I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 4

We worked with matrix inverses, seeing how the Gauss-Jordan algorithm can be used to calculate the inverse of a $3\times 3$ matrix. We solved a matrix equation.

Here find a corrected Example 2 from p. 39. In class, I made a slip in the third frame. The row operations are the same.

$\displaystyle A^{-1}=\left(\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 5 & 4 \\ 3 & 6 & 5\end{array}\right)$.

We also had our second Maple lab.

## Week 5

We will see how linear systems can be written as matrix equations, and solved using matrix inverses. Then we will talk about determinants, and perhaps push towards the end of Chapter 1.

## Linear Algebra: 20% Test

Will take place Wednesday 14 March, in Week 7.

## Maple Catch Up

If you have missed the first lab you have two options: either download Maple onto your own machine (instructions may be found here) or come into CIT at another time to use Maple.

Go through the missed lab on your own, doing all the exercises in Maple. Save the worksheet and email it to me.

## Assignment 1

Assignment 1 has a hand-in time and date of 14:30 Friday 2 March (Week 5) and has been given out.

Careful in your “collaboration” — don’t take an explanation, etc. from another
student unless it makes sense to you: otherwise you are not going to get the benefit out of completing this assignment.

## Week 4

We continued our work on Chapter 2 — the method of undetermined coefficients for solving linear odes — by looking at the case of external forces. We had two tutorials.

## Week 5

We will hopefully finish off our work on the method of undetermined coefficients: we will start the second tutorial (Thursday) only when we finish Section 2.4 (what if we know the initial conditions).

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Assessment

Considering our progress, I have decided to swap the positions of the first and second assessments. This time last year I had Section 1.2.3 completed but have (necessarily) slowed down this year. Last year Section 1.2.3 was tested both in the first (written) assessment and in the second (VBA) assessment.

It is more important that Section 1.2.3 is tested in the written component therefore the decision to switch the assessments.

Due to this change, the information in Sections 3.6 and 3.7 is now out of date.

The following is the proposed assessment schedule:

2. Week 7, 20 % In-Class Written Test, More Info in Week 5
3. Week 11, 20% Second VBA Assessment, More Info in Week 9

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of the assessment. The following is the proposed layout of the assessment:

### Q. 1: Numerical Solution of Initial Value Problem [80%]

Examples of initial value problems that might be arise include:

• Damping

$\displaystyle \frac{dv}{dt}=-\frac{\lambda}{m}v(t)$;           $v(0)=u$

• The motion of a free-falling body subject to quadratic drag:

$\displaystyle \frac{dv}{dt}=g-\frac{c}{m}v(t)^2$;           $v(0)=u$

• Newton Cooling

$\displaystyle \frac{d\theta}{dt}=-k\cdot (\theta(t)-\theta_R)$;           $\theta(0)=\theta_0$

• The charge on a capacitor

$\displaystyle \frac{dq}{dt}=\frac{E}{R}-\frac{1}{RC}q(t)$;           $q(0)=0$

Students have a choice of how to answer this problem:

• The full, 80 Marks are going for a VBA Heun’s Method implementation (like Lab 3).
• An Euler Method implementation (like Lab 2), gets a maximum of 60 Marks.

You will be asked to write a program that takes as input all the problem parameters, perhaps some initial conditions, a step-size, and a final time, and implements Heun’s Method (or Euler’s Method): similar to Exercise 1 on p. 114 and also Exercise 1 on p.109 (except perhaps implementing Heun’s Method).

If you can write programs for each of the four initial value problems above you will be in absolutely great shape for this assessment.

### Q. 2: Using your Program [20%]

You will then be asked to use your program to answer a number of questions about your model. For example, assuming Heun’s Method is used, consider the initial value problem (3.7) on p. 105.

1. Given, $v_0=0.2$, $m=3$, $\lambda=1.5$, $h=0.01$, approximate $v(0.3)$.
2. Given, $v_0=0.4$, $m=30$, $\lambda=1.5$, $h=0.1$, investigate the behaviour of $v(t)$ for large $t$.
3. Given $v_0=0.2$, $m=0.1$, $\lambda=1.5$, $h=0.5$, $T=10$, run the Heun program. Comment on the behaviour of $v(t)$. Run the same program except with $h=0.05$. Comment on the behaviour of $v(t)$.
4. Given, $v_0=0$, $m=3$, $\lambda=1.5$, $h=0.1$, $T=2$, run the Heun program. Comment on the behaviour of $v(t)$.

## Week 4

We jumped forward and looked at Heun’s Method in the 09:00 class. We went back then and looked at the Three Term Taylor Method in the afternoon. We stated in the afternoon that Heun’s Method gives the same answer as the Three Term Taylor, and without the need for implicit differentiation.

In VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

In May 2017 I wrote down some problems that I hoped to look at in my study of random walks on quantum groups. Following work of Amaury Freslon, a number of these questions have been answered. In exchange for solving these problems, Amaury has very kindly suggested some other problems that I can work on. The below hopes to categorise some of these problems and their status.

## Solved!

• Show that the total variation distance is equal to the projection distance. Amaury has an a third proof. Amaury suggests that this should be true in more generality than the case of $\nu$ being absolutely continuous (of the form $\nu(x)=\int_G xa_{\nu}$ for all $x\in C(G)$ and a unique $a_{\nu}\in C(G)$). If the Haar state is no longer tracial Amaury’s proof breaks down (and I imagine so do the two others in the link above). Amaury believes this is true in more generality and says perhaps the Jordan decomposition of states will be useful here.
• Prove the Upper Bound Lemma for compact quantum groups of Kac type. Achieved by Amaury.
• Attack random walks with conjugate invariant driving probabilitys: achieved by Amaury.
• Look at quantum generalisations of ‘natural’ random walks and shuffles. Solved is probably too strong a word, but Amaury has started this study by looking at a generalisation of the random transposition shuffle. As I suggested in Seoul, Amaury says: “One important problem in my opinion is to say something about analogues of classical random walks on $S_n$ (for instance the random transpositions or riffle shuffle)”. Amaury notes that “we are blocked by the counit problem. We must therefore seek bounds for other distances. As I suggest in my paper, we may look at the norm of the difference of the transition operators. The $\mathcal{L}^2$-estimate that I give is somehow the simplest thing one can do and should be thought of as a “spectral gap” estimate. Better norms would be the norms as operators on $\mathcal{L}^\infty$ or even better, the completely bounded norm. However, I have not the least idea of how to estimate this.”

## Results to be Improved

• I have recently received an email from Isabelle Baraquin, a student of Uwe Franz, pointing out a small error in the thesis (a basis-error with the Kac-Paljutkin quantum groups).
• Recent calculations suggest that the lower bound for the random walk on the dual of $S_n$ is effective at $k\sim (n-1)!$ while the upper bound shows the walk is random at time order $n!$.  This is still a very large gap but at least the lower bound shows that this walk does converge very slowly.
• Get a much sharper lower bound for the random walk on the Sekine family of quantum groups studied in Section 5.7. Projection onto the ‘middle’ of the $M_n(\mathbb{C})$ factor may provide something of use. On mature reflection, recognising that the application of the upper bound lemma is dominated by one set of terms in particular, it should be possible to use cruder but more elegant estimates to get the same upper bound except with lighter calculations (and also a smaller $\alpha$ — see Section 5.7).

## More Questions on Random Walks

• Irreducibility is harder than the classical case (where ‘not concentrated’ on a subgroup is enough). Can anything be said about aperiodicity in the quantum case? (U. Franz).
• Prove an Ergodic Theorem (Theorem 1.3.2) for Finite Quantum Groups. Extend to Compact Quantum Groups. It is expected that the conditions may be more difficult than the classical case. However, it may be possible to use Diaconis-Van Daele theory to get some results in this direction. It should be possible to completely analyse some examples (such as the Kac-Paljutkin quantum group of order 8).This will involve a study of subgroups of quantum groups as well as normal quantum subgroups and cosets.
• Look at a random walk on the Sekine quantum groups with an $n$-dependent driving probability and see if the cut-off phenomenon (Chapter 4) can be detected. This will need good lower bounds for $k\ll t_n$, some cut-off time.
• Convolutions of the Random Distribution: such a study may prove fruitful in trying to find the Ergodic Theorem. See Section 6.5.
• Amaury mentions the problem of considering random walks associated to non-central states (in the compact case). “The difficulty is first to build non-central states (I do not have explicit examples at hand but Uwe Franz said he had some) and second to be able to compute their Fourier transform. Then, the computations will certainly be hard but may still be doable.”
• A study of the Cesaro means: see Section 6.6.
• Spectral Analysis: it should be possible to derive crude bounds using the spectrum of the stochastic operator. More in Section 6.2.

## Future Work (for which I do not yet have the tools to attack)

• Amaury/Franz Something perhaps more accessible is to investigate quantum homogeneous spaces. The free sphere is a noncommutative analogue of the usual sphere and a quantum homogeneous space for the free orthogonal quantum group. We can therefore define random walks on it and the whole machinery of Gelfand pairs might be available. In particular, Caspers gave a Plancherel theorem for Gelfand pairs of locally compact quantum groups which should apply here yielding an Upper Bound Lemma and then the problem boils down to something which should be close to my computations. There are probably works around this involving Adam Skalski and coauthors.
• Amaury: If one can prove a more general total variation distance equal to half one norm result, then Amaury suggests one can consider random walks on compact quantum groups which are not of Kac type. The Upper Bound Lemma will then involve matrices $Q$ measuring the modular theory of the Haar state and some (but not all) dimensions in the formulas must be replaced by quantum dimensions. The main problem here is to define explicit central states since there is no Haar-state preserving conditional expectation onto the central algebra. However, there are tools from monoidal equivalence to do this.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Maple

We will have our second Maple Lab next week (Week 4: 21 March 2018).

If you have missed the first lab you have two options: either download Maple onto your own machine or come into CIT at another time to use Maple.

Go through the first lab on your own, doing all the exercises in Maple (Exercises 1, 2, and 3). Save the worksheet and email it to me.

If you have never done Maple before you might want to do a Lab or two with me before catching up.

## Week 3

We did some examples of matrix arithmetic and looked at the concept of a matrix inverse.

We had our first Maple Lab

## Week 4

We will continue working with matrix inverses, seeing how the Gauss-Jordan algorithm can be used to calculate the inverse of a $3\times 3$ matrix.

We will have our second Maple lab.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Assignment 1 – Warning

Assignment 1 has a hand-in time and date of 14:30 Friday 2 March (Week 5) and has been given out.

This assessment is worth 15% of your final grade. The corresponding question on the final exam is worth 24.5% of your final grade. This assessment is designed to aid your understanding of this topic and go on to score well in the final exam.

Careful in your “collaboration” — don’t take an explanation, etc. from another
student unless it makes sense to you: otherwise you are not going to get the benefit out of completing this assignment.

## Postponed Lecture

There will be no class this coming Monday, 19 February. This class will instead take place at 13:00 on Tuesday, 20 February: in B212.

## Week 3

We looked at applications to temperature distribution, where the Jacobi Method is used to find approximate solutions to a diagonally dominant linear system. We started work on Chapter 2 — the method of undetermined coefficients for solving linear odes.

In tutorial you worked on applications of linear systems (flow networks and plates).

## Week 4

We will continue our work on Chapter 2 — the method of undetermined coefficients for solving linear odes.

Amaury Freslon has put a pre-print on the arXiv, Cut-off phenomenon for random walks on free orthogonal quantum groups, that answers so many of these questions, some of which appeared as natural further problems in my PhD thesis.

It really is a fantastic paper and I am delighted to see my PhD work cited: it appears that while I may have taken some of the low hanging fruit, Amaury has really extended these ideas and has developed some fantastic examples: all beyond my current tools.

This pre-print gives me great impetus to draft a pre-print of my PhD work, hopefully for publication. I am committed to improving my results and presentation, and Amaury’s paper certainly provides some inspiration is this direction.

As things stand I do not have to tools to develop results as good as Amaury’s. Therefore I am trying to develop my understanding of compact quantum groups and their representation theory. Afterwards I can hopefully study some of the remaining further problems mentioned in the thesis.

As suggested by Uwe Franz, representation theoretic methods (such as presented by Diaconis (1988) for the classical case), might be useful for analysing random walks on quantum homogeneous spaces.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 3

We finished our work on Taylor Series, and used it to analyse the errors when using the Euler Method.

In VBA we should all have been able to finish Lab 2 (certainly up to automatic selection, p.108). Those who did so started on a Newton Cooling program. Students who have not completed these two programs (damper and Newton Cooling), are advised to work on them outside class.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Maple

We will have Maple next week (Week 3: 14 March 2018).

## Week 2

We did one more example of Gaussian elimination and then we introduced matrices as linear maps and discuss some of their properties.

For those of you interested in the why when it comes to matrix multiplication, have a look here.

We hoped to have a Maple class but the software was not on the machines in C128.

## Week 3

We will continue working with matrices and perhaps introduce matrix inverses.

We will have our first Maple lab.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 2

We finished looking at Gaussian Elimination (including Gaussian Elimination with Partial Pivoting — absolutely necessary if you are rounding.).

If you download Maple (see Student Resources), there is a Maple Tutor that is easy to use and will help you. Open up Maple and go to Tools -> Tutors -> Linear Algebra -> Gaussian Elimination.

We looked at applications to network flows: traffic and pipes.

## Week 3

We will look at applications of linear systems to temperature distribution.