Well… I have an inkling that because dual groups satisfy what I would call the condition of abelianness (under the ‘quantisation’ functor), all (quantum) subgroups are normal… this is probably an obvious thing to write down (although I must search the literature) to ensure it is indeed known (or is untrue?). **Edit: **Wang had it already, see the last proposition here.

Let be a the algebra of functions on a finite classical (as opposed to quantum) group . This has the structure of both an algebra and a coalgebra, with an appropriate relationship between these two structures. By taking the dual, we get the *group algebra**,* . The dual of the pointwise-multiplication in is a coproduct for the algebra of functions on the dual group … this is all well known stuff.

Recall that the set of probabilities on a finite quantum group is the set of states , and this lives in the dual, and the dual of is , and so probabilities on are functions on . To be positive is to be positive definite, and to be normalised to one is to have .

The ‘simplicity’ of the coproduct,

,

means that for ,

,

so that, inductively, is equal to the (pointwise-multiplication power) .

The Haar state on is equal to:

,

and therefore necessary and sufficient conditions for the convergence of is that is *strict. *It can be shown that for any that . Strictness is that this is a strict inequality for , in which case it is obvious that .

Here is a finite version of Freslon’s result which holds for discrete groups.

*Let be a probability on the dual of finite group. The random walk generated by is ergodic if and only if is not-concentrated on a character on a non-trivial subgroup .*

Freslon’s proof passes through the following equivalent condition:

*The random walk on driven by is *not *ergodic if is *bimodular*with respect to a non-trivial subgroup , in the sense that*

.

Before looking at the proof proper, we might note what happens when is abelian, in which case is a classical group, the set of characters on .

To every positive definite function , we can associate a probability such that:

.

This is Bochner’s Theorem for finite abelian groups. This implies that positive definite functions on finite abelian groups are exactly convex combinations of characters.

Freslon’s condition says that to be not ergodic, must be a character on a non-trivial subgroup . Such characters can be extended in ways.

Therefore, if is not ergodic, .

For , we have

,

dividing both sides by yields:

.

As , and , this implies that is supported on characters such that, for all :

,

such that . The set of such is the annihilator of in , and it is a subgroup. Therefore is concentrated on the coset of a normal subgroup (as all subgroups of an abelian group are normal).

This, via Pontragin duality, is not looking at the ‘support’ of , but rather of . Although we denote , and when is abelian, is a group (unnaturally, of characters) isomorphic to . Is it the case though that,

gives the same object in as

?

Well… of course this is true because .

We could proceed to look at Fagnola & Pellicer’s work but first let us prove Freslon’s result, hopefully in the finite case the analysis disappears…

*Proof: *Assume that is not strict and let

.

There exists a unitary representation and a unit vector such that

Cauchy-Schwarz implies that

.

If is not strict there is an such that this is an inequality and so is colinear to , it follows that .

This implies for and :

,

and so is closed under multiplication. Also and so and so is a subgroup. It follows that is a character on , which is not trivial because is not strict.

I don’t really need to go through the third equivalent condition. If coincides with a character on a subgroup , for

,

and so is not strict

Now let us look at the language of Fagnola and Pellicer. What is a projection in ? First note the involution in is . The second multiplication is the convolution. This means projections in the algebra are symmetric with respect to the group inverses and they are also idempotents. They are actually equal to Haar states on finite subgroups.

I think periodicity is also wrapped up in Freslon’s result as I think all subgroups of dual groups are normal. Perhaps, oddly, not being concentrated on a subgroup means that the positive function (probability on the dual) is one on that subgroup…

Well… let us start with irreducible. Suppose fails to be ergodic because it is irreducible. This means there is a projection such that that (and support less than ?)

Let us look at the first condition:

.

What now is the support of ? Well… some work I have done offline shows that the special projections, the group-like projections, correspond to to for a subgroup of . If is reducible, it is concentrated on such a quasi-subgroup, and this means that coincides with a *trivial *character on . In terms of Fagnola Pellicer, .

Now let us tackle aperiodicity. It is going to correspond, I think, with being concentrated on a ‘coset’ of a non-trivial character on …

Well, we can show that if is periodic, there is a subset such that for all . We can use Freslon’s proof to show that is in a subgroup on which .

Now what I want to do is put this in the language of ‘inclusion’ matrices… but the inclusions for cocommutative quantum groups are trivial so no go…

We can reduce Freslon’s conditions down to irreducible and aperiodic: not coinciding with a trivial character, and not coinciding with a character.

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- The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
- Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
- For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

- have I answered the question that
__was asked__? - does my answer make sense? If no, say so, and then try and fix your solution.
- check your answer (e.g. if you are looking at a general true, look at a special case; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

You are invited to give your feedback on my teaching and this module here.

We will finish our study of Graph Theory by looking at Eulerian graphs, Hamiltonian graphs, and Dirac’s Theorem.

We had the test on Tuesday.

On Friday we had a look at this graph, a Chapter 4 question:

We will have two review lectures on Monday and Tuesday in the usual times and venues and tutorials as normal. Tutorials on Week 13 at the usual times and venue. As long as there are not too many people in a tutorial (max 18), you can attend both tutorials if you want.

We will go through the MATH6055 Winter 2018 paper, and then I will answer your questions if there are any. If there are none I will help one-to-one. Usual class times and locations.

Have been emailed to you. I will have the tests with me in the Tuesday and Wednesday classes.

These are not always found in your programme selection — most of the time you will have to look here.

Autumn 2019 is relevant. Two sample papers here:

Both Sample Exams should be considered under the following understanding:

__This sample has been drafted to give you an idea of the MATH6055 LAYOUT and is no indication of the specifics, difficulty or length of any individual questions.__

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>

- The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
- Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
- For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

- have I answered the question that
__was asked__? - does my answer make sense? If no, say so, and then try and fix your solution.
- check your answer (e.g. for a fitted curve or beam function, input values and see do they make sense; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

You are invited to give your feedback on my teaching and this module here.

I am not sure what is going on with Leonard’s classes but I will be present this Friday, tomorrow 6 December for tutorials in A243L (11:00) and A213B (12:00).

On Monday we finished off Chapter 4 by looking at Error Analysis. Better exercises than the book here (including corrections to the sheet handed out in class). We then had over two and a half hours of tutorial time for lectures, and another tutorial on Friday.

The exam is on the morning of Friday Week 13 which means that I cannot give ye (Friday) tutorials in Week 13.

In the lecture slots (Monday, Wednesday, Thursday), I will go through Q. 1-2, 4 of Sample Paper I. Then I will answer any questions. If there are none I will help one-to-one. Usual class times and locations.

Any remaining time can be given over to tutorials, including questions about Q. 3 of Sample Papers I and II.

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

These are not always found in your programme selection — most of the time you will have to look here.

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>

- The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
- Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
- For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

- have I answered the question that
__was asked__? - does my answer make sense?
- check your answer (e.g. differentiate/antidifferentiate an antiderivative/derivative, substitute your solution into equations, check your answer against a rough estimate, or what a picture is telling you, etc)

You are invited to give your feedback on my teaching and this module here.

Have been emailed to you along with final CA results.

We looked at centroids of laminas and centres of gravity of solids of revolution.

We had one and a half classes of tutorial time.

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

- Monday 16:00
- Tuesday 09:00
- Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

- Monday at 09:00 in B180
- Monday at 17:00 in B189
- Wednesday at 10:00 in F1. 3

The exam is on Friday 13 December. Past exam papers (MATH6040 runs in Semester 1 and Semester 2) may be found here.

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

Please feel free to ask me questions about the exercises via email or even better on this webpage.

These are not always found in your programme selection — most of the time you will have to look here.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>You are invited to give your feedback on my teaching and this module here.

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

If you go into Canvas, and go into MATH6055 and the ‘Algebra’ unit, you might see online practise questions for Test 2.

On Monday we half-finished the Examples of Functions (we will finish this off on Friday) mini-chapter before starting the final chapter, the easy chapter, on Network (Graph) Theory.

We will finish our study of Graph Theory by looking at Eulerian graphs, Hamiltonian graphs, and Dirac’s Theorem.

The Test is on Tuesday.

We will have two review lectures on Monday and Tuesday in the usual times and venues and tutorials as normal.

We will go through the MATH6055 Winter 2018 paper, and then I will answer your questions if there are any. If there are none I will help one-to-one. Usual class times and locations.

Two sample papers here:

Both Sample Exams should be considered under the following understanding:

__This sample has been drafted to give you an idea of the MATH6055 LAYOUT and is no indication of the specifics, difficulty or length of any individual questions.__

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

]]>

You are invited to give your feedback on my teaching and this module here.

I am not sure what is going on with Leonard’s classes but I will be present this Friday and next Friday for tutorials in A243L (11:00) and A213B (12:00).

These have been sent to you. Comments to follow.

We had a tutorial dedicated to differentiation on Monday.

We tried to look at a Probability and Statistics Word Cloud on Wednesday. It was a disaster as my computer froze… anyway:

My answer: *when and are *independent.

My answer: *Always*

My answer: *bell-shaped curve.*

A difficult one. My answer: *Every probability/area under a bell curve can be calculated by transforming the area to a curve, and calculating the area ‘there’.*

My answer: *To *infer *things about the population, e.g. the population mean.*

My answer: *An interval that we believe the population mean is in (with a certain confidence).*

On Wednesday and Thursday we had a look at more general Taylor Series: not just near , before doing a revision of partial differentiation, and linking at error analysis.

On Monday we will finish off Chapter 4 by looking at Error Analysis. We will then have over two and a half hours of tutorial time for lectures, and another tutorial on Friday.

The exam is on the morning of Friday Week 13 which means that I cannot give ye (Friday) tutorials in Week 13.

In the lecture slots (Monday, Wednesday, Thursday), I will go through Q. 1-2, 4 of Sample Paper I. Then I will answer any questions. If there are none I will help one-to-one. Usual class times and locations.

Any remaining time can be given over to tutorials, including questions about Q. 3 of Sample Papers I and II.

]]>

You are invited to give your feedback on my teaching and this module here.

Possibly early next week. I am a little behind in my corrections after being sick last week though. Definitely Thursday.

We had our test on Monday, then will looked at completing the square, and work on Tuesday and Thursday.

Here is some video of revision antidifferentiation.

We will look at centroids of laminas and centres of gravity of solids of revolution. Any spare lecture time will be given over to tutorial time.

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

- Monday 16:00
- Tuesday 09:00
- Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

- Monday at 09:00 in B180
- Monday at 17:00 in B189
- Wednesday at 10:00 in F1. 3

The exam is on Friday 13 December.

Please feel free to ask me questions about the exercises via email or even better on this webpage.

With regard to *TRANSPOSITION** *(pages 86-97 of the manual), please fill out this survey.

October 14 you took a quiz as part of the Transposition Project that the Mathematics Department is undertaking in an effort to improve our teaching.

You will have another 15 minute quiz on Monday.

If you do not have an internet ready device, or did not do the first quiz, you may leave class early.

Thank you again for your participation.

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

If you go into Canvas, and go into MATH6055 and the ‘Algebra’ unit, you might see online practise questions for Test 2.

We worked with logarithms and started a quick look at Examples of Functions (which we needed Algebra to talk about).

On Monday we will finish the Examples of Functions mini-chapter before starting the final chapter, the easy chapter, on Network (Graph) Theory.

We will finish our study of Graph Theory by looking at Eulerian graphs, Hamiltonian graphs, and Dirac’s Theorem.

The Test is on Tuesday.

We will have two review lectures on Monday and Tuesday in the usual times and venues and tutorials as normal.

We will go through the MATH6055 Winter 2018 paper, and then I will answer your questions if there are any. If there are none I will help one-to-one. Usual class times and locations.

]]>

See my email of 12 November regarding the Friday tutorial split for the rest of the semester.

Tomorrow we go back to the normal room of A213B for the 12:00 tutorial.

Hopefully I can get these to ye within the week. I have some ‘man flu’ at the moment but hopefully that goes away.

We had an extra tutorial on Monday. Most people concentrated on the Sample Question 3s on Probability and Statistics.

We finished off Chapter 3 — and spoke about the Bad and Good and Bad News — before we began Chapter 4 with a Revision of Differentiation and had a look at Maclaurin Series.

Oh: one thing — I never told ye what a p-value was (Sample Paper I Q. 3 (c) B ii.). It is the same as the level of significance, the probability of making a Type I Error… the answer is 5%.

We will have a tutorial dedicated to differentiation on Monday.

We might look at a Probability and Statistics Concept MCQ on Wednesday.

On Wednesday and Thursday we will have a quick look at more general Taylor Series: not just near , before doing a revision of partial differentiation.

On Monday we will have yet another extra tutorial.

We will finish off Chapter 4 by looking at Error Analysis, including rounding error. We might have time on Thursday for an additional tutorial.

The exam is on the morning of Friday Week 13 which means that I cannot give ye (Friday) tutorials in Week 13.

In the lecture slots (Monday, Wednesday, Thursday), I will go through Q. 1-2, 4 of Sample Paper I. Then I will answer any questions. If there are none I will help one-to-one. Usual class times and locations.

Any remaining time can be given over to tutorials, including questions about Q. 3 of Sample Papers I and II.

]]>