November 2005 - mathfun@mailman.xmission.com

RE: [math-fun] Abramowitz & Stegun
by Jud McCranie 28 Nov '05

28 Nov '05

At 03:08 PM 11/27/2005, you wrote: >Simon Plouffe has a scanned copy of A&S online at > >http://www.lacim.uqam.ca/~plouffe/articles/Abramowitz&Stegun.pdf > >It's huge, almost 1000 pages of images. Very handy to have a >local copy downloaded. I've got the hardback book. Somewhere.

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Re: [math-fun] Abramowitz & Stegun
by Simon Plouffe 28 Nov '05

28 Nov '05

Bonjour, Oui, le document est libre de droit le NBS l'a mis à la disposition du public c'est pour ça que j'ai pu mettre ma copie. Je l'ai fait en pdf et le document a élé OCRisé, c'est plus lourd que le deja-vu mais il y a la possibilité de rechercher. Le scan a été fait à 300 DPI. Si je peux me permettre, celui qui est sur le net à l'adresse qu'on a donné n'est pas d'aussi bonne qualité que ma version, en passant ce livre que j'ai scanné était ma propre copie, il y a des notes ici et là. Simon Plouffe simon.plouffe(a)gmail.com ps : je demeure maintenant à Lyon

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Re: [math-fun] Desperate seeking help with with DTFT (Discrete Time Fourier Transform)
by dasimov＠earthlink.net 28 Nov '05

28 Nov '05

Fred wrote: << . . . The "delta" is not a Kronecker delta, but an infinite spike whose integral is nonzero (here pi): this is an example of a "distribution", a generalisation of the classical notion of "function" which permits such limiting cases to be discussed compactly. . . . >> Thanks. (A slip of the fingers; I meant Dirac delta. Yes, indeed, a linear functional on the vector space of functions, given by delta(f) = f(0). I'm only half as confused as I seem to be.) I'm almost sure you're right that the author is "by convention" setting 1/(1-exp(iw)) to 0 where w = 2k*pi, and then adding explicitly the "singular part", a periodic delta function. --------------------------------------------------------------------- I'm thinking, f(w) = 1/(1-exp(iw)) = -2i exp(-iw/2) / sin(w). So maybe somehow integral_a^b(f(w)) = (1/2)*(2 pi*i) * (Res(f(w)) at w=0) can be used to express the "singular part" of f(w) ??? --Dan

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[math-fun] Desperate seeking help with with DTFT (Discrete Time Fourier Transform)
by dasimov＠earthlink.net 27 Nov '05

27 Nov '05

I hope some math-funner can help me with a DFTF question: The DTFT of a complex doubly-infinite sequence x := {x[n] in C | -oo < n < oo} is given by X(w) = sum over all integers n of x[n]*exp(i n w). A textbook I'm teaching from states that if x[n] is defined as the "unit step sequence": x[n] = 0, n < 0; x[n] = 1, n >= 0 by definition, X[w] =sum_n=0^oo exp((i n w), and that by some mystical procedure, (***) X[w] = 1/(1 - exp(iw)) + pi * sum over all integers n of delta(w + 2*pi*n) (where delta = Kronecker delta function). I don't know how to interpret this, since of course in the normal sense X(w) doesn't converge for any w. But in some generalized function sense, can someone please explain equation (***), and if possible at least hint at a rigorous foundation (just in case it has one). Thanks, Dan

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[math-fun] semi-regular matchstick graphs
by Erich Friedman 25 Nov '05

25 Nov '05

greetings. a matchstick graph is a planar graph whose edges are unit line segments. the smallest 2- and 3-regular matchstick graphs are easy to find. http://www.stetson.edu/~efriedma/mathmagic/1205/d23.gif the smallest known 4-regular matchstick graph (credited to H. Harborth and M. Timm) is more surprising. http://www.stetson.edu/~efriedma/mathmagic/1205/d4.gif there is no 5-regular matchstick graph. i'm interested in matchstick graphs whose vertices have two different degrees. what's the smallest matchstick graph with vertices of degrees m and n (m<n) ? has anyone considered this problem before? i think i know the answers for m=1, m=2, and m=3 n<=12. let f(m,n) be the minimum number of matchsticks necessary. f(1,n) = n f(2,n) = 3n/2 for n<=10 even f(2,n) = 2n-5 for n>=12 even f(2,n) = 3n-4 for n<=9 odd f(2,n) = 4n-13 for n>=11 odd f(3,4) = 14 f(3,5) = 16 f(3,n) = 5n-18 for 6<=n<=12 i am particularly interested in possible answers for m=4. surely someone can do better than i have. http://www.stetson.edu/~efriedma/mathmagic/1205/d4,5.gif http://www.stetson.edu/~efriedma/mathmagic/1205/d4,6.gif the same question is perhaps more interesting (and definitely more challenging) if we require the matchstick graph to have equal numbers of vertices of degrees m and n. let g(m,n) be the minimum number of matchsticks necessary. my best results are: g(1,2) = 3 g(1,3) = 6 g(1,4) = 20 g(1,5) = 156 (these come from adding one edge to each vertex of a regular graph) g(2,3) = 5 g(2,4) = 9 g(3,4) = 21 again, the ones i'm most interested in are the larger ones. can someone improve these? can anyone find any such graph with m=2 n=7? how about m=4 n=5? http://www.stetson.edu/~efriedma/mathmagic/1205/dd2,5.gif http://www.stetson.edu/~efriedma/mathmagic/1205/dd2,6.gif http://www.stetson.edu/~efriedma/mathmagic/1205/dd3,5.gif http://www.stetson.edu/~efriedma/mathmagic/1205/dd3,6.gif any improvements, comments, or references would be appreciated. erich friedman stetson university

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[math-fun] April 1 comes late this year
by Richard Guy 23 Nov '05

23 Nov '05

When I tried 1, 2, 18, 26 on OEIS I got no hits, but 1, 3, 8, 9 yielded my full quota of 30. Alas, I fear that they are both finite in the present context, so won't make it into the Hall of Fame when Neil returns from his well-earned rest (though I don't believe that he ever takes one). They are the integer values of $\prod_{i=1}^n \frac{\sigma n}{n}$ and the values of n which make this an integer, where \sigma n is the sum of divisors function. How did I come to be thinking about this? Did someone mention it recently? Season's greetings R.

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[math-fun] Zeroes of Riemann Zeta and Estimates of Prime-counting Functions
by Robert Baillie 23 Nov '05

23 Nov '05

Can anyone recommend a relatively easy discussion of how the location of zeroes of the Riemann zeta function affect estimates of prime-counting functions? Specifically I have two questions. 1. What would happen if there were zeroes on (for example) the line Re(s) = .75? 2. We know that the first umpteen billion zeroes are on Re(s) = 1/2, with Im(s) < T. As we compute more zeroes, this bound T gets larger. How exactly does this improve estimates for Pi(x), Theta(x), and Psi(x) ? Bob Baillie

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[math-fun] New error-correcting codes ...
by Guy Haworth 23 Nov '05

23 Nov '05

Not my field, but an answer from a UoReading colleague whose it is: "The answer is essentially 'Yes'. These new codes do indeed allow comms links to get very close to the Shannon limit. The practicality of real-time decoding is a bit arguable for some of the latest developments but LDPC and Turbo Codes are already in widespread use." Guy

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[math-fun] turbo codes and such
by Chris Landauer 22 Nov '05

22 Nov '05

hihi, all - i remember the articel rich is talking about, so it was in wired or discopver or ieee spectrum or computer or someting liek that i'll look for it (i saved it) more soon, cal

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[math-fun] new error correcting codes
by Schroeppel, Richard 22 Nov '05

22 Nov '05

I recently came across an article, perhaps in Science News, praising recent developments in error-correcting codes, particularly Turbo Codes and Asymmetric Low Density Parity Check Codes. Supposedly, these come close to the Shannon bound on maximum achievable data rate (through an error-prone channel with known error probability), while also being practically decodable with reasonable resources. The hype included statements that all of the 20th century work on ECCs was now obsolete, etc. Question for the ECC folks among us: How much of this is real? Is there a true breakthrough? Rich rcs(a)cs.arizona.edu

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