Beware of Tamil (Mega) Serials

Today in my house we were to supposedly watch a serial, but apparently watched advertisements. In half an hour's time allocated for a serial, the advertisement took away 20 minutes, remaining was the serial. To add to the frustration, the story covered was just 2 scenes in which people were repeating the same dialogue in different ways to while away the time. This is the state of tamil serials which run in almost every house from 9-11PM. Though I would not claim that I was not a victim of it, I have come out of it. Even if I want to follow a serial, I see once in a month which is more than sufficient frequency to follow them. There is a serial named "Anandham" (meaning happiness) in which ironically people seldom laugh or be happy.

I would seriously suggest people who have not watched and now reading this blog to keep away from these serials.

Eigen values are not a good measure of Positive definiteness of a matrix

A matrix cannot be concluded as to whether it is positive definite or not using the eigen values alone.This can be proved using a counter example which is as follows.

This matrix has a has eigen values 1 and 2. but for x = 1, y =1 the value of the X'AX is -97

This counter example clearly proves that eigen values cannot be used to conclude whether a matrix is positive definite or not. Nevertheless it can be used when the matrix is symmetric.

PS: Discussion with T.R. Maruthi, MS scholar, EE Dept, was the impetus for this blog. Our discussion helped me know about this.

Information Theory and Thermodynamics, A parallelism

The word which strongly connects Thermodynamics and Information Theory is Entropy. In Information theory, the Gaussian distribution (figure on the right side) is the distribution with the maximum entropy which is defined as 'S' below.

This also means that minimum amount of information (mean and variance) is sufficient to describe the distribution completely without knowing any further information. Addition of any further information to the system will only reduce the entropy. Similarly, in thermodynamics as the system tends to equilibrium, the amount of information required to describe the system reduces and sufficient to know the average properties of the system.

In a process, as the number of contribution to noise(with any kind of distribution) increases, the overall noise tends to Gaussian distribution, which is a consequence of central limit theorem. This physically means that as more and more noise is added to a system, the amount of information with which one can characterize the noise reduces and the description has to be done with a minimum assumption on the noise. A thermodynamics counterpart of number of noise would be the number of molecules with which one will have to characterize the system. As the number of molecules increases, the amount of information which one has to keep track of increases and consequently one will be left with no choice than to work with macroscopic properties which allows to describe the system with least information.

One more interesting analogy is central limit theorem and second law of thermodynamics. central limit theorem says addition of noises makes the overall noise tend to gaussian, that is move to a maximum entropy. Second law of thermodynamics also says that any process moves towards maximum entropy.

One of the classic papers to read about this is 'Information theory and Statistical Mechanics' by E.T. Jaynes (physical Review, vol 106, pg. 320).

A basic understanding of Central limit theorem can be got from wikipedia!

I would like to thank prof. Shankar Narasimhan whose lectures on 'Multivariate data analysis' was a preeminent factor in my attempt to write this blog.

Humility personified..

I was reading the biography of one of the greatest physicist and a nobel laureate, S.Chandrasekar. In a letter to his brother from England, he wrote, "Do you think, that just because one read a little calculus and conics when his equals did not read a little statistical mechanics which others of his age did not, happened to publish a few papers, Do you think that he is better than his friends, or Do you imagine he has in any case intrinsic merit? You are woefully mistaken if you think so... "

This was the man who published papers in the Proceedings of the Royal Society and in the Philosophical Magazine in the second year of the undergraduate studies, without any special guidance. He believes that advances in science are not made by publishing papers, or making a discovery but by earnest study and a lot of work. He stands as a personification of humility.

PS: This is my first blog. I would be very happy to receive suggestions and comments which will help me in improving my writing skills and spread my horizon of knowledge.