August 2010 - mathfun@mailman.xmission.com

[math-fun] A real, live, practical use for Lambert's W-function
by Henry Baker 26 Aug '10

26 Aug '10

FYI -- For those who 1) have a mortgage; that 2) isn't underwater; and 3) still have a job & can qualify for refinancing: "Optimal Mortgage Refinancing: A Closed Form Solution" http://www.economics.harvard.edu/faculty/laibson/files/Mortgage%20refinanci… http://www.nber.org/papers/w13487 An actual mortgage refinancing calculator based on the formulae from this paper: http://zwicke.nber.org/refinance/index.py

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[math-fun] Morpion Solitaire: new world records!
by Christian Boyer 26 Aug '10

26 Aug '10

Morpion Solitaire: NEW WORLD RECORDS in August 2010, both 5T and 5D games, by Chris Rosin. - 172 moves at the standard 5T game, an extraordinary record o beating for the first time the famous 34-year-old record done by hand by Charles-Henri Bruneau, 170 moves published by Science & Vie in 1976 o greatly improving the results obtained by computer, the previous 5T computing record being of "only" 146 moves - 82 moves at the 5D game, two more moves than the previous record of Tristan Cazenave obtained in 2008 Look at the "August 2010" news of www.morpionsolitaire.com Christian Boyer.

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[math-fun] generalized sphericons
by Bill Gosper 23 Aug '10

23 Aug '10

YOW, http://www.flickr.com/photos/sbprzd/3138091584/in/set-72157604502314691/, http://www.flickr.com/photos/sbprzd/3074973521/in/set-72157604502314691/ is the symmetrical one I was about to waste hours unpeeling! It looks a bit too cylindrical to stretch a horsehide sphere over. Have you tried the (heptagonal) third image in http://www.tweedledum.com/rwg/sphericons7.PNG, or better(?) yet, http://www.tweedledum.com/rwg/sphericon6.png , which has no flat face? --rwg I guess "rind" is a better name than "sphericon". Especially if we make them out of papyrus. Seb>And here is a version with a spherical image printed on it: >http://www.flickr.com/photos/sbprzd/3135520252/in/set-72157604502314691/ >and they can indeed be printed and glued: >http://www.flickr.com/photos/sbprzd/3074973521/in/set-72157604502314691/ >Cheers, >Seb

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[math-fun] Space-filling fabric
by Jason 23 Aug '10

23 Aug '10

http://credentiality2.blogspot.com/2010/06/space-filling-fabric.html I just got back my Hilbert space-filling fabric from spoonflower, a site that'll print any image onto a variety of fabrics. Space-filling fabric is suitable for completely filling any plain spots on your walls, or for topological application. Spoonflower will sell you some of your own: http://www.spoonflower.com/fabric/246090 (I don't think I get any commission if that happens) It's kind of pretty (if rather garish) as a wall hanging, but beyond that I haven't yet figured out quite what to use it for. Someday if I have time, I think it'd be interesting to try out a more restricted color space that clashes less. I'd also buy a few yards of a Gosper fabric for anyone who will generate the corresponding image and source code. I do find it amusing that the fabric is spun from more-or-less one dimensional thread, although I don't see any options on spoonflower for specifying that the manufacturing process must preserve continuity and come from a single thread.

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Re: [math-fun] generalized sphericons
by Dan Asimov 23 Aug '10

23 Aug '10

Maybe "rhind" ? --Dan RWG wrote: << I guess "rind" is a better name than "sphericon". Especially if we make them out of papyrus. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] generalized sphericons
by Bill Gosper 23 Aug '10

23 Aug '10

I gurgled:> Yow, there are totient(n)/2 of these for every star polygon(n/d), i.e. >(totient(n)/2)^2 for every n. The nine for n=7: >www.tweedledum.com/rwg/sphericons7.PNG . No, for even n there are more, because there are two kinds of mirror symmetry. Oh, foo, I used a needlessly unsymmetrical sphericon to make the baseball, Back to the drawing board. --rwg

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[math-fun] Oh, no, a one-piece baseball cover
by Bill Gosper 23 Aug '10

23 Aug '10

Here we go again. www.tweedledum.com/rwg/baseball.png . The cusps are the endpoints of the cut (= the conical points on the translucent solid (= convex hull)). We should check it by cutting one out of a rag. For a unit radius sphere, the distance between the parallel sections of the cutout is 2 sin 4 pi/5 . The angle they span is sin(pi/10) pi (!), and the angle spanned by the elbows is sin(3*pi/10) pi. --rwg For your calculating pleasure: 12/5/(3-sqrt 5).

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[math-fun] Superseeker defends itself
by N. J. A. Sloane 23 Aug '10

23 Aug '10

Fred Lunnon said: > I fed A180238 to gfun via Maple9 --- and strange to say, gfun replied > straight away "FAIL". So as I suspected, it is nonsense; and there appear > to be bugs both in the way superseeker is to utilising the gfun package, > and its output of the results; while the documentation leaves something > to be desired (like, some documentation)! Superseeker would say that WFL was not familiar with gfun, which has a lot of commands. Look: s1:=[1,3,9,27,75,189,447,951,1911,3621]; (that is A180238) > listtoalgeq(s1,y(x),[lgdegf]); 2 3 [-27 + 27 y(x) - 9 y(x) + y(x) , lgdegf] The result may be nonsense, but Superseeker feels an obligation to report it, since once in a while it finds useful things. As for documentation, the help file (send an empty email to superseeker(a)research.att.com) describes what it does. It states that it uses gfun, but of course does not show the large set of descriptions of all the gfun commands. And as I said earlier the full Superseeker code can be seen on the OEIS web site. Best regards Neil PS The web page cite.html (on the OEIS site) mentions quite a few results that were found with the aid of Superseeker.

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[math-fun] Billiard word counts (was "Interleavings of three arithmetic progressions")
by Fred lunnon 22 Aug '10

22 Aug '10

Called earlier "interleavings of arithmetic progressions", billiard sequences can be viewed equivalently as generated by discrete approximation to a line in m-dimensional space; by a weightless ball bouncing against the walls of an m-dimensional hypercube; or by a point translated continuously through a suitably partitioned m-dimensional torus T^m. In each case, we ask for function f_m(n) counting the number of m-symbol words of length n occurring as factors of possible billiard sequences. Allan Wechsler posed the case m = 3. I have now had a look at m = 4, for which I find f_4(n) = 1, 4, 16, 64, 244, 856, 2776, 8356, 23032, 59200, 142624, 324484, 696256, 1422436, 2779900, when n = 0,...,14. [OK, now do your worst, Michael!] Setting n = 14 in the expression log(f(n)/f(n-1)) / log(n/(n-1)) gives 9.04, suggesting to the credulous optimist that perhaps f_4(n) = O(n^9). The program does its best to avoid inequality analysis by first projecting each candidate sequence onto the axis hyperplanes, yielding m new (m-1)-symbol sequences which are necessarily also billiard, and can be validated from a precomputed list for dimension m-1. If any of these fails, the candidate fails; and if only one candidate remains after n-th symbols are attached to a valid (n-1)-length sequence, there is still no need for inequality analysis --- the ball cannot avoid bouncing next against some wall pair! The projection criterion is surprisingly effective: without inequality analysis, when m = 3 it catches every dud word for n < 10; when m = 4, for n < 12. I find it frustrating that it can't be beefed up just a little more, since inequality analysis of those cases remaining slows the Maple program by a factor of 10. A (notionally slightly weaker) version of this projection constraint involves projecting instead onto planes, yielding instead m_C_2 2-symbol sequences, one on each pair of the m symbols. If a candidate is already known to be valid, it may be reconstructed from just m-1 of these binary projections --- say, just those involving symbols (1,2),...,(1,m). But it is already known that there are O(n^3) of the latter, from which it follows immediately that f_m(n) <= O(n^(3m-3)). The numerical results suggest that degree 3(m-1) is actually sharp! Finally, a taste of the difficulty to be expected in beefing up projection to achieve sufficiency is provided by the following example: the one-way infinite (and easily made two-way?), ultimately period-6 ternary word 1121231123 (112123)^oo is not billiard (by inequality analysis on its length-10 prefix); though all three of its projections 11212112(11212)^oo, 1113113(1113)^oo, 22323(223223)^oo are binary billiard words (aka Sturmian). Extraordinarily long impasses are possible: the above's length-31 prefix 1121231123 1121231121 2311212311 2 may be followed by the foreign length-85 suffix 3121123121 1231213121 2131212131 2132112132 1121321132 1211321211 3212131211 3212113212 13121 yielding a non-billiard length-116 ternary word, all of whose projections are billiard words, but which cannot be further extended without invalidating some projection. Any prospect that the projection criterion might be strengthened via larger samples is quashed by these examples. Another apparently promising test is to check that the candidate, whose length-(n-1) prefix was drawn from a precomputed list, has length-(n-1) suffix also a member of the same list. Surprisingly, this test appears to reject no additional candidates besides those already caught by projection: is there a (not terribly exciting) theorem here? Fred Lunnon On 7/26/10, Fred lunnon <fred.lunnon(a)gmail.com> wrote: > On 7/25/10, Michael Kleber <michael.kleber(a)gmail.com> wrote: > > Data update: the sequence through n=30 is > > > > 1,3,9,27,75,189,447,951,1911,3621, > > 6513,11103,18267,29013,44691,67251,98547,140865,197679,272799, > > 370659,497403,658371,859863,1110453,1420527,1799373,2260161,2815401,3479235, > > 4269279 > > > > Looks more n^5ish than n^6ish, at a casual glance. > > > evalf(log(4269279/3479235)/log(30/29)); # 6.036081126 > evalf(log(3479235/2815401)/log(29/28)); # 6.033051457 > > No way --- it's order n^6 !! > > > > The last term took something like 11 hours to count, so I think it's time to > > give Mma a break. > > > > > > --Michael > > > Sterling effort. > > I've been trying hard to upstage it by finding an explicit formula, > but am beginning > to understand that the 3-D case may well qualitatively harder than the 2-D. > > There is a literature about generalisations of Sturmian sequences from > 2-D / 2-symbol alphabet to n-D, which connects them to MDCF's. > This particular generalisation is called "billiard words" (or > "billiards words") --- > the "words" here being what we might call infinite sequences, and "factors" > finite words occurring within them. > > One reference which surveys these is the 2003 preprint available on the web: > Laurent Vuillon "Balanced Words". > A detailed study (also available on the web) is > NICOLAS BEDARIDE "CLASSIFICATION OF ROTATIONS ON THE TORUS T^2" > > The following may well be very relevant, but I can't get at them from home: > > Jean-Pierre Borel "How to build billiard words using decimations" > RAIRO-Theor. Inf. Appl. Volume 44, Number 1, January-March 2010 59--77 > > Jean-Pierre Borel "A geometrical Characterization of factors of > multidimensional Billiards words and some Applications" > Theoretical Computer Science 380 (2007) 286--303 > http://hal.archives-ouvertes.fr/hal-00465586/fr/ > > The particular difficulty afflicting billiards --- in comparison with, > say, the more > common Arnaud-Rauzy generalisation --- is their nonlinear complexity: > apparently, > the number of distinct factors of length n in any given trajectory of > cubical billiards > (Allan's original problem) is in general n^2 + n + 1 . > > > Fred Lunnon >

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Re: [math-fun] Billiard word counts (was "Interleavings of three arithmetic progressions")
by N. J. A. Sloane 21 Aug '10

21 Aug '10

Yes, I am on the math-fun mailing list! The lgdegf search does refer to log deriv of a generating function If you go to the foot of any OEIS page, and click on Superseeker, you will see more about what Superseeker does, including all the source code It uses the Maple gfun package to search for generating functions. See for example pictor.math.uqam.ca/~plouffe/articles/gfun.pdf Neil

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