Researchers chip away at Smale's 7th unsolved problem in mathematics
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Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#2but Wikipedia shows a huge table of values.it's just that we don't have a name or certain constructions for values above 12
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#3I actually was asked this problem in a programming interview. This is first I've read that it is an unsolved problem, which definitely makes me chuckle.
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#4but Wikipedia shows a huge table of values.it's just that we don't have a name or certain constructions for values above 12
Those are the best known configurations, we don't have proofs that they are the lowest possible.
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#5Needless to say, there has been a lot of work on this subject from a number of different directions, and any particular new paper is unlikely a priori to be a revolutionary breakthrough. Look at Google Scholar or a similar citation graph and you can find thousands of papers about it. (Some key words: “spherical cap discrepancy”, “Fekete points”, “Riesz energy”, “logarithmic potential”, obviously also “Thomson’s problem”, etc.)
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#6but Wikipedia shows a huge table of values.it's just that we don't have a name or certain constructions for values above 12
Those are the best known configurations, we don't have proofs that they are the lowest possible.
Isn't this an optimization problem that can be solved with computer to solve a high degree equation?
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#7Earlier quoted context omitted.
Those are the best known configurations, we don't have proofs that they are the lowest possible.
Isn't this an optimization problem that can be solved with computer to solve a high degree equation?
Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally intractable.
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#8I actually was asked this problem in a programming interview. This is first I've read that it is an unsolved problem, which definitely makes me chuckle.
How did you answer?
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#9Earlier quoted context omitted.
Isn't this an optimization problem that can be solved with computer to solve a high degree equation?
Not sure. For n points on a sphere, you're trying to prove a global minimum a very bumpy function over 2n-dimensional space. It may become computationally intractable.
And even if you can find a numerical solution to arbitrary precision, at best you only get arbitrarily close to the mathematically optimal solution (at best because a purely numerical approach cannot rule out that it misses a very small, but high peak in the function to be optimized)
As an extreme example, for n=2 and using binary floating point, you probably will find the mathematically correct solution, but there's no way to tell numerically whether your answer is off by a fraction of your floating point's epsilon value.
That's probably not important to physicists who want to know an answer, but it is for mathematicians.
What I find very surprising is (from https://en.m.wikipedia.org/wiki/Thomson_problem):
"Numerical solutions for N=8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square and pentagonal, respectively."
I would like to see the visualizations of the better solutions for n=20 (wikipedia links to the one for n=8) and/or hear a heuristic argument as to how that can happen.
Re: Researchers chip away at Smale's 7th unsolved problem in mathematics
#10Correction: The article states that Thomson's problem has only been solved for 2, 3, 4, 6, and 12 charges. However, in 2013 the 5-electron case was solved by Richard Schwartz. Here's the paper:
http://www.tandfonline.com/doi/abs/10.1080/10586458.2013.766...