Transposition Project – Part II

October 22 you took a quiz as part of the Transposition Project that the Mathematics Department is undertaking in an effort to improve our teaching.

You will have another 15 minute quiz on Monday.

If you do not have an internet ready device, or did not do the first quiz, you may leave class early.

Thank you again for your participation.

Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, will take place on Friday 30 November in the usual lecture venue of B214.

All Chapter 3 material is examinable, but the test will be similar to the sample test.

There will also be a question on the growth rate of functions: the relevant info to answer these questions are on p. 81, and they are exercises 18, 19 on p. 79.

Here is a sample test.

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 10

We introduced and studied the properties of uses of logarithms. We began the chapter on Network Theory by looking at the Bridges of Konigsberg Problem.

Week 11

In Week 11 we will explore Network Theory, or rather Graph Theory, in more depth. We will look at digraphsconnectednessdegreewalks, and trees.

Test 2 will be on Friday.

Assessment 2

Assessment 2 is on p.136.

• The hand in deadline is 16:00 this Monday 26 November 2018.
• Hand it at tutorial today — or 13:00 on Monday. Otherwise drop it into my office, A283. I should be here Monday 11:00-13:00, 14:00-15:00. Outside these times there should be someone in A283 who can let you in and leave your assignment on my desk.
• Hand in whatever you have done by the deadline: work handed in late will be assigned a mark of zero.
• Email me your Excel work.
• Print off a hard copy of your excel and submit this with any other written work.
• Further instructions in the manual.

Week 10

We began Chapter 4 with a Revision of Differentiation, and on Wednesday you had a tutorial to get up to grips with calculating derivatives.

We went on to look at at Maclaurin Series.

Week 11

We will look at more general Taylor Series: not just near $a=0$ — and also for functions of several variables.

Test 2

The 15% Test 2 will take place at 16:00 on Monday 26 November, Week 11, in B263. There is a sample test in the notes, p.146. Chapter 3: Differentiation is going to be examined. A Summary of Differentiation (p.144): you will want to know this stuff very well. You will be given a copy of these tables

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 10

We looked at completing the squarework and centroids.

A good revision of integration/antidifferentiation may be found here.

Week 11

We have our test on Monday.

On Tuesday we will have a tutorial for a lecture. Those who are not strong on antidifferentiation will be given revision exercises from MATH6015.

On Thursday we will look at centres of gravity of solids of revolution.

Week 12

On Monday and Tuesday we will have extra tutorials: you will be invited to either work on integration and/or matrices; the choice will be up to you.

On Thursday we will finish off the module by doing an extra example of a centre of gravity of a solid of revolution.

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 E15
• Monday at 17:00 in B189
• Thursday at 12:00 in E4

Study

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

Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, will take place on Friday 30 November in the usual lecture venue of B214.

Here is a provisional sample test. If we don’t cover something by 20 November inclusive it won’t be on the test: this is so you will have two full tutorials before the test. This will be clarified on Tuesday 20 November.

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 quadratics, and studied exponents; with a first look at exponential equations and exponential functions.

Week 10

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

Assessment 2

Assessment 2 is on p.136. It has a hand-in time of 16:00 Monday 26 November.

As suggested in class, I would advise you to — if possible — complete this assignment early if you can, freeing up time in your tutorial to get work done on Chapter 3: Probability & Statistics.

Week 9

We completed Chapter 3 by looking at Sampling and Hypothesis Testing. We had a tutorial for a lecture on Wednesday.

Week 10

We will begin Chapter 4 with a Revision of Differentiation and go on to look at then at Maclaurin Series and Taylor Series. We might hold an extra tutorial for a lecture.

Test 2

The 15% Test 2 will take place at 16:00 on Monday 26 November, Week 11, in B263. There is a sample test in the notes, p.146. Chapter 3: Differentiation is going to be examined. A Summary of Differentiation (p.144): you will want to know this stuff very well. You will be given a copy of these tables

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 9

We looked at applications of partial differentiation to differentials and error analysis. We started Chapter 4 on (Further) Integration by looking at Integration by Parts.

A good revision of integration/antidifferentiation may be found here.

Week 10

We will look at completing the square and work.

Study

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

Student Resources

In a recent preprint, Haonan Zhang shows that if $\nu\in M_p(Y_n)$ (where $Y_n$ is a Sekine Finite Quantum Group), then the convolution powers, $\nu^{\star k}$, converges if

$\nu(e_{(0,0)})>0$.

The algebra of functions $F(Y_n)$ is a multimatrix algebra:

$F(Y_n)=\left(\bigoplus_{i,j\in\mathbb{Z}_n}\mathbb{C}e_{(i,j)}\right)\oplus M_n(\mathbb{C})$.

As it happens, where $a=\sum_{i,j\in\mathbb{Z}_n}x_{(i,j)}e_{(i,j)}\oplus A$, the counit on $F(Y_n)$ is given by $\varepsilon(a)=x_{(0,0)}$, that is $\varepsilon=e^{(0,0)}$, dual to $e_{(0,0)}$.

To help with intuition, making the incorrect assumption that $Y_n$ is a classical group (so that $F(Y_n)$ is commutative — it’s not), because $\varepsilon=e^{(0,0)}$, the statement $\nu(e_{(0,0)})>0$, implies that for a real coefficient $x^{(0,0)}>0$,

$\nu=x^{(0,0)}\varepsilon+\cdots= x^{(0,0)}\delta^e+\cdots$,

as for classical groups $\varepsilon=\delta^e$.

That is the condition $\nu(e_{(0,0)})>0$ is a quantum analogue of $e\in\text{supp}(\nu)$.

Consider a random walk on a classical (the algebra of functions on $G$ is commutative) finite group $G$ driven by a $\nu\in M_p(G)$.

The following is a very non-algebra-of-functions-y proof that $e\in \text{supp}(\nu)$ implies that the convolution powers of $\nu$ converge.

Proof: Let $H$ be the smallest subgroup of $G$ on which $\nu$ is supported:

$\displaystyle H=\bigcap_{\underset{\nu(S_i)=1}{S_i\subset G}}S_i$.

We claim that the random walk on $H$ driven by $\nu$ is ergordic (see Theorem 1.3.2).

The driving probability $\nu\in M_p(G)$ is not supported on any proper subgroup of $H$, by the definition of $H$.

If $\nu$ is supported on a coset of proper normal subgroup $N$, say $Nx$, then because $e\in \text{supp}(\nu)$, this coset must be $Ne\cong N$, but this also contradicts the definition of $H$.

Therefore, $\nu^{\star k}$ converges to the uniform distribution on $H$ $\bullet$

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist $\nu\in M_p(G)$ that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of $e\in\text{supp}(\nu)$.

So we have some questions:

• Is there a proof of the classical result (above) in the language of the algebra of functions on $G$, that necessarily bypasses talk of points and of subgroups?
• And can this proof be adapted to the quantum case?
• Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, will take place on Friday 30 November in the usual lecture venue of B214.

Here is a provisional sample test. If we don’t cover something by 20 November inclusive it won’t be on the test: this is so you will have two full tutorials before the test.

I will give you a copy of the sample today, Friday 9 November. 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 8

In Week 8 we finished talking about equations and started studying quadratics.

Week 9

In Week 9 we will finish talking about quadratics and begin studying exponents.

Assessment 1 – Results

I will have the assignments with me tomorrow if you want to see your work.

Assessment 2

Assessment 2 is on p.136. It has a hand-in time of 16:00 Monday 26 November.

As suggested in class, I would advise you to — if possible — complete this assignment early if you can, freeing up time in your tutorial to get work done on Chapter 3: Probability & Statistics.

Week 8

We looked at the Poisson distribution, the Normal distribution, and started discussing Sampling.

Week 9

We will complete Chapter 3 by looking at Sampling and Hypothesis Testing. We may have an extra tutorial during one of the lectures.

Test 2

The 15% Test 2 will take place at 16:00 on Monday 26 October, Week 11, in B263. There is a sample test in the notes, p.146. Chapter 3: Differentiation is going to be examined. A Summary of Vectors (p.144): you will want to know this stuff very well. You will be given a copy of these tables

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 looked at Implicit Differentiation and Partial Differentiation. If you are interested in a very “mathsy” approach to curves you can look at this.

Week 9

We will look at applications of partial differentiation to differentials and error analysis. We might start Chapter 4 on (Further) Integration. A good revision of integration/antidifferentiation may be found here.

Study

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