (Zoom Tutorial in Week 7)

Week 11  – 25% Differentiation Test (Zoom Tutorial in Week 10)

Week 14 – 25% Integration Test (Zoom Tutorial in Week 13)

## Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

### Assessment Corrections

I cannot promise at this point when these will be completed. I am having a hardware issue at the moment which means that I might have to get the ball rolling for ye by correcting the Written Assessment first. Watch this space.

## Week 8

### Lectures

Some theory background for Chapter 2:

Intro to PDEs and Laplace’s Equation (39 minutes)

The first part of this on finite differences is important for the rest of the semester. The second, a handwaving derivation of Laplace’s Equation. It should give you an idea of what a PDE is, should help consolidate some “thermo”-type stuff, but is theoretical background.

There is a longer, more in-person (old) version of the above material: Intro to PDEs (less than 20 minutes), and Derivation of the Laplace Equation (40 minutes).

### Theory Exercises and Q & A

You might want to focus on labs that you didn’t get to look at yet (as well of course as Lab 6)

## Week 9

I have linked here to last year’s videos but will be recording fresh videos. How different they will be I am not too sure.

## Week 10

### Lectures

Perhaps of the order of 1.5 hours of lectures on the Heat Equation. We will then be finished the lecture material.

### Labs

We will do our last lab in Week 10.

## Assessment

This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

You have to put regular time to work on Chapter 3:

1. because without doing so you could be very, very lost on 35% Assignment 3, and
2. because if you are going into Level 8 Structural Engineering it will be assumed that you are competent with the Chapter 3 material

You need to get cracking on Chapter 3.

## Assignment 1 & 2 Corrections

I realise that ye are still waiting for these. Hopefully ye will not be waiting too long for these but I am not in a position to make any promises at this point.

## Week 8

### Lectures

You could need about an hour and three quarters to watch these:

### Exercises

If you are up to date on the Week 7 exercises, put some time into:

• p.127, Q.1-11

Submit work for Canvas feedback by Monday 22 March for video feedback after Tuesday 23 March.

## Outlook

In Weeks 9 and 10 you spend all your time doing Chapter 3 Exercises/Assignment 3. This is actually four weeks with the Easter break — and it is my intention to continue providing learning support throughout the Easter break.

Section 3.5 Systems of Differential Equations. is important for next year but I have messed up and so it will not be examined. I will record lectures and fill in the notes for ye though.

Weeks 11 and 12 are given over to Chapter 4. Perhaps of the order of 2 hours of lectures on Double Integrals in Week 11 and of the order of 2 hours of lectures on Triple Integrals. If I see that students have submitted Assignment 3 early I may record these earlier than Weeks 11 and 12.

## Assignments

35% Assignment 1 on Chapter 1 — due end of Week 5, 28 February.

15% Assignment 2 on Chapter 2 — due end of Week 7, Sunday 14 March

35% Assignment 3 on Chapter 3 — due end of Week 11, Sunday 25 April will be released soon

15% Assignment 4 on Chapter 4 — due end of Week 13/14/15, (7/14/21 May) tbd

## Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

I have been working for a number of months on a paper/essay based on this talk (edit: big mad long draft here). After the talk the seminar host Teo Banica suggested a number of things that the approach could be used to look at, and one of these was orbits and orbitals. These are nice, intuitive ideas introduced by Lupini, Mancinska (missing an accent), and Roberson introduced orbitals.

I went to Teo’s quantum permutations tome, and Chapter 13 (p.297) orbits and orbitals are introduced, and it is remarked that, where we are studying quantum permutation groups $\mathbb{G}< S_N^+$, a certain relation on $N\times N\times N$ is believed not to be transitive. This belief is expressed also in the fantastic nonlocal games and quantum permutations paper, as well as by Teo here.

One of the things that the paper has had me doing is using CAS to write up the magic unitaries for a number of group duals, and I said, hey, why don’t I try and see is there any counterexamples there. My study of the $\widehat{S_3} led me to believe there would be no counterexamples there. The next two to check would be $\widehat{S_4} and the dual of the quaternion group $\widehat{Q}. I didn’t get called JPQ by Professor Des MacHale for nothing… I had to look there. OK, time to explain what the hell I am talking about.

I guess ye will have to wait for the never-ending paper to see exactly how I think about quantum permutation groups… so for the moment I am going to assume that you know what compact matrix quantum groups… but maybe I can put in some of the new approach, which can be gleaned from the above talk, in bold italics. A quantum permutation group $\mathbb{G}\leq S_N^+$ is a compact matrix quantum group whose fundamental representation $u^{\mathbb{G}}\in M_N(C(\mathbb{G}))$ is a magic unitary. The relation that was believed not to be transitive is:

$(j_3,j_2,j_1)\sim_3 (i_3,i_2,i_1)\Leftrightarrow u_{j_3i_3}u_{j_2i_2}u_{j_1i_1}\neq 0$,

that is the indices are related when there is a quantum permutation $\varsigma$ that has a non-zero probability of mapping:

$(\varsigma(j_3)=i_3)\succ(\varsigma(j_2)=i_2)\succ (\varsigma(j_1)=i_1)$. (*)

This relation is reflexive and symmetric. If we work with the universal (or algebraic) level, then $e\in\mathbb{G}$ will fix all indices giving reflexivity, if a quantum permutation $\varsigma\in\mathbb{G}$ can map as per (*), it’s reverse $\varsigma^{-1}:=\varsigma\circ S$ will map, with equal probability of $\varsigma$ doing (*):

$(\varsigma^{-1}(i_3)=j_3)\succ(\varsigma^{-1}(i_2)=j_2)\succ (\varsigma^{-1}(i_1)=j_1)$,

so that $\sim_3$ is symmetric.

Now, to transitivity. We’re going to work with the algebra of functions on the dual of the quaternions, $F(\widehat{Q}):=\mathbb{C}Q$. Working here is absolutely fraught what with coefficients $i$ and $-1$ and elements of $Q$ of the same symbol. Therefore we will use the $\delta^g$ notation. Consider the following vector in $F(\widehat{Q})$:

$\displaystyle (u^{\widehat{\langle j\rangle}})_{,1}:=\frac{1}{4}\left[\begin{array}{c}\delta^1+\delta^j+\delta^{-1}+\delta^{ij}\\ \delta^1+i\delta^{j}-\delta^{-1}-i\delta^{-j} \\ \delta^1-\delta^{j}+\delta^{-1}-\delta^{-j} \\ \delta^{1}-i\delta^{j}-\delta^{-1}+i\delta^{-j}\end{array}\right]$.

This vector is the first column of a magic unitary $u^{\widehat{\langle j\rangle}}$ for $\widehat{\langle j\rangle}\cong \widehat{\mathbb{Z}_4}\cong \mathbb{Z}_4$, and the rest of the magic unitary is made by making a circulant matrix from this. Do the same with $k\in\widehat{Q}$, another magic unitary $u^{\widehat{\langle k\rangle}}$, and so we have $\widehat{Q} via:

$u^{\widehat{Q}}=\left(\begin{array}{cc}u^{\widehat{\langle j\rangle}} & 0 \\0 & u^{\widehat{\langle k\rangle}} \end{array}\right)$.

Now for the counterexample: $u^{\widehat{Q}}_{67}u^{\widehat{Q}}_{41}u^{\widehat{Q}}_{87}\neq 0$ so  $(6,4,8)\sim_3 (7,1,7)$ and $u_{78}^{\widehat{Q}}u_{14}^{\widehat{Q}}u_{78}^{\widehat{Q}}\neq0$ so $(7,1,7)\sim_3(8,4,8)$, but $u^{\widehat{Q}}_{68}u^{\widehat{Q}}_{44}u^{\widehat{Q}}_{88}=0$ so $(6,4,8)$ is not related to $(8,4,8)$ and so $\sim_3$ is not transitive.

That $u^{\widehat{Q}}_{68}u^{\widehat{Q}}_{44}u^{\widehat{Q}}_{88}=0$ is a bit of algebra, and I guess the others are too… but instead we can exhibit states $\varsigma_2,\,\varsigma_1\in S(F(\widehat{Q}))$ such that $\varsigma_2(|u^{\widehat{Q}}_{67}u^{\widehat{Q}}_{41}u^{\widehat{Q}}_{87}|^2)>0$ and $\varsigma_1(|u_{78}^{\widehat{Q}}u_{14}^{\widehat{Q}}u_{78}^{\widehat{Q}}|^2)>0$ instead. The algebra structure of $F(\widehat{Q})$ is:

$\displaystyle F(\widehat{Q})=\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus M_2(\mathbb{C})\subset B(\mathbb{C}^6)$.

Define $\varsigma_2$ to be the vector state associated with $\xi_2:=(0,0,0,0,1/\sqrt{2},1/\sqrt{2})$. Then:

$\|u^{\widehat{Q}}_{67}u^{\widehat{Q}}_{41}u^{\widehat{Q}}_{87}(\xi_2)\|^2=\frac14$.

$\varsigma_2\in\widehat{Q}$ is a quantum permutation such that:

$\mathbb{P}[(\varsigma_2(7)=6)\succ (\varsigma_2(1)=4)\succ (\varsigma_2(7)=8)]=\frac{1}{4}$.

Similarly the vector state $\varsigma_1$ given by $\xi_1:=(0,0,0,0,0,1)$ has

$\|u^{\widehat{Q}}_{78}u^{\widehat{Q}}_{14}u^{\widehat{Q}}_{78}(\xi_1)\|^2=:\mathbb{P}[(\varsigma_1(8)=7)\succ (\varsigma_1(4)=1)\succ (\varsigma_1(8)=7)]>0$.

Now, classically we might expect that $\varsigma_2\star \varsigma_1$ (convolution) might have the property that:

$(\varsigma_2\star\varsigma_1)(|u^{\widehat{Q}}_{68}u^{\widehat{Q}}_{44}u^{\widehat{Q}}_{88}|^2)>0$,

but as we have seen the product in question is zero.

Edit: The reason this phenomenon happens is that $u_{11}^{\widehat{Q_8}}$ and $u_{55}^{\widehat{Q_8}}$ are projections to random/classical permutations! I have also found a counterexample in the Kac-Paljutkin quantum group for similar reasons.

In the paper under preparation I think I should be able to produce nice, constructive, proofs of the transitivity of $\sim_1$ and $\sim_2$, constructive in the sense that in both cases I think I can exhibit states on $C(\mathbb{G})$ that are non-zero on suitable products of $u_{ij}$, using I think the conditioning of states:

$\displaystyle\varsigma\mapsto \frac{\varsigma(u_{ij}\cdot u_{ij})}{\varsigma(u_{ij})}$.

There is also something here to say about the maximality of $S_N. All must wait for this paper though (no I don’t have a proof of this)!

### Written Assessment 1 – Week 7

Next week’s Zoom will be Monday 15:00 instead of Tuesday.

25% Written Assessment 1, based on Weeks 1-5, so everything up to p.73.

It will be a one hour assessment, but I am going to give ye 15 minutes grace, as well as 15 minutes to upload. The test will run therefore from 09.30 to 11.00, Tuesday 9 March. It is open book — you can use your manual, any Canvas materials, as well as Excel/VBA.

The questions mentioned below are only a guide to the content not the actual questions.

40% of the marks will be numerical methods (Euler, TTT, Heun); finding approximations like p.34, Q.1-3, p.37, Q.1-3, p.48, Q.1-4 (non-Excel parts of Q.4), p.60, Q.4-7.

30% of the marks will be numerical analysis; understanding these methods and their errors like p.34, Q. 4, p.38, Q.4-6. p.49, Q.5-8,

15% will be neither; like p.17, Q. 1-2, p.40, Q.1-3, p.60, Q.1-3

15% will be Runge-Kutta, like p.72, Q.1-2.

Academic Dishonesty will not be accepted and suspected breaches, such as communication with others during the assessment, will be pursued in line with this policy (Links to an external site.)

## Week 7

### Lectures

Just a very short lecture (time is taken up by Written Test 1):

Goal Seek for Boundary Value Problems

## Week 8

I have linked here to last year’s videos but will be recording fresh videos. How different they will be I am not too sure.

We will look at Intro to PDEs (less than 20 minutes), and then watch the Derivation of the Laplace Equation (40 minutes).

## Assessment

This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

## Week 7

### Lectures

Some revision and some new material, need about two hours to carefully watch:

### Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises.

When you are happy with Chapter 2 (Weeks 4-6) you can try:

• p. 103, Q. 1-3
• p. 115, Q. 1-3

Additional Exercises: p. 116, Q. 4-7

Submit work for Canvas feedback by Sunday 14 March for video feedback after Monday 15 March.

## Matrices Test — Week 8, 19:30 16 March

We will have a Zoom Tuesday 8 March at 20:00 for any questions that ye would like to ask about this assessment. This tutorial will be recorded in the cloud.

## Week 8

We will look at Related Rates and then look at Implicit Differentiation. If you are interested in a very “mathsy” approach to curves you can look at this.

I have live video of the above material here.

## Week 9

We will look at partial differentiation and its applications to error analysis.

Looking further ahead, a good revision of integration/antidifferentiation may be found here.

## Assessment Schedule

Week 5 – 25% Vectors Test

Week 8 – 25% Matrices Test (Zoom Tutorial in Week 7)

Week 11  – 25% Differentiation Test (Zoom Tutorial in Week 10)

Week 14 – 25% Integration Test (Zoom Tutorial in Week 13)

## Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

You have to put regular time to work on Chapter 3:

1. because without doing so you could be very, very lost on 35% Assignment 3, and
2. because if you are going into Level 8 Structural Engineering it will be assumed that you are competent with the Chapter 3 material

You need to finish off Chapter 2 and Assignment 2 ASAP (only worth 15%) and then get cracking on Chapter 3.

## Week 7

### Lectures

You could need three hours to watch these 115 minutes lectures:

### Exercises

You need to finish off Assignment 2 and then start looking at Chapter 3 Exercises:

• p.105
• p.115, Q. 1-5

Submit work for Canvas feedback by Monday 15 March for video feedback after Tuesday 16 March.

## Outlook

Week 8 we are going to finish off the material for Assignment 3.

Then in Weeks 9 and 10 you spend all your time doing Chapter 3 Exercises/Assignment 3. This is actually four weeks with the Easter break — and it is my intention to continue providing learning support throughout the Easter break.

Weeks 11 and 12 are given over to Chapter 4.

## Assignments

35% Assignment 1 on Chapter 1 — due end of Week 5, 28 February.

15% Assignment 2 on Chapter 2 — due end of Week 7, Sunday 14 March

35% Assignment 3 on Chapter 3 — due end of Week 11, Sunday 25 April will be released when ye have enough material to complete

15% Assignment 4 on Chapter 4 — due end of Week 13/14/15, (7/14/21 May) tbd

## Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

## Week 6

### Lectures

Very little by way of lectures this week, so that you can put the bulk of your time towards catching up on Chapter 2

### Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises.

Assuming you have a handle on the exercises from Week 4 and Week 5 you can try:

• p. 84, Q.1-3
• p.94, Q. 1-6

Additional Exercises: p. 96, Q. 1-10

Submit work for Canvas feedback by Sunday 7 March for video feedback after Monday 8 March. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

## Matrices Test — Week 8, 19:30 16 March

The open-book assessment will be designed to be done in about 45 minutes, however you will be given one hour to complete the assignment along with an additional 15 minutes to upload your work. 45 minutes means about one and a half exam questions (see MATH6040 matrices questions (Links to an external site.) ( (usually matrices are Q. 2, sometimes Q. 1)) to get an idea of how long one exam question is). Please contact me if the timing is an issue.

The assessment is based on Chapter 2. There will be five questions broken into parts (a), (b), and (c). Some/most of the part (c)s should be easier than in the Vectors Test. Some of the part (a)s will be a little harder.

Additional practise questions (beyond the manual) may be found by looking at past MATH6040 exam papers (Links to an external site.).

Academic Dishonesty will not be accepted and suspected breaches, such as communication with others during the assessment, will be pursued in line with this policy (Links to an external site.)

We will have a Zoom Tuesday 8 March at 20:00 for any questions that ye would like to ask about this assessment. This tutorial will be recorded in the cloud.

## Week 7

We will do a quick revision of differentiation.  If you want to look ahead here are two videos:

Then we will look at Parametric Differentiation.

## Outlook

For most students, Chapters 1 and 2 are easier and you will want to do well on them. Things are going to get a little harder for the rest of the semester and you will want to try and do homework regularly.

## Assessment Schedule

Week 5 – 25% Vectors Test

Week 8 – 25% Matrices Test (Zoom Tutorial in Week 7)

Week 11  – 25% Differentiation Test (Zoom Tutorial in Week 10)

Week 14 – 25% Integration Test (Zoom Tutorial in Week 13)

## Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

## 25% Vectors Test, 19.30 Tuesday 23 February

This is just repeating some the information from Week 4:

The open-book assessment will be designed to be done in about 45 minutes, but you have an hour and 15 minutes including time for uploading.

The assessment is based on Chapter 1. The questions into parts (a) (easy), (b) (medium), and (c) (hard). You might be advised to do all the parts (a) and (b) first, try and get as close to 70% as possible with those, and then leave the parts (c) to the end. Otherwise you might waste time doing parts (c) when there are easier and more marks available in parts (b) and particularly (a).

Academic Dishonesty will not be accepted and suspected breaches, such as communication with others during the assessment, will be pursued in line with this policy.

We had a Zoom tutorial: it is in the cloud.

## Week 5

### Lectures

As promised very few lectures so that if you have been a little behind on Chapter 2 already you can catch up after Test 1

### Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises.

Assuming you are ready for the Vector Assessment, and have a handle on the Week 4 exercises try

p.79, Q.1-4

Submit work for Canvas feedback by Sunday 28 February for video feedback after Monday 1 March. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

## Week 6

We will finish Chapter 2 by talking about determinants and Cramer’s Rule.

## Outlook

In Week 7 we will start differentiation. For most students, Chapters 1 and 2 are easier and you will want to do well on them. Things are going to get a little harder for the rest of the semester and you will want to try and do homework regularly.

## Assessment Schedule

Week 5 – 25% Vectors Test

Week 8 – 25% Matrices Test

Week 11  – 25% Differentiation Test

Week 14 – 25% Integration Test

## Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

In theory you are supposed to spend seven hours per week on MATH6040: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH6040 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are three hours of MATH6040 slots on your timetable that you can use if you want. Even with just three hours you should have at least one hour to try and submit exercises for video feedback.

## 25% Vectors Test, 19.30 Tuesday 23 February

The open-book assessment will be designed to be done in about 45 minutes, however you will be given one hour to complete the assignment along with an additional 15 minutes to upload your work. 45 minutes means about one and a half exam questions (see MATH6040 vectors questions (usually vectors are Q. 1, sometimes Q. 2) to get an idea of how long one exam question is). Please contact me if the timing (19.30 Tuesday 23 February) is an issue.

The assessment is based on Chapter 1. As it is an open book assessment, I have decided to split the questions into parts (a), (b), and (c).

The parts (a) are easy, and worth 40% of the total mark. The parts (b) are of medium difficulty, and are worth 30% of the total mark. The parts (c) are more difficult and worth 30% of the marks. You might be advised to do all the parts (a) and (b) first, try and get as close to 70% as possible with those, and then leave the parts (c) to the end. Otherwise you might waste time doing parts (c) when there are easier and more marks available in parts (b) and particularly (a).

Additional practise questions (beyond the manual) may be found by looking at past MATH6040 exam papers.

Academic Dishonesty will not be accepted and suspected breaches, such as communication with others during the assessment, will be pursued in line with this policy. To make life easier for me in this regard your assessment will be student-number-personalised. In addition by submitting you will be pledging that you will undertake the assessment in good faith.

We will have a Zoom Tuesday 16 February at 20:00 for any questions that ye would like to ask about this assessment. This tutorial will be recorded in the cloud.

## Week 4

### Lectures

There are 116 minutes of lectures here. You should need about three hours to watch these (I recommend 50% extra time for pausing/rewinding)

Some deeper discussion here: Why do we multiply matrices like we do? Why can’t I divide by zero?

### Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises.

Assuming you are ready for the Vector Assessment, try

• p.66, Q. 1-4
• p.70, Q.1-2
• p.73, Q.1-3
• Additional Exercises, p.66, Q.1, p.73, Q.4-5

Submit work for Canvas feedback by Sunday 21 February for video feedback after Monday 22 February. Ideally you don’t submit work that you are certain is correct, but instead submit work you need help and feedback with. Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

## Week 5

You will have your test. In lectures, we will look at Linear Systems. We won’t do too much so you have time to revise either Chapter 1 or Week 4 exercises.