Wednesday, June 06, 2007

Maths Graffiti (?)

Apologies for the size of that picture, it was unfortunately taken using a phone. What you see is a wall of my bedroom (keep this picture in mind). I never realised how fun it was taking of wallpaper... I mean it requires no thought whatsoever, and it's rather mechanical. Good workout for the arms though! Yes, my mum is unrelenting. I'm lazy. So she claims. Not a good combination. You see I want to take things nice....and......slowly. However in the way I might be obsessed about maths and football, my mum is about interior designing etc. She's great at what she does, (don't tell her I said that!) but since I've got holidays now and well since this is my room, I'm having to push my weight around.

Although I was most unhappy when my mum packed up all my books and folders, it was necessary (according to her). Having spoken to Milo and Bella today, they were most shocked to learn that I had not got started on any of the books! I did confess to being a slow reader, but how does one find time to read?! I really need to get some definition in my daily routine. Being awake at this time isn't helping! However I have a plan. You see tomorrow- actually today(!) I'm going to be going into University in the morning (i.e. a few hours :( ). This is my only option since I no longer have a bus pass.... . Initially I decided to go in to return some books to the library, however upon some thinking I have decided not to return any at the moment! Seriously which undergraduate student is going to go to the library during the summer? And some of the books that I have were published in 1967- so hopefully no one will want them! It's a heavy load you see..

The other reason of course is to hand in the form for the course units which I want to do. I have decided to choose the following now (thanks to the comments here), and must likely will not be changing my mind (although we must leave this open since we can't for see any life changing moments to come!). They are Metric Spaces, Algebraic Structures 2, Discrete Mathematics, Introduction to Geometry, Calculus of Several Variables and Propositional Logic. Bella is doing three of them as well (geom, discrete, calc) as well as Numerical analysis and an external finance one. So the only module which the Tweenies are not doing is propositional logic. Boo hoo.

Now since I'm going to be awake early toady, I'll be tired early and so should go to sleep earlier. Still following? In this way I hope to start sleeping earlier and waking up at a reasonable time! (Whether this plan works is an entirely different question!)

So back to the wall. We're going to put wallpaper on the wall, and so that means that whatever I write on the wall now will be covered. Now there is only one thing which I can possibly write, and will do so. However, the readers and the commenter's of this blog (i.e. you!) have always given me reason to be positive about my maths, and helped me out on quite a few occasions. You could say that my mathematical learning and knowledge has been somewhat influenced positively by Blogistan (eg videos about maths writing, book recommendations, and other interesting maths from various blogs). I fear naming and pin pointing stuff since I don't want to test my poor memory!

Back to the wall again! Due to the reason I've mentioned about, call it A, say (:D) I'm giving you a once in life time opportunity. Tell me when to stop the hyperbole, but you might not want to take up this opportunity, but then think of what you might be missing out on! (Erm.. don't ask what, since I'm still thinking about this). So to the point(!), whatever you want me to write on that wall I will do so. However the catch- whatever maths you want me to write on that wall will be written. You want the proof for infinitely many primes, post and your wish will be my command! I'm definitely going to be putting the definition for null sequences (one can grow to like it :o ) amongst other things, but I have 3.5 walls (one's at a weird angle) so don't hold back. I don't even insist that I understand the material. I have to mathify (i.e. prettify) my room from within. So even when the wallpaper has been put on, I'll know what I'm surrounded by. :D

Pretty sad, I've got to admit, but opportunities like this are rare. If I ask really nicely i.e. beg would you suggest somethings? (I won't write your names if you don't want to!) But there has to be a proof, theorem, or anything in maths which makes you feel all 'serene', happy, joyful, ecstatic? (shift F7 on word might give you more adjectives) Whatever that thing is, let me know and it'll be appearing in my 'art gallery'. :o I'll even promise to write any mechanics down. ;)

Convinced, or do you just not have anything in maths which makes you feel like you're on cloud 9? If so then you could still suggest maths that you feel is 'important'. If you're worried about whether this is 'graffiti', then don't. I have 'planning permission' from the powers that be and they liked the idea of me being surrounded by maths. Dare I go further exaggerate this and say, you have guardian angels which you never see.... :D (it's early!)

So, I'm pretty excited by this and so unfortunately this post is littered with explanation marks, but please don't make me cry. Sniff. (You didn't see any onions...) I'll definitely be posting pictures of the outcome, but please do hurry. My mums hoping to get my room complete for my birthday- I'm thinking that this is her idea for my present!! Why do I feel like a three year old about this whole maths wall business. (I did suggest to my mum to stick white paper everywhere and then I could make my own maths wallpaper, but as you can tell that didn't go down to well!)

Once again alarm bells in my head are telling me not to be excited, but I could always write LaTeX stuff down if worse comes to worst. (But you wouldn't want that, would you?!) I'm quite enjoying all this room business. I'm going to visit a really old friend tomorrow- by really old I mean they've known me since primary school! Can't wait. I also believe that tomorrow is going to be my last visit to my office. :( More about that tomorrow- I don't want to change the mood of this post!

Some pictures if I may. My 'two' beds (I did say I supported United!):

(Notice the maths books near the pillow ;) )

I think it's bed time now. Although I'm sure I had a weird dream the other day, where I was with some undergrads, post grad students and lecturers in the mss building. :/ (I just remembered that now, but I remember the game Charades being played....) Yes definitely time for bed- I'm just trying to organize my thoughts for tomorrow. :/


steve said...

This formula was voted one of the best in mathematics (the top one being e^{i\pi}=-1 of course) so put that up. Easy to prove as well!

beans said...

Woo. Finally a response! (The e^{i\pi}=-1 has been put up already!).

Does that formula have a name? I'll be back. 8)

steve said...

It's an integral of a pancake function so called because it is very flat in the interval [0,1]. You can see its graph here, notice the scale on the y-axis.

There's more about it here.

beans said...

Did you mean that the pancake function was easy to prove?! Thanks for that link, it is indeed an interesting curve. It was also an interesting read, and well I'd forgotten about trying to divide! (Integration is not one of my strongest points. :o )

(Any more suggestions- it seems that I'll have to keep on bugging you!) Thanks. :)

steve said...

Did you mean that the pancake function was easy to prove?!

I meant that it is easy to prove that the integral is 22/7 - \pi. I hope you are able to do so. If I were to show you how you'd say "that was easy!"

More formulae? See here and here (particularly the definition of the Riemann zeta function and the fact that sum of 1/n^2 is pi^2/6. Then there is here. You should beable to fill up the walls with these!

beans said...

Yes thankfully I have done it now!! I had been making a stupid error yesterday- and stupidly had done the same thing incorrect on all occasions! (A good nights sleep always helps.) Is it bad that I'm over the moon having got the answer?! :o

Wow thanks a lot for them links! Dr. Coleman had told me to look up Stirlings forumla, so that makes it 7 from them 10 which I've heard/seen before. (Although I got lost in the proof of stirlings formula and since exams were looming I've yet to look at it again.)

My wall is going to look a treat. :D

Jake said...

Does that formula have a name?

If you meant e^(i*pi) = -1, then it is often called Euler's Identity.

In terms of other mathematical graffiti, I suppose one I would go for is the first Isomorphism theorem:

given groups G,H and a homomorphism \phi: G -> H
G/Ker(\phi) \cong Im(\phi)

It is a pretty basic result, but one that I have an affinity with really; it was what first got me quite interested in algebra and maths in general.

Another nice integral equality is the probability integral (easy to show by switching to polar coordinates):

\int_{-\infty}^{\infty} e^{-x^{2}} dx = \sqrt{\pi}

Jake said...

Sorr, I should have added: the trick is to consider the double integral
|| e^(-x^2-y^2) dx dy = I^2

where I was the original integral and then switch to polar coordinates to get I^2 = \pi

beans said...

Thanks a lot, I'll be sticking them all up tonight hopefully. (Under cover of darkness... :p )

(nice double integrals :D although the thought of them makes me shudder!)

Jake said...

(nice double integrals :D although the thought of them makes me shudder!)

LOL. Don't worry; that is the extent of my ascii art skills!

I have just been revising change of variable in multiple integrals actually; so I can empathise with your shuddering!

beans said...

LOL. Don't worry; that is the extent of my ascii art skills!

I seriously doubt that! :D

Changing to polar co-ordinates is the worst. All I can remember is that we had a substitute lecturer and he talked about the Jacobian a lot. (Hmmm, I porbably should go over this for the PASS sessions!)

Jake said...

Changing to polar co-ordinates is the worst. All I can remember is that we had a substitute lecturer and he talked about the Jacobian a lot. (Hmmm, I porbably should go over this for the PASS sessions!)

No! Changing to polar coordinates in double integrals is the easiest because the Jacobian determinant is always
r*(cos^2(theta) + sin^2(theta))=r
so you just need to remember 'r' and you are ok. In a more general change of variable; you are given the opportunity to make a mistake whilst calculating the Jacobian as well as the myriad of other opportunities to screw up whilst calculating multiple integrals!

beans said...

Ah I see you've fallen into the trap. :(

That may be easy, but the questions with polar co-ordinates are next to impossible. (well for me anyway!) I could never get them set up right, but yes once they're set nicely they're not too bad. :p)

I always try to do them with respect to x and y (if it's in that system) but one always tends to become more messy than the other so then I stop!

Jake said...

Quite a few of the past paper questions I have looked at are set up fairly nicely, so for instance, it gives you a function of x and y and then asks you to integrate over say the region
x^2 + y^2 < 2, y > 0, x < 0

i.e. the top left hand quadrant of a circle of radius 2

so it is fairly obvious that polar coordinates are the way forward.

beans said...

Since the maths departments have merged, we didn't really have that many past papers to work from. I used to do questions from the book we had to buy, so that explains the nastiness!

(I tend to get the angles wrong).