J.P. McCarthy: Math Page

MATH6055: Winter 2019, Week 9

November 15, 2019 in MATH6055 | Leave a comment

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.

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MATH7019: Winter 2019, Week 9

November 14, 2019 in MATH7019 | Leave a comment

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
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.

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.

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Irreducible and Periodic Positive Maps

November 14, 2019 in C*-Algebras & Operator Theory, Probability Theory, Research | Leave a comment

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.

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MATH6040: Winter 2019, Week 9

November 14, 2019 in MATH6040, MATH6040 - Day | Leave a comment

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..

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MATH6055: Winter 2019, Week 8

November 8, 2019 in MATH6055 | Leave a comment

Week 8

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.

Week 9

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

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MATH7019: Winter 2019, Week 8

November 8, 2019 in MATH7019 | Leave a comment

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.

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MATH6040: Winter 2019, Week 8

November 8, 2019 in MATH6040, MATH6040 - Day | Leave a comment

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.

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MATH6055: Winter 2019, Week 7

October 30, 2019 in MATH6055 | Leave a comment

Reading Week

“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.

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MATH7019: Winter 2019, Week 7

October 30, 2019 in MATH7019 | Leave a comment

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.

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MATH6040: Winter 2019, Week 7

October 30, 2019 in MATH6040, MATH6040 - Day | Leave a comment

Test 1

Results and Marking Scheme have been emailed to you.

Test 2

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

Outlook

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:

P. 95, Q. 1 (find b only), Q. 2-4

P. 85, Q. 1

P. 80, Q. 1-4

P. 84, Q. 1-2

P. 64, Q. 1-4 (remember BEMDAS — AD^T = A(D^T) not = (AD)^T)

If you are happy with these look at the differentiation revision: p.103, Q.1-2

You can email me at any time if you have any questions.

”

Week 7

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.

Week 8

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.

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J.P. McCarthy: Math Page

Test 2

Week 9

Week 10

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Tutorial Split

Assessment 2

Week 9

Week 10

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Introduction

Stochastic Operators and Operator Algebras

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Test 2

Week 9

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Week 8

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Assessment 2

Week 8

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Test 2

Week 8

Week 9

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Reading Week

Catch Up Lecture

Week 7

Week 8

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Assessment 2

Week 7

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Test 1

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Outlook

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