January 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Fracfac expansions (fractions in factorial base)
by Dan Asimov 12 Jan '12

12 Jan '12

Just to give credit where credit is due: Where I wrote "perhaps the first", that was entirely my wrong idea, not Jonathan Sondow's. Rich wrote: << Googling for 'e irrational' turns up a 2006 MAA article by Ed Sandifer that credits Euler in 1737 for proving the irrationality of e and e^2. The method is somewhat roundabout, proving the continued fractions from the Ricatti equation, and noting that the infinite CF -> irrational. My (admittedly rusty) memory is that the CF for tanh(x), and I1/I0, can be established directly by doing GCD-like steps on the power series for sinh/cosh; the subsequent steps in the GCD process have relatively clean power series. For full rigor, you'd need to prove some convergence results. But this makes the results a little less mysterious. I wrote: << I think I just found some article that claims this representation was invented by Cantor. I've read (in a Monthly article by Jonathan Sondow) that the fact Warren states below (which has a 0-line proof) was used by Cantor to give perhaps the first proof that e is irrational, as an immediate corollary. >> . . . >> ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] Fracfac expansions (fractions in factorial base)
by Dan Asimov 12 Jan '12

12 Jan '12

I think I just found some article that claims this representation was invented by Cantor. I've read (in a Monthly article by Jonathan Sondow) that the fact Warren states below (which has a 0-line proof) was used by Cantor to give perhaps the first proof that e is irrational, as an immediate corollary. --Dan Warren wrote: << . . . . . . the FracFac expansion of x terminates iff x is rational. . . . >> ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Peter Luschny's gamma function approximation-formulas collection
by Warren Smith 12 Jan '12

12 Jan '12

http://www.luschny.de/math/factorial/approx/SimpleCases.html Luschny describes efforts like Gosper's latest as "pimp my formula" :)

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[math-fun] Too beautiful to eat?
by Victor Miller 11 Jan '12

11 Jan '12

http://www.nytimes.com/2012/01/10/science/pasta-inspires-scientists-to-use-…

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[math-fun] How paper journals are killing science -- case in point: superluminal neutrinos
by Warren Smith 11 Jan '12

11 Jan '12

The CERN/OPERA team who claim neutrinos travel faster than light... this if true would completely overturn theoretical physics. But here is the thing. This calculation depends on a lot of details. For example, in their paper, they just say they measured the 700 km distance accurate to like 20 cm by means of "a geodesy campaign." Zero details provided. Now a full description of all that would maybe be 30 pages or perhaps even several 100, none of them really of any scientific interest by themselves. This would be unpublishable in a paper journal. They are total slaves to the "academia paper journal publication is all that matters," view. So they simply said a few sentences and stopped there. This was utterly inadequate for the purpose of determining whether they are correct. Of course, with electronic publishing, no such limitations exist. But they don't give a damn about that, they only care what is publishable in a paper journal. So in summary, this is an example of utter dysfunction on what may be the most important question in science, and it is directly caused by the brain-dead stone-age prejudice in favor of paper journals. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The function g(z) = e*(GAMMA(z+1/2)/sqrt(2*pi))^(1/z)... two amazing observations
by Warren Smith 10 Jan '12

10 Jan '12

here's a MAPLE script... E := exp(1); g := (z) -> E * (GAMMA(z+1/2)/sqrt(2*Pi))^(1/z); series( g(z), z = infinity, 50 ); #Now to overcome the fact MAPLE is a total idiot: subs(exp(-1/2)*exp(1/2)=1, %); subs(1/exp(-1/2)/exp(1/2)=1, %); subs(ln(1/exp(-1/2)/exp(1/2))=0, %); subs(ln(1)=0, %); subs(exp(-1)*exp(1)=1, %); subs(exp(1)*exp(-1)=1, %); subs(z = 1/R, %); simplify(%); #to get at last this SERIES... series( %, R, 48 ); 1/R -1/24*R +19/5760*R^3 -2561/2903040*R^5 +874831/1393459200*R^7 -319094777/367873228800*R^9 +47095708213409/24103053950976000*R^11 -751163826506551/115694658964684800*R^13 +... The first interesting amazing observation is, f only involves ODD powers of z (or of R=1/z). We can now convert f to a CONTINUED FRACTION. I find... g(z)/z = 1 - y/(24 + y/(10/19 + y/(181944/18029 + y/(59099062/376343621 + ... )))) where y = z^(-2) = R^2 The coefficients in floating point instead of exact rational form are coeff#......value ======......===== 1 ....... 24.0 2 ....... 0.5263157895 3 ....... 10.09174108 4 ....... 0.1570348445 5 ....... 7.183113315 6 ....... 0.08255654793 7 ....... 5.815365631 8 ....... 0.05303121627 9 ....... 4.991323711 10 ....... 0.03780359727 11 ....... 4.429552829 12 ....... 0.02873111220 13 ....... 4.016881221 14 ....... 0.02280569189 15 ....... 3.698138694 16 ....... 0.01868054686 17 ....... 3.442901964 18 ....... 0.01567044292 19 ....... 3.232881310 20 ....... 0.01339321448 21 ....... 3.056353642 22 ....... 0.01162041072 23 ....... 2.905423085 24 ....... 0.01020783673 25 ....... 2.774557162 ... The second interesting amazing observation is, all these coefficients are POSITIVE. The odd numbered and even numbered coefficients evidently behave differently. It looks vaguely as though they might be tending toward two positive constants in which case you could cut off the continued fraction and replace the rest with a periodic CF, whose limit L is known in closed form by solving a quadratic equation y/(a+y/(b+L))=L where in our case, the positive root clearly must be chosen. This also would prove convergence for all positive z by "Van Vleck's CF convergence theorem." Or they might be gradually shrinking toward 0. If they all are positive and shrink slowly enough toward 0 that their sum diverges, then again this would prove convergence of the CF for each positive z by "Van Vleck's CF convergence theorem" mentioned here http://en.wikipedia.org/wiki/Convergence_problem It looks to me like this happens (although that mere appearance proves nothing).

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[math-fun] Lower Limit Found For Sudoku Puzzle Clues
by Ray Tayek 10 Jan '12

10 Jan '12

http://idle.slashdot.org/story/12/01/08/1845258/lower-limit-found-for-sudok… Posted by <http://www.monkey.org/%7Etimothy/>timothy on Sunday January 08, @10:46AM from the 17-ain't-young dept. ananyo writes "An Irish mathematician has used a complex algorithm and millions of hours of supercomputing time to solve an important open problem in the mathematics of Sudoku, the game popularized in Japan that involves filling in a 9X9 grid of squares with the numbers 19 according to certain rules. Gary McGuire of University College Dublin shows in a <http://www.math.ie/McGuire_V1.pdf>proof posted online [PDF] that the minimum number of clues or starting digits needed to complete a puzzle is 17; <http://www.nature.com/news/mathematician-claims-breakthrough-in-sudoku-puzz…>puzzles with 16 or fewer clues do not have a unique solution. Most newspaper puzzles have around 25 clues, with the difficulty of the puzzle decreasing as more clues are given." --- co-chair http://ocjug.org/

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[math-fun] The function g(z) = e*(GAMMA(z+1/2)/sqrt(2*pi))^(1/z)... two amazing observations
by Warren Smith 10 Jan '12

10 Jan '12

Actually, assuming wikipedia's statement of Van Vleck's convergence theorem is correct, we see that a continued fraction y/(a + y/(b + y/(c + ... ))) if all the a,b,c... are positive, will obey 1) its even convergents converge 2) its odd ones do too 3) the two limits are the same if a+b+c+... = infinity all PROVIDED |arg(y)|<pi-epsilon. So in particular the continued fraction I devised for a Stirling-like gamma-function approximation, should converge everywhere in any complex z-halfplane Re(z)>epsilon>0 PROVIDED it is really true that all its coefficients keep on being positive and don't shrink too quickly. This positivity is a remarkable thing. I am not seeing a proof of it, there must be some deep reason for it and perhaps some quite general underlying theorems lurk saying that if you do certain things to continued fractions, their coefficients stay positive. (The other remarkable thing, that the series only involves odd powers, can be proven using methods in my paper that date back about 100 years; that's comparatively trivial.) Note that the odd and even convergents will provide bracketing bounds on the true limit (if z>0 is real) so you can easily tell when to stop working, you have enough precision. I would suggest to Munafo that he based on experiment compute a table saying how many terms of this CF are required for different z to get different levels of precision. Then we can assess how quickly it is converging and whether it is competitive with other gamma approximations.

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[math-fun] Archimedes
by Warren Smith 10 Jan '12

10 Jan '12

Arab scholar Abu al Rayhan Muḥammad ibn Ahmad al-Biruni (973-1048) claimed in an old manuscript that "Heron's" (10-70 AD) formula for the area of a triangle, was actually discovered over 300 years before by Archimedes (287-212 BC). I got this from Howard Eves: Intro to history of mathematics, page 124. (No surviving Archimedes writings support this.) Eves ("great moments in mathematics") and apparently the London Times 19 October 1965, says that in 1965, a steam shovel excavating a hotel in Syracuse found a stone on which was engraved a sphere inscribed in a cylinder. This was presumably the long-lost tombstone of Archimedes.

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[math-fun] (no subject)
by Bill Gosper 10 Jan '12

10 Jan '12

Robert> I only need it to work for real arguments. But if that were an issue I could stick with Lanczos. Any given set of Lanczos parameters work equally well on the entire half-plane Real[z]>0, and with the same guaranteed error bound. This article describes it really well: http://www.haoli.org/nr/bookcpdf/c6-1.pdf which is from: William H. Press et al., Numerical recipes in C: the art of scientific computing (2nd edition). Cambridge University Press, (1992) ISBN 0521431085. On Mon, Jan 9, 2012 at 12:06, Warren Smith <warren.wds(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=warren.wds%40gmail.com>> wrote: > Do you want it to work for complex or only for real argument?>> > I've been looking for a numeric Gamma method that gives 34 digits of> > precision suitable for quad-precision floating-point [2] although I would> > settle for 31 digits suitable for "double-double".> -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 -mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com --------------- Just two b*ggerfactors, da*n near double precision on z>9: http://gosper.org/munafacmu16.png Probably could be double prec for z>10. (Manually Remezed coefficients aren't finished, but mu won't change much.) I'm guessing quad will take eight b*ggerfactors. --rwg

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