J.P. McCarthy: Math Page

Why No Quantum Cyclic, S_3, or Alternating Groups

January 11, 2021 in Quantum Groups, Research, Research Updates | Leave a comment

I am currently (slowly) working on an essay/paper where I expand upon the ideas in this talk. In this post I will try and explain in this framework why there is no quantum cyclic group, no quantum $S_3$ , and ask why there is no quantum alternating group.

Quantum Permutations Basics

Let $A$ be a unital $\mathrm{C}^*$ -algebra. We say that a matrix $u\in M_N(A)$ is a magic unitary if each entry is a projection $u_{ij}=u_{ij}^2=u_{ij}^*$ , and each row and column of $u$ is a partition of unity, that is:

$\displaystyle \sum_ku_{ik}=\sum_k u_{kj}=1_A$ .

It is necessarily the case (but not for *-algebras) that elements along the same row or column are orthogonal:

$u_{ij}u_{ik}=\delta_{j,k}u_{ij}$ and $u_{ij}u_{k j}=\delta_{i,k}u_{ij}$ .

Shuzou Wang defined the algebra of continuous functions on the quantum permutation group on $N$ symbols to be the universal $\mathrm{C}^*$ -algebra $C(S_N^+)$ generated by an $N\times N$ magic unitary $u$ . Together with (leaning heavily on the universal property) the *-homomorphism:

$\displaystyle \Delta:C(S_N^+)\rightarrow C(S_N^+)\underset{\min}{\otimes}C(S_N^+), u_{ij}\mapsto \sum_{k=1}^N u_{ik}\otimes u_{kj}$ ,

and the fact that $u$ and $(u)^t$ are invertible ( $u^{-1}=u^t)$ ), the quantum permutation group $S_N^+$ is a compact matrix quantum group.

Any compact matrix quantum group generated by a magic unitary is a quantum permutation group in that it is a quantum subgroup of the quantum permutation group. There are finite quantum groups (finite dimensional algebra of functions) which are not quantum permutation groups and so Cayley’s Theorem does not hold for quantum groups. I think this is because we can have quantum groups which act on algebras such as $M_N(\mathbb{C})$ rather than $\mathbb{C}^N$ — the algebra of functions equivalent of the finite set $\{1,2,\dots,N\}$ .

This is all basic for quantum group theorists and probably unmotivated for everyone else. There are traditional motivations as to why such objects should be considered algebras of functions on quantum groups:

find a presentation of an algebra of continuous functions on a group, $C(G)$ , as a commutative universal $\mathrm{C}^*$ -algebra. Study the the same object liberated by dropping commutativity. Call this the quantum or free version of $G$ , $G^+$ .
quotient $C(S_N^+)$ by the commutator ideal, that is we look at the commutative $\mathrm{C}^*-$ algebra generated by an $N\times N$ magic unitary. It is isomorphic to $F(S_N)$ , the algebra of functions on (classical) $S_N$ .
every commutative algebra of continuous functions on a compact matrix quantum group is the algebra of functions on a (classical) compact matrix group, etc.

Here I want to take a very different direction which while motivationally rich might be mathematically poor.

Weaver Philosophy

Take a quantum permutation group $\mathbb{G}$ and represent the algebra of functions as bounded operators on a Hilbert space $\mathsf{H}$ . Consider a norm-one element $\varsigma\in P(\mathsf{H})$ as a quantum permutation. We study the properties of the quantum permutation by making a series of measurements using self-adjoint elements of $C(\mathbb{G})$ .

Suppose we have a finite-spectrum, self-adjoint measurement $f\in C(\mathbb{G})\subset B(\mathsf{H})$ . It’s spectral decomposition gives a partition of unity $(p^{f_i})_{i=1}^{|\sigma(f)|}$ . The measurement of $\varsigma$ with $f$ gives the value $f_i$ with probability:

$\displaystyle \mathbb{P}[f=f_i\,|\,\varsigma]=\langle\varsigma,p^{f_i}\varsigma\rangle=\|p^{f_i}\varsigma\|^2$ ,

and we have the expectation:

$\displaystyle \mathbb{E}[f\,|\,\varsigma]=\langle\varsigma,f\varsigma\rangle$ .

What happens if the measurement of $\varsigma$ with $f$ yields $f=f_i$ (which can only happen if $p^{f_i}\varsigma\neq 0$ )? Then we have some wavefunction collapse of

$\displaystyle \varsigma\mapsto p^{f_i}\varsigma\equiv \frac{p^{f_i}\varsigma}{\|p^{f_i}\varsigma\|}\in P(\mathsf{H})$ .

Now we can keep playing the game by taking further measurements. Notationally it is easier to describe what is happening if we work with projections (but straightforward to see what happens with finite-spectrum measurements). At this point let me quote from the essay/paper under preparation:

Suppose that the “event” $p=\theta_1$ has been observed so that the state is now $p^{\theta_1}(\psi)\in P(\mathsf{H})$ . Note this is only possible if $p=\theta_1$ is non-null in the sense that

$\displaystyle \mathbb{P}[p=\theta_1\,|\,\psi]=\langle\psi,p^\theta(\psi)\rangle\neq 0.$

The probability that measurement produces $q=\theta_2$ , and $p^{\theta_1}(\psi)\mapsto q^{\theta_2}p^{\theta_1}(\psi)\in P(\mathsf{H})$ , is:

$\displaystyle \mathbb{P}\left[q=\theta_2\,|\,p^{\theta_1}(\psi)\right]:=\left\langle \frac{p^{\theta_1}(\psi)}{\|p^{\theta_1}(\psi)\|},q^{\theta_2}\left(\frac{p^{\theta_1}(\psi)}{\|p^{\theta_1}(\psi)\|}\right)\right\rangle=\frac{\langle p^{\theta_1}(\psi),q^{\theta_2}(p^{\theta_1}(\psi))\rangle}{\|p^{\theta^1}(\psi)\|^2}.$

Define now the event $\left((q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right)$ , said “given the state $\psi$ , $q$ is measured to be $\theta_2$ after $p$ is measured to be $\theta_1$ “. Assuming that $p=\theta_1$ is non-null, using the expression above a probability can be ascribed to this event:

$\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]:=\mathbb{P}[p=\theta_1\,|,\psi]\cdot \mathbb{P}[q=\theta_2\,|\,p^{\theta_1}(\psi)]$
$\displaystyle =\langle\psi,p^{\theta_1}(\psi)\rangle\frac{\langle p^{\theta_1}(\psi),q^{\theta_2}(p^{\theta_1}(\psi))\rangle}{\|p^{\theta^1}(\psi)\|^2}$
$=\|q^{\theta_2}p^{\theta_1}\psi\|^2.$

Inductively, for a finite number of projections $\{p_i\}_{i=1}^n$ , and $\theta_i\in{0,1}$ :

$\displaystyle \mathbb{P}\left[(p_n=\theta_n)\succ\cdots \succ(p_1=\theta_1)\,|\,\psi\right]=\|p_n^{\theta_n}\cdots p_1^{\theta_1}\psi\|^2.$

In general, $pq\neq qp$ and so

$\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]\neq \mathbb{P}\left[(p=\theta_1)\succ (q=\theta_1)\,|\,\psi\right],$

and this helps interpret that $q$ and $p$ are not simultaneously observable. However the sequential projection measurement $q\succ p$ is “observable” in the sense that it resembles random variables with values in $\{0,1\}^2$ . Inductively the sequential projection measurement $p_n\succ \cdots\succ p_1$ resembles a $\{0,1\}^n$ -valued random variable, and

$\displaystyle \mathbb{P}[p_n\succ \cdots\succ p_1=(\theta_n,\dots,\theta_1)\,|\,\psi]=\|p_n\cdots p_1(\psi)\|^2.$

If $p$ and $q$ do commute, they share an orthonormal eigenbasis, and it can be interpreted that they can “agree” on what they “see” when they “look” at $\mathsf{H}$ , and can thus be determined simultaneously. Alternatively, if they commute then the distributions of $q\succ p$ and $p\succ q$ are equal in the sense that

$\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]= \mathbb{P}\left[(p=\theta_1)\succ (q=\theta_1)\,|\,\psi\right],$

it doesn’t matter what order they are measured in, the outputs of the measurements can be multiplied together, and this observable can be called $pq=qp$ .

Consider the (classical) permutation group $S_N$ or moreover its algebra of functions $F(S_N)$ . The elements of $F(S_N)$ can be represented as bounded operators on $\ell^2(S_N)$ , and the algebra is generated by a magic unitary $u^{S_N}\in M_N(B(\ell^2(S_N)))$ where:

$u_{ij}^{S_N}(e_\sigma)=\mathbf{1}_{j\rightarrow i}(e_\sigma)e_{\sigma}$ .

Here $\mathbf{1}_{j\rightarrow i}\in F(S_N)$ (‘unrepresented’) that asks of $\sigma$ … do you send $j\rightarrow i$ ? One for yes, zero for no.

Recall that the product of commuting projections is a projection, and so as $F(S_N)$ is commutative, products such as:

$\displaystyle p_\sigma:=\prod_{j=1}^Nu_{\sigma(j)j}^{S_N}$ ,

There are, of, course, $N!$ such projections, they form a partition of unity themselves, and thus we can build a measurement that will identify a random permutation $\varsigma\in P(\ell^2(S_N))$ and leave it equal to some $e_\sigma$ after measurement. This is the essence of classical… all we have to do is enumerate $n:S_N\rightarrow \{1,\dots,N!\}$ and measure using:

$\displaystyle f=\sum_{\sigma\in S_N}n(\sigma)p_{\sigma}$ .

A quantum permutation meanwhile is impossible to pin down in such a way. As an example, consider the Kac-Paljutkin quantum group of order eight which can be represented as $F(\mathfrak{G}_0)\subset B(\mathbb{C}^6)$ . Take $\varsigma=e_5\in \mathbb{C}^6$ . Then

$\displaystyle\mathbb{P}[(\varsigma(1)=4)\succ(\varsigma(3)=1)\succ(\varsigma(1)=3)]=\frac{1}{8}$ .

If you think for a moment this cannot happen classically, and the issue is that we cannot know simultaneously if $\varsigma(1)=3$ and $\varsigma(3)=1$ … and if we cannot know this simultaneously we cannot pin down $\varsigma$ to a single element of $S_N$ .

No Quantum Cyclic Group

Suppose that $\varsigma\in \mathsf{H}$ is a quantum permutation (in $S_N^+$ ). We can measure where the quantum permutation sends, say, one to. We simply form the self-adjoint element:

$\displaystyle x(1)=\sum_{k=1}^Nku_{k1}$ .

The measurement will produce some $k\in \{1,\dots,N\}$ … but if $\varsigma$ is supposed to represent some “quantum cyclic permutation” then we already know the values of $\varsigma(2),\dots,\varsigma(N)$ from $\varsigma(1)=k$ , and so, after measurement,

$u_{k1}\varsigma \in \bigcap_{m=1}^N \text{ran}(u_{m+k-1,m})$ , $u_{k1}\varsigma\equiv k-1\in\mathbb{Z}_N$ .

The significance of the intersection is that whatever representation of $C(S_N^+)$ we have, we find these subspaces to be $C(S_N^+)$ -invariant, and can be taken to be one-dimensional.

I believe this explains why there is no quantum cyclic group.

Question 1

Can we use a similar argument to show that there is no quantum version of any abelian group? Perhaps using $F(G\times H)=F(G)\otimes F(H)$ together with the structure theorem for finite abelian groups?

No Quantum $S_3$

Let $C(S_3^+)$ be represented as bounded operators on a Hilbert space $\mathsf{H}$ . Let $\varsigma\in P(\mathsf{H})$ . Consider the random variable

$x(1)=u_{11}+2u_{21}+3u_{31}$ .

Assume without loss of generality that $u_{31}\varsigma\neq0$ then measuring $\varsigma$ with $x(1)$ gives $x(1)\varsigma=3$ with probability $\langle\varsigma,u_{31}\varsigma\rangle$ , and the quantum permutation projects to:

$\displaystyle \frac{u_{31}\varsigma}{\|u_{31}\varsigma\|}\in P(\mathsf{H})$ .

Now consider (for any $\varsigma\in P(\mathsf{H})$ , using the fact that $u_{21}u_{31}=0=u_{32}u_{31}$ and the rows and columns of $u$ are partitions of unity:

$u_{31}\varsigma=(u_{12}+u_{22}+u_{32})u_{31}\varsigma=(u_{21}+u_{22}+u_{23})u_{31}\varsigma$

$\Rightarrow u_{12}u_{31}\varsigma=u_{23}u_{31}\varsigma$ (*)

Now suppose, again without loss of generality, that measurement of $u_{31}\varsigma\in P(\mathsf{H})$ with $x(2)=u_{12}+2u_{22}+3u_{33}$ produces $x(2)u_{31}\varsigma=1$ , then we have projection to $u_{12}u_{31}\varsigma\in P(\mathsf{H})$ . Now let us find the Birkhoff slice of this. First of all, as $x(2)=1$ has just been observed it looks like:

$\Phi(u_{12}u_{31}\varsigma)=\left[\begin{array}{ccc}0 & 1 & 0 \\ \ast & 0 & \ast \\ \ast & 0 & \ast \end{array}\right]$

In light of (*), let us find $\Phi(u_{12}u_{31}\varsigma)_{23}$ . First let us normalise correctly to

$\displaystyle \frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|}$

$\displaystyle\Phi(u_{12}u_{31}\varsigma)_{23}=\left\langle\frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|},u_{23}\frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|}\right\rangle$

Now use (*):

$\displaystyle\Phi(u_{12}u_{31}\varsigma)_{23}=\left\langle\frac{u_{23}u_{31}\varsigma}{\|u_{23}u_{31}\varsigma\|},u_{23}\frac{u_{23}u_{31}\varsigma}{\|u_{23}u_{31}\varsigma\|}\right\rangle=1$

$\displaystyle \Rightarrow \Phi(u_{12}u_{31}\varsigma)=\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ \Phi(u_{12}u_{31}\varsigma)_{31} & 0 & 0 \end{array}\right]$ ,

and as $\Phi$ maps to doubly stochastic matrices we find that $\Phi(u_{12}u_{31}\varsigma)$ is equal to the permutation matrix $(132)$ .

Not convincing? Fair enough, here is proper proof inspired by the above:

Let us show $u_{11}u_{22}=u_{22}u_{11}$ . Fix a Hilbert space representation $C(S_3^+)\subset B(\mathsf{H})$ and let $\varsigma\in\mathsf{H}$ .

The basic idea of the proof is, as above, to realise that once a quantum permutation $\varsigma$ is observed sending, say, $3\rightarrow 2$ , the fates of $2$ and $1$ are entangled: if you see $2\rightarrow 3$ you know that $1\rightarrow 1$ .

This is the conceptional side of the proof.

Consider $u_{23}\varsigma$ which is equal to both:

$(u_{11}+u_{21}+u_{31})u_{23}\varsigma=(u_{31}+u_{32}+u_{33})u_{23}\varsigma\Rightarrow u_{11}u_{23}\varsigma=u_{32}u_{23}\varsigma$ .

This is the manifestation of, if you know $3\rightarrow 2$ , then two and one are entangled. Similarly we can show that $u_{22}u_{13}\varsigma=u_{31}u_{13}\varsigma$ and $u_{22}u_{33}=u_{11}u_{33}$ .

Now write

$\varsigma=u_{13}\varsigma+u_{23}\varsigma+u_{33}\varsigma$

$\Rightarrow u_{11}\varsigma=u_{11}u_{23}\varsigma+u_{11}u_{33}\varsigma=u_{32}u_{23}\varsigma+u_{22}u_{33}\varsigma$

$\Rightarrow u_{22}u_{11}\varsigma=u_{22}u_{33}\varsigma$ .

Similarly,

$u_{22}\varsigma=u_{22}u_{13}\varsigma+u_{22}u_{33}\varsigma=u_{31}u_{13}\varsigma+u_{22}u_{33}\varsigma$

$\Rightarrow u_{11}u_{22}\varsigma=u_{11}u_{22}u_{33}\varsigma=u_{11}u_{11}u_{33}\varsigma=u_{11}u_{33}\varsigma=u_{22}u_{33}\varsigma$

Which is equal to $u_{22}u_{11}x$ , that is $u_{11}$ and $u_{22}$ commute.

Question 2

Is it true that if every quantum permutation in a $\mathsf{H}$ can be fully described using some combination of $u_{ij}$ -measurements, then the quantum permutation group is classical? I believe this to be true.

Quantum Alternating Group

Freslon, Teyssier, and Wang state that there is no quantum alternating group. Can we use the ideas from above to explain why this is so? Perhaps for $A_4$ .

A possible plan of attack is to use the number of fixed points, $\text{tr}(u)$ , and perhaps show that $\text{tr}(u)$ commutes with $x(1)$ . If you know these two simultaneously you nearly know the permutation. Just for completeness let us do this with $(\text{tr}(u),x(1))$ :

tr(u)\x(1)	1	2	3	4
0	–	(12)(34)	(13)(24)	(14)(23)
1	(234),(243)	(134),(143)	(124),(142)	(123),(132)
4	e	–	–	–

The problem is that we cannot assume that that the spectrum of $\text{tr}(u)$ is $\{0,1,4\}$ , and, euh, the obvious fact that it doesn’t actually work.

What is more promising is

x(1)\x(2)	1	2	3	4
1	–	e	(234)	(243)
2	(12)(34)	–	(123)	(124)
3	(132)	(134)	–	(13)(24)
4	(142)	(143)	(14)(23)	–

However while the spectrums of x(1) and x(2) are cool (both in $\{1,2,3,4\}$ ), they do not commute.

Question 3

Are there some measurements that can identify an element of $A_4$ and via a positive answer to Question 3 explain why there is no quantum $A_4$ ? Can this be generalised to $A_n$ .

MATH6055: Winter 2020, Week 9

December 1, 2020 in MATH6055 | Leave a comment

Outlook

Week 9

There is only a mini-chapter on examples of (real) functions. You should spend the rest of the time revising Chapters 3 and 4 (i.e. doing exercises). Some of you might have to catch up on the Week 8 lectures.

Week 10: 14 to 20 December.

There will be feedback on submitted exercises Wednesday, Thursday, and Friday.

I plan that we will have a number of Zoom tutorials (these are provisional):

Tuesday 14:00
Tuesday 15:00 C-X only
Wednesday 12:00 C-Y only
Thursday 12:00

There is a small chance that we will also have Monday 14:00 Zoom but will not know until closer to the day.

Week 11 – 40% Test 3

The final, 40% Test 3 on Functions and Algebra (chapters 3-5) will take place (provisionally) at 18:00-19:30 (perhaps with additional upload time) on Monday 21 December. There will be no more feedback on submitted exercises but there might be a 14:00 Zoom before the final Test.

30% Test 2 – Results

I had said that my aim was to have these to you before Wednesday 9 December. However there is a possibility now that the Test 2 results will not be released until after you sit Test 3. This is outside my control (watch this space).

Week 9

Lectures

Schedule about an hour and 20 minutes for these:

Examples of (Real) Functions I (25 minutes)
Examples of (Real) Functions II [18.30 on is Exercise Hints] (27 minutes)
Examples of (Real) Functions Summary (3 minutes)

Exercises

p.158-163, Q. 1-17 [Hints: [18.30-26.44]]

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MATH7019: Winter 2020, Week 11

December 1, 2020 in MATH7019 | Leave a comment

Outlook

Assignment 3 is up on Canvas and I hope to have Assignment 4 available quite soon. There is about an hour and a half of new material here (well, it is actually more-or-less MATH6040 material) and that is the lectures completed.

Week 11: Finish Assignment 3, watch all the Chapter 4 lecture material, and then try Chapter 4 exercises. Assignment 4 (20%, based on Chapter 4) will appear at some point.
Week 12: No new lecture material nor exercises. Will have daily Canvas assignments for students to get next day feedback on Chapter 4 exercises
Week 13: Finish Assignment 4 for Tuesday 22 December.

Assignment 2 – Results

It is my intention to have your results out by the end of the Week 10.

Week 11

Lectures

About an hour and a half here:

Partial Differentiation I (26 minutes)
Partial Differentiation II and Rounding Errors (27 minutes)
Rounding Error Example (8 minutes)

If you want a slightly slower re-introduction to partial differentiation I go over it a bit slower here (along with some more examples – this is from MATH6040).

Exercises

p. 171 (revision of partial differentiation)
p.176 – 177, Q.1-5. Additional exercises p.177, Q.6.

You can (carefully) take photos of your work and submit to whatever Canvas Exercises ‘Assignment’ is open.

Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Week 12

Revision/Working on Assignment 4. Will have daily Canvas Exercise Assignments for feedback.

Academic Learning Centre

Have you heard about CIT Maths Online (Links to an external site.) on Canvas? It’s full of helpful Maths and Stats resources, notes, quizzes and videos to help you throughout the whole year.

During this period of remote learning we will be using the Maths Online module on Canvas to offer Maths and Stats support to you and answer as many student questions as possible.

1-to-1 Maths appointments will be available through Zoom. There are 32 appointments available per week. Please log on to Maths online to see the most up to date resources and to book a 1-to-1 video call. Find out how to add it to your Canvas dashboard here https://studentengagement.cit.ie/alc/resources.maths (Links to an external site.).

Drop-in Maths is on every Tuesday from 12 – 2. Here you can drop in to ask our lecturer a quick Maths question.

If you have any other question about our remote Maths and Stats supports email Joy and Deirdre at academiclearning@cit.ie.

See the CIT Students tab above for further resources.

MATH7019: Winter 2019, Week 10

November 28, 2020 in MATH7019 | Leave a comment

Outlook

Assignment 3 is up on Canvas. There is about an hour and 20 minutes of new material here, similar next week (7-13 December) and no new material after that.

Week 9: If you have done the exercises from Week 8 and 9 you will be able to attack Assignment 3. There is some Chapter 4 lecture material and exercises.
Week 11: Finish Assignment 3, watch all the Chapter 4 lecture material, and then try Chapter 4 exercises. Assignment 4 (20%, based on Chapter 4) will appear.
Week 12: No new lecture material nor exercises. Will have daily Canvas assignments for students to get next day feedback on Chapter 4 exercises
Week 13: Finish Assignment 4 for Tuesday 22 December.

Assignment 2 – Results

It is my intention to have your results out by the end of the next week (6 December).

Week 10

Lectures

About two hours here.

Taylor Series I (25 minutes)
Taylor Series II (22 minutes)
Taylor Series III (23 minutes)
Taylor Series IV (13 minutes)

Exercises

p. 155, Q. 1-2
Show that the Maclaurin Series of $\sec x$ gives $\displaystyle \sec x\approx 1+\frac12x^2$ .
p. 162, Q. 1-5
p.165, Q.1-3. Additional exercises p.165, Q.4-5.

You can (carefully) take photos of your work and submit to the Week 10 Exercises.

Week 11

We will look at Error Analysis.

Academic Learning Centre

Have you heard about CIT Maths Online (Links to an external site.) on Canvas? It’s full of helpful Maths and Stats resources, notes, quizzes and videos to help you throughout the whole year.

During this period of remote learning we will be using the Maths Online module on Canvas to offer Maths and Stats support to you and answer as many student questions as possible.

Drop-in Maths is on every Tuesday from 12 – 2. Here you can drop in to ask our lecturer a quick Maths question.

If you have any other question about our remote Maths and Stats supports email Joy and Deirdre at academiclearning@cit.ie.

See the CIT Students tab above for further resources.

MATH6055: Winter 2020, Week 8

November 28, 2020 in MATH6055 | Leave a comment

Outlook

Week 8

As you will see below there is a lot of lecture material. It might be wise to break your learning into Exponents and Logarithms, and spend 3.5 hours on each. Start the clock running on Exponents, after you watch those lectures spend the rest of the 3.5 hours doing exercises on them, and then repeat the trick with logs.

That would be ideal but the good news is that we have all but finished the lecture material, there won’t be too many lecture videos in Week 9, there will be none in Week 10, and so there will be scope to catch up. However you will be better off keeping up as you will need time to revise Chapter 3 Functions also.

You can always submit whatever work you want to whatever is open on Canvas at the time (but please label properly with page and question number).

Week 9

We will complete the mini-chapter on examples of functions. You will spend the rest of the time revising Chapters 3 and 4 (i.e. doing exercises).

Week 10

There will be feedback on submitted exercises Wednesday, Thursday, and Friday.

I plan that we will have a number of Zoom tutorials (these are provisional):

Tuesday 14:00
Tuesday 15:00 C-X only
Wednesday 12:00 C-Y only
Thursday 12:00

There is a small chance that we will also have Monday 14:00 Zoom but will not know until closer to the day.

Week 11

The final, 40% Test 3 on Functions and Algebra will take place (provisionally) at 18:00-19:30 (perhaps with additional upload time) on Monday 21 December. There will be no more feedback on submitted exercises but there might be a 14:00 Zoom before the final Test.

30% Test 2 – Results

My aim would be to have these for you before Wednesday 9 December.

Week 8

Lectures – Exponents

Schedule about an hour and 50 minutes to watch these:

Exponents/Powers I (25 minutes)
Exponents/Powers II (25 minutes)
Exponents/Powers III and Logarithms I [9.35 to 12.40 of Exercise Hints] (24 minutes)

Exercises – Exponents

p.99, Q.11 (functions question that requires algebra)
p.128-129, Q. 1-10 [Hints: [9.35-12.40]]

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MATH6055: Winter 2020, Week 7

November 21, 2020 in MATH6055 | Leave a comment

You are advised to to spend seven hours per week on MATH6055. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent emailing questions, doing exercises, and catch-up/revision.

30% Test 2

The second 30% assessment takes place 6 pm Wednesday 25 November, and covers equivalence relations and Chapter 2. Relevant are the Week 3, 4 and 5 announcements on Canvas. Students who have been working on this material are in a good position and should be able to watch the Chapter 3 lectures while those who have not been working (i.e. doing and submitting exercises) are probably going to have to work on the Weeks 3-5 exercises.

Chapters 3 to 5 will be assessed in the final assessment in Week 10/11.

Zoom for Test 2

There will be two Zoom tutorials for Test 2 in Week 7:

Monday 23 November, 14:00
Tuesday 24 November, 14:00

The links have been sent via Canvas Mail. Students are invited to attend one or both tutorials where I will be answering student questions. The tutorials will be recorded.

Week 7

Lectures

Schedule about two hours and 15 minutes to watch this hour and a half of lectures.

Algebra: Intro and Basics (27 minutes)
Equations I (26 minutes)
Equations II (27 minutes)
Equations III (11 minutes)

Read the rest of this entry »

MATH7019: Winter 2020, Week 9

November 21, 2020 in MATH7019 | Leave a comment

I hope to get Assignment 3 up on Canvas by around or before Monday 30 November. There was a lot of lecture material in Week 8, not so much here.

Week 9: Finish watching the Weeks 7-9 lectures and give time to the Weeks 8 and 9 exercises.
Week 10: Assignment 3 (20%, based on Chapter 3) will appear. If you have done the exercises from Week 8 and 9 you will be able to attack this. There will be some Chapter 4 lecture material and exercises.
Week 11: Finish Assignment 3, watch all the Chapter 4 lecture material, and then try Chapter 4 exercises. Assignment 4 (20%, based on Chapter 4) will appear.
Week 12: No new lecture material nor exercises. Might have daily Canvas assignments for students to get next day feedback on Chapter 4 exercises
Week 13: Finish Assignment 4 for Tuesday 22 December.

Assignment 2

I plan to start corrections Thursday 26 November and it is my intention to release results.

Week 9

Lectures

Only a little over an hour here. Big focus on exercises.

Hypothesis Testing I (24 minutes)
Hypothesis Testing II (20 minutes)

Exercises

p. 150, Q.1-8

You can (carefully) take photos of your work and submit to the Week 9 Exercises.

Week 10

We will start looking at Taylor Series.

Academic Learning Centre

Have you heard about CIT Maths Online (Links to an external site.) on Canvas? It’s full of helpful Maths and Stats resources, notes, quizzes and videos to help you throughout the whole year.

During this period of remote learning we will be using the Maths Online module on Canvas to offer Maths and Stats support to you and answer as many student questions as possible.

Drop-in Maths is on every Tuesday from 12 – 2. Here you can drop in to ask our lecturer a quick Maths question.

If you have any other question about our remote Maths and Stats supports email Joy and Deirdre at academiclearning@cit.ie.

See the CIT Students tab above for further resources.

MATH7019: Winter 2020, Week 8

November 14, 2020 in MATH7019 | Leave a comment

I put few lectures (and no new exercises) into Week 7 but there are lots of lectures here and lots of new exercises. The faster you can (well-complete) Assignment 2 the better and it leaves a possible plan like (again I recommend 7 hrs per week… but managing your time is a difficulty for you to solve).

Green is coming up to Assignment 2, Orange to Assignment 3, and Red to Assignment 4

Week 8: Complete Assignment 2 and watch as much Week 7 and Week 8 lectures as possible. If ahead of things try some Week 8 exercises.
Week 9: There will not be as much lecture material, but we will finish Chapter 3. Finish watching the Weeks 7-9 lectures and give time to the Weeks 8 and 9 exercises.
Week 10: Assignment 3 (20%, based on Chapter 3) will appear. If you have done the exercises from Week 8 and 9 you will be able to attack this. There will be some Chapter 4 lecture material and exercises.
Week 11: Finish Assignment 3, watch all the Chapter 4 lecture material, and then try Chapter 4 exercises. Assignment 4 (20%, based on Chapter 4) will appear.
Week 12: No new lecture material nor exercises. Might have daily Canvas assignments for students to get next day feedback on Chapter 4 exercises
Week 13: Finish Assignment 4 for Tuesday 22 December.

Assignment 2

Assignment is on Canvas, and has a deadline the end of this week 8, Friday 20 November.

Week 8

Lectures

A lot here: you probably need three hours to watch these two hours and 22 minutes of lectures:

Normal Distribution I (28 minutes)
Normal Distribution II (27 minutes)
Normal Distribution III (21 minutes)
Sampling Theory I (27 minutes)
Sampling Theory II (24 minutes)
Sampling Theory III: Example (17 minutes)

Exercises

p. 134, Q. 1-7
p. 142-3, Q. 1-4

Additional Exercises

p. 134, Q. 8-9
p. 143, Q. 5-6

You can (carefully) take photos of your work and submit to the Week 8 Exercises.

Week 9

We will look at Hypothesis Testing.

Academic Learning Centre

Have you heard about CIT Maths Online (Links to an external site.) on Canvas? It’s full of helpful Maths and Stats resources, notes, quizzes and videos to help you throughout the whole year.

During this period of remote learning we will be using the Maths Online module on Canvas to offer Maths and Stats support to you and answer as many student questions as possible.

Drop-in Maths is on every Tuesday from 12 – 2. Here you can drop in to ask our lecturer a quick Maths question.

If you have any other question about our remote Maths and Stats supports email Joy and Deirdre at academiclearning@cit.ie.

See the CIT Students tab above for further resources.

MATH6055: Winter 2020, Week 6

November 13, 2020 in MATH6055 | Leave a comment

30% Test 2

The second 30% assessment takes place in Week 7 (week of 23 November, provisionally 6 pm Wednesday 25 November), and covers equivalence relations and Chapter 2. Relevant are the Week 3, 4 and 5 announcements on Canvas. Students who have been working on this material are in a good position and should be able to watch the Chapter 3 lectures while those who have not been working (i.e. doing and submitting exercises) are probably going to have to work on the Weeks 3-5 exercises.

Chapters 3 to 5 will be assessed in the final assessment in Week 10/11.

Zoom for Test 2

There will be two Zoom tutorials for Test 2 in Week 7:

Monday 23 November, 14:00
Tuesday 24 November, 14:00

I will send the links for these a little closer to the slots. Students are invited to attend one or both tutorials where I will be answering student questions. The tutorials will be recorded.

Week 6

Lectures

Schedule about an three hours to watch this 2 hours and ten minutes of lectures.

One-to-One, Onto, Bijective I (29 minutes)
One-to-One, Onto, Bijective II (29 minutes)
One-to-One, Onto, Bijective: End of Example 9 and Exercise Hints (10 minutes)
Compositions and Inverses I (29 minutes)
Compositions and Inverses II (24 minutes)
Compositions & Inverses Exercise Hints and Functions Summary (9 minutes)

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MATH7019: Winter 2020, Week 7

November 2, 2020 in MATH7019 | Leave a comment

I recommend continuing to aim for seven hours per week of MATH7019 but you may want to continue working on Chapter 2/Assignment 2 material before viewing the Week 7 lectures. There are no exercises for the Week 7 lectures.

Assignment 2

Assignment 2 is now available. It has a deadline the end of Week 8, Friday 20 November.

Week 7

Your focus at this time should be on Assessment 2: this includes catching up on Chapter 2 lectures and exercises, which were front-loaded into the Week 6 announcement.

Lectures

Not much here: 48 minutes which can be watched in about 70 minutes.

Random Variables I (27 minutes)
Random Variables II (20 minutes)

Exercises

Same as Week 6.

You can (carefully) take photos of your work and submit to the Week 7 Exercises.

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J.P. McCarthy: Math Page

Quantum Permutations Basics

Weaver Philosophy

No Quantum Cyclic Group

Question 1

No Quantum

Question 2

Quantum Alternating Group

Question 3

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Outlook

Week 9

Week 10: 14 to 20 December.

Week 11 – 40% Test 3

30% Test 2 – Results

Week 9

Lectures

Exercises

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Outlook

Assignment 2 – Results

Week 11

Lectures

Exercises

Week 12

Academic Learning Centre

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Outlook

Assignment 2 – Results

Week 10

Lectures

Exercises

Week 11

Academic Learning Centre

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Outlook

Week 8

Week 9

Week 10

Week 11

30% Test 2 – Results

Week 8

Lectures – Exponents

Exercises – Exponents

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30% Test 2

Zoom for Test 2

Week 7

Lectures

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Assignment 2

Week 9

Lectures

Exercises

Week 10

Academic Learning Centre

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Assignment 2

Week 8

Lectures

Exercises

Week 9

Academic Learning Centre

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30% Test 2

Zoom for Test 2

Week 6

Lectures

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Assignment 2

Week 7

No Quantum $S_3$