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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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[math-fun] Is Z^{>=} a rig?
by Mike Stay 03 Jul '03

03 Jul '03

I haven't been following the thread very closely, but I noticed you were trying to come up with a name for something defined over the naturals. Does this definition fit your structure? http://www.wikipedia.org/wiki/Rig_(algebra) -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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Re: [math-fun] Two problems related to recent topics
by asimovd＠aol.com 03 Jul '03

03 Jul '03

1. Suppose a cylindrical can C is inscribed in a unit cube so its axis lies along the main diagonal of the cube. Its radius can be anywhere between 0 (vol(C) = 0) and the radius of the circle inscribed in the hexagonal cross-section of the cube (vol(C) = 0). QUESTION: What radius maximizes the can's volume? 2. Let Z+ be the positive integers and let a > 0 be an irrational real. Define F: Z+ --> Z+ via F(n) = #( [n,n+1) int { K + L*a : K,L are in Z+} ). QUESTION: Find an asymptotic expression for F(n) as n --> oo. --Dan A.

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

03 Jul '03

Bill G. wrote: << Wouter suggested http://www.myphysicslab.com/collision.html . That's very nice, but I meant one 3D brick tumbling without collisions in 0g. >> I wrote software (for SGI machines, using GL) to do exactly this in 1987. I might be able to dig it up and it *might* be compatible with OpenGL, but I don't know what the chances are. (Maybe someone else on the list still has a copy of it handy?) Learned a lot about the Jacobi elliptic functions sn, cn, dn in the process, most of which I've forgotten by now. Most interesting thing was to be able to graphically duplicate with real-time computation what happens when one tries to flip a generic rectangular solid through the air 180 degrees about its middle axis. If you aren't familiar with this, I won't spoil your surprise. Go get a typical book, (WLOG size A x B x C with A < B < C), keep it from opening with a strong rubber band, and try to flip it through the air 180 degrees about its B axis. --Dan A.

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Re: [math-fun] NY State math exam flapment
by Dan Hoey 03 Jul '03

03 Jul '03

Henry Baker <hbaker1(a)pipeline.com> > From what I can tell, these tests are already doing what they are > intended to do: determine who can function in our society. In that case, the "Math A Regents exam" is not a mathematics test, so your remark is misdirected. This is a list about mathematics, not cultural integration tests. Not that your departure from the topic is comparable to the reek of gas about "flaming liberals" we got from the flaming ... whatever ... earlier today. That one belongs on a bus--they take any kind of rant you like there. Meanwhile, I'm intrigued by the straw problem. So far, the best I've done is to express the length as sqrt(tt+uu+vv) where (t,u,v) satisfy equations (x-t)^2 (tt+uu+vv) = 4 r^2 (uu+vv) (y-u)^2 (tt+uu+vv) = 4 r^2 (tt+vv) (z-v)^2 (tt+uu+vv) = 4 r^2 (tt+uu) in the box dimensions (x,y,z) and the straw radius r. Adding them up, I'm startled (possibly not for the first time) that any way you lean a disc of radius r on the floor against two walls, the center will be sqrt(2)r from the corner. Is there an easy geometrical proof? But I'm pretty much flummoxed about solving the whole system. Maybe I should have gone to high school in New York. Dan Hoey(a)AIC.NRL.Navy.Mil

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Re: [math-fun] Re: Z*(SqrtN)
by Dan Hoey 03 Jul '03

03 Jul '03

Marc LeBrun <mlb(a)fxpt.com> wrote > > I take it "Z*" is the nonnegative integers. > Yes. I've never much liked that notation either. Is there a better > one? I've also considered using Z0, N0 and even U (for unsigned) but Z* > seems pretty prevalent, despite its opacity. I like Z^(>=) (i.e. Z superscript greater-than-or-equal). Then you can use things like Z^(>1) for {2,3,...}. Let a countable number of flowers bloom! > >>=Henry Baker > >> "algebraic natural numbers" or "natural algebraic numbers" ?? > > Cute name, but I'd rather use that for the positive elements of > > Z(SqrtD) (or the nonnegative ones, in France and set theory). > The "non-negative algebraic integers" might be construed to include > the undesired Sqrt2-1, which is positive but has a negative > component. Yes, that's what I meant. I mostly like this definition of natural because it leaves useful definitions for "supernatural" and "subnatural". Of course, it's mostly because I get a kick out of naming "supernatural numbers". I imagine Knuth got a similar kick out of naming Conway's "surreal numbers". > > Then the elements of Z(SqrtD) with both components positive could > > be called "supernatural", and those with at least one component > > positive could be called "subnatural". > Isn't "natural" supposed to connote a strictly positive integer? In Germany and algebra, yes. In France and number theory, zero is natural. That's of course a gross (and possibly geographically incorrect) simplification of the zerophilia question, but basically you can't count on a particular meaning of "natural" unless you define it. For most natural purposes, it doesn't really matter--you can get a peano with or without the "Z" key, and you just have to transpose music written for the other kind. > I would think "natural algebraic numbers" would refer to only those > numbers with all components positive integers. I guess if you like N = Z^(>), that's a good way of thinking. In that case, I'd call Z^(>=) the "znatural numbers". Just be careful saying it, because people start slapping for mosquitos. > Assuming that, I guess it'd make (some kind of) sense to call the > superset got by adjoining the ("unnatural"?<;-) elements with zero > components the "supernatural algebraic numbers". I still like "super-" = both components, "sub-" = either component definition. Then if you want both components znatural, you're talking about the superznatural part of Z(sqrt2). Dan Hoey(a)AIC.NRL.Navy.Mil

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[math-fun] Mathematics testing
by Eugene Salamin 03 Jul '03

03 Jul '03

I have two suggestions for improving the art of mathematics testing. 1. Cultural bias. Formulate the questions culture-independently. For example "What is the length of the diagonal of a 7 by 11 by 13 inch rectangular solid?" 2. Incorrectly stated problems, incorrect official answer, ambiguous interpretations. Post the exam questions and multiple choice answers on the internet, so anybody can criticize them. Build up a huge inventory of questions, all publicly available. Badly worded or ambiguous questions are fixed or removed. Questions are at all levels of difficulty. Exam administrators choose exam questions from this validated library. Students can access the library as part of their exam preparation. Will students memorize answers instead of learning the concepts? Perhaps, but by the time the library reaches several million questions, even the laziest student will be motivated to learn algorithms and concepts. Design the process so that teaching mathematics and teaching to the exam become coincident. __________________________________ Do you Yahoo!? SBC Yahoo! DSL - Now only $29.95 per month! http://sbc.yahoo.com

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Re: [math-fun] The natural numbers start with 2
by Marc LeBrun 03 Jul '03

03 Jul '03

>=Willem Nienhuys <wsadjw(a)rw7.urc.tue.nl> > The natural numbers start with 2 (TWO). Didn't the ancient Greeks seriously take something like this position? Which reminds me to ask: How did those guys actually perform divisions? Was there some long-division like algorithm used in antiquity, or did they do something else? Of course a classical author might just state the result without showing the work, but there must be some leftover scraps of papyrus or something lying around...

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

03 Jul '03

I haven't had time to look at the test itself, so I will rely on your thoughts. Word problems are _always_ somewhat ambiguous; I think that that is the point of them. They test the student's ability to use common sense and abstract the important elements. That's why the easiest way to confuse student in word problems is to give them extraneous information. They keep trying to incorporate all the information, and all of the information isn't necessary. Are you telling me that in this era where _everyone_ practises for tests, using Kaplan, internet, whatever, that students aren't prepared for these kinds of tests? I keep hearing stories about teachers teaching _only_ for the tests. One would expect that under these conditions, the students would do well on the tests, but still not know any math. But we have an even worse situation -- they don't do well on the tests, and they don't know math, either. Perhaps students did better (as when we were in high school) when they didn't actually "study" for the tests. At 08:22 AM 7/3/03 -0400, William Thurston wrote: >On Thursday, July 3, 2003, at 12:35 AM, Henry Baker wrote: > >I looked at the exam on the web. There were many questions that to me >seemed poorly designed. This particular question as you paraphrased it >may be something that you think most people *should* be able to do, but >in reality I think few high school graduates actually master it, since, >in particular, solid geometry is generally not taught. The actual >question was about a juice container something like 2" x 4" x 5", and >it asked how long a straw would fit diagonally, to the nearest tenth of >an inch. A straw is not the same as a line, and the thickness of the >straw (not specified, of course, and *intended* to be disregarded) >would actually be likely to change the answer. People who've been >taught a particular mechanical formula for this particular question can >do it, but otherwise, it's likely to be a time-consuming and futile >exercise, compounded by trying to calculate square roots presumably >without a calculator.

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