I have now been working through GTB1 of M208 for the last three weeks and I still haven't finished it. I am beginning to wonder if the last two blocks of this course are tougher than the rest. I know that Christmas has slowed things down a bit, but usually I can complete a book in three weeks. One question was a corker. In Exercise 4.7 on p44, you are asked to find all the normal symmetry subgroups of regular hexagon. It sounds innocuous enough, but I ended up covering four sides of A4 with my answer.
GTB1 starts with some much needed revision of Group Theory. It is amazing what you forget when you have head full of analysis. Still, there are some things that are beginning to stick in my leaky brain.
The rest of the chapter starts to delve more deeply into conjugacy and normal subgroups. One essential idea is that conjugacy in symmetry groups represents symmetries that have a similar type. For example, when considering the square, the two reflectional symmetries which are associated with the lines of symmetry that pass through the corners of the square are of the same geometric type and are related by conjugacy. This is built up into the Fixed Point Theorem later in the chapter.
Another major section of the chapter is an exploration of the relationship between conjugacy and normal subgroups and this leads to four properties of subgroups that characterise normality. In the final section there is a look at infinite groups of 2x2 matrices.
I must say that it is tricky stuff. I find it hard to get all these abstract ideas ordered someway in my head so that they can be remembered. There is layer upon layer of ideas and whilst it is ok answering questions when you have just completed a topic, I can imagine that in an exam, it may be hard to come up with the right techniques for answering a random question.
Still, I am very glad that I have done so much advance work on this course as I will be able to spend time mastering the methods when the course actually starts. I am very much looking forward to my first assessment.
Tuesday 27 December 2011
Monday 12 December 2011
End of Analysis Block A
I have now completed the last of the books in the first analysis block of M208. This book was all about continuity in real functions, and as for the other analysis books, I really enjoyed working on it. In a nutshell, to determine whether a function f(x) is continuous at x=a, you have to ensure that, as x tends to a, f(x) tends to f(a). Of course, you need to consider the approach to x=a from both smaller and larger values of x.
I am already familiar with the idea of continuity having come across it in previous maths studies but this course deals with the subject in a much more rigorous way, using sequences in x to determine how f(x) behaves as x tends to a. One gripe that I have is that a major plank of the rest of the book, the Intermediate Value Theorem, is only proved for a special case and not proved in general. When a lot of M208 is devoted to carefully proving theorems it is sometimes surprising to find omissions, but I suppose that in some cases the proofs are too involved to reproduce for the course. The Intermediate Value Theorem is the basis of how we can define inverse functions for increasing or decreasing continuous functions.
Another gripe that I have is that it is sometimes difficult to judge the level of detail that is required in some of the answers to the exercises. Sometimes I find myself putting in too much detail and at others too little. Take Ex. 4.1 on p38 of AA4, for example. The solution says that f(n)=n2-1/n tends to infinity as n tends to infinity by the Reciprocal Rule. Is it really necessary to quote the Reciprocal Rule at this point? Isn't this obvious enough? In the examples in the text they manage fine without quoting this rule, so why suddenly do so here? There has to be some point at which you don't have to quote every theorem and rule that you have learned in M208 in order to prove something otherwise it is going to be extremely tedious/arduous. In fact, the solutions to the exercises are meant to be a guide as to how questions should be answered, but some rules, strategies and theorems are omitted in these solutions when convenient. So how are we to judge this?
I am already familiar with the idea of continuity having come across it in previous maths studies but this course deals with the subject in a much more rigorous way, using sequences in x to determine how f(x) behaves as x tends to a. One gripe that I have is that a major plank of the rest of the book, the Intermediate Value Theorem, is only proved for a special case and not proved in general. When a lot of M208 is devoted to carefully proving theorems it is sometimes surprising to find omissions, but I suppose that in some cases the proofs are too involved to reproduce for the course. The Intermediate Value Theorem is the basis of how we can define inverse functions for increasing or decreasing continuous functions.
Another gripe that I have is that it is sometimes difficult to judge the level of detail that is required in some of the answers to the exercises. Sometimes I find myself putting in too much detail and at others too little. Take Ex. 4.1 on p38 of AA4, for example. The solution says that f(n)=n2-1/n tends to infinity as n tends to infinity by the Reciprocal Rule. Is it really necessary to quote the Reciprocal Rule at this point? Isn't this obvious enough? In the examples in the text they manage fine without quoting this rule, so why suddenly do so here? There has to be some point at which you don't have to quote every theorem and rule that you have learned in M208 in order to prove something otherwise it is going to be extremely tedious/arduous. In fact, the solutions to the exercises are meant to be a guide as to how questions should be answered, but some rules, strategies and theorems are omitted in these solutions when convenient. So how are we to judge this?
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