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.