Kisi Aur Koi Bhi
Google has come up with a new English to Hindi translation service. In my opinion, it is a decent effort. Although, the quality of translation is not very impressive, yet. Indeed, it is not easy to make a computer understand and interpret natural language correctly.
Well, I was just looking for something fun to indulge myself in, and thought of translating this blog in Hindi using the above service. This is what resulted. Much fun was had.
Notice what the name of this blog appears as. Try reading some of the posts, especially the math ones. 
Oh, and don’t miss the names of the blogs on my blogroll. I particularly loved pehlaa siddhaanton, achche maTh, kharaab maTh and mEraa sikkaa pakshapaati! Go, figure out the last two. Lol!
PS: Wah! mEraa sikkaa pakshapaati! Sounds like a super-hit Hindi movie!
On Domain Stretching, Conditional Convergence and Absolute Convergence
Sometimes, an infinite series may not be as expressive as (or carry as much information as) the function it represents. In this post, we’ll mainly discuss this concept (and also look at conditional and absolute convergence briefly).
Consider the following infinite series: f(x) = 1 + x + x2 + x3 + x4 + x5 + …. Does this series ever converge? Try substituting 1/2 for x and see what happens. It converges, as we saw here. So, we write this as f(1/2) ~ 2. The twiddle or tilde sign here indicates that f(x) asymptotically tends to 2 at x = 1/2. Similarly, it is possible to prove the following: f(-1/2) ~ 2/3; f(1/3) ~ 3/2; f(-1/3) ~ 3/4 and so on. It is trivially true that f(0) = 1.
Let us now substitute 1 for x. We get f(1) = 1 + 1 + 1 + 1 + …. The series diverges at x = 1. It is obvious that the series diverges for x = 2, 3, 4, 5, … too. Observe the behaviour of f(x) when x = -1. We get f(-1) = 1 – 1 + 1 – 1 + 1 – 1 + …. If you take an even number of terms, then f(-1) = 0, and if you take an odd number of terms, then f(-1) = 1. We see that the partial sum of this series is definitely not becoming infinitely large. However, it is not converging either. This is considered a form of divergence. We can verify that f(-2), f(-3), f(-4), … also diverge. Loosely speaking, they seem to oscillate and go off to positive infinity and negative infinity at once.
We see that f(x) seems to have values only when x is between -1 and 1, exclusive. In other words, we have:
Observation 1: The domain of the function f(x) is from -1 to 1, exclusive.
Now, let us rewrite f(x) here and simplify it a bit.
f(x) = 1 + x + x2 + x3 + x4 + x5 + …
That is, f(x) = 1 + x (1 + x + x2 + x3 + x4 + … )
This implies f(x) = 1 + x f(x)
Therefore, we have f(x) = 1 / (1-x)
On simplification, we have:
Equation 1: 1 / (1-x) = 1 + x + x2 + x3 + x4 + x5 + …
Now, the question we ask is: Are the LHS and the RHS of this equation one and the same? Earlier in this post, we had discussed the value of the RHS for various values of x, viz. -1/2, -1/3, 0, 1/3, 1/2. You will see that the LHS concurs. However, they are not one and the same thing! They have different domains.
We can see that 1 / (1-x) has values everywhere except at x = 1. When we see this in contrast to Observation 1, we see that the domain of 1 / (1-x) is “stretched”. It includes the domain of the infinite series and more. This indicates that an infinite series sometimes defines only a part of a function. More appropriately, an infinite series might define a function over only a part of the function’s domain.
So, that was about “domain stretching” to uncover hidden properties of a function. I was sorely tempted to discuss this beautiful concept here. So, I had to make some room for it. 
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At this point, you are free to jump to Equation 2 below, from where our actual discussion of conditional and absolute convergence begins. Or, you can just stay on and see why Equation 2 is true.
On integrating Equation 1, we get:
-log(1 – x) = x + x2/2 + x3/3 + x4/4 + x5/5 + …
Therefore, log(1 – x) = – x – x2/2 – x3/3 – x4/4 – x5/5 – …
At x = -1, we get:
Equation 2: 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + … = log 2
Let us denote this series (the LHS of Equation 2) as S. So, S converges to log 2. However, for S to converge to log 2, there is a condition that needs to be satisfied: the terms have to be added in that order. If you add the terms in a different order, the series might either converge to a different quantity or diverge. For example, let us rearrange the terms in this series as follows:
1 – 1/2 – 1/4 + 1/3 – 1/6 – 1/8 + 1/5 – 1/10 – …
Or (1 – 1/2) – 1/4 + (1/3 – 1/6) – 1/8 + (1/5 – 1/10) – …
This is equivalent to 1/2 – 1/4 + 1/6 – 1/8 + 1/10 – …
This simplifies to 1/2 (1 – 1/2 + 1/3 – 1/4 + 1/5 – …) or 1/2 (S)
The rearranged series sums up to half of S!
Such series, whose limit depends on the order in which their terms are arranged, are said to be conditionally convergent. Those series that converge to the same quantity, no matter what order they are summed in, are said to be absolutely convergent.
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Other articles in this series: On Convergence, On Divergence
On Divergence
My earlier plan was to write a bit about the various types of convergence in this article. However, I think we would do better to understand divergence first. So, here goes. In the previous post, we saw an infinite series that converged. In other words, it exhibited limiting behaviour. (This infinite series was a geometric series with 1 as the first term and 1/2 as the common ratio. As a matter of fact, any geometric series, with a first term a and a common ratio r, converges. That is, the sequence of its partial sums has a limit. As we saw, for the geometric series in the previous post, the limit of the sequence of its partial sums is 2.) But, what of those infinite series that do not exhibit limiting behaviour towards any quantity? Such series (i.e. ones that are not convergent) are said to be divergent. The partial sums of a divergent series go on increasing without limit. Loosely speaking, the sequence of partial sums of a divergent series tends to infinity.
From the example infinite geometric series in the previous post, we can make the following observation:
Observation 1: If a series converges, then the individual terms of the series must tend to zero.
In that series, the tenth term is 0.001953125, the twentieth term is 0.0000019073486328125, the fiftieth term is (approx.) 0.00000000000000177636, and so on. Notice how the Nth term is tending to zero as N increases. So, Observation 1 seems true enough. And therefore, we can conclude that if the individual terms in a series do not approach zero, then the series diverges. However, the converse of Observation 1 is not true. If the individual terms of a series tend to zero, the series does not necessarily converge — it may diverge. A very good example of this is the following harmonic series: 1 + 1/2 + 1/3 + 1/4 + 1/5 + ….
In this harmonic series, the tenth term is 0.1, the hundredth term is 0.01, the millionth term is 0.000001, and so on. It is clear that as N increases, the Nth term in this series tends to zero. However, this series is known to diverge. And there happens to be a rather old but elegant proof of its divergence by Nicole d’Oresme, a French scholar (c. 1323 – 1382).
d’Oresme observed that (1/3 + 1/4) is greater than 1/2. Similarly, (1/5 + 1/6 + 1/7 + 1/8 ) is greater than 1/2. So is (1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16). And so on. By first taking two terms, then four terms, then eight terms, then sixteen terms, and so on, it is possible to group the series into infinitely many “blocks”, where each block adds up to a value greater than 1/2. No matter how many such blocks we consider, it is always possible to come up with the next, well-defined block. That is, there is always a value x > 1/2 waiting to be added, no matter how many blocks we have already added up. Loosely speaking, the sum of the entire series must therefore be infinite. That is, the sum of the series increases without limit. The series diverges. Quod Erat Demonstrandum.
This elegant proof by d’Oresme seemed to have been lost on the world for several centuries. Pietro Mengoli proved this result all over again in 1647, using a different approach. Forty years later, Johann Bernoulli proved it with yet another approach. Shortly after, Jakob Bernoulli came up with yet another proof! Neither Mengoli nor the Bernoulli brothers seemed to have known about d’Oresme’s fourteenth century proof. John Derbyshire asserts that d’Oresme’s proof remains the most elegant of all the proofs for this result, and is the one given in textbooks today.
PS: Johann Bernoulli was the father of Daniel Bernoulli (of Bernoulli’s Principle fame). Jakob Bernoulli (of Bernoulli Trial and Bernoulli Numbers fame) was Johann Bernoulli’s elder brother. That is one super-cool family, eh?
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Other articles in this series: On Convergence, On Domain Stretching, Conditional Convergence and Absolute Convergence
On Convergence
Currently, I’m reading John Derbyshire’s Prime Obsession. In this book, Derbyshire makes a very good effort to explain about the distribution of prime numbers and the Reimann Hypothesis in the layman’s language. Mathematics has been treated rather loosely at several places in the book, but you can forgive Derbyshire that. He has, in fact, tried to make mathematical concepts less mathematical and more intuitive in his book. The subject of the book is not central to the subject of this series of articles, though. In the next few articles, I am going to try and explain the concepts of convergence and divergence as simply as I can, using some examples from Derbyshire’s book. In this article, I’ll be talking about the concept of convergence.
Consider a finite series. For ex. 1 + 1/2 + 1/4 + 1/8. This series (essentially a sum) can be calculated precisely, because the number of terms in it is finite. The sum, in fact, is equal to 15/8 or 1.875. Any such finite series can be equated to a known quantity. However, when a series is infinite, i.e. it has infinitely many terms, precisely computing the sum is not possible — the series computed up to any large N can always be bettered by adding the (N+1)th term. In other words, it is not possible to equate the sum of an infinite series to a known quantity. So, the question is: can it be “approximated” at least? In other words, does it exhibit limiting behaviour towards some quantity? To put it in yet another way, does it tend toward some quantity? The answer is: Yes, sometimes. (At other times, you cannot zero in on a quantity for an infinite series at all. More on that in a later post.)
Consider the following infinite series now: 1 + 1/2 + 1/4 + 1/8 + 1/16 + …. Both, the finite series we saw earlier and this infinite series, have the same pattern of occurrence of terms (or progression). They are both geometric series, with 1 as the first term and 1/2 as the common ratio. Yet, these two series are entirely different in nature.
I know I have said that computing an infinite series is not possible. Even so, let us just start adding up the terms in the above infinite series. Let us see where it leads us. Up to four terms, the sum is 1.875, as we have seen earlier. The mathematical term of art for this is: the partial sum up to four terms is 1.875. Up to five terms, the sum is 1.9375. Up to six terms, 1.96875. Up to ten terms, 1.998046875. If you keep adding more and more terms like this, you will notice that the partial sum improves with the addition of more and more terms. However, you will also notice that the improvement in the Nth partial sum over the (N-1)th partial sum diminishes vanishingly as N increases.
Let us now take a “metrological” perspective of this — let us trace this infinite series on an imaginary six-inch scale/ruler. Let us assume that the Nth term in the series indicates the length (in inches, say) that we have to progress on the ruler. Assuming that we are at the zero mark to start off with, let us start moving along the ruler according to the value of each successive term. The first term is 1. So, on reading the first term, we progress to the 1-inch mark on the ruler. Since the second term is 1/2, we now move to the 1.5-inch mark. At the third term, we find ourselves at the 1.75-inch mark. And so on. Basically, at the Nth term, our progress on the ruler is half of that at the (N-1)th term. Hence, for infinitely large N, the progress on the ruler is infinitesimally small compared to the (N-1)th term.
As we can verify on the imaginary ruler, as more and more terms are added, the partial sum of the series gets closer and closer to the quantity 2 without ever equalling it. However, there is no limit to how close the partial sum of the series can get to 2. For any N, the Nth partial sum is closer to 2 than the (N-1)th partial sum. The larger N gets, the close the Nth partial sum gets to 2. For no value of N will the Nth partial sum be equal to 2, though. The mathematical term of art for this phenomenon is: the series asymptotically tends to 2. Loosely speaking, this means that the sum equals 2 at infinity. This is known as convergence. The series converges to 2.
We know that PageRank converges to the principal eigenvector of the modified adjacency matrix (L) of the Web. What this means is that the PageRank vector, no matter how many power iterations we conduct, will get painfully close to the principal eigenvector of L, but it will never be the principal eigenvector of L. However, there is no limit on how close the PageRank vector can get to the principal eigenvector of L.
There exist variants of convergence as well — absolute convergence and conditional convergence. (There are also pointwise convergence and uniform convergence, but I’m not yet fully equipped to explain them well.) But I think we’ll discuss those in another post, partly because I feel they might, by themselves, warrant a separate discussion and partly because my body is begging for some sleep right now.
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Other articles in this series: On Divergence, On Domain Stretching, Conditional Convergence and Absolute Convergence
Twenty20/IPL: Limited and Uncricketed
I’ve come across several articles and arguments that have praised the IPL for several reasons. For one thing, it has given professional cricketers (especially those from local and domestic circuits) an opportunity to make money. If players like Manpreet Gony, Ravindra Jadeja and Ashok Dinda had not had a tryst with IPL and had continued playing in the Indian domestic circuit alone, they probably would have become neither well-known for their talent nor rich.
The second good thing about IPL is the fact that it provides us entertainment. Now, let’s face it. We all like to be entertained. Only, entertainment in IPL is not guaranteed to be due to the game always. Sometimes, it is in the form of a controversy (like the Harbhajan-Sreesanth row). At other times, it is because of cute cheerleaders. Or even ridiculously dressed cheerleaders. Sometimes, our political leaders decide to have funny shouting matches in the parliament over the issue of cheerleaders. (Although, that hurts, because it is a criminal waste of the tax-payers’ money.) All kinds of entertainment there. Really. You can have a good laugh at least, if not enjoy the game.
The third good thing about IPL, as Amit Varma has pointed out, is that it brings out the bench strength of Indian cricket. With a few players like Gony, Jadeja, Ojha, et al. making a mark for themselves in this tournament, the future of Indian cricket does not seem bleak anymore.
There are several other reasons, probably, why IPL is good. However, they are not terribly important to my argument here. Sure, IPL/Twenty20 has a lot of positives. But is it cricket? The idea for this article emerged out of the dismal performance of the Bangalore team in the tournament so far. (I prefer to address the teams by the name of the city/state. I think it is simply too cocky to name your team Royal Challengers or Daredevils or Knight Riders. Ridiculously juvenile.) Everyone, including the Bangalore team owner himself (justifiably), has been asking the question why the team has not been able to perform well. Most people, again including the team owner (this time unjustifiably), have seemed to say that the Bangalore players are misfits. Bangalore should have had a better team composition, it seems. Sure, they are misfits. What kind of a batsman can’t score 9 runs an over, no?
But, you see, there seems to be a little, peanut-sized fallacy in that argument. We’re talking about a game where strategy and planning have absolutely no place. Good technique has no takers. You’re not a good batsman for playing breathtakingly beautiful cover-drives. You are a good batsman for hitting the ball out of shape. You’re not a good bowler for beating the outside edge with a peach of a delivery. (Eh, well, you’re never a good bowler. Never ever.) We’re talking about a game that has been wrongly modelled along the lines of football with respect to the entertainment factor. We’re talking about a game that is not what it claims to be. We’re talking about a game that is not cricket!
Make no mistake. I am not against IPL. I love Twenty20. (I absolutely loved watching Yuvraj Singh beat the living daylights out of Stuart Broad just because Andrew Flintoff pissed him off. The bowler was almost rechristened Stuart Not-Broad-Anymore. Er.. bad joke, I know.) Only, I am against Twenty20 being called cricket. Cricket is a game where strategy and planning matter a lot. Good technique is a must for survival in the game. Less importantly, but as a matter of fact, a game of cricket cannot be played in short durations of time owing to so many nuances. Just because Twenty20 involves wielding a bat, hurling a ball and chasing the ball (if it is not already lost before you’ve realize anything), it does not become cricket. What about the principles of cricket? What about the intrinsic nature of cricket? Are they present in Twenty20? The answer is a big, resounding NO!
(Warning: If you are one with warped notions of game-theory and are wanting to explain to me what is the objective of a game of cricket and how the same objective is served in Twenty20, then back off! Ever heard the words “art” and “craft”?)
As one of my friends put it, watching Dravid trying to hit a six irrespective of the merit of the delivery is painful. (I’m actually glad Tendulkar hasn’t played a single game so far. Even after he recovers from his injury, I’m hoping he fakes the injury and sits out.) I came across the following line in this Cricinfo article:
A Mercedes convertible is an object of envy out on the road, but a tadpole amid sharks if placed on an F-1 track.
The writer argues that the Bangalore team is not fit for “fast-lane cricket”. But there is no such thing as “fast-lane cricket”! No. Such. Thing. I believe the phrase “Twenty20 Cricket” is an oxymoron. You see, driving a Mercedes is not the same as driving an F-1 car. Syntactically, they are both acts of car-driving. But what about the semantics? They are entirely different crafts. Of course, classical cricketers like Dravid, Kallis, et al. can’t beat the bejesus out of bowlers every single delivery! (A whacko like Sehwag can do it. Or not.)
It is not cricketers like Dravid, Kallis, et al. who are misfits. It is the game of Twenty20 itself that is a misfit in the cricketing universe. It would do a lot of good to cricket to not compare it with Twenty20. In any way. Twenty20 has its own place, but that place is not in cricket. Conversely, cricket can never be Twenty20. (See, I’m fair!)
Twenty20 has its own merits. So it should definitely have its own place in sport. However, we (the public) need to learn to make the distinction between Twenty20 and cricket. Twenty20 has a corporate touch to it. If players fail there, the team owners are going to come out and fire CEOs, coaches and even captains. Players are going to be under pressure all the time. It is going to be like a regular corporate job. That’s the nature of the game. However, it is highly unfair to judge cricketing status based on happenings in Twenty 20. Somehow, we will have to learn to lead parallel lives as Twenty20 viewers and cricket viewers. One world should not collide with the other. Too idealistic? Try it. It’s possible. Just don’t be judgemental! How hard do you think that is?
Non-violence and Satyagraha
Dr. Arun Gandhi, grandson of Mahatma Gandhi and founder of the M. K. Gandhi Institute for Non-violence, in his 9 June 2002 lecture at the University of Puerto Rico, is said to have shared the following story with his audience. It is about non-violence, the awesome power it wrests, and its pervasive nature – it can be put to great use in almost every conceivable situation in life. It seems surreal but true. Here goes.
I was 16 years old and living with my parents at the institute my grandfather had founded 18 miles outside of Durban, South Africa, in the middle of the sugar plantations. We were deep in the country and had no neighbors, so my two sisters and I would always look forward to going to town to visit friends or go to the movies.
One day, my father asked me to drive him to town for an all-day conference, and I jumped at the chance. Since I was going to town, my mother gave me a list of groceries she needed and, since I had all day in town, my father ask me to take care of several pending chores, such as getting the car serviced. When I dropped my father off that morning, he said, ‘I will meet you here at 5:00 p.m., and we will go home together.’
After hurriedly completing my chores, I went straight to the nearest movie theatre. I got so engrossed in a John Wayne double-feature that I forgot the time. It was 5:30 before I remembered. By the time I ran to the garage and got the car and hurried to where my father was waiting for me, it was almost 6:00.
He anxiously asked me, ‘Why were you late?’ I was so ashamed of telling him I was watching a John Wayne western movie that I said, ‘The car wasn’t ready, so I had to wait, ‘not realizing that he had already called the garage. When he caught me in the lie, he said: ‘There’s something wrong in the way I brought you up that didn’t give you the confidence to tell me the truth. In order to figure out where I went wrong with you, I’m going to walk home 18 miles and think about it.’
So, dressed in his suit and dress shoes, he began to walk home in the dark on mostly unpaved, unlit roads. I couldn’t leave him, so for five-and-a-half hours I drove behind him, watching my father go through this agony for a stupid lie that I uttered.
I decided then and there that I was never going to lie again. I often think about that episode and wonder, if he had punished me the way we punish our children, whether I would have learned a lesson at all. I don’t think so. I would have suffered the punishment and gone on doing the same thing. But this single non-violent action was so powerful that it is still as if it happened yesterday. That is the power of non-violence.
Non-violence and Satyagraha are perhaps the two most phenomenal concepts that mankind was educated with in the 20th century. Personally, I would rate them as being more phenomenal than Gandhi himself. Till recently, I had always found myself very precariously placed as to my actual belief in these concepts. I didn’t think they had enough power to make any effect, yet I didn’t have the heart to give up on them. I’m sure that a lot of us would have experienced similar feelings.
However, there arose a situation last May, in which the passengers (I was one of them) of a Goa-Bangalore bus were subjected to oppression by the travel company. The situation required that we stand up for ourselves, and so we decided to do. But how exactly would we stand up for ourselves and fight, we didn’t quite know. Sub-consciously, I began thinking that non-violent demonstration was the way to go about it. Another passenger, a dude named Ajay Singhi, had also been thinking the same. The two of us managed to convince four others about the idea. Gradually, the idea spread to all passengers, and they were up to it. We all reached the travel company’s office together. There, we actually sat in a non-violent, noiseless “dharna”. We told the manager about our grievances, and insisted on not moving from there unless justice prevailed. Needless to say, we succeeded.
Later, after most of the passengers had left, the manager of the travel company’s office was to admit the following to Ajay and me: the fact that our protest was non-violent and silent had injured, and moved, them immensely. While we were protesting, the travel company had begun making covert preparations to bring in a few “security personnel”, in case we chose to get violent. And let’s face it. If we had got violent, we would have lost hands down. But non-violent protest is what won us the battle. Couple it with the fact that our protest was silent too. We didn’t create a furore about it. Not even the people in the neighbouring shop knew that there was a protest going on here. The manager said that they actually got scared of the way we were protesting. He also went on to say that beyond a point, it had become impossible for them look us in the eye!
And that is when the power of non-violence and Satyagraha struck me. Little else has had such a profound impact on me.
The crux of the matter is this: A protest, really, is an appeal for justice and truth. People who have a clear conscience, and thus have a strong case to demand justice for, are the only ones who can resort to non-violence. It is an act that requires courage, as Gandhi put it. It requires courage because the appeal for justice here is not to a court of law, but to the conscience of another man. It works. Big time.
Quote of the Day
He’s like a taxi-meter. He only goes in tens.
Kumble when asked about upstaging Murali. “He’s too hard to catch,” he concluded.
Simple is Profound
The profundity of reality goes hand in hand with the simplicity of its expression. The following quote by Gandhi is enough evidence of this.
Violence will be overcome by violence only when it is proved that darkness can be overcome by darkness.
Can darkness be overcome by darkness? No! Excellent argumentation.
If a fact is not expressed simply enough, its profundity is lost on us most of the time.
