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
> 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
[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.
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.
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
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
~~~~~~~~~~~~~~
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.
Thane wrote:
<<
1) I went to an excellent talk on this tonight. This is great stuff!
2) Here's the web site http://archimedespalimpsest.org/
. . .
. . .
>>
Fascinating. But the mysterious omission is no reference whatsoever
to any of it having been translated into English, as far as I can tell.
I certainly look forward to seeing what Archimedes meant to convey
in this document!
--Dan
