I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Catch-up Lecture

• 16:00 Wednesday 7 October B214

Timetable

Starting this week:

Environmental Engineering: Tutorial Monday 13:00, Lectures Wednesday 11:00, Thursday 10:00, Friday 10:00

Civil Engineering: Tutorial Friday 11:00, Lectures Wednesday 11:00, Thursday 10:00, Friday 10:00

Assessment 1

Assessment 1 has been emailed to all of you. The hand in date in Friday 16 October. Late handed in late shall be assigned a mark of zero.

• I want you to read the first page carefully. When you start getting down to actually doing the assignment, read the problems carefully
• the purpose of this assignment is to help with your understanding of the material and hopefully this will end up with you being proficient in its use and able to do well in the final exam
• you should start right now because you only have one week to complete the assignment. The hand in date is Friday 17 October.
• yes you can ask me questions about the assignment before, after and sometimes during class (if ye are working on a problem). Also you can ask me questions via email.
• if you are having problems using Microsoft Excel then you are to email me ASAP and arrange to visit me in my office. You will bring your work on a memory key and we will sort it out
• The assignment is worth 15%. The question on the final paper will be worth 25% of the 70% — which is 17.5% — on offer on the final paper. Do a good job with this assignment and you should do well on Q.1 on the paper and you might just have 32.5% in the bag before we go onto more technical stuff.
• MOST IMPORTANTLY — you are welcome to work together. All of you have different data so none of you will have the same answers. Therefore you cannot copy each other. The best way to learn is by teaching.
• Work submitted after 12:00 on Friday 17 October will be assigned a mark of zero. Hand up whatever you have on time.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test 1

The first 15% test will take place at 4 pm MONDAY 19 October in B263 (Week 6). You can find a sample in the course notes, after the section on rates of change I think. It is a test that could arguably take 42 minutes but I’ll give ye from 16.05 — 17.00. You will be given a copy of these tables. Don’t worry I’ll scribble out the “UCC”!

Note that the format will be the same of this.

1. Differentiation from First Principles
2. Tangent Lines
3. Differentiate by Rule
4. Differentiate by Rule
5. Differentiate by Rule
6. Rates of Change
7. Rate of Change/ Geometry of Graph,

but this is dependent on our progress in lectures.

Week 3

In Week 3 we covered everything in the notes up to the quotient rule.

Week 4

In Week 4 we will talk about another way of combining functions — composition — and how to differentiate them — the Chain Rule.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Assessment 1

Assessment 1 will be on the Wednesday of Week 4, 7 October 2015.

The following class groups will sit the assessment at 17:15 (arrive around 17:05):

• DNET1, WEB1, DCOM1-A

The following group will sit the assessment at 18:15 (arrive around 18:05)

• DCOM1-B

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 3

DCOM

In Week 2 we concluded our work for Assessment 1 by looking at percentage error. We started the Assessment 2 material by looking at VAT, Income Tax and Interest.

WEB/DNET

In Week 3 we looked at ratio and proportion, approximation and percentages.

Week 4

DCOM

We will finish off the section on percentages and then look at error. This will conclude our study of the Assessment 1 material. After this we will look at tax and interest.

WEB/DNET

We need to look at percentage error on Tuesday to conclude our look at the Assessment 1 material. After this we will look at tax and interest.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Catch-up Lectures

As you know there are no MATH7019 classes tomorrow, Friday 25 September.
The original proposal for Wednesday and Wednesday has been replaced with
• 16:00 Thursday 1 October A213B
• 16:00 Wednesday 7 October B214

Week 2

In Week 2 we covered linear least squares.

Week 3

With five classes this week we should almost complete chapter one.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Extra Tutorial

There will be no MATH6015 tutorial tomorrow morning, Friday 25 September.
This class will be rescheduled for 13:00 on Thursday 1 October in B185.
This means that there will be three tutorials next week. If you missed Monday’s tutorial and were planning on attending Friday I expect you to make two of next week’s tutorials.

Week 2

In Week 2 we learned how to differentiate with respect to first principles and also how to differentiate sums of differentiable functions.

Week 3

In Week 3 we will continue talking about the duality between algebra & geometry and perhaps begin to talk about differentiating some more functions.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

WEB1/DNET ONLY

There will be no 12:00 lecture tomorrow, Friday 25 September. The lecture will be rescheduled for:
• (This) Monday 28 September 11:00 in B149

Week 2

DCOM

In Week 2 we looked at Decimals — including long multiplication and division — Ratio & Proportion and approximation. We have just began to look at percentages.

WEB/DNET

In Week 2 we looked at Decimals — including long multiplication and division. We have just began to look at ratio and proportion.

Week 3

DCOM

We will finish off the section on percentages and then look at error. This will conclude our study of the Assessment 1 material. After this we will look at tax and interest.

WEB/DNET

We have four lectures this week so we will first catch up by finishing ratio and proportion and then talk about approximation. Then we will look at percentages and error and this will conclude our study of the Assessment 1 material. After this we will look at tax and interest.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Manuals

The manuals are available in the Reprographic/Copy Centre. Please purchase ASAP.

Week 1

In week one we spoke in general terms about curve fitting. We introduced Lagrange Interpolation and started talking about Least Squares curve fitting. We did one example where we fitted a line to some Carbon Dioxide Data.

Week 2

In Week 2 we will continue talking about Least Squares Curve Fitting.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Manuals

The manuals are available in the Reprographic/Copy Centre. You will need these notes for next week.

Week 1

In week one we spoke about the questions that the derivative was originally formulated to answer. Namely how do we find the tangent to a curve.

Week 2

In Week 2 we will look at calculating some derivatives.

Tutorials

Tutorials start properly this week:

• Monday 17:00 in B189
• Friday 09:00 in B185

As discussed on Thursday, you must make at least ONE of these classes every week.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Manuals

The manuals are available in the Reprographic/Copy Centre. You will need these notes for next week.

Week 1

We studied basic arithmetic, in particular we looked at the various number systems and the arithmetic of fractions in particular.

Week 2

In Week 2 we will look at Decimals — including long multiplication and division — Ratio & Proportion and probably percentages too.

Tutorials

Tutorials start properly in Week 2:

• DCOM1 Group A: Wednesday 09:00 in C212 (with Len O’Driscoll) and Thursday 09:00 in B185 (with Frances Wood)
• DCOM1 Group B with myself: Monday 12:00 in F1. 3 and Thursday 12:00 in B165
• DNET1 with myself: Monday 14:00 in B248 and Wednesday 15:00 in C212
• WEB1 with Ben O’Shaugnessy: Monday 16:00 in C212 and Thursday 15:00 in E7

Let $\mathbb{G}$ be a finite quantum group described by $A=\mathcal{C}(\mathbb{G})$ with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) $S^2=I_A$. Under the assumption of finiteness, there is a unique Haar state, $h:A\rightarrow \mathbb{C}$ on $A$.

Representation Theory

A representation of $\mathbb{G}$ is a linear map $\kappa:V\rightarrow V\otimes A$ that satisfies

$\left(\kappa\otimes I_A\right)\circ\kappa =\left(I_V\otimes \Delta\right)\circ \kappa\text{\qquad and \qquad}\left(I_V\otimes\varepsilon\right)\circ \kappa=I_V.$

The dimension of $\kappa$ is given by $\dim\,V$. If $V$ has basis $\{e_i\}$ then we can define the matrix elements of $\kappa$ by

$\displaystyle\kappa\left(e_j\right)=\sum_i e_i\otimes\rho_{ij}.$

One property of these that we will use it that $\varepsilon\left(\rho_{ij}\right)=\delta_{i,j}$.

Two representations $\kappa_1:V_1\rightarrow V_1\otimes A$ and $\kappa_2:V_2\rightarrow V_2\otimes A$ are said to be equivalent, $\kappa_1\equiv \kappa_2$, if there is an invertible intertwiner between them. An intertwiner between $\kappa_1$ and $\kappa_2$ is a map $T\in L\left(V_1,V_2\right)$ such that

$\displaystyle\kappa_2\circ T=\left(T\otimes I_A\right)\circ \kappa_1.$

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if $\{\kappa_\alpha\}_{\alpha}$ is a maximal family of pairwise inequivalent irreducible representation that $\{\rho_{ij}^\alpha\}_{\alpha,i,j}$ is a basis of $A$. When we refer to “the matrix elements” we always refer to such a family. We define the span of $\{\rho_{ij}\}$ as $\mathcal{C}\left(\kappa\right)$, the space of matrix elements of $\kappa$.

Given a representation $\kappa$, we define its conjugate, $\overline{\kappa}:\overline{V}\rightarrow\overline{V}\otimes A$, where $\overline{V}$ is the conjugate vector space of $V$, by

$\displaystyle\overline{\kappa}\left(\bar{e_j}\right)=\sum_i \bar{e_i}\otimes\rho_{ij}^*,$

so that the matrix elements of $\overline{\kappa}$ are $\{\rho_{ij}^*\}$.

Timmermann shows that the matrix elements have the following orthogonality relations:

• If $\alpha$ and $\beta$ are inequivalent then $h\left(a^*b\right)=0,$ for all $a\in \mathcal{C}\left(\kappa_\alpha\right)$ and $b\in\mathcal{C}\left(\kappa_\beta\right)$.
• If $\kappa$ is such that the conjugate, $\overline{\kappa}$, is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

$\displaystyle h\left(\rho_{ij}^*\rho_{kl}\right)=\frac{\delta_{i,k}\delta_{j,l}}{d_\alpha}.$

This second relation is more complicated without the $S^2=I_A$ assumption and refers to the entries and trace of an intertwiner $F$ from $\kappa$ to the coreprepresention with matrix elements $\{S^2\left(\rho_{ij}\right)\}$. If $S^2=I_A$, then this intertwiner is simply the identity on $V$ and so the the entries $\left[F\right]_{ij}=\delta_{i,j}$ and the trace is $d=\dim V$.

Denote by $\text{Irr}(\mathbb{G})$ the set of unitary equivalence classes of irreducible unitary representations of $\mathbb{G}$. For each $\alpha\in\text{Irr}(\mathbb{G})$, let $\kappa_\alpha:V_{\alpha}\rightarrow V_{\alpha}\otimes A$ be a representative of the class $\alpha$ where $V_\alpha$ is the finite dimensional vector space on which $\kappa_\alpha$ acts.