April 2009 - mathfun@mailman.xmission.com

Re: [math-fun] Music theory
by Dan Asimov 22 Apr '09

22 Apr '09

I have to admit, it's hard to believe there's that much to say about music theory that such a book could be worth $59.53 new. Of course, * notes are largely determined by frequency, i.e. a real number; * the human ear perceives a doubled frequency as "the same note", kind of; * it also perceives combinations of notes with frequency ratios being fractions with small denominators as pleasing. Beyond this, it's all experimentation with what sounds good. (On the other hand, there are some extremely complex musical traditions. Some (East) Indian music relies on a scale of 40+ notes, and there is much to be said about it. Though I wouldn't call it "theory".) --Dan Hilarie wrote: << I've come across a book of essays titled "Music Theory and Mathematics" which includes writing by Jack Douthett, John Clough, et al. Is anyone here familiar with their work? >> _____________________________________________________________________ "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] Music theory
by Hilarie Orman 22 Apr '09

22 Apr '09

I've come across a book of essays titled "Music Theory and Mathematics" which includes writing by Jack Douthett, John Clough, et al. Is anyone here familiar with their work? Hilarie

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Re: [math-fun] Bijection between the sets of even and odd subsets of an even-sized set
by Dan Asimov 20 Apr '09

20 Apr '09

All the places MathSciNet finds this paper, I can download it, but the downloaded .pdf file (even when it was a downloaded .ps file that was then converted to .pdf on my computer) was riddled with some kind of glitches. (They made pages look like a blacktop road with a dashed white line down the middle.) --Dan << "Producing New Bijections from Old" by David Feldman and me (published in Advances in Mathematics, volume 113 (1995), pages 1-44) >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Bijection between the sets of even and odd subsets of an even-sized set
by James Propp 19 Apr '09

19 Apr '09

Dan Asimov writes: >P.S. I meant to mention that someone else brought up this example in >math-fun some years ago. That could have been me. >But I don't recall knowing exactly what "natural" meant in this context. It means invariant/equivariant under a relabelling of the elements of S. The article "Producing New Bijections from Old" by David Feldman and me (published in Advances in Mathematics, volume 113 (1995), pages 1-44) goes into more detail about this. In this particular case, the upshot is: If a set S comes equipped with a specified subset T of odd cardinality (which is automatically true if S itself has odd cardinality), then there is a bijection f between the odd-cardinality and even-cardinality subsets of S that is equivariant with respect to the permutations of S that preserve T (e.g. the map that sends every subset of S to its symmetric difference with T), and conversely, if a set S comes equipped with a bijection f between the odd-cardinality and even-cardinality subsets of S, then there is a subset of S of odd cardinality that is invariant under all the permutations of S that respect f (e.g., f of the empty set). Jim Propp

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[math-fun] Biijection between the sets of even and odd subsets of an even-sized set
by Dan Asimov 18 Apr '09

18 Apr '09

Given a finite set S of size n, let E denote the set of its subsets with even size and let O denote the set of its subsets of odd size. Then it's easy to see that #(E) = #(O). (Expanding the LHS of (1-1)^n = 0 does the trick nicely.) When n is itself odd, there is a simple natural bijection between E and O: just map any subset of S to its complement. But when n is even, there doesn't seem to be anything nearly so natural. (If we set S = {1,...,n} and each subset X = {a,b,...,c} where a < b < ... < c, then one can certainly list the elements of E by size and, within each size, by lexicographic ordering, and likewise for O. This defines a bijection E -> O. But that's less natural than just taking each subset to its complement.) QUESTION: Exactly how is "natural" defined so that complementation is a natural bijection E -> O for n odd, but there is no known natural bijection E -> O for the case of n even? Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Biijection between the sets of even and odd subsets of an even-sized set
by Dan Asimov 18 Apr '09

18 Apr '09

Very nice. And it's clear why that doesn't work for #(S) = odd. --Dan Andy wrote: << On Sat, Apr 18, 2009 at 2:47 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > Given a finite set S of size n, let E denote the set of its subsets with > even size and let O denote the set of its subsets of odd size. Then it's > easy to see that > > #(E) = #(O). > > (Expanding the LHS of (1-1)^n = 0 does the trick nicely.) > > When n is itself odd, there is a simple natural bijection between E and O: > just map any subset of S to its complement. > > But when n is even, there doesn't seem to be anything nearly so natural. There is no "natural" bijection, that is one that is invariant under a permutation of the elemnts of S. Proof: What does the bijection map the empty set to? An arbitrary choice is required here. >> _____________________________________________________________________ "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] P.S. Re: Biijection between the sets of even and odd subsets of an even-sized set
by Dan Asimov 18 Apr '09

18 Apr '09

P.S. I meant to mention that someone else brought up this example in math-fun some years ago. But I don't recall knowing exactly what "natural" meant in this context. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] If We Have Free Will, Then So Do Electrons
by Dan Asimov 17 Apr '09

17 Apr '09

Mike Stay wrote: << Obligatory reference: http://www.overcomingbias.com/2008/04/zombie-movie.html >> That's utterly hilarious!!! The funniest thing I've read in months. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] vitamin drop experiment
by Dan Asimov 12 Apr '09

12 Apr '09

Hmmm. Assuming a cylindrical bottle-interior, let it be centered on the z-axis with its bottom at z=0. WLOG we can normalize the (empty) bottle weight to 1, and assume its center of gravity to be at z=h. Then there is a constant C such that the weight of the liquid, filled to a height of z=t, is Ct in normalized units. Thus the center of gravity of the bottle filled to height t is z(t) = (1*h + Ct*(t/2)) / (1 + Ct) z'(t) = ((1+Ct)*(Ct) - (h + (C/2)t^2)(C)) / (1 + Ct)^2, which is 0 when Ct^2 + 4t - 2h = 0, or t = (-1 + sqrt(4 + 2hC)) / C. So the minimum center of gravity is at h + (-1 + sqrt(4 + 2C*h))^2 / 2C z = -------------------------------- sqrt(4 + 2C*h) Unless, as usual, I made a mistake. --Dan rwg wrote: << . . . a math-fun tie-in is to find, given the bottle weight and density of the "fluid", the fill depth which minimizes the height of the center of gravity. >> _____________________________________________________________________ "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] Weeks Between Easters
by Hans Havermann 12 Apr '09

12 Apr '09

Long before even one round of the 5700000-year Gregorian Easter cycle is up, we will likely no longer use that formulation, assuming even that mankind survives that long. Nevertheless, as a mathematical entity - there it is, straddling the boundary between religious necessity and astronomical reality. When two separate Easters are one year apart, they can be 50, 51, 54, or 55 weeks apart. When they are 3 years apart, there are 5 weekly possibilities. When they are 10 years apart, there are 6 weekly possibilities. When they are 125 years apart, there are 7 weekly possibilities. Can there ever be 8 weekly possibilities?

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