On Bitcoins

For now, the bitcoin gini index  is as divergent from the ideal as can be. More on what makes this tick to follow soon..

Shanon Entropy and Information Gain

Shanon's Information Gain/ Entropy theory gets applied a lot in areas such as data encoding, compression and networking. Entropy, as defined by Shanon, is a measure of the unpredictability of a given message. The higher the entropy the more unpredictable the content of the message is to a receiver.

Correspondingly, a high Entropy message is also high on Information Content. On receiving a high Entropy/ high Information Content laden message, the receiver has a high Information Gain.

On the other hand, when the receiver already knows the contents (or of a certain bias) of the message, the Information Content of the message is low. On receiving such a message the receiver has less Information Gain. Effectively once the uncertainty about the content of the message has reduced, the Entropy of the message has also dropped and the Information Gain from receiving such a message has gone down. The reasoning this far is quite intuitive.

The Entropy (& unpredictability) is the highest for a fair coin (example 1.a) and decreases for a biased coin (examples 1.b & 1.c). Due to the bias the receiver is able to predict the outcome (favouring the known bias) in the later case resulting in a lower Entropy.

The observation from the (2-outcomes) coin toss case generalizes to the N-outcomes case, and the Entropy is found to be highest when all N-outcomes are equally likely (fair).

Queuing System and Little's Law

On its way..

Triangular Matrix

There are two special kinds of square matrices (n X n matrix) known as the Upper Triangular Matrix & the Lower Triangular Matrix.

Upper Triangular Matrix - Elements below the main diagonal are all zero.
     i.e. M[i,j] = 0, for i > j

Lower Triangular Matrix - Elements above the main diagonal are all zero.
     i.e. M[i,j] = 0, for i < j

where,

Main diagonal - The diagonal running through M[1,1] to M[n,n] for a nXn square matrix.

NP, NP-Hard & NP-Complete

NP: One that has a non-deterministic, polynomial time solution. (Mind you, it is still polynomial time). The other way to define it is, given a solution (certificate), one can verify correctness of the solution using a deterministic Turing Machine (TM) in polynomial time.

NP-Hard: A hard problem - at least as hard as the the hardest problem known (& unsolvable) so far.

NP-Complete: ( NP ) <intersection> ( NP-Hard ).

NP & NP-Hard are different, and overlap only in certain cases when they are NP-Complete. Many NP-Hard as a result, are not NP-Complete.