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[math-fun] Re: Number derivative
by James Propp 17 Mar '04

17 Mar '04

>Alexandru Buium (Univ. New Maxico Albuqquerque) invented >such a thing. See his papers.. Couldn't find his papers on the web, but let me guess: d12/d2 = (d/d2) (2)^2 (3) = 2 (2) (3) = 12? Jim Propp

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Re: [math-fun] Re: Number derivative
by lkmitch＠att.net 17 Mar '04

17 Mar '04

Right, linearity is not preserved. In the paper (pdf available online), there's a discussion of differential equations under this notion; I guess its utility will depend on one's perspective. Kerry -- lkmitch(a)att.net www.fractalus.com/kerry > Then we have 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear. > Does this notion lead to something further, or is it a dead end? > > Gene

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Re: [math-fun] Re: Number derivative
by lkmitch＠att.net 17 Mar '04

17 Mar '04

It's (or at least, one flavor of it) based on the product rule: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. It only holds for natural numbers, due to the fundamental theorem of arithmetic. One can show that 1' = 0, and p' = 1 for any prime p. Then, for example, 6' = (2x3)' = 2'x3 + 2x3' = 1x3 + 2x1 = 5. You can find more details here: Ufnarovski, V., "How to Differentiate a Number," Journal of Integer Sequences, vol 6, 2003. Kerry Mitchell -- lkmitch(a)att.net www.fractalus.com/kerry

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Re: [math-fun] Number derivative
by Mike Stay 17 Mar '04

17 Mar '04

His site doesn't mention it, and google doesn't turn up anything for the query buium "number derivative" Any idea which papers? Other key words? John McKay wrote: > Alexandru Buium (Univ. New Maxico Albuqquerque) invented > such a thing. See his papers.. > > John -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] Number derivative
by MCKAY＠vax2.concordia.ca 17 Mar '04

17 Mar '04

Alexandru Buium (Univ. New Maxico Albuqquerque) invented such a thing. See his papers.. John

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[math-fun] kissing/number/derivative
by Daniel Asimov 17 Mar '04

17 Mar '04

1. News flash -- I just read in the MSRI newsletter that Oleg Musin has extended methods of Delsarte to prove that the kissing number in 4 dimensions is 24. (For some time it was known that it had to be either 24 or 25.) This means the kissing number is now known only in dimensions d = 1 (2), 2 (6), 3 (12), 4 (24), 8 (240), and 24 (196560). * * * 2. Just read that the 2004 Science Talent Search winners have been announced, and among the top 10 is a student whose project is about the "number derivative". QUESTION: What is the number derivative? Dan asimov(a)msri.org

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[math-fun] a search for a mathematical expression using a large database
by Simon Plouffe 13 Mar '04

13 Mar '04

Hello everyboy, I made an experiment over a period of several months using my inverter (the home version with 610 million constants). I took the latest values of the CODATA 2002 NIST table of physical constants and tried a vast experiment to find any reasonable mathematical expression for those ratios. http://physics.nist.gov/cuu/Constants/Table/allascii.txt That data is from december 2003. I used many simple and naive models to try to find anything, any possible expression as long as it is simple, short and easily explained. Here are the results : http://www.lacim.uqam.ca/~plouffe/Search.htm Important note : that article (preprint) is an exercice in numerical analysis and an attempt to find a mathematical and simple expression and NOT any attempt to any physical theory. My knowledge of physics is naive and probably outdated. I have no idea if this is making any sense in the real physical world. It is only <the best possible mathematical expression> that could exist for those numbers that have been found using what I believe are appropriate tools. The tables I used are the ones on the Inverter and a set of specialized tables constructed from the OEIS and my own tables that are not yet public. There are 2 main findings : First I discovered a weakness in the PSLQ, LLL or integer relations algorithm that only exist for a specific type of numbers. Second, I propose a set of at least 12 values among the 28 known values (actually there are 14 + inverses). In other words, I have a mathematical expression for 12 of the 14 values. These expressions are all generated by 1 number only. The second finding is related to the first as explained in the article. The article (preprint) as been submitted to a known periodical. Simon Plouffe

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Re: [math-fun] Nonrepeating sequence with 3 digits
by Jeffrey Shallit 12 Mar '04

12 Mar '04

Yes. This is a variant of the Thue sequence. The easiest example is the fixed point of 2 -> 210 1 -> 20 0 -> 1 You can see this in my book _Automatic Sequences_ with Allouche, exercise 33, p. 27. Jeffrey Shallit

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[math-fun] PDF article of ; a search for a mathematical expression using a large database
by Simon Plouffe 12 Mar '04

12 Mar '04

Sorry, some people can't read the fonts ? Here is the document in PDF http://www.lacim.uqam.ca/~plouffe/Search.pdf Simon Plouffe

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[math-fun] Re And you think You have problems ...
by N. J. A. Sloane 12 Mar '04

12 Mar '04

> >> I have a 1Mbit file; I'm willing to add 100K additional bits >> for error correction. What should I do? > >Perhaps you could add six bits of error correcting code for each block >of 64 bits. That provides room for a code of Hamming-distance three, so >you'd get single-error correction in each block. (But double-errors >would be corrected wrong.) > >-- >Don Reble djr(a)nk.ca > We can do a little better than that! If we have length 2^20 = 1048576 and 100000 check bits, we have rate .09536. So by the Gilbert-Varshamov bound there is a binary linear code with minimum distance 12792, which will correct any error patterm with up to 6396 errors. Neil Sloane

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