October 2009 - mathfun@mailman.xmission.com

Re: [math-fun] Which letters are used as variables in which languages?
by Dan Asimov 06 Oct '09

06 Oct '09

Andy wrote: << After all, English mathematics routinely uses i as a variable in subscripts, and as the square root of -1, and that never seems to be a problem. >> I don't have any info about your other comments, but I can confirm that in English mathematics where the (or a) square root of -1 plays an important role, the subscript i (for an integer) is usually avoided. --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] Which letters are used as variables in which languages?
by Andy Latto 05 Oct '09

05 Oct '09

I was writing on another list about the mathematician's use of the subjunctive, in phrases like "Let G be a group" or "Soit G un groupe". I wasn't sure whether "groupe" was masculine or feminine, so I googled "Soit G une group" and "Soit G un groupe", and got 214 and 252,000 hits, respectively, so that answered that. But I was amused that Google asked me "Did you mean 'Soit L un groupe'?" And sure enough, "Soit L un groupe" gets 5 million hits. Apparently the conventions of what variables are used to represent what sorts of entities varies from English to French. English speakers typically use G because it's the first letter of group, and if there's a second group to talk about we call it H, since that's the next letter of the alphabet. Any idea why the French use "L"? What letter to other languages use? And is there any mathematics written in Greek, and if so, do they have to avoid using pi or phi (and what other letters? gamma, often used for Euler's constant?) as variables? Or is it sufficiently clear from context? After all English mathematics routinely uses i as a variable in subscripts, and as the square root of -1, and that never seems to be a problem. -- Andy.Latto(a)pobox.com

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[math-fun] alternative proof of the infinity of primes
by rcs＠xmission.com 05 Oct '09

05 Oct '09

Theorem: An infinity of primes. Proof: Assume the contrary, that there are only P primes. Consider the set of integers up to N that are representible as products of the P primes (allowing repeated prime factors). The size of this set is O((logN)^P), which grows more slowly than N. QED. Restatement: If there were only a finite number of primes, the set of fully-factorable D-digit numbers would only be polynomial in D. I'm amused that the argument doesn't depend on properties of divisibility, but rather on the assoc-comm property of multiplication, and that the product XY is much bigger than X and Y. [extra credit: find more words with 5 I's. serious XC: turn this into a lower bound on pi(N).] This argument would have been perfectly comprehensible to Fermat, though he might have had trouble formalizing it. But the idea of arguing based on rate-of-growth, comparing polynomial(D) to exponential(D), seems modern. So I wonder when this proof was discovered. Rich

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[math-fun] clumpy heapsort tree
by Steve Witham 05 Oct '09

05 Oct '09

Wondering whether anyone recognizes this binary tree: 0 1 2 3 6 9 12 4 5 7 8 10 11 13 14 15 30 45 ... 240 16 17 31 32 46 47 18 21 24 27 33 36 39 42 48 51 54 57 19 20 22 23 25 26 28 29 34 35 37 38 40 41 43 44 49 50 52 53 55 56 58 59 ... (To be clear, 15 and 30 are daughters of 4, 45 and 60 are daughters of 5, and so forth.) I would think the OEIS-appropriate version was 0,1,2,3,6,9,12,4,5,7,8... or 1,2,3,4,7,10,13,5,6,8,9... Those aren't in there, but I'm not sure that's the right way to encode the tree. The parent function (starting from child 1) is 0,0,1,3,3,1,6,6,2,9,9,2,12,12,4,15,15... What I'm really wondering is whether someone's done heapsort with this arrangement before. --Steve

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Re: [math-fun] minmax question
by Steve Witham 02 Oct '09

02 Oct '09

>From: "David Wilson" <davidwwilson(a)comcast.net> > >Let f(n, k) be the number of binary min and max operations required to find >the kth largest of n integers. > >One way to find f(n, k) is to take the min over all of the maxes of subsets >of k elements in A. This proves f(n, k) <= n C(n-1, k-1) - 1. Or if k > n/2, take the max of the mins of (n-k)... >Is it possible to do better? [mentions of linear median/select] (& later) >I'm not talking about an algorithm, but a fixed expression. You can take a sorting network as in http://en.wikipedia.org/wiki/Sorting_network and convert it to the kind of expression you're looking for, by replicating every reused expression (and sub...expression). In that way, the problem converts to optimizing sorting networks for a new cost function. --Steve

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[math-fun] minmax question
by David Wilson 02 Oct '09

02 Oct '09

Let f(n, k) be the number of binary min and max operations required to find the kth largest of n integers. One way to find f(n, k) is to take the min over all of the maxes of subsets of k elements in A. This proves f(n, k) <= n C(n-1, k-1) - 1. Is it possible to do better?

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[math-fun] Lego orrery design
by Mike Stay 02 Oct '09

02 Oct '09

I'm thinking about the problem of designing gear trains for a lego orrery which I'd like to be fairly accurate. The standard gear sizes are 8, 16, 24, 40, so I have ratios of 1:2:3:5 to work with. I figure that since lg(3) (where lg is log base 2) is irrational, you can get arbitrary accuracy out of a gear train. I think I need a + b lg(3) + c lg(5) ~= lg(period), with a, b, c small. Is LLL the right tool for solving this kind of thing? -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

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