Thursday, December 06, 2007

Tiredness, inequalities and tiredness again

This week has really flown, but still a day remains, about which I keep reminding myself. On Tuesday I had gone to sleep at 3am, and yesterday at 1am. Both these instances due to Real Analysis and me playing catch up. Last night it had been more of a case of going through the problem sheet again, seeing if things made sense a second time round. Some did, which made me lose any sense of time, and others were sent to my list of questions.

Today I really could not get out of bed. I struggled in a bad way, wishing that I had already missed a lecture this semester. (If you miss one then you don't feel too bad about missing another. However, I only ever miss one due to unforeseen circumstances, and not being able to get out of bed is a lame excuse.) It was Numerical analysis at 9am. In my zombie like state, with no breakfast to keep my insides warm, I managed to get to my lecture.

Surprisingly, parts did make sense (for a finite amount of time)! However, once the lights were dimmed due to the screen being used I felt myself leaving the lecture room. My eyes began drooping, but nevertheless I stayed strong and held on for another five minutes.

Now here comes the bizarre part. At 10am I was in the AT building, in search for an empty class. I found one which was nice and warm, with the lights off too! I first waited to see if a class was scheduled to take place, and eventually went and sat in the furthest corner -- unseen. Taking my coat off, I created for myself a pillow. Setting the alarm on my phone, I said goodnight to the world!

The bizarre thing is that no one woke me up whilst I slept! I don't know whether or not anyone walked into that room, but such was my tiredness that I probably didn't realise. It was indeed a refreshing sleep, and when my alarm sounded I proceeded to press the snooze button. But alas, at 10:40am, I could no longer press the snooze button again and so dragged myself upright - it was time for a cup of tea (and then a lecture of course!) Thankfully I felt normal again (but once again for a finitely small period).

Today I was sat for a while contemplating things. It was rather a weird moment, for when one does sit and think about things which they haven't done so, strange worrying facts come to light. I had a lonely moment so to speak, and it is with me now as well.

During lunch I conversed with many people, and one of them being my PT. He told me how to correctly pronounce Cauchy's name, and well in want of not sounding like a fool I will do so! (My personal vendetta against Cauchy feels upset by this). Weirdly enough that darn fellow kept on propping up in conversations today. Indeed one lecturer said something about Complex analysis being named after Cauchy! My insides vanished at this thought - then it would be impossible to escape him.

I dare not delay mention of inequalities (some of you are probably grinding your teeth impatiently!) As you all know, I absolutely am awful at manipulating inequalities. Last year in sequence and series (and indeed other modules) I had this problem; and even now, a year on I STILL have a problem. College was where it all started.

Because of my mind freeze when it comes to this topic, many mathematicians have felt like using my head as a punch bag! (Well for other reasons too, but this being the main one). I know that BR, and DC would very much like to do such a thing, but what can I do? I honestly can't explain why I cannot manipulate the darn things.

The question in the sheet had been to show that |\tan b - \tan a | \ge |b-a|, using the Mean Value Theorem.

So, I started the problem: Let f(x) = tan be continuous on [a,b] and differentiable on (a,b), then there exists a c \in (a,b) such that:

\displaystyle \frac{f(b)-f(a)}{b-a}= f'(c).

Using f(x)= tan x, (and missing a few many steps) we get to,

|\tan b - \tan a|=\displaystyle \frac{|b-a|}{|\cos^2 c|}.

We know that |cos^2 c| is less than or equal to 1, hence the above simplifies to give:

|\tan b - \tan a| \le \displaystyle |b-a|.

As you can see my answer was not of the required form -- the inequalities were the wrong way round. Initially I couldn't see anything wrong with my argument, and even asked DC whether or not there was a typo in the sheet! He did get slightly miffed though (in a manner of speaking), because on another question I had also used modulus signs and hence my argument had all been wrong. I did annoy him today due to this, but I was feeling very frustrated (and tired) myself. I probably caught him at the wrong time, but he told me that I will have to somehow see it for myself, for me to understand what I keep on doing wrong. (The other question was resolved though).

I didn't get another chance to look at the question until later, for it was time for the example class. In the example class I said said one stupid thing, and then another to myself, but I survived to tell the tale! I am being very "bad" and going to the example class without having tried all the problems myself beforehand, but they are a good chance for a recap (or so I reassure myself). After the example class I did another potentially stupid thing, but no one was hurt -- well it was slightly funny too! Now PS today did something which I think he shouldn't have. Next Thursday I will rectify this, but still it was something I should be doing -- not the other way round!! (All you need to know is that tea is involved, as alway.) There was something that I have been meaning to ask PS, but would you believe it, I forgot. It has been on my mind for a few days now, but today I had to forget! (Well I have the final Thursday to ask this, but only if I remember). Many students give me weird looks because I talk to lecturers. I have noticed this a lot this week, but who cares. I like Thursdays!

After my converstaion with PS (cool as always) I did some errands and then returned to the "inequality problem". It must have been the tea that I drank, but I saw something differently:

\displaystyle |cos^2 c| \le 1 \Rightarrow \displaystyle \frac{|cos^2 c|}{|cos^2 c|} \le \frac{1}{|cos^2 c|}

This is what we wanted i.e. \displaystyle \frac{1}{|cos^2 c|}  \ge 1, which will give us the answer that |\tan b - \tan a | \ge |b-a|!

I feel different (read angry, foolish and apologetic to DC for the tantrum I threw!) because of this. Inequalities are evil and tricky, which is just me saying that I am dumb when it comes to them. Most likely, I haven't learnt from this mistake. More care is needed when I tackle inequalities because I seem to take things for granted. Less than or equal to seems to be OK, but when things flip around and signs change, I fall in the trap. Is there any way round this? Now for a comment for the future: my bad of manipulation of inequalities is going to prove fatal in my future mathematical life. Hows that for a depressing future! Actually I might as well add taking the modulus to that -- I always seem to want to do that too.

It wasn't only DC that I annoyed today, but I have been annoying Dr. W for her abstract (and two other people)! Unfortunately for me, I think I have pushed all those named to their limits (helped convergence?). Hopefully I will be forgiven for my bug like nature by tomorrow, but I dare not find out. (I seem to have that... ability to drive everyone insane, and to their limit, which I take isn't always a good thing! Not on purpose though...)

Now tiredness is once again sweeping through me; my plans to do Cauchy analysis(!) have been thwarted, and maybe if I wake up at 5/6am I will do some then. I have had enough of real analysis... well not really, because I would love to finish the few remaining questions and start integration, but self control is needed. It's stupid how sometimes even though something gives us unnatural pain, we still enjoy it and go back to it. (Taylor series make me want to vomit though, and series in general!)

I will end with a quote from a book, which is namely meant to calm my nervous and panicky self, as the magnitude of my "falling behind" stands as a painful reminder. Indeed, I think this quote should reassure all of us who have sometimes felt "thick" compared to friends and others.

"Those individuals who seem to just "know" how to solve maths problems have simply spent enough time exploring the mathematical landscape, to have developed a good sense of the terrain."

Hopefully(!) I will mention more of that book as I finish reading it, but it is about time I went and spent some quality time in this landscape. (Preferably without being tired, but with six-ish hours tomorrow I better not get my hopes to high).

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