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,

]]>

A group is a well-established object in the study of mathematics, and for the purposes of this post we can think of a group as the set of symmetries on some kind of space, given by a set together with some additional structure . The elements of *act *on as bijections:

,

such that , that is the structure of the space is invariant under .

For example, consider the space , where the set is , and the structure is the cardinality. Then the set of all of the bijections is a group called .

A set of symmetries , a group, comes with some structure of its own. The identity map , is a symmetry. By transitivity, symmetries can be composed to form a new symmetry . Finally, as bijections, symmetries have inverses , .

Note that:

.

A group can carry additional structure, for example, compact groups carry a topology in which the composition and inverse are continuous.

Given a group together with its structure, one can define an algebra of complex valued functions on , such that the multiplication is given by a commutative pointwise multiplication, for :

.

Depending on the class of group (e.g. finite, matrix, compact, locally compact, etc.), there may be various choices and considerations for what algebra of functions to consider, but on the whole it is nice if given an algebra of functions we can reconstruct .

Usually the following *transpose *maps will be considered in the structure of , for some tensor product such that , and , is the group multiplication:

See Section 2.2 to learn more about these maps and the relations between them for the case of the complex valued functions on *finite *groups.

Quantum groups, famously, do not have a single definition in the same way that groups do. All definitions I know about include a coassociative (see Section 2.2) comultiplication for some tensor product (or perhaps only into a multiplier algebra ), but in general that structure alone can only give a quantum *semi*group.

Here is a non-working (quickly broken?), meta-definition, inspired in the usual way by the famous Gelfand Theorem:

A quantum group is given by an algebra of functions satisfying a set of axioms such that:

- whenever is noncommutative, is a
virtualobject,every commutative algebra of functions satisfying is an algebra of functions on aset-of-pointsgroup, andwhenever commutative algebras of functions , asset-of-pointsgroups.

A *set-of-points *group is a just a group, and as we said above, no matter what class of complex valued functions we take, is going to be commutative when the multiplication is pointwise.

A *virtual *object in contrast does not exist as a set, but is given merely via its -satisfying algebra of functions . It can not be a set of points, else the multiplication would be commutative (if the multiplication to be considered pointwise multiplication).

Recall for commutative , and thus , so . Not all algebras of functions on quantum groups have antipodes defined on the entire algebra. For example, the antipode on (an algebra of functions on a) *locally compact quantum groups *is only densely defined. This is an issue of topology, but even *algebraic compact quantum groups, *which do not *a priori* involve topology, have algebras of functions that do not necessarily have an involutive antipode:

.

If a quantum group has an involutive antipode, let us assume on the whole algebra for the purposes of this post, it is called *Kac*, otherwise it is *non-Kac*.

As it provides a class with an example of a non-Kac quantum group, we can talk about the algebra of *regular* functions, , on an algebraic compact quantum group . The axioms are, more or less:

- The algebra is a unital *-algebra,
- and there is a comultiplication, antipode and counit, such that
- is a *-homomorphism, and the structure maps satisfy:
- , and
- there exists an invariant Haar state , that is equal to integration against the Haar measure, , , whenever is a set-of-points compact group

Here the group axioms are written in the language of the category ** **(structure maps are morphisms, relations are commutative diagrams), and is the functor composition of the free functor before the dual endofunctor.

As an example of choices and considerations, it would be considered a defect of the theory that a Haar state is assumed.

Now, for each there is a famous algebraic compact quantum group called , and whenever , the quantum group is non-Kac. For , is a virtual object, and intuition is fairly thin on the ground as there is no set of points group.

Before we start we must collect some facts about antipodes. In the finite commutative case, the Haar state is just:

,

averaging the function. The commutative compact (think “almost finite”) case is also averaging the function: we will think of the Haar state on a general as averaging the function, and write for the Haar state.

For the algebra of regular functions, , on an algebraic compact quantum group , the antipode is a unital *-antimorphism. The Haar state is tracial exactly when .

A *finite quantum group *is given by an algebra of functions that has the same axioms as algebraic compact quantum groups except if the algebra is assumed to be a -algebra, the existence of the Haar state can be derived rather than assumed. Choices and considerations!

As an example, consider the group ring with:

The “pointwise” multiplication is , and is noncommutative as soon as is nonabelian (if is abelian say hello to Pontryagin). We can check that is the algebra of functions on a finite quantum group, which we denote .

I have learnt that a lot of weird stuff that happens for quantum groups already happens for dual groups (see Perhaps the most surprising…, third paragraph, p.4). Now what doesn’t happen for dual groups, either in the finite case or indeed the various infinite cases, is that they become non-Kac.

However, the purpose of this post is to try and explore this question of intuition, for non-involutive antipodes, by trying to say what it would mean if or were non-Kac. Of course this is nonsense because both these quantum groups ARE Kac… but we’ve been loose enough up to this point let us keep going.

Let be a finite group. Note that:

.

If , then this implies that there exists a symmetry such that .

It implies that:

while still there exists and such that

.

Can we make *any *sense of this? Not on the levels of set-of-points groups… but let us imagine. I end up with the following. The quantum symmetries in a non-Kac quantum group force upon quantum spaces a path dependence, and thus a direction of time. We must start somewhere, say the quantum space has initial configuration . Act on this space now with a quantum symmetry , and then . This cannot bring us precisely back to , because if it did we could act again with and then we would have … instead we must go to a state that can be identified with , but it is not exactly the same as — it has a different path history.

Non-Kac also implies that if we act on with , we do not get a state that can be identified with . But we still want inverses and therefore has an inverse such that but .

Does this has anything at all to do with “God given time?”. Is it a coincidence that we are (possibly) led to this kind of path history as soon as we leave the Kac condition, a condition that guarantees that a certain modular automorphism group is trivial?

Saying that the Haar state is tracial is saying that all functions commute on average:

,

however, even with , I cannot find anything even nonsensical to say from here.

We can however see a link between these for dual groups.

Consider . The Haar state on is given by:

.

Note that we have:

.

If were tracial,

and we might conclude that and so would be Kac. On the other hand, if it were not tracial we might find a witness to .

There are no conclusions here. What we *can *say is as follows:

There are non-Kac algebras that satisfy axioms such that if a commutative algebra satisfies , it is the algebra of complex functions on a set of points group . Woronowicz[Theorem 1.4]shows that the non-Kac algebras have a non-trivial modular automorphism group.

It must go back to choices and considerations. Should a non-Kac algebra be considered an algebra of functions on a quantum group? What do you think?

An expert could tell us about choices and considerations. I don’t think it is possible to understand on a intuitive level what non-Kac says about the symmetries.

A sting in the tail is as follows. The self-adjoint elements of a can be considered quantum mechanical physical observables. Perhaps for all observables ,

?

This might somehow imply that symmetries such that cannot be detected by observables. This is not the case. There is an element such that, where :

.

Indeed for just when is *-linear… but this happens exactly when is Kac. Ouch!

]]>

“Any and all work, submitted at any time, will receive feedback.”

“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.”

60 minute, 25% Further Integration Test, 19:30 Tuesday 12 May 2020

This test will examine Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 4 from that exam. The question comprises 25% of that exam, and that translates to 30 minutes of exam time. I will extend this to 45 minutes and allow 15 minutes to upload.

Here is video based on Q. 4, on p.226 of your manual.

The material for this test was covered in Week 10, Easter Week 1, and Easter Week 2 (lectures and exercises).

If you have any further questions on these topics, I am answering emails seven days a week. Email before midnight tonight to be guaranteed a response Monday 11 May. I cannot guarantee that I answer emails sent on Monday (although of course I will try).

60 minute, 10% Vectors Test, 19:30 Tuesday 19 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 25% of that exam, and that translates to 30 minutes of exam time. I will extend this to 45 minutes and allow 15 minutes to upload.

Here is a video based on Q. 1, on p.222 of your manual.

Chapter 1 Exercises may be found on:

- p.29
- p.39
- p.46

You can submit work for feedback by midnight Saturday 16 May to *Vectors Exercises *on Canvas. After this, email before midnight Monday 18 May to be guaranteed a response Tuesday 19 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

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“Any and all work, submitted at any time, will receive feedback.”

“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.”

90 minute, 40% Test, 11:00 Monday 11 May 2020

This test will examine Chapter 2, Section 3.6, and Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 2, 3 (d), and 4 from that exam. The question comprises 40% of that exam, and that translates to 48 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

Here is tutorial video for this test.

Section 3.6 and Chapter 4 material can be found in Week 10, Easter Week 1, and Easter Week 2 (lectures and tutorial).

Chapter 2 Exercises that you should be looking at include:

- p.86, Q. 1-4
- p.91, Q. 1-7
- p.86, Q. 5-6 (harder)

I have developed the following for you to practise your differentiation which you need for Chapter 2 here:

If you have any further questions on these topics, I am answering emails seven days a week. Email before midnight tonight to be guaranteed a response Monday 11 May. I cannot guarantee that I answer emails sent on Monday (although of course I will try).

90 minute, 20% Linear Systems Test, 11:00 Thursday 21 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 35% of that exam, and that translates to 42 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

Here is video based on Q. 1 from the Summer 2019 paper on the back of your manual.

Chapter 1 Exercises may be found on:

- p.28
- p.38
- p.44
- p.51

You can submit work for feedback by midnight Monday 18 May to *Linear Systems Revision Exercises *on Canvas. After Monday, email before midnight Wednesday 20 May to be guaranteed a response Thursday 21 May. I cannot guarantee that I answer emails sent on Thursday (although of course I will try).

“Any and all work, submitted at any time, will receive feedback.”

“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.”

The final assessment, based on Lab 8, takes place 11:00, Tuesday 12 May.

If you have not yet done so, you will undertake the learning described in Week 10. Perhaps you should also look at the theory exercises described in Week 10.

The final assessment will ask you to do the following:

Consider:

(1)

For one (i.e. or or ) or all (i.e. general ), use equations (2) and (3) to write (1) in the form:

i.e. find the temperature at node at time in terms of the temperatures at the previous time . You may take .

so if you want to send on this work for feedback please do so.

If you submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Submit work — VBA or Theory, catch-up or revision — based on Week 10 to the *Lab 8 VBA/Theory Catch-up/Revision II *assignment on Canvas by midnight 9 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 11 May to be guaranteed a response Tuesday 12 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

*Keep in mind at all times:*

“Any and all work, submitted at any time, will receive feedback.”

60 minute, 25% Further Integration Test, 19:30 Tuesday 12 May 2020

This test will examine Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 4 from that exam. The question comprises 25% of that exam, and that translates to 30 minutes of exam time. I will extend this to 45 minutes and allow 15 minutes to upload.

If you need to catch up on Week 10, Easter Week 1, and Easter Week 2 (lectures and exercises) do so ASAP.

Any exercises you do can be submitted to Integration Exercises II on Canvas by Saturday 9 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 11 May to be guaranteed a response Tuesday 12 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

Here is video based on Q. 4, on p.226 of your manual.

If possible, submit 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.

Further exercises may be found under the CIT Exam Papers portal, under MATH6040, under MATH.

The 25% Integration Assessment will take place Tuesday evening 12 May.

After this you will be invited to do revision on Vectors. Here is a video based on Q. 1, on p.222 of your manual.

As outlined in an announcement:

60 minute, 10% Vectors Test, 19:30 Tuesday 19 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 25% of that exam, and that translates to 30 minutes of exam time. I will extend this to 45 minutes and allow 15 minutes to upload.

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*K**eep in mind at all times:*

“Any and all work, submitted at any time, will receive feedback.”

ASAP you need to catch up on Week 10, Easter Week 1, and Easter Week 2 (lectures and tutorial) for:

90 minute, 40% Test, 11:00 Monday 11 May 2020

This test will examine Chapter 2, Section 3.6, and Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 2, 3 (d), and 4 from that exam. The question comprises 40% of that exam, and that translates to 48 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

Here is tutorial video for this test.

Chapter 2 Exercises that you should be looking at include:

- p.86, Q. 1-4
- p.91, Q. 1-7
- p.86, Q. 5-6 (harder)

I have developed the following for you to practise your differentiation which you need for Chapter 2 here:

Any exercises you do can be submitted to Week 13 Exercises by midnight Friday 8 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Sunday 10 May to be guaranteed a response Monday 11 May. I cannot guarantee that I answer emails sent on Monday (although of course I will try).

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.

The 40% Test on Chapters 2, 4, and Section 3.6 will take place Monday 11 May.

In the form of the Test 1 trust pledge, instructions, and tables, practical information for Test 1 may be found here.

After this you will be invited to do revision on Linear Systems. Here is video based on Q. 1 from the Summer 2019 paper on the back of your manual.

The 20% Linear Systems Test will take place Thursday 21 May.

90 minute, 20% Linear Systems Test, 11:00 Thursday 21 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 35% of that exam, and that translates to 42 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

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They’re neither slick, perfect, nor as good as I would like them to be, but I am prepared to give some time every week to answering HL LC Maths student questions.

The videos are labelled in the descriptions, so if you are looking for, say, Q. 5 you can flick through the videos until you find the question you are looking for (e.g. Q. 5 starts at 17.05 here).

All students are looking for help: but perhaps the student I am best placed to help is a student (eventually) going for a H1 who needs something to be explained in more depth, or to give the thought process behind attacking a more challenging problem.

Students can ask me questions on whatever platform they want and I will try and address them. If I get no questions I will just tip away at these exam papers.

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