You are advised to to spend **seven** hours per week on MATH7019. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. Also catching up on material we have already covered, and doing exercises. At the time of writing, tutorials consist of you emailing questions and getting feedback on submitted work, but this is subject to change.

Schedule about two and half hours to watch these 104 minutes of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Beams: Variables (28 minutes)
- Beams: Loads I (28 minutes)
- Beams: Loads II; and Review of Quadratics (26 minutes)
- Simply Supported Beams I (22 minutes)

You need to schedule about four and half hours to work on this week’s exercises.

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning.

**p. 72 (instructions) for p. 73, Q. 1-5****p. 89, Q. 1**

You can (carefully) take photos of your work and submit to the Week 4 Exercises those images on Canvas before midnight Sunday 18 October. The intention would be that after 09:00 Monday 19 October someone (I am going on two weeks paternity leave at some stage) will download all student work and reply with feedback.

If possible, 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.

We will then look at more examples of simply supported beams before moving onto fixed-end beams.

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there ith *specific *questions.

Assessment 1 has a provisional hand-in of the end of Week 4, start of Week 5. Once the class list is established I can send on the student-number-personalised assessment.

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

]]>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 doing exercises, emailing questions, and submitting work. At the time of writing, the mechanism for submitting work is not yet set up.

Schedule about three hours to watch these two hours of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Laws of Sets and Illustrations of DeMorgan’s Laws (25 minutes)
- Inclusion/Exclusion Principle; and some difficult notational stuff (14 minutes)
- Cartesian Products (21 minutes)
- Tuples: Passwords and Bit Strings (7 minutes)
- Relations (22 minutes)

Here is the first tranch of notes if you have not purchased nor printed off a manual

- Relations: Examples and Reflexivity (27 minutes)

I recommend spending four hours on this week’s exercises.

I do not at the time of writing have the mechanism in place for submitting work.

You can (carefully) take photos of your work. If possible, convert the images to 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.

Ordinarily I would encourage you to ask questions via email at any time but at present this mechanism is not in place.

Do/attempt:

**p. 32, Q. 10-11****p. 33, Q. 12-14****p. 38, Q. 1-5****p. 39, Q. 6****p. 43, Q. 1-2**

**Additional/Harder Exercises:**

**p.34, Q. 26****p. 39, Q. 7-8****p. 43, Q. 3-6**

In Week 3 we will finish Chapter 1 by looking more at relations and their properties. We will start our study of Chapter 2: Functions. A function is a special type of relation.

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there ith *specific *questions.

At the time of writing, the assessment schedule has not been finalised, but we are considering two 50% tests.

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

]]>Some Facetime with me: click here.

I will be on paternity leave until Monday 12 October. My colleague Dr Mark Hartnett (mark.hartnett@cit.ie) will answer any email queries that you have.

The lectures are being delivered via pre-recorded lectures. As you will see, the lectures use a manual that contain all the lecture material, via gaps that are filled in during lectures, and exercises. I tend to use a number of colours during lectures, and pencil, so you might want to consider ordering some of these:

In a sliding scale from best to worst, in my opinion, here are your options for using this manual. There are other options but I cannot recommend them. If you do option one you have all your notes in one place and can follow the lectures as if you were in the classroom.

- Email copy.centre@cit.ie and tell them you want to order a bound copy of
*MATH6055 Manual Winter 2020*. The manuals can be collected from Reprographics beside the Student Centre. Note that this is a cash-free area so you will need to put the appropriate amount of funds on your student card. At the time of writing I do not know the cost but it will be of the order of €13. This seems like a lot of money for a manual but with all the materials (including exercises, summaries, etc) it comes to 165 pages and provides a comprehensive resource for this module. - Print off the manual at home or somewhere else. Click here to find a copy.
- I am going to scan and email the completed slides. You can keep these somewhere for your notes. You could print these or keep digital copies. Here is the first tranch of notes.

You are advised to to spend **seven** hours per week on MATH6055. This should comprise of however long is recommended to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. At the time of writing, the mechanism for submitting work is not yet set up.

Schedule about three hours to watch these two hours of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Introduction and Number Sets (21 minutes)
- Set Notation and Venn Diagrams (26 minutes)
- Subsets, Binary Strings, Set Operations (27 minutes)
- Set Operations: Examples (24 minutes)
- Truth Tables (24 minutes)

I recommend spending four hours on this week’s exercises.

I do not at the time of writing have the mechanism in place for submitting work.

You can (carefully) take photos of your work. If possible, convert the images to 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.

Ordinarily I would encourage you to ask questions via email at any time but at present this mechanism is not in place.

Do/attempt:

**p. 31, Q. 1-3****p.32, Q. 4-7****p.33, Q. 15-21****p.33, Q. 22-24**

**Additional/Harder Exercises:**

**p.34, Q. 25, 27**

We will talk more about Laws of Sets, the Inclusion/Exclusion Principle, and we will hopefully start talking about Cartesian Products.

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there ith *specific *questions.

At the time of writing, the assessment schedule has not been finalised, but we are considering two 50% tests.

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

]]>You are advised to to spend **seven** hours per week on MATH7019. This should comprise of however long is recommended to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. At the time of writing, tutorials consist of you emailing questions and getting feedback on submitted work, but this is subject to change.

Schedule about an hour and a quarter to watch these 51 minutes of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Non-linear Laws and Linearisation (14 minutes)
- Log-Linear Least Squares: Example I (19 minutes)
- Log-Linear Least Squares Example II & Curve Fitting Summary (15 minutes)

Here are Chapter 1 slides if you have not purchased or printed off the manual.

You need to schedule about two and a quarter hours to work on these exercises.

**p.41, linearise.****p.49, Q.1-7**.

Schedule about an hour and a quarter to watch these 54 minutes of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Beams: Intro and Calculus Review (23 minutes)
- Intro to Differential Equations (4 minutes)
- Impulse and Step Functions (27 minutes)

You need to schedule about two and a quarter hours to work on these exercises.

**p. 59, Q. 1-2.****p.65, Q.1-3.**

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner. I am not sure exactly what will happen with these questions while I am on paternity leave… hopefully someone will take these questions for you.

You can (carefully) take photos of your work and submit to the Week 3 Exercises those images on Canvas before midnight Sunday 11 October. The intention would be that after 09:00 Monday 12 October someone (I am going on two weeks paternity leave at some stage) will download all student work and reply with feedback.

If possible, 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.

We will plough into Chapter 2, looking at simply supported beams.

*specific *questions.

Assessment 1 has a provisional hand-in of the end of Week 4, start of Week 5. Once the class list is established I can send on the student-number-personalised assessment.

You are advised to to spend **seven** hours per week on MATH7019. This should comprise of however long is recommended to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. At the time of writing, tutorials consist of you emailing questions and getting feedback on submitted work, but this is subject to change.

Schedule about three hours to watch these 126 minutes of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Revision: Lines, Differentiation, Optimisation (18 minutes)
- Least Squares: Theory (19 minutes)
- Least Squares: Example I (22 minutes)
- Least Squares: Practise and Cramer’s Rule (19 minutes)
- Least Squares: Example II (21 minutes)
- Least Squares: Example III (23 minutes)
- Correlation (4 minutes)

Here are Chapter 1 slides if you have not purchased or printed off the manual.

You need to schedule about four hours to work on this week’s exercises.

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner. I am not sure exactly what will happen with these questions while I am on paternity leave… hopefully someone will take these questions for you.

- p.34, Q. 1-4
- p.37, Autumn 2015

Additional (Harder) Exercises:

- p.22, show that and are both positive.
- p.26, repeat the page 22 analysis for :

Partially differentiate this with respect to , , , solve equal to zero, to find the equations in the middle of p.26.

You can (carefully) take photos of your work and submit to the Week 2 Exercises those images on Canvas before midnight Sunday 4 October. The intention would be that after 09:00 Monday 5 October someone (I am going on two weeks paternity leave at some stage) will download all student work and reply with feedback.

If possible, 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.

We will start looking at non-linear models.

*specific *questions.

Assessment 1 has a provisional hand-in of the end of Week 4, start of Week 5. Once the class list is established I can send on the student-number-personalised assessment.

Some Facetime with me: click here.

The lectures are being delivered via pre-recorded lectures. As you will see, the lectures use a manual that contain all the lecture material, via gaps that are filled in during lectures, and exercises. I tend to use a number of colours during lectures, and pencil, so you might want to consider ordering some of these:

In a sliding scale from best to worst, in my opinion, here are your options for using this manual. There are other options but I cannot recommend them. If you do option one you have all your notes in one place and can follow the lectures as if you were in the classroom.

- Email copy.centre@cit.ie and tell them you want to order a bound copy of
*MATH7019 Manual Winter 2020*. The manuals can be collected from Reprographics beside the Student Centre. Note that this is a cash-free area so you will need to put the appropriate amount of funds on your student card. At the time of writing I do not know the cost but it will be of the order of €15. This seems like a lot of money for a manual but with all the materials (including worked examples, summaries, etc) it comes to about 187 pages and provides a comprehensive resource for this module. - Print off the manual at home or somewhere else. Click here here to find a copy.
- I am going to scan and email the completely slides. You can keep these somewhere for your notes. You could print these or keep digital copies. Here is Chapter 1.

You are advised to to spend **seven** hours per week on MATH7019. This should comprise of however long is recommended to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. At the time of writing, tutorials consist of you emailing questions and getting feedback on submitted work, but this is subject to change.

There is probably less than two and half hours here for Week 1: this might be nice to ease yourself back into things, but if you are hungry for more material feel free to jump into Week 2 (see Canvas announcements).

Schedule about 80 minutes to watch these 50 minutes of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Intro to MATH7019 (15 minutes)
- Intro to Curve Fitting (9 minutes)
- Lagrange Interpolation (27 minutes)

You need to schedule about an hour to work on this week’s exercises.

You can (carefully) take photos of your work and submit to the Week 1 Exercises those images on Canvas before midnight Sunday 27 September. The intention would be that after 09:00 Monday 28 September someone (I am going on two weeks paternity leave at some stage) will download all student work and reply with feedback.

*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.

Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner. I am not sure exactly what will happen with these questions while I am on paternity leave… hopefully someone will take these questions for you.

**Do p. 13, Q. 1-3. Additional exercise: p.13, Q.4.**

We will start talking about Least Squares curve fitting.

The Student Resources tab above contains some information about calculators.

Here is a list of some allowed and not allowed calculators.

If you have to purchase a calculator, my recommendation is that you purchase something like a Casio fx-83GT PLUS. This might be available in the CIT shop.

*specific *questions.

See here for more.

]]>

When the Jacobi Method is used to approximate the solution of Laplace’s Equation, if the initial temperature distribution is given by , then the iterations are also approximations to the solution, , of the Heat Equation, assuming the initial temperature distribution is .

I first considered such a thought while looking at approximations to the solution of Laplace’s Equation on a thin plate. The way I implemented the approximations was I wrote the iterations onto an Excel worksheet, and also included conditional formatting to represent the areas of hotter and colder, and the following kind of output was produced:

Let me say before I go on that this was an implementation of the Gauss-Seidel Method rather than the Jacobi Method, and furthermore the stopping rule used was the rather crude .

However, do not the iterations resemble the flow of heat from the heat source on the bottom through the plate? The aim of this post is to investigate this further. All boundaries will be assumed uninsulated to ease analysis.

Consider a thin rod of length . If we *mesh* the rod into pieces of equal length , we have *discretised *the rod, into segments of length , together with ‘nodes’ .

Suppose are interested in the temperature of the rod at a point , . We can instead consider a sampling of , at the points :

.

Similarly we can *mesh *a plate of dimensions into an rectangular grid, with each rectangle of area , where and , together with nodes , and we can study the temperature of the plate at a point by sampling at the points :

.

We can also *mesh *a box of dimension into an 3D grid, with each rectangular box of volume , where , , and , together with nodes , and we can study the temperature of the box at the point by sampling at the points :

.

How the temperature evolves is given by *partial differential equations*, expressing relationships between and its rates of change.

Consider some real-valued function . Its derivative at — its rate of change with respect to — is defined as

Another way of thinking about this is that we have the following limit as :

,

which implies — by the definition of a limit — that for sufficiently close to zero,

.

This is known as the forward difference approximation, and here

,

will be used.

There is also a backward difference approximation (comes from approaching from below):

,

which will be used in the sequel. We will not worry at this time about how accurate or otherwise these approximations are although if we know our calculus, we know they should be good if the second derivative is small near .

Similarly, as is the derivative of , i.e.

,

we have

.

Now use the backwards forward difference approximation on and :

.

The *equilibrium* temperature distribution on an (insulated) rod, connected to heat sources at and , of temperatures and , is given by the solution of the differential equation:

, , .

This can be derived by considering a small length element , and considering that, for equilibrium, the heat flowing into must equal the heat flowing out of (else the heat — and therefore the temperature — in is increasing/decreasing).

As a differential equation, it is very easy: it can be antidifferentiated directly. It is probably worth bringing in a small amount of geometric intuition at this point.

Recall that the derivative of with respect to is not only the rate of change of with respect to , but also slope of the tangent of . Therefore, the second derivative of the temperature with respect to distance is the rate of change of the slope of (the tangent to) , and, as Laplace’s Equation says, for equilibrium, this is zero, i.e. the rate of change of slope doesn’t change, that is the slope is a constant, that is we have a line (for various technical reasons — let alone physical ones — the second derivative being zero doesn’t give a *piecewise*-line curve (derivative undefined at jumps)).

Therefore the temperature changes uniformly from to , giving solution:

.

For example, for , and :

Things are a little more complicated on a plate (uninsulated except at the edges) as the equilibrium temperature distribution is now the solution of a *partial *differential equation, known as Laplace’s Equation:

, given.

Consider a grid point with temperature :

Approximate the second derivatives at using forward differences:

and

.

If we assume that , and substitute these approximations into Laplace’s Equation, we find an equation whose solution approximates :

.

Multiply both sides by and solve for to find the Laplace Finite Difference Equation:

This approximate equation echoes something called the *mean value property for harmonic functions.*

Starting with an approximation to each of the , say all equal to zero, substituting these into the right hand side of this equation gives a *better *approximation. Iteratively feeding these approximations into the right hand side of this equation gives successively more accurate solutions to the this equation, and this iterative scheme, known as the Jacobi Method, converges to the exact solution of this Laplace Finite Difference Equation.

The *transient* temperature distribution on an (insulated) plate, connected to heat sources at , is given by the solution of the differential equation, known as the *Heat Equation*:

, , given.

This can be derived by considering a small area element , and considering the relationship between the nett heat flowing into and the resulting change in temperature.

As ,

,

that is the temperature does not change, the temperature distribution reaches *steady *or *equilibrium state*. Note if the left hand side is equal to zero, the equation reduces to Laplace’s Equation.

This means that if you solve the Heat Equation on a plate using (appropriate) finite differences, the temperatures converge to the same solutions as those of Laplace’s Equation (all else being equal).

The transient temperatures of the plate is a function:

.

We mesh up the domain using but also, for time, . Write

Now use finite differences to approximate the heat equation at :

.

Via , this can be written as:

,

in short, a formula for the temperature at at time , in terms of the temperature at that point — and the temperatures at the four adjacent points — *at the previous time. *This is the Heat Finite Difference Equation.

Now, is a system parameter. The dimensions of the plate are also a system parameter, and essentially is determined by the number of internal gridpoints to be considered. The time step is basically a free choice.

Note if

,

then the Heat Finite Difference Equation reads:

,

exactly the same as the Laplace Finite Difference Equation!

Therefore, suppose we have a plate with internal gridpoints spaced a distance apart, and the plate material parameter is . Suppose we are interested in the steady state temperature distribution, and our initial approximation to the steady state distribution is given by , then using the Laplace Finite Difference Equation, which converges to the steady state solution, also gives the transient temperatures of a plate with initial temperature but at time steps of:

.

This means that when using the Jacobi Method (without over-relaxation) to approximate the steady state, the intermediate approximations do have physical meaning.

]]>A is for atom and axiom. While we build beautiful universes from our carefully considered axioms, they try and destroy this one by smashing atoms together.

B is for the Banach-Tarski Paradox, proof if it was ever needed that the imaginary worlds which we construct are far more interesting then the dullard of a one that they study.

C is for Calculus and Cauchy. They gave us calculus about 340 years ago: it only took us about 140 years to make sure it wasn’t all nonsense! Thanks Cauchy!

D is for Dimension. First they said there were three, then Einstein said four, and now it ranges from 6 to 11 to 24 depending on the day of the week. No such problems for us: we just use .

E is for Error Terms. We control them, optimise them, upper bound them… they just pretend they’re equal to zero.

F is for Fundamental Theorems… they don’t have any.

G is for Gravity and Geometry. Ye were great yeah when that apple fell on Newton’s head however it was us asking stupid questions about parallel lines that allowed Einstein to formulate his epic theory of General Relativity.

H is for Hole as in the Black Hole they are going to create at CERN.

I is for Infinity. In the hand of us a beautiful concept — in the hands of you an ugliness to be swept under the carpet via the euphemism of “renormalisation”…

J is for Jerk: the third derivative of displacement. Did you know that the fourth, fifth, and sixth derivatives are known as Snap, Crackle, and Pop? No, I did not know they had a sense of humour either.

K is for Knot Theory. A mathematician meets an experimental physicist in a bar and they start talking.

- Physicist: “What kind of math do you do?”,
- Mathematician: “Knot theory.”
- Physicist: “Yeah, Me neither!”

L is for Lasers. I genuinely spent half an hour online looking for a joke, or a pun, or something humorous about lasers… Lost Ample Seconds: Exhausting, Regrettable Search.

M is for Mathematical Physics: a halfway house for those who lack the imagination for mathematics and the recklessness for physics.

N is for the Nobel Prize, of which many mathematicians have won, but never in mathematics of course. Only one physicist has won the Fields Medal.

O is for Optics. Optics are great: can’t knock em… 7 years bad luck.

P is for Power Series. There are rules about wielding power series; rules that, if broken, give gibberish such as the sum of the natural numbers being . They don’t care: they just keep on trucking.

Q is for Quark… they named them after a line in Joyce as the theory makes about as much sense as Joyce.

R is for Relativity. They are relatively pleasant.

S is for Singularities… instead of saying “we’re stuck” they say “singularity”.

T is for Tarksi… Tarski had a son called Jon who was a physicist. Tarksi always appears twice.

U is for the Uncertainty Principle. I am uncertain as to whether writing this was a good idea.

V is for Vacuum… Did you hear about the physicist who wanted to sell his vacuum cleaner? Yeah… it was just gathering dust.

W is for the Many-Worlds-Interpretation of Quantum Physics, according to which, Mayo GAA lose All-Ireland Finals in infinitely many different ways.

X is unknown.

Y is for Yucky. Definition: messy or disgusting. Example: Their “Calculations”

Z is for Particle Zoo… their theories are getting out of control. They started with atoms and indeed atoms are only the start. Pandora’s Box has nothing on these people.. forget baryons, bosons, mesons, and quarks: the latest theories ask for sneutrinos and squarks; photinos and gluinos, zynos and even winos. A zoo indeed.

We didn’t even mention String Theory!

The author gives the definition and gives the definition of a (left, quantum) group action.

Let be a compact matrix quantum group and let be a . An (left) *action *of on is a unital *-homomorphism that satisfies the analogue of , and the Podlés density condition:

.

Schmidt in this earlier paper gives a slightly different presentation of . The definition given here I understand:

The *quantum automorphism group* of a finite graph with adjacency matrix is given by the universal -algebra generated by such that the rows and columns of are partitions of unity and:

.

_______________________________________

The difference between this definition and the one given in the subsequent paper is that in the subsequent paper the quantum automorphism group is given as a quotient of by the ideal given by … ah but this is more or less the definition of universal -algebras given by generators and relations :

where presumably all works out OK, and it can be shown that is a suitable ideal, a Hopf ideal. I don’t know how it took me so long to figure that out… Presumably the point of quotienting by (a presumably Hopf) ideal is so that the quotient gives a subgroup, in this case via the surjective *-homomorphism:

.

_______________________________________

Let be a finite graph and a compact matrix quantum group. An action of on is an action of on (coaction of on ) such that the associated magic unitary , given by:

,

commutes with the adjacency matrix, .

By the universal property, we have via the surjective *-homomorphism:

, .

## Theorem 1.8 (Banica)

Let , and , be an action, and let be a linear subspace given by a subset . The matrix commutes with the projection onto if and only if

The action preserves the eigenspaces of :

*Proof: *Spectral decomposition yields that each , or rather the projection onto it, satisfies a polynomial in :

,

as commutes with powers of

Let . Permutations are *disjoint *if , and vice versa, for all .

In other words, we don’t have and permuting any vertex.

Let be a finite graph. If there exists two non-trivial, disjoint automorphisms , such that and , then we get a surjective *-homomorphism . In this case, we have the quantum group , and so has quantum symmetry.

*Proof: *Suppose we have disjoing with and .

The group algebra of can be given as a universal -algebra (related to something I am looking at, the relationship between universal -algebras and simplicity, note that is simple under suitable assumptions… this is about when a concrete -algebra is isomorphic to a univeral one, or when two universal -algebras are isomorphic).

Anyway, we have:

The proof plans to use the universal property (of ) to get a surjective *-homormorphism onto , and as the do not commute, this gives quantum symmetries.

Identifying with their permutation matrices, define:

.

Schmidt shows that these satisfies the relations of and thus we have a *-homormophism.

I thought such maps were automatically surjective… but obviously not. Schmidt shows that it is and so the rest follows

Consider so that and . The disjoint automorpshisms gives the famous surjective *-homorphism exhibiting the quantum nature of . This subgroup is therefore isomorphic to, well its algebra of functions, to .

Any pair of disjoint automorphisms give quantum symmetry,

]]>