Me Or My Maths: December 2007

Monday, December 31, 2007

A review of 2007

As seen in Michi's Blog, from each month, the first sentence of the first post. However as you will see, my "blog review" will be shorter (hence why I have linked the first post of the month too)!

The list doesn't exactly inspire one to stay tuned in does it?! Tonight I cannee be bothered linking any particular post, so I will save this for "the new year".

Whether you take my word for it or not, these past few days I have been a bean of a few words. Hence why I haven't really felt like blogging, and so without further ad (to those who care) "Happy New Year". Although to me it seems like another day, you don't want to be crying tomorrow. Or is it only my mother who used to warn me not to cry on New Year's day, otherwise I will be crying for the rest of the year! Anyway, see y'all tomorrow.

January
I spent a while trying to make this a 'cool' first post, however I've miserably failed.

February
Without a doubt, so far from my experiences at the University of Manchester, the only lectures so far that I've looked forward to have to be Dr Coleman's lectures.

March
Shockingly it's March today and, well that's not a good thing!

April
Some neat 'alliteration' to get this post started.

May
Firstly I must comment on what difference a good nights sleep does for a person, or for a bean.

June
I first want to link this post, due to which I am writing this.

July
Yes, I am indeed alive and that wave was with my right hand!

August
So, the time has come when a decision has to be made.

September
I ate some humble pie today and got some egg in the face!

October
Before I continue or should I say before I start, I want to say one thing.

November
"Dear.......

Yes I had received the email. ...

December
Whilst googling for pictures (for my first ever lecture), I came across something rather marvelous and... cool!

The Bernoulli Paradox

Start with:

$(-z)^2 = z^2$

where z is a complex number. Hence,

$2 \log(-z) = 2 \log(z)$

So,

$\log(-z) = \log(z)$ .

What is the fallacy?

Oh and congratulations to those who selected the option (b) in the previous post! :o

Sunday, December 30, 2007

Something "amusing"?

If I told you that I am in the process of writing an "amusing post" what would be your reaction?

(a) You nit wit - why aren't you revising?!! I don't want to see any silly post - get back to your books.
(a`) We are sorry to inform you, but amusing doesn't register when it comes out of your mouth. Please do not waste your time and indeed ours, and get on with some much needed revision.

(b) Erm... you said the same thing about that LaTeX post and we are still waiting for it. Oh and weren't you meant to post something about... HEY where are you running off too? Beans?! Hey stop!

(c) Amuse us then.
(c') I can't wait to read it - post the damn thing will you! Is it going to be as amusing as this one? Hurry up and finish it will you. This is torture... \hyperbole!

(d) *click on the ugly looking cross in the top right hand corner of your screen*

---

Yes - I have given up with stats, which explains the lightheadedness! Anyway, as you are musing over which option to choose (do enlighten me) I will say that I stopped typing it because I got confused between third person and first person. After some sleep I will re-attack it.

Also feel free to add an option! (a' and c' can be taken to equal to a and c respectively!)

If that hasn't amused you, then why not have a bash at:

$\int \sqrt{ \tan x} \;dx$ .

(If you want to post the answer then please don't! I am still struggling with that ugly thing, and am being very stubborn about it. Although if you manage to solve it, you can celebrate your success here. I am not that much of a scrooge person)!

Friday, December 28, 2007

My Way

We are allowed to have at least one lame song in our head, or are we not? Well this song (lamely) popped into my head this instance, and I thought of sharing it! (Aren't I so nice... but don't thank me just yet!)

Unfortnately I am still struggling with statistics. HOWEVER - even if I don't finish by the end of today, I will be starting complex analysis tomorrow. (Which I should have started today!) I am hoping that revising for complex analysis will be more relaxing in the sense that I am actually looking forward to it! My war with Cauchy has to continue and there can only be one winner *cue evil laugh*.

The song and indeed this post are a result of the z-distribution. (To be said in a spooky voice). In college I had "my way" of finding confidence intervals (for the Normal distribution). My way of doing things was systematic and we started with the information we had, and then moved along. No formulas existed. Formulas can be nice, but if you don't know where the formula came from, then it is hard to remember and use them.

In my notes, and indeed the book I am using, there is the following formula (which I would advise you to safely ignore). P( blah blah zkkdfsd) = 1 - \aplha.

Ignore it because I haven't really defined anything, but more information (if you are unfortunately interested) can be found here. My mind goes blank whenever I see that kind of notation. Today I thought to myself - "Why struggle with this when I can do it MY WAY" (hehe). So I persevered and eventually recalled my college way of doing things. And hey presto, my answer was the same as the one in the book! This was nice to see, and after much mindless staring (again) I realised that I had effectively done what the formula says. It takes longer doing it my way, but I think it is the more honest way of doing things.

The sad thing is that for my exam I will probably have to learn this formula and a few others (so I better start trying to understand them). Is stats making more sense now? NOPE! Why? Well, although I have a nice book which I am using, there is nothing remotely WOW about stats. Hence, on most occasions I can be found doodling or lost in thought about something or other. This is what has delayed my revision. Many people have told me to revise using the past paper as a guide. I should listen to them wise people but that is against my nature. (I.e. not doing something that I am told to do!)

They speak wise words indeed because past papers do give you a good indication of what to expect, and basically how to pass. My nature is such that I firstly struggle to understand the material, and then just before the exam I attempt the past paper and problem sheets. Doing this then causes me great panic because I find myself unable to answer the darn question. This I speak from experience. Last year and in all my previous "statistical years", I never tried understanding stats (for numerous reasons). Then why today am I wasting time on such an activity? I honestly don't know, apart from not wanting to get bad marks in stats! I feel extremely thick upon having done some revision. I can't believe how dumb I was in stats (not that I am no longer dumb, but at least now I know the difference between the population standard deviation and the sample standard deviation!)

That is enough about stats. I am shocked at the amount of coverage it has been getting in my blog! The shame... (Please don't tell anyone I know!)

I have an excruciatingly painful headache at the moment. It is causing me great distress and not allowing me to revise. What to do eh? I have two chapters left to complete, so maybe it is time I went to the "medicine cupboard". A couple of paracetamol, an ibuprofen and maybe some emflex won't do me any harm. OK fine, maybe I will just stick with the paracetamol. (I have never had an emflex before: "I wasn't allowed"!) Oh, and don't worry I am not addicted to these things. I only ever resort to such addictions when I am in (exponential) pain; or when the pain, although bearable, is a distraction (like now).

I should shut up. Yes - the most wisest thing that I have ever written, yet I am finding trouble doing so. I took a picture of Dr. E's book: Men of Mathematics, so I can tell him that I know who has it. At least then he won't think I have it!

Before I do shut up, here are the lyrics to "My Way". And you can listen to it here. Nice.

"And now, the end is here
And so I face the final curtain
My friend, I'll say it clear
I'll state my case, of which I'm certain
I've lived a life that's full
I traveled each and ev'ry highway
And more, much more than this, I did it my way

Regrets, I've had a few
But then again, too few to mention
I did what I had to do and saw it through without exemption
I planned each charted course, each careful step along the byway
And more, much more than this, I did it my way

Yes, there were times, I'm sure you knew
When I bit off more than I could chew
But through it all, when there was doubt
I ate it up and spit it out
I faced it all and I stood tall and did it my way

I've loved, I've laughed and cried
I've had my fill, my share of losing
And now, as tears subside, I find it all so amusing
To think I did all that
And may I say, not in a shy way,
"Oh, no, oh, no, not me, I did it my way"

For what is a man, what has he got?
If not himself, then he has naught
To say the things he truly feels and not the words of one who kneels
The record shows I took the blows and did it my way!

[instrumental]

Yes, it was my way"

Thursday, December 27, 2007

Weak Law of Large Numbers

Let $X_1,X_2,X_3,...$ be independent and identically distributed, each with mean $\mu$ and variance $\sigma^2$ . Then for any given $\varepsilon \text{\textgreater }0$ ,

$P(|\bar{X}_n - \mu| \text{\textless }\: \varepsilon }) \to 1 \quad \text{as}\; n\to \infty$ .

Proof?

Erm... well it has a lot of nasty probability involved, are you sure you want it? I didn't actually write that law for its own sake you know! (Proof will be given later, read *VERY LATER* but if you really want it then let me know! Alternatively you could try proving the above yourself. You might want to use Markov's inequality and take $g(\bar{X}_n)= (\bar{X}_n - \mu)^2$ and $k = \varepsilon^2$ .)

The real reason I wrote that law was to remind myself of something. Posts concerning negative numbers have been going around Blogistan, about this article. I have sat quietly because my own struggles with negative numbers is bizarre and still possibly exists (with respect to inequalities etc).

Yes - inequalities are indeed the reason for my post on this. I don't think I have a problem with deciding which number is bigger from -5 and -4, but today for the life of me I couldn't understand why -4 was bigger than -5! Whilst trying to understand the proof of the theorem (which I will mention in a while), the numbers -4 and -5 happened to be on my paper. As usual I talk to myself and asked myself which was bigger. Someone else happened to be listening to me at the time (a primary school kid) and so I asked her which was bigger. The kid answered -5. I queried at this choice and was told that 5 is bigger than 4. I told the child that she was incorrect, and then tried to give some silly explanation but ended up confusing myself!! Annoyed at myself I had apologised to the primary school kid before returning to the proof.

However that didn't stop me from talking to myself. Again I loudly asked, "Why is -4 bigger than -5?" (If you are wondering why the heck a doofus like me is doing a maths degree, then you are not alone in that thought!) I know that -4 is bigger but I just couldn't convince myself as to why; hence causing myself silly unwanted frustration. Finally I drew a vertical number line (as I used to in secondary school!). 5 was on top of 4 so 5 was bigger. -4 was on top of -5 therefore -4 was bigger. Yep - that was my lame explanation to myself but it knocked me out of my primary school behaviour and back to stats.

In statistics you have a lot of inequalities flying about, and it is crucial that you use the right sign. As you read above, we have Markov's inequality and another one is Chebyshev's inequality. (They give a bound for the probability - a bit like the estimation lemma in complex analysis(?)). Proofs that use these inequalities, ultimately require some manipulation of inequalities. Once again, like a couple of weeks ago, I couldn't see why the sign was the way it was (when I do post the proof you will know the exact details!)

Maybe I should have done this from day one, but today after much mindless staring, I (perversely) wrote -4 = 1 - 5. After which I wrote the inequality $-5 \ge -6$ . Hence -4 = 1 - 5 \ge 1 -6. That might seem very easy and you are probably asking the same question about my degree again(!), but the symbol used here was the greater than sign. I had once again been sticking the less than sign everywhere, and not understanding why it had been wrong. Going back to what I did allowed me to see why I had been wrong. In future maybe I should give every inequality that I have been given, numerical values just to get the sign right!

So how many times in the past few days has the question about my degree come up? I count four - two in this post and two yesterday. Is this due to statistics or should we be really worried?! (I think it's because of probability and stats though!) Seriously these past few days have left me wondering about whether I can do any maths whatsoever! Indeed I haven't been feeling like a mathematician.

I shouldn't really be up at this time but got talking to someone about things till 1:30am. Sometimes children are the best people to talk to, i.e. people younger than you! (I hope that I am still a child yet!) I mean those who have yet to experience what you have and are fearless in their claims. They have a certain freedom in their speech and any "advice" they give you is not harmful or bitter. Not recycled and like a waiter who serves food all the time (if you follow me). I think it is natural that if I was to give "advice" i.e. share my experiences, I will obviously also warn others of what not to do. Those who have not yet got those experiences i.e. those younger than you don't hear the warnings and don't see them. I think I am talking rubbish again, but as I sat and talked with someone younger than me I realised the freedom of being young. Impossible is actually nothing. And then you grow older and realise that if you do a certain thing it will hurt you, others etc and so you lose some amount of fearlessness.

Sometimes I wonder, would I prefer being a 12 year old (nit wit!) with no real worries in the world - no responsibilities; or would I prefer being a 19 year old who seems to over think! If I was 12 and aware of the exciting maths which I have been studying then sure thing! However it is a close call to make for I definitely do not want to go back to doing ratios and what not. Sigh. It's 2:44am and I am mumbling to myself. I was meant to be revising when I was talking earlier, and exclaimed this fact to my friend. "I was meant to finish stats today!" (A common sentence from my mouth nowadays). I was then told to stop worrying and to stop thinking that I can't complete it tomorrow or that I had to complete it by tomorrow. ( I didn't remind my friend that P(completion)=0). It is nice to be understood once in a while.

The demoralizing thing for me is that as I spend time understanding stats, I still don't get it. My problem has been that I am not revising but trying to understanding what has been written. However when I try to understand material I never work linearly through my notes, and actually use a gazillion books. I have been getting carried away recently and not remaining focused on the chapter at hand. This is what causes me to spend hours at times, on trivial things and sometimes more! (Time actually flies). I really want to understand the concepts properly though - not artificially. But sadly I have ran out of time. Tomorrow I must persevere and complete 6 lectures. One lecture takes me a more than an hour to understand as well..

---
Good morning! It seems that I feel asleep whilst typing that post. Well actually I thought of resting my eyes for a few minutes, and woke up to see the sun outside! After a quick read through that last paragraph I have come to a conclusion. One reason why I am really bad at statistics is because of my attitude towards the subject. Whenever it is mentioned, I don't pretend to hide my groans and pain. Honestly, I am unfortunately ruthless to this subject and my answer has always been because I can't understand it to a level of comfortableness.

The same applies to my complete miss hit with inequalities. As soon as I see a greater than sign my mind blows up and shuts down! Straight away I try to re-write it as a less than sign, then panic some and write something incorrect. Once I can get rid of this "dread" for inequalities I should be OK (or at least I hope to be). I feel silly because of this, but it is a serious problem. Inequalities are really powerful (and I have just remembered that some induction proofs make use of them). They are everywhere!! RUN FOR YOUR LIVES!

Enough drama. My attitude towards stats might never change, namely because I will never understand it and can't find anything "cool" about it. However I would love to be given a few extra days to try and understand it better. And for the record probability is MUCH better than statistics. So I was correct (for a change) in always writing "I hate stats". I hope though, that I am able to like it enough for my exam though.

And inequalities -- well if I can go for a year without making a blunder when dealing with them, I will consider myself cured!

Tuesday, December 25, 2007

Silly differentiation

Once upon a time last year I posted the most stupidest of things about logarithms. My brain had become saturated with mechanics was my excuse then; but it was indeed an embarrassing moment which made me ask the question about which degree I was studying for!

I am now giving you the wonderful opportunity for me to ask myself that question again (albeit reluctantly). I can't seem to differentiate. If anyone is reading this today then please do tell me what I have done wrong.

The following has to be differentiated and then evaluated at t=0:

$M(t) = \displaystyle \frac{e^{tb}- e^{ta}}{t(b-a)}$ .

You can use the quotient rule or write the t at the bottom to a negative power and use the product rule. (I feel awfully dumb writing this in the sense that I am unsure of the "validity" of what I am saying.)

So we differentiate using the quotient rule on this instance to get:

$\begin{align*} M'(t) &= \displaystyle \frac{t(b-a) \times (be^{tb}- ae^{ta}) - [(e^{tb}- e^{ta})(b-a)]}{t^2(b-a)^2}\\ & = \displaystyle \frac{t(be^{tb}- ae^{ta}) - (e^{tb}- e^{ta})}{t^2(b-a)}\\ \end{align*}$

The million dollar question - what have I done wrong? The first line by the way can be read as follows: "Bottom times the derivative of the top minus the top times the derivative of the bottom. All over the bottom squared." (I remember that thanks to Prof D from last year!)

The second line is my attempt at simplifying things, but what is the dumbest thing that you have spotted in my attempted differentiation? Please there has to be something. My mind is honestly blank - I don't even know what day of the week it is! All day I have been thinking it is Saturday today (and I have only had three cups of tea today). Actually talking about tea - I am missing Thursday tea time it seems. (Well I had a dream and it was a Thursday. I had been drinking tea and discussing my worries about not caring anymore with the usual person!) Ah well, roll on next semester I hope. Thursdays were a good day.

Anyway why do I ask for someone to tell me the mistake? Well if we stick t=0 into M'(t) then the angry bees start buzzing "we can't divide by zero". I need that horrible looking derivative to simplify to (a+b)/2 when t=0. Can you do that for me? Can you give me a baseball bat for my head?

I don't really have a million dollars but if you tell me what's wrong, I can then tell you the mathematics behind the differentiation. (Which I am sure you are all desperate to hear!!) Pretty please...? (I have even broken a rule of grammar there (I think) -- just so that I can go to sleep with a clear and unburdened mind!)

One week down and a finite amount left...

Writing helps? Well I don't know, does it? Will writing about how evil, disgusting and vile probability and statistics is help me? Will it make me revise 12 lectures in a day? So how does writing help then?

A week has gone by and my depressing post will prove to be scroogified (as expected). I don't know what's wrong with me. Why don't I care anymore? I seem to have stopped caring which is worrying. I sit downstairs and think to myself... "I will revise in 10 minutes." 10 minutes later I say the same thing again. This continues until it is 7pm and I think - dang the day is over! Why can't I revise?

I am not in exam and revision mode - that's what the problem is. How can I get myself in the right frame of mind? I know for a fact that stats is not helping one bit. Maybe I shouldn't have revised it first. Maybe I should have revised algebra first to get myself into that mode. It is because of stats and numerical analysis that I don't care anymore. They have numbed my mathematical senses. The feelings of delight and joy are rare. I feel like I am forcing the mathematical excitement nowadays. It has abandoned me.

Last year during my first semester exams I was in a similar state. I couldn't stand set theory and hated it as well as stats. Consequently I had a depressing exam period (which thankfully no one heard about!) My second semester though was infinitely better last year. Yes - my revision wasn't great but I enjoyed each and every module that I was studying. That enjoyment generated the excitement and energy and made me want to revise.

This is torture though. I DON'T WANT TO STUDY ANY STATS MODULES IN THE FUTURE! Yes you heard correctly - I don't, so why then do I have to do this bleeding module? The same applies for numerical analysis. Do they want to kill our mathematical delights and excitement? Do they want us to run away from the subject?

I honestly cannot look at my work and notes with the same enthusiasm as I once have. This is a temporary thing I hope, but it is worrying. My this year counts which puts the added pressure of doing well in stats this time (as opposed to last year). Everyone keeps on sayings things like "oh but you will be OK in the end." They are on auto pilot and I have stopped exclaiming that I am not revising.

The difference this year has also been that I have not kept up to date with my work as lectures progressed. My technique has always been to listen in lectures and then go home and try to understand. Once understood "file away" until revision time. The only module which I hope to have dodgily done this with is real analysis and areas of algebra. The other four put me in a very bad position indeed. I am not revising but learning the material now, which is what is killing my senses. I don't want to learn about bivariate distributions - they make me want to cry. I can't understand stats for the life of me. I have messed up really badly this semester.

Headaches have a funny knack of appearing as soon as I enter my room to revise. I am honestly going insane with worry because I can't and am not revising. My artificial understanding of all my modules is crippling me. What must I do? I have three weeks left to sort myself out. I have isolated myself from everyone because all they talk about is revision which reminds me of my sad plight. To see someone pretend to be happy when they are actually sad is another depressing thing - this I witnessed yesterday.

I am getting scared about something which is going to happen next year. I don't know what to do anymore. But then I remind myself, some things have to be done in life. They just have to be done - for Bob and Wendy. I wish for a series of unfortunate events to happen so that things get delayed. But then again, as someone pointed out to me - if I let things pan out the way they most likely will, I might end up doing what I discuss with PS. But the question I now ask myself is - do I still want to that considering my lack of care for my studies? My lack of care for most things?

How does one filter out everything in ones head? Maybe that is what the person meant when he said "writing helps". Pile of cack if you ask me. He was just making excuses for there not being a very good filtering system available. Maybe if you dilute everything and then ... whoops - let me check that with my chemistry teacher first. I find myself crumbling into nothing. Time flashes by and I am found running after it in panic. Sigh. What a wonderful day its going to be today.

To end this gloomy post, here is something (funny?) that I found on the Internet:

"Please accept with no obligation, implied or implicit, my best wishes for an environmentally conscious, socially responsible, low-stress, non-addictive, gender-neutral celebration of the Winter Solstice holiday, practiced within the most enjoyable traditions of the religious persuasion of your choice, or the secular practices of your choice, with respect for the religious/secular persuasions and/or traditions of others, or their choice not to practice religious or secular traditions at all.

Also, have a fiscally successful, personally fulfilling, and medically uncomplicated recognition of the onset of the generally accepted calendar year 2007, but not without due respect for the calendars of choice of other cultures whose contributions to society have helped make Europe great (not to imply that Europe is necessarily greater than any other country or is the only "Europe" in the western hemisphere), and without regard to the race, creed, color, age, physical ability, religious faith, choice of computer platform, or sexual preference of the wishee.

(By accepting this greeting, you are accepting these terms. This greeting is subject to clarification or withdrawal. It is freely transferable with no alteration to the original greeting. It implies no promise by the wisher to actually implement any of the wishes for her/himself or others, is void where prohibited by law, and is revocable at the sole discretion of the wisher. This wish is warranted to perform as expected within the usual application of good tidings for a period of one year, or until the issuance of a subsequent holiday greeting, whichever comes first, and the warranty is limited to replacement of this wish or issuance of a new wish, at the sole discretion of the wisher. Should tidings generally accepted as good not be experienced by the wishee during the warranted period, then this wish is null and void. This greeting has
no cash value.)"

And no - I didn't read it all myself! I am not that much of a misery gut so I hope everyone is having a nice Christmas break. (Bah Humbug!)

Saturday, December 22, 2007

Double Integrals (and some probability)

Rather than mentioning what I have not been doing, I will talk about what very little I have accomplished over the past four days! (I will save the "whinging" for another post.)

Double integration was taught in my first year as part of the calculus and vectors module. Initially I had difficulty understanding it but when it came to revising, things made a little more sense. I actually was able to evaluate a double integral last year - whatever the region of integration might be! (I had to know this to use Green's/Stoke's Theorems).

However during the first few stats lectures this semester, when double integrals were evaluated, my mind didn't register what was happening. Somewhere a part of me knew what to do, but I blindly assumed that it will all come back to me in due course. (Note: just because we have to evaluate double integrals in stats doesn't make it any nicer). I stupidly left it at that i.e. assumed that my last years knowledge of how to answer questions with double integrals would return magically. Clutching at straws I suppose.

Evaluating a double integral is "easy" (one could say). Well if the limits have been given and everything is hunky dory, then hopefully I can do the integration. What I can't do though is set the integral up. I had forgotten how to choose the limits of integration, which turns out to be important!

So why not make this a big deal. Over the past four days... yes four days I lie to you, I have been bravely (without a doubt!) struggling with mastering the art of finding limits for double integrals. A mouthful isn't it? Having (I hope) mastered this successfully, I proceed to kick myself with a certain venom! WHY couldn't I have done this before my stats coursework test? WHY?! Had I done so, I wouldn't have done as badly in my coursework! (Bad relative to the other tests and my own marker.) I got one question wrong due to my stupidity* but the rest of the marks were lost on evaluating double integrals. Hence why I will now attempt to solve the question here!

Question
The joint probability density function (pdf) of the random variables X and Y is given by:

$f(x,y)=\begin{Bmatrix} 2 & & 0 \le y \le x \le 1; \\0 & & \text{elsewhere}.$

a) Find the marginal pdf of X.

Solution
The marginal pdf of X is found by $f_x(x)= \int^{\infty}_{-\infty} f(x,y) \,dy$ . (The limits in this case are symbolic and will be determined by the region of integration).

Firstly we will (roughly!) sketch the region in order to deduce the limits:

The shaded in part (i.e. the yellow region) is the region of our integration. The integral tells us that we will be integrating with respect to y. So now to determine the limits, we draw a vertical rectangle (or line) through the shaded region on the diagram:

The red spot is where our pen should be. Unfortunately I can't give a concise mathematical explanation as to why we do this, but it is something along the lines that x is fixed and we want to see how y varies within this region. So we move upwards from the red spot until we hit the line y=0. We have to pass this line to enter the region and so it is the lower limit for this integral. Now that we are in the region we continue travelling along this green line until we hit the line y= x. We could carry on moving upwards until we hit the line y=1, but then we will be out of the region (and not answering the question). Hence when we hit the line y=x we stop and choose that to be our upper limit.

This gives us the integral $f_x(x)= \displaystyle \int^{x}_{0} 2 \,dy$ , which is left as an exercise for the reader! (I love saying that, but hopefully you get that this integrates to 2x!)

The question however asks for the pdf, and simply writing 2x is wrong. The correct answer (by elimination) then is:

$f(x,y)=\begin{Bmatrix} 2x & & 0 \le x \le 1; \\0 & & \text{elsewhere}.$

I didn't choose that option which is why it must be correct, for mine was wrong! I don't understand the reasoning for choosing $0 \le x \le 1$ , but then again somethings will never be understood. (I had "guessed" the other option with region $y \le x \le 1$ which was incorrect).

The question doesn't ask for the marginal pdf of Y, but I will mention it here. This time the integral will be $f_y(y)= \int^{\infty}_{-\infty} f(x,y) \,dx$ (with symbolic limits).

The region for the integration is the same, however to find the limits we now draw a horizontal line on the graph through the shaded region:

So we start at the big red spot and move towards the right on the green line. To enter the region we have to "hit" the line y=x first, which becomes the lower limit in the integral. Then we merrily continue our journey along the line until wham, we hit the line x=1 -- the upper limit. Thus our integral is $f_y(y)= \int^{x=1}_{x=y} 2 \,dx$ . (Once again left as an exercise!)

Now something is telling me that I might have written something horribly wrong! Call it a gut feeling, but have I got this all wrong? Do I need to call it a day, and find myself a nice farm? If I knew what the actual answer was then I would probably be more confident. Hence why I think you should take this post lightly! (I cant believe I am blogging about probability -- maybe that's why I am feeling slightly queasy and unsure about what has been typed? By the way, I realise that I mostly complain about stats but that includes probability. My module is called probability and statistics but it is easier to write just "stats". Maybe I should write "stats inc." to mean probability is included! I dislike both for the record.)

This post hasn't really been about double integrals, but nevertheless the method of finding the limits is similar. If the region we are integrating over is rectangular (i.e $a\le x \le b, \quad c \le y \le d$ ) then the integral is much nicer. It is only when the region is non-rectangular that things get messy. I think to validate my post title I will now do a proper double integral over a non-rectangular region! (And in future I will try to write the posts first and then choose the title).

Example
Evaluate the volume under the surface given by f(x, y) = x^2+ 0.5y, over the region bounded by the curves y = 2x and y = x^2.

Solution
Before we proceed a (dodgy) diagram is needed (with the region of integration shaded). Always sketch the region first.

At this stage of the game we are faced with a choice -- do we want to integrate with respect to x first or with respect to y? I will choose to integrate with respect to (w.r.t) y first, but if you choose to do otherwise, the answer should still be the same.

Since we have chosen to integrate w.r.t y first, we will be drawing a vertical line through the yellow region as in the example above. (I remember the direction of the line because the y-axis is vertical so the line should be vertical. If you choose to integrate w.r.t x first then draw a horizontal line.) Having drawn the line (on your paper!) we see that to enter the region we have to hit the curve y=x^2 first -- the lower limit; and to exit the region we have to hit the line y=2x -- the upper limit. So we now have our limits for the inner integral w.r.t. y.

For the outer integral (with respect to x), we need the points of intersection of the two curves i.e. the solutions to the equation 2x = x^2. Solving this gives us the limits x=0 and x=2 (I think we are meant to aim to make the outer limits as constants.) Thus our integral becomes:

$\int^{x=2}_{x=0} \int^{2x}_{x^2} \left( x^2 +\frac{y}{2} \right) \:dy dx$ .

Since I am faffing about as it is, here is the solution to the above double integral:

$\begin{align*} \int^{x=2}_{x=0} \int^{2x}_{x^2} \left( x^2 +\frac{y}{2} \right) \:dy dx & = \int^{x=2}_{x=0} \left[ x^2y +\frac{y^2}{4} \right]^{2x}_{x^2}\, dx \\*[2.ex] & = \int^{x=2}_{x=0} \left( x^2 + 2x^3 - \frac{5x^4}{4} \right)\, dx\\ & =\left[ \frac{x^3}{3} + \frac{x^4}{2} - \frac{x^5}{4}\right]^{2}_{0}\\ &= \frac{8}{3} \end{align*}$

As you can see, once set up the integration isn't too bad. It is easy to make a mistake, but last year when integrating w.r.t y, I sometimes wrote c instead of x as to avoid confusion. And then when I integrated w.r.t x, I wrote x again. (But with practice this stopped happening). The main thing I feel, is to have the correct limits. Actually I didn't get the stats (inc.) question above wrong because of my inability to integrate, but because I didn't understand the stats behind the pdf! (Although that can't be said for the other questions...)

I have been wanting to post a somewhat mathematical post for sometime; but boy does it take a while to dig them out. (This was meant to be my 30minutes break from marginal distributions, but has become a 2 hour break!) As you might have gathered, I am trying to revise stats even though my first exam is PDEs and vector calculus. I am thinking that I will try not to blog for a week and see how it goes. If I do blog then it means I have been revising which is a good thing! (You might need to read that sentence again). Many thanks to Steve for the cool looking LaTeX images in this blog, and thanks to paint for the horrible graphs!

Now to some nightmares about stats and probability.

EDIT: I think the title "limits of integrals" would be more suited to this post...

PS: This is a reminder to post about my dream. It was really weird and had the formula for the sum of numbers from 1 to n in it and my old school. Not really that weird maybe, because I have been thinking about that formula recently... Oh and if I do post this week the posts most likely will be short!

Friday, December 21, 2007

Your favourite Greek letter.

So which letter from the Greek alphabet is your favourite?After some serious thought, I have decided to choose two favourite letters: ${\Huge \eta}$ & ${\Huge \varepsilon}$ i.e. eta and epsilon.

Epsilon has been an old favourite of mine from last year, but eta is a new one. Naturally epsilon was chosen due to its common occurrence in analysis; and the phrase "Let $\varepsilon >0$ 0" alt="\varepsilon >0" style="vertical-align:-0pt;" /> be given" is quickly becoming another favourite of mine! (If I ever forget to write that in an analysis proof in the exam then I will say some very horrible words to myself...)

Eta on the other hand is different. It has no "mathematical connection" to being my favourite. I like it because I could write $\eta$ a million times on paper and still find it cool to write. It has a nice action to it... Whoops I am sounding sad again, but the pronunciation of it rolls of the tongue nicely too.

I like the way epsilon is said as well. Some people say "ep- sil- on" but I say "ep- sile-on" which has more authority to it (as it rightly deserves). Both these letters seem very royal (one could say!)

My worse letter at this moment in time (and possibly forever) is... $\xi$ (xi). A little birdie tells me that they are used a lot in applied mathematics! Not deliberately of course, but the horribleness of this letter is the only thing which can reflect the material. (*cue evil laugh*) I don't have any others that I "dislike" at the moment for I am waiting to read through my numerical analysis notes!

I might as well two other letters which I find cool as well (but not as cool as eta and epsilon): $\tau$ (tau) and $\varphi$ (phi). Tau is another royal sounding one, and phi is one which I could write for a long period of time.

Maybe your reasoning will have more mathematical back bone than my own- please feel free to enlighten (or embarrass) me! (But whatever you do, please don't say you like capital Gamma due to stats...please)!

PS: I really should stop making digs at applied mathematics, but seriously in my notes it's a xi here and another there. They are everywhere and not very nice to write as well. (You all know (I hope) that I don't really dislike applied maths that much. I have just tried to be diplomatic when mentioning my reason for my partial change in attitude towards it. Memories of sequences and series lectures last year still scar my thoughts, and part of this semester has been a repeat of that. Painful indeed.)

Thursday, December 20, 2007

Dangerous Knowledge - The Video

A while ago a programme came on TV called Dangerous Knowledge, which I made a big hoo haa out of (due to not being able to watch it). Sometimes one gets lucky and is presented with another chance to do what they previously weren't able to do. I am still unable to watch it yet (give me a spare hour please!) but now it is an easier task. (If it won't play below, the video can be found here).

That is the whole program I think, and here are some related videos.

(My lesson got cancelled - hurrah! Five hours sleep is definitely a health hazard. Oh and Noddy and myself are planning a prank today. Well it involves our trusted friend Scrooge, and if successful you will hear more about it. I think I had my timeout session this morning....)

The worst days of my life

I want university to start again. The sooner this holiday ends the better. If only next Monday lectures commenced.

On Monday I was found complaining about my exam timetable, and indeed the whole day was wasted due to my silliness of not revising until I had the timetable. Now my conclusion in the post title isn't a result of the timetable. On Monday night my cousin thought that her five(?) months old son had meningitis. This was at 11:30pm and so I ended up rushing to her house to babysit her other children, whilst she went to the hospital. I had taken my book with me, but still it had been a very tedious wait. At 2am they were still at the hospital and so I ended up crashing out at her house.

In the morning I was understandably very tired and was infuriated as she woke me up at 8 o clock! (The little tyke is in the all clear by the way). Annoyed at the inconsiderateness of herself, I tried to get back to sleep only to be woken up again. In the end I went home at 9ish and slept there for an hour. Then it was the dentists turn who I now owe 15 quid for a check up. So much for being students eh? I hadn't had time for breakfast, but straight after my check up chocolate had found itself in my mouth.

After the dentist it was hope hospital. I hate that place. I hated having to go there on a Tuesday just because I happened to be free since University has ended. I was at the hospital till 3ish, and finally got home at 4pm. The hospital further confirms my relief for not doing anything medicine related. Even if I am in excruciating pain, I would probably not go to the doctor or hospital. That is how much I hate them.

If things were bad up till now, here is where they got worse. At 6pm(ish) my using and abusing cousin rang again. It was her eldest sons production at school and she wanted to attend. Will someone be able to look after her other children? "NO" I had screamed in my head, yet I found myself at her house with the two tykes. This time I took some maths with me (and useless company).

The production was meant to last an hour. She came back at 9pm - taking the biscuit - and didn't even give a damn that her son had been bawling his eyes out. I am no baby expert, and later I was told that he was hungry! How the heck am I supposed to know this? For half an hour I had been holding him because he had been crying. As soon as I sat down he began bawling again. It was absurd. I was starving, and so were her children and she had gone shopping (which takes how long?) That is called taking advantage and once again taking the biscuit. (Shockingly the crying baby had eventually fallen asleep).

There you have my reason as to why people with children sometimes make me angry. She has a responsibility to her children at home too. My cousin is annoying if I may say so myself, and has a weird take on things. As discussions go, we had talked about university. She seemed to be of the opinion, "I am going to let my boys do whatever makes them happy." In exasperation I was screaming: "But you can't say that!" GAH. Sometimes it is pointless to argue with some people.

After I came home that night something happened which resulted in me going to sleep without any food. More of that in a minute but my early new years resolution is: I am NEVER going to babysit for her again. I don't want her appreciation and thanks, but she seems to think that whenever she shouts "babysitter" I will jump. You can't say no to her. One of her kids is a cool dude, but still the past two days have been enough to last me a lifetime. I didn't mind helping her, but as I said she abused the fact that we had an agreement. (That is an empty resolution for I can already see it breaking).

Now to the killer question - why haven't I posted for the past few day? Understandably Tuesday was a busy day, but I was meant to post my timetable on that night. I came home to learn that K9 has been installed (I refuse to link this but google k9 and scroll to the fourth object). It wasn't installed for me obviously, but for the younger members. However since no one knows about my blog this put me in a difficult situation indeed. I couldn't breathe fire at this revelation, for due to the current climate it was needed. But this has seriously limited everything I can do! I didn't know how it worked on Tuesday, and thought it logged every page that was accessed using the router; hence why I slept without eating on Tuesday (in anger)!

Today though I learn that you have to install it on the computer for it to monitor everything (please tell me this is so!) If not... then troubles a brewing. Due to this I don't know how often I will be blogging now, but I am angry because of the cause of this. Noddy couldn't understand why I was really angry, and I couldn't exactly say "I have a blog fool". If it wasn't for this blog I would be OK. Maybe it is a sign that I should ease posting for a while (pffft!) Indeed if I have to post at this time of the morning, a decrease in posting might just be a natural consequence of my current circumstances!

I did say "days of my life" and so I might as well mention today (i.e. Wednesday since I haven't slept yet).

Due to my forgetfulness things went wrong today. I hate the fact that whenever I make a mistake, someone has to point out that I study maths. What has that got to do with anything? Today was a day of realisations. I have no idea where I am going in life, but I know that I am going in the wrong direction. I tried correcting this today but failed. The sad thing in life is that no matter how much you try to make certain people happy, they won't ever be happy. You will break a leg in the process maybe, but still something will still not be right. Why bother then, with trying to make people happy, or a particular person? Emotions are something which I have a limited amount off. They are of a cyclic nature. I feel happy, sad, enthusiastic, drained and angry. Any other word like empty are just different words describing the same thing. Maybe I say this now because my current state can be described as wooden and empty?

I don't really know what I am rambling about, but a frown is safely nesting on my forehead. I read somewhere recently that when studying maths you should keep three problems in your head: A mega hard one, a medium one and an easy one. You should always try to relate anything new you read etc. to these three problems in your head. My future aspirations now have similar categories.

The impossible but most desired aspiration is what I sometimes discuss with PS - the condition on when he will stop reminding me about the "no order"(!) incident. I add the word impossible to this desire with a heavy heart. Life it seems has become full of sacrifices. This is another one of them which I might have to make, and maybe I should plan on making this sacrifice rather than believing it to be possible. That goes against my nature because still in my heart, I hope that I am able to do this impossible of tasks.

The medium one is connected to the impossible one but on a smaller scale, yet I will label it as impossible. Let us say that this is like being given an endless supply of tea for a year. (Whereas my impossible aspiration is tea for three years and more!) The trivial aspiration remaining is to pass my degree with a decent grade. This one is possible which is why it doesn't hurt my brain as much as the other two.

Are you convinced that I am crazy? Normality, as I have often said, is something which I think about and disregard. Namely because then I wouldn't be my crazy self. However now I want peace. I want everything which is causing my head to hurt to stop. I want everyone to stop having weird expectations of me, so when I can't meet them it doesn't matter. My bubble is slowly popping. Where did I go wrong? I have certain expectations of others, but why is it not a big deal when they let me down? Maybe everyone was having a bad day today? (Although as I said, I deserve most of the blame for forgetting two things).

Negativity is everywhere. Today was just horrible. Farmer they say, with plenty of donkeys. That is what I will be doing in a few years time. The first time I found this funny. The nth time I don't. It is important that I look outside the box. I wish that I had something physical to do at this moment in time, or something to distract myself from my depressing thoughts. I don't feel 19. Heck I don't feel seven today! I hate people who keep on reminding me about what I can't do considering my age.

I wonder what it would be like if I swapped places with someone for a day?

The volcano inside me is wanting to erupt. It needs to erupt rapidly rather than these slow bursts. But how does a volcano erupt without hurting anyone? I better get to sleep now (it's 4:45am) for I will be behind a wheel tomorrow. That is another thing which I have been doing this year, but not mentioning for childish reasons. Anyway, I will be talking about this henceforth for it has been something fun (and dangerous!) but worth mentioning. Playing computer games does not mean you can drive a real car. There I said it, but I have a cool instructor so maybe the damage won't be too bad tomorrow. (It's in the morning as well!)

Is a conclusion needed? Well the only thing that I can say is that I need to sort my head out so I can revise. My head doesn't need "sorting" in that way, but I need to process the three criteria for my aspirations. Indeed one of them should get me to start revising when processed, and the other two... well life is never going to be easy. My priorities are all over the place. Oh a final thing: I need to EAT!

Exam timetable - the actual one.

The actual exam timetable is as follows:
You might need to click on that to enlarge. Note to any Maths students on my course: do NOT assume that your room location is the same as mine. I have noticed that a number of exams are spread over different rooms. (Although you would be correct to assume that the date and time are the same!) Please check your student portal for your individual timetable, which will have your seat number too.

We received this table on Tuesday, much to my annoyance. However let us now compare and contrast the actual timetable, to my dream one!

Once again my dream one was:

Mon 14th Jan - STATS!
Fri 18th Jan - Algebra
Mon 21st Jan- Real & Complex Analysis
Fri 25th Jan - PDEs and Vector Calc.

And the real one can be written as:

17th Jan -PDE and Vector Calc
21st Jan - Algebraic Structures 1
22nd Jan - Real and Complex analysis
24th Jan- Probability and Statistics 2

It is quite bizarre how the order is similar! And hey, I "dreamt" that we had an exam on the 21st. It is not the worst of timetables (apart from the second week!) but what has been worser is my non-existent revision. It saddens me to say that, for by now I should have been steam rolling through stats. Maybe I needed to take a few days off, but how many times do I have to say that before it loses effect? The cogs are not in the right position and much energy is required to push them into place. Tomorrow I mean today is another wasted day, but Friday will be the start of an intense period. Whoops did I just say something ridiculous again? Let us just wait and see what happens. I have no one else but myself (and the rest of the world!) to blame when I mess up next month.

I was meant to do stats today but multiple integration popped up, so I went for a long break which lasted throughout the whole day. I hate them and what's worse is that triple integrals pop up in vector calculus! What joy. (I am revising that in the week before that exam, and hoping to start with stats). Why do I not seem to care as much anymore? I care about doing badly, but something is wrong with my attitude.

Monday, December 17, 2007

Exam timetable - the dream!

Mon 14th Jan - STATS!
Fri 18th Jan - Algebra
Mon 21st Jan- Real & Complex Analysis
Fri 25th Jan - PDEs and Vector Calc.

I looked up the dates just now, but the first exam was Stats on the first Monday, and PDEs on the last Friday! How convenient that there was a nice gap between each exam, and the exams were situated on the extreme days of the week.

The weird thing is that in my dream as I was accessing my student net, someone walked through the door. (Someone I know of course). As I woke up this morning and checked student net, I heard the doorbell. Bizarrely the person at the door was the person who I had just seen in my dream!!! I was most shocked to hear his voice, and before anyone says I was still dreaming, I have just got confirmation that it was indeed him. I don't mind the exam timetable from my dream if any one is wondering...

Sigh. Well I should be getting the real version this afternoon. I can't seem to revise until I have the timetable, and so I am taking my sweet time locating and sorting my notes!

Sunday, December 16, 2007

Peer Assisted Study Sessions (PASS)

I have decided that rather than having an exponentially long post about my doings this semester, I will break things into smaller chunks!

My reason for not mentioning PASS before hand was rather silly. I thought that from about 300 students, only 30 odd probably volunteer to be PASS leaders which somewhat narrows the range. Anyway, that is all in the past now. The advice "to hell with what everyone thinks" seems to be settling in. Let us not get distracted though!

So what is PASS? Well it is a scheme by The University of Manchester's maths department:

To support the first year student experience through collaborative exploratory discussion.
Enhance the learning experience and Personal Development of PASS leaders as well as for the first years
Improve academic performance and achievement and increase retention.

I will continue to "copy" the handout that was given to me (during PASS training sessions) for another two seconds:

"PASS is voluntary and intended to offer a safe, friendly place to help students to adjust to university life; improve study habits; enhance understanding of the subject matter etc."

Last year I volunteered (with the Tweenies) to become PASS mentors/leaders. Before term started we had to attend certain "training sessions" which gave us information about the scheme, and guidance on how to facilitate. It was rather interesting, for it gives you an insight on what not to do. For example, when a lecturer asks a question to the class and is met with a silence, what then should he do? He could say the answer, but rather he should remain silent for a few more seconds.

During this silence the students are first processing what has been asked; trying to formulate a response to the question; debating whether it is a ridiculous response or whether it is correct and then finally they give a reply. So they need that silence. If though after sufficient time has passed and there is still no reply, the teacher could try and rephrase the question or present a fact (etc).

As I was told this, my thoughts went back to my first year supervisions with my PT. He was rather brilliant, and I recall him doing the above. He used to ask a question and then look around the class. Most times out of ten I just looked at my paper, and "doodled" pretending to look busy! (Well I did sometimes think about the question but in the first six weeks of term my responses used to be ridiculous). Anyway, a long silence used to exist during which each of us either looked at one another and shrugged our shoulders, or glanced absently at our PT. If no one responded (i.e. Alan wasn't present!) my PT used to ask a different question and wait again. Or he would say the same thing differently, which most times resulted in someone going "Ah".

For someone like me, I felt this would be a difficult thing to do!! But when I mentioned the above to my PT he told me it comes with practise, and you shouldn't panic that no one is answering your question. Other things were mentioned in the training sessions, but lets not mention them here.

Our first meeting with our PASS group was when we took them on a tour of the campus in the first week of term. Although I haven't claimed the money yet, we were given £20 for this! (I can't find my NI number - but cash in hand would have been much blimmin' nicer and less of a hassle). This gave us a chance to informally meet each other, and for them to realise that I was slightly mad. (One person commented on how I managed to keep quiet during lectures, as we discussed talking in lectures. He was most amazed...) This group consisted of eight students and I decided to play a game to help remember names. (The key to this game was repetition and soon after I knew most of their names!)

This scheme is great in the sense it gives first years a chance to buddy up with second years. Not all first years might want this, but some really like it. The key word to PASS is that it is voluntary, and it is meant to be more of a discussion between a few friends.

Our first session was a full house, and so were the first few sessions. However, as the weeks went by the number of people attending rapidly decreased. On average 3.72 people attended! (Well it went from 8 to 7 to 6 .... and then went back up to 4 and hung about there for a while). This is slightly disappointing, but I can't complain. Our sessions were scheduled for Mondays at 10am - straight after our stats lecture, and very convenient for us. The first years started later on in the day, but on the first ever meeting we had all agreed on that time. After a number of weeks had gone by and attendance had started to decrease, it was suggested that the time be changed. Monday was the only convenient time for all of us, so we decided to move it forward by one hour. This did result in more people attending too, but I feel we could still have had it later.

I am disappointed at a few things, but overall I felt that the PASS sessions went well. They were very relaxed, and generally we ended on a positive note. Fizz, Bella and myself seemed to complement each others mathematical abilities too. Fizz and Bella are the stats and set theory gurus, so during them questions I used to join in with the questions! (Or make sure that proper sentences were written). I wasn't really a guru at anything, and probably didn't do much apart from talk! There was a quiet member of the group, and she probably understood something I had said to her. For on occasions she used to email me about stuff and so I used to help her during the sessions. The good thing was that quite a few of the times, the first years helped each other.

My policy on helping has probably been adapted from my lecturers and supervisors. I know how brilliant I found it, when DC somehow got me to answer questions in the sequence and series example class without telling me the answer. He used to either give me other similar examples or ask me about a completely unrelated thing, and then link it. I would have probably disliked it very much if he had just given me the answers; for by asking me questions, inevitably he was helping me ask myself the same questions in the future. I tried to do this during the sessions, but not as successfully as DC and my PT! I wanted the students to write, rather than me writing but sometimes that is not possible. PASS isn't about us giving the answers to the first year. We didn't even know the answers ourselves! Rather it is about us helping them to find the answers, by possibly asking them the right questions and the odd sentence.

I was very particular about this, and I don't think we ever used the solution sheets during the sessions. (Apart from the very first one when someone gave them to us!) PASS leaders are not expected to know the answers, but still I felt it unfair that we glance at the answer right under their noses. That is why maybe we used to feel a sense of accomplishment at the end of our sessions, because we had all together resolved the issues at hand.

A moment of "delight" that I can recall is helping the quiet student with the Euclidean algorithm backwards (and linear congruences). I struggled quite a bit with this topic myself you see, which is probably why I still remembered it. The student couldn't follow an example in her notes and I too confessed, that I couldn't see where the example in the notes followed from (although later on the lecturer told me that he had spotted the solution). As with standard notation in algebra, you have to be sufficiently experienced to spot things. Hence why I suggested that we start from scratch and do things the "long and nicer" way. In the end we got the same answer as the lecture notes which was a relief, but more importantly the student was then able to answer other similar questions.

I asked Bella for feedback (on PASS), but she gave me a diplomatic reply. My disappointment is due to the things which we didn't do and should have. An example: when you pull a muscle, sometimes you think you will walk it off. Or if you carry on like normal it will become better. Sometimes this works and other times it is rather painful but eventually works. I sort of neglected my duty to the PASS group in the same way you neglect to take care of your muscle. I think because the others didn't see the problem, I chose to ignore it to. The problem was namely the time, and even though we knew it was not great we ignored the problem and didn't consult with the group. Ah well, hopefully next semester I will be more on top of things.

This post seems to be everything about PASS which I suppressed during the semester!

During my first year I only attended one PASS session, but my point is that some students know what works for them. I already knew a few second year students, especially one who was always positive and encouraging (and still is as a matter of fact!); so I wasn't really looking for "mentors" to help me adapt to university life. I was mostly interested in whether or not the sessions would help my work, and maybe I was rash when after my first session I concluded otherwise. (I don't think the sessions coincide with my learning style. I have previously said that in my first six weeks I was seriously considering transferring onto another course, and PASS wouldn't have helped me. My motivation returned due to a lecturer, who I am greatly indebted too.)

Just because I didn't attend PASS doesn't mean its not a good thing. I just sort of know how I learn and groups don't really fit into that. That is why I can never actually work in example classes - I have to do the questions before hand. Always - and then get help with any problems I have.

Maybe I will be a "better" PASS person next semester, because the course material for the second semester was much nicer (i.e. no stats). And hopefully I won't brush any problems under the rug. Overall I will say it was good PASS first semester. I made a few first year friends, and one of them is even reading Marcus Du Sautoy's book a the moment!

I think that is all about PASS then. It wasn't too taxing - just an hour a week. But I hope that this can ease the tension I am having, about not doing anything during the year. Actually all this is just disguising the fact that I haven't done much maths. [Maybe I could mention more but I am hungry again. :( Next semester though, I will aim to talk about PASS sessions as they happen.]

PS: Note to the person who reserved a book which I have to return tomorrow - there are plenty more Kill Bill scenes in my head!! This is most annoying but I might actually make it less annoying by meeting a good friend from school. On Tuesday, after my dentist appointment, I have to accompany someone to the hospital. I thought this was meant to be my holiday? Oh, and Beans is back in town!

Did I say I was hungry?

My weekends are becoming very predictable. I drink lots of tea and don't eat much, and then stay awake till some absurd time. The bonus is (if all goes to plan), that I will not be woken up at some random early time. Yep, I am planning sleeping past midday. That isn't what I normally tend to do, but the idea is to get 7+hours of sleep and prepare myself for Monday. This is not a holiday for me, for this semester has been enough of a holiday in itself. Now it is crunch time, and I have to iron out the creases in my attitude.

I realise that I have gone on a lot about what I haven't been doing this semester, which has been rather silly. I haven't done certain things because I was doing other things! It's not like I did nothing... So tomorrow, maybe to brighten the mood around here I will talk about what I have accomplished this semester. Actually I mean, sometime after I have woken up today! Because of the silly issue I had during the year (of being "identified") I didn't mention certain things. But why spoil the surprise now, they will be mentioned in during the day.

A couple of hours ago, after yet another cup of tea, I decided to play Need for Speed. I think the best game of that series had to be Most Wanted. The police pursuits had been brilliant!! This one is rather... well, it doesn't have the feel of being a racing game. I enjoy playing games but games like this (tedious, one could say) are finished for completeness. I can't not finish it.

So as I was playing, my stomach was rumbling in the usual manner. Today though seemed to be a rare lucky day for me. Beautiful isn't it?
Well there were three of us, and none of us could separate the damn thing. (I confess to actually not wanting to disturb the shape!) Eventually, with some perseverance it was separated and we split it evenly into three. (I actually got more, because Agnes got sick after eating a few pieces and they then went into my stomach). Because it had been ages since I have posted some images, (due to the silly camera needing a different USB wire), in my excitement at seeing food I took a few photos.
You would think that I had never seen food in my life! My laziness (one could say) is going to prove disastrous one day. I normally never stop eating, so since I am full of excuses today (and most days!) I will blame everything on exam stress. That makes me feel much better, for it means that this is a temporary thing and soon I will return to my saturated self. (Well actually one reason why I am always eating is because this saturation isn't a very easy thing to obtain in my case.)

Looking at them pictures makes me feel hungry all over again, and reminds me of my dentist appointment on Tuesday (which might need postponing). By the way, no one had previously (i.e. last year!) answered my question about which word in maths contains a word related to dentists. To those who have still been thinking about this horrible word, you have 24 hours! I think 6 odd months has been enough, don't you agree?

I have been trying to organise my files (before revision commences) and my notes are in a sad state indeed. I am going to have a proper room clean tomorrow (damn- today!) to find the stray ones. That means I am not panicking just yet. (Hopefully I can get Milo to scan me the ones I can't locate).

My notes show a certain biasedness towards my most liked lectures. I have rough paper and nice paper. The rough one is for the example sheets and the nice one for my cool notes. For my numerical analysis lectures I have been using the rough paper - muhahaha. That is my way of getting back at numerical analysis. And even better, I have used my dodgy pen when writing the notes and not annotated using pencil. I write my notes so that I never have to copy them up again, but on this occasion these notes look like rough scribblings of a madman! And no, I am not going to copy them in neat (after locating two of them that is!)

I have been very badly behaved, and since I have found the camera phone I will try to take a picture of a particular numerical analysis' lecture notes! Steve won't be too happy though... I forgot the begin\{array} command again! Surprisingly my stats notes have been written on nice paper and using the nice pen. Maybe I felt that was the only way I could resist crunching them up and damaging them? Actually the stats notes (that I have located!) are fairly structured - I just can't understand them though. (No surprises there).

It is so much easier to post when one doesn't have the stress of "I shouldn't be sat here." And it is even easier when one should be sleeping!! 5pm + 7 looks OK, but 5 +9 would be even nicer. How time flies. I guess it is about time I hit the sack. (I am in a buzz light year mood: too infinity and beyond.)

Saturday, December 15, 2007

Damn Student net!

We were meant to get our exam timetables on Friday, and now they tell us that we will be getting them on Monday. On Monday it will be Tuesday and Tuesday it will be Wednesday (and so on). Maybe they might forget to give us the timetables and we will be excused from sitting any exams?

This is most annoying because an exam timetable allows one to structure their revision. Or at least start their revision!! I am hoping like mental that the stats exam is right at the end after a weeks gap (like last year). My gut though, is telling me that the first exam will be the stats one... eugh. The stats books in my room are making me feel queasy. [On the topic of timetables, we got our provisional timetable for the second semester. Never mind three hours in a row, I will be having four hours in a row on Thursdays and then another hour after a break! I hope that gets changed, but I don't mind the no 9am start on a Monday (I can accept it on a Thursday and Friday though).]

A random thing: digestion amazingly starts when one sees food and their mouth waters. During this process, in the saliva enzymes are secreted. Isn't that just fantastic? And food travels down your oesophagus in 5 seconds! Then in the stomach gastric juices get to work on it and churn your food into something called chyme (I think). Because of stomach acids being very acidic(!), your stomach has to produce layers of mucus to protect it. Then the rest happens- (small intestine, absorption, large intestine etc.)

Isn't that just amazingly amazing? I wonder, should I do the module called maths in biology (or something like that)? I honestly have a fascination with the way the human body works, and confess to watching a programme about it the other week. Gruesome programme was how others termed it, but they were just squeamish when they were showing the vocal cords. It is basically a string! So when people talk or sing, muscles in the back of their mouths control the "plucking" of this string (like a guitar string). Before I go on about the lungs, brain nose etc I will shut up. (But if anyone wants to know about the ears do let me know...please!)

I think I mentioned the stomach (rather randomly) because I am hungry. What can I say, the post before was exhausting! (I can't help myself: the stomach is the size of a fist, and can expand rather remarkably but for that it needs food!) Damn me now too - forget Biology and concentrate on maths fool. (talking to myself again).

End of semester depression

Is it normal to feel slightly depressed, that the semester has ended?

The second semester has gone really fast. Too fast for my liking, for I haven't really appreciated it in the mathematical sense. Yesterday, at 4:45pm I could still be found in the maths building. (Actually I had a reason to be there, but Dr. E remarked on whether or not I have a home to go to!)

I was actually waiting for Prof. D, and when I finally found him, I mentioned to him about my reluctance to leave and go home. This was most weird, because during the day I had been rather excited about it being the last day of term. Anyway, I feel "less depress" today but still I would rather term hadn't ended. Why? Well then that would mean that exams are not looming around the corner!

Yes - my depression was closely linked to the fact that exams are looming, and they are going to be one of my most difficult ones (namely because of stats and numerical analysis). This year counts towards my final degree, and I think 40% of my second year contributes to the overall final award. [I have another non-trivial but important reason as to why I want to do well] I did say I was going to start revision today (HA HA) but my brain and body have been recovering from the dynamics of this week.

When I got up this morning, my muscles were all clenched and rough, and some are still rather sore. I had done a lot of running around on Friday, for different reasons; and on Thursday carrying the 13 horrible stats and Numerical analysis books home had been excruciatingly painful. (Not just physically painful, but mentally too!) On the bus it had been most awkward for there hadn't been anywhere to sit. And when a seat did become available, my books had it. The problem with the books was that each had an element that I needed, and I am a sucker when it comes to maths books. {I bought a book last Sunday, and the damn fool has still not sent it. I am getting most vexed out by this, for it has been a while since I last bought a book.}

Friday wasn't really an intense day (like normal). We just had revision lectures and proved the Fundamental Theorem of Algebra in complex analysis (which I unfortunately didn't follow at all, but more about that in a minute).

In the Numerical analysis lecture we had discussed last years past paper. It is very important to write definitions down is what I learnt from the lecture. That is all. Write everything down but try to write the right thing! The exam format is as follows: We were taught the course by two lecturers - the first half being vector calc. and first order pdes, and the second being the numerical analysis and second order pdes. There will be three questions for each half, and we have to answer at least two from both sections. They have given us a choice for the final question - how very nice of them!

Now the lecturer (I think) advised students to do the easy questions first. My advice though is the opposite! Do the stinking hard questions first - do them straight away whilst your brain is slightly fresh. I say this with confidence and this is what Professor Abrahams told us last year. His advice proved to be the difference in my final result for mechanics last year. My friends had all come out of that exam, not having completed at least one question (apart from what you can't do). On the other hand, due to Prof. A's advice, I had attempted everything that I could.

There is a period in the exam when you have a slight dip. This happens all the time, when you get bored (maybe) and start looking around. You fidget with your pens for a while, stretch your legs and basically annoy everyone. Your concentration is the lowest at this point and there is no sense in doing the difficult question at this point. Leave the easy or do-able questions for this dip, for then your brain won't need as much energy and should be on auto pilot. (Of course some exams don't have "do-able" questions - in that case my friends, we are doomed! I.e. the real analysis exam!)

So when the lecturer was giving us that advice, to my friends I was screaming NO! Maybe do a slight easy one first, to get you rolling. But if you leave the hardest till the end, I am sure that you will be struggling to complete it. Prof. A's advice is what made me pass mechanics, and so I hope it helps me in other exams. Or does anyone have any other tips of techniques for exams?

After the lecture I headed to the AT building for some "unfinished business". I did something... naughty though. I was on the second floor and saw PS walking, so I proceeded to walk behind him. I think he was lost in thought, for he didn't sense my presence! So I carried on slowly tiptoeing behind him, until he got to his office. Even then, as he turned his key, he hadn't sensed me, so at the opportune moment I shouted "BOO". Haha, OK that is childish, but I did pretty well to not have been detected! [PS's is OK though, if anyone was concerned.] I have done the same to DC too (i.e. shouting BOO!) even though he claims that I wouldn't be able to walk behind him like that without being sensed. Pfft. I am as swift as an elf(?) - well I can't think of anything else.

I enjoy "giving" if that makes sense. My Dad always told us that don't give people an opportunity to ask you for help. I try to do this most times, but I enjoy showing appreciation towards something I found great and enjoyed. It is very easy to be critical. Where am I going with this? I don't know to be honest. One finds it hard to explain this feeling of gratitude that one sometimes wishes to express! I do so in the most trivial of ways, but find it hard to give explanation. I guess it is just a way of saying thank you!

Whoops, that aside was just that - an aside!

After my Monsters inc moment, I went to see my vector calculus lecture to enquire about the example class (which I hadn't attended). The lecturer had arranged for an extra two example classes, but on Thursday. Due to my adventures in the UMIST campus I had missed the previous one and the questions done had been important (I was told). They won't be going on-line, which had been my question. Humbug.

Then it was the complex analysis lecturers turn who I asked for a book recommendation. I made a real big blunder too.
"There is this book by a guy called Priestley, but is it any good?"
"It's a woman - Hillary Priestly!"
"But I thought it was Maurice Priestly! I know there is a guy called that - honestly!"
"Yes, he's at Manchester but he hasn't wrote that book. It was a lady."

Dang. Egg on my face indeed!

During the revision lecture for Algebra, I could be found banging my head against the table. Why? Well it was a revision lecture and we were going through the past paper. A question asked for the order of an element from S_{something!}. The answer was 3 which I said correctly (wow that makes two in a week). However later on there was another question on permutations, and this time one was required to write the cycles as disjoint cycles, and then proceed. PS said: "Whatever you do, don't write that this has no order!" I don't think he's going to ever let me forget that, and indeed after the lecture he put a condition on when he will stop saying it! (I.e. never!) No order does sound very dumb though. It is either infinite order or "a order". The positive out of this is that I (hopefully) will never write that dumb thing again. No order - unbelievable.

We had tea (WLOG) after the lecture, and I have decided not to have tea anymore. It depends on how needy I am, but I can always have it after half an hour. Brilliant plan wouldn't you agree? (You see otherwise, due to the condition attached on this tea business, I will always be hoping that there is no change in the till! ) PS is one of the coolest lecturers at Manchester, and sorry first years but you will not have him now unless you do Galois Theory in your 3rd or 4th year! (He's not teaching linear algebra next semester.)

Tea time over, I had some lunch and killed time until the meeting at 2pm. I didn't know where the room was, so I waited for DC so I would not get lost. He however had something else to do so I went to the first floor and stood looking lost. Thankfully though, I spotted RWT and she saw my needy face and allowed me to follow her to the meeting room! It was a Zzzzz type of meeting. Well I haven't got any printing credit and so hadn't printed all the documents out. Even if I did have credits I wouldn't have printed anything out, for I shouldn't be required to do so. The person next to me allowed me to share his papers (cheers), but since we were sharing I couldn't zone out and just read them... I mean I couldn't skim through the rest of the document whilst obviously listening. [I was awake during the first hour or so].

Now here is the silly part. I had a lecture at three and the meeting was going to finish at 4pm. I was going to leave at 3pm, for why miss my first ever lecture on the last day of term? During the meeting I had been keeping an eye on my neighbours watch. It was a nice big one too. But during the meeting he had obviously shifted his position (as we all had) and so I had lost sight of the watch. Hence my thoughts on what time it was had vanished.

It was a weird meeting for I was the only undergraduate there (the other one hadn't turned up!) When I had first said something (amongst all these adults) my voice had quivered slightly and I know that I spoke extra fast. But after that little hiccough, my heart had relaxed and so I became physically relaxed too. (The Head of School had been there too and other people who I didn't know!) Fern had been sat on the side opposite me, and we had caught each others eyes at some stage- hers rolling, which resulted in us both stifling our laughter. (The meeting did seem to drag). Anyway, it was when I was saying something that my neighbour shifter his position again and I caught sight of the time. Damn- it was 3:10pm and I was late. Without continuing my discussion, I excused myself and rushed to my lecture.

Naturally I was late, but the person in front of me had marvellous hand writing so I copied what I missed from his paper without him knowing. Well until I prodded his arm and asked him whether he had written n! or m!. This lecture ended early too, after we had finished the proof of the theorem.

Due to the meeting just before, I had been in a discussion mood. Milo and Fizz asked me about the meeting and we talked about one element of it, and my personal view on this matter. This led to a heated debate and unfortunately I was in an argumentative mood. Normally if someone says something and I disagree, I let it pass or just state my view and shut up. On this day though I was in the funny debating moods and met every point with another point. I was not being very sensitive towards Milo's viewpoint, but if I am selling you some chocolate I am not going to agree with you when you say its dodgy! I am going to try and persuade you otherwise and convince you to buy it. Thankfully for us, a third person gave a resolving solution and we changed topic. [I hate it though when I am sometimes expected to not argue my point. I do get hyper, but I can't be expected to monotonously state my view point. But yes, if I know that someone else is unrelenting too, maybe I should back off. In my defence I was fresh with questions etc from the meeting.]

And so ended my "third semester" at university, as we all went our separate ways. Bella confident having done most of her revision. Milo to get ready for work and me to the AT building! The end of this semester reminds me that three more semesters are left, before I leave. :(

I passed on the example class at 4pm for I had to see Prof D and DC. (Note to self: we cannot differentiate at the end points because we can't take limits at end points. Why, because we need to have both sided limits to be equal for the limit to exist!) I also discovered this other thing from DC, which I am going to make sure that I post. It nicely uses my (diminishing) knowledge from linear algebra. DC has a wall of shame in which consists of a "good luck with your new job" card and a letter. There is a particularly good reason for this, connected to the time when DC deliberately didn't clean part of black board in G51, even after we pointed it out to him. He had then walked out boldly saying: "sack me then", and so... what could one do when given this challenge? It is a sad thing indeed, but I am sure he will still be around to fulfill his Galois Group role! (It wasn't my fault for he did the same thing twice). *

Anyone noticed the cyclic nature of this long post, as I am sat with nothing else to do? My conversation with Prof. D was as always fruitful, after which I left with a heavy heart. You would think that something bad had happened, from the way I was dragging my feet! (Which I normally never do).

I think a part of me has "evolved" with regards to blogging anonymously. I haven't posted as much this semester, due to various reasons and my work along increasing exponentially from last year. I may even boldly state that I had it easy last year\{certain things}! There was a silly reason for this dip in posting, but I think I have gotten over that now. I made a few enemies (cue dramatics music) but I think that is more due to a misunderstanding that people have of me. Once which they will probably never correct. I need myself a cup of tea now. With BLUE top milk - hurrah!

*I haven't explicitly confirmed anything, so don't go telling anyone incorrect things!!!

Me Or My Maths