June 2019 - mathfun@mailman.xmission.com

Re: [math-fun] Draft of my June 17, 2019 essay
by Keith F. Lynch 15 Jun '19

15 Jun '19

James Propp <jamespropp(a)gmail.com> wrote: > I seem to recall that for the last half of W. S. Gilbert?s life, he had to > avoid using the word ?never? in public because of all the wags who would > jump in with ?What, never?? > Was ?What, never?? the first viral catch-phrase? Hardly ever. For instance "What a shocking bad hat!" was popular decades earlier.

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[math-fun] Pronunciation question
by James Propp 14 Jun '19

14 Jun '19

According to the IPA rendering that appears at the beginning of the webpage for the mathematician DeBruijn ( https://en.m.wikipedia.org/wiki/Nicolaas_Govert_de_Bruijn) and the page on Dutch pronunciation that it links to, the final vowel in his surname should be pronounced like the “ou” in the Scottish English pronunciation of the word “house”. But I don’t speak Scottish English (though I occasionally speak a parodic version based on Monty Python skits and other less-than-definitive phonological resources). Is there a wensite that will let me type “Scottish English” into one field and “house” into another so I can hear what the vowel sounds like? I’ll be saying “DeBruijn” in a video and I want to get it as close to correct as I can. ... Though I’ve been advised that when it comes to pronouncing Dutch vowels it’s impossible for Americans to get it right, so the important thing is to say it in an American accent lest you mislead people into thinking your pronunciation is correct! Thanks, Jim Propp

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Re: [math-fun] Draft of my June 17, 2019 essay
by Keith F. Lynch 14 Jun '19

14 Jun '19

Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > Was your dad a miner? No, but he was a minor (decades before I was born). And, believe it or not, he never once visited the sun, with or without pick and shovel. Nor Titan either. Have you read Mr. Propp's draft? My response to his draft was in the same spirit as the draft.

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Re: [math-fun] Draft of my June 17, 2019 essay
by Keith F. Lynch 13 Jun '19

13 Jun '19

James Propp <jamespropp(a)gmail.com> wrote: > http://mathenchant.org/050-draft2.pdf One advantage of moving the moon closer to preserve total eclipses, rather than building a wall on the moon for the purpose as you propose, is that it would also get rid of the need for leap seconds, as it would restore the rotation rate our planet had when the length of the cesium ("atomic") second was established, i.e. it would Make Seconds Great Again. Another advantage is that it could be done without the hassle of traveling to the moon. All we'd have to do is build lots of tidal power stations here on Earth and run them in reverse, putting energy into them to undo decades of tidal drag. Keep in mind that traveling to the moon would be harder today than it was during the Apollo project, since the moon is further away now (by about six feet). I don't know if there is any fuel on the moon, but there is on Mars. That meteorite from Mars that was found in Antarctica may or may not have contained fossils, but it definitely contained polycyclic aromatic hydrocarbons, which are better known as petroleum. No, I'm not proposing a trip to Mars. I'm proposing that we fuel our society by mining oil-bearing Mars meteorites from Antarctica. When we run out of those, we can siphon methane ("natural gas") from the lakes on Saturn's moon, Titan. The total length of hollow tubing manufactured since the beginning of recorded history would probably reach Titan if it was all put end to end. If not, we could move Titan by building lots of tidal power stations on Saturn. Of course burning oil or natural gas consumes oxygen. What happens when Earth starts to run out? It's been claimed that nowhere else in the solar system is there any free oxygen. That's not true. There's one place where there's far more free oxygen than on our planet: The sun. There's plenty of oxygen there, and it's nearly all free, i.e. almost none of it is chemically combined with any other element. The sun also conveniently located, just one AU away, ten times closer than Titan. So we should mine the sun for oxygen.

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[math-fun] Symplectification of Polya Walk Integrals
by Brad Klee 12 Jun '19

12 Jun '19

Let generating function F_d(z) = Sum a_{d,n} z^n generate the counts a_{d,n} of 2*n-step, origin-returning walks on d-dimensional simple-cubic lattice. It is not difficult to prove the following: F_d(z) is the volume integral of a 2*(d-1)-dimensional algebraic variety with symplectic structure, and S_{d-1} permutation symmetry. See below for a the construction in Mathematica, and a few helpful references. --Brad SymplecticGeo[nDim_] := ReplaceAll[Subtract[s^(nDim), Factor@Times[s^(2*nDim), 1 + Total[Plus[ ChebyshevT[2*nDim, X[#]/Sqrt[s]], I ChebyshevU[2*nDim - 1, X[#]/Sqrt[s]] Y[#]/Sqrt[s] ] & /@ Range[nDim]], 1 + Total[Plus[ ChebyshevT[2*nDim, X[#]/Sqrt[s]], -I ChebyshevU[2*nDim - 1, X[#]/Sqrt[s]] Y[#]/Sqrt[s] ] & /@ Range[nDim]]]], s -> (1/nDim) Plus[ Total[X[#]^2 & /@ Range[nDim]], Total[Y[#]^2 & /@ Range[nDim]]]] VolInt[nDim_, Terms_] := TrigReduce[Normal[ Series[Cancel[D[v /. Solve[z == TrigReduce[ SymplecticGeo[nDim] /. { X[n_] :> v^(1/2/nDim) Cos[t[n]], Y[n_] :> v^(1/2/nDim) Sin[t[n]]} ], v][[1]], z]], {z, 0, Terms}]]] /.Cos[_]->0 Column[VolInt[#, 5] & /@ Range[5]] Out[]:= 1 + 4 z + 36 z^2 + 400 z^3 + 4900 z^4 + 63504 z^5 1 + 6 z + 90 z^2 + 1860 z^3 + 44730 z^4 + 1172556 z^5 1 + 8 z + 168 z^2 + 5120 z^3 + 190120 z^4 + 7939008 z^5 1 + 10 z + 270 z^2 + 10900 z^3 + 551950 z^4 + 32232060 z^5 1 + 12 z + 396 z^2 + 19920 z^3 + 1281420 z^4 + 96807312 z^5 Expand[SymplecticGeo[1]] Out[]:= X[1]^2 + Y[1]^2 - 4*X[1]^4 - 4*X[1]^2*Y[1]^2 http://mathworld.wolfram.com/PolyasRandomWalkConstants.html http://demonstrations.wolfram.com/EdwardssSolutionOfPendulumOscillation/ https://oeis.org/A287318 https://oeis.org/A307618

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Re: [math-fun] Draft of my June 17, 2019 essay
by Dan Asimov 11 Jun '19

11 Jun '19

The first name of the author of "How to Lie With Statistics" is Darrell. —Dan

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[math-fun] Draft of my June 17, 2019 essay
by James Propp 11 Jun '19

11 Jun '19

As usual, I try to compensate for my limitations as a thinker and writer by getting input from you guys (and others) in advance. Suggestions of all kinds are welcome. Those of you who find my tendency to slip into glibness irritating might find this month's essay more irritating than usual, but I'm trying to find a style that is readable and funny without being annoying, and if I've failed I want to know about it! The current draft is at http://mathenchant.org/050-draft2.pdf Thanks, Jim Propp

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[math-fun] Polynomial bijections?
by Dan Asimov 11 Jun '19

11 Jun '19

Let R denote the ring Q or Z. (Not the reals!) Then R^n denotes the cartesian product R x ... x R (n times). Suppose f : R^n —> R^n is a polynomial mapping of the form f(x_1, ..., x_n) = (P_1(x_1,...,x_n), ..., P_n(x_1,...,x_n)) where each P_j is a polynomial in the x_1, ..., x_n over the ring R. Question: --------- (*) If f is a bijection, does that imply that each P_j is linear ? --------- The polynomial bijections of R^n —> R^n (R = Q or Z) form a group. Just in case there are any non-linear ones: What is this group? E.g. for Z^2 or Q^2. It is known that if the ring is the complex field C, there do exist non-linear polynomial bijections C^2 —> C^2 of form f(z,w) = (P(z,w), Q(z,w)). Of course the same question (*) can be asked about any ring at all. —Dan

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[math-fun] Paley-13 is Graceful
by Ed Pegg Jr 11 Jun '19

11 Jun '19

https://math.stackexchange.com/questions/3250051/is-paley-13-a-graceful-gra… has a picture of a graceful embedding. --Ed Pegg Jr

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[math-fun] 1, 2, 3.
by Bill Gosper 09 Jun '19

09 Jun '19

Is apparently the complete sequence of solutions to PrimeQ[1+Hyperfactorial@x]. (Checked through a couple of hundred.) (Unconvincing) probabilistic arguments "predict" finiteness of prime subsequences of (sufficiently) rapidly growing sequences, but {1, 2, 3} seems a little too pat. Maybe RCS can conjure one of his magic case arguments. The complete list of prime 1+BarnesG seems to be 1, 2, 3, 4, 5, 9, 16. (Checked through 278.) —rwg

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