[math-fun] Balanced mass transport in Z
I call to math-funsters' attention my recent MathOverflow post http://mathoverflow.net/questions/202165/transitivity-of-balanced-mass-trans... which has a math-fun-ish quality. One thing that I know, but that I don't prove in that post (though I do mention it in one of the two affiliated posts to MathOverflow) is that if mu is the measure that gives unit mass to every integer greater than or equal to 1, and nu is the measure that gives unit mass to every integer greater than or equal to 0, then nu cannot be obtained from mu by a single balanced-transport move, or by a finite sequence of such moves, or even by an infinite sequence of such moves convergent in the total variation metric. I pose this as a puzzle for math-fun, not only because it has a cute answer, but because I'm hoping it has more than one cute answer, and that one of you will find a different answer to this puzzle than the one that I know! (And it would also be nice if one of you proved or disproved transitivity, but that's a separate issue.) Jim Propp
To make it clearer why I consider this topic math-fun-ish, let me state a related puzzle in a self-contained and hopefully fun way. This is more or less the puzzle that I inadvertently posed in my original post (by rashly claiming that it could not be done!) and that Christian Remling solved. Suppose I give you the (infinite) mass distribution in which unit masses located at the odd positive integers, and I ask you to effectively move the mass at location 1 to location 0. You cannot simply slide the mass at location 1 one unit to the left; that would be too easy. The constraint I impose is that you must divide the infinite mass distribution into infinitely many pieces, each supported on an interval of width 10 (or some larger number of your choosing), and then re-allocate the mass in each piece, in a fashion that preserves both the total mass and center of mass of each piece. You are allowed to subdivide mass. For instance, you could split the mass at every odd positive integer 2n+1 and send half the mass to 2n and the other half to 2n+2. If you do this for all n, you will obtain a distribution in which 1/2 a unit of mass is located at 0 while 1 unit of mass is located at each of the even positive integers. But that does not achieve the desired target distribution. It is not hard to achieve the target distribution by employing several of these moves in succession; but can you get there using just one such move? Christian figured out how to do that; can you? What neither of us knows how to prove (or at least I don't, and if Christian does he hasn't told me!) is whether you can in fact effectively move ANY finite number of the unit masses wherever you like, just by dividing the measure into pieces of bounded length (subdividing the individual masses as you like) and re-allocating the mass in each piece, in a fashion that preserves both the total mass and center of mass of each piece. Can any of you help? Jim Propp On Mon, Apr 6, 2015 at 12:54 PM, James Propp <jamespropp@gmail.com> wrote:
I call to math-funsters' attention my recent MathOverflow post
http://mathoverflow.net/questions/202165/transitivity-of-balanced-mass-trans...
which has a math-fun-ish quality.
One thing that I know, but that I don't prove in that post (though I do mention it in one of the two affiliated posts to MathOverflow) is that if mu is the measure that gives unit mass to every integer greater than or equal to 1, and nu is the measure that gives unit mass to every integer greater than or equal to 0, then nu cannot be obtained from mu by a single balanced-transport move, or by a finite sequence of such moves, or even by an infinite sequence of such moves convergent in the total variation metric.
I pose this as a puzzle for math-fun, not only because it has a cute answer, but because I'm hoping it has more than one cute answer, and that one of you will find a different answer to this puzzle than the one that I know!
(And it would also be nice if one of you proved or disproved transitivity, but that's a separate issue.)
Jim Propp
This question and the questions it raises are extremely interesting!!! I can't help wondering about the variant of this question where the space on which the locally-finite atomic measure is defined is the real line: (-oo,oo) . And maybe instead of defining a relation in terms of [a discrete sequence of certain kinds of moves — and the limit when such a sequence converges in a certain metric]: Perhaps instead define a nice topology on (say) {locally-finite atomic measures with countable support} (from the total variation metric? the Wasserstein metric? the compact-open topology on functions with locally-finite countable support from R^oo := Prod_{k=1,oo} (-oo,oo) to the nonnegative reals [0,oo) ?) . . . and then define ~ by mu ~ nu <=> there is a continuous curve of such measures starting at mu and ending at nu. --Dan
On Apr 6, 2015, at 9:54 AM, James Propp <jamespropp@gmail.com> wrote:
I call to math-funsters' attention my recent MathOverflow post
http://mathoverflow.net/questions/202165/transitivity-of-balanced-mass-trans...
which has a math-fun-ish quality.
One thing that I know, but that I don't prove in that post (though I do mention it in one of the two affiliated posts to MathOverflow) is that if mu is the measure that gives unit mass to every integer greater than or equal to 1, and nu is the measure that gives unit mass to every integer greater than or equal to 0, then nu cannot be obtained from mu by a single balanced-transport move, or by a finite sequence of such moves, or even by an infinite sequence of such moves convergent in the total variation metric.
I pose this as a puzzle for math-fun, not only because it has a cute answer, but because I'm hoping it has more than one cute answer, and that one of you will find a different answer to this puzzle than the one that I know!
(And it would also be nice if one of you proved or disproved transitivity, but that's a separate issue.)
participants (2)
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Dan Asimov -
James Propp