Tuesday, January 22, 2008

Jake said...

I had gotten myself into a confusion over the notation and even now my mind is still prickling about this. If G=, then H= is a subgroup of G. But then the question I need myself to answer is what is a^s.

I wouldn't worry about things like H normally being a subgroup of G because the context will always state what is a subgroup of what etc. and if you are introducing a subgroup yourself, you can always just use a different letter.

a^s (for an integer s) on the other hand refers to the repeated application (extended by associativity) of the group operation to a according to the following rules:

s > 1: s-1 applications (s a's)
s = 1: a
s = 0: identity element
s < 0: (a^|s|)^(-1)

With regards to the Riemann integral, remember that if f is monotonic on [a,b] then it is continuous on [a,b]
Then, assume WLOG
f(a)<= f(b)
then f(a)<= f(x) <= f(b)
so f is bounded and you can use continuity and Riemann's criteria to show that it is Riemann integrable.

The choice of questions is very personal really, I don't think per se that the complex analysis ones were neccesarily easier, it depends on what you know best and feel most comfortable with. e.g. I missed the lectures on the Residue theorem and calculating residues and real integrals etc. so the day before the exam I read up on it and did loads of examples so it was fresh in my mind and those questions came pretty quickly to me so I chose to do three complex analysis ones but if the real analysis was fresher in your mind then the sensible thing was to do three of those questions. The complex analysis ones were risky in a way because more so then the real analysis ones, the latter part of the question often relied heavily on using your answers from the earlier parts (especially the one on integrating functions of the form Q(cos x, sin x) around the unit circle)

Anyway, I am sure you have done better than you think in your exams, I'll even let you know what I get in probability and stats and that will probably make you feel a whole lot better!

Good luck for later!

Beans said...

Hi Jake,

You're right about context. I think I am still not as "abstract" as I would like to be. But hey, got algebraic structures 2 to look forward too!

I had a vague idea of the monotonic question, but because I hadn't looked at it at all, thought best not to risk it! I felt it was worth 10 odd marks, but on the other hand the other question wasn't completed so it wouldn't have mattered.

Had I actually revised more for complex analysis I would probably be saying the same. That's the very reason why I only did two complex ones- I could only answer certain parts of the question!

Lol, I don't think it's about doing good, but about not doing as well as I would have liked. The quote of the day best explains my situation. Not one single paper that I sat, did I fully complete. That's why I am not very happy. Sigh. I don't think the stats paper was that bad! (And I doubt that would make me feel better! :p) I would rather we not be told our results this year... "they got lost due to an administrative fault"?!

No more stats is what is giving me comfort!