## Test 2

On Chapter 3, on at 5 pm (not 4 pm) Monday 25 November, Week 11 in Melbourne Hall. You will have one full hour.

The BioEng2B tutorial will take place from 4 pm sharp to 4.45 pm in B263 on that day.

Chapter 3: Differentiation is going to be examined. A Summary of Differentiation (p.147-8): you will want to know this stuff very well. You will be given a copy of these tables

There is a sample test on p. 149 of the notes.

I strongly advise you that 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.

Between tutorials and private study you really should aim to have completed as least the following:

• P. 113, Q. 1-6 (not 5c or 6iii)
• P. 119, Q. 1-4
• P. 127, Q. 1-4
• P. 138, Q. 1-3
• P. 146, Q. 1-4
• The Sample Test

There are more questions in most of these exercises.

If you are having any problem, take a photo of your work and email me your question.

## Week 10

We started Chapter 4 on (Further) Integration with a revision of antidifferentiation, and had a look at Integration by Parts. We used implicit differentiation to differentiate inverse sine.

## Week 11

We have our test on Monday, then we will look at completing the square, and work on Tuesday and Thursday.

## Week 12

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.

## Week 13

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.

## Test 2

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.

## Week 9

In Week 9 we finished talking about equations (and quadratic equations) and began studying exponents. We saw at the very end of Friday’s lecture that if we define a function:

$2^{\Box}:\mathbb{R}\rightarrow \mathbb{R}^+$,

by $2^{\Box}(x)=2^x$, then we define $\log_2:\mathbb{R}^+\rightarrow \mathbb{R}$ as an inverse function:

$\log_2:=(2^{\Box})^{-1}$.

## Week 10

We will introduce more and study the properties of uses of logarithms.

## Tutorial Split

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

## Assessment 2

Three things must be submitted:

• a soft, digital, copy of your Excel file MATH7019A2 – Your Name
• a hard copy of your Excel file MATH7019A2 – Your Name

The soft copy is to be be submitted on CANVAS.

Combine the second two elements into one report. Ideally the Excel work for Problem A near the written work for Problem A, etc. If this isn’t easy, maybe just put all the written work at the front, and all the Excel work at the back.

The assignment can be handed up in any class before Monday 18 November (inclusive). I expect most to hand it up on Monday:

• Monday, 18 November 13:00 in B228,

Otherwise drop the assessment to my office A283. I will be here Monday 18 November, until 16:00 sharp (which is the deadline).

Work assigned late will be awarded a mark of zero. Hand up what you have on time.

## Week 9

We had an extra tutorial on Monday.

We continued to slowly work our way through Chapter 3 by looking at Sampling and Hypothesis Testing.

## Week 10

We will have an extra tutorial on Monday.

We will finish off Chapter 3 — and talk about the Good and Bad News — before we begin Chapter 4 with a Revision of Differentiation and go on to look at Maclaurin Series and Taylor Series.

In my pursuit of an Ergodic Theorem for Random Walks on (probably finite) Quantum Groups, I have been looking at analogues of Irreducible and Periodic. I have, more or less, got a handle on irreducibility, but I am better at periodicity than aperiodicity.

The question of how to generalise these notions from the (finite) classical to noncommutative world has already been considered in a paper (whose title is the title of this post) of Fagnola and Pellicer. I can use their definition of periodic, and show that the definition of irreducible that I use is equivalent. This post is based largely on that paper.

## Introduction

Consider a random walk on a finite group $G$ driven by $\nu\in M_p(G)$. The state of the random walk after $k$ steps is given by $\nu^{\star k}$, defined inductively (on the algebra of functions level) by the associative

$\nu\star \nu=(\nu\otimes\nu)\circ \Delta$.

The convolution is also implemented by right multiplication by the stochastic operator:

$\nu\star \nu=\nu P$,

where $P\in L(F(G))$ has entries, with respect to a basis $(\delta_{g_i})_{i\geq 1}$ $P_{ij}=\nu(g_jg_{i^{-1}})$. Furthermore, therefore

$\nu^{\star k}=\varepsilon P^k$,

and so the stochastic operator $P$ describes the random walk just as well as the driving probabilty $\nu$.

The random walk driven by $\nu$ is said to be irreducible if for all $g_\ell\in G$, there exists $k\in\mathbb{N}$ such that (if $g_1=e$) $[P^k]_{1\ell}>0$.

The period of the random walk is defined by:

$\displaystyle \gcd\left(d\in\mathbb{N}:[P^d]_{11}>0\right)$.

The random walk is said to be aperiodic if the period of the random walk is one.

These statements have counterparts on the set level.

If $P$ is not irreducible, there exists a proper subset of $G$, say $S\subsetneq G$, such that the set of functions supported on $S$ are $P$-invariant.  It turns out that $S$ is a proper subgroup of $G$.

Moreover, when $P$ is irreducible, the period is the greatest common divisor of all the natural numbers $d$ such that there exists a partition $S_0, S_1, \dots, S_{d-1}$ of $G$ such that the subalgebras $A_k$ of functions supported in $S_k$ satisfy:

$P(A_k)=A_{k-1}$ and $P(A_{0})=A_{d-1}$ (slight typo in the paper here).

In fact, in this case it is necessarily the case that $\nu$ is concentrated on a coset of a proper normal subgroup $N\rhd G$, say $gN$. Then $S_k=g^kN$.

Suppose that $f$ is supported on $g^kN$We want to show that for $Pf\in A_{k-1}$Recall that

$\nu^{\star k-1}P(f)=\nu^{\star k}(f)$.

This shows how the stochastic operator reduces the index $P(A_k)=A_{k-1}$.

A central component of Fagnola and Pellicer’s paper are results about how the decomposition of a stochastic operator:

$P(f)=\sum_{\ell}L_\ell^*fL_{\ell}$,

specifically the maps $L_\ell$ can speak to the irreducibility and periodicity of the random walk given by $P$. I am not convinced that I need these results (even though I show how they are applicable).

## Stochastic Operators and Operator Algebras

Let $F(X)$ be a $\mathrm{C}^*$-algebra (so that $X$ is in general a  virtual object). A $\mathrm{C}^*$-subalgebra $F(Y)$ is hereditary if whenever $f\in F(X)^+$ and $h\in F(Y)^+$, and $f\leq h$, then $f\in F(Y)^+$.

It can be shown that if $F(Y)$ is a hereditary subalgebra of $F(X)$ that there exists a projection $\mathbf{1}_Y\in F(X)$ such that:

$F(Y)=\mathbf{1}_YF(X)\mathbf{1}_Y$.

All hereditary subalgebras are of this form.

## Test 2

On Chapter 3, on at 5 pm (not 4 pm) Monday 25 November, Week 11 in Melbourne Hall

The BioEng2B tutorial will take place from 4 pm sharp to 4.45 pm in B263 on that day.

Chapter 3: Differentiation is going to be examined. A Summary of Differentiation (p.147-8): you will want to know this stuff very well. You will be given a copy of these tables

There is a sample test on p. 149 of the notes.

I strongly advise you that 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.

Between tutorials and private study you really should aim to have completed as least the following:

• P. 113, Q. 1-6 (not 5c or 6iii)
• P. 119, Q. 1-4
• P. 127, Q. 1-4
• P. 138, Q. 1-3
• P. 146, Q. 1-4
• The Sample Test

There are more questions in most of these exercises.

If you are having any problem, take a photo of your work and email me your question.

## Week 9

We looked at partial differentiation and its applications to differentials and error analysis.

I had hoped to do a Concept MCQ for Chapter 3 but we kind of ran out of time… I would ask ye to complete this (on your own) before Monday.

## Week 10

We will start Chapter 4 on (Further) Integration. A good revision of integration/antidifferentiation may be found here, but this material will be gone through in Monday’s lecture..

## Week 9

In Week 9 we will finish talking about equations (and quadratic equations) and begin studying exponents.

## Assessment 2

Assessment 2 is on p.137. The relevant Excel files have been emailed to you. It has a hand in date/time of 16:00 Monday 18 November, Week 10.

Start ASAP.

## Week 8

We did more work on Chapter 3; we looked at the Normal distribution, and we started discussing Sampling.

## Week 9

We will continue to slowly work our way through Chapter 3 by looking at Sampling and Hypothesis Testing.

We will have an extra tutorial on Monday.

There will also be a new arrangement with tutorials next week — watch your email.

## Test 2

On Chapter 3, on at 5 pm (not 4 pm) Monday 25 November, Week 11 in Melbourne Hall

The BioEng2B tutorial will take place from 4 pm sharp to 4.45 pm in B263 on that day.

Chapter 3: Differentiation is going to be examined. A Summary of Differentiation (p.147-8): you will want to know this stuff very well. You will be given a copy of these tables

There is a sample test on p. 149 of the notes.

I strongly advise you that 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.

## Week 8

We finished looking at Related Rates and then we looked at and finished Implicit Differentiation.

I have video of the above material here.

We then start looking at Functions of Several Variables.

## Week 9

We will look at partial differentiation and its applications to differentials and error analysis.

We might start Chapter 4 on (Further) Integration. A good revision of integration/antidifferentiation may be found here.

If you have time to look at MATH6055 over the Reading Break, may I suggest that you look at the Functions Exercises:

P. 61, Q. 1-11, 13
P. 54, Q. 1-4
P. 46, Q. 1-5
P. 41, Q. 1-2, 3 (a), 4, 5 (a)
If you have any questions please email me, perhaps with a photo of your work.

## Catch Up Lecture

13:00 this coming Wednesday 6 November, in B217.

## Week 7

In Week 7 we started delving more into algebra

## Week 8

In Week 8 we will start talking about equations.

## Assessment 2

Assessment 2 is on p.137. The relevant Excel files have been emailed to you. It has a hand in date/time of 16:00 Monday 18 November, Week 10, and you can already do all the questions.

Start as soon as reasonable: perhaps during this Reading Week.

## Week 7

We had a tutorial on Monday on Static Beam Equations. On Wednesday we looked at an MCQ on Static Beam Equations.

Then we looked at the Three Term Taylor Method and began Chapter 3 on Probability and Statistics. We looked in particular emphasis on Mutual Exclusivity and Independence.

Our Wednesday lecture was only 30 minutes long.

## Week 8

We will do more work on Chapter 3; perhaps we will start looking at the Normal distribution. We will not have a tutorial this Monday.