Bharat Rakshak

Here is some serious discussion going on about P Vs NP (Terrence Tao et al.)

http://rjlipton.wordpress.com/

Moved the thread to Tech & Econ forum for more visibility.

AmberG Do I have your e-khat?

Edited...

About those rumors I posted ...oh well.. Fields Medals awards have been announced. Kiran kedlaya and Manuj Bhargava's name weren't there. Though, Ngo was in the list of four.

(Four names are: Ngô Bảo Châu, (Vietnam/France), Elon Lindenstrauss (Israel),
Stanislav Smirnov (Russia) and Cédric Villani (France))

For the interested ICM link is at :http://www.icm2010.org.in/welcome
(There are links for prize announcements, etc and live streaming for lectures etc in case any one is interested)

Amber:

Being a math major and having studied math both in the US and India, I know there are really good pure mathematicians in India. However, the applied math community is not even up to standards compared to the US. In my opinion, it is the applied math crowd that "brings down" pure math community, I mean the visibility of the pure mathematicians gets obscured by the sub-standard applied math people in India.

Also, thanks for the Agrawal's link - it was good to hear his perspective on Deolalikar's work

I feel that Vinay should not have rushed through his "proof". It was ridiculous to see the media frenzy. Almost all major newspapers declared he had solved the problem. As you may know these things should not be rushed. Andrew Wiles got it wrong after he declared the FLT (Fermat's Last Theorem) in a UK conference. Perelman did the same in the case of Poincare's conjecture, but it turned out to be successful in his case, although there were controversies with Cao filling in some details. Vinay could have just waited. I don't see any great rush for solving this problem. Even if there was a rush, I am sure his proof would have been different - as I believe no two people can obtain the same proof at the same time!

Pratibha: promote study of mathematics to spread scientific temper


Like this image (More at if you like the posters - http://www.icm2010.in/icm-2010-posters)

AmberG, Did you get my e-mail? What do you think of the attachement?

^^^No I did not. Which email address did you sent to?

shephard
I printed from last post.

Will send again.

Ancient use of big numbers baffles experts

Hyderabad, Aug. 20: Centuries before modern European mathematicians formalised the concept of infinity, ancient Indians dabbled in numbers so gigantic that their motivations have left even 21st century mathematicians baffled.

A series of Indian texts dated from about 1000 BC to the 2nd century BC described mystifyingly large numbers, some with no links to reality, delegates at the International Congress of Mathematicians 2010 under way have said.

...
...
The Yajur Veda, which some scholars believe was composed around 1400 BC, has a line that invokes successive powers of 10 up to a trillion, Plofker said in a special paper circulated at the congress.

“Such vast numbers could have had no practical application in the context of Vedic culture,” she wrote.

The allure of large numbers is also visible in the epic Mahabharata that describes a cosmic time span over four billion years long, constructed out of 360-year periods, said Plofker, who told The Telegraph she was drawn to ancient Indian mathematics after opting for a Sanskrit course because it had a script unfamiliar to her.

...
...
One Buddhist text from the early first millennium refers to what Plofker has described in her paper as “sequences of numbers that could be conceived only through mind-boggling comparisons” such as the number of grains of sand in the beds of a hundred million rivers.

Another Jain text from the early first millennium refers to a time period of two raised to the power of 588.

Some scientists believe observations of nature may have driven the abstract thinking that led to large numbers.

....
...

Plofker!

^^^Thx. As said before in this thread, Highly recommend Kim Plofker's book, there was a post by Svenkat.
link here
(I recommended it too) The book is:
Mathematics in India ...

As said before, David Mumford, (winner of the Fields Medal),has written a six-page essay on the book, which captures, concisely, the major achievements of Indian mathematics. There is the use of Pythagoras’ famous theorem, Pāṇini’s rules of Sanskrit grammar and recursion, Pingala, whose study of Sanskrit verses led to the binary notation and the development of Pascal’s famous triangle,binomial coefficients, and Madhava of Sangamagramma and the genius of the Kerala School etc.. etc..very nice
Here is that link again: http://www.ams.org/notices/201003/rtx100300385p.pdf

^^^Also to add, her talk:
"Indian rules, Yavana rules: foreign identity and the transmission of mathematics"
On August 26th at 11:30 AM if you are unable to travel to Hyderabad, can be watched live.
(This and many other sessions - Just check the ICM site (Link:
http://e-program.icm2010.in/live-streaming.php

The importance of mathematics

http://www.hindu.com/2010/08/21/stories ... 671200.htm

AmberG ji,

For us madrassa math majors,where is the soln,(hints,no help ) of your Aug 15 problem.

^^^ Let me just take y = 1 + n^19 + n^47
For any value (n>1) it is divisible by (1+n+n^2) hence not prime.
Easy to prove, since (n^3 - 1) = (n-1)(n^2+n+1), is divisible by ( n^2+n+1)
(If I call k=n^2+n+1, i see n^3==1 (mod k)
Hence, n^19 = n (mod k ) (because x^3==1 hence x^18==1 etc ...)
and n^47 = n^2 (mod k) ( 47 divided by 3, leaves a remainder 2)
so y == 1+n+n^2 == 0 (mod k) hence y is divisible by k.
QED.

(One could have used complex number, w = (1 + i sqrt(3))/2 ... and easy to see that w^3=1) we see both w and w^2 are roots so y is divisible by (n-w)(n-w^2))

*****
Similarity if one just notice that if you divide by 5 and only see remainders,
8 leaves remainder 3, 19 leaves remainder 4, 47 leaves 2 and you have x^1 (1 leaves remainder 1) hence 1+n+n^8+n^19+n^47 is divisible by (1+n+n^2+n^3+n^4)
(which is also (n^5-1)/(n-1))
HTH

Svenkat, Ramana, SwamyG ...

Worth watching Kim Plofker's lecture at ICM 2010 : (See Link below)

Indian rules, Yavana rules: foreign identity and the transmission of mathematics

(She talks about few case studies dealing with western (Yavan) perception of Indian Mathematics and Indian perception of western mathematics in historic times.

(This is about an 1 hr video of her lecture)

Amber G. wrote:^^^Thx. As said before in this thread, Highly recommend Kim Plofker's book, there was a post by Svenkat.
link here
(I recommended it too) The book is:
Mathematics in India ...

As said before, David Mumford, (winner of the Fields Medal),has written a six-page essay on the book, which captures, concisely, the major achievements of Indian mathematics. There is the use of Pythagoras’ famous theorem, Pāṇini’s rules of Sanskrit grammar and recursion, Pingala, whose study of Sanskrit verses led to the binary notation and the development of Pascal’s famous triangle,binomial coefficients, and Madhava of Sangamagramma and the genius of the Kerala School etc.. etc..very nice
Here is that link again: http://www.ams.org/notices/201003/rtx100300385p.pdf

BTW look at the maps of India projected in various slides .. which are not incorrect like some others you see..

X posting from Ancient warfare thread...

Quote by Sanku:

>>>If that was so differential equations should have been invented in the war torn North India and not the relatively peaceful Kerala in the 15th Century.

I read somewhere that the infinite series was developed in Kerala in the 14th Century and that was the basis for Newton's invention of Calculus...but was calculus invented in Kerala before Newton??

geeth wrote: I read somewhere that the infinite series was developed in Kerala in the 14th Century and that was the basis for Newton's invention of Calculus

Nilakantha did some wonderful work on astronomy, and though the pieces are definitely there, I don't think Calculus was explicitly developed.

adityaS wrote:

geeth wrote: I read somewhere that the infinite series was developed in Kerala in the 14th Century and that was the basis for Newton's invention of Calculus

Nilakantha did some wonderful work on astronomy, and though the pieces are definitely there, I don't think Calculus was explicitly developed.

AdityaS: Yes, Nilakantha and Kerala school did some wonderful work. The reference you gave about what is known as Madhava-Gregory ( "Leibniz-Gregory" as your wiki article quoted) series is, generally credited to Madhava (Madhava of Sangamagrama - also from Kerala) who was born almost a hundred years ago (He died about 20 years before Nilakantha was born) .. Actually Nilakantha in his book credits that particular series to Madhava.

David Mumford (Fields medal!) says "It seems fair to me to compare him (Madhava) with Newton and Leibniz" ... Yes there is quite a bit of development of Calculus there. Fortunately "Ganita-Yukti-Bhasa" written in Malyalam by Jyesthedeva (written around 1540 ?) survives (unfortunately I can't read Malyalam ) but it was recently (2008) translated into English by K Sarma (costs only about Rs 1000 in India but about $200 here in US )

For example this is from Sarma's book (English translation of Jyesthedeva of Madhava's work etc. pp 97). showing in fairly accurate way

Here, multiply the repeated summation of the Rsines by the square of the full chord and divide by
the square of the radius. . . . In his manner we get the result that when the repeated summation of
the Rsines up to the tip of a particular arc-bit is done, the result will be the difference between the
next higher Rsine and the corresponding arc. Here the arc-bit has to be conceived as being as minute
as possible. Then the first Rsine difference will be the same as the first arc-bit. Hence, ifmultiplied by
the desired number, the result will certainly be the desired arc.

(He is always clear about which formulas are exact and which formulas are approximations - Explicitly notes things like integral of x^n is (x^(n+1)/(n+1) ... and correctly (and beautifully) showing derivatives and integrals of trigonometric functions ... few centuries before Newton ... Although )

Also, many now very seriously opine that this (the writings of th e Kerala school) was / (may have been) transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Kochi at the time. As a result, it may have had an influence on later European developments in analysis and calculus ..(Many articles such as 2001 -Journal of Natural Geometry by Almeida, John and Zadorozhnyy write about this in detail) ... though no translation of this work has been found/survived in Europe then)

FWIW - As mentioned before, one of the best book i have seen is Kim Plofker's Mathematics in India.

The differential equation for the sine function, in finite difference form, was
described by Indian mathematician-astronomers in the fifth century (Aryabhata) (see Plofker, section 4.3.3) (Describes the finite difference equation satisfied by samples sin(n. delta(θ)) of sine function)

I have read/studied Bhaskara's Surya sidhanta and it is quite obvious that he certainly knew (and mentions it) the derivative of sine is cos etc..

Chapter 7 of Plofker's book describes Kerala school and here it talks quite a bit about calculus etc..If you have not, it is a real treat to feast on the riches described in Plofker's book or Ganit-yukti-bhasa...

To end, not the slightest interest in dissing Newton and others (they were certainly great).. but here is what David Mumford summarizes :

The discovery of the finite difference equation for sine led Indian mathematicians eventually to the full theory of calculus for polynomials and for sine, cosine, arcsine, andarctangent functions, that is, for verything connectedtothe circle and sphere that might be motivated by the applications to astronomy. This workmatured over the thousandyear period in which the West slumbered, reaching its climax in the work of the Kerala school in the fourteenth to sixteenth centuries. I won’t describe the full evolution but cannot omit a mention of the discovery of the formula for the area and volume of the sphere by Bh¯askara II. Essentially, he rediscovered the derivation found in Archimedes’ On the Sphere and the Cylinder I. That is, he sliced the surface of the sphere by equally spaced lines of latitude and, using this, reduced the calculation of the area to the integral of sine. Now, he knew that cosine differences were sines but, startlingly, he integrates sine by summing his tables! He seems well aware that this is approximate and
that a limiting argument is needed but this is implicit in his work. My belief is that, given his applied orientation, this was the more convincing argument. In any case, the argument using the discrete fundamental theorem of calculus is given a few centuries later by the Kerala school, where one also finds explicit statements on the need for a limiting process, like: “The greater the number
[of subdivisions of an arc], the more accurate the circumference [given by the length of the inscribed
polygon]” and “Here the arc segment has to be imagined to be as small as one wants. . . [but] since
one has to explain [it] in a certain [definite] way, have said [so far] that a quadrant has twenty-four
chords.”

AmberGji,

responding very late,though I read up your solution long time back.I felt very dissappointed by the proof.It is so non-constructive in the sense I cannot check the divisibility of 1+n+n^8 +...n^47 by 1+..n^4 for even n=2.Any comments.

Ramanaji,
May I humbly ask what this corner is doing in the Tech and Economic forum though I guess there must be some reason if you think this deserves publicity but cannot fathom what it is except maybe to show off that we 'Hindoo fanatic' jingoes are also 'intellectually' inclined which does not stand up to scrutiny based on the interesting but noway profound prblems discussed here notwithstanding some formidable minds posting here.

I found an interesting elementary problem.Want to share it.

A school has 1,000 students and 1,000 lockers, all in a row. The lockers all start out closed. The first student walks down the line and opens each one. The second student closes the even-numbered lockers. The third student approaches every third locker and changes it state. If it's open, he closes it; if it's closed, he opens it. The fourth student does the same, and so on, through 1,000 students. How many lockers end up open?

svenkat wrote: responding very late,though I read up your solution long time back.I felt very dissappointed by the proof.It is so non-constructive in the sense I cannot check the divisibility of 1+n+n^8 +...n^47 by 1+..n^4 for even n=2.Any comments.

When n=2, the expression is divisible by 31, (when n=3, it is divisible by 121 (or 11) etc)) which is easy to check. You can use a multi-precession calculator or just remember that 2^5 (or 32) gives remainder of 1 when divided by 31.

(Thus, When divided by 31 ..2^8 will give remainder of 2^3; 2^19 will give remainder of 2^4; and 2^47 will give remainder of 2^2 and thus remainder for the whole expression will be 1+2+2^3+2^4+2^2 or 31 IOW it is divisible by 31)

May I humbly ask what this corner is doing in the Tech and Economic forum though I guess there must be some reason if you think this deserves publicity but cannot fathom what it is except maybe to show off that we 'Hindoo fanatic' jingoes are also 'intellectually' inclined which does not stand up to scrutiny based on the interesting but noway profound prblems discussed here notwithstanding some formidable minds posting here.

FWIW - I made that request... sure many/most (?) items here are just for fun; some are similar to "Nation on March" type but, IMO, quite a bit discussion about (say ICM, P vs NP etc) are in no way less "profound" and very relevant to main theme (security etc) of our nation. It is no accident that many of "useless" type problems posted here have been taken from (or appeared in) contests sponsored by NSA, Defense Department, AEC (India) types....

A school has 1,000 students and 1,000 lockers, all in a row. The lockers all start out closed. The first student walks down the line and opens each one. The second student closes the even-numbered lockers. The third student approaches every third locker and changes it state. If it's open, he closes it; if it's closed, he opens it. The fourth student does the same, and so on, through 1,000 students. How many lockers end up open?

Since, obviously, only square numbers have odd number of factors the answer is 31

AmberGji,
Thanks for responding.That makes things clearer.Still I dont think even for n=2 or 3,one can do a direct check ie by an ordinary calculator.But modulo arithmetic makes it clear.

I think it is unfair you should post the answer that too making a statement with half a line proof. I am sure atleast one or two will enjoy the problem.

^^^^
I'd encountered the problem before, except that the doors were 1..100. Was working on writing some code to do this and was trying to figure out the logic when I suddenly realized that square numbers are the only ones with odd # of factors. Didn't need to write code after that.

Amber G: Interesting, thank you. I will see if my discount bookstore has a copy of Plofker.

Here's a fairly old problem from the IMOs that is easy to state, but not so easy to solve:

Determine all pairs (a,b) of positive integers such that ab^2+b+7 divides a^2b+a+b.

adityaS - Ramanaji graciously sent me a copy of the Plofker's book . Also if you have time, check out her special lecture at ICM (I gave the link of 1 hour video a few posts above) about the communications between Indian mathematicians and Yavana (West) mathematicians.

As to your problem, Can I ask which IMO was this? (Looks relatively easy, in my opinion, for IMO -One can easily guess (Don't peek if you don't want to) (... easy ones like a=b=7 .. one can work out b=7n, and a = nb; and apart from that b=1 (a=11, or 49 ) these are the only solutions (fairly easy to prove), If I am not careless.. (.)

Added Later: Okay Checked my archives.. Found out which IMO.. (Proposed by David Monk UK) (Nice problem)

AmberG< have you come across a book "Street Fighting Maths" by Sanjoy Mahajan of MIT? Its very practical book on applying college level maths to solve everyday problems. Send e-khat if you want a dekko.

^^^ Thanks. I have not seen the book before so will take a look. Found that pdf is available at:
http://mitpress.mit.edu/books/full_pdfs ... matics.pdf
Thanks again.

BTW .. heard that MIT open courses on net will no longer remain free...

Pity! Guess belt tightening hits all.

Hey Amber G.

Can you please suggest some magazines which publish articles (or book reviews) about recent changes in Mathematics for non-experts like myself (I am a science graduate but not a Mathematician).

I have noted that the Science and Nature magazines do not publish much about Mathematics.

Thanks.

abhishek_sharma wrote:Hey Amber G.

Can you please suggest some magazines which publish articles (or book reviews) about recent changes in Mathematics for non-experts like myself (I am a science graduate but not a Mathematician).

I have noted that the Science and Nature magazines do not publish much about Mathematics.

Thanks.

Hi -
Yes, I have also not seen good (relatively in my opinion) math articles in Scientific American or such magazines...Of course, there are a few good math sites and open courses from many universities (eg IIT, MIT etc) .

One thing you may like to check out Terence Tao's blog What's New
Prof Tao (Fields Medalist) is not only considered one of greatest living mathematician (In many fields), he is also famous for being a good teacher, and writer for non-specialists.. This could be a good start for other sites too.

HTH

Book review: This are is very important due to bad economy!

Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss, "Laws of Small Numbers: Extremes and Rare Events,3 Edition"
Springer Basel | 2010 | ISBN: 3034800088 | 509 pages |

Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as excedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation. Reviews of the 2nd edition: "In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field." David Stirzaker, Bulletin of the London Mathematical Society "Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook." Holger Drees, Metrika


This book provides a nice introduction to the mathematical theory of rare events and extreme values. It takes the point process approach to rare events using the empirical point process defined by the crossing of a high or low level to employ an approximate Poisson process. This approach has broad applicability and the authors use it to consider rare and extreme events for stochastic processes including Gaussian processes, Poisson point processes, extreme value models and nonparametric regression.
This book is noteworthy for its mix of theory and application and for the introduction of the software XTREMES. Chapter 6 is devoted to explaining how to analyze data for extremes using the software XTREMES. A diskette is included with the text with version 1.2 of XTREMES that is an interactive program that can run on MS-DOS operated computers. The text is very much up-to-date on the theoretical developments as of 1994.

Okay. Thanks Amber G.

Benoît Mandelbrot, Novel Mathematician, Dies at 85

http://www.nytimes.com/2010/10/17/us/17mandelbrot.html

^^^ I don't know if many here are familiar with Mandelbrot Competition (for high school - to find talent).
If you are (or have kids who are) in high school, interested in Math, you may enjoy it. My kids did. Check out mandelbrot.org to see if the school is interested. (It has both individual level and team level competition). Apart from AMC competition (leading to AIME, National Olympiad, and International Olympiad), I recommend this (along with perhaps USAMTS).

Mandelbrot, BTW, was also very well known for his popular teaching and inspiring math in others. (Most people know him for Fractal Geometry and famous figures like:

David Mumford Awarded Top U.S. Honor

..one of the leading mathematicians in the United States, was granted the presidential National Medal of Science earlier this month. President Obama will award the medal, which is the highest scientific honor in the country, at a White House ceremony later this fall.

You may recall, University of Hyderabad's wish to honor Field Medalist Mumford was thawed by GOI's red tape. He was supposed to get Honorary degree from UoH, along with World Chess Champion Anand around ICM. (GOI kept wondering if Anand is Indian citizen for months even when UoH has sent a copy of his passport etc... never mind that he always played under Indian flag and no other )

BTW- small tidbit - Chandra (S. Chandrasekhar) was awarded this medal (in 1967 ?) by Johnson.. (His award of Padma Vibhushan came a year or two later)

(Also check out, my post here on Aug 20 regarding Prof Mumford and, if you haven't yet checked out- check out:
Review of Mathematics in India

What about the Infosys Prize for Khare in math ? No one seem to have mentioned it anywhere in BRF. Strange isn't it when we rally against Nobel's motive, we do not seem to recognize our own achievements being celebrated by our own prizes.

svenkat wrote: also 'intellectually' inclined which does not stand up to scrutiny based on the interesting but noway profound prblems discussed here notwithstanding some formidable minds posting here.

How about some light tones then?

Two calculus jokes from:

http://www.workjoke.com/mathematicians-jokes.html

1. A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying "I differentiate you!"

One day he met a new patient; and true to form he stared at him andsaid "I differentiate you!", but for once, his victim's expression didn't change. Surprised, the mathematician marshalled his energies, stared fiercely at the new patient and said loudly "I differentiate you!", but still the other man had no reaction. Finally, in frustration, the mathematician screamed out "I DIFFERENTIATE YOU!"
The new patient calmly looked up and said, "You can differentiate me all you like: I'm e to the x."

------

2.

The functions are sitting in a bar, chatting (how fast they go to zero at infinity etc.). Suddenly, one cries "Beware! Derivation is coming!" All immediately hide themselves under the tables, only the exponential sits calmly on the chair.

The derivation comes in, sees a function and says "Hey, you don't fear me?"
"No, I'am e to x", says the exponential self-confidently.
"Well" replies the derivation "but who says I differentiate along x?"


-----

And a third along the same lines, which I heard from someone:

3. During a function conference, some functions of different nationalities are sitting in a bar, chatting and generally hanging out. While everyone is thus schmoozing, someone notices a lanky function sitting alone in a corner. The lone function is approached and asked, "Why are you sitting here all alone and gloomy? You must get out and speak with people, you should come out and integrate with us". The function replies: "What difference would that make? I am e^x".