July 2006 - mathfun@mailman.xmission.com

[math-fun] Q-factorials
by Schroeppel, Richard 29 Sep '06

29 Sep '06

I've been playing with Q-factorials: QF(0) == 1 QF(N+1) == QF(N) * (1 + Q + Q^2 + ... + Q^N). When Q=1, this gives QF(N) = N!. The degree of QF(N) is Triangle(N-1) = N(N-1)/2. Q-binomials seem to be polynomials: QB(A,B) = QF(A+B) / (QF(A) QF(B)). The degree of QB(A,B) is AB. The degree of QB(A,B,C) = QF(A+B+C) / (QF(A)QF(B)QF(C)) is AB+AC+BC. Etc. Perhaps we can use some of our standard bag of tricks for defining QF(N) for non-integer N. Put QF(Z) ~= Q ^ [ Z(Z-1)/2 ] for RealPart(Z) large, and divide by (Q^Z-1)/(Q-1) to go from Q(Z) to Q(Z-1). There's a set of functional equations similar to X! (X+1/2)! = (2X+1)! * stuff. Presumably there's also a reflection equation with QF(Z) QF(-Z) = more-stuff that looks like pi Z / sin(pi Z). Rich

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P.S. Re: [math-fun] Chinese Mathematicians Prove Poincare Conjecture
by dasimov＠earthlink.net 08 Sep '06

08 Sep '06

P.S. I did not see this in the Ximhua press release, but the Asian Journal of Mathematics says (in the link provided by Ray) that the article in question finishes the proof of the Thurston Geometrization Conjecture (of which the Poincare conjecture is just one of many important consequences). Up to now, a few media (Scientific American in 1994, New York Times in the past year) have claimed the Poincare conjecture has been proved, although no respectable professional journal has reported any such thing (only comments like "time will tell.") And not even media outlets have reported that the Geometrization conjecture was proved. Hmm. I guess time will tell. --Dan

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[math-fun] A sequence puzzle
by franktaw＠netscape.net 07 Aug '06

07 Aug '06

Here are the first 20 terms of a sequence: 1,2,6,12,60,3,21,168,504,2520,27720,4,52,364,5460,21840,371280,1113840,21 162960,5 The puzzle is to determine the rule of formation. Franklin T. Adams-Watters

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[math-fun] update on DLMF, E(Abramowitz & Stegun)
by Schroeppel, Richard 06 Aug '06

06 Aug '06

~~~~~~~~~~~~~~ From: Dan Lozier <lozier(a)nist.gov> Subject: Looking Ahead to the DLMF The Digital Library of Mathematical Functions has been a much bigger job than any of the project participants expected. I am pleased to report that all of the chapters are in the final stages of editing and validating. The process of selecting a publisher for the print edition has been mapped out according to NIST and US Government procurement rules. The procurement will be competitive among qualified mathematics publishers. In addition to the print edition, the DLMF will be distributed free from a public Web site at NIST. The Web address is http://dlmf.nist.gov. A sample chapter on Airy functions has existed on this Web site since the beginning of the project. A tremendous amount of work has been done on the Web site since then to improve its "look and feel." A new sample chapter on the gamma function is being prepared. It will be on the Web site by the end of the summer. The DLMF project is modeled after the Handbook of Mathematical Functions, edited by Abramowitz and Stegun, and published in 1964. It is being constructed by the National Institute of Standards and Technology with substantial funding contributions by the National Science Foundation. More than 50 individuals are contributing to the project as paid authors and validators. The staff at NIST consists of another dozen or so people.

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[math-fun] Kaplansky's Rat Theorem
by Bill Dubuque 02 Aug '06

02 Aug '06

Google Books reports Kaplansky has classified rodent tumors http://books.google.com/books?q=kaplansky%27s+theorem+rodent --Bill Dubuque

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[math-fun] Has someone repealed the laws of physics?
by Mike Speciner 31 Jul '06

31 Jul '06

Science News reports in "Out of Sight" [15 July 2006, vol 170 no 3 p 42]: "Electromagnetic radiation typically exerts a repulsive pressure on objects, so it should, in theory, attract metamaterials. Also, the well-known Doppler shift, in which the frequency of electromagnetic radiation from an object increases as the object approaches, is expected to reverse for metamaterials. Experiments have yet to confirm those two predictions." IMHO, attractive pressure from EM radiation would violate conservation of energy and momentum, while reverse Doppler shift would violate relativity. Am I confused, or should Science News be embarassed? --ms

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[math-fun] asinh (hyperbolic arc sine) identities ?
by Henry Baker 31 Jul '06

31 Jul '06

Hyperbolic arc sine function is an interesting function which is 1-1, linear at the origin and logarithm towards +-oo. I'm looking for interesting identities involving this function. In particular identities for things like asinh(sinh(x)+c), c constant asinh(sinh(x)*c), c constant asinh(sinh(x)+sinh(y)) asinh(asinh(x)*asinh(y)) Also, identities that allow asinh(x) to be computed recursively in terms of itself (shifts up or down). If these don't work so well, then identities involving acosh or atanh, but neither of these functions are particularly interesting due to their reduced domains.

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[math-fun] MathPath Math Camp
by Henry Baker 29 Jul '06

29 Jul '06

Congratulations to Funster Conway on "MathPath" summer math camp for kids! It got a first-page article in today's (Sat 7/29) Wall Street Journal. Here's the MathPath web site: http://www.mathpath.org/ Here's the entrance exam (not easy!) in pdf format: http://www.mathpath.org/Quiz/quiz.2006.pdf ---- I had great fun in the 1960's with the NSF Summer Science Training Program math course when I was in high school, and it sounds like these camps are filling this important void.

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Re: [math-fun] Calculating pi
by dasimov＠earthlink.net 27 Jul '06

27 Jul '06

Jason writes: << . . . Let's see if I can write it as a formula instead of an algorithm: n n 4 ___ ___ --- \ \ pi ~ n*n * / / 1 if sqrt(x*x+y*y) < n --- --- x=0 y=0 Does that look about right? . . . >> Yes. In fact one of the old classic math books (Geom. & the Imagination? What is Mathematics? Math. & the Imagination?) uses this observation to obtain the series pi/4 = 1 - 1/3 + 1/5 - 1/7 + . . . . --Dan

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[math-fun] The mathemagician and pied puzzler: a collection in tribute to Martin Gardner
by Thane Plambeck 25 Jul '06

25 Jul '06

Funsters might be interested to know that I just finished a little project for Tom Rodgers and AK Peters, Ltd. to put the entire 1999 book The mathemagician and pied puzzler: a collection in tribute to Martin Gardner online, as separate PDF files, one per article (or one big one, if you have the patience to download it). You can find it here http://g4gardner.pbwiki.com Please let me know if there are any problems with it -- Thane Plambeck http://www.plambeck.org/ehome.htm

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