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.

# Diaconis-Van Daele Fourier Theory

I will be giving this talk with some of the fourth and fifth class pupils at Gaelscoil an Ghoirt Alainn, Mayfield in a few weeks.

A few historical and actual inaccuracies to keep things simple!

We define the transfer function of a black box model as

$\displaystyle H(s)=\frac{Y(s)}{X(s)}$,

where $X(s)$ is the Laplace transform of the input and $Y(s)$ is the Laplace transform of the output. This yields:

$Y(s)=H(s)\cdot X(s)$.

$H(s)$ is a rational function and therefore has zeroes and roots (removable poles and roots are removed) and therefore so does $Y(s)$:

$\displaystyle Y(s)=K\cdot \frac{\prod_{j=1}^m(s-z_j)}{\prod_{i=1}^n(s-p_i)}$.

These $\{z_j\}$ are the zeroes of $Y(s)$ and the $\{p_i\}$ are the poles of $Y(s)$This means that we have (assuming $m) a partial fraction expansion:

$\displaystyle Y(s)=\sum_{i=1}^n\frac{C_i}{s-p_i}$.

Applying the Inverse Laplace transform we have an input:

$y(t)=\sum_{i=1}^n C_ie^{p_it}$.

These $p_i$ are complex numbers and depending on their nature we get different behaviours.

A positive real pole is a pole $p\in\mathbb{R}$ such that $p>0$. This corresponds to an output $e^{pt}$ which tends to infinity as $t\rightarrow \infty$. This is divergent or unstable behaviour.

A negative real pole $p=-a$ with $a>0$ corresponds to an output $\displaystyle e^{pt}=e^{-at}=\frac{1}{e^{at}}\rightarrow 0$ as $t\rightarrow \infty$. This is convergent or stable behaviour.

A zero pole $p=0$ yields an output $e^{pt}=e^{0\cdot t}=e^0=1$ which is a constant output (which is considered stable).

A purely imaginary pole $p=k i$ is, via Euler Formula, corresponds to oscillatory behaviour:

$e^{pt}=e^{ikt}=\cos(kt)+i\sin(kt)$.

A genuinely complex pole $p=a+ik$ ($a>0$) with a positive real part (in the right-half plane) corresponds to the following output

$e^{pt}=e^{(a+ik)t}=e^{at+ikt}=e^{at}e^{ikt}=e^{at}(\cos(kt)+i\sin(kt))$.

While the $e^{ikt}$ component is oscillations, the $e^{at}$ goes to zero so we get behaviour that looks like:

Note that this behaviour is unstable.

When we have complex pole $p=-a+ik$ ($a>0$) with a strictly negative real part (a pole in the left-half plane), then we have $e^{pt}=e^{-at}(\cos(kt)+i\sin(kt))$. Note that we have oscillations but modulated by a decreasing $e^{-at}$. This is the motion of an underdamped harmonic oscillator:

This is inherently stable behaviour.

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 2

You will get a chance to view your paper in the tutorials of Week 13.

P.163 & 171 the Q. 3s can be done together.

## Week 12

We finished off the course by talking about centroids of laminas and centres of gravity of solids of revolution. There are marking schemes to Q.4 questions on pages 158, 168 and page 179. I have also done out some more examples of questions for 4 (c) on centroids and centres of gravity — I will give these to you in Week 13.

## Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture slots:

• Tuesday 11:00 and 15:00
• Thursday 11:00

In the normal rooms.

Note you won’t get a question like Q.3 (c) but I have a replacement question done out.

We will also have tutorial time in the tutorial slots. You can come to any or both tutorials.

• Monday at 14:00
• Thursday at 14:00

Again in the normal rooms.

## Study

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

## Student Resources

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.

## Assignments 2

I hope to have the corrections finished tomorrow. If not it will be Monday but no later.

## Week 12

We finished the last chapter on Multiple Integration including triple integrals.

## Week 13

We will first go over the sample and then have tutorial time.

## Student Resources

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.

## Continuous Assessment Summary

The results of the individual components of the CA are lower down — in particular Quiz 11.

Unless you won or came second in the league, you are identified by the last four digits of your student number. QPP is your QPP out of 20, MM are your Maple Marks out of 100 (Maple is worth 10% so say 80% gives you 8 Marks for your final grade). GPP is your Gross Percentage Points for CA (out of 30). PP is your Passing Percentage for the final paper — what you need on the final paper to pass. Finally FP is your First Percentage — what you need on the final paper to get a first: 70%.

 S/N QPP MM GPP PP FP Kelliher 20.0 93.2 29.3 15.3 58.1 Kiely 19.8 94.3 29.2 15.4 58.2 3281 19.9 89.8 28.9 15.9 58.7 5527 20.0 88.6 28.9 15.9 58.8 8416 19.7 84.1 28.1 17.0 59.8 8403 18.1 89.8 27.1 18.5 61.3 4198 18.5 85.2 27.0 18.5 61.4 6548 18.5 79.5 26.4 19.4 62.3 1864 17.2 86.0 25.8 20.3 63.1 8478 17.2 83.5 25.5 20.7 63.6 8556 17.5 76.1 25.1 21.3 64.2 7878 16.6 77.0 24.3 22.4 65.3 5546 16.1 77.3 23.8 23.1 66.0 2567 15.2 84.1 23.6 23.4 66.3 8603 15.7 77.3 23.4 23.7 66.6 1852 14.4 88.6 23.3 23.9 66.8 8455 14.4 78.4 22.2 25.4 68.3 2859 14.0 78.4 21.8 25.9 68.8 7950 10.4 63.4 16.7 33.2 76.1 4775 9.2 75.0 16.7 33.3 76.1 9464 8.6 60.0 14.6 36.2 79.1 7209 5.5 45.0 10.0 42.8 85.7 5553 1.3 45.0 5.8 48.9 91.7

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.

## Assignments 2

Worked Solutions here… corrections really should be well completed by the end of next week.

## Week 11

We began the last chapter on Multiple Integration with double integrals.

## Week 12

We will finish the last chapter on Multiple Integration including triple integrals.

## Week 13

We will first go over the sample and then have tutorial time.

## Student Resources

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.

## Continuous Assessment

You are identified by the last four digits of your student number unless you are winning the league. The individual quiz marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes — now this includes missed quizzes — before I was doing the best non-zero but now I am including the zeros if a zero is in your best eight), MPP is your Maple Percentage Points for the biweekly lab, MT your mark on the Maple Test and MM your Maple Marks (as a percentage). GPP is your Gross Percentage Points (for best eight quizzes and Maple). Most of the columns are rounded but column 11, for quiz ten, is correct — as is GPP.

The Maple Test went very poorly… I have sent ye on worked solutions — please see the Remarks therein.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP MT MM GPP Kelliher 3 3 3 3 2 3 3 3 3 2.5 100 20.0 7.5 1.8 93.2 29.3 Kiely 2 3 2 3 3 3 2 3 2 2.4 98 19.7 7.5 1.9 94.3 29.1 5527 2 3 3 3 3 3 3 3 3 2.4 100 20.0 7.5 1.4 88.6 28.9 3281 2 3 3 3 2 3 3 3 2 2.4 99 19.8 7.5 1.5 89.8 28.8 8416 2 1 2 3 3 1 2 3 3 2.5 97 19.4 7.5 0.9 84.1 27.8 8403 2 1 2 3 1 2 2 3 2 1.8 89 17.8 7.5 1.5 89.8 26.8 6548 2 1 2 3 2 2 2 3 3 2.4 91 18.2 7.5 0.5 79.5 26.1 4198 0 1 2 3 2 3 3 3 2 1.1 86 17.2 7.5 1.0 85.2 25.7 8478 2 2 2 2 0 3 2 2 2 1.8 85 17.0 7.5 0.9 83.5 25.3 1864 1 2 3 2 1 1 2 3 1 2 82 16.4 7.5 1.1 86.0 25.0 7878 2 2 2 2 2 2 2 2 2 0.6 83 16.6 7.5 0.2 77.0 24.3 8556 1 1 2 2 0 2 3 3 1 2.1 81 16.3 7.5 0.1 76.1 23.9 2567 2 2 2 1 2 2 2 2 2 1.1 76 15.2 7.5 0.9 84.1 23.6 8603 1 2 2 2 0 2 3 2 2 0.8 78 15.7 7.5 0.2 77.3 23.4 1852 1 1 2 2 2 2 1 1 2 2.3 72 14.3 7.5 1.4 88.6 23.2 5546 0 0 1 1 2 2 3 2 3 2.2 73 14.6 7.5 0.2 77.3 22.3 8455 0 1 1 1 2 2 2 1 2 1.5 64 12.8 7.5 0.3 78.4 20.6 2859 2 1 0 1 2 2 2 3 0 0.5 61 12.1 7.5 0.3 78.4 19.9 7950 0 0 1 0 2 1 3 3 0 1.3 51 10.1 6 0.3 63.4 16.4 4775 1 0 1 0 0 1 2 2 1 0.4 44 8.7 7.5 0.0 75.0 16.2 9464 1 1 2 1 1 1 2 0 0 0 43 8.6 6 0.0 60.0 14.6 7209 2 1 2 1 0 0 0 0 0 0 28 5.5 4.5 0.0 45.0 10.0 5553 0 1 0 0 0 0 0 0 0 0 7 1.3 4.5 0.0 45.0 5.8

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 2 & Continuous Assessment Results

Unless you excelled in the CA, you are identified by the last four digits if your student number.

The fourth column is your CA marks out of 30. The fifth column is the number of attendance warnings you have. PP is the pass percentage you must get on the final paper to pass. FP is the First Percentage you must get on the final paper to get a first (70%).

Note that the top six students had no attendance warnings. The average of the students with no or one attendance warning was 68 but the average of the students with two or more attendance warnings was 36.

You will get a chance to view your paper in the tutorial of today, 30 April and the tutorials in Week 13. Note if I have adjusted your Test 1 or Test 2 result (you will have received an email about this), this will not be reflected here (but will be on the CIT System).

Top six

 S/N Test 1 Test 2 CA Ex 30 AW PP FP Toomey 96.1 91.4 27.7 0 17.6 60.5 Sologub 94.1 91.4 27.4 0 18.0 60.9 Timmons 88.2 97.1 27.3 0 18.1 61.0 2259 82.3 97.1 26.4 0 19.4 62.3 4183 86.3 91.4 26.2 0 19.7 62.6 5667 90.2 80.0 25.1 0 21.2 64.1 1332 72.5 94.3 24.5 1 22.1 64.9 9832 80.4 65.7 21.6 1 26.3 69.2 7459 60.8 85.7 21.5 1 26.4 69.2 0039 72.5 68.6 20.8 0 27.4 70.3 0132 92.2 42.9 20.1 0 28.5 71.4 1223 47.1 77.1 18.2 0 31.1 73.9 7604 68.6 48.6 17.3 1 32.4 75.2 7398 49.0 60.0 16.1 0 34.2 77.1 5686 58.8 45.7 15.4 1 35.1 77.9 4953 84.3 11.4 14.3 4 36.7 79.6 4912 19.6 77.1 14.1 0 37.0 79.8 N/A 60.8 34.3 14.1 0 37.0 79.9 2888 52.9 37.1 13.3 2 38.1 81.0 1326 60.8 20.0 12.0 0 40.0 82.8 7255 33.3 45.7 11.6 0 40.5 83.4 2317 33.3 31.4 9.5 4 43.5 86.4 1999 31.4 31.4 9.3 3 43.9 86.8 8849 52.9 0.0 7.9 2 45.8 88.7

P. 147 Q.2, 11; P. 55, Q.2 (a), (b)

## Week 11

We got as much of the first two sections done in our two lectures this week.

## Week 12

Finish off the course.

## Week 13

There should be a an exam paper at the back of your notes — I will go through this on the board and we will also have tutorial time.

## Study

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

## Student Resources

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 Classes

…are finished; that is no Wednesday classes anymore.

## Test 2

Test 2 will take place on Tuesday 28 April at 11 in B214. There is a sample in the notes (after P.140). Before or after the sample there is a quick summary of the chapter — you need to know all these formulae and ideas.

Because the test is so close I am going to say that the question on partial differentiation will definitely come from P.132, Q. 1(a), 2(i)-(iii). Also the question on error analysis will be one of Q. 2,3,4 on P.139.

P. 132 Q.1; P. 139, Q.3

## Week 10

We finished off partial differentiation, applied to it error analysis.

## Week 11

We have one chapter to do in five lectures which isn’t ideal but I have cut some examples so we will be O.K. I am going to try and get as much of the first two sections done in our two lectures this week.

## Week 12

Finish off the course.

## Week 13

There should be a an exam paper at the back of your notes — I will go through this on the board and we will also have tutorial time.

## Study

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