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During the past week, while at Hawaii for the IEEE 802.11 interim, I happened to glance this NY times article. The story is about a New Hampshire professor Yitang Zhang coming out with a recent proof on establishing a finite bound on the gap between prime numbers.  While browsing the details, there are more details emerging as several pieces of blog and articles and reviews are being written (and some are still being written). Now, looks like the claim is more or less accepted by pundits in the field, and termed as a beautiful mathematical breakthrough. As an outsider sitting with curiosity, I feel nice to scratch the surface of this new finding.

The subject surrounding this news is number theory, prime numbers to be precise. The question of interest is on the gap between adjacent prime numbers. We know that $2$ and $3$ are prime with a gap of $1$, but this is truly a special case and unique per definition. The gap between $3$ and $5$ is 2. Similarly $5$ and $7$ differ by $2$. One may have thought that, the gap between successive primes go up as we flee along the number line. Not quite. For example, we can see that there are a lot of pairs with a gap of 2.  The easy ones are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139) and the list goes on. It was conjectured that there are infinitely many such pairs, but the proof of that is not quite as easy as yet! It is known that there are precisely $808,675,888,577,436$ below $10^{18}$, but infinity is still a lot far from $10^{18}$! An interesting quest was to really prove that there are infinitely many twin primes, but this still remain as an open conjecture.

Now the new discovery by Zhang is not quite proving the twin conjecture, but a close relative of that. Twin conjectures are strictly about prime pairs separated by $2$. A related question is, how about prime pairs $p$ and $q$ which are separated by $k$ where $k$ could be a finite number. When $k=2$, then we have the  special case of the classical twin prime case. Can we at least prove mathematically that there exists infinitely many primes such as $(p,q=p+k)$ for some $k$. If so,  what is the smallest $k$ where this holds true? Zhang now has a proof that for $k$ as small as $70$ million. Mathematically, if we denote $p(n)$ is the $n$th prime, then the new claim says (stated crisply in the paper abstract),

$\lim_{n \to \infty} {} \left(p_{n+1}-p_{n}\right) <70 \times 10^{6}.$

$70$ million is still a large gap, but as Dorian Goldfeld says, is still finite and nothing compared to infinity! In future, it is not unlikely that  we may get to see this gap coming down and perhaps to the best case of $k=2$. Who knows?

The result is still interesting, even to general interesting folks like us. This kind of says that, the gap between prime numbers is worst case bounded by a finite number. If we really plot the prime numbers, then we will see a saturation like behavior!  Like many other things at asymptotic (for example, the eigenvalues of a large random matrices exhibit very interesting properties, when the size goes to infinity), things at infinity may exhibit some charm, after all!

The  paper is accessible here, but as expected the proof is hard (for me at least). Hopefully we will have some diluted explanation of its essence from experts in the coming days. Already,Terrence Tao had this sktech, a couple of weeks ago on his google+ feed. Over the last week or so, finer details on the new break through are still emerging. Terry Tao also has initiated an online Wiki collaboration in an effort to improve upon from this work (For experts that is, not for me!).

Congratulations Professor Zhang.

The first mail  this morning (from Nihar Jindal)  brought this very sad news that Tom Cover has passed away. A giant in this field who contributed immensely to many flavours of Information theory will be missed. Everything he touched had class written all over, gracefulness, simplicity, elegance and all the more depth.

A tremendous loss! His legacy will continue.

For a generation of kids like me, growing up in the 80s and 90s in India, Anant Pai, popularly known as Uncle Pai was one of the most influential figure. Not because of his personality or the aura in public life, but the sheer creativity in transpiring the richness in the Hindu mythological stories to us in the form of children stories. The Amar Chitra Katha stories from him, not only had improved our knowledge on the many epic stories and its variants, but also brought the curiosity in the fairy tale world to young minds. It is with immense sadness I passed the day hearing his sad demise, last week.

For me, there has never been a second thought on what the best ever love poem and the poet are. It is the one and only Elizabeth Browning and her beautiful poem How do I love thee (See below. A beautiful reading by Helen Mirren is here in youtube).The Browning couple stands tall when it comes to some of the all time toppings in literary romantic poems. I remember my wife (then my fiancee) sharing a piece of Hindu Sunday literary supplement which had this poem. I have the sonnet etched back in my mind, even now!

How do I love thee? Let me count the ways.
I love thee to the depth and breadth and height
My soul can reach, when feeling out of sight
For the ends of Being and ideal Grace.
I love thee to the level of everyday’s
Most quiet need, by sun and candle-light.
I love thee freely, as men strive for Right;
I love thee purely, as they turn from Praise.
I love thee with a passion put to use
In my old griefs, and with my childhood’s faith.
I love thee with a love I seemed to lose
With my lost saints, — I love thee with the breath,
Smiles, tears, of all my life! — and, if God choose,
I shall but love thee better after death.

Ok, why did this pop up now? Well, in UK, apparently there was an opinion poll on the best one liner; the love one liner that is. Mind you, it is not the full fledged poem, or for that matter a full sonnet or stanza itself, just a one linear. Here are the 10 most popular according to the survey. Just ahead of the Valentines day, a good time pass! Happy reading and Happy Valentines day folks!

1. ‘ Whatever our souls are made of, his and mine are the same’ – Emily Bronte
2. ‘If you live to be a hundred, I want to live to be a hundred minus one day so I never have to live without you’ – A A Milne
3. ‘But soft! What light through yonder window breaks? It is the east and Juliet is the sun’ – Shakespeare ‘Romeo and Juliet’
4. He was my North, my South, my East and West, My working week and my Sunday rest, My noon, my midnight, my talk, my song; I thought that love would last forever: I was wrong’ – W.H. Auden
5. ‘You know you’re in love when you don’t want to fall asleep because reality is finally better than your dreams’ – Dr. Seuss
6. ‘ When you fall in love, it is a temporary madness. It erupts like an earthquake, and then it subsides. And when it subsides, you have to make a decision. You have to work out whether your roots are become so entwined together that it is inconceivable that you should ever part’ – ‘Captain Corelli’s Mandolin’
7. ‘Grow old along with me! The best is yet to be’ – Robert Browning
8. For you see, each day I love you more. Today more than yesterday and less than tomorrow’ – Rosemonde Gerard
9. ‘But to see her was to love her, love but her, and love her forever’ – Robert Burns
10. ‘I hope before long to press you in my arms and shall shower on you a million burning kisses as under the Equator’ – Napoleon Bonaparte’s 1796 dispatch to wife Josephine.

This topic may have been discussed and debated a thousand times. I remember seeing one such report, not so long ago in EE times. The Economist has just placed a new one as well. The title is ” Why doing a PhD is often a waste of time”. Pretty interesting nevertheless.  Anyway, I personally do not think that it is all waste. On the other hand there are a lot of things out to learn in the process. Eventually it is a matter of personal choice!

On the Christmas day, out of blue I bumped across an old archive of Robert Fanos’s interview (oral history). Beautiful one.

After Neruda (one of the greatest poet of all time, in my book), the first name come to me from Peru is Vargas Llosa. South American writers are always special, for their sheer brilliance in writing. Gabríel Garcia Márquez, Llosa’s contemporary from the Spanish speaking continent,  is more known (who is one of my all time favourite), but the latter is no less in literary genius. So, a well deserved Nobel laureate. New York Times has published a nice report on this year’s literature Nobel.

The P versus NP has popped up again in to the mainstream. The HP scientist Vinay Deolaikar has recently prepared a manuscript which claims, it is after all P ≠ NP. Greg’s blog seems to be the one which first broadcasted this to general public. I have not had much time to digest the proof claims or counter arguments in detail. Here is a fascinating series of arguments discussing the proof claims. I am going to do some digesting on this in days to come! We will wait for the experts to have their say. I am going to relish this discussion.

The Iceland volcano Eyjafjallajokull (or rather more correctly Eyjafjallajökull) has produced a spectacular view stemmed from the surface of those beautiful white mountains and then went on to stage the show on the air, all alone. The amazing thing is that, the volcano this time made sure that everyone in the world pay attention to it. It didnt allow anybody to spoil the show. Air traffic itself was ceased for over a  week. The question of to fly or not to fly immediately popped up and it was on the not side. Believe it or not, I was never scared of a volcano. Quite frankly I didn’t buy the argument that it was something worth worrying about. But this time, I paid a lot of attention to it. Again, not really had worried about it, but was overwhelmed to learn about the kind of impact a natural event such as this can make even when not really causing any serious casualties directly.

It is somewhat ironic that, the day to celebrate love is not named after Cupid, the Roman god of love. His mother goddess Venus and father Mercury are not considered either.  The Greek mythical folks cannot be happy either. Eros, the Greek god of love or his mother Aphrodite, the goddess of love are not the ones remembered by the lovers of this century. In the Hindu mythology, encomium is poured over to Kama (or Kamadeva), but who listens when it comes to naming the modern love day? All the accolades instead went to a saint who to his innocence did not really had any fun himself when it came to love, but he was generous enough to facilitate the young lovers.

One of the legend of St. Valentine’s day go like this. Valentine was a priest who served Rome during the third century. Emperor Claudius II decided to bring in a law to outlaw marriages. His claim was that, single men, without wives and families make better soldiers. The priest Valentine, apparently was not quite ready to bulge to this idea of Claudius. In those days, of course you don’t challenge a ruler in public. How powerful democracy is. We are lucky, don’t we?

Anyway, defying Claudius, Valentine continued to secretly perform marriages for young lovers. When Valentine’s actions were discovered, the king ordered that he be executed. The martyr Valentine became one of the most popular saints in centuries to come in Europe, especially in France and England.

Valentine day of the modern world has surely made St. Valentine proud for his worthy sacrifice. After all, it was all for a good cause. Love is beautiful. It is up-to the people to decide, what way they want to celebrate. There is nothing as beautiful than seeing people in love.  The very sign of love is pleasing to the eyes. Let love relish.

Happy Valentine’s day.

I’ve stumbled upon to this nice article (titled Principles of Effective Research) (I saw it through Mahdi’s  blog) by Michael Nielson (yes, the same Nielson who is known for his work on Quantum computation: the author of the popular (technical) book Quantum computation and quantum information). Even though, the title states “effective research”, I thought the underlying methods are applicable to any work, not merely research work. If you trust my word, then many of the points jotted down by Nielson applies well to life and work in general. It appears that he is writing a book on “The future of science“. I am looking forward to that too!

Oh boy, what did we see this evening at Wimbledon? A grandslam final, filled with nothing short of a breathtaking drama. A near neck to neck battle between king Federer and a fabulously charged up Roddick. Guess who was watching that epic cliffhanger? The emperor of that piece of grass strip in central London! No point in guessing the name: Pistol Pete Sampras. Sampras was visiting Wimbledon after 2002, perhaps just to witness another great champion Federer get past him in the number of grandslam titles. What an occasion! Unbelievable tennis on display when blue sky topped the roof in clean light. I feel for Rodick here. This was ‘the chance’, he had at hand: and truly well he deserved, one must say. I for one had written him off yesterday, even though he had played great tennis in the semi final to beat British hope Andy Murray. From one Andy to another Andy, the other finalist name changed, breaking the great British hope, since Henman (Well, Henman was not really a realistic hope, when Sampras was taking a stroll down the Wimbledon park).  I was expecting Fedex to just roll over him in the title clash. But alas! Didn’t he give Fedex  a run for his crown?

In the end, Federer had that extra epsilon, call it luck or experience.  He was there on that center court final stage on every single year for the last seven summers. Last year he lost it only by a whisker to the Spaniard Nadal. Federer truly deserved to be the grand-slam record holdert. He is the best player on the circuit and he is so very effortless, athletic and passionate.  The great man is a beauty and indeed is a treasure to this great game.   I cant have enough praise on the way he played tennis over the years.  He is so very smooth and graceful. A touch of Lara, Tendulkar or Dravid in cricket. I really was feeling a lot low when Sampras retired in 2002, but the Swiss has indeed made up that void since then.  A humble soul Federer typify the Swiss people I guess. So gentle and an amazing role model to the new generation. I really hope that he gets a few more grand slams titles.

Turning back to the losing finalist, I can imagine how hard it would be to be an Andy Roddick who narrowly missed the crown by perhaps one or two moments of marginal shots.  Sometimes sport can be so cruel! In the end winner takes it all and it is agonizing. It must be hard to be a second at that level. But then, that is what it takes it to be the best in the world. Only thin air make the separation. It is courage and wisdom at times to grab that silver line.  Grabbing is secondary, seeing it in the first place is what separates the best from the next best. After all, it is not easy to get there. Isn’t life beautiful?

On  a sunny Lausanne morning, I woke up much later than usual. A game of cricket last evening had its due share in settling my body parts and indirectly in this wake up delay as well. I was all excited to re start working on the one sided set constraint problem which I pondered about a little the other day. After a routine coffee, decided to check Indian newspapers on line and the first news said Kamala Suraiyya’s passed away. To most of us, especially the ones associated with Kerala, she is the one and only Madhavikutty known to outside world as Kamala das. A name change and a religion hop didn’t really bother a secular Malayali. However the truth remains that, she was easily one of the most misread, misinterpreted writers of this generation.

I have not read a lot of Madhavikutty’s major works. That is a shame, I must accept. But I remember reading many short stories of hers, published in magazines, newspaper supplements and weeklies. One of the stories I still remember is  Punnayoorkulam where she touchingly depicts the life of a poor servant. Other short stories instantly coming to my mind are chandana marangal (Sandal wood trees, ചന്ദന മരങ്ങള്‍) and Pakshiyude maNam (Smell of a bird, പക്ഷിയുടെ മണം).  Her story telling style was unique; most notably with her precise and careful selection of words. It is incredible that she could write so well in both Malayalam and English. Not many people know that she was nominated for Nobel prize in 1984.  Unfortunately I didn’t get to know much of her English works other than a collection of short stories titled Padmavati the Harlot and Other Stories , which I happened to read sitting inside a book store in Trivandrum during the summer of 1996. That was an experience of some sort. I didn’t have money to buy books then. I used to spend a lot of time inside the book store (thankfully they allowed that) and spend nearly the whole day there. In two days of a week I could read quite a lot. I had restricted visits to two days a week to pretend that I was not exploiting that facility. Nevertheless, over a period, I had befriended with some of the shop guys and they politely let me enjoy this habit, realizing that I was a mere student who couldn’t afford to buy anyway.

Her life and works were dragged into so much controversy. I am not sure whether that helped her to increase the readership. I personally think she was an incredible writer who didn’t need these controversies to claim fame and readership. Her autobiography and the frank style of telling stories created public attention. Perhaps it came at a time when it was unusual for an Indian woman to be that open to express her emotions and life. I do not know much into that controversy, other than learning that it had some. I did not read her autobiography either to judge whether it had some explosive presentation of vivid emotions of a woman. Anyhow, such was her life. Some people would remember purely because of such controversies. Sadly many would have failed to realize the pure writing talent of such a bilingual writer, one of the best Kerala produced. Her death will surely create a void, considering that the language writing has become so thin these days.

Many people in Kerala were surprised when she converted her religion in the last stretch of her life. It was not because of the religion she chose to or the one she was born into. It was more because of the fact that she chose to give importance to latching onto a religion for keeping piece with her life. Anyway, that was her personal choice and everyone accepted it, period.

Madhavikutty’s departure is a big loss. May her soul rest in peace. I leave you with this documentary on Kamala das by Ignou:

I’ve finished reading the memoirs of Walter Rudin. It was a quick read for a few hours. His autobiography is titled The way I remember it, published by AMS in the history of mathematics series.  It wasn’t particularly interesting, to say the least. From a mathematician who wrote excellent books on functional analysis and several others,  I was expecting a much better story. Of course one cant write an imaginary story in an autobiography, but then the incidents in his life is pretty much the story of any European intellectual during the war days. The best I liked is the one from Karl Popper. However, I could connect many incidents from Rudin’s life, primarily because of the geography. There is a chapter on his days in Switzerland, which also touched upon Lausanne. That part for once enthused me! Was wondering how Lausanne would have been 70 years ago! If you are completely unaware of the life in Europe around the WW period, then this will give you a perspective.  Like many scientific minds of that era, he had a long route to the United States. He discusses the path and family traits of that journey, in a somehat uncomplicated language.

In his autobiography, Rudin has discussed some of his contributions to mathematics as well. That part appeared a little informative, but technical read. If you know his work already, you would connect it nicely.  I particularly liked the chapter on Function Theory in the Unit Ball of Cn.

In all, not a book I would recommend, unless you are a Walter Rudin fan and knows his contributions in much more detail. However, this may be a motivating read for a young school kid aspiring to be a mathematician. Why did I say that? I don’t know! Don’t ask me why either!

Stumbled upon the news on New York times: it is about a new search engine being developed by some former Google folks. First there is excitement when it comes to a startup idea when you know that they know how it is to be confronting their former employers in business. Anyway, the new engine is called cuil (pronounced just like ‘cool’). I am all for new ideas. Hopefully we are into better search engines. Since these folks are also from Google, you can expect a certain Google standard guaranteed. Google undoubtedly changed the search engine business, by simply scaling the internet to a level hitherto unimagined. Yet again, a Stanford connection to a new startup. Tom Costello and his wife Anna Patterson (former Google architect) surely will know this business better than us (correction, better than me to say the least).

If their motto of producing a more appropriate search engine, bettering Google, then we should feel happy and proud of this adventure. Surely Google cant relax either. In all it is a win win for the world. A preliminary look at the search engine game me a good feel. I am not sure whether the change in appearance (after being stuck and used to Google search for so long) gives me this impression. Anyway I look forward to see their progress.

I leave it to you to try out for a comparison. I did a Cuil on “compressed sensing” and found this where as a google of “compressed sensing” displayed this. Google displayed the search result as a list (rows) where as the Cuil results to a tabular form. Too early to say anything discrete, but I am going to try the new one as well. Google is by far the fastest (at the moment).

Felt a little sorry to hear that Art Buchwald passed away. The first memoirs of his humour columns which used to appear in the last page of “The Hindu” Sunday (and later on Mondays) newspaper, came calling once again. These days, those columns are hardly seen in Indian newspapers, but thanks to Internet, I still get to read some of his, very occasionally though. One of the recent one, I found very interesting was in the “The Washington Post” column [1] of his, titled “When the going gets tough, the war gets going”. In his own style he could sent the right message to the folks responsible. I thought that was very apt and brief to send a clear message to those proponents of war. Well, fact can be harsh and unpleasant, but the truth is that millions of innocent people pay the price for fault not of theirs. To be courages to tell the truth is wisdom in itself.

Interested by his writing, I was once curious to know his background. My intention was to find out, how and why he turned into writing humorous articles. Often, I hear that comedians and humourists and writers who write such articles have a very difficult life story to tell. In a way Buchwald too had his difficult childhood. His mother was in a mental hospital/asylum for a good part of his childhood. It is heard that he couldn’t see his mother for about 30 years or so. My heart goes to a child with such a terrible childhood. Perhaps, his intense sorrows made him to cover up all those and present a brighter spot to millions of readers all over the world, through his columns.

This morning, I came across with the news [2] of his demise. His unique styled articles will be a big miss from now on. But he has made a mark through his brilliant short columns. His death wasn’t entirely unexpected (I guess he was critically ill in early 2000), but the news of his sad demise had created an irreplaceable void in the literary circles.