Please find a provisional plan for Weeks 8-13. I do not yet have any certainty around assessment. At this time it is my intention to deliver week-on-week up to 19 April, and then have three weeks of review after this. Of course all plans are provisional in the current environment.

Week 8 to Sunday 22 March

Lectures

There are about 110 minutes of lectures. You should schedule 3 hours to watch them and take the notes in your manual. You need this extra time above 110 minutes because you will want to pause me.

I am trying to front load some of the lecture material so that you can complete the assignment if you so wish. In subsequent weeks there will be less time on the video lectures.

Tutorial

You need to schedule four hours to work on the following exercises. This is comprised of the one hour you would have normally and the three non-contact hours you should be doing all the time.

You can (carefully) take photos of your work and submit to the Week 8 Ungraded Tutorial Work those images on Canvas before midnight Sunday 22 March. After 09:00 Monday 23 March I will download all student work and reply with feedback.

If possible, 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.

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.

Start with:

  • p.133, Q. 1-10

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I thought I had a bit of a breakthrough. So, consider the algebra of a functions on the dual (quantum) group \widehat{S_3}. Consider the projection:

\displaystyle p_0=\frac12\delta^e+\frac12\delta^{(12)}\in F(\widehat{S_3}).

Define u\in M_p(\widehat{S_3}) by:

u(\delta^\sigma)=\langle\text{sign}(\sigma)1,1\rangle=\text{sign}(\sigma).

Note

\displaystyle T_u(p_0)=\frac12\delta^e-\frac12 \delta^{(12)}:=p_1.

Note p_1=\mathbf{1}_{\widehat{S_3}}-p_0=\delta^0-p_0 so \{p_0,p_1\} is a partition of unity.

I know that p_0 corresponds to a quasi-subgroup but not a quantum subgroup because \{e,(12)\} is not normal.

This was supposed to say that the result I proved a few days ago that (in context), that p_0 corresponded to a quasi-subgroup, was as far as we could go.

For H\leq G, note

\displaystyle p_H=\frac{1}{|H|}\sum_{h\in H}\delta^h,

is a projection, in fact a group like projection, in F(\widehat{G}).

Alas note:

\displaystyle T_u(p_{\langle(123)\rangle})=p_{\langle (123)\rangle}

That is the group like projection associated to \langle (123)\rangle is subharmonic. This should imply that nearby there exists a projection q such that u^{\star k}(q)=0 for all k\in\mathbb{N}… also q_{\langle (123)\rangle}:=\mathbf{1}_{\widehat{S_3}}-p_{\langle(123)\rangle} is subharmonic.

This really should be enough and I should be looking perhaps at the standard representation, or the permutation representation, or S_3\leq S_4… but I want to find the projection…

Indeed u(q_{(123)})=0…and u^{\star 2k}(q_{\langle (123)\rangle})=0.

The punchline… the result of Fagnola and Pellicer holds when the random walk is is irreducible. This walk is not… back to the drawing board.

I have constructed the following example. The question will be does it have periodicity.

Where \rho:S_n\rightarrow \text{GL}(\mathbb{C}^3) is the permutation representation, \rho(\sigma)e_i=e_{\sigma_i}, and \xi=(1/\sqrt{2},-1/\sqrt{2},0), u\in M_p(G) is given by:

u(\sigma)=\langle\rho(\sigma)\xi,\xi\rangle.

This has u(\delta^e)=1 (duh), u(\delta^{(12)})=-1, and otherwise u(\sigma)=-\frac12 \text{sign}(\sigma).

The p_0,\,p_1 above is still a cyclic partition of unity… but is the walk irreducible?

The easiest way might be to look for a subharmonic p. This is way easier… with \alpha_\sigma=1 it is easy to construct non-trivial subharmonics… not with this u. It is straightforward to show there are no non-trivial subharmonics and so u is irreducible, periodic, but p_0 is not a quantum subgroup.

It also means, in conjunction with work I’ve done already, that I have my result:

Definition Let G be a finite quantum group. A state \nu\in M_p(G) is concentrated on a cyclic coset of a proper quasi-subgroup if there exists a pair of projections, p_0\neq p_1, such that \nu(p_1)=1, p_0 is a group-like projection, T_\nu(p_1)=p_0 and there exists d\in\mathbb{N} (d>1) such that T_\nu^d(p_1)=p_1.

(Finally) The Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group is ergodic if and only if the driving probability is not concentrated on a proper quasi-subgroup, nor on a cyclic coset of a proper quasi-subgroup.

The end of the previous Research Log suggested a way towards showing that p_0 can be associated to an idempotent state \int_S. Over night I thought of another way.

Using the Pierce decomposition with respect to p_0 (where q_0:=\mathbf{1}_G-p_0),

F(G)=p_0F(G)p_0+p_0F(G)q_0+q_0F(G)p_0+q_0F(G)q_0.

The corner p_0F(G)p_0 is a hereditary \mathrm{C}^*-subalgebra of F(G). This implies that if 0\leq b\in p_0F(G)p_0 and for a\in F(G), 0\leq a\leq b\Rightarrow a\in p_0F(G)p_0.

Let \rho:=\nu^{\star d}. We know from Fagnola and Pellicer that T_\rho(p_0)=p_0 and T_\rho(p_0F(G)p_0)=p_0F(G)p_0.

By assumption in the background here we have an irreducible and periodic random walk driven by \nu\in M_p(G). This means that for all projections q\in 2^G, there exists k_q\in\mathbb{N} such that \nu^{\star k_q}(q)>0.

Define:

\displaystyle \rho_n=\frac{1}{n}\sum_{k=1}^n\rho^{\star k}.

Define:

\displaystyle n_0:=\max_{\text{projections, }q\in p_0F(G)p_0}\left\{k_q\,:\,\nu^{\star k_q}(q)> 0\right\}.

The claim is that the support of \rho_{n_0}, p_{\rho_{n_0}} is equal to p_0.

We probably need to write down that:

\varepsilon T_\nu^k=\nu^{\star k}.

Consider \rho^{\star k}(p_0) for any k\in\mathbb{N}. Note

\begin{aligned}\rho^{\star k}(p_0)&=\varepsilon T_{\rho^{\star k}}(p_0)=\varepsilon T^k_\rho(p_0)\\&=\varepsilon T^k_{\nu^{\star d}}(p_0)=\varepsilon T_\nu^{kd}(p_0)\\&=\varepsilon(p_0)=1\end{aligned}

that is each \rho^{\star k} is supported on p_0. This means furthermore that \rho_{n_0}(p_0)=1.

Suppose that the support p_{\rho_{n_0}}<p_0. A question arises… is p_{\rho_{n_0}}\in p_0F(G)p_0? This follows from the fact that p_0\in p_0F(G)p_0 and p_0F(G)p_0 is hereditary.

Consider a projection r:=p_0-p_{\rho_{n_0}}\in p_0F(G)p_0. We know that there exists a k_r\leq n_0 such that

\nu^{\star k_r}(p_0-p_{\rho_{n_0}})>0\Rightarrow \nu^{\star k_r}(p_0)>\nu^{\star k_r}(p_{\rho_{n_0}}).

This implies that \nu^{\star k_r}(p_0)>0\Rightarrow k_r\equiv 0\mod d, say k_r=\ell_r\cdot d (note \ell_r\leq n_0):

\begin{aligned}\nu^{\star \ell_r\cdot d}(p_0)&>\nu^{\star \ell_r\cdot d}(p_{\rho_{n_0}})\\\Rightarrow (\nu^{\star d})^{\star \ell_r}(p_0)&>(\nu^{\star d})^{\star \ell_r}(p_{\rho_{n_0}})\\ \Rightarrow \rho^{\star \ell_r}(p_0)&>\rho^{\star \ell_r}(p_{\rho_{n_0}})\\ \Rightarrow 1&>\rho^{\star \ell_r}(p_{\rho_{n_0}})\end{aligned}

By assumption \rho_{n_0}(p_{\rho_{n_0}})=1. Consider

\displaystyle \rho_{n_0}(p_{\rho_{n_0}})=\frac{1}{n_0} \sum_{k=1}^{n_0}\rho^{\star k}(p_{\rho_{n_0}}).

For this to equal one every \rho^{\star k}(p_{\rho_{n_0}}) must equal one but \rho^{\star \ell_r}(p_{\rho_{n_0}})<1.

Therefore p_0 is the support of \rho_{n_0}.

Let \rho_\infty=\lim \rho_n. We have shown above that \rho^{\star k}(p_0)=1 for all k\in\mathbb{N}. This is an idempotent state such that p_0 is its support (a similar argument to above shows this). Therefore p_0 is a group like projection and so we denote it by \mathbf{1}_S and \int_S=d\mathcal{F}(\mathbf{1}_S)!

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Today, for finite quantum groups, I want to explore some properties of the relationship between a state \nu\in M_p(G), its density a_\nu (\nu(b)=\int_G ba_\nu), and the support of \nu, p_{\nu}.

I also want to learn about the interaction between these object, the stochastic operator

\displaystyle T_\nu=(\nu\otimes I)\circ \Delta,

and the result

T_\nu(a)=S(a_\nu)\overline{\star}a,

where \overline{\star} is defined as (where \mathcal{F}:F(G)\rightarrow \mathbb{C}G by a\mapsto (b\mapsto \int_Gba)).

\displaystyle a\overline{\star}b=\mathcal{F}^{-1}\left(\mathcal{F}(a)\star\mathcal{F}(b)\right).

An obvious thing to note is that

\nu(a_\nu)=\|a_\nu\|_2^2.

Also, because

\begin{aligned}\nu(a_\nu p_\nu)&=\int_Ga_\nu p_\nu a_\nu=\int_G(a_\nu^\ast p_\nu^\ast p_\nu a_\nu)\\&=\int_G(p_\nu a_\nu)^\ast p_\nu a_\nu\\&=\int_G|p_\nu a_\nu|^2\\&=\|p_\nu a_\nu\|_2^2=\|a_\nu\|^2\end{aligned}

That doesn’t say much. We are possibly hoping to say that a_\nu p_\nu=a_\nu.

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DME2C Lab 5: Runge-Kutta

DME2C are invited to do Lab 5: Runge-Kutta remotely.

I have set up an (ungraded) assignment on Canvas that you (DME2C student) can submit your Lab 5 work. After Tuesday 09:00 I will download all the submissions and give feedback on each.

Recall our overall framework for our programmes in MATH7016:

  • Define variables
  • Give initial values
  • Loop that
    • Prints current variables (of interest)
    • Calculates next variables

For Runge-Kutta, as usual there will be an x and y variables, but also a number of k_i variables which represent (estimates of) the slope at various points between the current and next value.

The loop must calculate the various k_i BEFORE calculating the next and values. The next value is given as:

y_{i+1}=y_i+h\cdot \phi(k_1,\dots,k_n),

where \phi(k_1,\dots,k_n) is a weighted average of the slopes k_i. There is one for Euler’s Method (slope at previous), two for Heun’s (slope at previous and slope at predicted next), and we also look at ‘Common’ RK3 which uses three k variables, k_1,\,k_2,\,k_3, and ‘Classical’ (state-of-the-art) RK4 which uses four.

All of Lab 5 is to be done in VBA. Problem 4 is missing the formula:

y_{i+1}=y_i+h\cdot \phi(k_1,\dots,k_4),

the relevant formula is on p. 74. Also the sign of the derivative is wrong (and I have the rocket fuel being ejected too quickly… use

\displaystyle\frac{dv}{dm}=-\frac{5}{m}.

DME2C: Concept MCQ 5

I want to keep the (ungraded) MCQ league going — I have pledged €35 of my own cash for first (€20), second (€10), and third (€5), and it would be great to keep it going until the ‘end of the season’. The same three names have been leading the league for a few weeks so maybe they might falter now.

Well anyway, DME2C can enter MCQ5 by emailing me their selection (ABCDDB or whatever) before Tuesday 09:00.

Week 8

Luckily enough from my point of view, because of St Patrick’s Day, I have already recorded lectures for next week.

First watch Goal Seek for Boundary Value Problems (less than 20 minutes).

Then you will be in a position to do Lab 6: Boundary Value Problems.

Between Tuesday 17 March and Monday 23 March, you will be able to submit your work to Canvas, from which I will give you individual feedback.

After this watch the rest of the first playlist, watch Intro to PDEs (less than 20 minutes), and then watch the Derivation of the Laplace Equation (40 minutes).

I will also send on an MCQ6 to keep the league going.

Week 7

In the Tuesday 09:00 class we had Written Assessment 1.

The lab was based on Runge Kutta methods.

In the 12:00 lecture we did a (written) Shooting Method example.

In VBA we will also have MCQ V.

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Week 7

We continued looking at “The Engineer’s Transform” — the Laplace Transform.

We continued looking at partial fractions and then the inverse Laplace Transform.

We had one tutorial Wednesday PM where we worked on partial fractions.

Week 8

It is my intention to record lectures at home. I have a camcorder and a whiteboard in my home office.

I am going to, more or less as soon as possible, record at home enough lectures for you to be able to do the second assignment. This means I want to push forward up to p.132.

I expect ye to watch the lectures and to spend some time doing exercises. How much time you devote to exercises is up to you, but in theory you are supposed to spend 7 hours per week on MATH7021 between classes and independent learning.

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Tuesday 17 March 2020, Week 8

This class is postponed to the same time and room, but Wednesday 18 March.

Week 7

In Week 7 we did a matrices Concept MCQ and then did a quick revision of differentiation:

Then we looked at Parametric Differentiation.

We had no tutorial time

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Quasi-Subgroups that are not Subgroups

Let G be a finite quantum group. We associate to an idempotent state \int_Squasi-subgroup S. This nomenclature must be included in the manuscript under preparation.

As is well known from the GNS representation, positive linear functionals can be associated to closed left ideals:

\displaystyle N_{\rho}:=\left\{ f\in F(G):\rho(|f|^2)=0\right\}.

In the case of a quasi-subgroup, S\subset G, my understanding is that by looking at N_S:=N_{\int_S} we can tell if S is actually a subgroup or not. Franz & Skalski show that:

Let S\subset G be a quasi-subgroup. TFAE

  • S\leq G is a subgroup
  • N_{\int_S} is a two-sided or self-adjoint or S invariant ideal of F(G)
  • \mathbf{1}_Sa=a\mathbf{1}_S

I want to look again at the Kac & Paljutkin quantum group \mathfrak{G}_0 and see how the Pal null-spaces N_{\rho_6} and N_{\rho_7} fail these tests. Both Franz & Gohm and Baraquin should have the necessary left ideals.

The Pal Null-Space N_{\rho_6}

The following is an idempotent probability on the Kac-Paljutkin quantum group:

\displaystyle \rho_6(f)=2\int_{\mathfrak{G}_0}f\cdot (e_1+e_4+E_{11}).

Hence:

N_{\rho_6}=\langle e_1,e_3,E_{12},E_{22}\rangle.

If N_{\rho_6} were two-sided, N_{\rho_6}F(\mathfrak{G}_0)\subset N_{\rho_6}. Consider E_{21}\in F(\mathfrak{G}_0) and

E_{12}E_{21}=E_{11}\not\in N_{\rho_6}.

We see problems also with E_{12} when it comes to the adjoint E_{12}^{\ast}=E_{21}\not\in N_{\rho_6} and also S(E_{12})=E_{21}\not\in N_{\rho_6}. It is not surprise that the adjoint AND the antipode are involved as they are related via:

S(S(f^\ast)^\ast)=f.

In fact, for finite or even Kac quantum groups, S(f^\ast)=S(f)^\ast.

Can we identity the support p? I think we can, it is (from Baraquin)

p_{\rho_6}=e_1+e_4+E_{11}.

This does not commute with F(G):

E_{21}p_{\rho_6}=E_{21}\neq 0=p_{\rho_6}E_{21}.

The other case is similar.

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Back before Christmas I felt I was within a week of proving the following:

Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group G is ergodic if and only if \nu is not concentrated on a proper quasi-subgroup, nor the coset of a ?normal ?-subgroup.

The first part of this conjecture says that if \nu is concentrated on a quasi-subgroup, then it stays concentrated there. Furthermore, we can show that if the random walk is reducible that the Césaro limit gives a quasi-subgroup on which \nu is concentrated.

The other side of the ergodicity coin is periodicity. In the classical case, it is easy to show that if the driving probability is concentrated on the coset of a proper normal subgroup N\lhd G, that the convolution powers jump around a cyclic subgroup of G/N.

One would imagine that in the quantum case this might be easy to show but alas this is not proving so easy.

I am however pushing hard against the other side. Namely, that if the random walk is periodic and irreducible, that the driving probability in concentrated on some quasi-normal quasi-subgroup!

The progress I have made depends on work of Fagnola and Pellicer. They show that if the random walk is irreducible and periodic that there exists a partition of unity \{p_0,p_1,\dots,p_{d-1}\} such that \nu^{\star k} is concentrated on p_{k\mod d}.

This cyclic nature suggests that p_0 might be equal to \mathbf{1}_N for some N\lhd G and perhaps:

\Delta(p_i)=\sum_{j=0}^{d-1}p_{i-j}\otimes p_j,

and perhaps there is an isomorphism G/N\cong C_d. Unfortunately I have been unable to progress this.

What is clear is that the ‘supports’ of the p_i behave very much like the cosets of proper normal subgroup N\lhd G.

As the random walk is assumed irreducible, we know that for any projection q\in 2^G, there exists a k_q\in \mathbb{N} such that \nu^{\star k_q}(q)\neq 0.

Playing this game with the Haar element, \eta\in 2^G, note there exists a k_\eta\in\mathbb{N} such that \nu^{k_\eta}(\eta)>0.

Let \overline{\nu}=\nu^{\star k_\eta}. I have proven that if \mu(\eta)>0, then the convolution powers of \mu\in M_p(G) converge. Convergence is to an idempotent. This means that \overline{\nu}^{\star k} converges to an idempotent \overline{\nu}_\infty, and so we have a quasi-subgroup corresponding to it, say \overline{p}.

The question is… does \overline{p} coincide with p_0?

If yes, is there any quotient structure by a quasi-subgroup? Is there a normal quasi-subgroup that allows such a structure?

Is \overline{p} a subgroup? Could it be a normal subgroup?

As nice as it was to invoke the result that if e is in the support of \nu, then the convolution powers of \nu converge, by looking at those papers which cite Fagnola and Pellicer we see a paper that gives the same result without this neat little lemma.

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Week 6

We finished off Chapter 2 on Monday and then had some tutorial time.

Wednesday, after an Undetermined Coefficients MCQ, we started looking at “The Engineer’s Transform” — the Laplace Transform. We looked at the first shift theorem, and how the Laplace Transform interacts with differentiation. We started looking at partial fractions.

Only those of us who finished Assignment 1 early got much tutorial time with Chapter 2. I will hope that we will get more tutorial time on Chapter 2 at a later date, but until then, I have developed the following for you to practise your differentiation which you need for Chapter 2 here:

It might not be the worst idea in the world to devote some time to this over Easter.

Assignment 1 – Results and Remarks

Have been emailed to you. Some remarks with common mistakes.

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