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.

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:

,

by , then we define as an inverse function:

.

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

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

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 — especially those of us who struggled in the test.

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

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

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
- your written work

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.

We had an extra tutorial on Monday.

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

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.

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

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 — especially those of us who struggled in the test.

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

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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.

Consider a random walk on a finite group driven by . The state of the random walk after steps is given by , defined inductively (on the algebra of functions level) by the associative

.

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

,

where has entries, with respect to a basis . Furthermore, therefore

,

and so the stochastic operator describes the random walk just as well as the driving probabilty .

The random walk driven by is said to be *irreducible *if for all , there exists such that (if ) .

The *period *of the random walk is defined by:

.

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 is not irreducible, there exists a proper subset of , say , such that the set of functions supported on are -invariant. It turns out that is a proper subgroup of .

Moreover, when is irreducible, the period is the greatest common divisor of all the natural numbers such that there exists a partition of such that the subalgebras of functions supported in satisfy:

and (slight typo in the paper here).

In fact, in this case it is necessarily the case that is concentrated on a coset of a proper normal subgroup , say . Then .

Suppose that is supported on . We want to show that for . Recall that

.

This shows how the stochastic operator reduces the index .

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

,

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

Let be a -algebra (so that is in general a virtual object). A -subalgebra is *hereditary *if whenever and , and , then .

It can be shown that if is a hereditary subalgebra of that there exists a projection such that:

.

All hereditary subalgebras are of this form.

A stochastic operator is *irreducible *if there exists no proper hereditary -invariant -subalgebra of .

Classically, the proper hereditary subalgebras are algebras of the form with . In the case of random walks on groups, we need only look at subgroups rather than all subsets.

Fagnola points out that the appropriate generalisation in the noncommutative case is not merely a subalgebra but a hereditary subalgebra. Fagnola gives some references I must see how this result interacts with my results on irreducible random walks on (finite) quantum groups. Looking ahead there is a choice between equivalent definitions:

- no proper -invariant hereditary subalgebras, and
- no non-trivial –
*subharmonic*projections,

and as they are equivalent, and Fagnola and Pellicer go on to prove an ergodic theorem and so this is perfect… although I also read that Fagnola and Pellicer say that this is not new, and they are primarily interested in the result for completely bounded maps… another paper title *Spectral Properties of Positive Maps on* *-algebras *is referenced by Fagnola and Pellicer

Fagnola and Pellicer prove that:

*A stochastic operator is irreducible if and only if for all projections ,*

They also show that the spectrum of , , the unit disk.

*A stochastic operator with a non-trivial fixed point is *not *irreducible *

Constant functions are certainly eigenvectors of eigenvalue one. If there is a non-constant fixed points we have a second eigenvalue equal to one. If is irreducible, this is not the case.

*An irreducible stochastic operator has a unique faithful invariant state *

In the (finite) quantum group case, this is the Haar state .

At this point Fagnola and Pellicer ask that be a Schwarz Map. In the random walk case, the stochastic operator is in fact completely positive and so Schwarz (proof in a paper in preparation).

Let be an irreducible stochastic operator. A *partition of unity *(F+P say resolution of the identity) is called -cyclic if the hereditary subalgebra is mapped to the hereditary by . If there exists such a partition such that , then the random walk given by is *periodic.*

Classically, we have a partition of , and the random walk bounces around the in a cyclic way, and the support of will be concentrated on such subsets. For the case of a random walk on a group, , and .

For random walks on (finite) quantum groups:

*Let be an irreducible stochastic operator for a random walk on a finite quantum group. A partition of unity is cyclic for if and only if:*

.

Furthermore:

*Let be an irreducible stochastic operator for a random walk on a finite quantum group. The following are equivalent:*

*a -cyclic partition of unity**contains all the -th roots of unity*.

This provides a non-commutative generalisation of the standard ergodic theorem. Think in terms of eigenvalues. If there exists a -cyclic partition of unity with , then and so we do not have convergence.

Therefore, for convergence, one must be irreducible AND have only a trivial -cyclic partition of unity. Assume this in the prequel. Theorem 3.7 says that is a finite subgroup. Therefore there cannot be an irrational such that .

Could there be a rational such that ? The spectrum is a group and so if , all the -th roots of unity are in the spectrum and so, by assumption …

Therefore if there is no non-trivial -cyclic partition of unity, then and the convergence of follows.

Idea: take a random walk on a (finite) quantum group. Assume that the driving probability is not concentrated on a quasi-subgroup. Assume that the stochastic operator has a cyclic partition of unity with . Construct some kind of an object such that the projections in the partitions of unity are the identities on the cosets, and that the support of lies on one of them… the cyclic group to appear.

We have a zoo of results. From quantum to classicial, from spectral to algebraic, from sets to groups.

The fundamental result in this analysis is spectral. At this point we define a stochastic operator to be a *completely positive *and *unital *operator on a -algebra . Examples include the stochastic operator of a random walk on a compact quantum group and the stochastic operator of a classical Markov chain. We do not always need *completely bounded* for these results.

As we are primarily interested in

irreducible + aperiodic = ergodic,

and the usual thing is to establish irreducibility and then look at aperiodicity *assuming *irreducibility. In a paper in preparation, I sometimes study periodicity not assuming irreducibilty. Perhaps assuming irreducibilty might help my look at periodicity. Let us not look at reducible + aperiodic (although I am fairly sure) we can restrict such stochastic operators to ergodic ones.

This is basically finite dimensional linear algebra. Let be a stochastic operator.

- We have that . In particular, .
- If is a simple eigenvalue, then is said to be irreducible. A non-trivial fixed point
*should*give rise to non-faithful invariant states to which can converge to. - Assuming irreducibility, if , then converges (aperiodic). Otherwise does not converge (periodic).
- In the case of irreducible + aperiodic, converges to :

.

This is the topic of the paper of Fagnola and Pellicer.

- .
- A stochastic operator is irreducible if for all projections , or . A reducible stochastic operator has a projection such that (recall ). This
*should*correspond to a non-faithful invariant state . - Assuming irreducibility, a stochastic operator is
*aperiodic*if and only if the only partition of unity such that

,

is the trivial . Otherwise does not converge.

- In the case of irreducible + aperiodic, for all , , where is the unique faithful -invariant state, and is called
*ergodic.*

This is the topic of my paper in preparation. For any random walk driven by , the stochastic operator:

,

and the support projection of is the the smallest projection such that .

- is irreducible if and only if the is NOT supported on a proper-quasi-subgroup (if the support projection is a group-like projection not equal to ).
- Assuming irreducibility…. this is where we are stuck but we can use the language of Fagnola and Pellicer.
- In the case of irreducible + aperiodic, for all , , where is the Haar state, and the random walk driven by is
*ergodic**.*

For a stochastic operator

- is irreducible if and only if for all , there exists such that

- Assuming irreducibility, a markov chain is aperiodic if for all , the state ‘s period:

.

- In the case of irreducible + aperiodic, the distribution converges to a unique stationary, strict in the sense that as :

For any random walk driven by

- is irreducible if and only if the is NOT supported on a proper-subgroup
- Assuming irreducibility, the random walk is aperiodic if is not supported on the coset of any proper normal subgroup.
- In the case of irreducible + aperiodic, , where is the uniform distribution, and the random walk driven by is
*ergodic**.*

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.

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.

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..

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

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.

]]>In Week 8 we will start talking about equations. On Wednesday we had a catch up lecture where we looked ahead to quadratic equations and quadratic functions.

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

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on ~~Tuesday 26 November~~ Tuesday 3 December in the usual lecture venue of D160.

I will draft a sample test and have it for ye by the end of next week. 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.

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

]]>

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.

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

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.

]]>

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.

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.

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.

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

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

”

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

In Week 7 we started delving more into algebra

In Week 8 we will start talking about equations.

Test 2, based on Chapter 3: Algebra, will take place around the end of Week 10, start of Week 11. About two weeks after we finish Chapter 3.

]]>

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.

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.

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.

]]>

Results and Marking Scheme have been emailed to you.

On Chapter 3, is planned for **5 pm (not 4 pm) Monday 25 November. **Please contact me if this poses a problem for you.

For *most *students, Chapters 1 and 2 are easier and you will want to do well on them. Things are going to get a little harder for the rest of the semester and you will need to attend class regularly and prepare well for Test 2.

You have already been recommended to use your Reading Week:

“*If you plan on doing some work on MATH6040 over the Reading Week, may I recommend looking at Chapter 2 exercises:*

”

We started Chapter 3 with a review of differentiation followed by a look at Parametric Differentiation.

Here are two videos if you missed this material:

We mentioned briefly the Reuleaux Triangle and it’s use in the “Harry Watt Square Drill Bit”. See here for an animation of how this works.

We will finish looking at Related Rates and then look at Implicit Differentiation. If you are interested in a very “mathsy” approach to curves you can look at this.

On Chapter 3, is planned for **5 pm (not 4 pm) Monday 25 November. **Please contact me if this poses a problem for you.

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