Test Tuesday 20 April 19:30 to 20:45. Five questions, one for each of the five sections in Chapter 3.

You can find old (e.g. Chapter 3) videos on my YT channel here. (Links to an external site.)

One more video that is actually Chapter 3 material (and the recording was a little messed up so no live writing… a new version will be in the Week 11 lectures.):

Revision of integration and integration by parts.

- Revision of Integration (Links to an external site.) (44 minutes)
- Integration by Parts (Links to an external site.) (36 minutes)

Here are last year’s lectures of the same material:

- Revision of Antidifferentiation: Direct and Manipulation (Links to an external site.) (27 minutes)
- Revision of Antidifferentiation: Substituion (Links to an external site.) (22 minutes)
- Integration by Parts (Links to an external site.) (30 minutes)

I recorded a lot of this material before in a live lecture: press here to watch live version (Links to an external site.) (84 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.

Try:

- p. 165, Q.1-13
- p. 171, Q.1-5

Additional Exercises: p. 171, Q. 6-9

Submit work for Canvas feedback by Sunday 18 April for video feedback after Monday 19 April.

Perhaps of the order of 1.5 hours of lectures on completing the square (Links to an external site.), and work (Links to an external site.).

Perhaps of the order of 1.5 hours of lectures on centroids (Links to an external site.) of laminas and centres of gravity of solids of revolution (Links to an external site.).

I will continue to provide learning support.

**Week 11 ** – 25% Differentiation Test *(Zoom Tutorial in Week 10, after Easter)*

**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 (Links to an external site.) for information on the Academic Learning Centre, etc.

]]>Have finally been completed. Please check your grade and if you have any queries please don’t hesitate to email me.

Written Assessment 2 will not be entirely unlike Written Assessment 1. The Week 12 Zoom will be Monday 26 April 15:00 instead of Tuesday 27 April.

25% Written Assessment 1, based on Weeks 6-10, so everything from p.74 to 106.

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 27 April. **It is open book — you can use your manual, any Canvas materials, as well as Excel/VBA.

40% of the marks will be for Section 1.9

30% of the marks will be for Section 2.1

40% of the marks will be for Section 2.2

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.) (Links to an external site.).

- Heat Equation I (Links to an external site.) (19 minutes)
- Heat Equation II (Links to an external site.) (36 minutes)

This (Links to an external site.) explains the connection between the Heat Equation and the Jacobi Method approximations to the Laplace Equation

There is a longer, more in-person (old, last year) version of the above material. How different they are I am not too sure.

- Derivation of Heat Equation (Links to an external site.) (26 minutes)
- Finite Differences for the Heat Equation (Links to an external site.) (29 minutes)
- Heat Equation Finite Differences Example I (Links to an external site.) (11 minutes)
- Heat Equation Finite Differences Example II (Links to an external site.) (16 minutes, continued from above, the dog interrupted)

Q&A on Tuesday 13 April as normal: p.104, Q.1-3

No more lectures, no more labs: you will focus on VBA Assessment 2. Q&A on Tuesday.

No more lectures, no more labs: you will focus on Written Assessment 2. Q&A on **Monday.**

Study should consist of

- doing exercises from the notes
- completing VBA exercises

Please see Student Resources download 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*

These have finally been released. Apologies for the delay. Any queries do not hesitate to contact me.

No lectures: all MATH7021 time to be invested into Chapter 3 and Assignment 3. See Week 11, below, if you have completed Assignment **3** and are hoping to look ahead.

**As I send this, there are two slots for feedback over Easter with deadlines of today, 5 April, for feedback, tomorrow, 6 April, and 12 April for feedback 13 April.**

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

- p.127, Q.1-11

Additional exercises p.130, Q.12-19.

Submit work for Canvas feedback by Monday 19 April for video feedback after Tuesday 20 March.

Weeks 11 and 12 will be given over to Chapter 4. Perhaps of the order of 2 hours of lectures on Double Integrals (Links to an external site.) in Week 11 and of the order of 2 hours of lectures on Triple Integrals (Links to an external site.). Students who have already submitted Assignment 3 can look at the old lectures. The material is exactly the same… for the sake of of doing honest work I will be rerecording for Weeks 11 and 12, but if you want to get ahead you can watch these:

**Week 11:**

- Revision of Concept of Integration (Links to an external site.) (24 minutes)
- Concept of Double Integration (Links to an external site.) (23 minutes)
- Double Integral Examples (Links to an external site.) (29 minutes)
- Polar Coordinates for Double Integrals (Links to an external site.) (20 minutes)
- Double Integrals over Circular Regions: Examples (Links to an external site.) (13 minutes)

**Week 12:**

- Integration over a 3D Box (Links to an external site.) (12 minutes)
- Integration over a Cylinder (Links to an external site.) (15 minutes)

Re: Section 3.5 Systems of Differential Equations (Links to an external site.); important for next year, won’t be examined this semester but will get ye the notes and lectures before the end of year (so as to have them as a reference for next year).

**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

Please see Student Resources (Links to an external site.) for information on the Academic Learning Centre, etc.

]]>So anyway some problems and brief thoughts. I tried and failed to resist the urge to use the non-standard notation and interpretation used in this new paper… I guess this post is for me… if you want to understand the weird “ is a quantum permutation” and ““, etc you will have to read the paper.

So the big long crazy draft of has some stuff about why there is no quantum cyclic or alternating groups but these are arguments rather than proofs. A no-go theorem here looks as follows:

A finite group hasno quantum versionif whenever is a quantum permutation group with group of characters of equal to , then .

I know the question for is open… is it formally settled that there is no quantum ? Proving that would be a start. A possible strategy would be to construct from and a quantum permutation like a character on that is in the complement of in . The conclusion being that there can be no quantum permutation in that is is commutative. There might be stuff here coming from alternating (pun not intended) projections theory that could help.

From April 2020:

My proof of the Ergodic Theorem leans heavily on the finiteness assumption but a lot of the stuff in that paper (and there are many partial results in that paper also) should be true in the compact case too. How much of the proof/results carry into the compact case? A full Ergodic Theorem for Random Walks on Compact Quantum Groups is probably quite far away at this point, but perhaps partial results under assumptions such as (co?)amenability might be possible. OR try and prove ergodic theorems for specific compact quantum groups.

What I would be interested in doing here is seeing can I maybe use the language from the Ergodic Theorem to prove some partial results in this direction. The analysis is possibly a little harder than I am used to. What I might want to show is that if is an idempotent state (I am not sure are there group-like projections lying around, I think there are maybe here), and its support projection in , that the convolution of quantum permutations such that , that . This would almost certainly require the use of a group-like projection. Possibly restricting to to quantum permutation groups we would then have a good understanding of non-Haar idempotent “quasi-subgroups”:

A quasi-subgroup of a compact(permutation?)quantum group is a subset ofthat (for )contains the identity, is closed under reversal, and closed under the quantum group law.A quasi-subgroup is a quantum subgroup precisely when it is closed under wave function collapse.

The other thing that could be done here would be to refine two aspects of the finite theory. One, from the direction of cyclic shifts, in order to make the definition of a cyclic coset more intrinsic (it is currently defined with respect to a state), and two to further study the idea of an idempotent commuting with something like a “finite order” deterministic state (see p.27).

For , there is no intermediate quantum subgroup . That is there is no comultiplication intertwining surjective *-homomorphism to noncommutative such that there is in addition there is another such map .

This well-known conjecture is that there is no intermediate quantum subgroup at *any *.

Let be the Kac–Paljutkin quantum group of order eight and consider the quantum permutation , the state space of the algebra of functions on , . Where and , . Where and (by abuse of notation) the Haar state of in . Where , consider the idempotent state on :

.

There are three possibilities and all three are interesting:

— this is what I expect to be true. If this could be proven, the approach would be to hope that it might be possible to construct a state like on *any *compact quantum group. The state has a nice “constraint” property: I can only map 1 or 2 to 3 or 4 and vice versa. A starting idea in the construction might be to take the unital algebra generated by a non-commuting pair . Using Theorem 4.6, there is a *-representation such that for some we have a representation and from this representation we have a nice vector state that is something like . Well it doesn’t have the “constraint” property… one idea which I haven’t thought through is to condition this lovely quantum permutation in a clever enough way… we could condition it to only map to or some … but that seems to only be the start of it.

for — this would obviously be a counterexample to the maximality conjecture.

is a non-Haar idempotent — this is a possibility that I don’t think many have thought of. It wouldn’t disprove the conjecture but would be an interesting example. This might be something non-zero on e.g. but zero on some strictly positive .

I can’t really describe this so instead I quote from the paper:

It could be speculated that the dual of a discrete group could model a particle “entangled” quantum system, where the -th particle, corresponding to the block , has states, labelled . Full information about the state of all particles is in general impossible, but measurement with will see collapse of the th particle to a definite state. Only the deterministic permutations in would correspond to classical states.

So I want to read Schmidt’s PhD thesis and maybe answer the question of whether or not there is a graph whose quantum automorphism group is the Kac–Paljutkin quantum group. Also see can anything be done in the intersection of random walks and quantum automorphism groups.

I want to able to extract results on locally compact quantum groups to compact quantum groups.

This is a bit mad… bottom of p.35 to p.36.

Maybe take and define a Markov chain on using a quantum permutation. So for example you measure with to get . Then measure and iterate. The time taken to reach or some other or maybe hit all of … this is a random time . Is there any relationship between the expectation of and the distance of the convolution powers of to the Haar state. Lots of things to think about here.

More mad stuff. See Section 8.5.

If has a finite set of finite-order generators it is a quantum permutation group. What about if there are only infinite order generators? I *guess *this isn’t a quantum permutation group (subgroup of some ) but maybe using Goswami and Skalski such a dual can be given the structure of a quantum permutation group on infinite many symbols. At the other end of the scale… are the Sekine quantum groups quantum permutation groups?

From April 2020:

*Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac?*— No interest in this by me at the moment*Following Urban, study convolution factorisations of the Haar state.*— ditto.*Examples of non-central random walks on compact quantum groups*— Freslon and coauthors have cornered the market on interesting examples of random walks on compact quantum groups… I don’t think I will be spending time on this.

Note that Simeng Wang has sorted: *extending the Upper Bound Lemma to the non-Kac case.* There are a handful of other problems here, here, and here that I am no longer interested in.

Click here for pdf.

]]>

Extending the Upper Bound Lemma to the non-Kac case. As I speak, this is beyond what I am capable of. This also requires work on the projection and quantum total variation distances (i.e. show they are equal in this larger category).

After that post, Simeng Wang wrote with the, for me, exciting news that he had proven the Upper Bound Lemma in the non-Kac case, and had an upcoming paper with Amaury Freslon and Lucas Teyssier. At first glance the paper is an intersection of the work of Amaury in random walks on compact quantum groups, and Lucas’ work on limit profiles, a refinement in the understanding of how the random transposition random walk converges to uniform. The paper also works with continuous-time random walks but I am going to restrict attention to what it does with random walks on .

For the case of a family of Markov chains exhibiting the cut-off phenomenon, it will do so in a window of width about a cut-off time , in such a way that , and, where is the ‘distance to random’ at time , will be close to one for and close to zero for . The *cutoff profile *of the family of random walks is a continuous function such that as ,

.

I had not previously heard about such a concept, but the paper gives a number of examples in which the analysis had been carried out. Lucas however improved the Diaconis–Shahshahani Upper Bound Lemma and this allowed him to show that the limit profile for the random transpositions random walk is given by:

Without looking back on Lucas’ paper, I am not sure exactly how this works… I will guess it is, where and , and so:

,

and I get , and on CAS. Looking at Lucas’ paper thankfully this is correct.

The article confirms that Lucas’ work is the inspiration, but the study will take place with infinite compact quantum groups. The representation theory carries over so well from the classical to quantum case, and it is representation theory that is used to prove so many random-walk results, that it might have been and was possible to study limit profiles for random walks on quantum groups.

More importantly, technical issues which arise as soon as disappear if the *pure* quantum transposition random walk is considered. This is a purely quantum phenomenon because the random walk driven only by transpositions in the classical case is periodic and does not converge to uniform. I hope to show in an upcoming work how something which might be considered a quantum transposition behaves very differently to a classical/deterministic transposition. My understanding at this point, in a certain sense (see here)) is that a quantum transposition has fixed points in the sense that it is an eigenstate (with eigenvalue ) of the character . I am hoping to find a dual with a quantum transposition that for example does not square to the identity (but this is a whole other story). This would imply in a sense that there is no quantum alternating group.

The paper will show that the quantum version of the ordinary random transposition random walk and of this pure random transposition walk asymptotically coincide. They will detect the cutoff at time , and find an explicit limit profile (which I might not be too interested in).

I will skip the stuff on but as there are some similarities between the representation theory of quantum orthogonal and quantum permutation groups I may have to come back to these bits.

At this point I will move away and look at this Banica tome on quantum permutations for some character theory.

Banica talks about looking at the law of the main character,

,

where is my own notation. In the classical case it is known that as , , and in the quantum case it can be shown that , the free Poisson distribution.

*The characters of corepresentations, given by:*

*behave as follows, in respect to the various operations:*

*In addition if then*

*The moments of are*

*When , the law of is a real measure, supported by .*

The second part here is most important at the moment.

*The quantum groups with have the following properties.*

*The moments of fix are the Catalan numbers:*

*The character “fix” follows the Marchenko-Pastur law (related to free Poisson).* I might denote it .

*The fusion rules for irreducible representations are the same as for :*

*The dimensions of the representations are as follows, with *

.

In proving this, Banica exhibits via a recurrence that:

I just need to check have we that … actually this follows from the fusions rules. So, the characters of the irreducible representations satisfy

,

Furthermore, by the fusion rules, they commute, and so the -algebra generated by them is commutative. At this point we will do some work with Brannan to better understand what is going on with such commutative -algebras. This necessitates working with .

Let be the universal -algebra generated by the elements of a unitary matrix of self-adjoint , i.e. such that:

.

The representation theory was studied by Banica:

*For any fixed ** there is a maximal family of irreducible representations of , labelled by non-negative integers with the the properties that is the trivial representation, is the fundamental representation . The representations are self-conjugate and satisfy the fusion rules:*

*For , , and for :*

,

*where is defined by . The characters are self-adjoint and satisfy*

.

*This implies that the characters commute and the central *-algebra generated by the characters is generated by and . The spectral measure of relative to the Haar state (might have to return to this) is Wigner’s semicircle law:*

*Let . Then .*

It’s hilarious how bad I am at this stuff… I just spent 25 minutes trying to prove that … I guess I got there in the end via Murphy Theorem 5.1.11… there is a (pure) state . Also . Hit both of these with to find:

.

Now we move on

*Proof:* So, from that

,

therefore . The universal property gives a surjective -homomorphism onto . We have , and Brannan says that in fact the spectrum of fix in this algebra of functions is the whole of (nice argument here). Brannan argues that the spectrum of any element of a -algebra contains the spectrum of its image, we have that

Let denote the sequence of polynomials defined by initial conditions , , and the recursion:

.

Think these are , for Chebyshev of second kind.

*Let * *be the unital -algebra generated by the irreducible characters. Then there is a -isomorphism defined by:*

OK, I think I understand this central algebra a little better. What I want to do now is show that the characters for have some nice properties, and related to these . I don’t think there is any issue using these… but should the restriction be to rather than ? Surely. I know that in the dual of the dihedral group the spectrum of fix is [0,4] and it isn’t hard to suppose that that can be stretched to… well now… the paper continues to use [0,4]. Presumably that is enough. Nice to have recognised the [0,4] from some other works I am looking at.

*FTW work with [0,4] because in the reduced algebra this is the spectrum of fix.*

_________________________

Back to FTW. If I rewrite the orthogonal quantum group character recurrence as:

And if I write the recurrence for the Chebyshev polynomials of the second kind:

If we give these are an argument and input we get the orthogonal group character recurrence. Now we are looking for the recurrence for the quantum permutation group characters:

Struggling again I once again consult work of Brannan (p.13)… euh I’m an idiot… that was not difficult.

Right, so the quantum permutation characters are

.

I am now going to focus on the quantum random transpositions (incidentally Freslon gave a talk on this topic here).

Now I have to back and look at Freslon’s first random walk paper.

Consider a pure state on . Restricted to the central algebra a pure state is still pure. And pure states on an abelian algebra are evaluation at points of the spectrum:

.

For the spectrum in the universal version (this is surely the same as the enveloping C*-algebra?) is so that for all there is a central state on defined by:

.

OK, I think the idea is to say, OK, we could define the state on the central algebra as … and that could be written as . Instead Freslon chooses rather than in the central algebra as a representative of , but … and of course we know about .

Now as a central state, and via the Upper Bound Lemma,

The power comes from the fact that the upper bound lemma involves a trace and there is a lying around whose trace is …

Let us jump back to and consider … none the clearer on this… I get:

…

OK, I have cleared up the issue… I thought *had *to be a typo, but no it is correct.

Now! Finally we have that this other state (which we must explore after this calculation):

* *So in the language of my paper in preparation, is a quantum permutation with fixed points… the question for me would we have, for the cyclic vector in the GNS representation (all sorted now)?

Is ? I find:

.

OK, now I understand the confusion. It is the case that and both are simply evaluation at … but is only notation and not the same as and so indeed.

We will end with some analysis.

Any state on has a unique extension to a state on . Such a state is an element of the Fourier-Stieltjes algebra, the topological dual of . Denote the norm by . We also have the topological double dual which I denote by . It can be shown then that:

,

where are the orthogonal projections in . Then this is used to define the total variation distance.

Now we turn to absolute continuity. Define an inner product on by . Complete to and embeds through left multiplication into . The weak closure of the embedding of in is called and is a vNa. *If * extends to a normal bounded map on , then becomes an element of the *Fourier algebra *which is the Banach space predual of of , and, crucially (quoting from a heavy LCQG paper of Brannan and Ruan):

,

which further implies, if and states on the algebra of regular functions extend to the vNa , that the total variation norm can be given as a supremum over projections in . Cool.

]]>I have a bit of a backlog of corrections but hopefully I can get these to ye before Easter Sunday.

Final two sections of Chapter 3:

- Partial Differentiation I (31 minutes)
- Partial Differentiation II (38 minutes)
- Error Analysis using Differentials (30 minutes)
- Error Analysis Example (5 minutes)

Here are last year’s lectures of the same material:

- Functions of Several Variables (12 minutes)
- Partial Differentiation: Theory (25 minutes)
- Partial Differentiation: Examples (11 minutes)
- Higher Order Partial Derivatives (21 minutes)
- Partial Differentiation Tutorial (9 minutes)
- Differentials (10 minutes)
- Propagation of Errors (7 minutes)
- Propagation of Error Examples (25 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.

Try:

- p. 143, Q. 1-5 [Note these exercises are
*interleaved –*there are questions here from earlier sections in Chapter 3] - p. 150, Q. 1-6 [Note these exercises are
*interleaved –*there are questions here from earlier sections in Chapter 3]

Additional Exercises: p. 143, Q. 6, p. 151, Q.7-8, 9-10

Submit work for Canvas feedback by Sunday 28 March for video feedback after Monday 29 March.

I will be providing learning support over Easter.

Looking further ahead, to after Easter, a good revision of integration/antidifferentiation may be found here. Here is some video of revision of antidifferentiation.

Perhaps of the order of 1.5 hours of lectures on starting Chapter 4 on (Further) Integration with a revision of antidifferentiation, and a look at Integration by Parts. We will use implicit differentiation to differentiate inverse sine.

Perhaps of the order of 1.5 hours of lectures on completing the square, and work.

Perhaps of the order of 1.5 hours of lectures on centroids of laminas and centres of gravity of solids of revolution.

**Week 11 ** – 25% Differentiation Test *(Zoom Tutorial in Week 10, after Easter)*

**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.

]]>I cannot promise at this point when these will be completed. It is my intention to have these corrected before the end of Week 9. Watch this space.

I will give proper notice for VBA Assessment 2 by the end of Week 9 and for Written Assessment 2 during Easter.

- Laplace’s Difference Equation I (21 minutes)
- Laplace’s Difference Equation II (22 minutes)
- Heat Flux Density (4 minutes)

There is a longer, more in-person (old, last year) version of the above material:

- Laplace’s Difference Equation (26 minutes)
- Mean Value Property (12 minutes)
- Laplace’s Difference Equation Examples (29 minutes)
- Laplace’s Equation Irregular Boundary (9 minutes)
- Calculating Heat Flux Density (7 minutes)

See VBA Lab 7

Q&A on Tuesday as normal:

p.97, Q. 1,2 and p. 98 exercises

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.

- Derivation of Heat Equation (26 minutes)
- Finite Differences for the Heat Equation (29 minutes)
- Heat Equation Finite Differences Example I (11 minutes)
- Heat Equation Finite Differences Example II (16 minutes, continued from above, the dog interrupted)

None. Perhaps additional tutorial time in Week 11 for Written Assessment in Week 12.

We will do our last lab, Lab 8 in Week 10. VBA Assessment 2 in Week 11.

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 6-10

Study should consist of

- doing exercises from the notes
- completing VBA exercises

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 get cracking on Chapter 3.*

I am experiencing a hardware issue at the moment and it is making it difficult to get corrections done.. My promise is to complete corrections of both Assignment 1 and Assignment 2 before the end of Week 9. Sorry for the delay.

No lectures in Week 9. If you are behind please put big time into Chapter 3, catching up.

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

- p.127, Q.1-11

Additional exercises p.130, Q.12-19.

Submit work for Canvas feedback by Monday 29 March for video feedback after Tuesday 30 March.

In Week 10 you will also spend all your time doing Chapter 3 Exercises/Assignment 3. 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 during Easter once I am caught up on corrections.

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 (or student could look at the old lectures, the ones that say “integrals” here)

**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

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

]]>Now that it is complete, although I really like all its contents (well except for the note to reader and introduction I spilled out very hastily), I can see on reflection it represents rather than a cogent piece of mathematics, almost a log of all the things I have learnt in the process of writing it. It also includes far too much speculation and conjecture. So I am going to post it here and get to work on editing it down to something a little more useful and cogent.

EDIT: Edited down version here.

]]>