I have been bumbling along for the last two and a half weeks making slow progress with book AB2 on differentiation. I have got past the hard part of trying to understand the proof that the Blancmange function is nowhere differentiable. I did actually get stuck and resorted to asking for help from my tutor, Alan, but by the next morning (having slept on it) I realised it was all staring me in the face. Since then I have been learning about Rolle's Theorem and the Mean Value Theorem which I have come across before (but not in so much detail). It seems that as I inch closer and closer to the end of this course, so I seem to take longer and longer to finish a book.
I had a tutorial last Saturday and Alan managed to cover the whole of the first group theory block in an hour and a half (half an hour to say what was covered and an hour to go through some old TMA questions). Impressive! I keep thinking I will be able to sit back and relax and enjoy the ride, but I keep finding that there are bits that I think I don't know. I find myself thinking "how the hell do you do that again?" and a surge of panic rushes through me. It keeps me on my toes.
I am still making progress with the TMA's. TMA04 is done and handed over and I have done the first question of TMA05 which I quite enjoyed. I love the challenge of an interesting question.
I am disappointed that the OU is removing the Diploma qualification that I am working towards. Usually I would have had quite a few years to complete this. Now that I have to start this in a year or so, it is almost certainly going to put the nail in the coffin for my future study with the OU. I saw the diploma as a stepping stone, something to achieve and an encouragement for me to continue with my studies. It would have been a useful qualification to get, but now that I have to pass it by the end of 2014, I can't see myself having the energy to do the other course I need, MST209. M208 has nearly finished me off as it is. The dividing line between enjoying the course and finding it a burden has worn very thin and now I am being pressured into the next step it is pushing me into the 'no' mode. There are, after all, plenty of other avenues for my interests in maths which do not have strict timescales and exams and I suspect that when this course finishes in October, I will be off to explore these instead. It is the OU's loss!
One of the avenues I would like to get back to is my study of prime numbers. Before I started work at the OU I had done what I think now is some good work at understanding how to compute pi(n) - the number of primes less than n. One of the first things I think I will do is to try and explain what I did in a blog like this. It was all very interesting. Nothing new to anyone else I am sure, but it was an interesting exercise for someone who knows little about the subject and who enjoyed working it out for himself!
Thursday 29 March 2012
Monday 26 March 2012
Interesting stuff
I have learnt a couple of interesting things in the last few days. The first is an answer to my question "Can we argue somehow that the Blancmange function is continuous without using the epsilon-delta definition of continuity" (see my previous post). The answer is yes and I am very pleased that my hunch was correct!
One of the tutors on the OU forum (Steve Meyer) has kindly answered this question for me. Basically, as B(x) is constructed from an infinite sum of continuous functions of the form (1/2n)s(2nx) and |(1/2n)s(2nx)|≤1/2n+1 then, as the infinite sum of 1/2n+1converges, the M-test shows that B is continuous without using epsilon-delta. There is a paper by John Kennedy which describes these ideas.
The second thing that I have learnt is quite amusing. I was trying to answer a question about the symmetries of a 3d object and got very stuck. Most of the symmetries were obvious but there were a few that weren't. I remembered how to algebraically find the difficult ones but I needed to be able to describe them geometrically and I couldn't. I made a model out of cardboard to help me and asked my wife and her brother if they could see what the difficult symmetries were. After an evening of trying we had to admit defeat and give up. I later found out that these difficult symmetries do not have a simple geometric description (such as a reflection in a plane). It was just as well I didn't spend any more time banging my head against a brick wall! Lesson learnt.
One of the tutors on the OU forum (Steve Meyer) has kindly answered this question for me. Basically, as B(x) is constructed from an infinite sum of continuous functions of the form (1/2n)s(2nx) and |(1/2n)s(2nx)|≤1/2n+1 then, as the infinite sum of 1/2n+1converges, the M-test shows that B is continuous without using epsilon-delta. There is a paper by John Kennedy which describes these ideas.
The second thing that I have learnt is quite amusing. I was trying to answer a question about the symmetries of a 3d object and got very stuck. Most of the symmetries were obvious but there were a few that weren't. I remembered how to algebraically find the difficult ones but I needed to be able to describe them geometrically and I couldn't. I made a model out of cardboard to help me and asked my wife and her brother if they could see what the difficult symmetries were. After an evening of trying we had to admit defeat and give up. I later found out that these difficult symmetries do not have a simple geometric description (such as a reflection in a plane). It was just as well I didn't spend any more time banging my head against a brick wall! Lesson learnt.
Saturday 17 March 2012
Bother!
In the last few days I have learned that the OU is doing away with one of the qualifications I was working towards. This is the Diploma in Mathematics D23 and consists of the two second year pure and applied courses M208 and MST209. I am doing M208 this year and I thought I might do MST209 at some point, but probably later rather than sooner.
Well now I will have to pass MST209 by 31st December 2014 which means I will have to start it by October 2013 at the latest. I know this is year and a half away but this is still too soon for my liking. I intended to take a big break and then contemplate if I really have the motivation to complete the second course. Now I have no choice if I want the diploma. Botheration!!
More on this later.
Well now I will have to pass MST209 by 31st December 2014 which means I will have to start it by October 2013 at the latest. I know this is year and a half away but this is still too soon for my liking. I intended to take a big break and then contemplate if I really have the motivation to complete the second course. Now I have no choice if I want the diploma. Botheration!!
Sunday 11 March 2012
All downhill from now on?
I have just completed book AB1 from the final analysis block and I am 3/4 the way through TMA04. I feel that the clouds are beginning to lift and some way off, but now visible, is the finishing line! Perhaps this is an odd thing to say when the course is just 6 weeks old but after going through a difficult patch after working on M208 for nearly a year, it is a nice feeling when you begin to feel that things may be downhill from now on.
I didn't find the last part of GTB3 on the counting theorem too bad. The video that goes with the course was a big help and the examples that they discussed were interesting.
Well AB1 - yuk. The first couple of sections are ok on limits and asymptotic behaviour of functions, but then came the big elephant in the room, the epsilon-delta definition of continuity and the continuity of strange functions. I can't say that I really understand the epsilon-delta definition of continuity at all well yet. I managed the exercises and worked through the proofs, but I think a lot more practise is going to be required before I can master this stuff.
The strange functions are just a p in the a. I did wonder why it was necessary to prove that the Blancmange function B(x) was continuous when it is constructed from a sum of continuous functions. They showed that B(x) is convergent for each x in R by the Comparison Test, so why isn't this and the Sum Rule for continuous functions enough to say that B(x) is continuous? I don't know.
This may also sound odd but light relief has been to work through TMA04 on the first analysis block. Read a bit of AB1, feel totally baffled, and then for light entertainment find out if a sequence or series in TMA04 converges. Bliss in comparison.
It seems that I made the right decision when it comes to the TMAs. I decided not to agonise too much over my mathematical language and to trust my own instincts. This seems to have been justified as I have had the remainder of TM01 back from my tutor and he seems happy enough with what I have done. Although I argued successfully enough to score 100% on this TMA, there were one or two places where I didn't quite get to grips with the argument and overcomplicated things. I think this accurately reflects the topics that I will need to work on in revision. At least I didn't waste lots of time wondering whether I should be using the word 'hence' rather than 'it follows' or whether an implication should be an equivalence! Trust the gut!
I didn't find the last part of GTB3 on the counting theorem too bad. The video that goes with the course was a big help and the examples that they discussed were interesting.
Well AB1 - yuk. The first couple of sections are ok on limits and asymptotic behaviour of functions, but then came the big elephant in the room, the epsilon-delta definition of continuity and the continuity of strange functions. I can't say that I really understand the epsilon-delta definition of continuity at all well yet. I managed the exercises and worked through the proofs, but I think a lot more practise is going to be required before I can master this stuff.
The strange functions are just a p in the a. I did wonder why it was necessary to prove that the Blancmange function B(x) was continuous when it is constructed from a sum of continuous functions. They showed that B(x) is convergent for each x in R by the Comparison Test, so why isn't this and the Sum Rule for continuous functions enough to say that B(x) is continuous? I don't know.
This may also sound odd but light relief has been to work through TMA04 on the first analysis block. Read a bit of AB1, feel totally baffled, and then for light entertainment find out if a sequence or series in TMA04 converges. Bliss in comparison.
It seems that I made the right decision when it comes to the TMAs. I decided not to agonise too much over my mathematical language and to trust my own instincts. This seems to have been justified as I have had the remainder of TM01 back from my tutor and he seems happy enough with what I have done. Although I argued successfully enough to score 100% on this TMA, there were one or two places where I didn't quite get to grips with the argument and overcomplicated things. I think this accurately reflects the topics that I will need to work on in revision. At least I didn't waste lots of time wondering whether I should be using the word 'hence' rather than 'it follows' or whether an implication should be an equivalence! Trust the gut!
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