February 2016 - mathfun@mailman.xmission.com

[math-fun] A problem about finite sets
by Victor Miller 09 Sep '16

09 Sep '16

Say that a finite set of integers, A, has *property P* is there are two non-empty disjoint subsets S and T of A, such that sum(S) = sum(T) (where sum(X) is the sum of the elements of X). 1) Show that every 10 element subset of {1,...,100} has property P (this is fairly easy). 2) Is it true that every 9 element subset of {1,...,100} has property P? Victor

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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] math puzzles for very young enthusiasts
by Fred Kochman 08 Mar '16

08 Mar '16

One of my daughters writes, One of my co-workers recently bought Smullyan's Lady or the Tiger book for his daughters, but he said that it seems to be too difficult so far. (To be fair, his oldest daughter is only 8 years old.) His kids seem to love mind-bending puzzles though - do you know of any logic books that would be better for really young kids? Any suggestions? Probably "logic" could be broadly interpreted. Thanks, Fred Kochman

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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] Droste fail
by Bill Gosper 29 Feb '16

29 Feb '16

https://en.wikipedia.org/wiki/2011_Tōhoku_earthquake_and_tsunami#/media/Fil… --rwg

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Re: [math-fun] black hole merger accompanied by gamma ray burst ??!!
by Warren D Smith 28 Feb '16

28 Feb '16

> APG: > I went to a talk at the CMS by Bruce Allen, director of the Max Planck > Institute: > > http://www.talks.cam.ac.uk/talk/index/64415 > > He attributed the most probable explanation to the gravitational waves > exciting a nearby dust cloud, causing it to resonate gamma rays. > > This is consistent with it being several orders of magnitude weaker > than the gravitational source. --so the dust cloud would be about 0.4 light seconds away from the holes to explain the 0.4 second delay gravitons-->photons. As opposed to us at around 1 billion light years distant seeing distance distortions of one part in 10^21. Cloud would be "excited" by the gravity wave altering distances. The distance-altering factors would be of order 10^(-21) * (1 billion years)/(0.4 seconds) = 10^(-4). If you squeeze an atom by 1 part in 10000 it is not going to produce energies anywhere near gamma rays. More like infrared. If you squeezed an atomic nucleus by a factor of 2, then I might believe that'd yield gamma rays. But not 1 part in 10000? If you squeezed a much nearer atomic nucleus, only, say, a light-millisecond away from the holes (100 miles), then you might well be able to get gamma rays. But not with a 0.4 sec delay; they ought to reradiate instantly. So the question is: can squeezing atomic nuclei by 1 part in 10000 cause gamma rays? I doubt it. Nuclei are in an energy min. Small distortions from that min should increase energies by amount QUADRATIC in the distortion. Factor 2=1+1 distortion should yield energies comparable to the nuclear binding energies of order 10 MeV/nucleon or comparable to the proton self-energy of order 1 GeV. So factor 1+1/10000 distortion should provide energies of at most 1 GeV / 10000^2 which is 0.1 eV, which is diddlysquat compared to gamma rays. Or if you viewed it as some probability of converting nucleus to excited state then the probability ought to be of order 1/10000^2, which seems not enough to yield the observed photon luminosity? I mean, only a very tiny part of space is occupied by nuclei, and so only a very tiny fraction of gravity wave could be converted to photon energies in this way, and so, it seems really hard to believe in this whole hypothesis.

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[math-fun] Propp's one-dimensional packing question
by Warren D Smith 28 Feb '16

28 Feb '16

J.Propp: One-dimensional packing of monomers with forbidden distances Given a finite set S of positive integers, and a positive integer N, let F(N,S) be the largest possible cardinality of a subset of {1,2,…,N} no two of whose elements differ by a number in S. What is known about the computational complexity of F? WDS: You can solve problems of this kind by "dynamic programming." Let T be the maximum member of S. The runtime and will be O(n)*C^T and the space usage O(C^T) for some constant C>1. For each n=1,2,....,N: tabulate the largest possible cardinality of a subset of {1,2,3,...,n} meeting both the forbidden distance conditions and the "boundary condition" specified by a given subset of {n+1, n+2, ..., n+T}. Note, each loop iteration's table is computed from previous table. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] black hole merger accompanied by gamma ray burst ??!!
by Warren D Smith 28 Feb '16

28 Feb '16

Fermi satellite reports that starting 0.4 seconds after the 0.1-second-long black hole merger detected by LIGO, a burst of 10-10000 KeV gamma rays lasting 1 second happened. It is thought to have come from very roughly the same part of the sky. If it came from the same source, then its energy was about 5 earth masses converted into gamma photon energy. The black hole merger released about 3 solar masses of graviton energy. But another satellite failed to detect the gamma burst. http://gammaray.nsstc.nasa.gov/gbm/publications/preprints/gbm_ligo_preprint… If the two holes had been orbiting for aeons they should be pure vacuum and incapable of producing any high energy (or even moderate energy) photons. If there was (say) a jupiter orbiting one of the holes, one would not expect it to get swallowed in 1 second, it would have happened at least about 1 day later, I'd think. Abraham Loeb proposed both holes were created inside the same huge star (evidently over 61 solar masses) then spiralled into each other soon after creation. Other matter in the star got heated and glowed in gamma rays for 1 sec before shutoff because everything got swallowed. His proposal is here: http://arxiv.org/abs/1602.04735 This seems dubious to me. Wouldn't the outside of the star block the gamma rays for longer than 1 second? Wouldn't it not even know until a speed-of-sound delay later? Why would two holes form inside a star in a huge departure from spherical symmetry? That's one hell of a big star, do stars of such high masses even exist that are ready to collapse? (And it would need to be rotating very rapidly, even stranger.) Certainly if they do exist, they are very rare. But, it is hard to think of any competing explanation. Maybe the answer is this all was just a huge coincidence and the two signals are unrelated. The INTEGRAL satellite team has published saying their satellite says there was no gamma burst, at keast in their 75-2000 KeV range, suggesting it all was bullshit: http://arxiv.org/abs/1602.04180 -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Packing complexity question
by James Propp 27 Feb '16

27 Feb '16

Responses to my MO post http://mathoverflow.net/questions/232165/one-dimensional-packing-of-monomer… haven't taught me anything. Do any of you know (or can any of you figure out) how difficult packing problems of this kind are, from a complexity standpoint? If S is held fixed, then F(n,S) is polynomial time, but that's not very interesting; I want to know what happens as S varies. Jim Propp

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