October 2010 - mathfun@mailman.xmission.com

[math-fun] Bell System Technical Journal is now on-line
by Eugene Salamin 15 Oct '10

15 Oct '10

The complete set of issues of the Bell System Technical Journal, which was published from 1922 to 1983 has been placed on-line and is freely accessible at ( http://bstj.bell-labs.com/ ). -- Gene

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Re: [math-fun] Almomorphism
by Phil Carmody 14 Oct '10

14 Oct '10

> Almomorphism > > This note is about "almost homomorphisms" -- Almomorphisms. > We already use these: I'm assigning a name, and suggesting we > might get some milage from formalizing our use. > An almomorphism can be "almost" in a few ways. > It might usually be true, or sometimes be true, or be true more > often than chance; or it might be nearly correct, most of the time. > > Example A: > Residues modulo A and modulo B, with A < B < A+small. > Then addition modulo A is frequently the same as addition modulo B. > The answer c+d (mod A) is the same as c+d (mod B) when there's > no wraparound, i.e. c+d < A. This happens about half the time for > randomly chosen c & d. Looking at multiplication, the no-wraparound condition > becomes cd < A. This only happens with frequency about logA/A, which is > somewhat more likely than would be expected by chance. Both of those wraparound conditions are also apply to the almomorphism between the integers modulo A and rational integers. For example, any fixed-size integral type in typical computer languages where overflow is non-catastrophic. (Wrapping is typical, but saturation still satisfies the conditions.) Phil

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Re: [math-fun] Fwd: Alf van der Poorten
by Dan Asimov 13 Oct '10

13 Oct '10

Fred wrote: << His book on Fermat's Last Theorem was fascinating --- I was most reluctant to return it to its owner, and this notice reminds me to go out and just buy the thing, assuming it remains in print. >> For a mere £76.48 (or cheaper used) you can get it here: < http://www.amazon.co.uk/Fermats-Wiley-Interscience-Canadian-Mathematics-Mon… > --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Almomorphism
by rcs＠xmission.com 13 Oct '10

13 Oct '10

Almomorphism This note is about "almost homomorphisms" -- Almomorphisms. We already use these: I'm assigning a name, and suggesting we might get some milage from formalizing our use. An almomorphism can be "almost" in a few ways. It might usually be true, or sometimes be true, or be true more often than chance; or it might be nearly correct, most of the time. Example A: Residues modulo A and modulo B, with A < B < A+small. Then addition modulo A is frequently the same as addition modulo B. The answer c+d (mod A) is the same as c+d (mod B) when there's no wraparound, i.e. c+d < A. This happens about half the time for randomly chosen c & d. Looking at multiplication, the no-wraparound condition becomes cd < A. This only happens with frequency about logA/A, which is somewhat more likely than would be expected by chance. Example B: Floating point numbers, as a model of the reals. (Within the represented range) floating point operations are a reasonably close image of the real numbers being modeled. Example C: Rounding of numbers. Rounding maps ranges of numbers into single values. The results of arithmetic operations on the single values are frequently close to the result of rounding the operation applied to the original operands. We have a lot of knowledge about how rounding affects answers, and when it is correct, or slightly wrong. For example, if c and d are rounded to C and D, and the relative error of the rounding is at most 10%, and c and d are positive, then C+D (i.e. round(c)+round(d)) is within 10% of c+d. Example D: The projective line, and the reals. The reals have no infinity, but arithmetic works on all the other finite values. These examples show that the question of when the results are valid is complicated, but we know a lot of the answers. Rich

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[math-fun] Fwd: Alf van der Poorten
by rcs＠xmission.com 12 Oct '10

12 Oct '10

Alf van der Poorten has died. He did a fair amount of work with Somos sequences. Jeff Shallitt has a tribute on his blog, at http://recursed.blogspot.com/2010/10/alf-van-der-poorten-1942-2010.html and there's a notice at http://www.austms.org.au/tiki-read_article.php?articleId=110 Rich ----- Forwarded message from gerry.myerson(a)mq.edu.au ----- Date: Mon, 11 Oct 2010 20:43:38 -0500 From: g m <gerry.myerson(a)mq.edu.au> Reply-To: Number Theory List <NMBRTHRY(a)listserv.nodak.edu>, g m <gerry.myerson(a)mq.edu.au> Subject: Alf van der Poorten To: NMBRTHRY(a)listserv.nodak.edu I regret to inform the Number Theory list of the passing of Alf van der Poorten. http://www.austms.org.au/tiki-read_article.php?articleId=110 -- Gerry Myerson (gerry.myerson(a)mq.edu.au) ----- End forwarded message -----

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Re: [math-fun] P.S. Re: Fwd: Alf van der Poorten
by Dan Asimov 12 Oct '10

12 Oct '10

Hit the wrong keys, sorry. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] P.S. Re: Fwd: Alf van der Poorten
by Dan Asimov 12 Oct '10

12 Oct '10

<< The most impressing magic square I know is a 4x4 that has an amazing number of subsets adding to 34. >> Oops, that was a silly thing to say. I mean *symmetrical subsets*. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] €8,000 (~$10,000) prizes on magic squares
by Dan Asimov 12 Oct '10

12 Oct '10

Christian writes: << Six months after the press release below, look at the October 2010 news of www.multimagie.com: the current status is given, with the most interesting results. 6900 euros (~8500 dollars) and 10 bottles of champagne still to win! . . . >> Before putting effort into solving any of these enigmas, I'd like to know that there is adequate incentive. So I wonder if Christian could please tell us the brand of champagne. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Jim's question for Conway
by James Propp 12 Oct '10

12 Oct '10

I wrote: >I'll send Thurston an email to see what else I can learn about >where the problems originated and how they spread. Thurston says he heard about them from Conway! ?!? Jim Propp

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[math-fun] €8,000 (~$10,000) prizes on magic squares
by Christian Boyer 11 Oct '10

11 Oct '10

While magic squares have been known and studied for many centuries, it is surprising that for certain types of magic squares we still do not know today which are the SMALLEST possible! In an effort to make progress on these unsolved problems, twelve prizes totaling €8,000 (~$10,000) and twelve bottles of champagne are offered for the solutions to twelve enigmas. Details here: www.multimagie.com/English/MagicSquaresEnigmasE.pdf Christian Boyer. www.multimagie.com/indexengl.htm

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