M338 TMA04

So I managed to finish the final TMA for M338 and will post it off tomorrow. I have to say conceptually this unit and block have been the hardest course I have ever done. However things improved from my initial wtf about block C at the start of the month to the stage where I'm begining to see how some of the definitions hang together and what they can be used for.

So an outline of the TMA

Question 1 was a question about whether or not subsets of a set with a given topology were disconnected or not. This involved the union of Z (the set of all integers) with N the set of all integers > 1. Also whether or not the topology is path connected. I think I managed to answer most of this question

Question 2 was concerned with the notion of compactness essentially an extension of the concept of finiteness to topological spaces. A set is compact if for each cover of a set X (ie a collection of sets wich contains X) a finite subcover can be found. It's a generalisation of the idea that whilst a closed interval of the real line [a,b] can be covered by a set of finite intervals an open interval (a,b) cannot be, as there will always be a gap between a and any finite interval of the form a-1/n as a is not in the interval. Despite a slightly counterintuitive part I think I got the first part correct. The second part of question was quite straightforward essentially sketching two sets and showing that that their union was closed and bounded.

The first part of Question 3 Involved Cauchy sequences and it was good to get back to something a bit more concrete. A Cauchy sequence of functions is one for which given an epsilon > 0 and integers n,m > N the distance function between a sequence of the form an is such that d(an,am) < epsilon whenever n,m > N
This was relatively straightforward. The second part of the question involved showing that a union of sets was closed and compact. This involved a lot of epsilon delta and upper bounds of sequence type arguments it felt like the more trickier parts of the analysis parts of M208 but I think I got there in the end

Question 4 involved the contraction mappting theorem and its use to find the size of an interval containing the zero of f. There was a little bit of calculus here but not enough for my tastes. The contraction mapping theorem has an application to the proof of the existence of solutions to a given type of differential equations but alas this part is not assessed.

Question 5 involved similiarity transformations for Fractals and is related to the contraction mapping theorem. As there were similar examples in the exercise book to the question, it wasn't to difficult to solve the problem. In fact question 5 is probably the easiest question of the whole course.

So all in all, I think I've done reasonably well I may not have phrased my answers quite correctly and I certainly cant claime to understand all the chain of definitions and proofs that this intricate subject requires but I feel reasonably confident about the TMA and the ability to answer the exam questions with a bit of practice. But again jsut as with M337 and M208 I can only feel I've scratched the surface of this subject and would appreciate the time to revisit it again.

The past few weeks have been quite intensive as I've been to quite a few concerts at the Edinburgh Festival. I'll give a review of the concerts in the next week or two. But there is no rest for the wicked as I have to do the TMA for MST324 withing the next week. I'm still plugging away at my scales but due to the festival and finishing the TMA for M338 not as much as I could be.

  

M338 Block C

Ok momentary panic over. I've decided after  the tutorial to stick with the Topology. I get the impression that we are all struggling with it. Even our tutor Alan says he's not sure about some bits of it. Anyway the tutorial clarified some issues or at least showed how to go about tackling questions. My mate Neil showed a brilliant insight into one of the questions involving whether there was a path in Z between all elements of Z for  a topology which consisted of {0,z} U (Subsets of even numbers). Our tutor had showed that there was indeed a path between odd and even numbers and was about to go through the same tortuous reasoning to show a path between the even numbers and a path between the Odd numbers. When Neil interjected and said that if there is a path between an odd and an even number then there is also a 2 step path to an even number or odd number if P(a,b) is a path from a to b where a is odd and b is even there is also another path
P(b,c) from b an even number to c an odd number so there is also a path from a to c which is the sum
P(a,c) = P(a,b) + P(b,c) and as a and c are both even or odd then this shows that there is a path for all elements of Z. Anyway it's battling on with the TMA this week. I feel a bit more confident and hope to finish by the end of next week. One of the problems as Alan admitted is that the course material gives very few examples to illustrate the definitions and there does seem to be little or no motivation for them. I will probably do M303 the new pure maths course when it comes on the scene even though it is an excluded combination as it will help me consolidate my meagre understanding of the course material for M338, also as I wont be doing the group theory course M336 then its a chance to cover that also M303 contains an introduction to groups rings and fields. So I will put this into my third (!) open degree. I'll be concentrating on Music for the next 4-5 years so don't think I want to embark on postgraduate work for either maths or philosophy. I want to do at least the new philosophy 3rd level course and there is going to be a new third level music course appearing on the scene as well. I might also include the two third level astrophysics courses and the second level Chemistry course we'll see.

The Festival starts for me on Wednesday and I'm going to quite a few of the concerts in the Usher Hall I will also be taking a few days off. I'll give reviews of the concerts I'll be going to over the next few weeks. The itinery for this week is

Wednesday   15th  Tristan and Isolde
Thursday       16th Syzmanowski Symphony No 1 and Brahms No 1 this includes a performance with Nicola Benedetti
Friday           17th Syzmanowski Symphony No 2 and Brahms No 2
Saturday       18th Syzmanowski  Symphony No 3 and Brahms No 3
Sunday         19th Syzmanowski  Symphony No 4 and Brahms No 4

I must confess I don't really know the Syzmanowski Symphonies and so I'm looking forward to getting to know these pieces.

Anyway I may bump into some of you who read this blog there. I'll usually be in the Traverse theatre bar after the concerts.

The piano practice is coming on I'm doing at least 1/2 hour per day before I get to work, This week I want to consolidate what I have done so far namely scales and broken chords for C major, D major and G major. The left hand and right hand parts for two of the Grade 1 pieces separately and a rather pathetic attempt to put them together and Chapters 1 - 4 of Fanny Waterman's book volume 1.

Next post I'll tell you how the Key system of Western music and the circle of 5ths can be reduced to modulo arithmetic.

Do I or don't I

Well just like other people who indulge in  this OU maths blogging lark I'm in a bit of a dilemma vis a vis whether or not I should continue with a course. The course in question is M338. So far I've done hardly anything for the last TMA and I really can't motivate myself to continue wading through definition after definition wondering what the point of it all is. On the other hand I've come so far 3 TMA's all with a respectable grade two passes and it's only 1 more TMA, the last part of which looks relatively straightforward and I guess the exam wont ask anything too challenging  It's just getting a feel for the definitions of compactness and connectedness which I'm really stuck on. So it would seem on the face of it stupid to give up now. A grade 2 pass should be within my grasp. I have a tutorial next week and so I'll leave it till then.

Life would so much simpler if I jacked it in but it would also be an admission of defeat.

On a brighter note I seem to be slowly getting through the keyboard/piano practice and can do (single handedly) a reasonable attempt at the scales of C major, G major and D major. Also my broken chords seem to be coming together. The left hand is a bit more tricky especially for scales ascending. It's not just a question of playing the correct notes but also making sure the fingering is correct. I'm slowly beginning to develop an intuition when something is not quite right for example in the scale of G major which has one sharp F# when using the left hand the second finger must always hit the black key. If it doesn't then you know you have your fingering wrong and it can be quite frustrating at times. I'm manging to put in about 1/2 an hour before I go to work and about an hour when I get back something seems to have been unleashed here. If I carry on this rate with suitable guidance I would hope to put myself forward for the grade 1 exams in March of next year and ideally Grade 2 in the summer and Grade 3 in winter certainly 1 want Grade 1 and 2 by the end of 2013.  I will also do the Grade 5 theory exam (a prerequisite for going beyond grade 5 practical) in 2013 probably after I've completed my  OU music course which covers grade 5 and a lot more. 

I've decided to only do 1 OU maths course per year from October 2012 and due to the timing of the presentations it is going to have to be Fluids and mathematical methods for 2012 and Number theory and Logic for 2013, which has been given another years leave of absence before the doors finally close.  I really wish the open university could be a bit more clearer as to when the last presentation of a course is going to be. I felt bounced into registering for M381 as it was going to be it's last presentation but then all of a sudden they change their minds.

I wonder if certain friends (I'm talking about you Duncan) might be tempted to do number theory  in 2013 as well. Anyway Fluids and mathematical methods along with music and piano practice should be more than enough to keep me busy for 2013.

Finally for those who are interested I got my third TMA for MST324 in the mid sixties as expected. If I get a distinction for the final one and  do well in the exam I still qualify for at least grade 2 and maybe even distinction. That incidently is another reason for me to contemplate abandoning topology as I would only have one exam to concentrate on.