In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises, and assignments if you are ready for them. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

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. You might want to complete Assignment 1, or submit more Chapter 1 exercises for feedback, before doing Chapter 2 exercises.

p.62, Q. 1-2

p.70, Q. 1-7

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

We will finish Chapter 2.

15% Assignment 2

Based on Chapter 2, should be available some time in Week 4, for submission in Week 7.

Study

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

Student Resources

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

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

As soon as I send this I am putting the finishing touches to your first assessment — it might be here by tomorrow, Wednesday 3 February.

If you want a deep dive into the temperature of a plate stuff there is a post here which explains — in great detail — how the ‘iterations’ are actually physically meaningful.

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: you may want to put time into the Week 1 and 2 exercises if you did not do so previously.

p.55, Q. 1-4

Submit work for Canvas feedback by Monday 15 February for video feedback after Tuesday 16 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 4

We will go deeper into Chapter 2

35% Assignment 1

Should be released Wednesday 3 February.

Study

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

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 MATH7021 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.

Week 2

Lectures

You should need about two hours to watch these (I recommend 50% extra time for pausing/rewinding)

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.

p.27, Q. 1-13 (you might have some of these done in Week 1)

p.38, Q.1-5

Additional Exercises, p.38, Q. 6-1

You may need the formula that the area of a triangle is equal to .

Submit work for Canvas feedback by Sunday 7 February for video feedback after Monday 8 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 3

We will look at the applications of vectors to work and moments

25% Test 1 in Week 5

I will probably design student-number-personalised assessments. Watch this space: official notice in Week 3.

Study

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

Student Resources

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

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material, then complete the lab, and if there is any time left over do the suggested theory exercises.

It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7016 slots on your timetable but remember that half an hour is taken up with the weekly Q&A. Learners should decide for themselves whether the weekly Q&A is helping their learning.

Week 2

Lectures

There are about 82 minutes of lectures. You could schedule about 2 hours and 10 minutes to watch them and take the notes in your manual. You 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 (starting Tuesday 2 February)

(Last year with the strike I recorded the same stuff in a classroom)

The Q&A is dedicated to answering questions but do not hesitate to contact me with questions at any time. My usual modus operandi is to answer emails every morning.

Lab

Once you have watched the lectures you should attempt VBA Lab 2.

Theory Exercises and Q & A

p.34, Q.1-4

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

Week 3

We will do more on Taylor Series and the Euler Method. When we have that done we will look at Huen’s Method.

In VBA we will finish off our Euler Method Lab.

Assessment

I really have not yet thought about assessment but this is what I am thinking at the moment. 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

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..

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

Week 2

Lectures

You should need about two and half hours to watch these

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.

p.30, Q. 1-11 (you might have some of these done in Week 1)

p.42, Q. 1-3

p.48, Q. 1-3

Additional Exercises, p.33, Q. 12-22

Submit work for Canvas feedback by Monday 8 February for video feedback after Tuesday 9 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 3

We will finish Chapter 1 and make a start to Chapter 2.

35% Assignment 1

Watch this space — I might release it before I have a definite due date.

Study

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

Student Resources

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

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7021. We have voted for the OFFLINE model: my recommendation is that you watch the lecture material, then spend whatever other time you have for MATH7021 on exercises. It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7021 slots on your timetable that you can use if you want.

Week 1

Lectures

We had one hour of Live Zoom. This should be recorded in “Zoom” on the left there. Updated annotated notes.here

You should watch this hour and 23 minutes of lecture video this week:

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.

It might be the case that you need to see more examples (in Week 2 lectures… probably available by Friday evening) before attempting these. On the other hand, if you can do these already you are in a great position in MATH7021.

p.30, Q. 1-11

Additional Exercises, p.33, Q. 12-22

Submit work for Canvas feedback by Monday 1 February for video feedback after Tuesday 2 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 2

We will finish the Gaussian Elimination examples and begin to look at applications of linear systems to traffic and pipe flow.

35% Assignment 1

I may release Assignment 1 before I know the hand-in date… hopefully there will be some coordination on this quite soon.

Study

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

Student Resources

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

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH6040. I know this might not be possible for everyone. My recommendation is that you watch the lecture material (n/a in Week 1 if you attended the live lecture), then spend time doing exercises — how long will depend on the…

Delivery Model Vote

Please read here for more on this. Vote here BEFORE 12:00 Thursday 28 January.

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material (n/a in Week 1 if you attended the live lectures), then complete the lab, and if there is any time left over do the suggested theory exercises.

Week 1

Lectures

We had two hours of Live Zoom. These should be recorded in “Zoom” on the left there.

Where you find the time to do your VBA Lab 1 is up to you but I recommend using the two hours on your timetable. If you are serious about MATH7016 you might need to spend more than two hours.

Theory Exercises and Q & A

p.17, Q.1-2

Q & A on these and any other MATH7016 topic next Tuesday 12.30 (waiting room open 12.25).

I am currently (slowly) working on an essay/paper where I expand upon the ideas in this talk. In this post I will try and explain in this framework why there is no quantum cyclic group, no quantum , and ask why there is no quantum alternating group.

Quantum Permutations Basics

Let be a unital -algebra. We say that a matrix is a magic unitary if each entry is a projection , and each row and column of is a partition of unity, that is:

.

It is necessarily the case (but not for *-algebras) that elements along the same row or column are orthogonal:

and .

Shuzou Wang defined the algebra of continuous functions on the quantum permutation group on symbols to be the universal -algebra generated by an magic unitary . Together with (leaning heavily on the universal property) the *-homomorphism:

Any compact matrix quantum group generated by a magic unitary is a quantum permutation group in that it is a quantum subgroup of the quantum permutation group. There are finite quantum groups (finite dimensional algebra of functions) which are not quantum permutation groups and so Cayley’s Theorem does not hold for quantum groups. I think this is because we can have quantum groups which act on algebras such as rather than — the algebra of functions equivalent of the finite set .

This is all basic for quantum group theorists and probably unmotivated for everyone else. There are traditional motivations as to why such objects should be considered algebras of functions on quantum groups:

find a presentation of an algebra of continuous functions on a group, , as a commutative universal -algebra. Study the the same object liberated by dropping commutativity. Call this the quantum or free version of , .

quotient by the commutator ideal, that is we look at the commutative algebra generated by an magic unitary. It is isomorphic to , the algebra of functions on (classical) .

every commutative algebra of continuous functions on a compact matrix quantum group is the algebra of functions on a (classical) compact matrix group, etc.

Here I want to take a very different direction which while motivationally rich might be mathematically poor.

Weaver Philosophy

Take a quantum permutation group and represent the algebra of functions as bounded operators on a Hilbert space . Consider a norm-one element as a quantum permutation. We study the properties of the quantum permutation by making a series of measurements using self-adjoint elements of .

Suppose we have a finite-spectrum, self-adjoint measurement . It’s spectral decomposition gives a partition of unity . The measurement of with gives the value with probability:

,

and we have the expectation:

.

What happens if the measurement of with yields (which can only happen if )? Then we have some wavefunction collapse of

.

Now we can keep playing the game by taking further measurements. Notationally it is easier to describe what is happening if we work with projections (but straightforward to see what happens with finite-spectrum measurements). At this point let me quote from the essay/paper under preparation:

Suppose that the “event” has been observed so that the state is now . Note this is only possible if is non-null in the sense that

The probability that measurement produces , and , is:

Define now the event , said “given the state , is measured to be after is measured to be “. Assuming that is non-null, using the expression above a probability can be ascribed to this event:

Inductively, for a finite number of projections , and :

In general, and so

and this helps interpret that and are not simultaneously observable. However the sequential projection measurement is “observable” in the sense that it resembles random variables with values in . Inductively the sequential projection measurement resembles a -valued random variable, and

If and do commute, they share an orthonormal eigenbasis, and it can be interpreted that they can “agree” on what they “see” when they “look” at , and can thus be determined simultaneously. Alternatively, if they commute then the distributions of and are equal in the sense that

it doesn’t matter what order they are measured in, the outputs of the measurements can be multiplied together, and this observable can be called .

Consider the (classical) permutation group or moreover its algebra of functions . The elements of can be represented as bounded operators on , and the algebra is generated by a magic unitary where:

.

Here (‘unrepresented’) that asks of … do you send ? One for yes, zero for no.

Recall that the product of commuting projections is a projection, and so as is commutative, products such as:

,

There are, of, course, such projections, they form a partition of unity themselves, and thus we can build a measurement that will identify a random permutation and leave it equal to some after measurement. This is the essence of classical… all we have to do is enumerate and measure using:

.

A quantum permutation meanwhile is impossible to pin down in such a way. As an example, consider the Kac-Paljutkin quantum group of order eight which can be represented as . Take . Then

.

If you think for a moment this cannot happen classically, and the issue is that we cannot know simultaneously if and … and if we cannot know this simultaneously we cannot pin down to a single element of .

No Quantum Cyclic Group

Suppose that is a quantum permutation (in ). We can measure where the quantum permutation sends, say, one to. We simply form the self-adjoint element:

.

The measurement will produce some … but if is supposed to represent some “quantum cyclic permutation” then we already know the values of from , and so, after measurement,

, .

The significance of the intersection is that whatever representation of we have, we find these subspaces to be -invariant, and can be taken to be one-dimensional.

I believe this explains why there is no quantum cyclic group.

Question 1

Can we use a similar argument to show that there is no quantum version of any abelian group? Perhaps using together with the structure theorem for finite abelian groups?

No Quantum

Let be represented as bounded operators on a Hilbert space . Let . Consider the random variable

.

Assume without loss of generality that then measuring with gives with probability , and the quantum permutation projects to:

.

Now consider (for any , using the fact that and the rows and columns of are partitions of unity:

(*)

Now suppose, again without loss of generality, that measurement of with produces , then we have projection to . Now let us find the Birkhoff slice of this. First of all, as has just been observed it looks like:

In light of (*), let us find . First let us normalise correctly to

So

Now use (*):

,

and as maps to doubly stochastic matrices we find that is equal to the permutation matrix .

Not convincing? Fair enough, here is proper proof inspired by the above:

Let us show . Fix a Hilbert space representation and let .

The basic idea of the proof is, as above, to realise that once a quantum permutation is observed sending, say, , the fates of and are entangled: if you see you know that .

This is the conceptional side of the proof.

Consider which is equal to both:

.

This is the manifestation of, if you know , then two and one are entangled. Similarly we can show that and .

Now write

.

Similarly,

Which is equal to , that is and commute.

Question 2

Is it true that if every quantum permutation in a can be fully described using some combination of -measurements, then the quantum permutation group is classical? I believe this to be true.

Quantum Alternating Group

Freslon, Teyssier, and Wang state that there is no quantum alternating group. Can we use the ideas from above to explain why this is so? Perhaps for .

A possible plan of attack is to use the number of fixed points, , and perhaps show that commutes with . If you know these two simultaneously you nearly know the permutation. Just for completeness let us do this with :

tr(u)\x(1)

1

2

3

4

0

–

(12)(34)

(13)(24)

(14)(23)

1

(234),(243)

(134),(143)

(124),(142)

(123),(132)

4

e

–

–

–

The problem is that we cannot assume that that the spectrum of is , and, euh, the obvious fact that it doesn’t actually work.

What is more promising is

x(1)\x(2)

1

2

3

4

1

–

e

(234)

(243)

2

(12)(34)

–

(123)

(124)

3

(132)

(134)

–

(13)(24)

4

(142)

(143)

(14)(23)

–

However while the spectrums of x(1) and x(2) are cool (both in ), they do not commute.

Question 3

Are there some measurements that can identify an element of and via a positive answer to Question 3 explain why there is no quantum ? Can this be generalised to .

There is only a mini-chapter on examples of (real) functions. You should spend the rest of the time revising Chapters 3 and 4 (i.e. doing exercises). Some of you might have to catch up on the Week 8 lectures.

Week 10: 14 to 20 December.

There will be feedback on submitted exercises Wednesday, Thursday, and Friday.

I plan that we will have a number of Zoom tutorials (these are provisional):

Tuesday 14:00

Tuesday 15:00 C-X only

Wednesday 12:00 C-Y only

Thursday 12:00

There is a small chance that we will also have Monday 14:00 Zoom but will not know until closer to the day.

Week 11 – 40% Test 3

The final, 40% Test 3 on Functions and Algebra (chapters 3-5) will take place (provisionally) at 18:00-19:30 (perhaps with additional upload time) on Monday 21 December. There will be no more feedback on submitted exercises but there might be a 14:00 Zoom before the final Test.

30% Test 2 – Results

I had said that my aim was to have these to you before Wednesday 9 December. However there is a possibility now that the Test 2 results will not be released until after you sit Test 3. This is outside my control (watch this space).

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