August 2006 - mathfun@mailman.xmission.com

[math-fun] Dilworth's theorem
by Fred W. Helenius 09 Aug '06

09 Aug '06

At 07:04 PM 8/8/2006, Dan Asimov wrote: > Dilworth? Robert Dilworth (1914-1993) was a Caltech mathematician who was a pioneer in lattice theory. His name is attached to a fundamental result: Let P be a partially ordered set. A subset of P in which any two elements are comparable is called a _chain_. A subset of P in which no two elements are comparable is an _antichain_. Dilworth's Theorem (1950): If P is finite, the minimum number of disjoint chains needed to cover P is equal to the maximum size of an antichain in P. -- Fred W. Helenius fredh(a)ix.netcom.com

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Re: [math-fun] Kaplansky's Obituary
by Daniel Asimov 08 Aug '06

08 Aug '06

Thanks, Michael R. & Bill D. Bill D. writes: << One can gain some idea of Kaplansky's work and its influence by searching books.google.com for "Kaplansky theorem". This yields 344 hits - more than most of his contemporaries. See the list below which includes a random sample of his contemporaries along with a mix of older luminaries for comparison. . . . . . . >> Dilworth? --Dan

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Re: [math-fun] Kaplansky's Obituary
by Michael Reid 08 Aug '06

08 Aug '06

> I don't think I saw any mention of it here, but Kaplansky passed away little > over one month ago, on June 25. Here's a link to a nice obituary . . . but > I wasn't able to extract specific results of his from this: > > < http://www.signonsandiego.com/uniontrib/20060714/news_1m14kaplansk.html> > > Actually, I'm curious -- if any cognoscenti would care to explain some > theorems he's famous for, I'd be very interested. i'm afraid i can't really do it justice, but there hasn't been much response, so i thought i'd try. one result of his i know is that a projective module over a (non-commumtative) local ring is free. it's possible that the finitely generated version predates him, and the general case is his contribution. also, his book "infinite abelian groups" is well-known. (oops, i see that's mentioned in the obit.) mike

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[math-fun] Re: Archimedes Palimpsest
by Steve Witham 07 Aug '06

07 Aug '06

>From: Thane Plambeck <thane(a)best.com> > >2) Here's the web site http://archimedespalimpsest.org/ Statistically improbable phrase: "lead lined hutch." --Steve

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[math-fun] corrected binary asinh function
by Henry Baker 07 Aug '06

07 Aug '06

[Sorry -- this needed some more edits.] Is anyone on this list aware of anyone who has studied the following functions: s2(x):=(2^x-2^(-x))/2 =sinh(x*log(2)) Generally, s2(x) acts like a sinh function, suitably fudged. as2(x):=sgn(x)*log(abs(x)+sqrt(1+x^2))/log(2) =asinh(x)/log(2) For x near zero, as2(x) is x/log(2) to a first approximation. For x very large, |as2(x)| approximates log(2|x|)/log(2) = log2(|x|)+1. s2(x)=-s2(-x) as2(x)=-as2(-x) Generally, as2(x) acts like the asinh function, suitably fudged. The cool thing about s2(n) is that its binary representation is a series of "1" bits exactly 2n bits long, with n-1 bits to the left of the binary point and n+1 bits to the right of the binary point. We can use linear combinations of these functions to create bit strings which are palidromes ! This bit representation also gives a bit more insight into the true nature of the sinh function. --- BTW, "asinh" now has a real paying job. I searched using Google and I found that astronomers are now using asinh instead of log to compute brightness of stars in a star catalog. I have advocated the use of asinh in audio & video processing to do "dynamic range compression" in a way which is somewhat smoother than current ad hoc methods and floating point methods. Another job is its use in genetic engineering & statistics to smooth data better than log when very small magnitudes are encountered, which would otherwise drive log nuts (i.e., very substantially negative). Asinh pops up in the calculation of Chebyshev filters in DSP's. Also asinh shows up in the design of certain magnetic coils.

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[math-fun] binary asinh function
by Henry Baker 07 Aug '06

07 Aug '06

Is anyone on this list aware of anyone who has studied the following function: as2(x):=(2^x-2^(-x))/2 =asinh(x)/log(2) For x near zero, as2(x) is x/log(2) to a first approximation. For x very large, |as2(x)| approximates log(2|x|)/log(2) = log2(|x|)+1. as2(x)=-as2(-x) Generally, as2(x) acts like the asinh function, suitably fudged. The cool thing about as2(n) is that its binary representation is a series of "1" bits exactly 2n bits long, with n-1 bits to the left of the binary point and n+1 bits to the right of the binary point. We can use linear combinations of these functions to create bit strings which are palidromes ! This bit representation also gives a bit more insight into the true nature of the asinh function. --- BTW, "asinh" now has a real paying job. I searched using Google and I found that astronomers are now using asinh instead of log to compute brightness of stars in a star catalog. I have advocated the use of asinh in audio & video processing to do "dynamic range compression" in a way which is somewhat smoother than current ad hoc methods and floating point methods. Another job is its use in genetic engineering & statistics to smooth data better than log when very small magnitudes are encountered, which would otherwise drive log nuts (i.e., very substantially negative). Asinh pops up in the calculation of Chebyshev filters in DSP's. Also asinh shows up in the design of certain magnetic coils.

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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] Positive integers mod primes
by Kerry Mitchell 06 Aug '06

06 Aug '06

Here's a sequence I've been playing with; is anything much known about it? For a positive integer n, find n mod p, where p is the largest prime not larger than n. Continue with that result, finding its residue mod its largest prime, etc., until you come down to 0 or 1. For example, 9 mod 7 = 2 and 2 mod 2 = 0, so the ninth element is 0. The sequence begins: 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1 and seems to have blocks of zeroes embedded between 1s. The sequence of the lengths of the blocks of zeroes begins: 2, 1, 1, 3, 1, 3, 1, 3, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2. For n = 9, there are two changes before the final result (9 to 2 and 2 to 0). The sequence of numbers of changes begins: 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1 and seems to be of the form of blocks of repeating numbers. The sequence of the lengths of these blocks begins: 1, 7, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, 2, 4, 2, 2, 4, 2, 4, 4. None of these four sequences is in OEIS. Is this interesting enough to add to OEIS? Kerry -- lkmitch(a)gmail.com www.fractalus.com/kerry

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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] archimedes documents revealed by X rays
by Simon Plouffe 05 Aug '06

05 Aug '06

This just came out, http://news.bbc.co.uk/2/hi/science/nature/5235894.stm They found a way to read a old document with x-rays. simon plouffe

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