Some of these problems have since been solved.

Consider a on a *finite* quantum group such that where

,

with . This has a positive density of trace one (with respect to the Haar state ), say

,

where 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 . Assume that (if , then for all and we have trivial convergence)

Therefore let

be the density of .

Now we can explicitly write

.

This has stochastic operator

.

Let be an eigenvalue of of eigenvector . This yields

and thus

.

Therefore, as is also an eigenvector for , and is a stochastic operator (if is an eigenvector of eigenvalue , then , contradiction), we have

.

This means that the eigenvalues of lie in the ball and thus the only eigenvalue of magnitude one is , which has (left)-eigenvector the stationary distribution of , say .

If is symmetric/reversible in the sense that , then is self-adjoint and has a basis of (left)-eigenvectors and we have, if we write ,

,

which converges to (so that ).

If is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers converge and thus so do the

Uses Simeng Wang’s . Result holds for compact Kac if the state has a density.

is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by

I believe I have a full 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:

- have I answered the question that
__was asked__? - does my answer make sense?
- 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)

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.

Monday is a bank holiday.

In the Wednesday 09:00 lecture we will work on the Summer 2018 Paper and hopefully finish it before the end of the Thursday 10:00 lecture.

If we finish the Summer 2018 paper early (unlikely), any extra time will be dedicated to one-to-one help.

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

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>

- 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:

- have I answered the question that
__was asked__? - does my answer make sense?
- 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)

with comments, should be emailed to you this week. Test 2 Marking Scheme here.

We had some tutorial time from 18:00 – 19:00 before the test for those who could make it.

We then looked at sections 4.2 (completing the square) and 4.3 (work).

We did not have time to complete Example 4 on p.171, nor section 4.4.2 (centre of gravity of solid).

Both what we did in class and what we didn’t have time for can be found here.

**Very Important Note: centres of gravity of solids do not appear on the summer exam paper. More information next week.**

If you want to do some work on Chapter 4, Further Integration before starting more general revision you could look at:

- P. 161, Q. 1-15 (revision of integration)
- P. 167, Q. 1-5 (integration by parts)
- P. 173, Q. 3-7 (complete the square — and similarly )
- P. 176, Q. 1-3 (work)
- P. 182, Q. 1-3 (centroids of 2D laminas)

There is an exam paper (Winter 2018) at the back of your notes — I will go through this on the board from 19:00-22:00 Tuesday night.

Past exam papers (MATH6040 runs in Semester 1 and Semester 2) may be found here.

Recall that this module is MATH6040: Technological Maths 201 and not MATH6015: Technological Maths 2.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>Has been corrected and results emailed to you.

Some remarks on common mistakes here.

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

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.

We will review the Summer 2018 paper.

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

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>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.

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

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.

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.

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

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

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

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.

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

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

We will review the Summer 2018 paper.

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>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.

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.

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.

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.

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 centroids and centres of gravity.

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

A couple of students approached me last night inquiring about solutions for exercises. Firstly I want to apologise to the students: I didn’t really give ye a proper explanation of why I am not going to be providing solutions to exercises.

A bit of background is that when students are working on questions on an evening, while they can email me questions, they are not going to get a response until the next morning. In reality sometimes I do answer emails promptly and one thing I should have said to the students is… try… try send an email. I am not getting many emails nor handed up work and it is up to students to take that opportunity.

Well anyway, the students were wondering could I provide solutions to the exercises in the notes.

What I didn’t really explain to the students is as follows: basically there are two approaches to learning – shallow and deep. I am actually going to meet my HoD today about the fact that it is my opinion that the cramming of so many topics into your two maths modules is almost forcing you to take a shallow approach.

Usually a shallow approach to learning in mathematics would consist of learning from examples: basically doing loads of questions out and “getting the method”.

My big problem with this approach to learning is that it doesn’t persist: this kind of learning might get you through an exam but the knowledge and skills you pick up using this type of learning is going to leave your head as fast as it enters.

What is far more beneficial is a deep approach to learning.

Usually a deep approach to learning in mathematics is driven largely by “understanding” rather than what might be called “method”. A deep approach will involve taking from lectures an understanding, however rough, of what is going on, and taking that understanding and using it to attack exercises. When you get answers correct, your understanding is probably sound, but when you get answers wrong, or don’t know where to start, your understanding isn’t what it needs to be. At this point you look back at the notes and yes, perhaps examples, to see where your understanding is lacking. Hopefully you pick up the understanding that is missing and in my opinion this is what learning is.

Having said all that, I said beforehand more-or-less than a deep approach to MATH6040 is difficult because you are under time pressure and there are so many topics. Therefore you are kind of forced into a shallow approach: I still think a deep approach is better but I appreciate that you are in a short term thinking situation where your aim to do well in exams.

This means there must be some kind of leeway. Where this leeway exists is in the fact that the notes contain loads of worked examples — the examples we complete in class as well as worked examples in the notes (see below).

Taking Chapter 3, into the five sections:

- parametric differentiation: 4 examples in lectures, one worked example with comments (p. 111)
- related rates: 4 examples in lectures, one worked example with comments (p. 117)
- implicit differentiation: 5 examples in lectures, one worked example with comments (p. 124)
- partial differentiation: 5 examples in lectures, one worked example with comments (p. 136), selected solutions to exercises are also included
- error analysis: 3 examples in lectures, two worked example with comments (p. 141)

If a student doesn’t want to try an exercise without a solution I would invite them to try examples we did in class or worked examples without looking at the solutions.

But my advice would be to struggle on with the exercises… and do email questions.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

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

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

In VBA we implemented same.

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.

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

- Second Order Problem Using Heun’s Method
- Heat Flux Density at a Point (p.101)
- 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 should consist of

- doing exercises from the notes
- completing VBA exercises

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

Please ask questions in the lab about questions you have gotten wrong.

]]>

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.

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.* *

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

In VBA we will implement same.

In the 09:00 class we will have a revision session, geared towards the 20% VBA Assessment 2.

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

Formulae will be provided in the VBA 2 Assessment.

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).

The VBA 20% Assessment 2 format will be as follows.

Specifically,

; , ,

for some , , and step-size determined by your student number.

I want Euler Shooting Method approximations to for .

You can use:

- An Excel Worksheet, or
- Excel’s
*Goal Seek,*or - A VBA program

but you have to use a Shooting Method (technically *Goal Seek* takes loads of shots so I am happy to call it a shooting method).

It is up to you to understand which method is easiest for you.

*Use a shooting method to solve the following with :*

, , .

*Solution: *The preliminary work is to turn this into a system of first order initial value problems. To do so introduce a new variable for the first derivative (as it happens is the shear).

Let together with the initial value .

If is the first derivative of with respect to then

so that we have

.

We have no initial value for so we just guess for the moment… say .

Please see the first worksheet of *Shooting Method for Bending Moment* (it will be emailed) for the implementation of Euler’s Method for the system:

; ,

; ,

The first shot with produced , an undershoot (we are trying to get ).

We try again with a larger , say . This produces an overshot of .

Now use the shooting method equation to find the correct :

.

Now see the worksheet where the Euler Method is run with this value and the resulting graph (I am happy with just the values but if you can input the graph). Note this value of yields as required.

(Usually in engineering we plot underneath the -axis… don’t worry about this.)

Very similar set-up to the previous except we don’t have to take any shots and instead ask Excel to try a load of shots.

See Worksheet 2 of *Shooting Method for Bending Moment.*

So perhaps just put as a placeholder.

Now do Goal Seek This produces and .

Again the set up is similar but we run the Euler Method via VBA.

See Worksheet 3 of *Shooting Method for Bending Moment* (or moreover the code behind the worksheet).

We have to take two shots and use the shooting method equation to get . Finally, we must run the program one more time.

Specifically,

; and

for some , , , and . These constants will be determined by your student number.

Use a Finite Difference Method with a mesh size (determined by your student number) [Sample: Lab 7, Problem 2], to produce approximations to for .

I will send on a worked example of this.

Specifically,

for the temperature at the point of a rectangular plate with boundary conditions given by , where is the boundary/perimeter of the rectangular plate .

The boundary temperature will be given in terms of your student number.

The above equation, Laplace’s equation, can instead by framed as the Mean Value Property which can be approximated using the ‘four adjacent gridpoint average’ once the rectangular plate is *meshed *using a square grid.

Sample Question: Lab 7, Problem 1,

The 40% Written Assessment will be broken up into two parts.

- Theory Element Tuesday 30 April 09:00 (30 minutes worth but given an hour).

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

The first part of the Test would take place at 09:00 . It would be designed to be easily completed in 30-40 minutes. It would be geared more towards theoretical questions.

The second part of the Test would take place in your VBA slot. I would have to tell you in advance what questions are coming up, e.g. maybe

- Q. 1 Second Order Problem Using Heun’s Method
- Q. 2 Euler Shooting Method
- Q. 3 Heat Equation

Each group would get questions with only minor variations from the sample questions. I will confirm this next week.

Study should consist of

- doing exercises from the notes
- completing VBA exercises

To add a bit of interest to the Ungraded Concept MCQs, I will keep a league table.

Unless you are excelling, you are identified by the last five digits of your student number. AW is the number of attendance warnings received.

Please ask questions in the lab about questions you have gotten wrong.

]]>

Week 8 was a disaster. We missed out on Monday due to St Patrick’s Day. We missed out on Wednesday morning because of my man flu. Thursday I had to go to Dublin for a funeral.

Week 9 we managed to get everything done so that you can now do Assignment 2.

We had two tutorials in Week 9

We have two additional classes:

**Monday 09:00, B189**

**Thursday 14:00, E6**

So we have six classes

Monday x 2

Wednesday x 2

Thursday x 2

The following classes will be tutorials:

Monday x 2

Wednesday 09:00

Thursday 14:00

Wednesday 14:00 and Thursday 10:00 will be lectures, focused on Systems of Differenial Equations.

Assignment 2 will have a hand-in time and date of 12:00 8 April: the Monday of Week 11. Assignment 2 is in the manual, P. 164. You need to start this assessment ASAP

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

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

We will review the Summer 2018 paper.

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