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With Scott in all smiles:-)
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 and are prime with a gap of , but this is truly a special case and unique per definition. The gap between and is 2. Similarly and differ by . 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 below , but infinity is still a lot far from ! 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 . A related question is, how about prime pairs and which are separated by where could be a finite number. When , 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 for some $k$. If so, what is the smallest where this holds true? Zhang now has a proof that for as small as million. Mathematically, if we denote is the th prime, then the new claim says (stated crisply in the paper abstract),
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 . 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.
Couldn’t stop giggling seeing this. Nice composition here!
Some folks seriously believed Chechans are from Czechoslovakia! Swaziland neighbors Switzerland? If you thought that was funny, but not as much as Chechans and Czechoslovakia:-)
Finally, I was there, but the 2 hour stunning drive from San Diego in the evening, through the picturesque terrain wast quite satisfying in the end. With the back injury backdrop, Federer was no where the fit champion he has been. The occasional graceful shots were in display last night, but by and large he was playing more from the shade than from the limelight. Nadal on the other hand was on the drive, even from the practice hit moments. The weather (as always) at Indian wells in the evening was superb and the atmosphere was blistering. The match unfortunately got over in less than 100 minutes. Always sad to see Federer losing this way, but then his legacy is not stemmed from a single match though. I wonder whether he will make it next year at 33. Hope he does!
I am here in Hawaii this week for the IEEE plenary. The view from the Hilton in Waikoloa village is pretty enthralling…
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!
- ‘ Whatever our souls are made of, his and mine are the same’ – Emily Bronte
- ‘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
- ‘But soft! What light through yonder window breaks? It is the east and Juliet is the sun’ – Shakespeare ‘Romeo and Juliet’
- 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
- ‘You know you’re in love when you don’t want to fall asleep because reality is finally better than your dreams’ – Dr. Seuss
- ‘ 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’
- ‘Grow old along with me! The best is yet to be’ – Robert Browning
- For you see, each day I love you more. Today more than yesterday and less than tomorrow’ – Rosemonde Gerard
- ‘But to see her was to love her, love but her, and love her forever’ – Robert Burns
- ‘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.
Over the weekend, I watched the new animation movie from Walt Disney. It is the fairy tale story of Stuepnsil. A really nice movie. Quite different from many of the animation movies of the past.
On the Christmas day, out of blue I bumped across an old archive of Robert Fanos’s interview (oral history). Beautiful one.
After all, there are trees on earth’s terrain. Why not on Google earth? Google just don’t delay it any further. Now we can see trees with Google earth 6. Amazing view of SFO, right here. As they say, it is here as it is there. Now we truly have the world in front of our eyes. The beautiful 3D world is now ready.
Via Lance’s blog, I came across this hilarious prize known as Ig Nobel prize. The term “Ig” stands for “Ignoble”! The prize is apparently given to something which may appear to be funny, but has some serious reasoning behind. In other words, these are peculiar awards given to something which”‘first make people laugh, and then make them think”. Quite amazing huh?
I am yet to explore a lot on this. Lance listed one very interesting one. I find it extremely noteworthy! Robert Faid of Greenville, South Carolina, farsighted and faithful seer of statistics, got the Ig Nobel prize for calculating the exact odds (710,609,175,188,282,000 to 1) that Mikhail Gorbachev is the Antichrist.I wonder how he arrived at this magical number! Didn’t Faid know how to play a game in the stock market then?
Wikipedia has an interesting entry on this topic. Would you believe, the young Russian physicist Andre Konstantinovich Geim who just won this years Physics Nobel for his work on graphene had also won the Ig Nobel in 2000! Quite amazing.
A cool presentation I liked there is about the evolution of the universe tagged against the timeline since the big bang. It goes to show how fast things moved in the beginning; yet how slowly it took to get into this fabulous shape (whatever is known as of today) that we live in. No doubt this is a continual process of marvel.
Even though, the season-3 of the hilarious TV episode is being broadcasted for sometime. I didn’t have had a chance to view any of them on TV yet, but the other day, I checked youtube for some snippets. May be because of the two seasons of serials, I didn’t see anything impressive in the new ones. Not bad by any means, but it all started sounding a bit too repetitive now. I have been an ardent fan of the first two season episodes. Nothing made me laugh broader than watching some of those hilarious picks of Dr. Sheldon and gang.
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.
After a long gap (over six months or so), I finally played some tennis again. Much to my surprise, I wasn’t all that rusty in spite of the long layoff from any major sporting (barring some recent treks and once in a while cricket) activities during this period. After a few hits, my serves started holding and I slowly felt the rhythm. I started enjoying this beautiful game once again! In the Swiss heat, it was almost unbearable at times to absorb the hot air from the synthetic surface. On Friday and Saturday’s I played for two hours each until late evening. Yesterday, after the game we took dip in the Lake Geneva near the UNIL sports center arena at St.Sulpice. I never felt a better swim than this before. Such was the feeling of taking a clean water swimming after a good game of sports. It was getting darker and a swim between 2100 to 2200 on the fading summer light in the foothills of the Alps was simply amazing. I just cant compare a place to this amazing Lake Geneva region. Quite a place this is. After the swimming I seemed to have regained all vigour to play a few more games. Had there been floodlights, we were on for a few more perhaps! They say winning is an elixir for eternal youthfulness, but did I feel that swimming in lake Geneva comes close to that?
While, standing in the lake with chest level water and overlooking the Alps mountains, it reminded me of the photos of saints in olden days taking a morning yama’s in the Ganges overlooking the Himalayas. I’ve never been to Ganges, but for once I could perhaps feel a sense of their state of mind. I felt like singing one of those Yesudas classical songs, standing with half immersed body. I don’t quite remember whether I did one. I was in a state of fulfilment sort to say!
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?
Shrini Kudekar yet again showcased his creativity and acting skills on the eve of Dinkars public defense. Here is the video.
Today, during the evening chat, Emmanuel Abbe threw an interesting question: Whether the sum of square roots of consecutive binomial coefficients converge to some closed form! That is, . We tried a few known combinatorics tweak, but no meaningful solution arrived. We were also wondering whether this has some asymptotic limit, but that too did not yield anything. A quick check on Mathematica wasn’t helpful either. Now the question is: Does this sum yield some closed form expression.
While playing with this sum in Mathematica, I found that for the sum of squares of binomial coefficients, there is a nice simple closed form.
I was toying with a proof. It turns out that, the proof is extremely simple and is a one line tweak of the Vandermonde identity . Simply substitute and we have the results on table. The natural question then would be: Is there a generalization for for any . Ofcourse now for it is trivial.
Apparently, it turns out that, there is no closed form expression for a general (all) . There are some interesting divisibility properties of these sums. An interesting account of that is addressed by Neil Calkin (Factors of sums of powers of binomial coefficients).
At the moment, I get a feeling that sum of fractional powers of binomial coefficients is not trivial. May be there is no closed form. May be not!
Here is a list of some of my favourite Indian commercial ads. Thanks to youtube, I get to see them again! The adhisive brand “Fevicol”has produced some inredible and funny ads. Most of their ads stuck on to the viewers mind. Among the other funny ads, I liked camlin erasurs and marker ones. I better dont say too much here. As they say, the fun in an ad is best viewed and chilled out! Have a look and enjoy the fun and appreciate the creativity of these fabulous ad makers.
The old Ericsson mobile phone ad (I guess 1996). The concept of a “small” phone back in 1996 perhaps is too outdated for today. All boils down to Moore’s law!
Rimii Sen and Aamir Khan did a nice job here in this Bengali accented conversation. The coke ad is one of the better ads from the cool drink folks.
Fevicol: Simply superb ad from the popular adhesive brand.
Naukri.com’s famous Hari Sadoo funny ad:
How about this one. To be this is too good an ad from Camlin.
The Peugeot ad used to appear in Channel 4 in UK. It was an incredible ad. I am glad that this is there for viewing in youtube. Superb one.
The title might mislead you. So, let me clarify upfront. I am not on a mission to self appraise. I am to talk about the autobiography of ‘Noerbert Wiener’, titled ‘I am a Mathematician’. This is a piece of book I am reading currently. Since I have heard a lot of stories about Wiener and having known some (percentage is minuscule!) of his work, the presentation of the book didn’t provide disappointment. Rather, it is a very very interesting sketch of his life, put in his own style.
I mentioned about stories being heard about him. There are many of them. I am not saying this candidly, because I hardly checked the authenticity of such tales. Nevertheless, I get ready to laugh everytime, I begin to hear anything about him. The mathematical work of this once child prodigy is very well known and is treasured. His wit and absentmindedness are quite unique. Some of the anecdotes, I have heard about him are;
1.During one of these trips down the hallway at MIT, Wiener was interrupted by several of his students who talked to him for several minutes about what they were working on. After the conversation had ended, Wiener asked one of them “Could you please tell me, in which direction was I traveling when you stopped me?” One of them replied, somewhat confusedly, “You were coming from over there [gesturing] this way [gesturing].” Wiener replied, “Ah, then it is likely that I have already had lunch. Thank you.” and continued down the hallway to his office. (A somewhat similar story is attributed to Einstein as well. As far as I heard, this is when Claude Shannon was giving a lecture at Princeton. It was well attended. Einstein made a back door visit when Shannon was in full stream. Shannon obviously noted Einsteins coming in, chatting with someone in the last row and the leaving soon. The curious Shannon (after the lecture) went to the folks to whom Einstein seemed talking. To Shannon’s surprise, Einstein was apparently asking them ‘where the tea was served’.)
2: After several years teaching at MIT, the Wieners moved to a larger house. Knowing her husband was likely to forget where he now lived, Mrs. Wiener wrote down the address of the new house on a piece of paper and made him put it in his shirt pocket. At lunchtime, an inspiring idea came to the professor, who proceeded to pull out the paper and scribble down calculations, and to subsequently proceed to find a flaw and throw the paper away in disgust. At the end of the day, it occurred to Wiener that he had thrown away his address. He now had no idea where his home was. Putting his mind to work, he concocted a plan: go to his old home and wait to be rescued. Surely Margaret would realize he was lost and come to pick him up. When he arrived at the house, there was a little girl standing out front. “Excuse me, little girl,” he asked, “would you happen to know where the people who used to live here moved to?” “It’s okay, Daddy,” the girl replied, “Mommy sent me to get you.” (Decades later, Norbert Wiener’s daughter was tracked down by a mathematics newsletter. She said the story was essentially correct, except that Wiener had not forgotten who she was.)
Description on the image: Norbert Wiener with Amar Bose (Bose audio fame) and Lee (the early MIT pioneers): Source of this image is  http://www.siliconeer.com/past_issues/2005/January2005-Files/jan05_bose_archive.jpg
This picture is taken on 2004, May 03. The day has no big significance, but for the first time, I was there watching an English premier league soccer game live. Thanks to Jeff Torrance who managed to get an extra executive ticket, I could avail a feel of this fabulous experience of live soccer game in Europe and that too in England. Jeff, a huge Chelsea fan was so thrilled to get into the Chelsea gate, through the team restaurant. The entry to the stadium gave me an experience, that I never had before. Throughout the journey from Cambridge to London and then to Sanford bridge we had quite a lot of laugh pulling Cyrian for his Irish jokes and what not!