M338 TMA02 Back

So got this back reasonably pleased top end of grade 2. Made a silly mistake in the last question and as predicted didn't do all that well in question 2. My tutor says I should be pleased as it was a hard TMA anyway going to take a week off from OU maths till Monday of next week hopefully giving it a bit of a rest will help my subconscious digest it all.

MST324 TMA02 (Nearly finished my pudding)

Well after finally eating my meat for M338 I've nearly finished my pudding for MST324. What a joy (says he optimistically) The questions were on the wave equation and Fourier transforms.

Question 1 Involved using D'Alembert's solution to the wave equation to solve a particular boundary problem.

Question 2 (The best)  involved solving the inhomogeneous damped wave equation for a particular function using seperation of variables. This is a showcase question consolidating all the techniques we have covered so far. The steps are

1) Write the function u(x,t)  as a product  of two functions f(x)g(t)
2) Substitute these into the partial differential equation and this gives two separate ordinary differential equations in both f(x) and g(t) in terms of a constant .
3) Solve the homeogeneous  differential equations for f(x) and g(t) using the appropriate boundary conditons
4) Use the inhomogeneous function on the RHS of the partial differential equation to obtain particular integrals for f(x) and g(t).
5) The resulting solution is then u(x,t) = f(x)g(t)

This is maths at it's best, some of the steps might be a bit tricky but I think it's really cool how it all hangs together.

Question 3 Another question on a solution of a partial differential equation by separation of variables. This was another nice question and a bit simpler than question 2

Question 4 Testing your knowledge of the properties of Fourier transforms based around a question involving the recurrence relationships of Hermite Polynomials. This was OK but a bit tricky in parts and I still have to do the last bit.

So overall I'm quite pleased and this TMA has been a joy to do, there was none of the head scratching associated with M338, but I can see the last bit of ice cream in the tub and it will soon be back to struggling with the many definitions of continuity for Topological spaces and so forth. Anyway it looks like I'm not the only one to have struggled with the TMA for M338. According to my friend Duncan who is doing M208 this year, Alan my tutor for M338 has said that it is taking him 2 hours per question per person to mark the scripts. So if the tutors find it difficult it's not surprising that we do.

M338 TMA2 away

Well for better or worse (probably worse) TMA02 is away. I have no real confidence that I understand what is going on. However for interest here is a summary of the questions and my response. Part 1 had to calculate d(x,y) a so cakked distance function  for various x y

Question 1 a distance function  is defined and we have to show that it is a metric ie we have to show

M1 d(x,y) >= 0 with equality only if x = y

M2 d(x,y) = d(y,x) ie d(x,y) is symmetric

M3  Finally show that d(x,z) >= d(x,y) + d(y,x) (ie, the triangle inquality)

The metric had 2 forms depending on whether or not given two points in R^2 (x1,y1) (x2,y2) y2 = x2 or not

First part just asked  you to show  that you can apply the definition correctly think I did OK on this.

Second part show that the metric satisfies M1 and M2 of the definition of a metric

M1 is simply that d(x,y) >= to 0 with equality only if x = y. A bear trap here is to neglect to show that if

d(x,y) = 0 x = y think I managed to negotiate this sucessfully. Again this seemed relativiely sttaightforward

(In what follows e means element of )

Then one had to show that given a,b e R^2,  that |a1-b1|+ |a2-b2| < d(a,b) again this seemed straightforward

But this was supposed to be a hint to be used in proving the triangle inequality ie to prove that d satisfied M3

I seemed to prove that M3 was satisified directly without using the hint so there is a nagging doubt that I missed some cases.

Then one had to sketch 2 open balls of d for various d(x,y) an open ball is such that d(a,r) < r for a e R

where a is the centre of the open ball, This is a generalisation of the definition of an open interval

This was relatively straightforward although conterintuitive for the second ball. I came to the conclusion that there was no open Ball centered on (1,1) with radius 1/2 which satisfied the second condition of the metric,

logic tells me I'm correct but my instinct doesn't. Antyway I couldn;t fault my logic so it stands for better or worse.

It was then desired to show that d(x,y) was not metrically equivalent to the Euclidean metricd2(x,y)  The euclidean metric is simply the disance between two points Here I totally failed. The definition of two metrics d1 and d2 being equivalent is that it is impossible to find two real numbers m and M such that

                       md1(x,y) < d2(x,y) < M d1(x,y)

for all x,y e R^2 and we were supposed to use a solution to the first part as a hint to show this. Unfortunately all I could demonstrate was that given d2 and d1 it was indeed possible to find two number m and M such that the above inequality was satisfied. I'm looking forward to Alan's solution to put me right (watch this space).

So for question 1 reckon I've got about 75% of the full marks at least I've explained my confusion

The first part of Question 2 introduced a definition of some open sets and we had to show that they formed a topology on R the first part.

A topology T on a set X is a collection of subsets of X satisfying  the following axioms

T1 the topology must include the empty set and the set  X

T2 the topology T must be such that any two intersections of the subsets of T must also be an element of T

T3 the topology T must be such that any unions of subsets of T must also be an element of T

For finite toplogies it is sufficient to demonstrate this for any two subssets of X which are elements of T. 

The elements of T are called open sets of X, their complements are closed sets of X. 

At first sight the definition of an open set of X may seem totally arbitrary but T2 and T3 guarantee a form of closure. 

 I reckon I got this correct. The second part introduced a new definition of left continuity and we had to show that a specific function was left continous. I think I got most of these parts correct,

Finally we had to show that given a function is both left continuous and monotonically increasing that the funcition was indeed continuous on R.

The background to this is that, so far the purpose of M338 has been to generalise the M208 definition of continuity to reformulate it in terms of whether or not the inverse of f(a) gives rise to an open set of X. As my mastery of the various terminology is still a bit hazy I'm not convinced I got anything resembling a perfect answer to this part of the question.

Question 3 tested whether or not you understand basic definitions such as closure, boundary, exterior interorior or the 'density' or 'non density' of certain finite sets given a topology T defined  on those sets

As there were no explicit examples (unlike M208) this involved a) understanding some basic definitions and b) how to apply them. I think I got there in the end but was a bit of a struggle to say the least. Also when defining a map from a set X to X if the function exclude a certain element a then the inverse of f(a) = 0 the empty set, But this is still part of the topology of T. So the ambiguity when testing for continuity is whether or not f(a) has an inverse mapping onto the topology, I decided that it did as 0 is still an open set of T but I can see arguments for the other point of view. So again confident that I got about 75% of this question correct.

Question 4 involved sketching certain sets finding their exterior and interior and boundary and deciding whether or not they were open or closed or neither. I think I got most of this correct.

So overall I might be pushing the boundary between grade 2 and grade 1 for this one but if I get above grade 2 that will only be due to the genoristiy or otherwise of Alan's marking. In terms of really understanding the subject I'm still only on block A2. I neeed over the next 4 weeks to get to grips with A3 and A4 before moving onto block B which admittedly looks a bit more straightforward.  So yes I'm being pushed out of my comfort zone, but as my mate Neil said I wanted a 'brain fuck' and I think I've got this in spades. Obviously my confidence has been quite shattered and as a caution for those who think this is the ultimate in pure maths, it should be noted that a course in Topology, along the lines of M338 usally forms a second year course in most brick universities. Anyway I've eaten most of my main course so for the next two weeks, I'll be having my pudding of MST324 I can't wait.

Obviously I will have to try and consolidate my extremely limited understanding of the first part of M338 as well so it will be like eating porridge with salt as opposed to sugar. Still KBO (keep buggering on ) as Churchill is alleged to have said.

Hand being Forced

Sometimes events just force your hand. As some of my followers know I had planned eventually to study composistion via the Open College of Arts.

http://www.oca-uk.com/distance-learning/music

 However as I have been getting bogged down in Topology I had intended to postpone this till October, which would have given me time to pay all fees upfront and also get Topology out of the way. However they have just announced that their fees will be increasing by 60% for students who register after June 1st 2012. However they will keep a TA arrangement for those who register before then. So it looks like I'll be registering for the course in the next month or two,  taking advantage of their installment plan. I have enough savings for their deposit, it also means less beer for  the next 6 months but that's probably a good thing.

In a way I'm quite excited, but it means something is going to have to give over the next few years, I really can't study Maths, Philosophy and Music to anything like the depth I want to simultaneously. As my maths is important, especially as I have a long term ambiton to become an OU maths tutor, and I want to do some more undergraduate courses, especially M347 mathematical statistics and M336 groups,  before I embark on the MSc.  I think philosophy will have to take a back seat,for now. I will continue to do one or two courses via Edinburgh University every year and there are also Geoffrey Klempner's Pathway modules

 http://www.philosophypathways.com/programs/pack.html

but it looks like I will have to abandon my plan of doing AA308 philosophy of the mind and wait till the replacement course comes in within the next couple of years.

So Provisional Plan is

                        Maths                       Music                      Philosophy
  Current          M338, MS324         Comp 1 (June)         Philosophy of Arts (E/U)
  Oct 2012       MST326 M381
  June 2013                                      Comp 2                    TBD
  Oct 2013      M347  M336
  June 2014                                      Composition of
                                                        Extended                  TBD
                                                        Piece (Comp 3)
  Oct 2014      Start MSc                                                  OU 3rd level course

So if all goes well I could have composed my first symphony by June 2015. Not that it will ever get performed.

I had let my music Studies slide so it's good that I'm being forced to bring them back Still need to finish TMA02 for topology before doing anything else.

Added 2 pm
Have found a blog of someone who has been through the OCA music experience
http://www.fionacoulter.com/blog/

Schubert 1797 - 1828

As a relief from all the angst about Pure maths and Topology which seems to be getting quite a few of us down. Radio 3 has decided to spend the whole of this week playing continuous Schubert.

http://www.bbc.co.uk/radio3/

 Schubert is amongst my favourite composers. Some years ago I wrote this little biography and I share it with my friends


http://dl.dropbox.com/u/16049029/Schubert_bio_cdf.pdf

For those who don't know Schubert then you don't know what you've been missing. I would recommend starting with the last 3 piano sonatas preferably played by Alfred Brendel.

http://www.amazon.co.uk/Schubert-Piano-Works-Franz/dp/B00000417C/ref=sr_1_1?ie=UTF8&qid=1332792862&sr=8-1

The last quartets and String quintet by the Lindsay quartet

http://www.amazon.co.uk/Schubert-Late-String-Quartets-Quintet/dp/B0001Z2RSW/ref=sr_1_1?s=music&ie=UTF8&qid=1332792942&sr=1-1

Finally the three great song cycles with Fischer Dieskau and Gerald Moore.

http://www.amazon.co.uk/s/ref=nb_sb_noss?url=search-alias%3Dpopular&field-keywords=Fischer+Dieskau+schubert+song+cycles+

Anyway I hope you enjoy exploring Schubert as much as I have. Take the opportunity to listen to Radio 3 this week especially the main evening concerts. I've just finished hearing a performance by Paul Lewis, one of Alfred Brendel's pupil's of the two A major sonata's.

It's quite surprising just how quickly time flies I can't believe it's almost 10 years since I wrote the short biography. Hopefully it wont be another 10 years before I write another one.

That sinking feeling

Help, I think topology must be one of the densest subjects to get into. I have to say I'm really struggling with this one as it's quite alien to anything I've come across. Still I'm not alone that does seem to be the consensus of a number of people on the forum.

I think part of the problem, is that one is dealing with general relations between functions, which aren't specified at all. In the analysis sections of M208 there was usually a function which was specified and it was possible, no matter how strange the function to say something about it. For example, the notorious Blancmange function of M208, yes proving it's continuous is a real pain in the neck (See the discussion I've been having with Duncan on this )

http://matrices-reloaded.blogspot.co.uk/2012/03/all-downhill-from-now-on.html

but at least you had something to get your teeth into. The function is defined, I more or less know the epsilon delta defintion of continuity and how to test if a function is differentiable or not. OK I might not grasp all the details of the proof but I can get the gist and I hope finally to consolidate my understanding with a few pints with Duncan next week.

With topology it's quite different, the definitions are about relations, between relations, between relations (alright I exaggerate a bit). It's very often definitiion 1, definition 2, combine definition 1 and 2 to get theorem 1. Definition 3 definition 3.1 Definition 3 and Definition 3.1 gives us theorem 2 and on and on, unfortunately there is no real road map as to where all this is going so it seems like wading through treacle.

Having said that, I think I've finally developed some sort of strategy to cope with this course. (a bit too late for the second TMA due in a couple of weeks time), but I think the only way to get a handle on this course is to try and learn the defintions by heart, at least in the first instance, testing yourself at odd moments during the day once those are grasped then you can begin to follow the theorems. This seems to have worked so far for unit A2. At first I was totally baffled, by the defintion of continuity for metric spaces, in terms of balls, (Indeed one can legitimately say topology or at least metric is a load of balls !!) A ball is essentially a generalisation of an  open interval around a point in two or higher dimensional space. However having written it out a few times and more or less learnt it off by heart, it now begins to make sense and I can move on. Hopefully I've not left it too late to catch up. Fortunately we have a tutorial on Saturday and that should give me confidence to tackle the TMA over the next week. In the mean time I'll keep plodding away.

On a brighter note, I got my first TMA back from the Waves course, grade 1, but I dropped a few marks because of silly sign errors and also I had totally missed out a small section of one of the questions. TMA02 of this course is tempting, but as the school teacher said in the Pink Floyd's the Wall, "You can;t have your pudding until you've had your meat, how can you have your pudding if you don't eat your meat? "

So I have to eat the meat of TMA02 for the topology course before I can have my pudding of the waves course.

Finally in a slightly amusing way, I've left my small mark in Edinburgh. I was out drinking with some colleagues from work in the Blue Blazar a local pub just off Lothian Road, when a new guy who's just joined went to the toilet. He came up and said "Thats what I really like about Edinburgh, where else would you see Maxwell's equations scrawled next to some graffiti slagging off Hibs supporters". What is really amusing is that I had scribbled those on the toilet door about six years ago after a rather boozy Christmas do. I was amazed that they are still there.

Hopefully the mist will begin to clear vis a vis topology and I can't wait till my pudding but I must be careful not to gobble it all down quickly otherwise I'll drop some marks for silly mistakes.

Progress

Well sorry for not posting for a while anyway progress of sorts has been made on M338 I got my first TMA back and did reasonably well but lost a few marks for taking short cuts. Also there was some difficulty in notation hopefully the wrinkles will be ironed out in time for the exam. Have to admit some of this course seems like wading  through treacle and it's difficult to see the point of a lot of it. Still just keep carrying on I know a knowledge of topology is an essential pre-requisite if one wants to understand things like the singularity theorems of General Relativity proved by Penrose and Hawking so mustn't get too disheartened.

Also finished the first assignment for MS324 which was on the whole a joy to do I'm definitely an Applied mathematician. The questions were essentially a review of mathematical techniques one should be aware of with one or two wrinkles thrown in. The first question was a question on Partial Fractions and an integration based on taking the limit of the integral to infinity. This involved some tricky manipulations of Logartihms but I think I got there. The second question was a straightforward and pleasurable solution of an inhomogeneous second order differential equation. The third question was a straightforward but tedious question on multivariable calculus and finding the stationary points. What made this question a bit tedious was the fact that the nature of the points varied for a wide range of conditions and you had to be careful not to miss out any combinations. The final question was on the deformation of springs more of a modelling question than a mathematics question I think I got most of it correct but I admit the very last part seemed badly phrased so I'm not sure if I got it correct or not. Still it's only 3 marks so should be on target for a good stab at this.

Both courses haven't really got started yet I really hope I can penetrate the fog of definition and theorem that is plaguing the topology units but at the minute I really can't see the wood for the trees. Conversely I know that the first real block of MS324 is on one of my favourite topics namely the wave equation and it's solutions I and feel reasonably confident I can polish off the next TMA in the next two weeks. The problem is it would be like a kid who eats his dessert before his main course instead of leaving it to last but what the heck.

As a final point at the risk of overloading people I've found yet another book on mathematics as applied to physics.

http://www.amazon.co.uk/Mathematics-Classical-Quantum-Physics-Dover/dp/048667164X/ref=sr_1_1?ie=UTF8&qid=1330880090&sr=8-1

Although quite old it covers all the branches of mathematics (apart from Topology) and their direct relevance to physics. It has quite an accessible introduction to the Hilbert space formulation of quantum mechanics stresses the key analogy between geometric and vector spaces which is so important to physics as well as having lots of concrete and challenging examples. I intend to try at least some of the questions which involve the separation of variables of pde's in spherical coordinate systems something not really covered in MS324 but which plays an important part in both classical and quantum physics. Anyway nice to see a book covering both classical and quantum physics in the same volume. A bit less abstract than say Szekres

http://www.amazon.co.uk/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_sc_2?s=books&ie=UTF8&qid=1330880446&sr=1-2-spell

but not just another recipe book, Szekres and Byron complement each other nicely in my honest opinion.