July 2003 - mathfun@mailman.xmission.com

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

05 Jul '03

Bill: I did some reading & searching about these kinds of issues several years ago. Answer: triple systems _are_ highly unstable, and _do_ fly apart. 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. 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 taffee (sp?). I think that the star systems that we see in reasonable quantities are those which _have_ managed to remain "stable" over a reasonable period of time. So to find the "unstable" ones, you have to look quickly before they fall apart. There's been some progress in improving simulations to preserve important quantities -- e.g., energy, angular momentum, etc. I don't know whether it's possible to _guarantee_ such preservation to a large number of digits, though. Hopefully, someone on this list will give a better report on the state of the art of conservation of "constants of integration" in modern simulation algorithms. I have a small simulation on my Palm Pilot (called "asteroids" or some such). I downloaded it from one of the Palm Pilot freeware download sites. (BTW, I'd love a Palm Pilot version of the original MIT space war, if anyone is aware of such a thing.) ------ At 11:52 AM 6/27/03 -0700, R. William Gosper wrote: >http://www.freewebz.com/vitaliy/triApplet/triGrav.html >is an interesting three-body simulation applet with colored, fading >trails. Unfortunately, you don't get to play with the initial >conditions, which are equal (point) masses, zero net momentum, zero net >angular momentum (wrt origin), fixed or slightly randomized initial >positions, and slightly randomized initial velocities. Chaos quickly >magnifies the initial perturbations into qualitatively different runs. > >The bodies spend some of their time "braiding", but most of their time >approximating two "two body problems", with a pair of bodies in a tight >orbit simulating a double mass about which the third body describes a >larger orbit that frequently crosses the screen boundaries. However, >due to the zero initial angular momentum, these larger orbits are highly >eccentric and terminate after only one period in a sort of "elastic >collision" which shuffles the orbital relationships. > >Sooner or later you have to restart because one of these >collisions leaves two bodies in such a tight, low energy orbit >that the liberated kinetic energy imparts the third body with >escape velocity, the tight pair flying off in the opposite >direction. (Occasionally, there is not quite escape velocity, > and the bodies return after several minutes.) > >So why don't triple star systems fly apart, too? One obvious >difference is that "collisions" are vastly less frequent in 3-space >than 2-space. Another is that heavenly masses are typically highly >unequal (e.g., Sun, Earth, Moon). Another possibility is spurious >energy buildup in the simulation from accumulated Runge-Kutta errors. >A sensitive test of this would be if we could initialize to the >recently discovered (barely) stable braiding orbit. > >But I think the main source of instability is the absence of net >angular momentum. My guess is that the author nulled out the >angular momentum because stability is visually boring compared >to braiding and "colliding", even at the cost of impermanence. >Or perhaps someone has found a chaotic triple star that proves >me wrong? >--rwg >PS: Does anyone know of tumbling brick/asteroid animation on the Web?

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Re: [math-fun] NY State math exam flapment
by Henry Baker 05 Jul '03

05 Jul '03

The original article (Thompson, R.B. 1998. Designing a baseball cover. College Mathematics Journal 29(January):48) was pretty good. Unfortunately, it does not appear to be online anymore, since the widely referenced mathsoft site is gone. The author apparently works for the U. of Arizona, but he doesn't have it online either. Perhaps someone on this list can get Dr. Thompson to post it on his web site? At 11:01 AM 7/4/03 -0400, Bernie Cosell wrote: >On 4 Jul 2003 at 7:25, Henry Baker wrote: > >> I hope that you know enough about baseball to appreciate the >> elegance of the baseball's seam: >> >> http://www.sciencenews.org/sn_arc98/4_11_98/mathland.htm > >What property of the 'seam' problem for the baseball precludes just an >equator for the seam as a possible solution? Is there some "minimization >of necessary stretch" or other criterion that drives the more complicated >shape? > > /Bernie\ > >-- >Bernie Cosell Fantasy Farm Fibers >mailto:bernie@fantasyfarm.com Pearisburg, VA > --> Too many people, too few sheep <--

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Re: [math-fun] NY State math exam flapment
by Henry Baker 05 Jul '03

05 Jul '03

At 12:54 PM 7/4/03 -0400, Bernie Cosell wrote: >On 4 Jul 2003 at 9:13, Henry Baker wrote: >> At 11:11 AM 7/4/03 -0400, William Thurston wrote: >> >20. How many different five-member teams can be made from a group of >> >eight students, if each student has an equal chance of being chosen? >> >(1) 40 (2) 56 (3) 336 (4) 6720 >> > >> > I think the correct answer is not among these choices: 1 It's very >> >poorly phrased, being neither colloquial English or typical >> >mathematical/combinatorial English. >> >> I agree that there is a reading that would give the choice "1", but since that >> choice is not available, you have to assume that that interpretation is not >> intended. > >Could you elaborate on how to get the choice '1'? Hmm...well... I guess >you could say that you can only make *ONE* five-man team, but you could >do it in 56 different ways... Is that what you had in minde? Yes. Once you had made one team, you don't have enough people left over to make another. BTW, the "equal chance" bit is a red herring. The probabilities don't matter, so long as none of them are zero. I think that this is thrown in there to indicate that you should put on your "combinatorial & probability" hat for this question.

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Re: [math-fun] NY State math exam flapment
by asimovd＠aol.com 04 Jul '03

04 Jul '03

Two comments. One, the stem-and-leaf plots were invented by the late statistician & topologist John Tukey and described in his book Exploratory Data Analysis. If it's not part of the curriculum in NY state, then I find it moronic that this was put on the exam; if it is part of the curriculum, then it seems quite reasonable. Second, having recently created a standardized test under contract, and learned what level of mathematical savvy many of the people in this business have, I strongly suspect that "each student has an equal chance of being picked" was added to "How many teams of 5 person can be chosen from a group of 8 students?" solely because the question writer actually thought it was relevant. --Dan A.

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Re: [math-fun] NY State math exam flapment
by Henry Baker 04 Jul '03

04 Jul '03

Interspersed comments, below: At 11:11 AM 7/4/03 -0400, William Thurston wrote: >I thought I'd add a little more raw material from the exam. I'd attach >the PDF version if it were allowed, but instead I'll just type in a >couple more questions: > >14. If the expression 3 - 4^2 + 6/2 is evaluated, what would be done >last? > (1) subtracting, (2) squaring, (3) adding, (4) dividing > > I guess it's easy for us to guess what they want --- but it seems >silly, and also it assumes you disregard the commutative rule for >addition. It's significant for compilers to analyze computations and do >them in efficient, rather than most obvious orders. I agree that this is treacherous, especially because they don't use parentheses. There have been computer languages in which right-to-left has been guaranteed, instead of the classroom left-to-right. I would have been happier if they had left out subtracting, as there is an additional ambiguity about a-b vs. a+(-b). >20. How many different five-member teams can be made from a group of >eight students, if each student has an equal chance of being chosen? >(1) 40 (2) 56 (3) 336 (4) 6720 > > I think the correct answer is not among these choices: 1 It's very >poorly phrased, being neither colloquial English or typical >mathematical/combinatorial English. I agree that there is a reading that would give the choice "1", but since that choice is not available, you have to assume that that interpretation is not intended. >21. The student scores on Mrs. Frederick's mathematics test are shown >on the stem and leaf plot below: > 4 | 3 > 6 | 0 5 5 7 9 > 7 | 2 5 6 8 9 9 9 > 9 | 0 1 2 5 9 > >Find the median of these scores. > > --this is a new one for me. I'm clueless, even seeing the key, which >I'm not printing. Ditto. It would be interesting to see how many Fields medal winners would be able to understand this one.

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[math-fun] triple star
by R. William Gosper 04 Jul '03

04 Jul '03

http://www.freewebz.com/vitaliy/triApplet/triGrav.html is an interesting three-body simulation applet with colored, fading trails. Unfortunately, you don't get to play with the initial conditions, which are equal (point) masses, zero net momentum, zero net angular momentum (wrt origin), fixed or slightly randomized initial positions, and slightly randomized initial velocities. Chaos quickly magnifies the initial perturbations into qualitatively different runs. The bodies spend some of their time "braiding", but most of their time approximating two "two body problems", with a pair of bodies in a tight orbit simulating a double mass about which the third body describes a larger orbit that frequently crosses the screen boundaries. However, due to the zero initial angular momentum, these larger orbits are highly eccentric and terminate after only one period in a sort of "elastic collision" which shuffles the orbital relationships. Sooner or later you have to restart because one of these collisions leaves two bodies in such a tight, low energy orbit that the liberated kinetic energy imparts the third body with escape velocity, the tight pair flying off in the opposite direction. (Occasionally, there is not quite escape velocity, and the bodies return after several minutes.) So why don't triple star systems fly apart, too? One obvious difference is that "collisions" are vastly less frequent in 3-space than 2-space. Another is that heavenly masses are typically highly unequal (e.g., Sun, Earth, Moon). Another possibility is spurious energy buildup in the simulation from accumulated Runge-Kutta errors. A sensitive test of this would be if we could initialize to the recently discovered (barely) stable braiding orbit. But I think the main source of instability is the absence of net angular momentum. My guess is that the author nulled out the angular momentum because stability is visually boring compared to braiding and "colliding", even at the cost of impermanence. Or perhaps someone has found a chaotic triple star that proves me wrong? --rwg PS: Does anyone know of tumbling brick/asteroid animation on the Web?

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Re: [math-fun] NY State math exam flapment
by Henry Baker 04 Jul '03

04 Jul '03

But you're actually making an important point. These tests _do_ measure cultural issues. So what? If the issue is to get along in _this_ society -- i.e., go to college, get a job, etc. -- then cultural issues _do_ matter. Rather than trying to eliminate them from the test, let's put them back into the curriculum. Let's take everyone to a baseball game, so that they know what "America's pastime" is all about. This business of trying to protect immigrants from American culture is B.S. I recall several Vietnamese immigrant kids who managed to learn enough English in 2-3 years to _win_ the spelling bee! The best time to learn culture is when you're a kid -- it comes effortlessly. >From what I can tell, these tests are already doing what they are intended to do: determine who can function in our society. So what if I went to France or Germany and couldn't perform on their math tests. If I were planning to live there, then I would expect to have to learn something about their language & culture. My brother-in-law studied physics in Germany, and they didn't provide any "special education" for him -- he had to learn German, and fast. A doctor friend went to medical school in Switzerland. Ditto. He had to learn French, and quickly. I don't see Saudia Arabia making any special provisions for non-Moslems. We're not doing people any favors through "race-norming" or "culture-norming", because the society at large isn't "race-normed" or "culture-normed". We're just putting off the inevitable day when they discover that they have to learn the same stuff as everyone else. At 02:39 PM 7/3/03 -0400, William Thurston wrote: >The main issue with "common sense" needed for word problems is: common >to who? It's quite hard to filter out the extraneous cultural issues >in word problems. I was on the ETS math GRE panel for a while. ETS >tries very hard to do it, and their tests are relatively high quality, >but still there are many cultural biases hard to eliminate. ETS does a >lot of "validity testing" which I don't think the regents exam is >subject to---i.e. no matter how hard you try to think from the student >point of view, you can't really tell how students might interpret or >stumble or misinterpret a question until you try it out on them. One >of the complaints about the regents exam is that it is only NY, so the >test writing companies aren't financially motivated to put much effort >into the test. It's not necessarily as predictable from year to year as >other standardized tests. > > In the juice container / straw instance: do the students in question >actually encounter juice containers like the ones described? Do they >interpret the question to mean you actually try to squeeze the straw >through the hole into the container? > >In other common sorts of questions, do the students actually play or >at least watch baseball? What is their common sense notion about the >typical size of a house? What variant of English do they use everyday? > I once had some students who were struggling with word problems in >calculus ... it turned out the issue was things like "needle and >thread". Even though they spoke fluent English they were from >Indonesia and terms like this which are everyday words for some people >were unfamiliar. The math itself was easy for them, and with a little >personal interaction they did well in the end.

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Re: [math-fun] NY State math exam flapment
by Henry Baker 04 Jul '03

04 Jul '03

I hope that you know enough about baseball to appreciate the elegance of the baseball's seam: http://www.sciencenews.org/sn_arc98/4_11_98/mathland.htm At 12:53 PM 7/4/03 +0200, Joerg Arndt wrote: >* Henry Baker <hbaker1(a)pipeline.com> [Jul 03. 2003 21:47]: >> But you're actually making an important point. These tests _do_ measure >> cultural issues. So what? If the issue is to get along in _this_ society -- >> i.e., go to college, get a job, etc. -- then cultural issues _do_ matter. >> Rather than trying to eliminate them from the test, let's put them back >> into the curriculum. Let's take everyone to a baseball game, so that they >> know what "America's pastime" is all about. > >Agreed. > >However, in math books I find the (not so rare) baseball examples >pretty annoying. It should in in almost all cases be possible to >find an analogue that doesn't use baseball. >(I am german & have absolute no clue about baseball). > >-- >p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0 >Life is hard and then you die.

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Re: [math-fun] triple star
by Henry Baker 04 Jul '03

04 Jul '03

http://quanta.kyunghee.ac.kr/physics/n-body%20problem/nbody.htm At 07:04 PM 7/3/03 -0700, R. William Gosper wrote: >Also, can anyone suggest a url detailing the Runge-Kutta >iteration for an n body problem (n small)? >--rwg

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[math-fun] misc
by Richard Schroeppel 04 Jul '03

04 Jul '03

There's another well-known concept of "positive" for algebraic numbers. I think it's named "totally positive", and it means the number is positive (in the reals) for all choices of embedding: I.e., all conjugates are positive. 1+sqrt2 fails here because the alternate embedding sqrt2 -> -1.414 gives a negative value (-.414). But 2+sqrt2 is totally positive. As is 2-sqrt2. It seems that totally positives are closed under + and *. I don't know what you do if there are complex embeddings. (cbrt2) [Puzzle: Identify interesting sets of complex numbers that are closed under + and *. (Right-half-plane works for +, fails for *.) Find a natural condition to exclude the trivial answer "start with any transcendental and close under +*".] ---- Two recent astronomy goodies: http://www.nso.edu/press/AO/ao.html Adaptive Optics View of the Sun Astronomers have developed the flex-mirror idea for a solar telescope. The hard problem is lack of sharp image contrasts to drive the servo-algorithm. The site has a couple of short movie clips of the surfce of the sun boiling, with convection cells a few hundred miles across. Must be some kind of magnetic viscosity. With a surfce gravity of 38G, you'd think the surface would be nearly flat. [Puzzle: Design a solar probe that will last for, say, a day. And figure out a way for it to communicate its measurements.] http://www.sciencedaily.com/releases/2003/06/030619075759.htm Speed of Gravity A few months ago, a group announced they'd measured the speed of gravity by looking at starlight bent by Jupiter's gravity. The above note is another group saying 'Tain't So. --- Hilarie and I have been enjoying the (solitaire) game Chromatron. "The first batch of puzzles is free." --- One of my favorite word puzzles is finding "consecutive" words, where the last letter is incremented to make a new word. CONVEX CONVEY TARN TARO TARP COMER COMES COMET MODEL MODEM CLOSER CLOSES CLOSET POSER POSES POSET (if we allow a bit of math jargon) The problem is to find more examples. The challenges are longer words, and longer sequences. I'd also like a true wraparound example, with Z incremented to A and the carry propagated to the next letter: OZ PA --- The GIMPS crowd (Mersenne prime search) recently had a false alarm. Someone's machine reported a final Lucas residue of 0. Woltman & co. insisted on keeping quiet until the double checks were completed, and the policy proved out. (I would shout it from the treetops. Oops.) Two different checking machines had nonzero final residues that matched each other. GIMPS errors happen: the GIMPS players run their machines hot (for speed), and O(1%) of the residues turn out wrong when double checked. And there's also the floating point used in the arithmetic, which occasionally has an undetected "overflow". [Floating point values are mapped to the nearest integer as part of the FFT multiplication.] But it's hard to imagine an arithmetic bug that would produce a 0 final residue (probability verging on infinitesimal). George has found another possible explanation, based on a NaN sneaking into the FFT somehow, and propagating. He's adding more checking of intermediate results to the program. --- New ECM (Elliptic Curve Method) factoring record: M997 (2^997-1) has the 57-digit divisor 167560816514084819488737767976263150405095191554732902607 found by Bruce Dodson. The cofactor is a probable prime. Rich rcs(a)cs.arizona.edu

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