[math-fun] refining the notion of density
Dan Asimov's remark "Finally, the optimality of a sphere packing is determined by the limit of its density in a ball B(R) of radius R, as R -> oo, when this limit exists, which is independent of the center of the ball. This means that any modification of a packing in a bounded region will have no effect on the limiting density." reminds me of a research program currently on my back-burner, awaiting some ideas to make it fully workable. It's a theory in which density is measured not by real numbers but by a richer non-Archimedean number system that allows one to simultaneously consider regular packings, line defects (which are infinitesimal compared to regular packings), and point defects (which are infinitesimal compared to line defects). In this theory, modifying a packing in a bounded region, by removing n spheres and replacing them by n' spheres, changes the "refined density" by an infinitesimal but non-zero amount proportional to n-n'. I can prove some of the basic lemmas in this theory, but what I really want to do in the first article on the theory is to show that, relative to my refined notion of density, the only densest packings of the plane by unit disks are the ordinary densest lattice packings. Maybe one of the existing proofs of Fejes Toth's theorem can be modified to work with refined density, but I don't know how to do this yet. Anyone interested in more details? Jim Propp
Are you using Conway Numbers as density values? ----- Quoting James Propp <jamespropp@gmail.com>:
Dan Asimov's remark
"Finally, the optimality of a sphere packing is determined by the limit of its density in a ball B(R) of radius R, as R -> oo, when this limit exists, which is independent of the center of the ball. This means that any modification of a packing in a bounded region will have no effect on the limiting density."
reminds me of a research program currently on my back-burner, awaiting some ideas to make it fully workable. It's a theory in which density is measured not by real numbers but by a richer non-Archimedean number system that allows one to simultaneously consider regular packings, line defects (which are infinitesimal compared to regular packings), and point defects (which are infinitesimal compared to line defects). In this theory, modifying a packing in a bounded region, by removing n spheres and replacing them by n' spheres, changes the "refined density" by an infinitesimal but non-zero amount proportional to n-n'.
I can prove some of the basic lemmas in this theory, but what I really want to do in the first article on the theory is to show that, relative to my refined notion of density, the only densest packings of the plane by unit disks are the ordinary densest lattice packings. Maybe one of the existing proofs of Fejes Toth's theorem can be modified to work with refined density, but I don't know how to do this yet.
Anyone interested in more details?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think there is a great opportunity to apply the Surreals to probability questions. E.g., in *some* sense, if you pick a random number from J = [0,1), once for each index in I = [0,1), then the probability of never picking 1/3 should heuristically be (1 - 1/c)^c (where c denotes the Surreal that is the least ordinal of cardinality 2^aleph_0). Then in the Surreals, (1 - 1/c)^c = 1/e, up to an infinitesimal. Then again, it's necessary to be careful about *what sense* is meant by this problem. For, using x |-> 2x there is a simple bijection from I = [0,1) to 2I = [0,2). This leads to the paradox that the probability of never picking 1/3 is both 1/e and 1/e^2. --Dan On 2013-06-23, at 11:04 AM, rcs@xmission.com wrote:
Are you using Conway Numbers as density values?
On Sun, Jun 23, 2013 at 11:52 AM, James Propp <jamespropp@gmail.com> wrote:
Dan Asimov's remark
"Finally, the optimality of a sphere packing is determined by the limit of its density in a ball B(R) of radius R, as R -> oo, when this limit exists, which is independent of the center of the ball. This means that any modification of a packing in a bounded region will have no effect on the limiting density."
reminds me of a research program currently on my back-burner, awaiting some ideas to make it fully workable. It's a theory in which density is measured not by real numbers but by a richer non-Archimedean number system that allows one to simultaneously consider regular packings, line defects (which are infinitesimal compared to regular packings), and point defects (which are infinitesimal compared to line defects). In this theory, modifying a packing in a bounded region, by removing n spheres and replacing them by n' spheres, changes the "refined density" by an infinitesimal but non-zero amount proportional to n-n'.
I can prove some of the basic lemmas in this theory, but what I really want to do in the first article on the theory is to show that, relative to my refined notion of density, the only densest packings of the plane by unit disks are the ordinary densest lattice packings. Maybe one of the existing proofs of Fejes Toth's theorem can be modified to work with refined density, but I don't know how to do this yet.
Anyone interested in more details?
Yes. Tangentially related thought: I want to say that I've seen two different types of non-Archimedean measures. You seem to be talking about the first kind, which only measures the size of integer-dimensional stuff; the second kind gives measures of the form sum_d H^d(X_d) epsilon^(-d) where epsilon is an infinitesimal, H^d is the d-dimensional Hausdorff measure, and X_d is the subset of X that locally looks d-dimensional. Certainly the first kind requires less machinery to define, though you do have to make some arbitrary choices -- if a point has measure epsilon, will [0,1], [0,1), or (0,1) get measure 1? The second kind is heavier weight but useful if you want to say things like "conditional on picking a point in X, the probability of picking a point in Y is the real part of m(X intersect Y) / m(X)". Open problem (to me) for the reader: given a real valued P(Y | X) such that P(X&Y | Z) = P(Y | X&Z) P(X | Z) for all X, Y and Z, when can we find a non-Archimedean valued P(X) such that P(Y | X) is always equal to the real part of P(Y&X) / P(X) ? -Thomas C
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Rich S. asks:
Are you using Conway Numbers as density values?
No; my density "values" are a different sort of thing; more along the lines of elements of the ordered ting |R[epsilon] where epsilon is a formal indeterminate satisfying epsilon < r for all r > 0 in |R. Or at least, that's what I _think_ they're like. My values come from a somewhat abstract modding-out construction so I don't have a concrete picture of what they look like in general. Jim Propp
participants (4)
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Dan Asimov -
James Propp -
rcs＠xmission.com -
Thomas Colthurst