BR Maths Corner-1

[Raja Bose]

X-posting since it got posted in Nukkad by mistake...

Then why are there 9 serial numbers onlee, masquerading as important data?

Duh!!!. You were supposed to google / wiki for the "German Tank Problem" and ta-da, the answer pops out!

Nah, whats the fun in Googling for answers. Never heard of the German Tanki prablem but suspected it had something to do with population estimation hence the statement above . The problem with such problems is the naive inherent assumption of uniform distribution among deployed serial numbers and the fact that the sampling is representative, which doesn't always hold up in reality for sampling restricted to a short period of time. The usual estimation formula is this case is a function of the max. observed serial number and number of samples. The problem is the inherently high amount of weight given to the max. observed serial number and no weight is given to the values of the observed serial numbers.

In the specific case of the Asal Uttar battle, one has to know whether or not not all tanks have been committed to battle and pakis may not have deployed all tanks in the same time especially what is the sampling window. In this case, sampling window is governed by how the tanks got ambushed - there will be a bias due to the first 2 lines of tanks getting bogged down in the muddy fields of Punjab and ambushed by anti-tank gunners of IA, while rest of them on seeing such a debacle will probably downhill-ski paki style hence, there will be a skew in the observed serial numbers of tankis halaled so the usual MVUE formula in this case may not work depending on the skew. The skew itself will be determined as to how pakis chose tanks to use as lead scouts for cannon fodder and probing attacks - older ones with lower serial numbers then you will have a +ve skew and estimation using usual formula will fail, newer TFTA ghaazi ones with higher serial numbers then you will have a -ve skew but usual estimation will work becoz usual formula only cares for max. value and number of samples (not the value of samples themselves). If they had an ISI graduate who did SRSWOR across entire batch of tanks to choose the cannon fodder then also your usual estimation formula for population max. will work. But in reality, in case of attack on entrenched positions, it is the +ve skew scenario which might happen most often, throwing the estimates haywire and causing mucho pain since it will result in severe underestimation of Paki strength

One way to partially remedy the defect might be to include a weighted term which is a function of the skewness of serial numbers observed of halal'ed tanks and throw out the uniform distribution requirement. For example, If one observes a strong -ve skew and it is an attack on known enemy entrenched positions, then there is a higher likelihood of the actual number of tanks in inventory being much larger and the observed max. serial number being actually located close to the median or even below it (the bounds of the usual formula are [max.observed.serial.no, 2*max.observed.serial.no -1])

But in the end, we cannot depend on results of one battle especially with an enemy which doesn't control the production of tanks and only has a limited number of tanks at its disposal.

Thats why there are lies, damn lies and statistics . And a further reason why I have an inherent distrust of models (mathematical ones, not plastic or flesh ones).

[Amber G.]

Just curious, if any BRFite (or their kids etc) participated in US Math Olympiad or Indian Math Olympiad.

Just for fun, the following is a problem from India's INmo - try it if you like it (HIgh school, not too hard)

Given x,y,z are real, find all possible values of x,y and z if both the following eq. are satisfied.

(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)=xyz (eq 1)
and (x^4+x^2y^2+y^4) (y^4+y^2z^2+z^4)(z^4+z^2x^2+x^4) = (xyz)^3 eq(2)

How about x=y=z = 1/3 ? I think that is the only non trivial answer.

I got that since (x^2+xy+y^2) * ((x^2 - xy+y^2) = (x^4+x^2y^2+y^4) , if you divide the 2nd eqn by the 1st, you basically get ((y/x - 1/2)^2 + 3/4) * (((z/y - 1/2)^2 + 3/4)*((x/z - 1/2)^2 + 3/4) = 1

.

Nice. Yes, only non-zero ( actually one can easily see that apart from two of the three being equal to zero) real values are x=y=z=(1/3)

First one can easily see froin eq 2 that xyz >= 0 (because LHS is all positive - this means wrt (x,y,z) either all are positive or two are negative and one is positive) )
As shown above ((x^2+xy+y^2) * ((x^2 - xy+y^2) = (x^4+x^2y^2+y^4), so
(x^-xy+y^2)(y^2-yz+z^2)(z^2-zx+x^2)=(xyz)^2)
Let us take the case when all x,y,z are positive

if x is not equal to y
(x-y)^2 > 0 or (x^-xy+y^2 >xy
hence (x^-xy+y^2)(y^2-yz+z^2)(z^2-zx+x^2)> (xyz)^2
But that is contradiction, hence x=y (and similarly y=z )

In the case if x (or y or both) is negative, we still have, (assuming |x| is not equal to |y| ) |x^2-xy+y^2| > |xy|
and you still get the same conclusion as above.

putting x=y=z in eq1 we get the result as x=y=z=(1/3) and that's the only nonzero real solution.

[vina]

we cannot depend on results of one battle especially with an enemy which doesn't control the production of tanks and only has a limited number of tanks at its disposal

I know the problems behind the assumptions and all inherent bias problems. However, I would think that in a case like Pakiland-e-Baksheesh, this is easy right?. Like in Assal Uttar, the Pakis attacked with 100+ tanks and we managed to knock out the majority, the sample size as a percentage of total that the Amrikhans supplied will be pretty sizable and if the bulk are knocked out, the serial numbers will be a good mix anyway I think.

[Raja Bose]

^^Actually the percentage of total is just one (and lesser significant) part of the estimation - what is more important (and where skewness plays a role) is the max. observed serial number and the weight to be given to that number. For the usual estimation to work accurately, the max. observed serial number has to be very close to the max. serial number of entire population.

I guess the volatility argument is also what was exploited by the MIT fellas trying to win at Blackjack where they tried to reduce the uncertainty of which cards were going to pop out in order to maximize the expected amount of wins from casinos.

[Amber G.]

Just curious, if any BRFite (or their kids etc) participated in US Math Olympiad or Indian Math Olympiad.

FWIW:
An awards ceremony for the 12 USAMO winners was held yesterday in Washington, D.C. (U.S. Department of State building)
For this USAMO event (in the sequence of increasingly challenging mathematical contests starting with more than 200,000 in the first contest (AMC 10 and/or AMC 12) about 10,000 were invited to compete in the second contest and about 300 of these were invited to final USAMO ...from this result top students are invited to math training and US IMO team would be selected)

Small tidbit - usually there are a few Indian Americans in this group (going by previous years results) but this year there is none in top 12 (and only one in top 16) and in Junior USAMO there is one in top 13.
My recommendation to anyone in high school (6th to 12th grade) who is interested in Math should try AMC, if nothing just for fun. The contest is open to all students in usa (and similar contests in other countries).

[abhishek_sharma]

International math prize named after Leelavati

http://www.hindu.com/2010/06/09/stories ... 982200.htm

[Amber G.]

^^^ As noted before ()March 31 posts) this is during ICM where recipients of Fields Medal will also be announced.

[Amber G.]

FYI The Indian Math Olympiad Team consists of

Akashnil Dutta
Gaurav Patil
Akshay Degwekar
Anand Degwekar
Ronno Das
Satyaki Mukherjee

(US team is not chosen yet)

[Raja Bose]

FYI The Indian Math Olympiad Team consists of

Akshay Degwekar
Anand Degwekar

Birathers?

Good to see 2 bong brethren.

[Amber G.]

USA Team for IMO 2010
Evan O'Dorney
Allen Yuan
In Sung Na
Xiaoyu He
Ben Gunby
Calvin Deng

[abhishek_sharma]

A Math Problem Solver Declines a $1 Million Prize

http://www.nytimes.com/2010/07/02/science/02math.html

[Raja Bose]

^^^I thought they had offered him the Clay prize before, no? Poor fella is trying to pull a Grothendieck here.

[Amber G.]

Results of The Indian Math Olympiad Team this year:
Akashnil Dutta - Silver Medal (Won Silver in 2009 too)
Gaurav Patil - Silver Medal (Won Bronze in 2009)
Akshay Degwekar - Honourable mention
Anand Degwekar - Honourable mention
Ronno Das- Honourable mention
Satyaki Mukherjee - Bronze Medal

(Team Rank 36th)

US Team - 3 Golds (Evan O'Dorney, Xiaoyu He, Benjamin Gunby)
and 3 Silver (Calvin Deng, In Sung Na, Allen Yuan)
(Team Rank 3rd)

(China was 1st (6 Gold) , Russia 2nd, USA 3rd, Korea 4th -- But North Korea was disqualified,
TSP, 3 people (out of 6) had more than zero points, and 1 more than 4.

[Amber G.]

^^^ There is a news item here:
http://www.dnaindia.com/sport/report_in ... ad_1409146

BTW Gaurav missed the gold, and Anand D. missed the bronze by just a point or so..

[Amber G.]

Hmm.. this thread is inactive ...And since in 8/1947 was a special month ..

Find all x , such that

1+x+x^8+x^19+x^47 is prime.

[svenkat]

Any clues Amberji,expression is a prime for x=1.Beyond that,your assistance required.

[Amber G.]

Hint: Yes, it is prime for x=1 ONLY.. (tricky and cute but not hard to prove that for x>1 it is not prime)

Another Hint: (A much simpler problem - prove that 1+x^19+x^47 is prime only when x=1) (Again assuming, of course that x is a natural whole number)

[ramana]

Pop Sciecne article on

God's Number to solve Rubik's cube

[Amber G.]

X-posting , IMO important enough news and very significant if turns out to be verified.
Computer scientist Vinay Deolalikar claims to have solved maths riddle of P vs NP

Vinay Deolalikar (IIT Mumbai graduate, PhD from UCLA, at present at HP)
P vs NP is one of the seven millennium problems, (Clay Mathematical Institute has million dollar prize), and enormous prestige and honor awaits whoever solves it.

[Amber G.]

^^^I am no expert, but I did try to read/glance at the paper and talked/looked_at_blogs/emails_etc with others who are more knowledgeable..it looks like the work is significant but it may still be quite away from proof.

According to Terence Tao: ( Fields Medal - won International Math Olympiad medal at the age of 9 or 10 - youngest ever etc - quite a bright guy):

I think there are several levels to the basic question “Is the proof correct?”:

Does Deolalikar’s proof, after only minor changes, give a proof that P!=NP?
Does Deolalikar’s proof, after major changes, give a proof that P!=NP?
Does the general proof strategy of Deolalikar (exploiting independence properties in random -SAT or similar structures) have any hope at all of establishing non-trivial complexity separation results?
After all the collective efforts seen here and elsewhere, it now appears (though it is perhaps still not absolutely definitive) that the answer to #1 is “No” (as seen for instance in the issues documented in the wiki), and the best answer to #2 we currently have is “Probably not, unless substantial new ideas are added.” But I think the question #3 is still not completely resolved, and still worth pursuing (though not at the hectic internet speed of the last few days.)

Vinay Deolalikar has made few updates but still (according to blogs) standing by his proof.

Meanwhile another expert Neil Immerman thinks there could be fatal flaws in the proof, he has written to Vinay Deolalikar (and the letter starts with..

Dear Vinay Deolalikar,

Thank you very much for sharing your paper with me. I find your approach and your ideas fascinating, but I am afraid that there is currently a serious hole in your paper as I now describe....

(there is a wiki blog for all these)

[Amber G.]

India a world mathematics power, says professor Raghunathan
http://www.hindu.com/2010/04/01/stories ... 251200.htm

Rumors is that Kiran Kedlaya is in for Fields Medal. (Guys you heard it here first!). He is under 40, and is an invited speaker. People may recall I had talked about him in our math thread. He has won many IMO medals ... for those who don't know him, he is at present a prof at MIT.
(Of course, It is a guarded secret, and no one is suppose to know till the announcement is made)

Other I think may be Ngo or may be Lurie or Manjul Bhargava.. Lets us wait and see.

Time is quite near ... Any rumor in Hyderabad?
I'll add another Indian name here: Ashkay Venkatesh

[Amber G.]

According to Terence Tao: ( Fields Medal - won International Math Olympiad medal at the age of 9 or 10 - youngest ever etc - quite a bright guy):

I think there are several levels to the basic question “Is the proof correct?”:

Q1: Does Deolalikar’s proof, after only minor changes, give a proof that P!=NP?
Q2:Does Deolalikar’s proof, after major changes, give a proof that P!=NP?
Q3:Does the general proof strategy of Deolalikar (exploiting independence properties in random -SAT or similar structures) have any hope at all of establishing non-trivial complexity separation results?
After all the collective efforts seen here and elsewhere, it now appears (though it is perhaps still not absolutely definitive) that the answer to #1 is “No” (as seen for instance in the issues documented in the wiki), and the best answer to #2 we currently have is “Probably not, unless substantial new ideas are added.” But I think the question #3 is still not completely resolved, and still worth pursuing (though not at the hectic internet speed of the last few days.)

According to Tao's comment here
He updated the answers to #1 and #2 as now , "definitely NO" and #3 is "probably not".
Also: from New Scientist:
Tide turns against million-dollar maths proof
Vinay Deolalikar in the meanwhile, is said to be putting updates over this weekend to answer some questions...

[Raja Bose]

I guess any proof for a famous unsolved problem whether it's P<>NP or Fermat's Last Theorem is bound to run into huge doses of skepticism from the community. Wiles faced it and Deolalikar is facing it. Remember that even Wiles after his Steve Jobs-esque one last thing claim of having solved Fermat's Last Theorem had to subsequently deal with major flaws in his proof and only after sustained marathon efforts with one of his brilliant former students was able to get the correct proof. I hope Deolalikar does not get intimidated by the "Big Names" who are talking from authority and focuses on getting his proof correct.

[abhishek_sharma]

Step 1: Post Elusive Proof. Step 2: Watch Fireworks

http://www.nytimes.com/2010/08/17/science/17proof.html

[rohitvats]

I actually spent an hour going through the link to the blog Amber G. posted - and while I don't know anything about Maths, I did read through most of the comments to check the tenor and attitude.....and only thing I'd say is that it is down right hostile. From the way the so called Ace Mathematicians are trying to find loopholes to other subject matter experts.....there is somehow, a disdain in the commentary.

[Raja Bose]

^^That condescending tone is there as Deolalikar is not "one of them". It is true for any branch of research where the old fogies form Old Boys Club and actually stifle innovation in the name of "excellence" unless it comes from one of their peers or their connected ones. It is one of the worst aspects of academia today.

[rohitvats]

Hmmm...thanks for summing up. Yup, it makes sense. I'm actually miffed about the way the Chinese dude (tao) has taken it upon himself to to find faults with the proof. And from the enteries in that blog, I actually saw people pointing weaknessess in his faults.

Question to the people in know - this Tao chap is considered to some sort of genius. The age at which he did Phd and things like that are prominentlt mentioned. Plus some theorem he co-authored.

He is 35 (?) now. Tell me something - how much has been his contribution after acheiving all these early age trophies? Is the theorem (I hope that is what they call it) enough for him to be considered a genius for lifetime?

[abhishek_sharma]

I am not an expert in Mathematics but it is very difficult to assess the importance of papers written at the very top level (in any subject). Very few people even understand them.

Bias and prejudice does exist in academia. For example, see:

http://www.newyorker.com/archive/2006/0 ... 28fa_fact2

[Raja Bose]

Tao is a classical child prodigy and so are his brothers. They are born with those abilities (and in math generally you have to be). Ofcourse child prodigy doesn't equate to a pleasant personality or humility. In fact most brilliant people are quite unbalanced in their social interactions and other facets of their life. It is best to be balanced and normal In that aspect, I have 400% respect for Manjul Bhargava who is a child prodigy, accomplished in math and music (as is quite common for top mathematicians) yet is an extremely humble person and very down-to-earth.

A desi example of (slightly lesser) child prodigies would be Anish and Atish Das Sarma. Anish now works in Shammi Kapoor Co's research.

Irrelevant trivia: One of Tao's brothers was part of the Google Wave team in Oz and was widely featured during the initial hype.

[Raja Bose]

Bias and prejudice does exist in academia. For example, see:

http://www.newyorker.com/archive/2006/0 ... 28fa_fact2

That is why it is important not to get dazzled by someone's past accomplishments or pedigree when evaluating their work or their input to some problem. Many times such people due to their past brilliance are able to shepherd past complete BS without anybody raising an eyebrow. Hence, any evaluation must be dispassionate, rigorous and focused on the material not the man.

[abhishek_sharma]

Ofcourse child prodigy doesn't equate to a pleasant personality or humility. In fact most brilliant people are quite unbalanced in their social interactions and other facets of their life.

True. Gauss was very rude and impolite. He claimed to invent/discover non-Euclidean geometry. Some people believe that János Bolyai should get most credit for it. Similarly, Newton and Leibniz had their dispute regarding calculus. Freeman Dyson wrote that even Oppenheimer was rude and arrogant.

It is best to leave these smart guys alone and hope that they solve the problems of this society .

Probably OT: S. Chandrasekhar faced racism when he joined Univ of Chicago. But in that era, those attitudes were not uncommon.

[abhishek_sharma]

Bias and prejudice does exist in academia. For example, see:

http://www.newyorker.com/archive/2006/0 ... 28fa_fact2

That is why it is important not to get dazzled by someone's past accomplishments or pedigree when evaluating their work or their input to some problem. Many times such people due to their past brilliance are able to shepherd past complete BS without anybody raising an eyebrow. Hence, any evaluation must be dispassionate, rigorous and focused on the material not the man.

True. The problem is when a person like Einstein writes a paper on relativity, not many people can provide a rigorous evaluation. If they can't find an obvious mistake, they accept the paper.

Unfortunately, sometimes the reviewers may think that the subject of the paper is not useful. For example, George Akerlof wrote a paper on information asymmetry in Economics. The Quarterly Journal of Economics rejected it contemptuously. Many years later, he won a Nobel for it.

[Raja Bose]

^^Shannon is a famous example of the last.

[Amber G.]

FWIW: Few comments wrt Deolalikar's recent work with P vs NP
Can't believe a week has already passed (or only a week has been passed ) with so much activity..

The Synopsis of proof (only about 3 pages or so) can be seen at his web page at:
http://www.hpl.hp.com/personal/Vinay_De ... nopsis.pdf

However, today, he has taken down the manuscript and all drafts (there have been several drafts within last week - (about 3 I have seen) of the details from his webpage, some corrections/revisions were made by him in these drafts but, IMO, nothing major, and not all the answers (which I could follow) to the issues raised by others.

He still has his synopsis up, and the following comment by him is still there:

The preliminary version was meant to solicit feedback from a few researchers as is customarily done. It illustrated the interplay of principles from various areas, which was the major effort in constructing the proof. I have fixed all the issues that were raised about the preliminary version in a revised manuscript (126 pages); clarified some concepts; and obtained simpler proofs of several claims. This revised manuscript has been sent to a small number of researchers. I will send the manuscript to journal review this week. Once I hear back from the journal as part of due process, I will put up the final version on this website

A good place to look for serious (unfortunately it is more for mathematician's interest only) discussion is Suresh Venkatasubramanian's (The Geomblog): which is quite compressive.:

http://geomblog.blogspot.com/2010/08/po ... is-of.html
The link I gave before is pretty nice too (this and the above refer to each other anyway)

[Bade]

Has any math history buffs seen this one before ? These are scanned archives from Current Science.

Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion

A more easily readable form in one unit is available at http://www.physics.iitm.ac.in/~labs/amp ... ronomy.pdf

[Amber G.]

^^^ Bade and others - Talking about History of Math, Radha Charan Gupta, one who has done so much research about History of indian Math , is being honored in this ICM mega event tomorrow (or is it already 'today' in Hydarbad). He has been chosen for the prestigious Kenneth O May Prize by the International Commission for the History of Mathematics in 2009 and will receive the award from the president during the ICM. (There are few rumors, but we will know tomorrow who wins other awards)

If you are interested .. check out:
History of Indian Mathematics - India Gave Much More Than Zero
about this prize and background on Indian math and his work.

India not only gave the concept of zero to the world, but influenced many foreign mathematical traditions by its disocoveries. Much was not known until Radha Charan Gupta proved this by his immaculate research.

For his pioneering work he will be honoured at the International Congress of Mathematicians being held in Hyderabad during August 19-27, 2010. He is the first Indian to get this distinction--Kenneth O. May Prize.

Radha Charan Gupta, currently engaged in extensive research work at Ganita Bharati Institute in his native city Jhansi (Uttar Pradesh, India). It has been acknowledged that no scholar in the twentieth century has done more to advance widespread understanding of the development of Indian mathematics.

Gupta Expounded Cosmological Theories ...
He skillfully analyzed many unknown ingenious mathematical formulas in Sanskrit. He published several papers on the remarkable mathematical discoveries of the Jaina tradition, many of which had been almost inaccessible to anyone except specialists in Prakrit (an ancient Indian language). He also expounded many Jaina, Buddhist or Hindu cosmological theories. ... { I am pretty interested in Jaina work and have read quite a bit}

Prof. Gupta's major contributions include work on the history of development of trigonometry in India. ...
<snip> { Read more if interested}

Also may be of interest .. another award going to Indian origin person - The ICM will also award Leelavati Prize (for popularization of Math .. One of the thing Bhaskaracharya and his daughter (Leelavati) is known for)
(Ironic that many people do not even know how about Leelavati )
First Leelavati Prize Goes to Simon Singh

Also Vishy Anand will be there in Hydarabad too!
Will One Chess Player Trounce 40 Mathematicians ?

(At one time I thought I would attend the ICM event but it did not work out )

[Amber G.]

I made a request to admins to see what they think about moving this dhaga to T&E forum for more visibility.. what do others think?

[Yayavar]

^
Also may be of interest .. another award going to Indian origin person - The ICM will also award Leelavati Prize (for popularization of Math .. One of the thing Bhaskaracharya and his daughter (Leelavati) is known for)
(Ironic that many people do not even know how about Leelavati )

I remember NCERT mathematics (and chemistry was excellent too) text books with fondness. They always had a discussion of the history, and some background prior to each chapter. iirc, it was in 9th grade (or possibly 8th) it had the discussion on quadratic equations, a couple of problems from Leelavati. I found the book (in English) in the library and worked through a few problems.

[Amber G.]

FWIW: Comments about Deolalikar's work from renowned Prof. Manindra Agrawal (Of' Primes in P' fame!) .. it's about 10 minute audio:
http://dl.dropbox.com/u/1738449/podcasts/1708_pnp.mp3

Also FWIW - my personal perspective: when I heard the news, I was excited and started reading the paper (I also talked with others who are more expert in the field - but statistical mechanics type techniques which he used i understand a little).. pretty soon I found too much ambiguities (things not defined well and hard to understand for non experts almost to the point that I lost interest) .. this is at present, is feeling from most of the people I talked (or read technical comments).. Prof Agrawal's assessment is similar (and similar to, say, Tao's)

For those who have not heard of Prof. Agrawal, He (and two of his undergraduate students made world wide news - most major news paper like NY Times - had front page item, very rare for math related news - when they published their 'Primes in P' paper. The part about that beautiful paper was the work could be understand and appreciated by almost all who have background in math. The work was truly beautiful !
(They, (and IITK) deservedly got lot of credit, fame and recognition in a very short time. (If interested see, for example:
http://www.ams.org/notices/200305/fea-bornemann.pdf
or check out *many* wiki articles about the author or method (just google "primes in P" : like

http://en.wikipedia.org/wiki/AKS_primality_test

[Raja Bose]

I made a request to admins to see what they think about moving this dhaga to T&E forum for more visibility.. what do others think?

+1