Friday, May 08, 2009

End of an Era

Today is officially the last day, and indeed my last lecture of my University "career". Next week is a revision week, after which my exams will begin.

How am I feeling about this? Well yesterday during Hyperbolic Geometry, it seemed to hit home that everything is coming to an end. Today it feels slightly worse, but I know in my heart, that I am making the right decision.

At the moment I have one foot in one boat, and the other foot in another . Both these vessels are travelling at very fast speeds, and it is inevitable that I fall, as I have poor balance as it is. Thus both boats end up crashing. I propose that it is better for me to travel in one boat first, "become an expert" in guiding it, and then slowly make the jump again.

This decision may harm my mathematical aspirations, people claim, and I will become dumb in the subject which I already struggle with. However, I disagree. I won't lose my passion for Mathematics which is enough for me.

This will be a short post as I will leave the house in the next five minutes. I don't think that I could have wished for a better three years at Manchester. Yes, things did get hard for a while as I wasn't able to cope with sudden changes. However, that was part of my experience and hey, I'm smiling now! I don't really want to leave the people behind you see.

Last night, the teletubbies claimed that I was the worst Mathematician they knew (!) as I kept on saying that Maths was greater than Physics (and every other subject on the planet). I couldn't make F'(S) equal to zero, so was already frustrated, after which I deliberately annoyed Po and Tinky Winky, as they were revising Physics! I think I'm the only Mathematician that they know...

Anyway 10:45am means I must leave for the bus stop now!

Sunday, May 03, 2009

Nonsensical Theorem and Proof

Today I was in the mood for some Maths; and as I am home alone, I had no distractions etc. First up it was the Euler Product for the zeta function. I understood the sieve process, until the very two last lines which complete the proof. We take the sup and limits and then draw a nice black square. Well not very nice in my case, but at least I followed the main idea I consoled myself.

Next came lecture five - Derivatives of Infinite Product.

I understood the first paragraph, but then came the Theorem and its proof. Some generous lecturers offer rewards if students spot any mistakes in their notes. Although this proof and theorem is not something I originally wrote, I too will be generous and offer a reward if anyone can point out any corrections. However... to claim your reward, you have to ask for it in person! By the way I can on this occasion say that I copied exactly what was on the black board, as I remember the lecture very well (due to a small hiccup). So here goes:

Theorem
Let $f= \prod g_j$ with $g_j \in Hol(U), f \neq 0, g_j \neq 0$ in U, and $f = \prod g_{\lambda}$ (L.U). Then

$\frac{f'}{f} = \sum \frac{g_n'}{g}.$

Proof

$f_n = g_1 ... g_n \Rightarrow \frac{f_n'}{f} = \frac{g_1'}{g_1} + ... + \frac{g_n'}{g_n}.$

But $f \in$ Hol(U) and $f_n' \to f'$ by convergence lemma (2.2).

Let C be a circle in U, then $\exists \delta : | f(z)| \ge \delta$ "more than" 0,

$\exists N \text{ st } n \ge N \Rightarrow |f_n(z) | \ge \frac{\delta}{2}$.

$\Rightarrow | \frac{1}{f_n(z)} - \frac{1}{f(z)} | = \frac{|f_n(z) - f(z)|}{|f_n(z) f(z)| } \le \frac{2}{\delta^2} |f_n(z) - f(z)|.$

$\Rightarrow \frac{1}{f_n} \to \frac{1}{f}$ uniformly on C

Therefore $\frac{1}{f_n} \to \frac{1}{f}$ uniformly on each compact disc.

Therefore $\frac{1}{f_n} \to \frac{1}{f}$ (LU) $\Rightarrow \frac{f_n'}{f_n} \to \frac{f'}{f}$.
$\Box$

If you can't find any errors and understand the proof, some help would be nice! I'm being dumb, I know, but how does the conclusion in the proof help us?

Anyway, it's food time now! Erm - not fast food again... Pizza is quite healthy in my opinion, especially if it has some pineapples on it!