November 2002 - mathfun@mailman.xmission.com

[math-fun] Product integrals
by asimovd＠aol.com 07 Nov '02

07 Nov '02

Mike Stay (staym(a)datawest.net) asks: << sum is to integral as prod is to what? I.e. is there a name for exp(integral(log(f(x)),dx))? Where do I worry about branch cuts in that expression? >> These are often called "product integrals" and have application to the branch of probability called survival analysis. (As I gather you have already realized, given a positive function f defined on [a,b],the product integral of f over [a,b] is the limit as n -> oo of sum from 1 to n of f(x_i)^deltax_i, wherex_i and deltax_i are defined as in the Riemann integral, with the mesh of the partition going to 0 as n -> oo.) (Back in the days when I thought I was the first person to invent this idea, I was 16 and it was my Westinghouse project. As I found out when I got to college, Gerry Sussman thought of the very same idea for *his* Westinghouse project the same year. I still don't know who first invented these.) This can be used to calculate the *geometric* mean of a function over an interval; e.g., the GM of x over (0,1) is 1/e. These also work fine in the complex plane as well, as long as log is continuous and well-defined on the domain D of f. One way to ensure this is by picking D to be any simply-connected region of C - {0}. More generally, D can be chosen to be any simply-connected region of the Riemann surface for log. Once the domain has been chosen, there is a countable set of choices for which branch of log you prefer to use; they all differ of course by an integer multiple of 2 pi i. Just for fun, What is the geometric mean of all the complex numbers in the disk 0 < r <= 1, where we define log for -pi < theta < pi so that it's real for positive reals ??? --Dan

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[math-fun] Fibonacci pseudoprimes
by R. William Gosper 07 Nov '02

07 Nov '02

The only one between 2^35-10^7 and 2^35+15*10^6 is 2^35-6284047 = 34353454321 = 5300641*6481 . --rwg

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[math-fun] Prodigals (advertisement)
by Marc LeBrun 07 Nov '02

07 Nov '02

>="Mike Stay" <staym(a)datawest.net> > sum is to integral as prod is to what? > I.e. is there a name for exp(integral(log(f(x)),dx))? > Where do I worry about branch cuts in that expression? >="Richard Schroeppel" <rcs(a)CS.Arizona.EDU> > [editorial comment: Gosper calls these Prodigals. I don't think > they have any non-obvious proerties, but it is strange there's no > "official" name. --Rich] I must confess to coining "prodigal" for the "continuous product" (add 'em to get a "prodigal sum"<;-) You can worry about the branch cuts in that expression if you want, but usually I simply offer a prayer to Euler and blithely proceed, then go back later and check results. This may actually be unreasonably effective, because you can define the prodigal directly as a limit, analogous with the integral, instead of as a transform of an integral. (Martian mathematicians started with the prodigal, and then defined the integral by analogy!). While in some sense prodigals are "trivial", I advocate greater use of them because in certain situations they make things clearer and/or suggest interesting ideas that otherwise get lost in complicated expressions. Continuously compounded interest is naturally expressed as a prodigal, as are certain kinds of cumulative probabilities (consider a particle "deciding" whether to decay at each uncountable instant), for examples. Stirling's factorial approximation is basically the Bernoulli expansion of a prodigal. Note that instead of an arbitrary constant offset prodigals have an arbitrary constant scale factor (any application to big-O notation?). Speaking of notation, it's fun to write a "stretchy-P" for the prodigal, analogous to the "stretchy-S" of the integral: /\ |_/ qx | u | \/ (Easy to scribble, but how the heck do you get this graphic into documents?) Note that the "quintessential" qx, the analog of the differential dx, goes in the exponent. The inverse prodigal is of course 1/qx d(ln u)/dx du/(u dx) qu = e = e and there are naturally definite prodigals and the like. A provocative application is extending the idea of defining polynomials as the product of a countable number of zero monomials to defining functions as prodigals with uncountable numbers of "roots". (Note that, unlike polynomials, the function value need not be zero at any "root" point!) For example, x 1 qz x P (x-z) = ---------- 0 x-1 e (x-1) is the function defined by the unit interval "root segment". More generally, a function f(x) can be constructed as a prodigal over ALL "zeros" z (in the complex plane, say) weighted with multiplicity mu(z): mu(z) qz f(x) = P (x-z) Then we can view f and mu as "zero transforms" of each other (with suggestive analogy to Fourier convolution?). But that's about as far as I've managed to plod... maybe someone else will have fun prodding this further. "Enjoy"!

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[math-fun] Silver, purple rhymes
by Allan C. Wechsler 07 Nov '02

07 Nov '02

At 11:29 PM 11/6/02 -0700, you wrote: >I've moved the Math-Fun mailing list to another system that does >spam filtering. Everyone should have recently received a >"Welcome" message to the new list. The message includes a personal >password that you'll need to "manage" your mailing list options. >Interesting options include echoing of posts and a digest option. >[I think I've turned off the monthly password reminders.] >I think the initial setting is for single messages (non-digest) >and echoing ON. Another good feature is archiving. The URL for >reaching the archive in the message headers seems broken, but >you can get there from the generic list page at > > http://mailman.xmission.com/mailman/listinfo/math-fun > >The new posting address is > > math-fun(a)mailman.xmission.com > >I think sending to the old address (math-fun(a)cs.arizona.edu) >should still work, if I've got the forwarding right. > >The list is set to "members only" for posting. This will have two >annoying consequences: (a) if you post from a different address, or >(b) if someone else temporarily joins our discussion; then the >message is placed on hold for me to approve or reject. I'll try >to notice these and whitelist the extra addresses. > >The old reminders about preferring plain vanilla ascii, no Mime, >no attachments, pictures, binary files, etc. still apply. > >Undoubtedly there will be glitches; please report them to me at >my usual address below. With luck, this will be the last hand- >edited digest. > >Rich rcs(a)cs.arizona.edu > >%%%%%%%%%%%%%%%%%%%%%% > >From: "Mike Stay" <staym(a)datawest.net> >To: rcs(a)CS.Arizona.EDU >Subject: analysis question >Date: Tue, 5 Nov 2002 20:10:52 -0700 > >sum is to integral as prod is to what? > >I.e. is there a name for exp(integral(log(f(x)),dx))? Where do I >worry about branch cuts in that expression? >-- >Mike Stay >staym(a)datawest.net > >%%%%%%%%%%%%%%%%%%%%%% > >[editorial comment: Gosper calls these Prodigals. I don't think >they have any non-obvious proerties, but it is strange there's no >"official" name. --Rich] > >%%%%%%%%%%%%%%%%%%%%%% > >From: James Propp <propp(a)math.wisc.edu> >Date: Tue, 5 Nov 2002 23:51:15 -0600 (CST) >To: math-fun(a)CS.Arizona.EDU >Subject: PSL(2,F_7), SL(3,F_2), and the Fano plane > >What's a good way to see that PSL(2,F_7) is isomorphic to SL(3,F_2)? > >More concretely, what's a good way to see PSL(2,F_7) as being the >symmetry group of the Fano plane? > >(The Fano plane is the projective plane over F_2, so it's easy to see >why its symmetries are described by a linear group over F_2.) > >Also: What's the easiest way to see that the group in question is >simple? > >Jim Propp > >%%%%%%%%%%%%%%%%%%%%%% > >Date: Wed, 6 Nov 2002 10:12:56 -0800 (PST) >From: "R. William Gosper" <rwg(a)spnet.com> >To: math-fun(a)CS.Arizona.EDU >Subject: Silver, purple rhymes > >The CDROM version reveals that Webster's big old Third International has >all along defined CHILVER as a ewe lamb. And Roger Miller's heroic effort: > > Roses are red. > Violets are purple. > Sugar is sweet; > And so's maple syrple. Roses are red. Violets are purple. I won't shoot you, But Wyatt Earp'll. -A

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[math-fun] Testing
by ed pegg 07 Nov '02

07 Nov '02

I'm not sure if this works, so here's a test blather.... I rather like Isoptikon, a free Geometry program. It can be gotten for windows or Mac at http://itia.math.uch.gr/%7Epamfilos/#iso . Dan Hoey has shown the Reid hexagon is unique. A Reid polygon is a set of Golumb rulers with 0, 1, 2, ..., n-1 marks that measure every distance from 1 to BinomialCoefficent(n+2,3). 1,4,25,6,14,3 - 8,10,16,12,9 - 2,22,19,13 - 7,33,11 - 15,27 - 39 New Scientist magazine: By firing Beryllium-14 atoms at a carbon target, investigators seemingly got Beryllium-10 and tetraneutrons. --Ed Pegg Jr

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[math-fun] Test #3
by Richard Schroeppel 06 Nov '02

06 Nov '02

This is test #3. Rich

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[math-fun] test #2
by Schroeppel, Richard 06 Nov '02

06 Nov '02

Test message #2. Rich at Sandia

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[math-fun] test message
by Richard Schroeppel 06 Nov '02

06 Nov '02

This is a test mesage for the xmission math-fun list. Rich

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