December 2011 - mathfun@mailman.xmission.com

[math-fun] hyperelliptic curves and AGM?
by Warren Smith 30 Dec '11

30 Dec '11

>Victor Miller: This paper ( http://www.emis.de/journals/HOA/IJMMS/Volume12_2/245.pdf ) looks like something good to read. --WDS: Thanks, that was of some interest. He does two things. First he describes how any finite group G can be turned into a natural iteration on |G|-dimensional vectors that is symmetric under G. In the case where the vector elements are constant on a subgroup, and on its complement, this is really only a 2-dimensional iteration. He attempts to argue the limit of this 2D iteration has a nice log asymptote only in the Gauss AGM case (and another derived-from-Gauss case) and in all other cases has a nastier oscillatory*log asymptote. Second he unearths century-old work by Meissel on a certain 3D iteration. I point out: Meisel's iteration he discusses, can really be viewed as the k=2 special case (and Gauss's AGM is the k=1 case) of the following infinite family of mean-iterations. Let A=(A0,A1,A2,...,Ak) be a (k+1)-dimensional vector, which we shall iterate. Consider the polynomial(x) = (A0+x) * (A1+x) * (A2+x) *... *(Ak+x). which is monic of degree k+1. Let its coefficient of x^j be called Bj for j=0,1,...,k. The iteration is to replace Aj for all j simultaneously by Aj[new] = ( Bj / binomial(k+1, j) )^(1/(k+1-j)) Thus A0[new] = geometric mean, Ak[new] = arithmetic mean and Aj[new] for 0<j<k are other kinds of means. This seems a fairly natural iteration when you look at it in this way. Meissel and this paper argue rather incompletely convincingly that when k=2 the limit function has a certain irrational-power-of-logarithm asymptote. I guess this would offer the fastest known way to compute that particular power of log(x) from x... if there were any reason anybody wanted to do that. Being 3D, this iteration may offer more riches than that because there are a lot more kinds of asymptotes one can consider besides Meissel's. Warren D. Smith

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[math-fun] Explaining math to non-math people
by Henry Baker 30 Dec '11

30 Dec '11

I've seen a lot of videos of people trying to teach math ideas to non-math people, and many of these presentations fall completely flat for the non-mathematicians. For example, at some point, I'd like to be able to explain "fat-tailed" v. "thin-tailed" distributions to non-mathematicians, and this is not going to be easy. In particular, I'd like to show how the tail of exp(-x^2) is a lot thinner than exp(-x), which is a lot thinner than 1/x. Nassim Taleb ("The Black Swan") has probably tried the hardest to explain these ideas, coming up with the idealized countries of "Mediocristan" and "Extremistan" where thin-tailed and fat-tailed distributions (respectively) reign. But I still think that many (most?) non-mathematicians still get lost after hearing Taleb. http://en.wikipedia.org/wiki/The_Black_Swan_%28Taleb_book%29 http://www.fooledbyrandomness.com/ http://www.econtalk.org/archives/2009/03/taleb_on_the_fi.html --- Someone who used to work at NSF once told me a (probably apocryphal) story about testifying to Congress, where he said that some indicator number had to reach 10^20 for a project to be successful, but that the current best indicator that the project had demonstrated to date was 10^10. One of the Congressmen then said -- without a trace of irony -- "then we're halfway there". --- Has anyone here had any (successful) experience in explaining sophisticated mathematical concepts of this nature to non-mathematicians?

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Re: [math-fun] Gamma function (formulas collection)
by David Makin 30 Dec '11

30 Dec '11

Thanks Warren, will take a look to verify I got the UF version at least roughly correct - of course I had no simple way of checking it beyond the obvious positive integer values - which I seem to remember it got correct to a minimum of around 10 or 11 sf, not that great but good enough to play with.

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Re: [math-fun] Benford-type law for OEIS ???
by Dan Asimov 30 Dec '11

30 Dec '11

Actually it's virtually certain that the leading digits of OEIS terms of mathy sequences do *not* occur equally often. But maybe Benford's law holds without change. --Dan I wrote: << It might be interesting to know the statistics of leading digits of math-based sequences on OEIS. Maybe only a negligible fraction could be called non-math-based. But it might be good to exclude certain types of sequences, such as the decimal digits of a constant. And it might be also good to exclude low enough numbers, due to the high frequency of very low numbers. But I'm not sure what would be a best way to go about this. It may well turn out that were such a count performed, the fractions of 1,...,9 would be so close to equal that a reasonable hypothesis is that they're equally distributed. Opinions? >> ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Benford-type law for OEIS ???
by Dan Asimov 30 Dec '11

30 Dec '11

It might be interesting to know the statistics of leading digits of math-based sequences on OEIS. Maybe only a negligible fraction could be called non-math-based. But it might be good to exclude certain types of sequences, such as the decimal digits of a constant. And it might be also good to exclude low enough numbers, due to the high frequency of very low numbers. But I'm not sure what would be a best way to go about this. It may well turn out that were such a count performed, the fractions of 1,...,9 would be so close to equal that a reasonable hypothesis is that they're equally distributed. Opinions? --Dan << You can type stuff in the search box on oeis.org and it tells you how many "results found." But you cannot ask it to count the number of "*" (or +, -, /, ^, !) in its programs and formulas, at least not easily... thing #results found --------------------------------- 1 201449 2 177941 0 165928 3 155105 4 132137 5 116309 6 102769 7 91779 8 83837 9 76325 . . . . . . >> ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Gamma(p/q) values
by Warren Smith 30 Dec '11

30 Dec '11

Joerg Arndt: Just as quick copy and paste (suggest to start with the last one): J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.} Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.} Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.} Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.} ----WDS: Vidunas expresses all the Gamma(p/q) with q=60 or q=24 in terms of algebraic numbers and the Gamma values at just these ten p/q: 1/4, 1/8, 1/3, 1/15, 1/20, 1/24, 1/5, 2/5, 1/60, 7/60. (*) But he then notes that the first 6 among those 10 (i.e the first line) can be expressed in terms of the AGM. He also says all Gamma(p/120) can be expressed in terms of the following six: 1/40, 3/40, 7/40, 1/120, 7/120, 11/120. (**) So... focus your Riesian efforts on the Gamma(p/q) with p/q in the (*) and (**) lines! Vidunas at end also notes that Gamma(1/5) and Gamma(2/5) can be expressed in terms of two hyperelliptic integrals... but is there any fast AGM-like scheme for evaluating them? My earlier mathfun post on this topic was 25 Nov http://mailman.xmission.com/cgi-bin/mailman/private/math-fun/2011-November/…

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[math-fun] distribution of formula entities on OEIS
by Warren Smith 30 Dec '11

30 Dec '11

You can type stuff in the search box on oeis.org and it tells you how many "results found." But you cannot ask it to count the number of "*" (or +, -, /, ^, !) in its programs and formulas, at least not easily... thing #results found --------------------------------- 1 201449 2 177941 0 165928 3 155105 4 132137 5 116309 6 102769 7 91779 8 83837 9 76325 sqrt 16366 exp 5851 log 5301 cos 2853 sin 2627 factorial 2337 ln 1775 theta 1365 tan 1344 zeta 1214 eta 1157 Gamma 1094 Meijer 854 Dirichlet 853 sinh 837 cosh 788 arcsin+asin 695 arctan+atan 769 elliptic 604 Ei 583 hypergeometric(& variants) 555 asinh+arcsinh 419 arctanh+atanh 381 Beta 251 Li 238 AGM 231 Bessel 188 Si 153 Ai 108 arccos+acos 101 polylog 72 Klein 65 erf+erfc 59 Mathieu 54 Barnes 52 arccosh+acosh 48 Hurwitz 48 Lerch 25 Ci 20 Painleve 13 cbrt 11 erfi 8 Fresnel 4 Struve 4 pi 6910 Catalan 2574 Golden 790 Mascheroni 144 Glaisher 95 Apery 76 -- Warren D. Smith

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[math-fun] (1/8)!
by Bill Gosper 30 Dec '11

30 Dec '11

Here's a particularly simple one, given Gamma(1/4): Sqrt[Beta[1/8,1/8]]== 4*EllipticTheta[3, 0, 1/(E^(4*Sqrt[2]*Pi))]* Sqrt[Pi]/((2 - Sqrt[2])^(1/4) + 1/2^(1/8)) Not spectacular, but it's so simple, why not take three terms of theta[3]? {Sqrt[Beta[1/8, 1/8]], 4*(2/E^(16*Sqrt[2]*Pi) + 2/E^(4*Sqrt[2]*Pi) + 1)*Sqrt[Pi]/((2 - Sqrt[2])^(1/4) + 1/2^(1/8))} N[%, 72] {3.95668714346392620150868646519961135227029008333530237247887292531488627, 3.95668714346392620150868646519961135227029008333530237247887292531488354} --rwg

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[math-fun] the distribution of formulas on math-fun
by Warren Smith 30 Dec '11

30 Dec '11

It would be good to parse thru, e.g, the entire wolfram formula database, Weisstein encyclopedia, arXiv, or something. Instead I parsed thru the entire math-fun archives file billed as 62Mbyte using cheesy debatable counting hacks. Leaf node probabilities: 1 count=440650 2 count=408962 3 count=286967 4 count=243360 5 count=235382 6 count=243115 7 count=213293 8 count=202003 9 count=199254 (digit counts severely exaggerated due to header info) Pi or pi=3.14159 count=26225 Exp or exp or e=2.71828 count=9250 sqrt(5) or Phi or phi or golden or Golden=1.61803 count=5100 euler-mascheroni gamma=0.57721... count=3902 catalan=0.91596... count=152 zeta(3)=1.20205... count=71 zeta(5)=1.036927755.. count=25 Univariate Function counts: sqrt=4781 cos=1302 sin=1062 tan=348 ln=352 log=1354 factorial or anyhow "!" = 7111 Gamma=537 arctan=251 arcsin=778 zeta=373 excluding zeta(3) and zeta(5) arccos=154 cbrt=63 erf=20 Multi-variate function counts: elliptic=625 hypergeometric=560 AGM=264 Bessel=134 Theta=74 operators: * = 77725 + = 50149 ^ = 16768 -- Warren D. Smith

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[math-fun] identifying real numbers
by Warren Smith 29 Dec '11

29 Dec '11

REIS-ISHNESS: I think the Ries approach is promising but think it could use more development. (1) I think it needs a better notion of "complexity" or "entropy" of a formula (2) user wants to have more control. Re 1: The correct notion of complexity of a formula is basically this. A formula is a tree (e.g. 4+7*8 has + at root, children 4 and *, grandchildren 7,8). Now one could generate random formulas using a probabilistic tree-generator. But I think you need to use the "correct" probability distribution, based on real-life statistics, where "real life" basically means formulas arising in math books and papers. For example, I think irrational (and especially nonalgebraic) exponents of algebraic numbers basically never happen, and trig(x) happens only for x=rational+rational*pi, and GAMMA(x) only happens for x=rational. [You should keep track as you go of whether current numbers are integer, rational, algebraic, or transcendental.] Now assuming that is accomplished, any given formula-tree has a probability p of getting generated. Say the "entropy" of a formula is log2(1/p) in bits. We want Ries to investigate formulas roughly in order of decreasing p, i.e. increasing entropy. A "remarkably good" formula is one that achieves performance=log2(1/|error|) >> entropy. (Normally I'd expect performance=entropy roughly even for totally meaningless random reals.) It should print out the difference performance-entropy for each output. Re 2: Users often know a lot about what sort of formula is reasonable for their application, and there needs to be some kind of language allowing them to specify their class of formulas, thus modifying the probability distribution on formulas. Special functions are not at all a bad thing, they are a good thing, if controlled. There could be a file of parameters governing the probability distribution, which user could modify. CHEAP TRICKS: Finally, there are some extremely cheap tricks you can do. Given X, compute its regular continued fraction. If terminates, then X is rational. If periodic, then X is a quadratic irrational. It is peanuts to find all the best rational and also all the resulting quadratic-irrat approximations to X that thus-arise. Ditto for a few transformations of X such as ln(X), X/pi. Print out the ones that perform significantly better than happens for random reals. The idea of using PSLQ is harder and usually requires extended precision arithmetic so I do not consider it to be a "cheap" trick. Ries is aimed at avoiding extended precision. I think that is a worthwhile goal since often users are unable to compute their real to high precision. I suspect Ries should dominate the Plouffe "huge library" approach. That is since to consider N formulas, Ries uses memory of order sqrt(N) and compute time of order N*logN. Plouffe uses memory of order N and time of order log(N) but precompute time of order N. I suspect this Ries tradeoff will usually outperform the Plouffe approach on single searches. However, if trying to identify any of a large set of input numbers (almost all unidentifiable), then Plouffe approach should pay off. If trying to identify any of M reals, then Plouffe time M*log(N), beating Ries time M*N*log(N). However, Ries could by intentionally unbalancing its "meet in middle" approach achieve much better performance on the multi-identify problem. -- Warren D. Smith

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