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 , a certain relation on 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 led me to believe there would be no counterexamples there. The next two to check would be and the dual of the quaternion group . I didn’t get called JP**Q** 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 is a compact matrix quantum group whose fundamental representation is a magic unitary. The relation that was believed not to be transitive is:

,

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

. (*)

This relation is reflexive and symmetric. *If we work with the universal (or algebraic) level, then will fix all indices giving reflexivity, if a quantum permutation can map as per (*), it’s reverse will map, with equal probability of doing (*):*

,

**so that is symmetric.**

Now, to transitivity. We’re going to work with the algebra of functions on the dual of the quaternions, . Working here is absolutely fraught what with coefficients and and elements of of the same symbol. Therefore we will use the notation. Consider the following vector in :

.

This vector is the first column of a magic unitary for , and the rest of the magic unitary is made by making a circulant matrix from this. Do the same with , another magic unitary , and so we have via:

.

Now for the counterexample: so and so , but so is not related to and so is not transitive.

That is a bit of algebra, and I guess the others are too… but instead we can exhibit states such that and instead. The algebra structure of is:

.

Define to be the vector state associated with . Then:

.

** is a quantum permutation such that:**

.

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

.

Now, classically we might expect that (convolution) might have the property that:

,

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

In the paper under preparation I think I should be able to produce nice, constructive, proofs of the transitivity of and , constructive in the sense that in both cases I think I can exhibit states on that are non-zero on suitable products of , using I think the conditioning of states:

.

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

]]>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.).

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

Goal Seek for Boundary Value Problems

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).*

This is provisional and subject to change.

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

Study should consist of

- doing exercises from the notes
- completing VBA exercises

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

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

- Revision of Differentiation (21 minutes)
- Operators/Transforms (11 minutes)
- Parametric Differentiation I (34 minutes)
- Parametric Differentiation II (15 minutes)

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.

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.

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.

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.

**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)*

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

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

]]>*because without doing so you could be very, very lost on 35% Assignment 3, and**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.*

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

- Partial Fractions III (24 minutes)
- Inverse Laplace Transforms I (28 minutes)
- Inverse Laplace Transforms II (38 minutes)
- Inverse Laplace Transforms III (11 minutes)
- Laplace Transforms for ODEs I (14 minutes)

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

- p.105
- p.115, Q. 1-5

Additional Exercises: p116, Q.6-7

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

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.

**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

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

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

- Matrix Determinants (22 minutes)
- Cramer’s Rule (27 minutes)

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.

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.

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.

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.

**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)*

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

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

]]>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*.

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

- Linear Systems (26 minutes)

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

Additional Exercises: p. 79, Q.5-6

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.

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

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.

**Week 5 **– 25% Vectors Test

**Week 8 **– 25% Matrices Test

**Week 11 ** – 25% Differentiation Test

**Week 14 **– 25% Integration Test

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

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

]]>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.

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

- Matrix Arithmetic Theory (34 minutes)
- Matrix Arithmetic Examples (27 minutes)
- Matrix Inverses (37 minutes)
- Matrix Equations (18 minutes)

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

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.

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.

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

]]>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 is an issue.

The assessment is based on Chapter 1. As it is an open book assessment, it is my intention to make the test a little on the hard side (in terms of the questions I can ask from Chapter 1). I hope to have about 40% of the marks going for straightforward/easier stuff, 30% of the marks for slightly harder stuff, and about 30% of the marks will be harder again. That is an intention not a promise.

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 you will be pledging that you

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.

There are few lectures this week meaning that you should have time to do and submit exercises. You should need only about an hour and a half to watch these (I recommend 50% extra time for pausing/rewinding)

Work done by a Force (26 minutes)

Moment of a Force (29 minutes)

Vectors Short Summary (5 minutes)

Assuming you have the Week 1 and Week 2 exercises done (maybe do these first), try

- p.44, Q. 1-6
- Additional Exercises, p.46, Q.7-13

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

We will start looking at Chapter 2: Matrices. We will some examples of matrix arithmetic and look at Matrix Inverses — “dividing” for Matrices. This will allow us to solve matrix equations.

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

]]>Hopefully I have designed something in which you can take your time, get things right, get some good marks, and show off what you can do. You are advised to complete VBA Lab 3 first.

*I recommend watching the Week 6 lectures, but perhaps it might suit students to complete VBA Assessment 1 before looking at Week 6 Exercises. Do not let VBA Assessment slide very late as you will need time to prepare for Written Assessment 1. Really you should be looking to complete VBA Assessment 1 a few days before the due date. There is no lab in Week 6 in order to give you more time.*

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.

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.

**MORE INFO: Added 25 February**

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.).

There are 61 minutes of lectures. You should schedule about an hour and a half to watch them and take the notes in your manual. You might need this extra time above 61 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A.

- Boundary Value Problems (28 minutes)
- Shooting Method (33 minutes)

If you have not yet had a chance to look at Lab 4, but have questions, you can submit to Lab 4 Second Chance.

p.82, Q. 1-3

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

There will not be much in Week 7, just a short lecture on Goal Seek for boundary value problems.

This is provisional and subject to change.

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

Study should consist of

- doing exercises from the notes
- completing VBA exercises

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

]]>Hopefully I have designed something in which you can take your time, get things right, get some good marks, and show off what you can do. You are advised to complete VBA Lab 3 first.

*I recommend watching the Week 5 lectures, but perhaps it might suit students to complete VBA Assessment 1 before looking at Lab 4 and Week 5 Exercises. Do not let VBA Assessment slide very late as you will need time to prepare for Written Assessment 1. Really you should be looking to complete VBA Assessment 1 a few days before the due date. There will be no lab in Week 6 in order to give you more time.*

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.

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.

More information about practicalities in Week 6.

There are 82 minutes of lectures. You could schedule about 2 hours to watch them and take the notes in your manual. You might need this extra time above 82 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A.

- Second Order ODEs – Case Studies (37 minutes)
- Runge-Kutta Methods I (28 minutes)
- Runge-Kutta Methods II (17 minutes)

Lab 4 may not be up until 18 February, and can be attempted after watching the lectures above (submission not live until 20 February).

p.60, Q.1-7

p.72, Q.1-3

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

We will look at boundary value problems (in particular the Shooting Method and Goal Seek).

There will be no lab in order to give you time to complete VBA Assessment 1.

This is provisional and subject to change.

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

Study should consist of

- doing exercises from the notes
- completing VBA exercises

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

]]>