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.
I look forward to enlightenment tomorrow !!
ReplyDeleteStill think the epsilon delta definiton is worth getting to grips with.
As for three d symmetries I nearly did end up banging my head against a brick wall last year. I was saved by reminding myself of the systematic method that chemists use to deduce the symmetries of complicated molecules.
Also a big clue if you haven't already worked it out is that the number of direct symmetries must equal the number of indirect symmetries.
I think you will get bored after you've finished the M208 TMA's. So I suggest you look at some MST209 papers which I'll bring along tomorrow.
PS I'm still not a robot
I will bring some stuff along to look at tomorrow. This has been an interesting line of inquiry. I don't have a grasp of it all at the moment but it may be something to chase up once I have finished everything else.
DeleteYes, I dread to think what it is like when it gets more complicated. I can begin to see the importance of Group Theory. Yes, thanks for the hint, I had worked that one out.
Can't see myself getting bored. I am looking forward to getting back to my old activities of playing around with prime numbers, doing Cambridge problems, writing code, and reading all those juicy looking books in the library, and oh, having a play with MST209 as well, if I really feel that I can make the effort.
See you tomorrow.