Stationarity of a Markov source

Consider a discrete Markov source, ona finite alphabet set. Let the initial distribution be and the transition probability for the step be . When can we say that is stationary?
Clearly, the source has to be time invariant and thefore we need . For to be stationary, we need

etc, where is the distribution. But . Thus guarantees that all ’s have the same distribution. Now, consider, say, and .


It is clear that for the two joint distributions to be equal, is enough and therefore is sufficient.

 

Icing in the cake

The centenary celebrations of the Indian Institute of Science concluded this week. The centenary year saw many events- the centenary conference attended by Kalam, Dr Sam Pitroda’s speech, the centenary marathon run and a concluding ceremony precided over by the President Smt Prathibha Patil. To mark the end of the celebrations there was a violin concert by Shree Ganesh and Kumaresh. What should I say about the concert! Before the concert one of my friends asked me about the artists to which I replied, “Attend the whole concert. Your will remember your life as before the concert and after.”

The concert was in a big special Pandal made at the Gymkhana grounds. For once there was no issue of no-seats-to-late-comers in IISc . The acoustics was good. Accompanying the violinists were Phalgun and Krishnan on Mrudangam and Ghatam. They started with a short piece in Sahaana. It set the mood for the moment. It was followed by Mayamaalavagowla – their own composition called Ragapravaha – and then a Varnam in Kannada. Then they played Reethigowla, one of my favourite raagas. Later they played one more of their composition, in Naasika Bhooshani, followed by a sudden spurt of Kadhanakuthoohala. What then started as an RTP in Naatabhairavi later turned to a wonderful Raagamaalika. And then the percussion duet. At the time of vote of thanks the speaker did not resort to any cliches and the whole audience gave a standing ovation to the artists.

Ganesh and Kumaresh performed in my native place when I was ten. After that, I had to wait for another 11 years to listen to their next concert. They have been my favourites. Though I expected them to play some of their fusion compositions, they completely enthralled the audience just with carnatic classical.

Let there be more centenary celebrations. Why can’t time run faster !

 

Snippets

One of my profs made the following remark in the class:

“I hate late comers. They think that they can come a few minutes late and still follow the class. But then by induction they can skip the whole class.”

Clearly his reasoning has a mistake. The induction proof wont go through that ‘N’ for which he erases the black board !

My analysis professor mentioned the following comment made by Rudin : The whole of analysis is all about finding out when two operations commute ! (I could not find the source)

If one goes through the whole of Rudin’s books on Analysis – Principles of mathematical analysis and Real and Complex analysis – one finds only only two figures: The McGraw Hill logo on the front cover and the TMH logo on the back cover ! (Well Apostols book, for example, has more figures)

Mentioning to the fact that a random walk in one or two dimensions is recurrent but a three dimensional random walk is not, the great Kakutani made the following remark :

“A drunk man will find his way home, but a drunk bird may get lost forever.”

 

Large deviations – an example

The great Prof. Varadhan made a visit to the IISc on Febrary 13th. He gave a lecture at the IISc faculty hall on entropy and large deviations. The following example was interesting.

Consider a bug with limited energy trapped in the valley of a steep peak. It tries to scale the peak and reach the other side. Evey time it fails, it falls to the bottom of the valley. It changes its strategy and starts all over again. After a very large number of attempts, it succeeds and reaches the peak. As it goes down the other side, an observer on the other valley sees the bug coming down. He becomes curious, goes to the top of the peak and looks below to see how steep was the bug’s scale.

Now, having seen the bug, what can he conclude about the strategy adopted by the bug? Prof Varadhan made the following comment. The observer can surely conclude that the bug would have adopted the most efficient strategy. That has to be the case because as the number of iterations becomes very large, the probability of success is dominated by that of the best strategy and if a success ocurs it must come from the best strategy.

 

Self Inquiry

In about 1902 a person named Sri Sivaprakasam Pillai met Sri Ramana Maharshi at the Virupaksha cave in Thiruvannamalai and posed his doubts on self realisation. His questions and Ramana Maharshi’s answers (about 28 in number) have been published as the book ‘Who Am I ‘ . These questions and their answers can be read here. I would like to highlight below one question which explains how to practice self inquiry. It may be used as a method to channelize ones thoughts. Here is a photo of Ramana in his young age.

Qn: What is the means for constantly holding on to the thought Who am I ?

Ans: When other thoughts arise, one should not pursue them, but should inquire: ‘To whom do they arise?’ It does not matter how many thoughts arise. As each thought arises, one should inquire with diligence, “To whom has this thought arisen?”. The answer that would emerge would be “To me”. Thereupon if one inquires “Who am I?”, the mind will go back to its source; and the thought that arose will become quiescent. With repeated practice in this manner, the mind will develop the skill to stay in its source. When the mind that is subtle goes out through the brain and the sense-organs, the gross names and forms appear; when it stays in the heart, the names and forms disappear. Not letting the mind go out, but retaining it in the Heart is what is called “inwardness” (antar-mukha). Letting the mind go out of the Heart is known as “externalisation” (bahir-mukha). Thus, when the mind stays in the Heart, the ‘I’ which is the source of all thoughts will go, and the Self which ever exists will shine. Whatever one does, one should do without the egoity “I”. If one acts in that way, all will appear as of the nature of Siva (God).

 

Dividing the cake

Let us consider the following problem which is a popular example in game theory.

A mother wants to divide a cake between her two children. To make both of them happy she has to ensure that each one gets an equal share. That is she has to cut the cake into two equal halves. But the problems is that the cake does not have a regular shape. Hence what is equal to her eyes may not be equal to the eyes of her children. The consequences of an unequal division are imaginable. So how can she make both her kids happy?

The solution is to ask one of the kids to cut the cake into two and the other to choose the piece. It can be verified that this solves the problem. The interesting thing about the solution is that the mother was able to satisfy both the kids even without knowing what will make them happy.

Now if the mother has N kids what will she do?

What will happen if some kids form a collusion and try to get a bigger share for them ?

 

Reductio ad absurdum

Mathematicians often turn into poets. The joy induced by a mathematical argument often makes a mathematician go beyond theorems and proofs to praise it. I really liked this quote on reductio ad absurdum (proof by contradiction) which I found on Terence Tao’s blog . The existence of infinitely many primes has been proved by Euclid using this technique.

“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game ”.
(G.H. Hardy, 1877-1947)

 

Why Indians should celebrate Valentine’s day

Here is V Gangadhar’s explanation for why Indians should celebrate Valentines day. V Gangadhar writes the column ‘Slice of Iife’ in The Hindu newspaper on Sundays. Taken from the archives of Slice of life.

Manmadhan was in the habit of shooting arrows made of fragrant flowers at his victims, making them fall in love. Somewhat like Cupid of Greek legend. Unfortunately, Manmadhan once fooled around with Shiva, the Angry Young God, who was not known to be particularly romantic. On being struck by the romantic arrow, Shiva reacted strongly. He opened his third eye, discharged the requisite fire reducing poor Manmadhan to ashes. I do not know when this exactly happened but it would be nice to celebrate the martyrdom of Manmadhan as our version of Valentine’s Day.

 

Gmail adds colour

Gmail has now made available a wide variety of themes (more than 500 as far as I could count) for its users. Unlike their orkut counterparts these themes dont fill in your window but just give an ambience. The one that caught my attention was Phantasea designed by Mathew Paul. Its a dynamic theme – it changes its colours according to your place and time (takes one of three colours). Those who lose track of time in the internet now have a free wake up call. The one with predominantly blue shade looks very nice. Somehow my iGoogle page and Gmail are showing different shades of the theme though I have set the same time-zones for both ! Some varieties of this theme have been displayed in Mathew Pauls page.

I checked out some more themes. There are themes on Newyork, London, Paris and many other cities. Filmy themes are in general diasappointing. There is one for Tom cruise, Asin (!), Surya-Jo, Abhi-Ash, Katrina Kaif Amir Khan … The Rajnikanth theme could have had its music also to give a real feel for the fans of the superstar! The chroma Velveteen red featurse a nice shade of red. Nature lovers would like the monument valley, misty morning and natural details ; secret garden is really good. No theme on Sachin; There is one on the 64-cross-64 chess board which has been a source of inspiration as much for designers as for brainiacs.

Meanwhile if you are an artist for yourself you can create themes of your own and make it available for others.

 

The “Gaussian” Distribution

While reading through the achievements of the Kerala school of mathematics I came across the Madhava-Leibniz formula for computing the value of pi/4. Though the formula was discovered by Madhava its popularly known as Leibniz formula. This is an instance of Stiglers Law, which is the tendency of NOT attributing a discovery to its original discoverer.

Popular instances of the law include the United States of America, Halleys comet, Planck’s constant and, to my surprise, Gaussian distribution! I came to know that the Gaussian distribution was not first proposed by Gauss, a very late discovery for a communication engineer. The distribution was first proposed by de Moivre of the de Moivre-Laplace formula to approximate binomial distributions for large n. Now by recursive application of Stiglers Law can we say that De Moivre also was not the first proposer of the normal distribution ?