September 2015 - mathfun@mailman.xmission.com

[math-fun] open tree of life
by Warren D Smith 20 Sep '15

20 Sep '15

https://tree.opentreeoflife.org Great is the ballyhooing in the media, but it does not explain (that I noticed) how it works, how to use it. For example: what is the path from "humans" to "maize"? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] FunctionExpand knows no special values of DedekindEta.
by Bill Gosper 20 Sep '15

20 Sep '15

(The main simplifier rather annoyingly special-cases tau = i.) DedekindEta[quadraticsurd] comes out as algebraic*productGamma(rational)). E.g., DedekindEta[ I/10 - 1/5] == ((-1)^(1/12) Sqrt[2 + I] Gamma[1/4])/(2^(7/8) \[Pi]^(3/4)) DedekindEta[1/2 + I/Sqrt[6]] == ((-1)^(1/24) (-1 + Sqrt[2])^( 1/12) (-Sqrt[2] + Sqrt[3])^(3/8) Gamma[1/24])/( 2 2^(5/24) 3^(1/8) (-1 + Sqrt[3])^(1/4) Sqrt[\[Pi] Gamma[1/12]]) (There is strong evidence against any such formulas for, e.g., DedekindEta[I*(1/2 + 1/Sqrt[6])]].) The Gammas are predicted by known formulas. The algebraic is mostly predicted by ModularLambda, as I posted a few weeks ago. RootApproximant should take care of what's left. Thus it might be feasible to FunctionExpand DedekindEta on the fly. At present, I just look them up in a database that I expand as needed. Nonrigorous answers from RootApproximant may raise some eyebrows, although it's more reliable than the True and False returned by Equal. Perhaps the results can be double checked to $MaxExtraPrecision. I don't recall seeing a false positive survive a retry at higher precision. The radicals above were hand-simplified. Probably an on-the-fly eta simplifier would mostly give Root objects. Best might be a combination of hand-simplified table plus on-the-fly for the untabulated. Radical simplification is where computer algebra rejoins AI. At least until the creation of a miraculous algorithm. FunctionExpand knows two(?) special values of EllipticK, 1/2 and -1, with 0 and 1 automatic in the simplifier. This misses scads of special values at algebraics, e.g EllipticK[(-1 + Sqrt[2])^2] == (Gamma[1/8] Gamma[3/8])/( 8 2^(1/4) Sqrt[(-1 + Sqrt[2]) \[Pi]]) EllipticK[(1 + Sqrt[2])^2] == ((1 - I Sqrt[2]) Sqrt[(-1 + Sqrt[ 2])/\[Pi]] Gamma[1/8] Gamma[3/8])/(8 2^(1/4)) EllipticK[1/2 + Sqrt[-2 + Sqrt[5]]] == ((1 + Sqrt[5]) Gamma[1/20] Gamma[9/ 20])/(8 5^(1/8) (2 + Sqrt[5])^(1/4) Sqrt[2 \[Pi]]) EllipticK[1/2 - Sqrt[-2 + Sqrt[5]]] == ((1 + Sqrt[5]) Gamma[1/20] Gamma[9/ 20])/(8 5^(5/8) (2 + Sqrt[5])^(1/4) Sqrt[2 \[Pi]]) EllipticK[1/4 (2 - Sqrt[3])] == (3^(1/4) Gamma[1/3]^3)/(4 2^(1/3) \[Pi]) EllipticK[1/8 (2 - Sqrt[2] 3^(1/4))^2 (-1 + Sqrt[3])^2] == Gamma[1/4]^2/(2 3^(3/4) (-1 + Sqrt[3]) Sqrt[2 \[Pi]]) . . . Given a good DedekindEta capability, it might be possible to do these on the fly via EllipticK[(16 DedekindEta[\[Tau]]^8 DedekindEta[4 \[Tau]]^16)/ DedekindEta[2 \[Tau]]^24] == (\[Pi] DedekindEta[2 \[Tau]]^10)/( 2 DedekindEta[\[Tau]]^4 DedekindEta[4 \[Tau]]^4) Failing that, you can use the above plus a big DedekindEta database to create a big EllipticK database... to be also searched by ArithmeticGeometricMean, because ArithmeticGeometricMean[1, Sqrt[z]] == \[Pi]/(2 EllipticK[1 - z]) --Bill Gosper

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[math-fun] Happy Huntrodd's Day
by Brent Meeker 19 Sep '15

19 Sep '15

http://www.statslife.org.uk/history-of-stats-science/1783-huntrodds-day-cel… Brent

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[math-fun] Wanted: self-contained JavaScript for chip-firing
by James Propp 19 Sep '15

19 Sep '15

My next "Mathematical Enchantments" essay will be about chip-firing, aka the sandpile model on directed graphs. Or rather, it will be about the Polya urn model, which it will treat both randomly (using random walk on a small acyclic directed multigraph) and quasirandomly (using chip-firing on that same graph). I would really like the article to have, as a side-bar, a little JavaScript applet that will let readers play with chip-firing, to get a concrete feel for it. Would any of you (or your students) be able to create something like that in the next two or three weeks? It doesn't need to be versatile (in fact, versatility will be a minus if a plethora of buttons leads to a cluttered screen); it needs to be eye-catching and fun. Jim Propp

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[math-fun] The Erdos discrepancy problem
by Fred Lunnon 18 Sep '15

18 Sep '15

Terry Tao announces a solution to Erdös' discrepancy problem: given any infinite sequence of unit vectors in m-space, there is some d such that the subsequence comprising every d-th vector has partial sums of unbounded magnitude. See http://arxiv.org/abs/1509.05363 WFL

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[math-fun] Corey's denester found a messy heptanomial:
by Bill Gosper 18 Sep '15

18 Sep '15

DedekindEta[(3 I)/4] == ((1 + Sqrt[2] - Sqrt[2] 3^(3/4) + Sqrt[3] (-1 + Sqrt[2]) + 2 3^(1/4) (1 + Sqrt[2]))^(1/4) Gamma[1/4])/( 2 2^(1/48) 3^(3/8) (-1 + Sqrt[3])^(1/12) \[Pi]^(3/4)) PSLQ its log, along with every imaginable binomial "factor": DedekindEta[(3 I)/4] == ( Sqrt[1 + 3^(1/4)] (Sqrt[2] + 3^(1/4))^(3/8) (1 + Sqrt[3])^(5/24)Gamma[1/4])/( 2 2^(1/24) 3^(3/8) ((1 + Sqrt[2]) (Sqrt[2] + Sqrt[3]))^(1/4) \[Pi]^(3/4)) Still, the rhs looks disproportionately weird. --rwg

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[math-fun] Doomsday argument
by Warren D Smith 18 Sep '15

18 Sep '15

The doomsday argument, as Andy Latto hinted, seems equally valid for humans, and "intelligent life originating on earth", and "intelligent life restricted to earth" and "all descendants of a human" and... So?

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[math-fun] something linguistic
by Wouter Meeussen 18 Sep '15

18 Sep '15

I’m looking for couples of english words starting with ‘w’ that are pronounced in the same way, even written the same, but: the first is like “wh*” and the other “w*”. Bad examples: “were” and “where” do not qualify because the first “e” is pronounced differently “witch” and “which” do not qualify because of the extraneous “t” Motivation: in norther parts of England, the “wh” seems to be pronounced as “chw”, in a way similar to swedish “sj” or “sk”. That would mean that such pronounciation is the original one, and current BBC-english pronounciation is a ‘degradation’. Any perfect examples? I know this is not math, but some on this mailing list like word games and language fun too. (hé Bill?) Wouter.

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[math-fun] Correcting Wikipedia
by James Propp 18 Sep '15

18 Sep '15

This is slightly off-topic, but I know some of you have experience correcting Wikipedia (or trying to!), so this forum seems like the best place to ask: How do I fix mistakes at https://en.wikipedia.org/wiki/Jim_Propp? Also, I'm puzzled why entering "jim propp" into Safari on my iPhone brings up a page that says I was born in Yorkton, Canada. This (incorrect) information isn't on my home page or in the Jim Propp Wiki, so where is Safari getting it from? My father (who sometimes googles me from his laptop) brought the problem to my attention. Thanks, Jim Propp

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Re: [math-fun] Doomsday argument
by Warren D Smith 18 Sep '15

18 Sep '15

I was considering it for "intelligent life whose ultimate origin is on Earth" (ILWUOIOE) not "humans" (admittedly both are somewhat undefined) . If the births of intelligent lifeforms are ordered chronologically, then with probability P you are a member of the middle P fraction of all of them. That is the so-called "doomsday argument." So for example, with 95% chance, the total number of ILWUOIOE's there will ever be, is <20 times the total number so far. I simply am pointing out, this argument does NOT imply the last one will happen before 20T after now, where T is the timespan ILWUOIOEs have existed so far (which is perhaps T=30Kyears) -- because future ILWUOIOEs might have much longer lifespans. That incorrect argument is sometimes stated as though it were the "doomsday argument." In any case the doomsday argument is a comparatively unpowerful statistical argument, in the sense the confidence approaches 100% in a rather slow manner, not in an exponential manner like you usually encounter in the sort of statistics that makes me feel a whole lot better. So caveat emptor. I hope that if/when ILWUOIOEs do not all bite the dust, you will demand your money back. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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