Another day at the canteen :-

But this time my dinner was disturbed due to a freaky noise phat-phat-phat.

I looked up and saw that the black areas of the weird ceiling in which fans were put on were too small for some fans and they became so loose that their wings rattled and hit the boundaries of the empty space.

This gave me an idea. The space for the fans were square and the fans rotation is circular.

It struck to me  "Wow! I noticed a mathematical beauty" .:)

"You can't Square the circle." 

Which is in fact an ancient geometrical problem proposed by geometer s to construct a square with the same area as that of a given circle by only using compasses and a straight edge (ruler). 

Elementary mathematics tells us that the square will have a side of √Π r were r is the radius of circle.

But to construct such a square is impossible with a straight edge as √Π is transcendental (Meaning: There is no polynomial F(x) with rational coefficients such that F(Π = 0) .So one can't tell the exact value of Π .So the square you construct never has an area equal to the circle.

So "Squaring the circle "is used as a metaphor for "Doing the Impossible."

"It is easier to square the circle than to get round a mathematician."

-Quoted in H Eves In Mathematical Circles (Boston 1969).

While Eating food at the Canteen:

Ceiling......

This word reminds me of two things:

1.The Ceil function f(x) = [ x ].Reminds me of my habit of looking up when I don't understand something trivial.

2. Ceiling the roof of any room...

Whether it's due to my human tendency or not I tend to look up while thinking (as far as I can remember someone asked me when I was in 6th class "Why do you look up? Are  your test marks up there?").

Looking at the ceiling has always helped me to reconstruct my Ideas one of them being the following one:

At the end of  the day when things start cluttering up in my mind and the time is 8:30 pm I rush to the canteen to have my dinner.As usual when I look at the menu my hunger dies down and  the the extent to which I eat my food becomes an asymptotic to it.I started eating my food and Enakku pora pochhu.

As my mom always used to advise to look up in such cases I followed it.I looked up and saw the ceiling of the canteen,which shook me by its design

At once Rossmo's Formula struck to me. Why? Because the surface of the celing looks like the taxicab metric!

Where,

Pij is the probability that the criminal resides in the ith row and jth column

B = Radius of Buffer zone

(xn,yn) = Coordinates of points where crimes were committed

g, f are constants that are evaluated using experimental data ( just like Curve fitting).

C = No. of Crimes Committed

K = Constant of proportionality .Determined by using the fact that,

∑pij = 1. (Sum of all probabilities=1).

and ø = 0 if  B < |xi-xn| + |yi - yn|

            = 1 otherwise.

Yeww!  It's rather messy :( .
Let's simplify it  :-

Buffer zone:-

It is the region which is around the house of the criminal where he is less likely to commit a crime. It is of radius B with centre as the criminal’s house. 

In this case ǿ = 1 and the formula breaks down to:

This is true because as you move away from the buffer zone the criminal is less likely to travel a larger distance to commit the crime.It's like “Distance Decay”. Probability obeys the inverse distance relationship just like coulomb’s inverse square law of Electrostatics.

Case 2:

Let’s write the Rossmo’s Formula for region inside buffer zone:

In this case ǿ =0 and Rossmo’s formula breaks down to:

Once inside the buffer zone the criminal is alert he makes sure that he doesn’t commit a crime near to his house. So the probability increases radially outward. If the buffer zone is larger, then we have a greater chance of locating the hot zone,

So,

In Taxicab or Manhattan Geometry 2B is the maximum displacement the criminal can undergo where B is the radius of buffer zone (in this given metric it is a square region unlike Euclidean where it’s a circle).

The criminal thinking that he is clever he hides his crime by taking the routes other than displacements to hide his crime. So if for a crime the Manhattan distance d is less it not only means that it is less probable for criminal criminal’s residence but also he may have committed the crime cleverly by avoiding the shortest route.

What routes can he take by avoiding the shortest ones?

Well whatever the one route he chooses must be less than or equal to

Also:

If C1 and C2 are two crime sites within a buffer zone such that Manhattan distance of C1 is > Manhattan distance of C2,

Then probability for criminal residence is higher for C1 than C2 because,

(M.d)C1 > (M.d) C2

- (M.d) C1 < - (M.d) C2.

(2B-M.d)C1 < (2B-M.d) C2

[1/ (2B-M.d) C1] > [1/ (2B-M.d) C2]

[1/ (2B-M.d) C1] g > [1/ (2B-M.d) C2] g

From this we can arrive at,

(pij)C1 > (pij) C2 or,

And hence,

And so by considering both cases we arrive at Rossmo’s formula. Ta da! :)

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

                                                                                                   ~John Louis von Neumann

Math ..even when I'm unhappy...

People often ask me " Why do you watch so much of cartoons? "

( actually it's not so much of cartoons. It could be nicely put as some cartoon and a lot of anime .But it would be difficult for non-otakus to tell the difference between the two.)

The reasons to it are many but  one of the prime reasons to it (according to me) could be found by a retrospection of my childhood and that is due to my grandmother's huge list of serials.

Even today in a different way it makes me frustrated.I ,the other day was going through Lectures on Game theory and nothing made a clear sense due to the disturbance created by the high volume of the television.At one moment I got frustrated and kept the notes aside.To my mind came a idea to do an analysis of the reason that has made me unhappy through out my life....

Well here it is ,the analysis of my grandmother's serial watching over the ages :-

Let Y be the year in which  I am/was present and T_1 and T_2 be two non empty sets containing the serials watched by my grandmother in Tamil and Telugu languages respectively

T_1 = {S1,S2,S3,.........Sn}

T_2 = {S1’,S2’,S3’,.........Sn’}

L(x) is relation  that maps each serial 'x' to the  the language of the serial 'x'.Clearly

if we define L:T_1  → T_2 then  L(S1) ≠ L(S1').

τ(S) is a function that outputs the No. of episodes of the serial S aired  till the present date.

Let S be a non empty set containing all the serials watched irrespective of language i.e. 

S = T_1 ∪ T_2

If Y < 2011:

∀ X ∈ T_1  ∃ a unique Y ∈ T_2 such that 

τ(X) = τ(Y)  + t                               where        t >0  and t ∈ N (the set of natural numbers)

and the converse is not necessarily true i.e. the statement 

"∀ Y ∈ T_1 ∃ a unique X ∈ T_2" is not necessarily true.

In other words to say  the function   τ :T_1 → T_2  defined by τ(Y) = τ(X) - t  is not a Bijection.

If Y ≥ 2011 :

Let N be a non empty set containing a list of serials S" i where i≤4  such that  

L(S"i) ≠ L(X) ≠ L(Y)   where X ∈ T_1 and Y ∈ T_2

Γ(s) is a function that outputs the no. of episodes watched till present by my grandmother.

∀  S" i ∈ N  if  Γ(S" i)  >  4  then   delete the element S" i from N and add it to the set S i.e.

∀ S" i ∈ N if Γ(S" i) > 4   then  S" i ∉ N  , S"i ∈ S    ...

and that's what I could analyse and it took me 15 minutes to do and still neither the volume of television has reduced nor the serial has ended .But I hope that either I could perform a Fourier analysis on those sound waves to find a solution to the problem or learn some Game theory so that I could list the payoff's of mine and my grandmother's to find a strategic solution to this never ending game...:)...

"Go down deep enough into anything and you will find mathematics. ~Dean Schlicter"