October 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Raghavendra's algorithm for solving linear systems over finite fields
by Henry Baker 06 Jun '16

06 Jun '16

I can see applications in distributed/parallel implementations. You can do most of the checking & probably a lot of the randomization in parallel. At 09:02 AM 8/15/2012, Warren Smith wrote: >>Preliminary draft: >> http://www.eecs.berkeley.edu/~prasad/linsystems.pdf > >--I don't understand why anybody would want to use this algorithm >for any purpose. It's slower than everything else I can think of and also >randomized i.e. can fail. > >However... perhaps it might be useful if changed to be about finite >RINGS not fields. In rings, you cannot divide, hence gaussian elimination is >not usable. Hence this algorithm then has less competition. >(Even then I'm dubious, but then it'd have at least a hope.)

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[math-fun] 5 and 7 outcome fair dice
by Bill Gosper 06 May '16

06 May '16

Odd barrel dice have the problem that two faces are always uppermost, so how do you label them? Alternative to face-transitivity, there are fair dice which cannot rest on any face, and yet have one face uppermost. Spoiler: gosper.org/5&7dice.png --rwg I assume these are old ideas. Check out Wikipedia if you haven't seen spherical, 6 outcome dice.

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[math-fun] Computational effort
by James Propp 01 Mar '16

01 Mar '16

I don't know what the relevant units of computation are --- floating-point operations are the ones I've heard about most, but what if you're not using floating point? --- but I'm wondering if anyone has tried to quantify "how much computing" took place in (a) code-breaking at Bletchley Park, (b) the Manhattan Project, and (c) the Apollo space program. It seems likely to me that my laptop, running some silly number-crunching problem in Mathematica and returning an answer after six hours that I could probably have figured out in my head if I'd thought a little bit more before jumping in to write some code, has them all beat. What do you think? (And while we're on the subject: What's the canonical example of wasted computational effort? Isn't there some well-known old story of someone who had an early computer calculate the determinant of a singular matrix?) Jim Propp

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[math-fun] San Jose Math Decagon
by Bill Gosper 17 Feb '16

17 Feb '16

A couple of years ago Julian and I showed some animations to the San Jose Math Circle. gosper.org/decazoom.gif suggests why the area of a circle is the radius times half the circumference, and displays the continued fraction of a very lame approximation to π. --rwg

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[math-fun] xkcd
by rcs＠xmission.com 17 Dec '15

17 Dec '15

Today's XKCD cartoon has a nice collection of approximations. http://xkcd.com/1047/ There's the ancient expression 6 pi^5 for the mass ratio of the proton to the electron, and the more accurate (e^8 - 10)/phi. Robert Munafo & REIS are at least partially responsible. Rich

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[math-fun] Julian pimped my pump
by Bill Gosper 17 Dec '15

17 Dec '15

To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard

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[math-fun] A gallery of math art from Bridges 2015
by Alex Bellos 16 Nov '15

16 Nov '15

http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2… <http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2…>

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[math-fun] Bernoulli # convolution
by Bill Gosper 29 Oct '15

29 Oct '15

This can't be new--it's essentially the coefficient of x^n in coth(x) tanh(x): Sum[((-1 + 2^j)*BernoulliB[i - j]*BernoulliB[j])/((i - j)!*j!), {j, 2, -2 + i}]== -((-1 + 2^i)*BernoulliB[i])/i! but I don't see it in GKP's Concrete Math, nor DLMF. But that's probably because this is a poorly phrased special case of something. In particular, it has Bernoulli #s instead of Bernoulli polynomials. Anyone know better? Also, everybody knows the row sums of Pascal' triangle are 2^n, and the alternating row sums are 0^n, and the row sums of the Eulerian triangle are n!, but did you know the alternating row sums, 1, 1, 0, -2, 0, 16, 0, -272, 0, 7936,... A155585, are 2^n*EulerE[n, 1] = 2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1, 1]/(n + 1) ? (http://dlmf.nist.gov/26.14#E11) --rwg

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[math-fun] Octocog
by Fred Lunnon 29 Oct '15

29 Oct '15

This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible. Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges. Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun? Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry? Fred Lunnon

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[math-fun] Folding a band saw blade
by James Propp 27 Oct '15

27 Oct '15

I took a look at https://www.youtube.com/watch?v=f-7x8X6I3RA because I needed to fold a kid's camping tent --- a task that poses similar differential-geometric challenges (but without the sharp teeth). The solution Paul Sellers advocates uses a table or other external object. Is there an easy-to-describe hands-only two-person solution? Also: Is there a write-up anywhere of the mathematics behind folding a band? I'm guessing that it's the kind of situation where you can get a (2n+1)-to-1 fold for any n. But I don't even see how the 3-to-1 fold works, nor do I see how one can rigorously prove that a 2-to-1 fold is impossible. Come to think of it, I don't even know how to state the problem as a mathematical one. Jim Propp

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