Back in Singapore

I must apologise for the long absence from blogging. I am back in Singapore from Bristol for about 2 weeks and these 2 weeks are filled with much food! Hmm... Had most of the food I want so far still left with fried hokkien mee.

Anyway, I will be doing a UROPS project next semester. It will be on modal logic and its applications. For those who do not know what is modal logic, read up the books in the library.

I spent a fraction of these 2 weeks going out with my gf. It was very fun! Today it was even better! We played badminton with Sheena and Thye Heng together! haha Its so fun! My gf is not good in badminton but I will coach her! I hope we will treasure each other more after long absence from each other.

Anyway, My school attachment will start next week at Riverside Sec School! I look forward to it but at the same time a little anxious with it!

Am I stressed?

Notice that Alex (a lame friend of mine) wrote something like ... s-t-r-e-s-s-e-d ... Observe that this is the inverse of d-e-s-s-e-r-t-s...

It is very obvious that one is bad and one is sweet, or I termed it good... It seem to be that there is a clearly a bijective correspondence from {good,bad} to {stressed, desserts} via the function f: feelings ->metaphors. Please do not take this mathematically seriously. It is just a bored action of mine to pass time.

On a side note, I am left with one final exam here to do before going back to Singapore. I am rather feeling bad... not sure if its stressed...

Sorry for this post... that is of not much value.

Longing to go home

It certainly seems that there are a lot of criticisms going on regarding the content of my post. :) Ok, I shall not post the proof of the last bit of the theorem then. I shall leave it as a puzzle. I just finished my number theory exam yesterday. I think it should be all right. Anyway, I really lack the momentum to study for the final exam because of the following reasons (not in order of preference):

1) I want to go home
2) My stiff neck and shoulder aches are bothering me quite a lot and I am worried.
3) I think MVA is not going to be as smooth as number theory
4) I miss my gf
5) I miss my family

I hope my upcoming Norway trip will be a breather for me! I hope to get some fresh air and nice scenery. And also, I really quite worried about my chronic shoulder ache... I will see a doctor as soon as I reach Singapore. Can't wait for 14 June to come and I will be home... With that, I want to end with part of the lyrics from a well known National Day song.

Singapore, my homeland, this is where i belong...

Proof of theorem 65 in lecture notes

I shall state the full glory of Theorem 65 here:

Suppose that d is a positive integer that is not a perfect square, and the continued fraction expansion of sqrt(d) has convergents p_n/q_n and is ultimately periodic with period m. Then the only positive solutions (x,y) are given by (x,y)=(p_n,q_n) where n = lm - 1 where l is some interger corresponding to n being odd.

Proof:

Consider the equation x^2 - d*y^2 = 1. This can be transformed to (x+sqrt(d)*y)*(x-sqrt(d)*y)=1 => (x-sqrt(d)*y)=1/[(x+sqrt(d)*y)=1] =>x>sqrt(d)*y.
Whence, this yields 0
Thus we deduce that |sqrt(d) - (x/y)|<1/2*y^2.

We claim that if $theta in R$ and x/y is a rational number with (x.y)=1 satisfying |$theta$ - (x/y)|<1/2*y^2. then x/y is necessary a convergent to the continued fraction expansion of $theta$.

consider the equations

u*p_n + v*p_(n+1) = x
u*q_n + v*q_(n+1) = y

This set of equations do have an integral solution for (u,v) since
p_n*q_(n+1) + q_n*p_(n+1) = (-1)^(n+1).

If u or v =0, we got (x/y) = (p_(n+1)/q_(n+1)) and the coprimailty of x and y yields that v = $+ or -$ 1. Then the claim holds.

So if u and v are both not equal to 0 and that q_n -> infinity as n-> infinity, so we can choose an n such that q_nBut since u*q_n + v*q_(n+1) = y, u and v are of opposite signs and thus

|y*$theta$-x|=|u(q_n*$theta$-p_n) + v(q_(n+1)*$theta$-p_(n+1)| > |(q_n*$theta$-p_n)|.

Since |$theta$ - (x/y)|<1/2*y^2, it follows that |(x/y)-(p_n/q_n)|
This proves our claim.

Now let p_n/q_n be the convergent to sqrt(d).
Then we got the following:

sqrt(d) = (p_n*$theta$_(n+1)+p_(n-1))/(q_n*$theta$_(n+1)+q_(n-1))

whence,
q_n*sqrt(d) = (-1)^n/(q_n*$theta$_(n+1)+q_(n-1))

This leaves that n must be odd in order to have that Pell's equation.
(Check if n is even then there will be a contradiction)

Continue the proof some other day...

Feeling lazy...

Today I had my last lecture for number theory. But after that I feel so lazy. I am not sure what is the reason! Yesterday, I took some photos. However, my hands are shaky so the photos turned out to be very ugly. Nevertheless, I will post some of them here for criticising! (especially by ALEX).

Okay, thats about for photos. I should talk something about number theory today. In all, for this course, I have 65 theorems, lemmas, corollary in total. So obviously, to end the course, it will defintely be a theorem!!! So whats so 'big' about theorem 65? Basically, we learnt continued fractions and Pell's equation. Theorem 65 is the link between them. Consider the equation x^2 - dy^2 = 1, where d is a non square integer. Basically, this theorem says that if p_n and q_n are the convergents of the sqrt(d) then (x,y)=(p_n,q_n) are the solutions of the equation given certain conditions. First, n=lm-1 where l is a natural number and m is the period of the continued fraction of sqrt(d) and n must be an odd number. Seems complicated? Defintely not. Its fun! If you are feeling bored or nothing to do... maybe this can replace your sudoku puzzle when its so common nowadays. I am assure you that its more enjoyable than sudoku. To give yourself the fullest satisfaction, I will recommend you to prove this statement.

Anyway I got this friend who says that 'Life is short, make a fool of it while you can'. I will rephrase it... ' Life is short, make yourself and your family happy while you can and learn more mathematics too'!

A rainy then sunny day

Today I had 3 lectures, 2 sustainable development and 1 number theory. I have got a break between the lectures and I was at the royal fort gardens just outside the Mathematics Building. The weather's good and temperature is nice. I was inspired to take some photos of the small flowers around me. They look amazing. I am not sure if you can see these flowers in Singapore but I defintely have not noticed any before along NUS roads.

Ok, what I learnt today for number theory... diophantine equations, in particular, Pell's equation. Prof Trevor Wooley (http://www.maths.bris.ac.uk/~matdw/) is my idol really... He explained things really clearly and also with a sense of humour. Anyway Pell's equation is of the form x^2 - dy^2 =1 where d is a non negative integer and not a square. Basically, we went through 2 theorems. In all, what this lecture brings about is... there are infintely many solutions (x,y) for a fixed d which is not a perfect square. (I am not too sure if I remembered correctly because I didnt copied down any notes and waiting to print it...) The idea is to think of a fundamental solution and from there work out the congruence classes. And guess whats the end? ... The moral of this is once you get a solution, you basically get the rest of the solutions. How intrinsic it is... how amazing!!!

I hope I can attend more of Prof Wooley's lecture... though I miss Singapore too!!!

Multivariate Analysis and Number theory

Disclaimer: All the mathematical work here are not to be taken for truth since I am still a student only... and hence the title of my blog...

My title is What is Mathematics but I have not post anything mathematical at all. This is so absurd. So today I have decided to post something about mathematics.
To say something about the modules I am taking here - Multivariate analysis(MVA) and Number Theory (NT). It a misfortune when I register for the unit Multivariate analysis, I mistaken it to be multivariable analysis and it turns out to be totally different. But I only have myself to blame. I should have read the unit description more carefully.

In the morning I managed to catch up on chapter 2 and 3 of MVA. Basically, it is on principal components analysis. It is the usage of linear algebra (eigenvalues, eigenvectors, trace, etc) to aid in lowering the dimensionality of the entities being analyse. There are a few ways to analyse it. First we can look at the varience - covarience matrix or the correlation matrix. then work out the eigen values and eigenvectors. by using some steps, we can identify those PC that are of importance and significance. After which, depending on the entities, we can interpret those results. Quite cool right? I hope I am getting all these correct.

It quite wonderful that I only have 1 lecture today. It is a noon number theory lecture. I spent 45 mins to walk to school everyday (3km walk). Its seem mad but I feel its a good way to slim down and clear my mind. But anyway, I was early for lecture so I sat in the LT and look at the blackboard from the lecture in the previous class. It is very famililar. "universe of all sets", "well ordering principle", "well founded sets", "axiom of replacement" and such... ITS SET THEORY!!!
I took this module last semester and I hope it had not fade away from my head.

Thanks Dr Franklin.

But anyway for NT lecture today, I learnt about continued fractions!!! basically it is to express a real number, whether rational or not, as a continued fraction. If you got a irrational number, basically u get a recurring decimal, in this sense, will be a endless continued fraction!!! The alogrithm is not hard to undestand. Youy get to use the ceiling function, rationalise the fraction, inverting the fraction and such!!!

Ok I am tired now. I shall stop here today. I didnt go out to take any photos today thought the weather is nice... WHAT A WASTE!!!

Afternoon Walk

I had the morning doing my wiki project and also some multivariate analysis stuff. Therefore in the afternoon I decided to go for a walk along the clifton suspension. I realised I got to buck up on my photography skills so I took a few random shots! The weather was amazing today and its rather hot! about 11 deg celcius. You must be wondering how mad I am since it like always 30 deg celcius in Singapore. But mind you! I have been experiencing winter for the past 4 months... Anyway these are the few photos during the walk...