390,733 research outputs found
More distinct distances under local conditions
We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least (Formula presented.) distinct distances. Then V determines at least (Formula presented.) distinct distances, as n tends to infinity. © 2016 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelber
Formation Shape Control Based on Distance Measurements Using Lie Bracket Approximations
We study the problem of distance-based formation control in autonomous
multi-agent systems in which only distance measurements are available. This
means that the target formations as well as the sensed variables are both
determined by distances. We propose a fully distributed distance-only control
law, which requires neither a time synchronization of the agents nor storage of
measured data. The approach is applicable to point agents in the Euclidean
space of arbitrary dimension. Under the assumption of infinitesimal rigidity of
the target formations, we show that the proposed control law induces local
uniform asymptotic stability. Our approach involves sinusoidal perturbations in
order to extract information about the negative gradient direction of each
agent's local potential function. An averaging analysis reveals that the
gradient information originates from an approximation of Lie brackets of
certain vector fields. The method is based on a recently introduced approach to
the problem of extremum seeking control. We discuss the relation in the paper
Geometrical Ambiguity of Pair Statistics. I. Point Configurations
Point configurations have been widely used as model systems in condensed
matter physics, materials science and biology. Statistical descriptors such as
the -body distribution function is usually employed to characterize
the point configurations, among which the most extensively used is the pair
distribution function . An intriguing inverse problem of practical
importance that has been receiving considerable attention is the degree to
which a point configuration can be reconstructed from the pair distribution
function of a target configuration. Although it is known that the pair-distance
information contained in is in general insufficient to uniquely determine
a point configuration, this concept does not seem to be widely appreciated and
general claims of uniqueness of the reconstructions using pair information have
been made based on numerical studies. In this paper, we introduce the idea of
the distance space, called the space. The pair distances of a
specific point configuration are then represented by a single point in the
space. We derive the conditions on the pair distances that can be
associated with a point configuration, which are equivalent to the
realizability conditions of the pair distribution function . Moreover, we
derive the conditions on the pair distances that can be assembled into distinct
configurations. These conditions define a bounded region in the
space. By explicitly constructing a variety of degenerate point configurations
using the space, we show that pair information is indeed
insufficient to uniquely determine the configuration in general. We also
discuss several important problems in statistical physics based on the
space.Comment: 28 pages, 8 figure
Topological Signals of Singularities in Ricci Flow
We implement methods from computational homology to obtain a topological
signal of singularity formation in a selection of geometries evolved
numerically by Ricci flow. Our approach, based on persistent homology, produces
precise, quantitative measures describing the behavior of an entire collection
of data across a discrete sample of times. We analyze the topological signals
of geometric criticality obtained numerically from the application of
persistent homology to models manifesting singularities under Ricci flow. The
results we obtain for these numerical models suggest that the topological
signals distinguish global singularity formation (collapse to a round point)
from local singularity formation (neckpinch). Finally, we discuss the
interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure
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