Srinivasa Ramanujan: Indian mathematician

Ramanujan
Ramanujan

A leading Indian daily started a series on not so ordinary Indian people just before the Independence day on August 15th. On the d-day, ex Indian President APJ Abdul Kalam wrote a piece and talked about Srinivasa Ramanujan, one of the greatest mathematician of current times, from the land that created zero.

He is the person behind Ramanujan number, 1729 which is the smallest number to be a sum of two cubes in two different ways:

1729 = 9*9*9 + 10*10*10
1729 = 1*1*1 + 12*12*12

He was given books on advanced trigonometry which he mastered by the age of 13. While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan’s work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

Ramanujan is generally hailed as an all-time great mathematician, like Leonhard Euler, Carl Friedrich Gauss, and Carl Gustav Jacob Jacobi, for his natural mathematical genius. G. H. Hardy quotes: “The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems… to orders unheard of, whose mastery of continued fractions was… beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly-periodic function or of Cauchy’s theorem, and had indeed but the vaguest idea of what a function of a complex variable was…”. Hardy went on to claim that his greatest contribution to mathematics was discovering Ramanujan.

Share

Definition of a Sikh and Sikhism

Khanda

Who is a Sikh?

Please listen to my podcast on this topic:

Punjabi: Download
 

English Translation: Download
 

Share

Internet censorship and loss of privacy

Surveillance
Surveillance

An excellent essay by Bruce Schneider on government organized initiatives worldwide to reduce internet privacy and for censorship. There is also evidence that criminals can (mis)use such infrastructure for their own purpose. Please click here to read.
 
 
 
 
 
 
 
 
 
 
 

Every year brings more Internet censorship and control — not just in
countries like China and Iran, but in the United States, the United
Kingdom, Canada and other free countries.

The control movement is egged on by both law enforcement, trying to
catch terrorists, child pornographers and other criminals, and by media
companies, trying to stop file sharers.

It’s bad civic hygiene to build technologies that could someday be used to facilitate a police state. No matter what the eavesdroppers and censors say, these systems put us all at greater risk. Communications systems that have no inherent eavesdropping capabilities are more secure than systems with those capabilities built in.

Share

Recursion: my two cents

A lot has been written on the use of recursion in computer programming, yet it remains one of the least understood aspects – especially for beginners. Having visited the Wikipedia page on recursion, I believe the text is hard to understand, and the examples are forced: there is no reason to use recursion to solve Fibonacci series or calculate factorials.

Recursion means solving a problem by splitting it into smaller problems. If the problem is numerical, then splitting it into smaller numbers.

Consider the problem of creating all permutations of a character in a string. If ‘abc’ is input, the program should show ‘abc’,’acb’,’bac’,… and so on. How can we solve this problem?

Permute

We propose to write a function called ‘permute’ which creates permutations of the string that is passed to it. When I wanted to create this function, the first thing I did is to design the tree. Just doing this showed me a flaw in the initial design that I was planning, and later helped me with the debugging. The tree shows that each letter in the input string is extracted from the string one by one, and the rest of the string is passed again as parameter to the same function – the character removed is added to the prefix. Eventually the ‘string parameter’ becomes a single character at which time the permutation is printed.

Have a look at the code, and relate it to the tree:

#!/usr/bin/perl

print("Please enter a string to permute->");
$s = ;
chomp($s);

permute ($s,length($s),"","");

sub permute {
    my ($s,$l,$pref)=@_;
    my $i;

    if ($l > 1) {
        for ($i = 0; $i < $l; $i++) {
            $ch = substr($s,$i,1);
            $rest = substr($s,0,$i) . substr($s,$i+1,$l);
            permute($rest,$l-1,$pref . $ch);
        }
    }
    else {
        print $pref . $s . $sufx . "\n";
    }
}

As I said the first step in such recursive coding is to identify the tree. When I started out, I made the tree in an incorrect fashion. I split permute(abc) into a.permute(bc) and permute(bc).a. I felt that since the basic idea was to permute, this is how we should do it. However, doing this resulted in only four permutations at the bottom of the tree, instead of six as should be. This made me go back to the drawing board.

I want to end this on a humorous note for people who write recursive solutions to simple problems:

To loop is human, to recur - divine.

Share

Evolution of the Eclipse

1980 Eclipse
1980 Eclipse

Having witnessed 5 solar eclipses, one of which was a totality – over a period of 14 years, and having read about the one from 1980, I am going to try and chronicle how the eclipse viewing has changed over the past 29 years.

The photo at the top shows a normally busy income tax office square in an Indian city bereft of any people during the eclipse of 1980.

1980 1996 2009
TV Showed a movie to keep people indoors Showed a live telecast of the eclipse Let people just watch on their own, nothing special
People Stayed indoors mostly. Only the scientists watched the eclipse. It was quite a people event, the educated watched the superstitious didn’t Everyone watched! Right from the bathing sadhus to the city dwellers
Advise by the ophthalmologists Don’t watch! Not even with filters. Don’t watch! Not even with filters. Forgotten
Media Newspaper Newspaper / TV Newspaper / TV / Web / Flickr
Astrologers Spreading doomsday Spreading doomsday Spreading doomsday

timeline
In 1980, the eclipse was watched in 15 eclipse camps – it was a first in 82 years. Some interesting news items from that time:

  • No let up in rituals. Normally people try to ward off the harmful effects of the eclipse by putting pieces of ‘Durva’ into food items like milk, butter, ghee and drinking water.
  • With the repeated warnings of all the mass media still ringing in their ears, the elite class didn’t dare to have a direct look at the eclipsed sun.
  • ‘It was simply beautiful’ said an observer in Taita hills, Kenya. ‘We had 3 minutes 50 seconds of totality’. ‘The elders say we should go inside’ said Henry Kazungu 28, a member of the rat eating Giriama tribe. ‘But I want to stay out and see it. I want to be able to say later that this great thing happened in this year and I was there’.

Some news items from the 1995 eclipse:

  • The council of astrologers: “The sun signifies the rulers. Thus in the region where the total solar eclipse will be visible the heads of the governments will be adversely affected.”
  • With barely three days left for the solar eclipse, a serious controversy among scientists and ophthalmologists has put the West Bengal government in a quandary and the people in a fix.
  • One publication showed a photo of an elephant at Yamuna Ghat waiting patiently for the sun to be freed from its lunar embrace so that it could have a holy dip.
  • Unlike in 1980, millions of people watched the event last fortnight, turning it into a mass science festival.

From 2006:

  • Schoolchildren clapped and cheered as the first total eclipse in years plunged Ghana into daytime darkness, a solar show sweeping northeast from Barzil to Mangolia.

In 2009, the clincher was a photo in one of the national dailies showing the bathing sadhus wearing solar goggles and watching the eclipse. Flickr also made a world of difference: we could see various cultures in various countries trying to watch the eclipse, and interpreting it in their own ways.

Share