Welcome to my 4-D M-Space:- A Mathematical insight of my Life.: July 2014

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