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[math-fun] april 1 letter in Nature
by Thane Plambeck 31 Mar '04

31 Mar '04

Irrelevance, buzzwords, coffee: it's the virtual PI Sir - Several graduate students would like to suggest a modest proposal to the authors of the recent article "Functional genomic hypothesis generation and experimentation by a robot scientist" (Nature 427, 247-252; 2004). We've been doing a little research of our own and have generated an exciting hypothesis: we can build a Virtual Principal Investigator. The software platform will consist of a tiny amount of background knowledge about a completely different system from the one being actively investigated, a buzzword-invention engine, coffee-consumption apparatus, a grant-writing node and a cat-o'-nine-tails that will tie everything together. For much less money than an actual investigator, we created a machine that resides in an office, drinking coffee at a steady rate. At bi-monthly intervals, the Virtual Principal Investigator spits out grants that are derivatives of highly successful grants from the past. Between these time points, it will play an endless loop of inspirational messages and test the effectiveness of the whip as a method of controlling subservient organisms. Initial tests of our Virtual Principal Investigator have revealed no differences from our existing model of an investigative scientist, but we have yet to perform an experiment that was not naive or random. We look forward to discussing our results with you. Several non-robot scientists (names supplied) Department of Biological Chemistry and Molecular Pharmacology, Harvard Medical School, 240 Longwood Avenue, Boston, Massachusetts 02115, USA Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] Buying a PhD from a University that doesn't exist
by Thane Plambeck 30 Mar '04

30 Mar '04

Amusing research (123 slides!) by George Gollin http://www.hep.uiuc.edu/home/g-gollin/diploma_mills.pdf Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] Are you sharp
by Thane Plambeck 25 Mar '04

25 Mar '04

Once Bob Floyd saw a stanford newsgroup message sent by someone named Tuna Ertemalp. He immediately typed in a response that said "This is an obvious pseudonym. Aunt Palmtree, I have found you out!" Yet it was no pseudonym (I think Tuna works at Microsoft). Now I read in the introduction to Atiyah and MacDonald's Introduction to Commutative Algebra "We have provided a substantial number of exercises at the end of each chapter. Some of them are easy and some are hard. Usually we have provided hints, and sometimes complete solutions, to the hard ones. We are indebted to Mr. R. Y. Sharp, who worked through them all..." Aha! Caught them, I thought. But no, he's the current head of the math dept at Sheffield. Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] some exact formulas for integer sequences,
by Simon Plouffe 23 Mar '04

23 Mar '04

hello, (this is old material from 1993), I collected some of my notes about exact formulas for integer sequences. They are somewhat usefull since it can be used to compute the n'th term of a sequence. it uses [ ] , { } and some constants like exp(1). http://www.lacim.uqam.ca/%7Eplouffe/exact.htm note : most of those formulas do not appear in the current On-line Encyclopedia of integer sequences. By the way, can anybody do this : Can we find an EXACT formula for the n'th term of A001006 (Motzkin numbers), the only useful formula we have is a recurrence but with the index that grows with n which makes it difficult to compute for large n. http://www.research.att.com/projects/OEIS?Anum=A001006 is 1,1,2,4,9,21,51,127,323,835,2188,5798, ... and it grows roughly like 3^n. Simon Plouffe

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[math-fun] exact formulas for Motzkin numbers
by N. J. A. Sloane 23 Mar '04

23 Mar '04

Maybe Simon was looking at a very old copy of the OEIS. There are now several exact formulas for A001006. See the entry for more details. Here are two: %F A001006 a(n) = (-1/2) SUM (-3)^i C(1/2,i) C(1/2, j); i+j=n+2, i >= 0, j >= 0. %F A001006 a(n) = (1/(n+1)) * Sum_{i} (n+1)!/(i!*(i+1)!*(n-2*i)!) - Bernhart. NJAS

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[math-fun] weird continued fraction
by R. William Gosper 19 Mar '04

19 Mar '04

Benoit Cloitre sent me a draft containing a recurrence identity implying a convergent matrix product: [ 2 2 ] [ --------- --------- ] [ sqrt(%pi) sqrt(%pi) ] [ ] = [ %pi 2 %pi 2 ] [ sqrt(---) - sqrt(---) sqrt(---) - sqrt(---) ] [ 2 %pi 2 %pi ] inf [ sqrt(n + 1) ] /===\ [ ------------- 1 ] | | [ n sqrt(n + 2) ] | | [ ] , | | [ sqrt(n) ] n = 1 [ ----------- 0 ] [ sqrt(n + 2) ] giving the c.f. (if you'll pardon my ascii), _ /2 /3 2 /2 / - / - v V 1 V 1 ------ = ---- + ------------------ . pi - 2 1 /3 /4 / - / - V 2 V 2 ---- + ----------- 2 /4 /5 / - / - V 3 V 3 ---- + ---- 3 . . . Nontriangular 2x2 products usually diverge, and you get the c.f. as the limit of the top row over the bottom. If we can find decent asymptotics to c.f. numerators (and denominators), we might scale them out of the matrix to produce more interesting identities of which the c.f. is a projection (which, in the case above, conceals a sqrt(pi)). --Bill Gosper P.S., note the near "recurrence" of our old friend, pi/2+2/pi, the expected magnitude of a 3D rotation, and once-believed optimum "sofa problem" area.

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[math-fun] Re: Number derivative
by Mike Stay 18 Mar '04

18 Mar '04

You can take q-deformed number derivatives of field elements. Work in GF(p^n). Define [a] as (q^a - 1)/(q - 1). Let x be a generator of the field. D( x^a ) = ((qx)^a - x^a) / (qx - x) = (q^a - 1)/(q-1) x^(a-1) = [a] x^(a-1) If x^a + x^b = x^c, then [a] x^(a-1) + [b] x^(b-1) (qx)^a + (qx)^b - (x^a + x^b) = ----------------------------- qx - x So if q is such that (qx)^a + (qx)^b = (qx)^c, then we have (qx)^c - x^c = ------------ qx - x = [c] x^(c-1) and the derivative is linear. Setting q = x^(p^k - 1) will satisfy the equation, where 1 < k < n. Of course, you can do "differential equations" this way, too. For example, take GF(2^3) mod x^3+x+1; then x is a generator and D(x^2+x+1) = x^2+x+1. -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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Re: [math-fun] Number derivative
by Daniel Asimov 17 Mar '04

17 Mar '04

Thanks to John McKay's suggestion, I e-wrote Buium and this is what he replied: << I don't know if the project of the 2004 Talent search is related to the derivatives of numbers that I'm using. What I do is to take the derivative of a number x with respect to a prime number p as being the number "dx/dp":=(x-x^p)/p which is an integer due to Fermat's Little theorem. The fact that this is a good definition can be seen from a number of papers I wrote about it. The first of them appeared in Inventiones Math., 1995. Other relevant papers appeared in Duke Math J., 1996, Crelle J., 2000, etc. The precise references to the papers are listed on my web page www.math.unm.edu/~buium >> >From the wording of the PR squibs about this student's project, it seems likely to me that the number derivative she worked on is in fact Buium's. If any of you number theoretic folks care to take a look at Buium's papers to see just what the purpose of this definition is, and why it's called a "derivative", I'd like to hear what you have to say. --Dan

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RE: [math-fun] Experimental math workshop
by Schroeppel, Richard 17 Mar '04

17 Mar '04

Hilarie and I are going. I'll be talking about several short subjects, probably Modular Dilogs, PolyHypercubes, PolyPolygons, and maybe Post Tag. Rich -----Original Message----- From: Steve Gray To: math-fun(a)mailman.xmission.com Sent: 3/17/2004 5:20 PM Subject: [math-fun] Experimental math workshop Is anyone here going to this conference organized by David Bailey? It's in Oakland, March 29-30. I', reporting on my geometry experiments. Steve Gray <<ATT1166353.txt>> <<ATT1166354.txt>>

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[math-fun] Experimental math workshop
by Steve Gray 17 Mar '04

17 Mar '04

Is anyone here going to this conference organized by David Bailey? It's in Oakland, March 29-30. I', reporting on my geometry experiments. Steve Gray

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