August 2007 - mathfun@mailman.xmission.com

[math-fun] Doppler shift and relativity
by Eugene Salamin 24 Aug '07

24 Aug '07

Let's look at the Doppler shift in light due to the velocity v of an observer. The wave has a phase phi = 2 pi (k x - f t), f/k = c, where k = 1/wavelength, f = frequency. If we use the Lorentz transformation, x = gamma (x' + v t'), t = gamma (t' + v x/c^2), gamma = 1/sqrt(1 - v^2/c^2), and do the algebra, then phi = 2 pi (k' x' - f' t'), k' = gamma (1 + v/c) k, f' = gamma (1 + v/c) f, f'/k' = f/k = c. So we see that there is a Doppler shift in both wavelength and frequency that is 1 + v/c to lowest order. But look at what the ether theory predicts. [The ether is a hypothetical medium filling the vacuum whose vibrations constitute light.] Now we use the Galilean transformation x = x' + v t, t = t'. We get instead k' = k, f' = (1 + v/c) f, c' = f'/k' = (1 + v/c) c. There is no wavelength Doppler shift. This is peculiar. A hundred years ago, experimentalists like Albert Michaelson worked very hard trying to detect the motion of the Earth through the ether, looking for effects of order (v/c)^2 = 10^-8. Why does the existence of the easily measured v/c wavelength Doppler shift not trivially demolish the ether theory of light? Gene ____________________________________________________________________________________ Be a better Heartthrob. Get better relationship answers from someone who knows. Yahoo! Answers - Check it out. http://answers.yahoo.com/dir/?link=list&sid=396545433

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Re: [math-fun] Algebra I (RiP)
by Eugene Salamin 23 Aug '07

23 Aug '07

----- Original Message ---- From: Bernie Cosell <bernie(a)fantasyfarm.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Thursday, August 23, 2007 10:57:17 AM Subject: Re: [math-fun] Algebra I (RiP) A related amusing one is WRT temperatures: referring to something being twice as hot as something else [or more perplexing, half as warm or twice as cold]. /Bernie\ This happened in the R&D department where I work. A temperature controller was specified to be stable to +/- 5%. When I explained the problem, everybody understood, and it was changed to +/- so many deg C. Gene ____________________________________________________________________________________ Be a better Globetrotter. Get better travel answers from someone who knows. Yahoo! Answers - Check it out. http://answers.yahoo.com/dir/?link=list&sid=396545469

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Re: [math-fun] Algebra I (RiP)
by Dan Asimov 23 Aug '07

23 Aug '07

Bernie wrote: << A related amusing one is WRT temperatures: referring to something being twice as hot as something else [or more perplexing, half as warm or twice as cold]. >> Which is a bit like using the phrase "most unsuperlative" ; or one could ask which imaginary animal is "most nonexistent" ; and it could be quite useful to ask what of all things is the "most vacuous" , given how complicated a vacuum seems to be in modern physics. --Dan

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Re: [math-fun] Algebra I (RiP)
by Dan Asimov 23 Aug '07

23 Aug '07

<< I think the "five times less" is plainly a case of the general theory that one should be liberal in what one accepts, and stringent in what one produces. If you can understand it, fine. But try not to propagate the problem. >> In my view accepting such ill-formed locutions helps propagate the problem. --Dan

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Re: [math-fun] Algebra I (RiP)
by Dan Asimov 23 Aug '07

23 Aug '07

Michael wrote: << How do people feel about this construction, in which you use the phrase "five times less" to colloquially mean what a formal description would have to call "one fifth as much" or the like? This came up on a blog I read, in which commenters seemed about equally divided between "This is just fine, it's completely obvious what the construction means" and "that's literally meaningless, no reasonable person should produce such a statement." At the time I wondered what the breakdown would be among a mathematician audience, and this seems like a chance to find out. >> I feel this is just fine in colloquial writing, and not at all good in a formal report (e.g., a newspaper article). This is because newspaper articles et al. are well known for royally screwing up clarity in such matters. (They will commonly use a phrase like "150% more" whether they mean 3/2 as much or half again as much.) (Michael: This wasn't by any chance Mark Liberman's Language Log", < http://itre.cis.upenn.edu/~myl/languagelog/ >, was it?) --Dan

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OOPS, Re: [math-fun] Algebra I (RiP)
by Dan Asimov 23 Aug '07

23 Aug '07

This should read: This is because newspaper articles et al. are well known for royally screwing up clarity in such matters. (They will commonly use a phrase like "150% more" whether they mean 3/2 as much or 5/2 as much.)

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[math-fun] regular polygon sums
by Bill Gosper 23 Aug '07

23 Aug '07

Replying to: -------------------- |Date: Mon, 31 Jul 2000 00:13:00 -0700 (PDT) |From: "R. William Gosper" <rwg(a)spnet.com> |Message-Id: <200007310713.AAA23803(a)spnet.com> |To: math-fun(a)optima.CS.Arizona.EDU |Subject: atanh sum | |sum(atanh(tan(a)/tan(%pi*j/k+b)),j,1,k) = atanh(tan(a*k)/tan(b*k)); | | k | ==== | \ tan(a) tan(k a) | > atanh(--------------) = atanh(--------) . | / %pi j tan(k b) | ==== tan(----- + b) | j = 1 k | |Familiar? Expanding for small a gives the "regular polygon" sums for |the odd powers of cot, e.g., sum(cot(%pi*j/k+b),j,1,k) = k*cot(b*k); | | k | ==== | \ %pi j | > cot(----- + b) = k cot(b k) , | / k | ==== | j = 1 | |sum(cot(%pi*j/k+b)^3,j,1,k) = k*cot(b*k)*(k^2*csc(b*k)^2-1); | | k | ==== | \ 3 %pi j 2 2 | > cot (----- + b) = k cot(b k) (k csc (b k) - 1) . | / k | ==== | j = 1 ------------------------------ It may be possible to give closed forms for the sum over a period of any rationalfn(trig). A surprisingly simple result: ==== \ j pi k pi n > tan(---- + f) tan(---- + f) = - ( ) / n n 2 ==== 1<=j<k<=n A bit worse: n ==== \ j pi j pi cot(g - f) sin((g - f) n) > cot(---- + f) cot(---- + g) = n (------------------------- - 1) / n n sin(f n) sin(g n) ==== j = 1 Amazingly worse: n ==== \ pi j pi j pi j > cot(---- + f) cot(---- + g) cot(---- + h) = / n n n ==== j = 1 2 sin(h - g) cot(f n) sin(h n - g n) n (--------------------------------------- sin(g - f) sin(h - f) sin(g n) sin(h n) sin(h - f) cot(g n) sin(h n - f n) - --------------------------------------- sin(g - f) sin(h - g) sin(f n) sin(h n) sin(g - f) cot(h n) sin(g n - f n) + ---------------------------------------)/2 sin(h - f) sin(h - g) sin(f n) sin(g n) 2 2 cot(h - g) n cot(f n) sin((h - g) n) cot(h - f) n cot(g n) sin((h - f) n) + ------------------------------------- + ------------------------------------- 2 sin(g n) sin(h n) 2 sin(f n) sin(h n) 2 cos(g - f) cot(h n) cos(h - f) cot(g n) - (5 n + n + 6) (--------------------- - --------------------- sin(h - f) sin(h - g) sin(g - f) sin(h - g) cos(h - g) cot(f n) + ---------------------)/16 + (n + 3) (5 n + 2) sin(g - f) sin(h - f) cos(h - f) cot(h n) cos(h - g) cot(h n) cos(g - f) cot(g n) (--------------------- - --------------------- - --------------------- sin(g - f) sin(h - g) sin(g - f) sin(h - f) sin(h - f) sin(h - g) cos(h - g) cot(g n) cos(g - f) cot(f n) cos(h - f) cot(f n) - --------------------- - --------------------- + ---------------------)/16 sin(g - f) sin(h - f) sin(h - f) sin(h - g) sin(g - f) sin(h - g) 2 (5 n + 9 n + 6) (cot(h n) + cot(g n) + cot(f n)) + ------------------------------------------------- 8 2 cot(g - f) n sin((g - f) n) cot(h n) + ------------------------------------- 2 sin(f n) sin(g n) cos(g - f) cos(h - f) cos(h - g) - (--------------------- - --------------------- + ---------------------) sin(h - f) sin(h - g) sin(g - f) sin(h - g) sin(g - f) sin(h - f) 2 (15 n + 35 n + 18) cot(f n) cot(g n) cot(h n)/16 2 (15 n + 35 n + 18) cot(f n) cot(g n) cot(h n) + ----------------------------------------------, 8 from a harrowing derivation via 16-page intermediate results, including a quadrivariate generating fcn to circumvent a *nasty* cubic. This undermines my confidence in being able to automate such sums (and products). Here's the stringout if you want to test or simplify it: 'SUM(COT(%PI*J/N+F)*COT(%PI*J/N+G)*COT(%PI*J/N+H),J,1,N) = N^2*(SIN(H-G)*COT(F*N)*SIN(H*N-G*N)/(SIN(G-F)*SIN(H-F)*SIN(G*N)*SIN(H*N))-SIN(H-F)*COT(G*N)*SIN(H*N-F*N)/(SIN(G-F)*SIN(H-G)*SIN(F*N)*SIN(H*N))+SIN(G-F)*COT(H*N)*SIN(G*N-F*N)/(SIN(H-F)*SIN(H-G)*SIN(F*N)*SIN(G*N)))/2+COT(H-G)*N^2*COT(F*N)*SIN((H-G)*N)/(2*SIN(G*N)*SIN(H*N))+COT(H-F)*N^2*COT(G*N)*SIN((H-F)*N)/(2*SIN(F*N)*SIN(H*N))-(5*N^2+N+6)*(COS(G-F)*COT(H*N)/(SIN(H-F)*SIN(H-G))-COS(H-F)*COT(G*N)/(SIN(G-F)*SIN(H-G))+COS(H-G)*COT(F*N)/(SIN(G-F)*SIN(H-F)))/16+(N+3)*(5*N+2)*(COS(H-F)*COT(H*N)/(SIN(G-F)*SIN(H-G))-COS(H-G)*COT(H*N)/(SIN(G-F)*SIN(H-F))-COS(G-F)*COT(G*N)/(SIN(H-F)*SIN(H-G))-COS(H-G)*COT(G*N)/(SIN(G-F)*SIN(H-F))-COS(G-F)*COT(F*N)/(SIN(H-F)*SIN(H-G))+COS(H-F)*COT(F*N)/(SIN(G-F)*SIN(H-G)))/16+(5*N^2+9*N+6)*(COT(H*N)+COT(G*N)+COT(F*N))/8+COT(G-F)*N^2*SIN((G-F)*N)*COT(H*N)/(2*SIN(F*N)*SIN(G*N))-(COS(G-F)/(SIN(H-F)*SIN(H-G))-COS(H-F)/(SIN(G-F)*SIN(H-G))+COS(H-G)/(SIN(G-F)*SIN(H-F)))*(15*N^2+35*N+18)*COT(F*N)*COT(G*N)*COT(H*N)/16+(15*N^2+35*N+18)*COT(F*N)*COT(G*N)*COT(H*N )/8; --rwg PS, given that my Macsyma was last compiled for a 286, Mma's far greater capacity was a temptation, except that 5.2, at least, still lacks the equivalent of POWERSERIES for extracting the general term of (a certain derivative at a certain point of) that generating function. PPS, my former student aced his tests but skipped his homework, and is now repeating Algebra I. As soon as I can photocopy a few pages, I'll share some of the stupendous howlers I found on a quick perusal of his new text. Amazingly, it's *not* Prentice-Hall, so it must be that the "educators" actively seek texts by and for morons. --------------------------------- Luggage? GPS? Comic books? Check out fitting gifts for grads at Yahoo! Search.

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[math-fun] Re: a puzzle
by Richard Guy 22 Aug '07

22 Aug '07

John, Always good to hear from you. I'm sorry to ruin a good story, and I doubt if it'll make it into OEIS (but I send it to them, just in case) but the answer is probably pretty prosaic, and completely known only to the long-gone printers and binders of Academic Press in 1969. For 0 < n < 10, n+ appears on page 32n-43 n* ,, ,, 32n-35 except that 1+ doesn't appear page -11 = i (the flyleaf). Note that page -3 = ix. The pattern continues through 10 & 11, except that 10 appears only once, without star or plus. Incidentally, my copy was the complementary copy sent by Mordell to Davenport, which I obtained at DPMMS in Cambridge because Davenport had recently departed. I used to tease Mordell, because, when I was in Haifa in 1970, a suitcase was stolen from our Volkswagen van during a period of bombings. The suspicious suitcase was found in some bushes by the bomb squad. The only thing missing was a silk shirt that Louise had made for me. Mordell's book was still there, obviously of little value. Best to all, R. On Wed, 22 Aug 2007, jdb(a)math.arizona.edu wrote: > Dear Richard, > I wonder if you know about the mysterious markings at the bottom of certain > pages of Mordell's Diophantine Equations book. If one (or more) looks (look) > on pages 29, 53, 61, 85, 93, 117, etc.., one (etc.) sees mysterious numbers > like 5^+ or 3^* and one (etc.) wonders what in the damn hell they mean, > i.e., some of us. > > Since they increase as the book proceeds, it may be realted somehow to > printing. Andrew Bremner thinks they may be a rock climbing code, but is > unwilling to die for this idea. To be sure, Mordell was a rock climber. > > Do you know right of if this was a mysterious message to readers in the > know? > > Best, > John

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RE: [math-fun] Huge fraction
by Eric Angelini 22 Aug '07

22 Aug '07

Ok, I have it, sorry : Sum of first 1000 terms = 8.8065157488607 Sum of first 10000 terms = 11.3162897519777 Best, É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be(a)mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Eric Angelini Envoyé : mercredi 22 août 2007 18:04 À : math-fun Objet : [math-fun] Huge fraction Hello Math-Fun, I'm sorry to bother with this, but could someone compute this fraction? The result should be close to 4, I guess. (This is the sum of the first 10000 fractions of my previous "look and divide" diverging seq. The fractions are simplified immediately before computing the next term; thus the first "1" was "1/1" before simplification and the second "1/2" was "2/4"). Best, É. (there are 193 "return carriage" marks on my page and the fraction bar "/" is in yellow) http://www.cetteadressecomportecinquantesignes.com/Fraction.htm 0 1 1/2 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 3/23 3/26 3/29 3/32 3/35 3/38 3/41 3/44 3/47 3/50 3/53 3/56 3/59 3/62 3/65 3/68 3/71 3/74 3/77 3/80 3/83 3/86 3/89 3/92 3/95 3/98 3/101 4/105 ... etc. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Huge fraction
by Eric Angelini 22 Aug '07

22 Aug '07

Hello Math-Fun, I'm sorry to bother with this, but could someone compute this fraction? The result should be close to 4, I guess. (This is the sum of the first 10000 fractions of my previous "look and divide" diverging seq. The fractions are simplified immediately before computing the next term; thus the first "1" was "1/1" before simplification and the second "1/2" was "2/4"). Best, É. (there are 193 "return carriage" marks on my page and the fraction bar "/" is in yellow) http://www.cetteadressecomportecinquantesignes.com/Fraction.htm 0 1 1/2 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 3/23 3/26 3/29 3/32 3/35 3/38 3/41 3/44 3/47 3/50 3/53 3/56 3/59 3/62 3/65 3/68 3/71 3/74 3/77 3/80 3/83 3/86 3/89 3/92 3/95 3/98 3/101 4/105 ... etc.

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