Just some notes on the pre-print. I am looking at this paper to better understand this pre-print. In particular I am hoping to learn more about the support of a probability on a quantum group. Flags and notes are added but mistakes are mine alone.

Abstract

From this paper I will look at:

• lattice operations on $\mathcal{I}(G)$, for $G$ a LCQG (analogues of intersection and generation)

1. Introduction

Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with quasi-subgroups (the work of Franz & Skalski the correspondence was with pre-subgroups and group-like projections).

Let us show the kind of thing I am trying to understand better.

Let $F(G)$ be the algebra of function on a finite quantum group. Let $\nu,\,\mu\in M_p(G)$ be concentrated on a pre-subgroup $S$. We can associate to $S$ a group like projection $p_S$.

Let, and this is another thing I am trying to understand better, this support, the support of $\nu$ be ‘the smallest’ (?) projection $p\in F(G)$ such that $\nu(p)=1$. Denote this projection by $p_\nu$. Define $p_\mu$ similarly. That $\mu,\,\nu$ are concentrated on $S$ is to say that $p_\nu\leq p_S$ and $p_\mu\leq p_S$.

Define a map $T_\nu:F(G)\rightarrow F(G)$ by

$a\mapsto p_\nu a$ (or should this be $ap_\nu$ or $p_\nu a p_\nu$?)

We can decompose, in the finite case, $F(G)\cong \text{Im}(T_\nu)\oplus \ker(T_\nu)$

Claim: If $\nu$ is concentrated on $S$$\nu(ap_S)=\nu(a)$I don’t have a proof but it should fall out of something like $p_\nu\leq p_S\Rightarrow \ker p_\nu\subseteq \ker p_S$ together with the decomposition of $F(G)$ above. It may also require that $\int_G$ is a trace, I don’t know. Something very similar in the preprint.

From here we can do the following. That $p_S$ is a group-like projection means that:

$\Delta (p_s)(\mathbf{1}_G\otimes p_S)=p_S\otimes p_S$

$\Rightarrow \sum p_{S(1)}\otimes (p_{S(2)}p_S)=p_S\otimes p_S$

Hit both sides with $\nu\times \mu$ to get:

$\sum \nu(p_{S(1)})\mu(p_{S(2)}p_S)=\nu(p_S)\mu(p_S)$.

By the fact that $\nu,\,\mu$ are supported on $S$, the right-hand side equals one, and by the as-yet-unproven claim, we have

$\sum \nu(p_{S(1)})\mu(p_{S(2)})=1$.

However this is the same as

$(\nu\otimes\mu)\Delta(p_S)=1\Rightarrow (\nu\star \mu)(p_S)=1$,

in other words $p_{\nu\star \mu}\leq p_S$, that is $\nu\star \mu$ remains supported on $S$. As a corollary, a random walk driven by a probability concentrated on a pre-subgroup $S\subset G$ remains concentrated on $S$.

Slides of a talk given to the Functional Analysis seminar in Besancon.

Some of these problems have since been solved.

“e in support” implies convergence

Consider a $\nu\in M_p(G)$ on a finite quantum group such that where

$M_p(G)\subset \mathbb{C}\varepsilon \oplus (\ker \varepsilon)^*$,

$\nu=\nu(e)\varepsilon+\psi$ with $\nu(e)>0$. This has a positive density of trace one (with respect to the Haar state $\int_G\in M_p(G)$), say

$\displaystyle a_\nu=\nu(e)\eta+b_\psi\in \mathbb{C}\eta\oplus \ker \varepsilon$,

where $\eta$ is the Haar element.

An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so $b_\psi\geq 0$. Assume that $b_\psi\neq 0$ (if $b_\psi=0$, then $\psi=0\Rightarrow \nu=\varepsilon\Rightarrow \nu^{\star k}=\varepsilon$ for all $k$ and we have trivial convergence)

Therefore let

$\displaystyle a_{\tilde{\psi}}:=\frac{b_\psi}{\int_G b_\psi}$

be the density of $\tilde{\psi}\in M_p(G)$.

Now we can explicitly write

$\displaystyle \nu=\nu(e)\varepsilon+(1-\nu(e))\tilde{\psi}$.

This has stochastic operator

$P_\nu=\nu(e)I_{F(G)}+(1-\nu(e))P_{\tilde{\psi}}$.

Let $\lambda$ be an eigenvalue of $P_\nu$ of eigenvector $a$. This yields

$\nu(e)a+(1-\nu(e))P_{\tilde{\psi}}(a)=\lambda a$

and thus

$\displaystyle P_{\tilde{\psi}}a=\frac{\lambda-\nu(e)}{1-\nu(e)}a$.

Therefore, as $a$ is also an eigenvector for $P_{\tilde{\psi}}$, and $P_{\tilde{\psi}}$ is a stochastic operator (if $a$ is an eigenvector of eigenvalue $|\lambda|>1$, then $\|P_\nu a\|_1=|\lambda|\|a\|_1\leq \|a\|_1$, contradiction), we have

$\displaystyle \left|\frac{\lambda-\nu(e)}{1-\nu(e)}\right|\leq 1$

$\Rightarrow |\lambda-\nu(e)|\leq 1-\nu(e)$.

This means that the eigenvalues of $P_\nu$ lie in the ball $B_{1-\nu(e)}(\nu(e))$ and thus the only eigenvalue of magnitude one is $\lambda=1$, which has (left)-eigenvector the stationary distribution of $P_\nu$, say $\nu_\infty$.

If $\nu$ is symmetric/reversible in the sense that $\nu=\nu\circ S$, then $P_\nu$ is self-adjoint and has a basis of (left)-eigenvectors $\{\nu_\infty=:u_1,u_2,\dots,u_{|G|}\}\subset \mathbb{C}G$ and we have, if we write $\nu=\sum_{t=1}^{|G|}a_tu_t$,

$\displaystyle \nu^{\star k}=\sum_{t=1}^{|G|}a_t\lambda_t^ku_t$,

which converges to $a_1\nu_\infty$ (so that $a_1=1$).

If $\nu$ is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers $P_{\nu}^k$ converge and thus so do the $\nu^{\star k}$ $\bullet$

Total Variation Decrasing

Uses Simeng Wang’s $\|a\star_Ab\|_1\leq \|a\|_1\|b\|_1$. Result holds for compact Kac if the state has a density.

Periodic $e^2$ is concentrated on a coset of a proper normal subgroup of $\mathfrak{G}_0$

$e_2+e_4$ is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by $\langle e_1,e_3\rangle$

Supported on Subgroup implies Reducible

I have a proof that reducible is equivalent to supported on a pre-subgroup.

• 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:

2. does my answer make sense?
3. 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)

Week 12

On Monday, and Wednesday PM, we finished the module by looking at triple integrals.

The Wednesday 09:00 lecture was a tutorial along with most of Wednesday PM and the Thursday class.

• 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:

2. does my answer make sense?
3. 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)

Assignment 2

Has been corrected and results emailed to you.

Some remarks on common mistakes here.

Week 11

We had a systems of differential equations tutorial Monday and before looking at double integrals.

Week 12

We will look at triple integrals and then have one or two tutorials on. Possibly Wednesday 09:00 for double integrals and Thursday for triple integrals.

Week 13

We will review the Summer 2018 paper.

Study

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

Exam Papers

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

Test 2

Test 2, worth 15% and based on Chapter 3, will now take place Week 12, 30 April. There is a sample test [to give an indication of length and layout only] in the notes (marking scheme) and the test will be based on Chapter 3 only.

More Q. 1s (on the test) can be found on p.112; more Q. 2s on p. 117; more Q. 3s on p.125 and p.172, Q.1; more Q. 4s on p.136, and more Q. 5s on p. 143.

Chapter 3 Summary p. 144.

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

Homework

Once you are prepared for Test 2 you can start looking at Chapter 4:

• Revision of Integration, p.161.
• p.167, Q. 1-5
• p.182

Week 11

We had some tutorial time from 18:00-19:00 for further differentiation (i.e. for Test 2, after Easter).

We completed our review of antidifferentiation before starting Chapter 4 proper.

We looked at Integration by Parts and centroids.

For those who could not make it here is some video and slides from what we did after the video died.

Week 12

We will have some tutorial time from 18:00-19:00 for further differentiation.

We will have Test 2 from 19:00-20:05.

Then we will look at completing the square, centres of gravity, and work.

Week 13

We will look the Winter 2018 paper at the back of your manual.

CIT Mathematics Exam Papers

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

Week 10

We had two additional tutorials… actually four tutorials in total and two lectures; the lectures focused on Systems of Differential Equations.

Assignment 2

Assignment 2 now has a pushed back deadline of 12:00, 12 April: the Friday of Week 11. Assignment 2 is in the manual, P. 164. Usual warnings about copying apply.

Week 11

We will have a systems of differential equations tutorial Monday and then look at double integrals.

Week 12

We will look at triple integrals and then have one or two tutorials. Possibly Monday for double integrals and Thursday for triple integrals.

Week 13

We will review the Summer 2018 paper.

Study

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

Exam Papers

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

Test 2

Test 2, worth 15% and based on Chapter 3, will now take place Week 12, 30 April. There is a sample test in the notes and the test will be based on Chapter 3 only.

Homework Exercises

If you do any of the suggested exercises you can give them to me for correction. Please feel free to ask me questions about the exercises via email or even better on this webpage.

I recommend strongly that everyone completes P.102, Q.1.

After that you can look at:

• P.136, Q.1-3
• P. 143, Q. 1-4
• P. 125, Q. 1-4, P.172, Q.1
• P. 117, Q. 1-4
• P.112, Q. 1-5
• Sample Test 2, P.145

If you want to do more again, look at P.113, Q.6-9, P. 118, Q. 5-6, P. 125, Q.5. There is a Weekly Summary for the Chapter 3 Material on P.144.

If you read on there is some information below about solutions to these exercises.

Week 10

For those who were able to make it we had some tutorial time from 18:00-19:00 for parametric, implicit, and related rates differentiation. If you are really interested in understanding how does a curve have an equation, see here.

In class we looked at partial differentiation and error analysis. For those who could not make it, here is video of the partial differentiation material and here are the slides from the error analysis (the video died shortly after we started error analysis).

We only started a revision of Antidifferentiation to start Chapter 4 on (Further) Integration. I have this section completed here.

Week 11

We will have some tutorial time from 18:00-19:00 for further differentiation (i.e. for Test 2, after Easter).

We are under pressure for time but I have made the decision that we will be better off completing our review of antidifferentiation before starting Chapter 4 proper. This might put us under time pressure later on but I believe it is the correct thing to do.

We will look at Integration by Parts, completing the square, and work.

VBA Assessment 1 & Written Assessment 1 – Results

VBA Assessment 1 Results have been emailed and I hope to have your Written Assessment 1 Results to ye Wednesday or Thursday.

Week 10

We looked at finite differences for the Heat Equation. This completes the examinable written material.

In VBA we implemented same.

Week 11 — 2nd 20% VBA Assessment

I will be in B242 from 08:30 – 09:00 to help with any questions, ideally the p. 146 tutorial equations. This is extra time that I am making myself available but it is just an option for you.

This tutorial time will continue in B242 until 09:55.

In the 12:00 class we will have a revision session, geared towards the 40% Written Assessment 2.

To understand how your student numbers generate constants (see below) see this VBA Test 2 from 2017 (do not read this as a sample – it included e.g. the Heat Equation which you will not be examined on and the Laplace’s Equation might be slightly simpler than what ye will have).

Your VBA 2 Assessment will consist of three questions:

• shooting method
• finite differences; steady state temperature uninsulated rod (more P. 90)
• Laplace’s Equation

Formulae will be provided in the VBA 2 Assessment.

See last weeks’ Summary for more detail on the VBA 2 assessment.

Week 12 — 40% Written Test — After Easter

There will be no 12:00 class but I will be in B242 from 08:00 until about 08:40 for any last minute questions.

The 40% Written Assessment will be broken up into two parts.
• Theory Element Tuesday 30 April, Melbourne Rows E-G, 09:00 (30 minutes worth but given an hour).

It will be geared more towards theoretical questions. Please see P. 108-110. More questions p.84, Q. 3.

• Calculation Element in your Week 12 VBA time and lab, (45 minutes worth but given an hour and 45 minutes)

The second part of the Test will take place in your VBA slot. I have to tell you in advance what questions are coming up so let us say

1. Second Order Problem Using Heun’s Method
2. Heat Flux Density at a Point (p.101)
3. Heat Equation

Each group will get questions with only minor variations from the sample questions p. 111 (more Q. 1 on P.55).

Formulae will be provided in the Written Assessment 2.

Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

Final Concept MCQ League Table

Unfortunately with my illness this kind of ran of steam so this is the final standings.

VBA Assessment 1 & Written Assessment 1 – Results

I have been very ill over the last two weeks but have gotten some meds from the doctor and hope to get these back to you ASAP.

Weeks 8 & 9

I missed the 09:00 class on the Tuesday of Week 8 with illness.

In the afternoon, we did two examples: of the Shooting Method and of Finite Differences (for the temperature along a rod). Please see Shooting_and_FiniteDifferences_Examples.

In Week 9, we started looking at partial differential equations by looking at Laplace’s Equation.

In VBA, in Week 8 we had MCQ VI and we did the Boundary Value Problems lab.

In VBA, in Week 9 we did the Laplace Equation Lab (which also had some 1-d boundary value stuff). I will email on a VBA file of the 1-D finite differences problem.

This completes the examinable VBA material. The Heat Equation that we cover in Week 10 will not be examinable.

Week 10

We will look at finite differences for the Heat Equation. This completes the examinable written material.

In VBA we will implement same.