It's my birthday tomorrow and my friend mathematics wishes me mathematically in many ways:-

1) I'm 19 years now. 
The no. 19 is a happy prime.i.e. it is a prime and satisfies  the following property.

1|9     : (1)^2  + (9)^2 = 82
8|2     ;  (8)^2 + (2)^2 = 68
6|8     :  (6)^2 + (8)^2 = 100
1|0|0  : (1)^2 + (0)^2 + (0)^2 = 1! 
( Source: http://en.wikipedia.org/wiki/Happy_number)

2) ஓட்ட  வடை(Sambhar Vadai)  my  favorite dish:
I remember asking my mom to prepare Sambhar Vadai for my birthday which is inauspicious to prepare on occasions which are good such as birthdays but now I realize that as a wholly different thing,

It reminds me, the structure of a torus which topologically speaking is a product of two circles S X S in C^2.
 This has a consequence that topologically a coffee mug and Sambhar Vadai(Torus) a similar  as they are homoemorphic .Since the latter is my favorite the former is also my favorite drink. :)


3) The birthday problem :
" In a set of "n" randomly chosen people what is the probability that a pair of them will have the same birthday."
(Assuming the year is non leap year i.e. contains 365 days)
( As a budding Mathematician I wanted to study the generalized birthday problem but since the Density of my social network is tending to zero(to know more click here),it's better I stick on to the trivial case.:) )

Its clear that for n>365 probability that a pair will have same birthday is 1.

Solution:-
Let
n : Event that a pair of people will have the same birthday.
n# : Event that a pair of people will not have the same birthday.
P(A) denotes the probability of occurrence of event A.
_
P(A) denotes the probability of not occurring of event A(=P(n#) here)

we must have P(n) + P(n#) = 1

Since it's easier to compute P(E#) in this case we have

P(E) = 1 - P(E#) 

So,the required result is 
             _
p(E)=1-p(n)

In my case as the density of social network is low, it's guaranteed that p (E) will be low but let me wait till tomorrow and see.

4) Cake cutting Algorithm: 
I may or may not cut a cake for my birthday also I don't eat cake much ,maybe I  have nothing to say, but here's a video on how to mathematically cut a cake the right way:

An equation means nothing to me unless it expresses a thought of God.                                                                                    - Srinivasa Aiyangar Ramanujan.

My two language affinity   and  my embarrassment : 

It's uncommon and rare for me to interact with someone unless I really need to interact  or they interact with me  and of course when it involves mathematics :) 

But in one of such situations ,mainly when someone asks me "Where are you from?" and I reply "Andhra Pradesh ,Tirupati" and they ask me "so you know Telugu ,Please tell me the exact word for this -- (some random word)".

and at that instant of time(Time of embarrassment T_E)  my whole mind goes blank. I can't get anything related to it. I try to repeat the word several times in my mind, in English,.in Hindi .But even then no use of it. Strangely this happens due to my two language affinity.I would get the exact word for what he had asked in Tamil.

 and this gave me an idea to mathematically think about it.

Let us see my Language set,

L(x) = { Tamil,Telugu,English,Hindi} 

and x_1 - Tamil ,x_2- Telugu and respectively...

So when I speak something (which happens not to often) :
D(t) =  a(t)x_1 ^(x) + b(t)x_2^(y) + c(t)x_3^(z)+d(t)x_4^(w) be my discourse function .

So at t=T_E , b(t) =0

Some other constrains that creep in:

For t=Time at Home(as I speak in Tamil), b(t),c(t),d(t) = 0
As I have studied and wrote x_2(Telugu) , x_3(Hindi),x_4(English) clearly  y>z>w>x.

But finally axioms must hold true

so when I live in my 4-D M Space,

There is no time (the greatest enemy of my life)  .So D(t) = 0

But this is not true as The Axiom of 4-D M Space states that,

" When in 4-D M Space , D= M^(m)  where M is "The Language of Mathematics" and m is unpredictable."

"If a healthy minded person takes an interest in science, he gets busy with his mathematics and haunts the laboratory." ~W.S. Franklin

This perhaps was the first thing I proved myself as a student who didn't like math much in 9th class ......unknowingly rediscovered a proof of Pythagoras theorem:- 

Theorem ...It's something which is hard to propose and it's even more hard, in fact the hardest thing to prove (sometimes).As Arthur Eddington says,
"Proof is an idol before which a  pure mathematician tortures himself"


But back then I didn't even know what I was about to do in my life.Till then Math was just a part of my school curriculum.As most of the others ,I T.V.Karthik ,myself found Math as a green ,slime thing that looked ugly,tasted awkward and really  hard to digest .Mom used to advise me to study hard and score a 100 .But I used to get around 90 .Looking at the answer paper ,mom used to say" Ennappa? Thiripiyam goli aadditu vandhaya?".(Even today I write every math exam to my level best but never scored 100%)
It was a time when I hated Plane Geometry and found it nonsense. This very statement brings back memories especially this particular one:
" A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him threepence, since he must make a gain out of what he learns."."

But I came across Pythagoras theorem in 8th ,knew it as a result but not as a theorem and not even knew how to prove it.Yet the world as far I knew then viewed it as a great thing .

Teacher asks "Do you know the PYTHAGORAS THEOREM?"

A random kid in the class (definitely not me)  says " Yes.In a right angled triangle the square on hypotenuse is equal to ......." blah.blah.blah  the kid goes on and on even to prove it on the board.After a 10 min silence in the class everyone looking at the kid as if he came from outer space,class ends with the teacher ordering everyone to clap for the student.  ....................

My Memories fade out.

But a great thanks to my best friend in 9th class, I understood Plane geometry and my Mathematical journey began there which has been unstoppable till today.

For those interested in Pythagoras theorem's proof which I rediscovered as a student of 9th Standard,here it is:

My Memories fade in :

"I remember myself ,sitting in the front room ,with a blue pen in my left hand ,writing not on a white paper but on a brown envelope ,the above proof.I was excited ,my joy knew no bounds.I was happy for the first time in my life. But to my astonishment and deep regret I Google up  the result only to find that Pythagoras theorem was proved in the above manner by Jack Oliver in 1997."

Even today I don't know what was enigmatic in that proof but it gave me a push towards mathematics and it was just sufficient for me to set my life into motion mathematically.:)

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.  ~Albert Einstein, Sidelights on Relativity

Tired of waiting for getting my beard trimmed ,I decided to dive into my mathematical thoughts:-

It was a fine evening and I was tired of my schedule that day .I was sweaty and I was like what my mom would call as " எண்ணை  கு டம் ".My beard had grown, it started to itch me .I was irritated and decided to get it trimmed. The lazy me decided not to do it myself and so I went to the barber.

As usual someone was already up there and the barber said to wait for 20 min. I was tired already and to cool down I sat and thought "What hidden math could be in this place ?" I had the answer "Barber's Paradox"

Originally Bertrand Russell's Paradox but to popularize it ,he stated it as,

A city of people has one and only one barber  and he has been defined as

A barber is a person who shaves people and only those people who do not shave themselves.

Now the question is Should the barber shave himself or not?

If he shaves himself,he can no longer be called a barber.If he doesn't then how should he get shaved?

Russell's Paradox:- Does the set of all those sets which do not contain themselves contain itself?

This paradox is important as it couldn't be explained by George Cantor's theory of "What is a Set?" and it clearly showed that a strong Foundation of Set theory is needed.

Resolving the paradox:-

If one would think to resolve it , he or she would arrive at the conclusion that such a barber doesn't exist or The barber must be a women .etc.

Mathematically the problem boils down to ,

Let B = { x ε A : x ∉ x } .....(*) be a set

(Constructed using Axioms of Specification and Extension) .

Now the question is does Bε A ?

Solution:

Suppose B ε A 

Either B∈B or B∉B

then by (*) ,

i)If BεB we must have B ∉ B which is a contradiction so B ∉ A. 

ii)if B ∉ B then as B ε A ⇒ Bε B  which is a contradiction  B ∉ A.

So definitely B ∉ A. So we cannot conjure up a set by saying magic words like " Set of all the sets.."  or in other words 

 nothing can contain "Everything"

As I completed my analysis by thinking deep ,20 mins were completed and  it was my turn to get trimmed. A barber mentioned above may not exist but I was happy to have a barber who could shave me and save my time.

"As for everything else, so for mathematical theory: beauty can be perceived but not explained."

                                         —Arthur Cayley.