December 2014 - mathfun@mailman.xmission.com

[math-fun] Name of this transform?
by Mike Stay 18 Dec '14

18 Dec '14

What's the name of this transform? It basically treats the even part of the function as "real", the odd part as "imaginary", and rotates it by 90 degrees: T(f)(z) = z even(f)(z) - odd(f)(z)/z = z(f(z) + f(-z))/2 - (f(z) - f(-z))/2z It's somewhat related to the Hilbert transform, but that splits the function into positive and negative frequencies rather than even and odd part. -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Life on Mars? Latest grist
by Warren D Smith 17 Dec '14

17 Dec '14

1. The Curiosity rover detected time-varying methane concentration in Mars atmosphere, changing by factor>10 over a several-month period. They claim the usual "background" level is about 0.7+-0.25 parts per billion, but it has gone as high as 7.2+-2.1 ppb at least locally near Curiosity (these are 95% confidence intervals). Earth based telescope observations have been claimed to see Mars-wide changes in methane, spiking as high as 60 ppb. To put this in perspective, the total amount of methane (at background level) in the whole Mars atmosphere, is about 200 metric tons. Meanwhile the total amount in the Earth's atmosphere at any time is about 500,000,000 metric tons. 2. Also, it found chlorobenzene in a mudstone rock sample (something like dried clay). Chlorobenzene is the most abundant organic found in that sample (as well as the most complex; several chloro-alkanes were also found) with concentrations between 150 and 300 parts per billion. Chlorobenzene does not occur naturally on Earth but is present in pollution.

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[math-fun] Failed(?) puzzle about convex objects
by Warren D Smith 17 Dec '14

17 Dec '14

In N-dimensional space, we have a convex object A with unit volume. We scale it by x^(1/N) so it now has volume=x, and intersect it with convex object B. QUESTION: Is the volume of the resulting object, a convex (i.e. concave-down) function of x? When N=1, answer obviously is yes. Also, answer is yes for any N>=1 if A and B are suitable (same-oriented) hypercubes. Also if B is a halfspace. Or whole space. But answer is "no" for each N>=2 if A and B are suitable (different-oriented) hypercubes. So... that wasn't the right question. REVISED QUESTION: same thing, but demand both A and B contain the origin 0 inside. Nope, a counterexample is A=circle, B=square, N=2. So... doesn't seem like there is any nice theorem of this ilk. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] better bid - any way to game this site?
by Ray Tayek 17 Dec '14

17 Dec '14

http://betterbid.com/penny/HowItWorks.aspx lowest unique bid wins. thanks --- co-chair http://ocjug.com/

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[math-fun] Halp, Halp, I'm going senile.
by Bill Gosper 17 Dec '14

17 Dec '14

Yesterday, under the subject: Re: [math-fun] analytic function for new Airbnb logo?I blathered ------------ As I mentioned here before, as 0<q<1, the curve continuously deforms from a tiny "circle" to a big "+" sign, sweeping out a fairly convincing spaceplane: http://gosper.org/sst.png --rwg Two more mystery constants associated with this spaceplane: Its volume (4.231765644191597 ± ? (Mma 10.0.2 crashed when I asked for more digits), And its "fattest" (cross-sectional area, vs squarest) q, Out[87]= {q -> 0.43939962899109219485183075448731} which ISC guessed was In[81]:= 1/Log[3]^(7/4)/2^(1/12)/E^(3/5) (nice try). Pix: gosper.org/halphen.pdf The first graph (with strangely varying curvature) is cross sectional area vs q. ---------------------- But just 3.5 years ago (Date 2011-05-10 05:43, Subject Re: [math-fun] Redefine Halphen's constant?) I actually had the bleeping area and volume, and sent ------- For some reason, the original message >From rwg(a)sdf.lonestar.org Mon Aug 18 05:49:22 2008 is missing from my inbox, but the followup is there, and both are in the archive, so hopefully the rest of math-fun got both. These were about the family of closed curves described parametrically by (theta_2,theta_1}, which if 3D plotted for 0<q<.9, draws a "space plane" whose fuselage cross sections range from a tiny circle through a near square to nearly a plus-sign. ParametricPlot3D[{EllipticTheta[1, t, q], EllipticTheta[2, t, q],q}, {t, 0, 2*\[Pi]}, {q, 0, .9}, BoxRatios -> {1, 1, 1}] The q values maximizing area and "squareness" are respectively ParametricPlot[{{EllipticTheta[1,t, .4393996289909801220193544165615151], EllipticTheta[2, t, .4393996289909801220193544165615151]}, {EllipticTheta[1, t, .32810656687497825302548], EllipticTheta[2, t, .32810656687497825302548]}}, {t, 0, 2*Pi}] What's new tonight is an exact formula for the cross-sectional area 2 Pi EllipticThetaPrime[1, 0, q^2] (which can be written in Macsyma as 4 * pi * eta^3(q^4) ) and volume 2 pi^2 * sech(pi/sqrt(2)) . Before I found the area formula, Julian labored mightily to get 30 digits of the q maximizing area, difficult because Mma 8.0 seems to be missing the numerics: In[1854]:= Derivative[0, 1, 1][EllipticTheta][1, 69, .105] Out[1854]= (0,1,1) EllipticTheta [1, 69, 0.105] I'm puzzled by this apparently nondescript q and the corresponding maximal shape. I found no relations other than the Halphen's result . --rwg ---------------- (ISC: 0 for 2.) And Mma 10 is *still* missing the numerics! In[3]:= D[EllipticTheta[s, t, q], {t, 2}] == -4 q D[EllipticTheta[s, t, q], q] Out[3]= (0,1,0) EllipticThetaPrime [s, t, q] == (0,0,1) -4 q EllipticTheta [s, t, q] In[4] Pi N[% /. t -> -- /. s -> 1 /. q -> 0.69] 4 Out[4]= (0,0,1) 6.91574 == -2.76 EllipticTheta [1., 0.785398, 0.69] Vs my old Macsyma package: (c171) DIFF(THETA[S](T,Q),T,2) = -4*Q*DIFF(THETA[S](T,Q),Q); (d171) thetaderiv (t, 2, q) = - 4 q thetaderiv (t, 0, q, 1) s s (c172) SUBST([T = DFLOAT(%PI/4),S = 1,Q = 0.69],%); (d172) 6.91573338862223d0 = 6.91573338862223d0 Huge thanks to NeilB, Ashok Anumandla, and Babu Lella for (so far, partially) bringing up Macsyma 2.4 under XP under Yosemite! (But I'm still wondering why (d171) no longer auto-simplifies to a tautology.) If it weren't for the missing numerics, Mma would have numerous ways to exactly and manipulably express Halphen's and similar constants: InverseFunction[EllipticTheta^(0,0,1),3,3][1,\[Pi]/4,0] InverseFunction[EllipticTheta^(0,0,1),3,3][2,\[Pi]/4,0] InverseFunction[EllipticThetaPrime^(0,1,0),3,3][1,\[Pi]/4,0] InverseFunction[EllipticThetaPrime^(0,1,0),3,3][2,\[Pi]/4,0] Only two of these work: In[311]:= N[%, 22] Out[311]= (0,0,1) {InverseFunction[EllipticTheta , 3, 3][ 1.000000000000000000000, 0.7853981633974483096157, 0] (0,0,1) , InverseFunction[EllipticTheta , 3, 3][ 2.000000000000000000000, 0.7853981633974483096157, 0] , 0.3281065668749782530255, 0.3281065668749782530255} (Notice how it floated the exact subscripts 1 and 2, and now foolishly regards them as approximate.) The mail server's memory is nearly as bad as mine. I recovered the May 2011 text only thanks to a bounce of a copy sent to Steven Finch. More distressing: That mail also contained a bunch of denestings apparently debuted here, but half of them I don't even recognize: Apropos radical denestings, do you think these from math fun startled anybody? 1/5 3/5 3/5 2/5 1/5 2/5 3/5 1/5 1/3 - 2 3 + 2 3 + 3 + 2 (3 - 2 ) = ------------------------------------- 2/3 5 4/5 3/5 2/5 1/5 1/5 2 7 - 7 - 2 7 + 6 7 + 2 sqrt(8 - 7 ) = ----------------------------------- 5 1/5 sqrt(11 76 + 5) = 3/5 4/5 1/5 3/5 4/5 2/5 2/5 1/5 2 19 + 2 19 + 3 2 19 - 2 2 19 - 7 --------------------------------------------------------- 5 6/7 5/7 3/7 1/7 3/7 - 2 + 2 + 2 + 2 2 - 1 sqrt(4 - 1) = --------------------------------- sqrt(7) 5/6 1/6 2/3 1/6 5 2 sqrt(5) sqrt(4 +- sqrt(3) 5 ) = +- --------- -+ ------------ 3 sqrt(2) 3 1/3 5/6 1/6 1/6 5 2 5 2 + ------- +- --------- + ------- sqrt(6) 3 sqrt(3) 1/5 1/5 sqrt(4 - 3 ) = 3/5 4/5 3/5 2/5 2/5 4/5 1/5 1/5 - 2 3 + 3 + 2 2 3 - 2 3 + 2 --------------------------------------------------- 5 Five-termers are quite sparse. I've never been able to find a six-termer. --Bill Wow, who found these? I'd like to meet him! --rwg I just fixed a typo in the Macsyma notebook exported by Mma as http://gosper.org/thetpak.html ,(c575): The last thetaderiv_1 should be thetaderiv_2. This is hard to fix because Mma 10 can no longer read the Mma notebook full of bitmaps pasted in from the Macsyma notebook! (Macsyma never learned how to display formulas in html.) It is a scandal how many thetpak.html examples continue to elude Mma 10. --rwg

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Re: [math-fun] Black Hole Binaries can eject relativistic normal stars
by Henry Baker 15 Dec '14

15 Dec '14

At 11:49 AM 12/1/2014, Mike Stay wrote: >Frederic Pohl's book _The World at the End of Time_ includes this as a major plot point. Cool! I'll have to try to find this book. >On Mon, Dec 1, 2014 at 11:43 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> FYI -- Apparently, just like planets & planetoids can be ejected from solar systems, regular stars can be ejected from a system of binary black holes. >> >> I presume that "living" in a planetary system of such a star would provide time dilation relative to the rest of the universe. Of course, the event which accelerated the star in the first place might a) strip the star of all/most of its planets; or at least b) kill all life on any such planets that do manage to escape with the star. >> "In the context of astro-biology, SHS could spread life beyond the boundaries of their host galaxies [26, 27]." I forgot to say that one implication of b) is that since time is moving relatively slowly on the accelerated star, if all life is killed when the star is accelerated, there won't be much time for life to evolve on the accelerated star planetary system. So you now need 3 compounded miracles: for life to have already evolved on the star prior to ejection; for the star to be involved in a binary black hole system; and for life to survive the ejection. Good luck with that...

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Re: [math-fun] analytic function for new Airbnb logo?
by Bill Gosper 14 Dec '14

14 Dec '14

Henry>I wanted an approximation that was entire over the complex plane, so the straight lines and discontinuities would have to be fudged. If I take the image, convert it to black-and-white, and erode the boundaries to a single pixel width, I will get a "continuous" curve that goes around the origin twice. If I consider this curve in (r,theta) coordinates, I can now consider the Fourier transform. I'd like to reset the origin in such a way that the "fundamental" is zero -- i.e., the lowest frequency (which takes 2 revolutions) has a coefficient of zero. (There are other possible locations for the origin: the balancing point, the center of gyration, etc.) That having been done, I could alternatively use (log(r),theta) coordinates, which have some nicer properties, because it is the actual complex log function. Fourier analysis can approximate the discontinuous changes in curvature; hopefully modifying these coefficients a little will remove these discontinuities with a minimum of ringing. The sides of your rounded edge squares are impressively flat. <HGB At 03:48 PM 7/30/2014, Bill Gosper wrote: HGB> My approach would be to take the Fourier transform of the polar plot, rwg>Defined as piecewise linear|circular? and then simplify to a small number of coefficients. Of course, it will take 2 circuits around the origin to complete the figure. The sides are perhaps a little too flat, which will blow up the number of coefficients required. I guess if you transform [0,4pi) to [0,1) you could use some sort of Cheby polynomial approximation. It would be nice to be able to transform it back to p(x,y)=0, where p is a polynomial in x,y. At 02:56 PM 7/29/2014, Whitfield Diffie wrote: HGB>Anyone game for an analytic function to approximate Airbnb's new logo? They should manufacture a paper clip and give it out at trade shows. Whithttp://www.underconsideration.com/brandnew/archives/airbnb_logo_detail.… rwg> You can get some impressively straight segments with theta fcns: gosper.org/halph.pdf (10MB?!) or see http://gosper.org/thetpak.html (scrolled 69%). (.3281... is âˆš(one ninth constant), which I claim should be named Halphen's. It's the largest q for which the curve is convex.)) As I mentioned here before, as 0<q<1, the curve continuously deforms from a tiny "circle" to a big "+" sign, sweeping out a fairly convincing spaceplane: http://gosper.org/sst.png --rwg But a problem with this logo is that it switches curvatures discontinuously. <rwg Another possibility: modulate the arcspeed (slow down for curves, speed up for straightaways) before taking the Fourier.

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[math-fun] Intro to calculus book for an English major
by Bernie Cosell 14 Dec '14

14 Dec '14

A very smart friend has the misfortune of being an English major and so avoided essentially all math courses in college [many many years ago]. Any recommendations on a good "intro" book [I'd be helping as mentor/tutor]. A check on Amazon uncovers dozens of "Calculus made easy" and such books... I suppose I could just pick one at random... thanks! /b\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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Re: [math-fun] Intro to calculus book for an English major
by Bill Gosper 13 Dec '14

13 Dec '14

I suggest that your friend go the public library and browse the mathematics section for something interesting. -- Gene Presumably times have changed. ~1960, I worked in the Merchantville NJ Public Library, which doubled as the High School Library. It contained **ZERO** math books. The only math in the whole building was in old Britannica articles. (And Martin Gardner columns, but the SciAms were in storage.) --rwg

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Re: [math-fun] Intro to calculus book for an English major
by Bill Gosper 13 Dec '14

13 Dec '14

Your friend might enjoy it in vintage English: ... another edition of the *Methodus differentialis* [was published] in London in 1764; and a translation of the latter into English by Halliday in London in 1749. It's irritating that Stirling didn't "translate" it himself. But even without the risk of a second-rate translation, the pioneers in a field are hard to learn from. Their words don't quite mean the same as ours, and their understanding was sometimes less complete than we have come to possess. In any event, http://en.wikipedia.org/wiki/James_Stirling_%28mathematician%29 is fascinating. --rwg

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