8,066 research outputs found
Helly-Type Theorems in Property Testing
Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If is a set of
points in , we say that is -clusterable if it can be
partitioned into clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object . In this paper, as an
application of Helly's theorem, by taking a constant size sample from , we
present a testing algorithm for -clustering, i.e., to distinguish
between two cases: when is -clusterable, and when it is
-far from being -clusterable. A set is -far
from being -clusterable if at least
points need to be removed from to make it -clusterable. We solve
this problem for and when is a symmetric convex object. For , we
solve a weaker version of this problem. Finally, as an application of our
testing result, in clustering with outliers, we show that one can find the
approximate clusters by querying a constant size sample, with high probability
Jamming Percolation and Glass Transitions in Lattice Models
A new class of lattice gas models with trivial interactions but constrained
dynamics are introduced. These are proven to exhibit a dynamical glass
transition: above a critical density, rho_c, ergodicity is broken due to the
appearance of an infinite spanning cluster of jammed particles. The fraction of
jammed particles is discontinuous at the transition, while in the unjammed
phase dynamical correlation lengths and timescales diverge as
exp[C(rho_c-rho)^(-mu)]. Dynamic correlations display two-step relaxation
similar to glass-formers and jamming systems.Comment: 4 pages, 2 figs. Final version accepted for publication in Phys. Rev.
Let
A Transfer Matrix for the Backbone Exponent of Two-Dimensional Percolation
Rephrasing the backbone of two-dimensional percolation as a monochromatic
path crossing problem, we investigate the latter by a transfer matrix approach.
Conformal invariance links the backbone dimension D_b to the highest eigenvalue
of the transfer matrix T, and we obtain the result D_b=1.6431 \pm 0.0006. For a
strip of width L, T is roughly of size 2^{3^L}, but we manage to reduce it to
\sim L!. We find that the value of D_b is stable with respect to inclusion of
additional ``blobs'' tangent to the backbone in a finite number of points.Comment: 19 page
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