1,831 research outputs found
A remarkable periodic solution of the three-body problem in the case of equal masses
Using a variational method, we exhibit a surprisingly simple periodic orbit
for the newtonian problem of three equal masses in the plane. The orbit has
zero angular momentum and a very rich symmetry pattern. Its most surprising
feature is that the three bodies chase each other around a fixed eight-shaped
curve. Setting aside collinear motions, the only other known motion along a
fixed curve in the inertial plane is the ``Lagrange relative equilibrium" in
which the three bodies form a rigid equilateral triangle which rotates at
constant angular velocity within its circumscribing circle. Our orbit visits in
turns every ``Euler configuration" in which one of the bodies sits at the
midpoint of the segment defined by the other two (Figure 1). Numerical
computations by Carles Sim\'o, to be published elsewhere, indicate that the
orbit is ``stable" (i.e. completely elliptic with torsion). Moreover, they show
that the moment of inertia I(t) with respect to the center of mass and the
potential U(t) as functions of time are almost constant.Comment: 21 pages, published versio
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
A three-sphere swimmer for flagellar synchronization
In a recent letter (Friedrich et al., Phys. Rev. Lett. 109:138102, 2012), a
minimal model swimmer was proposed that propels itself at low Reynolds numbers
by a revolving motion of a pair of spheres. The motion of the two spheres can
synchronize by virtue of a hydrodynamic coupling that depends on the motion of
the swimmer, but is rather independent of direct hydrodynamic interactions.
This novel synchronization mechanism could account for the synchronization of a
pair of flagella, e.g. in the green algae Chlamydomonas. Here, we discuss in
detail how swimming and synchronization depend on the geometry of the model
swimmer and compute the swimmer design for optimal synchronization. Our
analysis highlights the role of broken symmetries for swimming and
synchronization.Comment: 25 pages, 4 color figures, provisionally accepted for publication in
the New Journal of Physic
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
- …