BR Maths Corner-1

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Amber G.
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Re: BR Maths Corner-1

Postby Amber G. » 03 Jan 2021 04:38

^^^ The PBS Nova special was about Ramanujan.
Nova (1974– )The Man Who Loved Numbers


"The Man Who Loved Only Numbers", as you said, is a biography of the famous mathematician Paul Erdős (by Paul Hoffman. It is written much (IIRC later 1990's. Both books are excellent.

(I can claim/brag a very high Erdős number :) ) ( Erdős number is the number of "hops" needed to connect the author of a paper with Paul Erdős. An author's Erdős number is 1 if he has co-authored a paper with Erdős, 2 if he has co-authored a paper with someone who has co-authored a paper with Erdős).
(Here is a very famous picture of Erdos with another great mathematician - then 10 year old - Terrence Tao)

Image

Vayutuvan
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Re: BR Maths Corner-1

Postby Vayutuvan » 03 Jan 2021 06:13

Amber G., you might know about the "happy ending problem" which was described in the book on Erdos. Let me pose it here.

Given five points in general position in a plane, i.e. no three points form a straight line, prove that there is always a convex quadrilateral.

The problem was given to Erdos and Szekeris by their friend Esther Klien. Both the men scratched their heads until Esther gave the solution.

It was the start of Ramsey Theory program by Erdos and Szekeris.

Erdos gave the problem that name because Szekeris and Esther Klein started a romantic relationship which ended in their marriage.

Amber G.
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Re: BR Maths Corner-1

Postby Amber G. » 04 Jan 2021 23:14

Honoring Ramanujan - (And talking about Bhargava etc):

Published today in RMS - Bhargava's class group cube has implications for black hole charges.
This new research published by Renowned Physicist Ashoke Sen and his team.
Image

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Ashoke Sen - as I hope most of the people know here as I have mentioned him here in brf, is likely to get Nobel some time - he won the Physics Breakthrough prize and Padma Bushan , (Along with Infosys prize, Dirac Medal, SS Bhatnagar Award, FRS, etc). (Not only that - he did his physics from IIT Kanpur and honored distinguished alum award and is/was prof at MIT :) )

Amber G.
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Re: BR Maths Corner-1

Postby Amber G. » 06 Jan 2021 00:23

Vayutuvan: I did not know about "happy ending problem" so read about it. Thanks.
--
There is a very nice article about Dyson, (along with John Conway, Ronald Graham) in New Yorker which I really liked.
The scientists who explored the world with their mind.

Three Mathematicians We Lost in 2020

Dyson, who I had as a prof had major impact on many in the world. As this article says "Freeman Dyson was a translator: he turned physics into math, and those subjects into English for the general public."

Dyson's adviser was Hardy, and he was one of the leading expert on Ramanujan's math. He switched his field from pure math to physics because, as he once said "Math is too messy".. (Story is Harish Chandra - famous Indian Mathematician - once told him that HC decided to switch from Physics to Math because "Physics is too Messy", Dyson told him "He is switching from Math to Physics because of the same reason).

Dyson, according to him, was quite proud of the fact that he (along with Wigner) "discovered" ML Mehta (who was just an obscure student in 50's ) and invited him to Princeton and they worked on Random Matrices, and topics like that (which has come back to fashion now)..and found that spectrum lines of complex nuclei followed math inspired by Ramanujan. (Personally I learned a lot and inspired a lot by these gurus). He was also the bridge between people like Feynman (with his diagrams) and Schwinger (with his rigorous Math) showing that mathematically those methods were equivalent. Dyson will be missed.

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Re: BR Maths Corner-1

Postby Mort Walker » 06 Jan 2021 01:15

There are a series of interviews on YouTube with Freeman Dyson talking about Richard Feynman. About math and physics.

I agree Dyson will be missed. Many of us have been missing Feynman for a long time too. He left too early.

Amber G.
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Re: BR Maths Corner-1

Postby Amber G. » 06 Jan 2021 02:21

^^^ There is a set of many "oral history" which are my favorites ... (Now they are available on you tube too).

There are a few where he talks about his invitation to Mehta and inviting him to Princeton..
https://www.youtube.com/watch?v=XgPcd1Fz3FA


***
Dyson and Feynman were very good friends - (Both were in Cornell around the same time with Bethe and had lot of adventures there)..One of trip both Feynman and Dyson have talked about a lot (in their auto biography as well as many other sources) was once they travelled across America in car.
(Feynman's famous "Surely you are Joking" talks about quite a few adventures of that journey).
Bethe, Feynman and Dyson - all were extremely quick (even when they were young students) in calculations or guessing/estimating/ complicated integrals etc.

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Re: BR Maths Corner-1

Postby sudarshan » 06 Jan 2021 02:32

The thing I know about Hans Bethe is that George Gamow once played a practical joke (which I remember reading either in Gamow's own book, or in somebody else's book when he was talking about Gamow). Apparently Gamow wrote a paper with one Alpher and Hans Bethe. And tried to pass it off as the "Alpher, Bethe, Gamow" paper. And then tried to persuade somebody else to be the fourth author, on condition that the guy changed his last name to "Delter." Paraphrasing, don't remember the exact story.

Hint: Greek alphabet, first four letters.

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Re: BR Maths Corner-1

Postby Amber G. » 06 Jan 2021 02:49

^^^ :) :) The above story is quite famous .. (I just checked it even has a wiki page -- https://en.wikipedia.org/wiki/Alpher%E2%80%93Bethe%E2%80%93Gamow_paper)..
- Ralph Alpher was Gamow's student, and they wrote a famous paper (actually now quite well known about origin of various elements inside stars), and Gamow humorously added Bethe's name (as H Bethe ( (in absentia) and this "alpha-beta-gamma' paper did get published. (Gamow said that since Bethe did not object and was quite helpful in discussions so his name was justified).

The story as you told is interesting as Gamow often joked that a later contributer Herman (who later did computer calculations based on that theory - stubbornly refuses to change his name to Delter.

Anyway - I HIGHLY recommend Gamow's books (he wrote dozens - all best sellers - Many text books but *many* popular books too) - If you can get "Thirty Years That shook Physics" where he has many such stories and interesting history of Physics. ..

I fondly remember visiting University of Colorado (where Gamow settled) had a huge display in their Physics department lobby - all covered with Gamow's published books - Some translated into many languages - I could see Gujrati, Marathi, Hindi etc..)..

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Re: BR Maths Corner-1

Postby sudarshan » 06 Jan 2021 06:01

Gamow seems to have been quite the clown. I remember a story of his, about some trip (sorry, details are hazy, read a couple of his books long ago) where he and family were going horse-riding. Gamow being like 6'3" and pretty big, had a hard time finding a horse to carry him. One was found though, rather ungainly, but big-boned and able to carry his weight. And Gamow promptly named it "Betelgeuse." Betelgeuse (also associated with nakshatra Arudhra) is a gigantic red star in the constellation of Orion, I believe the name is Arabic for "arm of Orion" or "hand of Orion." Gamow OTOH interpreted it as "shoulder of the giant" (the giant being Orion, and Betelgeuse being its left shoulder). He thought the name was appropriate for that horse, since the poor thing had the task of shouldering the giant (i.e., 6'3" Gamow).

Then years later when he visited that same spot, he saw that horse again, and out of curiosity asked one of the guides "what's that horse called?" And the guide replied "Battle Goose, sir." "What? Why Battle Goose?" "I don't know sir, somebody named it that way, and the name just stuck."

Then in one of the math problems in his book, where one had to find the location of a treasure, there was a gallows as a reference point. He used the Greek letter "gamma" as a symbol for the location of that gallows, since the capital Gamma in Greek "even looks like a real gallows." I thought that was pretty creative, his name also being "Gamow."

This seems to be going OT, so I'll stop. It should serve to get folks interested in his books though.

EDIT: Oh, forgot to ask. Is Gamow also the guy who once tried to submit a paper with the title "How to Cook a Helium Nucleus in a Potential Pot" - only to have the editor reject the title? So he was bemoaning the lack of sense of humor in that editor.

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Re: BR Maths Corner-1

Postby Vayutuvan » 06 Jan 2021 06:42

Amber G. wrote:Vayutuvan: I did not know about "happy ending problem" so read about it. Thanks.
--
There is a very nice article about Dyson, (along with John Conway, Ronald Graham) i


John Conway, one of my favorite mathematicians, succumbed to COVID sometime around May-June 2020. I forgot to post it here. He was in his mid-80s.

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Re: BR Maths Corner-1

Postby Vayutuvan » 06 Jan 2021 06:45

sudarshan wrote:Gamow seems to have been quite the clown.

(Paging @SriKumar, @chillarai, and @Mort Walker as well)

Now you are here, attempt the two problems I posed.

N faced polygon problem and the other is The Happy Ending Problem.

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Re: BR Maths Corner-1

Postby sudarshan » 06 Jan 2021 09:11

Vayutuvan wrote:Amber G., you might know about the "happy ending problem" which was described in the book on Erdos. Let me pose it here.

Given five points in general position in a plane, i.e. no three points form a straight line, prove that there is always a convex quadrilateral.

The problem was given to Erdos and Szekeris by their friend Esther Klien. Both the men scratched their heads until Esther gave the solution.

It was the start of Ramsey Theory program by Erdos and Szekeris.

Erdos gave the problem that name because Szekeris and Esther Klein started a romantic relationship which ended in their marriage.


I actually found this one rather easy, don't know if I made some major reasoning error (it's possible). I didn't look it up though, the below are my own thoughts on it.

Ee-e-nh, let's see now (as Bugs Bunny once said)....

Three non-collinear points in a plane will form a triangle, which is always considered convex. When you add a fourth point, which is not collinear with any any of the pairs of previous three points, the quadrilateral formed will be convex if each of the points are outside the triangle formed by the other three. Conversely, if the quadrilateral is concave, then one of the four points will be in the triangle formed by the other three (but not *on* the triangle, since the "no three collinear points" rule precludes that).

So of the five points, select any four. If these four form a convex quad, then the problem is over right there.

So we are concerned with the case where these four don't form a convex quad. Which means, one of them is inside the triangle formed by the other three.

So pick any triangle, and place a point inside it. It is to be shown that if we select a fifth point which is not collinear with any pair out of the previous four, then this fifth point will be such that, at least four out of the five will form a convex quad.

See the figure below. Three of the red dots form a triangle, and the fourth red dot is inside this triangle. The fifth point cannot lie on any of the green or black lines. So the fifth point has to lie within one of the pink, grey, or blue regions.

Image
Last edited by sudarshan on 06 Jan 2021 09:20, edited 1 time in total.

sudarshan
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Re: BR Maths Corner-1

Postby sudarshan » 06 Jan 2021 09:14

The important thing to note is, that any of the pink regions are equivalent with any of the other; any of the grey regions are equivalent with any of the other; and any of the blue regions are equivalent with any of the other.

I.e., by flipping or rotating the triangle, whatever is said about the case where the fifth point is in one of the pink/ grey/ blue regions, can be shown to apply to the remaining ones as well.

So simplify the above figure:

Image

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Re: BR Maths Corner-1

Postby sudarshan » 06 Jan 2021 09:17

Call this fifth point "5." In the above figure, put the fifth point in the pink region. 2, 3, 4, and 5 are guaranteed to form a convex quad (i.e., pick any three of these four points, it will instantly be seen that the fourth is outside the triangle formed by the other three).

Put the fifth point in the grey region, again, 2, 3, 4, and 5 are guaranteed to form a convex quad. (Again - pick any three of these four points, the fourth is instantly seen to be outside the triangle formed by the other three).

Put the fifth point in the blue region. Now, 1, 3, 4, and 5 are guaranteed to form a convex quad (same reasoning as above).

Hence proved. Did I miss anything?

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Re: BR Maths Corner-1

Postby Vayutuvan » 07 Jan 2021 10:57

If the fifth point is inside the triangle? I will check later unless you show that you already handled that case.

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Re: BR Maths Corner-1

Postby sudarshan » 07 Jan 2021 19:46

Vayutuvan wrote:If the fifth point is inside the triangle? I will check later unless you show that you already handled that case.


It's handled already, by this case:

sudarshan wrote:Put the fifth point in the blue region. Now, 1, 3, 4, and 5 are guaranteed to form a convex quad (same reasoning as above).


The blue region is inside the triangle.

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Re: BR Maths Corner-1

Postby Amber G. » 14 Jan 2021 05:05

Come listen to acclaimed author Robert Kanigel and renowned mathematician Ken Ono as they discuss The man who knew infinity and representing STEM in literature and cinema! Register now at: https://www.indiasciencefest.org/talks
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Amber G.
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Re: BR Maths Corner-1

Postby Amber G. » 14 Jan 2021 05:37

Sudarshan and others:
Adding some back ground to "Happy Ending Problem" :

Esther Klein brought this "puzzle" to George Szekeres and Paul Erdős. All were friends/mathematicians in their 20's (in mid 1930's)

"Given five points, and assuming no three fall exactly on a line, prove that it is always possible to form a convex quadrilateral — a four-sided shape that’s never indented (meaning that, as you travel around it, you make either all left turns or all right turns)."

This particular problem was not that hard. Esther Klein already knew the solution before she brought it to her friends and her friends were able to prove it for this particular case, but it started all thinking about general case. They were able to prove, for example it needed nine points to guarantee a convex pentagon.

Erdős and Szekeres proposed an exact formula for the number of points it would take to guarantee a convex polygon of any number of sides for n side : (2^(n–2) + 1). But this was a just a conjecture by Erdős - and as he did with many problems, he offered a $500 to anyone who could prove the formula was correct.

The problem is called "Happy Ending" because Esther Klein and George Szekeres fell in love, got married etc.
Erdős passed away in 1996 and Esther & George Szekeres in 2005 - Both passed away on the same day within hours of each other. Both were 90+ years old and 70 years have passed - The prize (now managed by others) is still there to be claimed.

From what I know, (have not read the latest) - The only other shape whose result is known is a hexagon, which requires at least 17 points. Some recently published papers by Suk and may be others provides nearly decisive evidence that the intuition that guided Erdős and Szekeres is correct -- it has given many interesting tools.

Here is a old picture of the three. (Around 1930) (Photo credit : Ronald Graham (Erdős)/Komal)
Image

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Re: BR Maths Corner-1

Postby Najunamar » 14 Jan 2021 20:43

Amberji, your statement does not include the coplanar clause which was in Vayutuvan's problem statement. It will be much more difficult to consider non coplanar cases no?

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Re: BR Maths Corner-1

Postby Amber G. » 14 Jan 2021 22:03

Najunamar wrote:Amberji, your statement does not include the coplanar clause which was in Vayutuvan's problem statement. It will be much more difficult to consider non coplanar cases no?

Nice to see you here, haven't seen you in this dhaga..:)

You raise an interesting point. But non-coplanar cases are actually "easier" for 5 points. (or more complex if you want to find general formula)

For example if one generalize this to say 3 dimension, where no four points are on the same plane then 4 points will always form a convex polygon. (Just like in a plane, 3 points will always form a convex polygon). Interestingly here (in 3 dimension) 6 points (where no 4 points are in the same plane) are enough to find 5 points so that when they are cyclic joined will be convex.

In "general" case of the above problem, in k dimension, if one chooses k+3 points, k+2 points can be joined. Erdos conjecture, a modified formula exist. Generally, for every d and k > d there exists a number m(d,k) such that every set of m(d,k) points in general position has a subset of k points that form the vertices of a "neighborly" polytope ..ityadi .. (I do not know a lot - in fact quite little, in graph theory but these problems - even simple looking problems become very hard)
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There are some good books ( eg Grünbaum's - Convex Polytopes) and plenty of material but not much in popular books I have seen.

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Re: BR Maths Corner-1

Postby Najunamar » 15 Jan 2021 00:42

Amberji thanks!, Yes, I was thinking about > 5 points or a generic octahedron (not regular)- in general for just 3D space but as you mentioned could also think of n-dimensional space too. Much obliged for the book suggestion - shall search in the public library system.

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Re: BR Maths Corner-1

Postby Amber G. » 16 Jan 2021 05:01

Meanwhile on Biden team - as a scientific adviser Eric Steven Lander - MIT professor, mathematician (won medal in IMO). Congratulations.
(Interesting to see, now both in US and India good scientists are getting listened too),

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Re: BR Maths Corner-1

Postby Vayutuvan » 16 Jan 2021 08:26

The book by Grunbuam is the authoritative book on Convex Polytopes.

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Re: BR Maths Corner-1

Postby Amber G. » 22 Jan 2021 12:33

Congratulations to Adam W. Marcus of EPFL, Daniel Alan Spielman of Yale University, and Nikhil Srivastava of University of California, Berkeley, winners of the 2021 National Academy of Sciences’ Michael and Sheila Held Prize for their work on the Kadison-Singer problem and Ramanujan graphs! (It's a $100,000 prize)


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