Wednesday, October 31, 2007

Library Fines

I mean local library fines - not the University ones. I had borrowed two books from my local library (maths books of course), one which was called Number Sense (I think!) and the other called "What is Mathematics?". I have been looking to buy the latter, but not yet had any luck; so I was rather pleased when I first came across it in the library.

The rules of the library: loans are for three weeks only, and you can only renew the books online twice. (So in total you can have the books for up to 9 weeks). After this period you have to return the books, but of course if they haven't been reserved you can take them out again. I know why they have these 'security measures' in place, but no one has taken them maths books out for a whole year! A whole year!

It would make life so much easier if we only returned books when someone reserves them. If they wanted to put a cap on the number of renewals, then make it up to a year. I mean, the most you can have a book for is a year, but if someone wants it then you have to return it.

For some reason, whenever I have had books overdue, I have only been charged a pence a day. 1p is the child rate if I remember correctly, but I was happy with this. Naturally sometimes I did forget to return books, but 10 days is only 10p, so it didn't really matter.

Then as the ten days become 20 or 30 days, a letter is sent home asking for the books. WHY, do they not send the letter straight away? (It can be argued whether they should be sending letters at all, but if they do, then why not as soon as the book is over due?) Do they deliberately wait a few weeks so the fines build up?

The books I had, had been renewed twice online. Online renewal means you don't really look at the date you have to return them, and forget that the books belong to the library. I fell into this trap. I have a mass collection of books: some my own, others the university's library and TWO from these belong to my local library. The other day I got a letter asking for them back. I wasn't worried, because if they have been three or four weeks overdue then that is only at least 28pence. It would never be an unreasonable amount of money, and I would just take the books out again.

Today, as I attended to a few matters, I decided to return the library books. Going to the front desk I gave the books to the man named Andy (shall we say). Whilst he did his job, I sheepishly muttered, "I think I have some fines to pay" taking my wallet out. (The 28p's add up to a few pounds at the end of the day). He grimaced and said, "Yeah you do: its £9.75 please!"

I choked back my smile, and put my wallet away. £9.75! How the heck did he manage to come to that total? I didn't know what else to do but glare at the maths books. I had to pay the fine to take any more books out, and I had wanted to take the second book out again. So slipping my hand into my pocket, I fished out my card and was lead to the other counter to pay. I hung my head in shame, and it was then that I noted Andy's name! As I entered my pin number I asked him, "How much are we fined for books being overdue for one day?".

"It's 15p a day".

Where's my 1p a day fine gone to! I slumped my shoulders, as you do in defeat after been caught doing something you shouldn't, and left after thanking Andy. I could have bought the book for that amount of money, was what was going through my head. Be warned fellow library goers - the libraries have increased their fines!

My university library is so much nicer in contrast! A few days before the loans are overdue, we are sent an automatic email reminding us to return the books. Then you proceed to the online renewal service, and then wait another three weeks for the email. You can renew books up to 9 times as well! (I know this since I have been renewing a book for the whole of last year, and they are wanting it back now).

This is not my first time being in 'trouble' with the library. I tend to always forget the date the books are due back, due to the fact that I always renew them online. I could say that I will never ever give a book late to the library, but then again I know that's a lie!

My few errands mean I haven't done any work today. Tomorrow is going to be the same, because I have to attend to a few more matters\{studying}. Therefore I must repeat (with a hint of desperation): Bernard - give me your damn watch!

Monday, October 29, 2007

"How old are you?"

When someone asks you that question, chances are that you might be in trouble. Or maybe you have done something which raises eyebrows, since it is not expected of someone 'your age'. Or you might be 21 and look 18, and the woman is reluctant to sell you alcohol. (I had written cigarettes first, but I think you can buy them at 16). Or maybe it is an 18-rated computer game? Whatever the reason, the question isn't always a welcome one.

I wouldn't say I was in trouble, but I was merely being reminded of this thing or concept known as age. (So I guess I did something which raised eyebrows!) An answer wasn't required in this instance of the question being asked. However, as always an answer I did feel inclined to give:

"Well it depends on which age you are after."

The natural response was a roll of the eyes (I take it a good thing that I am unable to do this damnest of things!) But when has that ever stopped me?

"Mathematically I am one years old. Mentally I am seven mod25. And, well by law I am now nineteen."

In college, as I once told Mr Hu of my mental age, he commented that my mathematical age will probably be more then seven. He gave an optimistic figure (which I will not reproduce!) but nonetheless it was definitely more than one. So why now does it seem to have gone back to one?

You see, my mathematical 'awakening' (so to speak) happened a year ago. Well the base case happened on the coming Monday last year. (I mean week 7 of the year). This was when we had a new lecturer for a course, and slowly my whole outlook and attitude to maths started changing. I began seeing maths differently, and started enjoying my degree. I had the rhythm for all modules\{stats}. That transformation in me at that stage of my life, is why I am sitting here now. But of course that was the base case! After the first semester had ended, I difficulty readjusted to the change in the second; but the wheels had already been set in motion by DC.

So I only see myself as mathematically one years old, because I feel before that my eyes were closed. I know mathematics is like a chain of knowledge. If I may lamely continue to use the 'base case reference', topics are built up from the base case. What I had learnt before my awakening is invaluable, since without that chain I wouldn't have actually got to the stage of wanting to study maths. Nevertheless, the magnitude of amazing maths which I see now, seems to make everything before hand seem insignificant. I will be having a 'Happy Mathematical Day' party on Monday.... depending on how the test goes that is! (You are all of course invited.)

As you can tell, a lot of thought has gone into this post! (Ach - you got me: I jest.)

If only it was true that when you start university you are one, and that you are only allowed to leave (if you want to) when you become a teenager (thirteen). I honestly can't imagine leaving university. That thought scares me. I always feel a certain reluctance to sometimes leave the building at 5pm, but I must! Today was the first time I thought about this, since I wondered what would become of 'the not to be mentioned thing' when I leave. :( I only got that far in my thoughts, for the prospect of not being there was too depressing. Life goes on and you become preoccupied with other things, but until you do become preoccupied you can imagine no such thing. I am having difficulty in setting my sights on a target. If I was to know what to aim for, then I would try my hardest to get there. Do I aim for what is impossible, and then whinge when I miss the target? Or do I aim for the possible, and then always ponder about what might have happened had I gone for the impossible?

The answer is staring me in the face. I could just about spit it out, but then as with all unfortunate scenarios, time runs out. "Pens down." calls the examiner and you feel the paper snatched away from you. You look dejectedly at your stationary, and hope that you have done enough. When I wanted to study maths, that is all I cared for. When I should have been doing chemistry or other homework, I didn't. Now I still want to do maths, but now things are no longer linear. They are piecewise continuous! I read somewhere that no matter what happens to the function (i.e. jumps etc), if it is periodic it can be represented by sines or cosines. This I believe is the motivation behind Fourier series (as I read a page a day!) The difficulty lies in trying to deduce or 'force' the function to be periodic. Not as simple as it sounds I'm afraid...

Now the mod25 was introduced when I read the quote:

The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished.'

However, I can now drop the mod 25 and reduce that to mod "some other number", where it is acceptable for one to act without thinking! Why? Well if I am going to be one mathematically on Monday, then I have a good 20 years remaining of my mathematical life! So maybe everything is not that bad after all? The question to ask is: if at 3 years of age I find myself no longer doing maths, will I ever get to 25? No probability can be stuck to that, which is probabilities way of getting its own back on me. Damn- I should have politely told it to chuck itself out of my window, and never come back!

I have been having a mixed time with statistics today. I need to go over my double integration for a start! My solutions are 'right' but I get them in a different way. This is most annoying, because I would have preferred for the solutions to follow from the definitions etc. But then again, I am probably the only student you will find who had forgotten how to compute E[X]. There seems to be a lot to do because I have honestly forgotten everything I once knew about stats. I also refuse to buy any stats books for my shelf, namely because apart from erm... bringing a grey cloud with them I will never use them again. EVER!

So my mental age is seven at the moment, and that is the way I like it for now. The grey hairs of turning eight are beginning to show, but once we stick a mod something in, all will be well.

And the 19 - well call me a law breaker on this occasion, but it is a no thank you very much from my part! I know why I prefer being seven and what not: certain responsibilities frighten me. At seven I can cope with them, but at 19 they are looming over my head. But I am sure that without this stupid things looming everywhere, I would most definitely not give a damn about whether I was seven or seventeen.

I think back to the wish I made last Thursday. Do people ever wish for realistic things, or do we consciously wish for things that we know we will never achieve? Or do them wishes serve as a comfort, that yeah, there is something that I would love to do. I guess some lucky people achieve their wishes, and it is my hope that someone is able to do the same.

Having told myself to get through five lectures of stats today, I will sadly report that I only managed three and problem sheet one. Not good enough since that means that tomorrow I have to go through seven, which is definitely not going to happen. Things here will return to a degree of normality soon. (Well the long posts are semi-returning).

So next time someone remarks on how old you are, make sure you ask them to specify whether they want your mathematical, mental or 'legal' age!

Saturday, October 27, 2007

How full is full?

Now when you say to someone that you are full, why do they tend to still offer you some more food or double check? I mean if you had a jug and it was full of water, no one would try putting more water in it because it is obviously full. Maybe it is because people can see the jug, whereas they can only ever dream about the 'fullness' of another's stomach. Don't ask me why I am talking about stomachs (apart from mine rumbling!) but the real 'fullness' I wanted to talk about was my week. This week that is! This week has been ridiculously full. Never in my life have I ever experienced this sensation of 'being full'. It has been 9am-5/6pm for the whole week. Today was the only day that I allowed myself a break from the 9am start, but 10am is not that better.

The question that I am asking myself is how 'well' have I juggled this fullness. The first fullness was my lectures obviously, and studies. I only managed to spend time on real analysis, calculus and algebra this week. Once again pde's, complex analysis and stats have lost out. However, it can clearly be seen that I am actually the real loser! I am still behind in everything, but have come to a decision. For the next week I promise not to look at my analysis stuff ... OK, I don't promise that since I know that I can do no such thing! I will just try to ease of the real analysis sounds much better. (I just need to complete the example sheet and section one of the notes, and then move onto section two which is rather non-trivial.)

What about algebra? Well I can't afford to 'not' spend time on this module, but I think since this story has been quite brilliant to follow, I won't have to spend 'as much' time on it. I sort of skipped half of problem sheet two, which will need finishing and the proofs from sheets one and three too. I have been a big dodo (as per usual): to prove that two sets A and B are equal, we have to show that A \subseteq B and B\subseteq A. It is enough to say that I wasn't trying to do such a thing!

Yes, this is my periodic review to see where I am sitting at the moment (apart from the chair). With calculus, if I can do integration and partial differentiation I am half way there. The other half is the more complex Fourier analysis and other coordinate systems. I need to learn the definitions and understand the picture in this module. However, since I have done 1.5 of the problem sheets, I only need to do 1.5 more. (Not as easy as it sounds I am afraid).

With complex analysis I am not sure where I am at. I can do the 'easy stuff', but I am still unable to say the definitions in my own words. This is namely due to the fact that definitions etc in this course are not as rigorous as analysis (well some aren't). Also because some proofs are not given, my understanding is slightly jumpy. However I have a book for that, and should hopefully be able to catch up and reach section 5. I have given myself permission to bang my books against my head, if in future I spend too long pondering over what the lecturer may have wrote on the board. Next time I will stick a question mark next to the section, and ponder over such sections later! You see I have been doing this a lot this year, and it bugs me sometimes. I fell behind in this module due to the OHP (well I need my excuse and that seems perfect). Anyway, all that is required here is for me to understand my notes and be able to do all the problem sheets without problems! (Very unlikely I know...)

All I can say about stats is the amount of hours I have put in is... 5! Ha - you all thought I was going to say zero right? OK, five is nothing to be proud of, since for everyone lecture I should be spending at least one hour of 'private study'. We have definitely had more than five lectures, so someone hasn't been doing what they should!

Finally pde's. I think I might have found another course I dislike as much as stats. Namely because I really don't follow the lectures. I have become used to really writing my own story out whenever I do work, or copy notes. I might have taken this, "Don't start a sentence with 'f: A->B is a function'" thing too seriously, but my notes and solutions have to be such that I can follow them at a later time. No such 'following' exists in my pde notes. We don't even write that many sentences! It's an equation here and another there (both numbered) and then somehow or other we get what we wanted. How very bizarre. I do try to write words everywhere, but this is not a very good story. My friends though feel completely different and like this story. I have honestly tried, but I just can't follow chunks of my own notes. I think they will soon make sense, i.e. the mechanical sense, which is the sad thing. Man I miss DH and DH, or should that have been (DH)^2!

My main reason for disliking certain applied courses is because the lecturers at times, always make a distinction between pure and applied maths. Now someone might say there is a difference and I won't disagree. However, why do people give the impression that applied maths is 'less rigorous' and maybe 'easier' than pure? I mean if I had an award for the most 'authentic' branch of mathematics, I would definitely give it to pure maths. Stats I would never consider! (sorry statisticians), and well applied mathematics doesn't seem to want this award. Reasons such as "this is applied maths" are given for proofs, which shouldn't be the case. A proof is a proof, whether it is stats, pure or applied maths you are talking about. Am I to believe that being an applied mathematician means I can just give sketchy proofs? I seem to be a complaining a lot this year when proofs are not given! That transformation still shocks me, and I bet DC is happy! (Humbug).

I am just picking on the proof thing, but do people do applied maths because they don't like proofs? Mechanics last year was excellent because we were shown what angular momentum is - not told what it is! That is what I want. I want to be shown what the heck pde's do. Funnily enough, the lectures in the first five weeks were meant to be showing us that.

So as you gather, I lot of work has to be put in on my part for that 'showing' to happen, hence the grumbling. I actually welcome my own opportunity to get some pde work done, so hopefully the grumbling will soon decrease. Oh, and it is not my intention to offend any applied mathematician! It is just that it can get a little frustrating when the reason for not doing something, is that this is applied maths.

Is that everything covered? I will summarise my to do list at the end, but now onto 'any other business' i.e. the other degrees of fullness.

I was given some more welcome advice today, "Don't spend too much time on "_ _ _ ". You have your studies too." Clearly I am no longer talking about my studies. I am behind on schedule for this thing, but it may be under control. Although I shouldn't be spending time on this next week, I know a considerable amount of time will be spent doing such a thing. Having said that, my aim should be to try to stop that. Sigh. I guess if ever I needed a time where I was able to exercise self-control it is now. Already I am losing in that department, but the suspense should make this a good game. I need one more name; a template; confirmation and then the flood works... I need the base case to be true, and only then will I be able to try and prove the induction hypothesis. (You know what I mean).

So that is the second degree of fullness, and the most annoying and pleasing one. Annoying because it is a big massive distraction, and pleasing because I am enjoying every minute of it. (But of course I am not enjoying it when I realise how much of a distraction it is, i.e. now!)

And finally, my other two duties. These are not too difficult, but just mean that I have to pencil a few dates in my non-existent diary, and hope to remember them! They fill a small cup of tea, rather than jugs and what not. Can potentially not taste nice, but hopefully I can cope with that. I have just noticed something unfortunate: the word hopefully has been used too many times! I can't think of another word, and can't be bother looking one up, so maybe that tells us something about my current state of mind at the moment. (Although desperate would be a better word).

So how full is full then? I really have no room for anything else, but with the stomach analogy, something always manages to find its way down.

My week has been rather horrible at times. I can put my finger on these times, but I am not sure whether or not they are finite? You see two of these horrible things will be infinitely horrible, and one is finite. The infinitely horrible ones are disjoint.

One is closely connected to a certain bet of some sort. This horribleness is more of a wanting, and the pain associated with it is due to never being able to achieve something. This is a recurring theme, but yesterday I found myself telling someone things I have never told anyone (not in a certain domain). I don't know whether the reason for that is embarrassment, or because I normally tend to accept these things and keep them to myself. Now that is proving difficult, and my acceptance is rebelling. I wanted at least someone to understand my function, and I hope it is understood. The conversation felt natural, and I would probably have never shut up! We talked about lots of things, and one thing that stuck out was how we are sometimes restricted by the actions of others.

This horribleness can potentially disappoint someone, apart from me that is, but there is no chance of anyone not being disappointed. So it is a lose lose situation for me. However, this is when the little 'bet' comes in. If I was to give a probability of such things happening, at the moment it would be zero. Sadly. Six words I hope to be able to here one day are: "For he's a jolly good fellow!" You meet some really cool people in life sometime.

The second infinite horribleness is more of an inevitableness. The difference on this occasion is that I have the power to make it finite. It is a trivial matter indeed, but I have been walking around with a fear hanging on my shoulders. Say that you have an empty jug and a half-full one. If someone was to ask you whether you have filled a jug up, would you be lying if you said no? Technically speaking that is true because you have one that is exactly empty, but ... there lies my problem -we are dealing with technicalities here. An invisibility cloak would be damn useful at time. I think being asked about the water jugs in private might induce a different response, since it is unfair of someone to ask another to respond in public.

And that was me being subtle! The finite horribleness is not really something horrible. It just means that I can't moan about postgraduate students from any example classes here! [nudge nudge wink wink] Not that I have any reason to do so (yet), but you never know about these things!! I am really hating the new building. And I am hating the uniqueness and existence theorems we did in ODEs! I have used them to prove that I am unique, and having pinched myself I deduced that I obviously exist. [This is an entirely different context as you have gathered]. Weirdly, previously it had been global existence and that had been perfect. It is local existence which is a pain in the neck. (But only because of the members not part of Wednesdays community).

The horribleness aside, I think the week was pretty great. Wednesday's lecture was good, Thursdays are always good because of the ultimate lecture i.e. the classic story. And today? Nearly perfect! I even gave a lecturer quite a fright - well it was a harmless 'boo', but in my menacing voice! Muhahah... ahem. Well it was so perfect that I am too tired to write about anything else (much to your relief!) Maybe tomorrow when I try to outline a 'to-do list'. I need me some sleep.

PS: The lecturer only bounced with the function f(x) =sin(pi/x) for a finite amount of time, and so fast that I am sure the bouncing around x=0 wasn't as bad as I am led to believe!

Wednesday, October 24, 2007

The Dame Kathleen Lecture II

Be there or be square. Actually you have to be square now, since you have no chance of being there! It is over now and this was the programme:

5.25: Sudoku, Latin squares, Geometry, and Hall's condition
Professor Peter Cameron - Queen Mary, University of London

Solutions to Sudoku puzzles form a special case of a type of combinatorial designs called gerechte designs, invented in the 1950's for designing experiments in agricultural research. Finding such designs includes such problems as finding a Latin square orthogonal to a given Latin square. Also, the problem of completing such a design from partial information was developed in statistics.

The experience of solving Sudoku puzzles suggests that finding such designs (with some values given) is hard. It is known that completing a partial Latin square is NP-hard. Some similar problems about gerechte designs are unsolved. The problem can be formulated to look like Hall's marriage theorem, but it is not known whether there are conditions analogous to Hall's which are necessary and sufficient for a solution.

Further conditions can be imposed. One variant, due to Robert Connelly, has particularly close connections to several topics in finite geometry: spreads, resolutions, and perfect codes.
And because I found Professor Peter Cameron cool, here is a mini-biography (stolen, I mean borrowed from the maths page!):

Peter Cameron was an undergraduate at the University of Queensland before coming to the UK and obtaining a DPhil from Oxford University in 1971 under the supervision of Peter Neumann. He held a position at Oxford before moving to Queen Mary, University of London, where he is Professor of Mathematics. He is an expert on combinatorics, permutation groups and the structures (for example, designs, graphs, codes and geometries) on which they act. He has authored approximately 250 publications, including several books, and has supervised over 30 doctoral students. In 2003 he was joint winner of the Euler Medal and this year his 60th birthday was marked by an international conference in Ambleside.


He has an Erdos number one too: cool again!

This particular Wednesday I was quite unusually feeling bouncy. (Not in the morning though since I had no breakfast: DAMN HALF TERM!) I laboured through my lectures (apart from the Real Analysis one of course, in which DC improvised quite remarkably! Well he did something different to his notes, but I was always confident that he wouldn't make us cross everything out in case we came to a dodgy conclusion!) I am not getting the material that we are doing now properly, since the bounded business is simply confusing: especially with millions of mini-theorems. But that being said, I was given some sound advice by DC: "Don't spend too much time on this course!" Human nature is such that you tend to spend more time on things that you enjoy, and find fun. Consequently, I have been spending too much time on analysis and algebra. I have spent zero time on stats, and zero amount of time on PDEs too, and a little time on complex and calculus. Then I wonder why in the world the work has been piling up!

That is an honest representation of how I have been working. I just love the analysis and algebra course, because I love the lectures that go with them. There is no question about that. I know DC made a fair point, (as I asked him to have a skim through my proof, and it is not everyday a lecturer tells you to 'study less' for their course!) but it is really difficult for me to sink my teeth into the other stinking courses at the moment. Yeah, that is how I feel about them. Well stats and PDEs especially, they are just confusing me at the moment.

Anyway! After the lectures and some lunch, I had to attend some formal matters. A divergence if I may. Another word that you might possibly want to use to describe me, is energetic or enthusiastic. I would like to think that I am enthusiastic in things that I set my mind to, since that is always a must in my case. Motivation and enthusiasm are directly linked. Because I am motivated in analysis and algebra, I am always found looking forward to them lectures, and go 'Awww' when they are over! However, this trait of mine is also one which many people might find annoying. There are certain degrees to this annoyance, but I make no apologies about being enthusiastic about the things I take part in. As I said before. No matter how unlikely this is, I would be reluctant to employ you if you were not enthusiastic. Actually, before this turns into the whole post, I will pick it up again some other time. A note though: if you find my being annoying, then you do know that you don't have to talk to me? If however I am annoying you for a reason, then please do tell me to stop! I don't get offended by things like that (most times)!

So back to this formal business. It was swings and roundabouts. I seem to be building many responsibilities for myself, and just added another one! So at the moment, on top of my studies (19hrs per week) I have three other responsibilities at university. I think if I manage to bring the studies up to scratch, I will most likely feel better about everything! (Slim chance of that happening though...)

After all this it happened to be 4pm-ish and the the Tweenies united. Well actually they all busied themselves with drinks, whilst I got my badge. However, after I got my badge it was my turn for a drink. From the wide(cough) choices available, I chose.... *drum roll please* TEA! Yes, it was one of the most delightful cups I have had in a while. I could actually accuse someone of 'spiking' my drink, but my jumping around before hand proves me wrong. I was actually quite excited about the lecture in my defence! My friends were rather amused to find me with two milk tubs (which were unfortunately green topped). I then got to work. No dilly dallying with the string: I used the spoon. I have to have my tea made in a certain way, I mixed it and mixed it again until it became the right colour, and then got rid of the tea bag. Then the two sugars went in, and finally the two milk tubs. After the first portion of milk had gone in, Bella exclaimed, "That is enough milk isn't it." But nope, it was still not the right colour, and so in went the second!

I have since had another cup at home, but it just didn't taste quite the same. If you ever have the pleasure (ahem!) of meeting me, and there happens to be a kettle nearby, you will be left wondering 'coffee, what's that!' Or so I hope.

Today I felt no fear. I felt comfortable and was happy at being there. I didn't feel awkward this time round, and didn't feel that I stood out amongst all them mathematicians. This was a welcome feeling. I think having all the Tweenies around was good too, but if I am being completely honest: if I had to go to the lecture on my own, it wouldn't have been a problem. I think it is wrong of me to ask the Tweenies to go in future. Well the reluctant members anyway! Jake and Fizz seemed up for it, but I think Milo and Bella would rather not have been there. I don't want them being there because I asked them to. Well they did have the option to decline... but having them there was better than not having them there! I would rather not 'oblige' them to come, since it is not fair of me to do so. Well maybe next year we will see what happens.

Did I mention how happy I felt? Jake, Fizz and myself had ended up discussing maths and what not, whilst Milo and Bella had found themselves sat at the tables. The hustle and bustle of being there, made me disregard the fact that I was naive undergraduate. Unsurprisingly, on the occasion I wasn't afraid of seeing my lecturers around, I didn't see many! Well I saw them, but didn't manage to speak to any (apart from one or two, who I think just watched as I was unable to stand still and talk at the same time).

There was a talk by someone in the atrium, but I didn't catch much of that. It was about the new building, and I just sighed! I am still sat in the middle about the building you see, and will not be moving for a while. The community feeling (I suppose) was immense. Being there just felt great; I felt like I belonged to something. In this community no weird looks exist, and no one calls you a freak for being there. That is indeed what I love about this community and that of this blog too.

After the five minute talk, we all stood (bounced) in the atrium, talking and having a drink. Eventually we all made our way to one of the larger rooms in the AT building for the lecture. I didn't look for any lecturer to provide cover for me, as I had last year. I think with things like this, there has to obviously be a 'first time' which prepares you for the other times. When everyone had settled, Prof. G invited the speaker and it was here when something changed within me (as I mentioned in the other post). Prof G. mentioned reading Professor Cameron's travel writing. It was in this moment, that yesterday famous mathematicians advice hit home, "To hell with what everyone thinks." echoed in my head. I did go into another cloud for a minute, but I think in the community present, my mathematical passion is not seen as perverse, which is a welcome change.

Onto the lecture now! (Finally I hear.)

Professor Cameron, as has been rightly said about him, is a fantastic lecturer. It was a rather pleasant lecture and very well delivered. My only problem is that had we started at 5ish, we could have gone to 6ish, and the end wouldn't have been slightly rushed. But then again, that want is a selfish want!

The lecture was about how ridiculous newspapers sound when they claim, "Sudoko's need only logic and reasoning. There is no mathematics involved." Quite a ridiculous claim as I am sure you will agree! To be honest, when I wasn't even a mathematician (i.e. college), to persuade my friends to have a go at them, I would say that there is no maths involved, since you can use letters instead - it's all about logic. Maybe that is why I laughed even harder, as I recollected this! Indeed, sudoko consist of or even are Latin squares, so letters would make perfect sense.

The mathematics behind sudoko make them all the more interesting, and magic squares seem to be floating about too. I mean, if I was to go and do a sudoko now I would firstly feel more 'Latin- squred informed', and secondly complete it with more satisfaction because of being slightly aware of the maths. I didn't bother too much which with reading the slides, but instead tried to listen. As soon as I did read a slide I lost track of the lecture (which was rather annoying for me).

In fear of remembering anything incorrectly, I will just try to briefly mention the lecture. I confess that some stuff did go whoosh over my head, and another bad habit of mine resurfaced. Having finally understand numbers in different bases, I became preoccupied 'checking' a square with numbers in base 3 (I think!) I evidently was remembering everything incorrectly because I wasn't getting the same answer. I think it was the square 00 which was said to be equal to 3. Anyway, I am most likely saying incorrect things here since I vaguely remember what it was. (I just remember the 00). If anyone else was at the lecture, please feel free to correct anything incorrect that I write.

However, I had a 'get in there' moment, when the word permutations was mentioned. Now, composition and decomposition of permutations is one thing that I can hopefully do! The word derangement was also mentioned, and it was one of the many words added to my mathematical vocabulary. Other words include: cosets, affine plane, F^4, quasigroups and many more which I will remember later, and hope to actually make sense of. I had the infamous shrek face when them words were mentioned, but I let it be.

It is remarkable how Professor Cameron seemed to be very well statistically informed! He built the story up from the very beginning, and there was mention of Euler and his conjecture too. I particularly liked the phrase Euler Spoilers, which described the people who spoiled Euler's conjecture! (Euler was only correct in his conjectures for n=6(?). ) I can't recall his conjecture I'm afraid. This post is rather disjoint, since different things are coming to me at different stages.

Magic squares are squares where all the rows, columns and diagonals all sum to the same number. Latin squares are sudoko puzzles: well if we take a 3x3 sudoko puzzle then its properties are such that you can only place the number one, once in the first row. It also must appear only once in the second and third rows, but not in the same colums as the other ones! An example would be:
1 2 3
2 3 1
3 1 2
The latin square is also known as the Latin-Greco square, but I am too lazy to write that every time. After Euler the story continued to how other peoples contributions and ideas. The 'just squares' idea was one I liked. I can't remember the mathematician who said it (behren??) but I found the idea of 'just mathematics' amusing. I remember the diagram with the fertilisers, and the idea about it, but can't write it properly in words. This somehow introduced the critical points/set, and a few other definitions.

I unfortunately missed a whole slide because of Bella, but all is forgiven! It was about the error code thing, and I honestly didn't catch a word. It seems that the application of these squares is rather impressive, and indeed they themselves are. Especially when the symmetry was considered. When we discussed the affine plane, I got slightly confused with all the coordinate business. I understood mod3, but I didn't really follow the coset and x_3=x_2=0 part and about the affine plane. Although admittedly that is a cool name!

As my jumps continue an example was highlighted, and I a theorem (the 'marriage theorem' Hall?) was discussed. I was amazed at it, because when solving Sudoko I have actually done what was done on the board. If in a sudoko (3, 6,7) can be written in three boxes and (2,3,6) in another. Then it is obvious that the 2 has to go in the fourth box, since 3 and 6 have to go into the first three. Next time I am doing a sudoko I will do so with a mathematical smile on my face!

It is best I stop soon, since the jumps are increasing. There was a whole comfortableness associated with the lecture, and I think the lecturer had a hand in that. The atmosphere was pleasant, and we ended at 6:15. Alarm bells were already ringing in my head, but I stayed until everyone left. (Dame Kathleen even said a few words too). I really would have preferred staying a little longer but had to dash. I actually did dash, and got weird looks from two people in the maths department. (A post grad student and the computer officer as they walked towards the maths building). I think what surprised them was the fact that I was running. If you must know in school sporting events I was the sprinter! I am OK when it comes to a quick dash and enjoy it, but I have let myself go when it comes to the long haul. I could do it at a casual pace, but sprinting it is for now.

Having said that, my legs are wooden but I love running. Last time someone said I should have snapped lecturers who nodded off, and indeed today I laughed to myself when I spotted one! It was only one on this occasion, but for just the smallest of time intervals. There were quite a few people with cameras, and thankfully the annoying woman from last time wasn't here. She had honestly never let go of the button, and the blinding flash had been most annoying. (Is it obvious that I have forgotten something?) Well whatever it is, it can wait another day! I really enjoyed being their today, and maybe that community will soon include another subset of people. The lecture was interesting and I have even got a cool name badge! I wonder whether the slides to the lecture will be available?

I have gone against my own restrictions and finished this post at 1:37am. The recurring words in my mind: who cares what others think. Nobody should make you change what you do. (Well that depends obviously, but on this occasion it holds). As long as you are happy, just keep on doing what it is you do. Hopefully more of these types of lectures will happen this year.

Tuesday, October 23, 2007

Maths Town

I heard this song in a movie the other day, and I felt like posting it today.

Frank SinatraThis Town


This town is a lonely town
Not the only town like-a this town
This town is a make-you town
Or a break-you-town and bring-you-down town

This town is a quiet town
Or a riot town like this town
This town is a love-you town
And a shove-you-down and push-you-'round town

[instrumental]

This town is an all-right town
For an uptight town like-a this town
This town, it's a use-you town
An abuse-you town until-you're-down town

This town is a losin' town
It's a miserable town
It's a nowhere town
And I am leavin' this town
You better believe that I'm leavin' this town
Man, it could never be uptown
It's bound to be downtown

[Fade]

This town, bye-bye
Bye, bye, bye, bye, bye-bye


I don't normally post songs or song lyrics, but surprisingly the substitution maths town makes perfect sense. It has a really nice feel to it that song does, and it has been playing in my head all day today. Well actually only one verse has, since that is all I heard of it! Is that a sad song, you decide? I don't think it is "sad", but it has a sad sound to it.

In perfect Lord of The Rings style, "I feel old Frodo". So begin the days when I look back on the holidays, and wonder about ever wishing them to end! Actually, I did want them to end, but I want them to start again now. "I feel old Frodo". But what to do? I have let things get on top of me, and I am losing my focus. I should be focused on my studies: on Hyperbolic geometry. As you have gathered, the steam engine that used to operate my posts has also become old. I know the real reason for this and I am ashamed for that reason.

As a famous mathematician said today, "To hell with what everyone else thinks." (in a manner of speaking!) Shame I am yet unable to adopt this philosophy yet. The advice I was given was actually very sensible, and appreciated. What is there to fear? "If you are going to do something you want, then what does it matter about what others think or say. It is not like you might see them again in a few years." So the advice continued. If I am being honest, I do tend to think like that at most times due to my "nature". Let us use the sandwich rule and let |me| \le weirdness. Then as x tend to 'me', + or -weirdness tends to weirdness, hence I tend to weirdness by the sandwich rule. [I can actually prove that now, and still remember my promise of posting it: such a beautiful theorem!] I know that is a load of mathematical 'bull', as a scouse friend of mine would have said. It has been one of them days (i.e. I did 1/n amount of work, as n \to infinity!)

I think when one cares about what others think, they are restricting their options and things that they can do in life. When will it stop? When will you do something without caring? Why does it matter to us at times, how we may be perceived by others? What am I scared of on this occasion? [Surprisingly that brought a smile to my face!] I swore today on numerous occasions - I was badly behaved. I cursed when the boiling water somehow missed the mug and went over me; when I was too busy trying to see whether my bus home had come, so that I crossed the road without looking... and a car was there! When I forgot to ... and so it continued. Tuesday's are infinitely worser than Mondays.

The bubble has to pop soon, and let us(i.e. me!) hope that it is a painless transaction. I will see you all tomorrow - 4pm sharp. Should be fun, but it is rather inconveniently timed for me... Anyone know of any mathematical life jackets out there? There is a back log on my posts, but last Fridays episode of 'Oscillating Friday's' has a technical difficulty, and will be running as soon as possible! I am reading quite an interesting book at the moment, and will be talking about that soon. Don't hold your breath though, for reading is somewhat difficult for me at the moment. And if anyone dared ask what book(!) then please please don't expect me to answer your question. (It's a maths book by the way!)

Tuesday, October 16, 2007

"The smell of coffee is ..."

Yes, even I am having trouble remembering the positive adjective that was used to describe coffee today! I spoke to DC and PB about tea and coffee today, and it seems that tea is the loser on this occasion; but I will loyally stand by its side defending it! How many people actually like the smell of coffee? What's there to like about it? I am tempted to actually 'taste' coffee, so that my not liking it is not based on its smell. But then again, your nose is one of your five senses. Actually I think DC might have been exaggerating the greatness of coffee, since tea can't actually be that bad!!

I have just made a shocking link. I didn't have any tea this morning, which might explain the paragraph above! Tea tea tea. It's just great. Now the post before, which was posted a moment ago was actually written yesterday. I wrote it as a 'draft email' but never got the opportunity to publish it. This post is about todays madness.

EDIT: I seem to have a back log of posts! This is about Tuesday, and posting this was inferior to my sleep it seems.

Where to begin? Ha - at the start of course! It is up to you to decide, whether or not having no tea is the reason for my ... 'aura' today. I just put it down to over doing it on the treadmill!

In todays real analysis lecture we defined continuity. I liked the definition of continuity because continuous functions are nice functions. Polynomials are continuous, so for some polynomial f(x), its limit as x tends to a is just f(a). See, I told you polynomials are nice. Rational functions are nice too, well if r(x) = p(x)/q(x), that is nice for all x but not when q(x) =0. Judging from todays lecture the proofs to show continuity may be 'nice' too. If we show that lim f(x)= f(a) (as x tends to a), then essentially we are done. NO MESSING AROUND WITH DELTA. Brilliant. I am, truth be told, not having a nice time with delta's at the moment. I know that the examples in lectures tend to be very easy compared to the example sheet questions. I know that the delta will be back, but at least in todays lecture it was hidden.

The problem with delta is, as I said before, we are now looking at an interval. I preferred looking at things as n got bigger and bigger. My problem (one of many!) at the moment, is not letting go of the smaller picture. I need to just let go. Today's lecture was brilliant as usual, and reminded me to go over some first year material (which I have no memory of!) Even though this course is tough, I seem to be enjoying it. As I said to someone, the challenge of trying to understand it is fun!

People were extremely rude in todays vector calculus lecture today. Please get the heck out of the lecture theatre, if all you want to do is talk. I was glad of the change of topic, but if this seven year old can button it, then the adults amongst us shouldn't have problems. What people don't realise is that what is being said is equally important. If the lecturer is just cleaning the board then a whisper is fine, but please have the sense to be quiet when the lecturer is talking. You can always copy the notes of your mates later...

Hmm, that seems to have made my day really short. It wasn't. Obviously 'the must not be named topic' has to have a mention too! A slight concern has come to my attention, and I am feeling nervous already. So nervous that I for a second thought to make it all stop. I still want it all to stop, but for selfish reasons. It won't stop because I won't let it, but I am becoming slightly panicky. After yesterday, I have been having a chicken little feeling. What with so many variables flying around... I think I need to take it easy. Ha - fat chance of that happening. I am somewhat indecisive at times, and now is one of them times. I am not regretting what I am hoping to do, but I feel that my bad habits from my past may start haunting me soon. There is so much that I wish to be able to do, but the damn delta's are having it their way. Come now, I said to take it easy!

I spoke to one of my best friends today. The only one who has known me since primary school! It was a nice blast from the past, and I learnt a few things. Having bumped into Prof. D again today, I have to take on board what he said. I have to catch up from where I have fallen behind from. Organisation is needed from my part and discipline. This week will be slightly hectic book, hopefully the discontinuities will soon end!

Quite strangely I even bumped into Mrs. H from college today! (My Biology teacher who has now retired). Now that was an even better blast from the past. You would think that upon seeing me, she would be running away, but she didn't! She also admitted to me that she does tend to run from other students too. (If you're still frowning, this means that I am not that scary after all!) I had sent an email to her but it had bounced, and my further maths teacher had then told me about the changes that have happened. I told Mrs. H about what I am hoping to do, and she gave me the thumbs up too. (She didn't look to surprised though).

I had written two long paragraphs about something, but I have gone onto delete them. Sometimes being me and being different from everyone can get... something or other. I think some people are more keen to look at the differences which separate us, rather than the similarities that may exist between us. I will always feel a certain disjointness from my fellow second year students (well the majority anyway). I feel that I have lost a 'race' that I entered, which is slightly upsetting. This is my tactic of preparing myself before the bad news hits me, so it is safe to ignore me. I wonder what it was like twenty or so years ago, when people were doing maths degrees, or any degrees then.

Meh, I might as well type todays happenings here. I didn't like Real Analysis today, namely because I am having a torrid time with a certain proof. The lecture was fantastic again, (a bit fast for 8 minutes but then became OK). I know the 'idea' behind what I am doing, but I want to do it my way. The wrong way! Actually I am having a torrid time with most of the proofs, and am getting definitions mixed up which is annoying. I need to spend a good couple of hours on analysis, which I am not managing to do. Somewhere I wish I wasn't that thick on the uptake, but then that is why I love maths. Not because I am thick initially, but the transaction from being thick to being less thick is great! And then I want to do 'further studies. If the likes of me can do such a thing, then there is hope for everyone. This negativity will flow out of my blood soon, but at the moment simple concepts are floating over my head. I am more annoyed because I am not putting the hours in myself. Not with any module.

That was the highlight of today: the real analysis lecture. Even though DC claimed to be feeling weak today, we still wrote a magnificent three pages of notes. I left him to wonder how many pages it would have been, had he not been 'feeling weak'. Right - I will shut up now, and have a nice cup of tea to drown the pile of miseries that have been piling up! As I learnt today: I am not the only one missing the mss building. (I really do hope that I didn't lose the 'race' though. Really badly hope that I didn't.) I need a filtering system to deal with everything at the moment. And yes, I am writing random things now! After the cup of tea, I am going to find myself a nice unbounded set in R.

Monday, October 15, 2007

Natural mumbers

I had gone to Macclesfield on the weekend. Yes - that dreaded place which results in me feeling funny upon returning home. I honestly believe that I have a bad chemical reaction when I go to a certain house. I detest hot/chilli/spicy food with a passion. Namely because I don't have the tongue for it. I think there is this one ingredient which I can't eat. Actually I know so, but since
food is mostly a mixture of ingredients I can't say which. Either I ate this silly thing, or someting was put in the food on the weekend!

I had made plans on going to bed early last night too, but found myself tossing and turning till past 1.30am. When I eventually asleep, I don't know. But I do know that one particular 'enactment' in my head was keeping me awake. I had forgot to reprogramme my 'internal' alarm clock, so woke up at 6am, only to hit the pillow again after much grumbling. Why this inane preamble: well I am already providing an excuse for my expected Monday behaviour!

Indeed with less sleep, my energy levels increase dramatically. Today, as I tried my best to be late for the 9am stats lecture I went into overdrive. I actually did try to be late, and had even gone to Blackwelss first! But alas, the powers that be know how much I love stats, and so wouldn't have it any other way but me turning up on time. Humbug. I think it is an achievement
indeed, that I caught the words correlation at the end. Without proof, I think we can deduce that I am having an awful time with stats already. Why is this one module being such an annoying pain?

Anyway, let us not talk about annoying things! Some time after the stats lecture, I had a certain task to do. Not one that I was keen on, but nevertheless I did it. The task was not physcially demanding, but I think I messed up. I am not 'great' at reading people, but sometimes you can tell when a smile is not genuine. But as I said, maybe I am looking too much into this and actually want a situation that didn't happen to have happened? I got slightly panicky and then botched up, but I think I did what I meant to (albeit in an idiotic way). Tomorrow will tell, but sometimes I hate not knowing how people actually feel about certain things. I mean, if I was to present you with an idea, and you had some reservations, I would prefer knowing them. GAH. As Fizz commented, I may be over-thinking slightly, but I don't want to step on someones toes or 'gatecrash someones party'.

Have I mentioned that Prof. D was great? Well I am saying it again. He was an awesome lecturer, and is a brilliant guy too. No one seems to be keen on the yellow in the new building (apart from DC and me!) but I guess I couldn't argue with Prof. D straight away! He's cool, and I miss his lectures. He even showed us some secret rooms in the new building... oooohhhh! (A bit like Harry Potter I suppose). I managed to speak to quite a few lecturers in the space of
forty-five minutes today *cue evil laugh*! I possibly annoyed 1.5, which is not bad for a Monday. Not bad.

I am hoping that having read up till this point, you might have gone away and hence will miss my embarrasing moment. Come four o clock, it was Algeb. structures. Given my euphoria in the morning, and the bouncing which resulted from this, I naturally went to the lecture with the same attitude. However, for the first time in my life I confess to having a bad algebra lecture! I honestly was shaking throughout and concentrating was difficult. When I did concentrate I was fine, but when I didn't life was tough. Normally if only one or two people are talking, you let it pass and it is easy to ignore. But today I couldn't ignore the talkers. I could hear the conversations and the mute button wasn't working. I wanted to walk out and then walk back in again: ready for action.

Now if that was bad, you might as well cover your eyes for what is coming up next. We were doing the order of the cyclic groups. I remembered from last year that the order of a is a natural mumber m where a^m= e (=the identity). Yep - so had began my mumber war! (I was writing mumber instead of number...)

The problem first began when I started trying to recall previous information. The proof rather vaguely slithered in and out of my mind. I had the wrong one in mind (since it had something to do with the division theorem), but that didn't stop me. After the proof I remembered another little example from last year. My confusion had started from when I had started thinking about the order of a perm, being the lcm of the order of its disjoint cycles. But to know that, you have to know that the order of a cycle is just its length. Give me a wall somebody! As I was getting excited by remembering the lcm thing, the lecture was moving onto Z_6. The order of 2 in Z_6 was asked. I thought, '2 doesn't have a multiplicative inverse in Z_6, so then it can't have an order'. "Infinite order", some fool replied (3 points for which fool!) PS shook his head, and said that Z_6 is finite so ... no, 2 can't have infinite order.

PS knew I had said that answer, so there was nothing left for me but to make a run for it! What would have been more embarrassing: running out embarrassed at the stupid answer, or just running out? I did bury my face into my notes, but I think, or rather I hope that PS knew that I was having a shaky lecture! As redemption I did clear the board at the end, which was fantastic indeed. I mean chalk dust flying everywhere, what could possibly be better? Well that was the second reason for doing that, the first was to say thanks. But you know something, I still really want my own blackboard. I have an A4 sized one (don't laugh), but the idea is to think big.

Boy do I miss the mss building (I will continue saying this for a while). I even said this to Prof. D today too! He gave me some nice advice, and I will see how things go. I am beginning to get worried now: all the Tweenies are way ahead of me in terms of work and understanding. I really need to find a way in which I can work at home. I think this Monday went rather well, wouldn't you agree?

Sunday, October 14, 2007

Complex Analysis I love thee.

Or should I have written that I love my complex analysis book? (Well I don't exactly love Complex Analysis yet you see!) This post is about yesterday, but unsurprisingly I was in no condition to type anything then. I unfortunately had only 4.5 hours of sleep the night before, (due to mysterious reasons indeed!) but still managed to survive all my lectures.

I had got home at sixish again, and after eating had really just crashed at 8pm. I had fallen asleep downstairs and had been unaware of happenings around me. Supposedly I was woken up to go upstairs, and although I am mocked at my response to this: ('Turn the light off'), I have no memory of saying such things!

Now from the information above it can be deduced that my day had been exhausting. I had stupidly on this occasion, decided to say hi to the mss building in the morning, when I had a lecture on the other side of the campus! I remarkably hadn't been late, but that meant a lot of huffing and puffing from my part. I have taken to 'driving' on the pavements, and I really think some sort of system should be introduced. There should be an 'over-taking' lane, where the fast walkers can walk; a slow lane for the slow walkers, and the general lane. I know this is not exactly ideal, but one can always dream. (It would be even cooler if you could only walk in one direction on pavements!) Whilst I am moaning about pavements, I might as well shout at cyclists. YOU ARE MEANT TO CYCLE ON THE ROAD - NOT ON THE PAVEMENT. Oh, and if you're cycling and it is a red light, then you are meant to STOP too.

That 'rant' sounded weird because I am not in a 'ranty' mood.

Anyway, our story starts at nine something on Friday morning. Light reading for me tends to be my course books; and on this morning, due to my disgraceful behaviour of falling behind in complex analysis, I was reading the recommended course book. Obviously I wasn't reading it like a fiction/story book, but I was trying to create a story from it. With any new course you have to familarise yourself with definitions. This year I am not leaving that to the end of term, but hoping to make this an ongoing process. The definitions in complex analysis are slightly 'dodgy' for me, so I found myself trying to say them in my own words. I read the book, talked to myself and then closed the book.

With the book closed, you could find me muttering the remarkable definitions to myself in English, trying to make sense of them. As I was busy talking to myself, I didn't notice who was nearby. A sudden voice woke me up, 'That seems to be an interestingly light read!' I looked around with a jolt. I found myself confirming what was said, and musing on how new courses tend to begin with lots of definitions. The person agreed and once again said something about the book looking interesting. Was I expected to diverge into a paragraph of how true that was the case? I didn't know how interesting the book was, so just mumbled something about complex numbers being rather cool. It seemed that we were both off to institutions of education. The man was a lecturer (if I recall correctly), but not in a mathematical or science course. I can't remember what exactly, but something to do with human....?

In this weird morning, I found myself facing a really pleasant gentleman indeed. He is the first random person that I have ever met, who hasn't become scared of my maths book, or maths! I found myself asking the obvious question about whether or not he was a mathematician. He (un)fortunately(!) replied that he had done a Physics degree in the 1960/70s (I can't remember properly). For a few minutes I forgot where I was, and amicably talked about applied maths etc. The man was now doing a Spanish degree whilst teaching too.

If you have been finding the conversation up till now weird, then prepare yourselves! We started discussing maths problems and then the kind kind kind gentleman, told me of one which had been sent to him some time ago. He said that it had taken him two weeks to solve it and had been quite difficult, and then proceeded to explain the problem. With 4.5 hours of sleep I didn't catch anything that was said, so I asked him if he could write it down for me. Quickly getting some paper from my bag, he jotted the problem down for me. Didn't I say he was a kind kind guy? I mean, how often do you meet random people who give you maths problems!!! He told me to have a go at it, and said that I should be able to solve it. I then hastily mentioned that although I love my maths, he shouldn't put any money on me solving the problem, for I am rather slow at it the same time. (Well I have problems moving off, but once in third gear things become nicer!)

The man then said that my lecturers might not even be able to solve it, but I hastily stopped that train of thought and told him that I have brilliant lecturers. This seemed to have convinced him, and he told me to go my lecturers if I had problems. I haven't had a chance to look at the problem yet, because since I have been home my bag has not been unpacked. It still is sat in the corner, and should be emptied tomorrow. So yes, I am sure that my lecturers will be able to solve it, and I will probably be asking them for assistance. (Also posting it here too).

Now at that moment I was still amazed at my luck, but then things got even more interesting. I told the man that he may regret ever getting me started, since I have a record for not shutting up! He replied that I have got some competition on this occasion, which had me in further shock. This guy liked to talk, and he claimed to be able to out talk me - just great. And so our conversation continued, me forgetting where we were and everyone else around us.

We continued talking about maths, and I told him about some of my lecturers. About how I loved taking notes, blackboards, and how I enjoyed talking to my lecturers (and joking around with them too). I told him a lot of other things as well, like wanting to see things pictorially at times, and he did the same. I think I probably beat him in this talking competition, but he then told me about his history. He's worked in the state, and even for the army too. This conversation was a result of me mentioning Fourier series. He said something about working with lasers, tanks etc. and something to do with the Masters programme.

Eventually he even managed to remember another maths problem, and he jotted that down for me too! Can you believe it - two maths problems in one day, and that from a complete stranger too. This time he was even more confident that I should solve it, but once again I highly doubt this. Now I could carry on about writing what we talked about, but it was just about maths and general things. Actually I recall telling him about the mss building and about how great it was due to the empty rooms it had, in which you could sit on your own and do maths. (He used to the same thing once upon a time, and said that the same mark was still there since his time at the university).

Before we had to go our separate ways, he asked me whether I would be there again next Friday. Even if I didn't intend to be there on the Friday, I most likely will be. He said that he will then tell me the building he meant, and check how I was getting on with the maths problems. And so that signaled the end of our meeting. I don't really care if I don't see him next Friday, but he was a friendly and pleasant man. I don't even know his name, but yes, it's all thanks to the Complex Analysis book! This one meeting is telling me to keep on reading my maths books!

Having wrote that much already, you are probably hoping that I stop now. But alas, having done nothing but sleep I feel all fresh and so will continue.

Unfortunately I didn't take that much in during the mornings two lectures. I think the PDE lectures and stats lectures go too slow. Now there is the nice casual pace like Complex Analysis, the slightly faster paced real analysis and algeb. structures lectures; and then you have lectures in reverse. I actually like the PDE lectures because the blackboard is used (HURRAH) and the lecturer makes the lectures fun. However, we never seem to do much during lectures and I am always left asking the question 'and...' after we finish. The stats one are unbearable because of the OHP. I honestly can't keep up with the copying and the listening - nothing went into my head.

My sleep deprived state could be the cause for this, but then why did the Algeb. Structures lecture make sense? Well not 100% sense, but I understood the subgroup criterion (and know to learn its theorem!) and really enjoyed it too. After the lecture I was left wondering about how cool it would have been if our degree ran over the course of 5 years. Or even better, the number of years was uncapped! I think PS is going to think I am weirder than weird now, but I honestly felt comfortable talking to him yesterday. It was a fun conversation. He has been really encouraging too, but I am still conflicted. Bella told me to focus on my main mathematical aim in life first, and then the back up plan second. I don't know: is it possible to do both at once? Twenty one seems rather young, but that is the whole point is it not? GAH. Did I mention how unsettling the future is? Especially for someone with an over active imagination like myself!

Sigh. Moving on, I befriended a joint honours student on Thursday and she is really nice. Someone from my college who was on a gap year last year, is also doing a maths degree this year! The cool thing about the new building is, as someone mentioned, you see and talk to a lot people. That is great. However, unlike staff and postgraduate students, we don't have our own 'offices', so doing work on your own is quite difficult. Also it is somewhat of a 'norm' for staff to be working in there offices, unlike for students. If you have been doing some work, then it is possible to escape from it and bump into many people; but you can't do the opposite. That being said though, I am confident that this week is going to be different and I might be able to do more work at home.

The complex analysis lecture was OK. I had a little tantrum because we had theorems, lemmas and what not thrown at us without proof! Yes - you read that correctly: I wanted the proofs! I spoke to the lecturer later on, and I guess his explanation made sense. (We are doing power series at the moment you see). The thing with proofs is that, as well as breaking lectures up into chunks, they allow you to digest the theorems. I didn't really digest any, but no proofs it is.

Now DC did something on Thursday which gave part of the game away. It is not game over yet, but soon will be. As things are developing, my paranoia is also playing up. Medication is futile: it is about facing my 'fears'. (Pfft!) So, although now I am thanking DC I think I was thrown into a panic initially about what happened. On Friday I decided to talk to, let's say cal*, about this mysterious object. cal had thanks to DC, began putting two and two together, but I though it best if I explained my motives. On Thursday I had subtly asked cal something, and on Friday cal told me that he had assumed that I had got two things confused. (And obviously rather than finding it funny, felt sorry for me I hope!)

I hadn't been confused at all, but was being mysterious. I talked a lot, and cal listened. I am sure that some zoning out did happen, but I did repeat a few things so that is OK. The bonus is that now cal is in so to speak as well. I sold my item. I think cal is cool too, and when I asked cal about what I could say to people who exclaim 'Maths' in a high pitched voice, I was told that cal doesn't normally like telling people that he's a mathematician! I can understand why too, but we have got to change this. I needed some dates from cal and they weren't nice. Although I was unsuccessful of trying to convince him to change them, I think our discussion to work around them might just work. I think it is safe to say that cal also knows that I am a 'giant freak'. Phew.

So my Friday ended on a positive note once again. It is quite weird, that rather than becoming depleted at 3pm on a Friday afternoon, I become filled with energy. I think this scares a few people, especially since I hadn't had any apples or bananas too.

* It's meant to be CAL and it stands for something. Have fun guessing!

Thursday, October 11, 2007

Oscillating Thursdays?

So what have we here: Thursdays have become Fridays it seems. As always when one has a 'tantrum' of some sort, they tend to feel sheepish later on. Blogging about such tantrums is surprisingly good, but in future I will just type such posts and not post them! (That will save me future embarrassment for sure). The basis for my outburst was valid I believe (although some may not agree), and even now the reason behind it can cause me some distress. However, on this occasion it is my resigned outburst. All I can do is play around with what I have, and hope for the best consequences due to what happened.

What happened indeed. Just something that I had hoped to happen next week sometime, when I was ready for it myself. This 'thing' is giving me unwanted stress because now I cannot 'predict' a certain outcome. One can never predict such things, but still... actually I will stop here, and just summarise using Maths (I have had enough of this problem for a day!) A set consisting of some objects existed. Now a subset within this set existed which had the power of expanding the set. However, the objects not in this 'other subset' had no such powers for the time being. Soon this whole set was going to be expanding universally, but before that could happen the set behaved badly: i.e. one object not belonging to the subset, entered it. Hows that sound?

Now onto other things. I was glad (not in the happy and gleeful way!) that I am not the only one having flu problems. All todays lectures were filled with many other people who weren't feeling too great - Fizz is feeling worse than I am! Thankfully I am on the mend now (since I no longer drag my feet anymore), so hopefully by mid-next week I will be fully back to normal. That is, I should be at least sleeping properly and hence doing my work, which will hopefully reduce my stress load.

After I recovered from my outburst, I realised something. I did mention being an unbounded spirit in my previous post, but I think this is not evident. I was lost in thought for quite a few parts of the day today, and during this thought I realised why some friction might have arisen. Due to my desire to study more at university (which is causing me to dislike the AT building, as time goes by) and involve myself with something else, 'fun' time has decreased. It is not understood that maths time and this other thing time is FUN time for me. I am loving every minute of it, and I even love complaining about stats if I may say so myself. GAH. Please don't think that I like being stressed out (so to speak) about such matters, because I don't. They are just thrown at me and I do not know how to respond, apart from throwing them in the cupboard.

This post is going to be long by the way, since I am in a mood for questions and thoughts at this moment. Since I have fallen behind in example sheets, I hadn't done all the questions on problem sheet 2 for algeb. structures. That is why I was annoyed at myself during the class, for not having done the questions. The second question I would have been able to do, but the first one would have been a friendly challenge. Thankfully only two questions were done, and so I can try to get more done for next week. I have learnt my lesson from copying solutions from the back of the book, or other people. I learnt it the hard way, but nevertheless it is a lesson I am grateful towards. I now resent the notion of looking at answers before doing the problems. Even if I am stuck, I hate looking at the answers and prefer asking my lecturers. The struggle is what it's all about! The book doesn't give you proper explanations or motivations at times.

But as Bella and Milo said, I am supposedly a 'passionate' person. Bella is passive as deduced by herself, and Milo sits in the middle of the fence. Bella is brilliant at motivating herself, and me not as much. My passion for maths is what is probably carrying me through my degree, which depends a LOT on my lecturers. So much for training myself to become slightly independent...


Now let us just ignore the top part of this post. My day significantly improved after Algeb. structures (as expected), but for more than one reason. It is the discussion that happened afterwards which is still ticking in my head. I spoke a lot of home truths today, some which very few people know. It wasn't 'deeply buried secrets', but more of my ambitions in life, and where I currently stand. I don't know whether you recall, but ages ago I mentioned something more important than me studying maths. (Yes something does exist I am afraid).

Today I found myself saying this. It is a sheepish aspiration one could say, but one that I have desired for quite some time: probably before I started my degree. I don't know why, but this thing is one thing that I really want to happen. I can't explain this, but as scary as this thing is, it is amazing too. I probably sounded really weird today, but this thing somehow conflicts with wanting to do further studies after hopefully graduating. It is possible to maybe do both at the same time, which is what Bella always tells me, but I shouldn't worry too much about that now. However, I don't think one can easily do both such tasks - it requires hard work. What would be the case if I was normal? I don't really like considering such things, but on cloud nine it is easy to plot ones life where external conditions tend to zero, so you are free to do as you please.

I felt comfortable talking about such things. The conversation felt proper. Sometimes it is hard for me to explain to others, about where I come from in certain situations. I probably didn't communicate well this time round, but nevertheless this conversation had me thinking. I shrugged as I walked away, about the endless choices that can be made. Nowadays thoughts of struggling with further studies no longer frighten me. Thanks to certain people, I look forward to that struggle. But if I want to do this other thing, for now my focus has to be on teaching. I have to force myself to think this way, and only aspire to do further studies further on (which will probably be more difficult and my position towards maths may change!)

Anyway, I no longer get depressed by such thoughts. Does one stay within ones limit, or does one try to diverge? (Excuse any appalling mistakes I am making in using maths language!) I mean, do I sit down and decide to reduce friction by doing teaching or fight for further studies. Damn me. Now that is depressing.

People find it weird that I can talk to my lecturers: I mean have proper conversations or joke about with them. Is that really weird? I mean I like my lecturers, they are cool people and mathematicians at the end of the day too! Last year I think I was scared of my fellow undergraduate students. I was scared because I used to love maths and... well no one knew about this. Last year I would always wait before approaching lectures, for everyone to leave. My classmates did make me slightly nervous. This year I am no longer (as!) afraid. I don't care what they want from their degrees, and how they want this. I also don't care that they think I am a freak, for having conversations and joking around with my lecturers. When talking to my lecturers, I don't feel the need to hide my mathematical passion. This is most pleasing, and one can also learn a lot.

So many thoughts are praying on mind at the moment, and thinking about the future is very depressing. Before university I wanted to do teaching. Not knowing what to do next is surprisingly unsettling. Where's that closet got to?

Before I end, I will say that I am glad that I talk to PS and other lecturers. (Although they probably think otherwise!) They have made a major difference to my university education and my ambitions. That is why I am keen for them to do something, but time will tell. (Developments have been happening, and now I just need to talk to one more person and the piece things together).

I hope tomorrow is not oscillating as well!

Wednesday, October 10, 2007

Who tried to kill \sout{Mr Black} .... with a car?

I was on the verge of losing my voice yesterday, but dived out the situation rather remarkably (if I may say so myself). OK, I had some help but nevertheless I am clinging to my voice, the way a child clings on to his mother. If my coughing etc. annoys you then I am sorry, but please note that it annoys me much more than it ever annoys you.

I was feeling particularly dozy this morning, but after a fantastic Analysis lecture, I will naively consider myself cured!

Yesterday many events occurred. The first of which was me acting really stupid or dumb; but seriously, real analysis makes me feel dumb more often than not. However, due to DC I am not worried about this dumbness, and hope that I will soon understand WHY we 'CHOOSE' things. It is to make life simpler, but it seems rather arbitrary at times. I just need to disassociate certain things learnt from sequence and series, because that is proving problematic.

For sequence and series, the first question we tend to ask is 'What happens as n gets bigger and bigger'. However, now we work in a 'deleted neighborhood' (pfft) i.e. we are seeing what happens
in an interval. I don't think I was having a good day yesterday too, but then
by magic after the lecture I felt great.

My real analysis lectures are simply brilliant. They have a really nice flow to them, and if yesterdays was brilliant, then I am at a loss as to how to define todays! Yesterday we proved the Sandwich rule for functions (amongst other things). I must confess that the sandwich rule is one of my favorite rules in this course; namely because whenever the beautiful theorem is mentioned, my stomach and mind have a reaction. This was noticed in last years example class,
but doesn't the word sandwich induce hunger? If you still have your eyebrows raised, then what say you about the Pancake function?! (Apologies for any stomach rumbles!)

The proof was neat, and one of my aims in life (cough) is to post it. (The weekend will tell.) There are always points in this lecture that go above my head, and I always complain to DC about inequalities used in the lectures. (Epsilon changes to inequalities, induce large changes in me!) However, I am really happy that everyday of the week, I have at least one lecture which has a
'feel great' factor to it. Even if it is a tough lecture, it is great: a classic story.

I have gone on about that lecture, but the lecturer and the lectures are fun! Lectures that I really enjoy make me happy, and we all know that a happy bean is a good bean!

Moving on: yesterday a lot of positive things happened, some particularly pleasing. Actually one thing has made me very happy, and nervous at the same time. I have to bite my tongue and do something, which I don't really want to, but have to! (Part of an agreement shall we say.) I supposedly(!) bounce {OK fine: this has been confirmed from last year too}, and the 'spring in my step' returned yesterday. It actually may have induced bounciness (not of the same sort) in others too, which is always pleasing to see. (I am not responsible for any accidents that may be a result of this).

I was told that 'maths is my hobby' yesterday as well, but this I will discuss in another post. Due to me writing this post over the course of two days, discontinuities have arisen: I can't remember a thing! Hence I will talk about today. (All you need to know about Tuesday is that DC's lectures are great, and I moved from A to A.5 regarding something. I also persuaded a few people to join me at A.5, and hopefully we can all get to B!)

Having already mentioned todays great Real Analysis lecture, I will just say that I found it particularly interesting. Namely because we showed lim_{x \to 1} \frac{\sin x}{x}= 1 in a really cool and new (to me) way (as well e^x \to 1 and other things). I also deliberately kept on saying something 'true' but not required. I think it is safe to say that although all the Tweenies may have enjoyed this lecture, other people might not have enjoyed us enjoying it so much! (If that makes sense.)

What does it imply if a lecturer tries to subtly kill you? Last year it was my PT, this year it was DC. Well I am sure that my PT didn't mean no harm; we were both running slightly late and crossed the road at the wrong time! At that time I had mentioned the prospect of being killed, but yesterday the lecturer himself said this! Most mystifying... I am thinking that the plan failed, but you wouldn't want to kill me now would you? (I don't think that is a question, if you follow me!) OK, fine I jest. It wasn't planned and deliberate; I have yet to drive DC to the point of no return, when he may have to do such a thing to reduce me. Maybe I am getting close. :D

My alternative would have been to either walk backwards, or go on a mini-action adventure, which consisted on walking on the road. (Nothing could have 'technically' happened to me if you were worried. I was on the inside of the road!) I would take the mini- adventure any day. I don't know what drives people to walk slowly or backwards. Walking is one of the forms of exercise that I get, so whats the point of negating this? (Note to any gullible readers like me: no one tried to kill anyone! It was just something that could have resulted from this 'mini-adventure'. I little bit of harmless ... fun.)

Whilst I am waffling away, I feel urged to ask something. BUT I won't. (I don't want this post to have any more discontinuities, and I can wait). What do I wish for: I wish to get better soon so that upon coming home, I don't crash and burn and actually do some work. Also, so that I can eat the darn American Hard gums! I wish that I can find the 'guts' to do something else too. (Have I missed anything?!)

Once again: mss building I need you. (In a true Nanny McPhee style!)

Monday, October 08, 2007

Phase two over

This morning as I walked to my lecture, I wanted it to rain. I really badly wanted it to rain. I wanted to see people scurrying everywhere - afraid of the rain. I wasn't afraid of getting wet, and didn't care that I was walking really slowly today. I wanted rain. Perfect for Monday's wouldn't you agree? Actually I felt very cold all morning, and the only thing that was missing was the rain. Talk about Monday morning depression eh?

I was sick in the morning, but thankfully due Wendy's magical formula, the friction in my throat decreased significantly. Not enough for me to munch on the American Hard gums, but the blackboard scraping sound was no longer there. This was all good news, but unfortunately this means onto phase three.

I got to the stats lecture on time (yes 9am), and it was semi OK. The technical bits and the horribly ugly example with the gamma function made no sense whatsoever, but I at least remembered the Poission distribution. Thankfully my incessant coughing didn't irritate many people, because quite a few people were coughing and sneezing too. It was quite cool when we all coughed in a sequence... (Yes, that was what I was thinking about during the lecture!)

What happened next? 'To the AT building' it was, and we attended to some business. Business over, Bella did some work whilst I tried my utmost to do the same. I couldn't. Hence, when Bella went to meet a friend, I actually tried to do some work (just basically read through my notes). In between, something else had happened which gave me reason to be excited. I will do my own little bit of persuasion tomorrow, but I am really happy at the way things could work out! [We'll see what my annoying presence can induce tomorrow.]

Highlight of my day: I have been freezing cold all day toady i.e. phase three. Being ill at this time is rather inconvenient. Already I am having difficulties in getting any work done, and now this! Bleh. The normal 'spring in my step' (so to speak), was in first gear for most of the day. I was walking too slowly for my own good, and was just dragging my feet. Once sat down, I really didn't want to get up again.

As you can probably tell, I am still in first gear. I actually struggled to keep up with the writing in Algeb. Structures today. Normally when PS writes on the board, he says what he writes, hence I don't always have to look at the board when writing the notes. Today I actually had to force myself to stop writing, so I could actually listen on a few occasions. I think I followed the lecture, but didn't keep up with it too well on paper. And because I am feeling slow, I had to keep on checking that I had copied correctly from the board. It was a good lecture though (as always) and I don't know any Sheffield fans so I can safely claim to having done the first thing! (It was really cold in that lecture too).

I think in the chemistry building, if the boards with the margins are used, then no one has any complaints. (I noticed this today). The guy in front of me was playing tetris on his phone for most of the lecture, and the people behind me and to the right, would not shut up. The funny thing is that the same group of people always sit behind. Always. And guess what? They talk a lot during lectures. (I am just trying to remember something that happened during the lecture, but can't remember what!)

Hmm, what to write next? I was waiting for the bus home at 6pm today, and when the bus finally came a person who I knew from college seemed to be on board. Strangely I can't remember the name, but this person knew my name. It was a five minute conversation, but I don't know if it just me, but when I said I do maths.... the bus went quiet! The eyes were diverted to outside the window, and well at least I got my chance to use my newly acquired knowledge of groups, and talk about (\mathbb{Z}, +). Maybe that explains why the eyes became slightly gazed and diverted to the window! I would also bet the loose change in my pocket, that this person even got of one stop sooner!

[I think I shouldn't use the addition of integers being a group as my 'wow factor' example. Any ideas on what I could use? I could try asking the 'trick' question about the smallest prime I suppose. Yes, I always tend to do this if you are wondering! But when someone exclaims (in a rather high pitched voice) 'Maths!', I can't but help explain how wonderful it is.]

EDIT: I just remembered what I forgot! Abel and Galois sort of worked on the same 'idea' (to do with quintic equations). Well if Galois is mentioned, then how can I not mention him! If only there were no exams and we went to lectures for the fun of it, and also learnt the material for the fun of it. I would love that. No stress whatsoever, and I wouldn't feel guilty if I was to read about a topic not on my syllabus.

Question for the reader (if you're interested!) Well actually I invite your ideas on what to do next: \displaystyle \lim_{x \to \infty} \ln \left(\frac{x+1}{x}\right )^x. . (Using limit rules we deduced that it is indeterminate, but what then? BTW, you only know about L'Hopitals rule, and haven't done a 'rigorous' course in analysis!)

I have real analysis to look forward to tomorrow, and having doing the example sheet for the calculus module, I am quite eager for that lecture too. (We ended the previous lecture without actually completing what we are meant to do when the interval is not [0, 2pi].)

I think somebody needs Wendy's magical formula again.

Sunday, October 07, 2007

Damn me...

... and my cough. Mrs H. from primary school (and weirdly another Mrs. H from secondary school), always used to say that I have a smokers cough. That is whenever I was spluttering my guts out, with a croaky voice to go with it too. I should have got my tonsils taken out as a kid but I never. I liked my voice being 'different', although it probably wasn't that different! It felt unique though.

I was scared that once out, I would no longer sound like myself. Mrs. S told me to tell the folks in year 3, but I 'forgot'. Due to my tonsils being 'messed up', my throat is messed up. I don't mind a blocked nose, a headache, or a broken back (actually ask me again about the back!) However, I hate the throat. I hate it. I have been weakly walking about the house today, and helping with some gardening. Well, I really didn't have a choice with that. I am the one who tends to shift heavy stuff from A to B, or just the one who does the donkey work as it had been called.

Imagine a blackboard. Now imagine my year 9 maths teacher, whose nails were unnaturally long (but they were her own). Next imagine her scraping the black board with her nails and the resulting horrible sound. (She did used to do that by the way!) That sound is what it feels like whenever I cough. Nasty. The worst bit is going to be tomorrow when I don't stop with the coughing in my lectures. Hence my public apology to anyone who I might annoy. I mean annoy you to the point where you want to strangle me, to shut me up. I don't like it as much as you do. I just hope it doesn't get that bad, where I actually struggle to breathe. (Happened in college in Mr. B's class. I got embarrassed because I wouldn't stop coughing, and then tried to hold my breath. Bad idea. That ended up making it much worse, and I had to leave the class for a while!)

The biologist in me wants to research more about this phenomenal issue. However the mathematician in me is whinging, because I have done close to zilch today. Well I read my algeb. structures notes and tried doing some questions from the problem sheet, but that was a problem too. I am not fully 'ill' yet if that makes sense. The 'smokers cough' though, gives me an indication of what is to come. Keeping oneself busy whilst being under the weather sometimes works too. Tomorrow should be good then I hope, and I will try my best to keep my head down! (Ha.) I am more annoyed because it has been ages since I ate some American Hard gums, and I bought some today. Damn: I better save them for a rainy day.

Am I too ill for a 9am stats lecture? I already know the answer to that, but I always kid myself with thoughts of missing lectures! The amusing thing for others is that situations like these, can lead to me losing my voice! I can envisage the delight on certain lecturers and peoples faces if that happens, but a word of warning: losing my voice has never stopped me from talking. The croaking is just plain more annoying, and so disappear the happy faces!

My plan: honey, water, sleep and then wake up early to do the work I was meant to do today. Which of them I am actually going to do, I will leave you to guess. Hint: waking up early sounded problematic as I typed it!

'So beans how was your weekend?'
'I watched some football, gardening, lots of coughing, and yeah... maybe something called maths.'

Now does it make sense about why I have to work at university? It is not only because of the distractions at home, but because of another non-trivial *cough!* reason (which is really annoying). By the way, I have spied the Dame Kathleen lecture being scheduled for the 24th October (but don't quote me on that). This time I will definitely be going with S={Milo, Jake, Fizz, Bella, Beans}. I might need to work a little on convincing Bella, but the whole idea is to make sure that the \sout{group} set is as large as possible. I think it is safe to say that I have learnt my lesson from last year, but that being said, I am no longer 'afraid' in the way I was, and can on this occasion confidently say that I would be happy with the set S={Beans} going. At least now I know not to hide from my lecturers, as I did last year! Anyone else want to come?

Saturday, October 06, 2007

Yes or no?

Today I slowly coaxed myself into sorting my folder out. I am ashamed to admit that that is all I have done today. I haven't even finished sorting it out yet because I got side tracked. My notes for Vector Calculus have become rougher and rougher, and in them a lot of the results have been left for us to prove and confirm. That is why I decided to copy one page of the lecture notes in neat, and attempt to get the result that the lecturer gave us. I have been semi-succesful, and just need to check one during the example class.

Whilst I was in the mood for integration, I decided to do the example sheet for the class on Monday. The question is as follows:

\displaystyle \int^{2\pi}_{0} \sin (mx) \sin (nx) dx

where m, n \in \mathbb{Z} .

One way to do this is by parts, but that is long winded and it is easy to make mistakes. I managed to succesfully do it by parts, but having just completed question one, 'a trick' was fresh in my mind. Hence why I am now attempting the above question in a different way, using 'the trick'. That is by using complex exponentials and then 'playing around' and extracting the imaginary part.

We begin by re-writing the integral as: \int e^{i(mx)}e^{i(nx)} dx = \int e^{ix(m+n)} dx . (I have dropped the limits of the integral for the time being).

Having only started complex analysis (and not having defined integration yet!) I am tentatively attempting this question. (Google hasn't been much of a help). Your job is to tell me whether what I have written is ridiculous or not! Not that difficult a job I hope.

We begin by letting \displaystyle u=x(m+n)\text{ so } \frac{du}{dx} = m+n. \text{ Thus } dx= \frac{du}{m+n} .

This gives:

  \begin{array}{cccc} \displaystyle \int e^{ix(m+n)} dx & = & \displaystyle \frac{1}{(m+n)} \int e^{iu} du \\*[3ex] &=& \displaystyle \frac{e^{iu}}{(m+n)i}  \\*[3ex] &=&  \displaystyle  -\frac{ie^{iu}}{(m+n)}  & \text{ using } 1/i= -i \\*[3ex] &=& \displaystyle -\frac{i(\cos u + i\sin u)}{(m+n)} \\*[3ex] &=& \displaystyle -i\frac{\cos u}{(m+n)} - \frac{ \sin u}{(m+n)} \end{array}

EDIT: (It's meant to be plus the real part).
The imaginary part is required, which is: \displaystyle \frac{\cos u}{(m+n)}. .

Hence, if we now insert the limits we get the required answer:

\displaystyle \left[ \frac{\cos (m+n)x}{(m+n)} \right]^{2\pi}_{0}=0. .

How does that look? It seems a bit 'dodgy' to me; well the actual integration with the i involved does. Having LaTeXed this, I can't decide which method is worse than which! (I really like my work spread out, as you may have noticed.)

These integrals define my day. I was rather pleased when I managed to dig out question one though, since it was slightly tricky. The lecturer had given us a hint (humbug), which I had forgotten. I was going to sheepishly email him to ask him what he told us, but after I had written out the first few sums of the series, something clicked. You can try the question if you want:

Let S(x) =\displaystyle \sum_{n=1}^{\infty} a^n \sin(nx) , ( mod a is less than 1). Show that \displaystyle S(x) = \frac{a \sin x}{1 - 2a \cos x +a^2}.

Back to the folder now. What joy.