BR Maths Corner1
Re: BR Maths Corner1
^^^ 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 coauthored a paper with Erdős, 2 if he has coauthored a paper with someone who has coauthored a paper with Erdős).
(Here is a very famous picture of Erdos with another great mathematician  then 10 year old  Terrence Tao)
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 coauthored a paper with Erdős, 2 if he has coauthored a paper with someone who has coauthored a paper with Erdős).
(Here is a very famous picture of Erdos with another great mathematician  then 10 year old  Terrence Tao)
Re: BR Maths Corner1
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.
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.
Re: BR Maths Corner1
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.

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 )
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.

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 )
Re: BR Maths Corner1
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.

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 Corner1
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.
I agree Dyson will be missed. Many of us have been missing Feynman for a long time too. He left too early.
Re: BR Maths Corner1
^^^ 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.
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.
Re: BR Maths Corner1
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.
Hint: Greek alphabet, first four letters.
Re: BR Maths Corner1
^^^ 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 "alphabetagamma' 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..)..
 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 "alphabetagamma' 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..)..
Re: BR Maths Corner1
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 horseriding. 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 bigboned 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.
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.
Re: BR Maths Corner1
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 MayJune 2020. I forgot to post it here. He was in his mid80s.
Re: BR Maths Corner1
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.
Re: BR Maths Corner1
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.
Eeenh, let's see now (as Bugs Bunny once said)....
Three noncollinear 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.
Last edited by sudarshan on 06 Jan 2021 09:20, edited 1 time in total.
Re: BR Maths Corner1
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:
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:
Re: BR Maths Corner1
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?
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?
Re: BR Maths Corner1
If the fifth point is inside the triangle? I will check later unless you show that you already handled that case.
Re: BR Maths Corner1
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.
Re: BR Maths Corner1
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
Re: BR Maths Corner1
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 foursided 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)
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 foursided 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)
Re: BR Maths Corner1
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?
Re: BR Maths Corner1
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 noncoplanar 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)

There are some good books ( eg Grünbaum's  Convex Polytopes) and plenty of material but not much in popular books I have seen.
Re: BR Maths Corner1
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 ndimensional space too. Much obliged for the book suggestion  shall search in the public library system.
Re: BR Maths Corner1
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),
(Interesting to see, now both in US and India good scientists are getting listened too),
Re: BR Maths Corner1
The book by Grunbuam is the authoritative book on Convex Polytopes.
Re: BR Maths Corner1
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 KadisonSinger problem and Ramanujan graphs! (It's a $100,000 prize)
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