June 2011 - mathfun@mailman.xmission.com

[math-fun] old series reversion article
by Bill Gosper 27 Jun '11

27 Jun '11

Enormous thanks to Stephen M. Jones for resurrecting my Lisp Machine files, and to Quad for converting this one <http://gosper.org/newsrope.pdf> to LaTeX. (Slightly updated from a 90s Macsyma Newsletter.) --rwg Speaking of latex, HOUSEPAINTER NEUROPATHIES

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[math-fun] Spirograph fun
by Dan Asimov 26 Jun '11

26 Jun '11

I was just playing with the (very awkward) computer graphics program "Grapher" on my Mac, reproducing one of the first computer programs I ever wrote -- around 35 years ago -- that draws iterated cycloids produced by rolling circles on circles on circles on circles. By playing with the 8 parameters of 4 radii and 4 frequencies -- especially the latter -- one can of course drastically change the look of the picture. I was very suprised that one combination of frequencies resulted in a picture of a bagel with a square hole, the inner envelope of the iterated cycloid. (The outer envelope has 4-fold symmetry but is far curvier than a square.) This was particularly unexpected since the frequencies were 23, 70, 162, and 162 [sic]. So I wonder, how could one predict, from these parameters, striking features of the resulting shape like this square inner envelope. --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] Question inspired by nothing in particular
by Dan Asimov 26 Jun '11

26 Jun '11

I think by taking an extreme case one can see that geodesics are not typically planar. E.g., consider an ellipsoid of revolution, with semiaxes 1, 1, and 10^9. Near the xy-plane this will be very close to a vertical (circular) cylinder. So the geodesic from (1,0,0) to the point nearest (-1,0,1) will be very close to a helix. --Dan << A and B are points on an ellipsoid E. S is the shortest path on the ellipsoid's surface between A and B . Does S always lie in a plane? >> Sometimes the brain has a mind of its own.

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[math-fun] Planet Mercury Is Looking Less Boring
by Henry Baker 18 Jun '11

18 Jun '11

FYI -- 'A third idea is that a giant impact early in MercuryÂs history knocked off a large fraction of the planet. ÂThis is the model that is still in the running,Â Dr. Nittler said.' [Why does such an impact necessarily have to be "early" in Mercury's history?] http://www.nytimes.com/2011/06/17/science/space/17mercury.html June 16, 2011 Close Up, Mercury Is Looking Less Boring By KENNETH CHANG For years, many planetary scientists did not express much curiosity about Mercury, which looked gray and cratered Â a slightly larger version of the Moon. But data released Thursday from NASAÂs Mercury Messenger spacecraft, which entered orbit around Mercury in March, is painting a more vibrant picture of the solar systemÂs innermost planet. ÂMercury ainÂt the Moon,Â Ralph L. McNutt Jr., the missionÂs project scientist, said at a NASA news conference on Thursday. Among the new findings: Some of MercuryÂs topography is not seen anywhere else in the solar system Â rimless pits, for instance Â and its mineralogy is vastly different from the MoonÂs, whose rocks have much less potassium. Scientists already knew that Mercury has a magnetic field, while the Moon does not. The latest batch of data includes the clearest pictures yet of MercuryÂs polar regions, plus readings of the elements in its crust, which have helped scientists rule out some theories about the planetÂs origins. Mercury Messenger has also discovered that the planetÂs magnetic field is stronger in its northern hemisphere than in its southern, which hints at something odd in the structure of its molten core. The new information could reveal how Mercury formed and changed over the 4.5-billion-year history of the solar system, which in turn could help astronomers understand the panoply of Earth-size planets around other stars and the possibility of conditions friendly for life on them. NASAÂs Kepler telescope has discovered dozens of possible Earth-size planets, but its observations can determine little beyond their size. ÂWe have in our solar system four experiments in how four Earthlike planets evolve once they form under slightly different conditions,Â said Sean C. Solomon, the principal investigator for Mercury Messenger, referring to Mercury, Venus, Earth and Mars, the four rocky planets of the inner solar system. ÂWhat weÂre learning is that each of those experiments had an extraordinarily different outcome,Â Dr. Solomon said. ÂAnd one of those experiments we live on. So it really behooves us, in a very general way, to understand how Earthlike planets form and evolve and operate.Â One of the mysteries is why the iron core of Mercury is unusually large, extending out three-quarters of the way to the surface. EarthÂs core, by contrast, extends a little more than halfway. One idea was that Mercury was originally larger, and the young Sun was so intense that the radiation stripped away the outer layers, leaving behind the Mercury seen today. But that theory predicted low abundances of certain elements like potassium that would have easily evaporated in the intense heat. Mercury Messenger measured ample amounts of potassium. ÂWe can rule out this kind of model,Â said Larry R. Nittler, a staff scientist at the Carnegie Institution of Washington and a member of the science team. Another theory is that Mercury formed out of metal-rich meteorites. By and large, the observed composition of the surface rocks does not fit with that theory either, although a variation could still prove the correct explanation, Dr. Nittler said. A third idea is that a giant impact early in MercuryÂs history knocked off a large fraction of the planet. ÂThis is the model that is still in the running,Â Dr. Nittler said. ÂThere are probably going to be many more models devised before we have an answer on this.Â High-resolution images of craters revealed irregular pits, ranging in width from several hundred feet to a few miles. The pits do not have rims like craters produced by impacts, leading the scientists to speculate that they were etched by some unstable material that evaporated quickly when exposed at the surface. Mercury MessengerÂs one-year mission around the planet is only one-quarter done, so more information about the planet will be forthcoming. Already, Dr. Solomon said, the orbiter has dispelled the misconception that Mercury is a boring place, and that what NASAÂs Mariner 10 spacecraft saw in 1974 and 1975 during three flybys was all that was to be seen. ÂSome even in the planetary community, after the Mariner 10 mission, placed a low priority on returning a spacecraft to Mercury on the grounds that it was very much like the Moon, weÂd been to the Moon,Â Dr. Solomon said. ÂIt was an example, to use a phrase coined by a very famous space scientist, of Âone of the burnt-out cinders of the solar system.Â And it is anything but that.Â

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Re: [math-fun] Exact N-body simulations?
by Henry Baker 17 Jun '11

17 Jun '11

After Gosper & Schroeppel found some papers of Prof. Donald Greenspan (1928-2010) via Rob Corless, it quickly became clear that Greenspan's entire career was spent developing a theory very similar to the one I was advocating. The best paper of his that I was able to find on the web (quickly) regarding the N-body problem is this one: Greenspan, Donald. "Discrete Mathematical Physics and Particle Modeling". Advances in Electronics and Electron Physics, Vol. 63, pp. 189-268. On pp. 190-202, he goes through the development of a set of _difference equations_ for the 3-body problem in 2-D that conserve mass, energy, linear momentum and angular momentum. He discusses that for each time step of the iteration he needs to solve 12 equations in 12 unknowns. "Such systems are solved easily by the Newton-Lieberstein method (Liberstein, 1960; Greenspan, 1974d)..." "Then the Newton-Lieberstein iteration formulas for this system are ... in which omega is an overrelaxation factor in the range 0<omega<2 and each partial derivative is evaluated at the same point as is the function in the numerator of the term in which the partial derivative occurs. Computer implmentation is always practical because no matrix inversion routines are required". (?????) I haven't yet gone through the development to see how complex these equations are (e.g., whether they require polynomial root-finding), but Greenspan goes a very long way in the direction that I wanted to go. Perhaps the distance to Hilbert's Tenth Problem isn't very great at all: if we can take an arbitrary instance of Hilbert's Tenth Problem and construct an N-body problem of the Greenspan sort to solve it, then we are home. Greenspan apparently did a lot of the work for this paper with Robert LaBudde. LaBudde, Robert A., and Greenspan, Donald. "AN ENERGY CONSERVING MODIFICATION OF NUMERICAL METHODS FOR THE INTEGRATION OF EQUATIONS OF MOTION". Internat. J. Math. & Math. Sci. Vol. 10 No. 1 (1987) 173-179. Much of Greenspan's work during 1968-1978 is available at the U. Wisconsin CS Dept's technical reports web site: http://www.cs.wisc.edu/techreports/

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[math-fun] Incomplete Beta identity?
by Neil Bickford 16 Jun '11

16 Jun '11

Bill and I just found this beta identity and don't recognize it: Beta[1-x,n,n]+Beta[x,n,n]==2^(1-2 n) Beta[1/2,n] --Neil Bickford

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[math-fun] $10K for solving a math equation
by Ray Tayek 14 Jun '11

14 Jun '11

http://joshjames.com/NewGig " ... *The $10,000 prize for solving the math equation and puzzle will only be awarded to one person, as determined at our sole discretion. The recipient will be the first person who we determine has found the answer key, signed a non-disclosure agreement, and then finished the equation and puzzle and demonstrated proof of his or her solution after obtaining the key. ... " anyone recognize this? --- co-chair http://ocjug.org/

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[math-fun] Puzzle, mostly by Julian
by Bill Gosper 12 Jun '11

12 Jun '11

We have In[735]:= # == Normal[Series[#, {x, 0, 23}]] &[Tan[Sin[x]] - Sin[Tan[x]]] Out[735]= -Sin[Tan[x]] + Tan[Sin[x]] == x^7/30 + 29 x^9/756 + 1913 x^11/75600 + 95 x^13/7392 + 311148869 x^15/54486432000 + 10193207 x^17/4358914560 + 1664108363 x^19/1905468364800 + 2097555460001 x^21/7602818775552000 + 374694625074883 x^23/6690480522485760000 Without using a computer, prove or disprove: The terms of the infinite series are nonnegative. --rwg Hey--get your hands off that keyboard! Hey--you're reading the screen! Oops, OK, you can read the screen. (J: cc Varun?)

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Re: [math-fun] periodic nag
by Bill Gosper 11 Jun '11

11 Jun '11

Professor Manuel Kauers has just polished this off in a pretty little .nb whose pdf is http://gosper.org/FibonacciSum.pdf . You can get Christoph Koutschan's HolonomicFunctions.m from http://www.risc.jku.at/research/combinat/software/ after getting a password from Peter Paule <Peter.Paule(a)risc.uni-linz.ac.at>. --rwg On Sun, Apr 17, 2011 at 2:12 PM, Bill Gosper <billgosper(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote: > Can anybody prove:> Sum[k/n*Binomial[-n, n - k]*Fibonacci[k + 1], {k, 1, n}] == 1 ?> I was almost surprised when it defeated Julian last nite.> --rwg> He wasn't really defeated--just lazy. He typed Sum[k/n*Binomial[-n, n - k]*Fibonacci[k + 1], {k, 1, n}] and got 1 ----------------- Binomial[-n, -1 + n] (5 + 3 Sqrt[5]) n 1 ((7 + 3 Sqrt[5]) Hypergeometric2F1[2, 1 - n, 2 - 2 n, - (-1 - Sqrt[5])] - 2 2 2 Hypergeometric2F1[2, 1 - n, 2 - 2 n, -----------]) 1 + Sqrt[5] and said "Use A=B". I did, and after a major struggle, found this expression to satisfy In[561]:= Equal[x[n + 2], (((17*n^2 + 37*n + 18)*x[n + 1] - 4*(2*n + 1)*(2*n + 3)*x[n])/((n + 2)*(n + 3)))] Out[561]= 2 -4 (1 + 2 n) (3 + 2 n) x[n] + (18 + 37 n + 17 n ) x[1 + n] x[2 + n] == ---------------------------------------------------------- (2 + n) (3 + n) which is solved by x[n]->1. As marvelously inelegant as JPGrossman's solution. --rwg

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Re: [math-fun] Exact N-body simulations?
by Henry Baker 11 Jun '11

11 Jun '11

You missed my point, entirely. I'm talking about integer and/or rational coordinates, not real numbers, so even the 2-body case boils down to drawing conic sections on an integer lattice -- a problem that took computer graphics people tens of years to fully come to grips with. The whole point of mentioning PoincarÃ© was to acknowledge that we weren't going to "solve" a >3-body problem -- merely simulate it. This is no more, and no less, than simulating a Turing Machine computation instead of solving the Halting Problem. I just wanted to figure out a way to simulate such a computation more accurately. It also occurred to me since yesterday that the better proof of unsolveability would be to reduce the 2D N-body (in integers) problem to Hilbert's Tenth Problem, which was found to be unsolveable in 1973; "Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem I've looked at a lot of papers about N-body simulation codes using floating point arithmetic, and one of the biggest problems is to preserve the various conservation laws during the simulation. Even the slightest losses or gains in one of the conservation laws leads to non-physical behavior over a large number of iterations. These non-conservations are a bigger problem than accuracy if you are looking for the characteristics of long-term behavior -- e.g., whether one or more bodies is ejected from the system & never comes back. BTW, in my description, I talked about the representation of the _position_ coordinates, but not the representation of the _velocity_ coordinates. These, too, would have to be integers or rational numbers. While it probably makes the most sense to represent the position coordinates rather directly, the best representation of the velocity coordinates is very much up in the air. One might want to represent integral _inverse_ velocities, e.g., instead of the velocities themselves. At 08:27 PM 6/10/2011, Cordwell, William R wrote: >The planar Newtonian 2-body problem is solvable exactly by switching to the center of mass coordinate system. It is known that the general N > 2 body system is not solvable exactly. > >In the CM coordinate system, the two momentum vectors are opposite, so assuming planar should be no restriction. > >----- Original Message ----- >From: Henry Baker [mailto:hbaker1@pipeline.com] >Sent: Friday, June 10, 2011 07:18 PM >To: math-fun <math-fun(a)mailman.xmission.com> >Subject: [math-fun] Exact N-body simulations? > >All this talk of Gaussian integers reminded me of a problem that I'd dearly love to see solved. > >Consider N bodies, each with integer masses. Start them in initial positions in a 2D (for the >moment, there's plenty of complexity in 2D) which are on integer coordinates. The attractive >Newtonian gravitation force is proportional to m*m'/(dx^2+dy^2), so this force is rational. >We need to compute new positions for each of the bodies in such a way that all the conservation >laws are preserved exactly (if this is possible), yet the number of bits required doesn't grow >too quickly (see below). > >We need to preserve the total energy, we need to preserve momentum & angular momentum, & we >might like to preserve information (the system is reversible). > >Note that we don't necessarily require that delta-t be constant, although if it isn't, we may >need to have another counter to represent the "clock" -- i.e., the global time. > >I don't know how to do this even in the 2-body case, but the Minsky circle hack may provide >some guidance -- if we could get the two bodies to each draw a Minsky circle while revolving >around one another, then we'd have a good start on the problem. > >In addition to solving the problem of finite arithmetic, we also need to figure out what >happens in collisions. Taking a cue from Fredkin, we should probably have 100% elastic >collisions in order to ensure reversibility. > >How does this square with PoincarÃ© ? Well, I'm assuming that my N-body problem for finite >N is Turing complete, so with enough bodies and enough "tape" (empty space), we should be >able to compute any computable function. Minsky already provides one potential representation >with his "counter machines". So we want a reasonably "efficient" Turing Machine (or Counter >Machine) simulation so that we aren't wasting more than a fixed fraction of the bits required >to represent the state (N pairs of integers or N gaussian integers). > >Has anyone attempted this sort of exact simulation before?

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