August 2007 - mathfun@mailman.xmission.com

Re: [math-fun] How we "do mathematics"
by gale＠Math.Berkeley.EDU 06 Aug '07

06 Aug '07

> Dan, I'M AFRAID I didn't express myself clearly.I wasn't asking for a solution but rather about the mechanics of finding it. Did either of you use pen or pencil or paper at any point? I had to draw two pictures before I could see what the equations had to be. Maybe you were able to do all these things in your head in which case my speculations are off base. [SPOILER] > > David writes: > > << > It's often said that among the sciences, supporting mathematics is a great > bargain because unlike other sciences it does not depend on using advanced > and expensive equipment. This is true. of course, but I claim mathematics > is nevertheless technology dependent on a a quite marvelous invention > called a pencil (or perhaps a pen, or whatever Euclid and Pythagoras > used). > Try to imagine how one could do mathematics if there were no such devices. > Their usefulness is as a means of storage. If we couldn't write things > down > we'd have to keep everything in our heads at the same time which is hard. > Example. An extra credit problem in my high school algebra book went > something like this. > > Two racers run around a circular track at constant rates. If they run in > opposite directions (one clockwise, the other counter clockwise) they meet > every minute. If they run in the same direction they meet every hour. Find > the ratio of their speeds. > > If you don't already know the problem maybe some of you can do it in your > heads but if you can't, save the paper you used and see how many symbols > you wrote down? How many pictures you drew? >>> > > Let's see: > > 1 / f-s = K > 1 / f+s = 1 > f-s = 1/K > f+s = 1 > f/s = 2f / 2s = 1+1/K / 1-1/K > = K+1 / K-1 > = 61/59 > > which makes 61 symbols. (This seems a lot > more than ought to be necessary.) > > --Dan > > _______________________________________________ > 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] How we "do mathematics"
by Eric Angelini 06 Aug '07

06 Aug '07

> it's the kind of thing that throws people because it > sounds like there's not enough information. Old Boniface he took his cheer, Then he bored a hole through a solid sphere, Clear through the center, straight and strong, And the hole was just six inches long. Now tell me, when the end was gained, What volume in the sphere remained? Sounds like I haven't told enough, But I have, and the answer isn't tough! ---- > There also is a meta-theoretic answer to this puzzle. > Assume the puzzle can be solved. Then it must be solvable > with a hole of any diameter, even zero. But if you drill a > hole of zero diameter that is six inches long, you leave > behind the volume of a six inch diameter sphere. http://www.faqs.org/faqs/puzzles/archive/geometry/part1/

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RE: [math-fun] How we "do mathematics"
by Eric Angelini 06 Aug '07

06 Aug '07

> lots of math contest experience and high school teaching (...) A (very) old one I like : Two Death Valley marathonians friends complete their run together and seek shadow in their tent nearby the finish line. The first one had prepared a fresh 5-liter tank of water in a cooling box and his friend a similar 3-liter one. They were about to start drinking when a journalist enters the tent desperately asking for water. OK, the water is divided into three equal parts -- and drunk. The thankful journalist insists to pay his share and leaves 8 dollars to be split between the two friends. How? ---- (...) and high school teaching experience, do this kind of stuff in our heads

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Re: [math-fun] How we "do mathematics"
by Dan Asimov 05 Aug '07

05 Aug '07

[SPOILER] David writes: << It's often said that among the sciences, supporting mathematics is a great bargain because unlike other sciences it does not depend on using advanced and expensive equipment. This is true. of course, but I claim mathematics is nevertheless technology dependent on a a quite marvelous invention called a pencil (or perhaps a pen, or whatever Euclid and Pythagoras used). Try to imagine how one could do mathematics if there were no such devices. Their usefulness is as a means of storage. If we couldn't write things down we'd have to keep everything in our heads at the same time which is hard. Example. An extra credit problem in my high school algebra book went something like this. Two racers run around a circular track at constant rates. If they run in opposite directions (one clockwise, the other counter clockwise) they meet every minute. If they run in the same direction they meet every hour. Find the ratio of their speeds. If you don't already know the problem maybe some of you can do it in your heads but if you can't, save the paper you used and see how many symbols you wrote down? How many pictures you drew? >> Let's see: 1 / f-s = K 1 / f+s = 1 f-s = 1/K f+s = 1 f/s = 2f / 2s = 1+1/K / 1-1/K = K+1 / K-1 = 61/59 which makes 61 symbols. (This seems a lot more than ought to be necessary.) --Dan

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[math-fun] Letter from Grozny
by Fred lunnon 05 Aug '07

05 Aug '07

I wonder if any other math-funners have been the recipients of a curious example of mathematical spam, recently deposited in my in-box with the customary ominous thud. Among other things, its covering gloss observes that the (putative) author has spent some 30 years perfecting his solution to the "problem of finding exponents of prime numbers", and invites the reader's opinion of the attached 47-page treatise on the topic. Painstaking inspection of this document eventually reveals that the "exponent" under discussion is (minus) the p-adic valuation m! [the maximum power of p dividing factorial m]. I have on reflection decided that it might be both kinder and wiser to refrain from suggesting that he consult Graham, Knuth & Patashnik's "Concrete Mathematics", which I believe quotes a theorem of Legendre from 1830 to the effect that this exponent equals (m - s_p(m))/(p - 1), where s_p(m) denotes the sum of the digits of m in base p. But I can't help feeling that it is a great shame he could not have devoted just a few weeks of his 30 years to reading this wonderful book, from which he might just have actually learnt some beautiful and useful mathematics; instead of which, he seems frozen in a state of perpetual numerical infantilism. "A little knowledge is a dangerous thing ..." ----------------------------------------------------------------------- Oddly, I had also been contemplating exactly the same problem for the previous few days; or at any rate an extension of it. Several authors have investigated divisibility properties of Stirling numbers of the second kind S(n, m); but what about the first kind, s(m, n)? Since s(m, 0) = m!, it's reasonable to enquire whether Legendre's classic result might generalise. The motivation for looking at this stems from divisibility of Bell numbers B(n). From the combinatorial definitions, B(n) = \sum_m S(n, m), for n >= 0. And extending the Stirling number recurrence S(n, m) = S(n-1, m-1) + m*S(n-1, m) backwards reveals the little-known law S(-n, -m)*s(m, n) = (-1)^(n+m), permitting both kinds to be defined for all n,m [but don't tell Maple, which insouciantly returns zero for all negative first arguments]. Feeding back to the first equation then suggests defining B(n) = \sum_m |s(m, -n)|, for n < 0. The sum is plainly infinite in the usual metric, but converges p-adically for all p, with m terms giving in practice nearly m/(p-1) base-p digits. Now B(n) is already known to be periodic mod p^t for n >= 0; with the extension, the periodicity appears to extend to all n. Using the fact that S(n, n-k) (extended) is polynomial in n of degree 2*k, B(n) can be extended to a continuous function of a continuous variable; p-adically this function is almost-periodic. Fred Lunnon [03/08/07]

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[math-fun] How we "do mathematics"
by David Gale 04 Aug '07

04 Aug '07

It's often said that among the sciences, supporting mathematics is a great bargain because unlike other sciences it does not depend on using advanced and expensive equipment. This is true. of course, but I claim mathematics is nevertheless technology dependent on a a quite marvelous invention called a pencil (or perhaps a pen, or whatever Euclid and Pythagoras used). Try to imagine how one could do mathematics if there were no such devices. Their usefulness is as a means of storage. If we couldn't write things down we'd have to keep everything in our heads at the same time which is hard. Example. An extra credit problem in my high school algebra book went something like this. Two racers run around a circular track at constant rates. If they run in opposite directions (one clockwise, the other counter clockwise) they meet every minute. If they run in the same direction they meet every hour. Find the ratio of their speeds. If you don't already know the problem maybe some of you can do it in your heads but if you can't, save the paper you used and see how many symbols you wrote down? How many pictures you drew?

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Re: [math-fun] Letter from Grozny
by Eugene Salamin 04 Aug '07

04 Aug '07

----- Original Message ---- From: Fred lunnon <fred.lunnon(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Saturday, August 4, 2007 6:37:26 AM Subject: Re: [math-fun] Letter from Grozny On 8/4/07, David Wilson <davidwwilson(a)comcast.net> wrote: > Most people seem aware of their mathematical limitations, but a minority of > circle-squarers, crackpots, and isolates of varying proficiency convince > themselves that they are breaking new ground when in truth they are usually > either retreading the beaten path or forging a new path to La La Land. I've > had run-ins with Archimedes Plutonium and the Smarandache crowd and others > of this sort, and my advice is not to acknowledge them. The benefits of > dispossessing these people of their cherished illusions do not justify the > effort. Marc has no doubt studied this phenomenon in more detail than I have. For what it's worth, I'd say that most people are driven to study mathematics --- or to any other intellectual endeavour --- by a mixture of three motives: to indulge their curiosity; to earn approval from others; and (maybe) to control their environment. The people encountered on math-fun will as a rule be motivated by curiosity; the crackpots by approval. It's the difference between wanting to learn maths and just wanting to be "a mathematician". It follows that, in the unlikely event that one manages to put a serious dent in their self-image, they will respond by attempting to reinforce it with an ever-intensifying barrage of self-justification. [Or just possibly worse.] Which is, of course, exactly what happens. Fred Lunnon I've had encounters with several relativity deniers. Invariably, they fail to understand what the theory is saying, instead attacking a strawman that they presume to be relativity. Initially assuming they are genuinely interested in understanding relativity, I try to help them get onto the right path, but that has never succeeded. Sometimes the denier has a good engineering intuition, but it fails to generalize to the concepts needed for relativity. These people will devote a tremendous amount of time and effort into refuting relativity, or proving their own alternate theory, than they would need to learn the correct theory in the first place. My most recent encounter, a well known inventor, is stuck on the idea that a light wave has a frequency. While not disagreeing with the existence of a Doppler shift, he believes that relativity is saying that when an observer measures the frequency, the observation alters the frequency of the light. But how could the observer alter the frequency? So relativity is nonsense. He just can't adjust his intuition to realize that there is no inherent frequency, only measured frequencies. It was particularly difficult to carry on a dialog with this guy because he would just cherry pick what parts of relativity are true and which are false. He didn't appreciate that it is a body of linked logical inferences. Fortunately today, thanks to denier Petr Beckmann, we now have Galilean Electrodynamics, a Journal and a mutual admiration society of deniers and crusaders against relativity and quantum mechanics. They are not just a bunch of random idiots, in fact, they are largely physics PhDs and professors. Now I can politely brush off these troublemakers by pointing them towards this organization. Is there a similar organization for mathematical crackpots? Gene ____________________________________________________________________________________ Choose the right car based on your needs. Check out Yahoo! Autos new Car Finder tool. http://autos.yahoo.com/carfinder/

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[math-fun] surprising mathematics, Lissajou
by Simon Plouffe 04 Aug '07

04 Aug '07

Hello, this is about a certain <plug-in> which works with winamp, winamp being the music player for mp3 files. That plug-in is used when you choose to display those graphics that follow the rythm of any music that plays at the same time. One in particular is surprisingly nice : Milkdrop. The nice and very interesting thing is that the source code is fully OPEN, in other words ,what you see can be modified, reprogrammed, and even explored. Among all the ones I have seen (including itunes), that one is particularly elegant and very beautiful. if you want to see the code : (for pc's) just go to : C:\Program Files\Winamp\Plugins\Milkdrop all the generators are there. Some of them are quite intricate, I counted 547 source files, one of them has 774 lines of instructions! this is an orgy of formulas and the result is very nice to see on a screen. Then... can this be exploited to show nice things ? It uses Lissajous curves generated with cos and sin and simple transforms, see here for some examples : http://www.google.com/search?source=ig&hl=en&q=lissajou&btnG=Google+Search For those not knowing what I am talking about : You need to install WINAMP, it comes with a series of plugins for displaying the music, the plugin is milkdrop and can be activated with CONTROL-SHIT-K when a music is playing. For a briefing on that milkdrop plugin you can go here : http://www.nullsoft.com/free/milkdrop/screenshots.html the ones showed there are just some examples, in the 'automatic' mode (just leave it playing , no need to do anything), it displays a different pattern every 15 seconds. To change the pattern just press the SPACE bar. to display infos : press either F1, F2, F3, F4 to go out of that mode press esc. have fun! simon plouffe

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Re: [math-fun] Re: Letter from Grozny
by Marc LeBrun 03 Aug '07

03 Aug '07

> Harrumph! should have read > s(m, 1) = (-)^(m+1)*(m-1)! > Just testing you. Everybody failed. WFL Oh drat! I thought it might be s(m+1, 1) = (-)^m m! but was too shy to write in the margin.

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[math-fun] Re: Letter from Grozny
by Schroeppel, Richard 03 Aug '07

03 Aug '07

(mainly to Fred Lunnon) I got a similar letter, but didn't look beyond the accompanying text message. If you want to reply, (a difficult decision), then suggest either GKP, or Hardy&Wright, or maybe Melzac. I wonder what list he's working from? Rich

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