15,275 research outputs found
Point configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in
equilibrium under all force laws (that act between pairs of points with
strength given by a fixed function of distance). It is straightforward to show
that every sufficiently symmetrical configuration is balanced, but the converse
is far from obvious. In 1957 Leech completely classified the balanced
configurations in R^3, and his classification is equivalent to the converse for
R^3. In this paper we disprove the converse in high dimensions. We construct
several counterexamples, including one with trivial symmetry group.Comment: 10 page
Point configurations that are asymmetric yet balanced
A configuration of particles confined to a sphere is balanced if it is in
equilibrium under all force laws (that act between pairs of points with
strength given by a fixed function of distance). It is straightforward to show
that every sufficiently symmetrical configuration is balanced, but the converse
is far from obvious. In 1957 Leech completely classified the balanced
configurations in R^3, and his classification is equivalent to the converse for
R^3. In this paper we disprove the converse in high dimensions. We construct
several counterexamples, including one with trivial symmetry group.Comment: 10 page
An asymptotic existence result on compressed sensing matrices
For any rational number and all sufficiently large we give a
deterministic construction for an compressed
sensing matrix with -recoverability where . Our
method uses pairwise balanced designs and complex Hadamard matrices in the
construction of -equiangular frames, which we introduce as a
generalisation of equiangular tight frames. The method is general and produces
good compressed sensing matrices from any appropriately chosen pairwise
balanced design. The -recoverability performance is specified as a
simple function of the parameters of the design. To obtain our asymptotic
existence result we prove new results on the existence of pairwise balanced
designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201
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