July 2003 - mathfun@mailman.xmission.com

Re: [math-fun] odometer word game
by asimovd＠aol.com 06 Jul '03

06 Jul '03

hideous hideout

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Re: [math-fun] odometer word game
by Life1ine＠aol.com 06 Jul '03

06 Jul '03

Rich: HAL > IBM ;) RTW

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[math-fun] Annoyed
by David Wilson 06 Jul '03

06 Jul '03

A027424 annoys me. I keep thinking there must be a fast way to compute this sequence, but I can't find it. Another way to state my frustration: What is A027424(10^9)?

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RE: [math-fun] A027424
by Schroeppel, Richard 05 Jul '03

05 Jul '03

For those who haven't guessed, A027424(N) is the number of distinct integers in an NxN multiplication table. According to the accompanying text, (info from Pomerance) Erdos showed it's o(N^2), and Linnik & Vinogradov got O(N^2/(logN)^c) for some c>0; and there's a finer result in a 1988 book by Hall & Tenenbaum. An easy lower bound is to consider primes P > N/2, times anything < N. This gives N * (N/2logN) - ((N/2logN)^2)/2, after subtracting double counting of P*P'; or roughly N^2/2logN. I'd expect the large corner of the table, near N, to have mostly unique numbers (except for reflection); this might also give a lower bound. For computing the exact number, you might be able to work out the count of new numbers in row N, from the factorization of N. The work for the factors of each N is small, maybe logN or loglogN. Rich -----Original Message----- From: Allan C. Wechsler [mailto:acw@alum.mit.edu] Sent: Saturday, July 05, 2003 5:09 PM To: math-fun Subject: [math-fun] A027424 Is it not the case that a number appears in the N x N multiplication table precisely when none of its prime factors exceed N, and the number itself does not exceed N^2? Not that I'm claiming this helps to calculate A027424(N). -A _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Tumbling rigid body
by asimovd＠aol.com 05 Jul '03

05 Jul '03

Gene wrote: << The proper place to introduce elliptic functions is in a course in complex variables, >> I'm moved to mention that there are several opverlapping meanings for "elliptic functions". There are the Jacobi elliptic functions cn, sn, dn, so named since they were originally designed to compute the arclength of an ellipse. Cf. < http://mathworld.wolfram.com/JacobiEllipticFunctions.html>. There are also the more general just plain "elliptic functions", which are any doubly periodic meromorphic (analytic except for poles) functions on C, of which the Jacobi elliptic functions are a special case (as are the Weierstrass elliptic functions). Cf. <http://mathworld.wolfram.com/EllipticFunction.html>. (The general elliptic function can be thought of as a nonconstant homolorphic map from a Riemann surface of genus 1 to one of genus 0 (a torus to a sphere).) I'm not sure why these more general functions are called "elliptic". (Maybe the ellipse arclength functions just kept getting generalized and the name stuck?) In any case, a whole lot more about the elliptic integrals that underlie elliptic functions can be found at < http://mathworld.wolfram.com/EllipticIntegral.html>. --Dan

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Re: [math-fun] odometer word game
by asimovd＠aol.com 05 Jul '03

05 Jul '03

I wrote: << Coolest of all would be long odograms of common words that have no root in common. >> Here I meant to say "odograms of long common words that have no root in common." But if it's a long odogram (lots of consecutive words), so much the better! So ... Coolest of all would be long odograms of long common words that have no root in common. --Dan

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Re: [math-fun] triple star
by asimovd＠aol.com 05 Jul '03

05 Jul '03

Bill G. (rwg(a)spnet.com) writes: << Furthermore, if you run the "likely story" for a while, it will usually (eventually) fly back apart as in the forward time case. So the apparent time disambiguation seems to be an illusion due to insufficient duration of observation. ? >> Yes, this is in synch with the Ehrenfest urn model of thermodynamics. (Start with 2 urns, one with 50 white balls and the other with 50 black balls. At each time step, pick a ball at random from each urn and interchange them.) Initially, the balls almost certainly get more disordered. Wait long enough (very long!!!) and they'll return to their original state. But in fact at any state of reasonably low entropy (once they've recovered from their initial conditions), there will be no difference between the states in forward time vs. those in backward time -- belying the flat-out statement that "entropy increases". Even projecting the minimal-entropy initial state backwards in time will be statistically identical to projecting it forwards, of course, by symmetry. "Entropy increases" is not a law of physics, but rather a consequence of initial conditions. In our universe it can be traced back to our local stars' having "advanced" rather than "retarded" radiation, or more accurately to their radiation being emanated toward us in the same direction that we perceive ourselves to grow older. (So if there are stars with "retarded" radiation, i.e., being emanated in the time direction we call negative, their local life forms would see those stars just as we see ours, timewise.) --Dan

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Re: [math-fun] odometer word game
by asimovd＠aol.com 05 Jul '03

05 Jul '03

Now that I get it, I think that Odometer is a neat word game. My favorite examples are words that don't appear to have a root in common. Coolest of all would be long odograms of common words that have no root in common. --Dan

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Re: [math-fun] Tumbling rigid body
by Henry Baker 05 Jul '03

05 Jul '03

I'm curious as to why elliptic functions seem to be relegated to the if-we-have-time-this-semester sections of textbooks. 1. Are elliptic functions useful for closed form solutions of things other than tumbling bricks? 2. What specific feature about tumbling bricks would give a clue that elliptic functions were needed in its solution? At 09:34 AM 7/4/03 -0700, Eugene Salamin wrote: >I've written a paper which reviews the exact mathematical solution for >the tumbling motion of torque-free rigid body whose three principal >moments of inertia are different. This has all been known for over 100 >years, and is adapted from Whittaker's "Analytical Dynamics". It's an >orgy of elliptic functions and theta functions. I'll be delighted to >email the paper to anyone who wants it. It is a Microsoft Word >document. > >--- "R. William Gosper" <rwg(a)spnet.com> wrote: >> Wouter suggested http://www.myphysicslab.com/collision.html . >> That's very nice, but I meant one 3D brick tumbling without >> collisions in 0g. >> Also, can anyone suggest a url detailing the Runge-Kutta >> iteration for an n body problem (n small)? >> --rwg

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Re: [math-fun] triple star
by Henry Baker 05 Jul '03

05 Jul '03

My intuition, which is nowhere near a proof: All 2-body collisions/singularities can be replaced by perfectly elastic bounces, which don't lose any information. Three-body collisions/singularities cannot be "regularized" at all, and hence would appear to "lose information". Since all non-collisions are approximations (more or less) to singularities, it would appear that something much more complex is going on to either insert new information, or delete existing information. I think that one way to attack this problem is via information flow. At 12:28 PM 7/5/03 -0700, R. William Gosper wrote: >Henry>Answer: triple systems _are_ highly unstable, and _do_ fly apart. > >But not always. There was a recent claim that three equal masses can >"braid" forever, even with perturbations as "large" as 10^-5. > >> As you >>point out, the "stable" ones are hierarchical -- there's such a vast scale >>difference between the masses, that the small perturbations take a long time >>to build up to something meaningful. > >What I wish I could try is two equal masses in tight circles simulating >a double mass making a circle of r/2 (r large) with the third mass at >r, with the signs of "both" (all three) angular velocities equal. Even >assuming small initial noncircularities and nonplanarities, do you think >this system will eventually decay? I guess we have to make the Newtonian >approximation, else even the two body problem decays.* > >How many bodies are necessary/sufficient to disambiguate time? E.g., if >we observe sixteen swarm around for a while, whereupon fifteen crystallize >into a motionless triangle while the sixteenth departs like a shot, our >suspicions might be aroused, especially if it was the white one. > >Reconsider now the low angular momentum case, which we see usually flies apart >fairly quickly. This seems to disambiguate the sign of time, even though the >physics is reversible. I.e., if you time-reverse a "fly-apart" event, two >stars in tight orbit are cruising along when this rogue star comes out of >nowhere and "collides inelastically" with the pair, converting its kinetic >energy to the gravitational potential of the bound triple system. A likely >story. Yet that is exactly what happens in the "almost fly-apart" scenario >where the single and the pair don't quite escape, and eventually recombine. >Furthermore, if you run the "likely story" for a while, it will usually >(eventually) fly back apart as in the forward time case. So the apparent >time disambiguation seems to be an illusion due to insufficient duration >of observation. ? > >>There's another problem with stars that isn't true of point masses -- you can >>bring point masses arbitrarily close together, but when you do this with stars, >>they get pulled apart like taffy. > >And if there's no lower limit on how tightly two can orbit, there is no limit on >the energy with which the third can be ejected. >(Moments after I typed this, the applet obliging exhibited its most violent >ejection in my experience--two bodies nearly superposed, with no visible >orbital motion, shot off to the left, the third twice as fast to the right, >leaving afterimagess about a quarter inch apart. Minor puzzle: Why >was there no noticeable vertical vibration in the tight pair proportional >to their perceptible horizontal separation? I can think of three >explanations: >1. A stroboscopic coincidence; >2. Only one body moves during a given time step; >3. The bodies were displayed superimposed, but at slightly different > times, tricking my motion-tracking into seeing an illusory spread. > (I failed to check the afterimages before their dimness surpassed mine.) > --rwg > >*Who did this to Pope? >Nature and nature's laws lay hid in night, >God said, "Let Newton be," and all was light. > >It did not last; the devil howling "Ho! >Let Einstein be!" restored the status quo.

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