4,490 research outputs found
A Result About the Density of Iterated Line Intersections in the Plane
Let be a finite set of points in the plane and let be
the set of intersection points between pairs of lines passing through any two
points in . We characterize all configurations of points such that
iteration of the above operation produces a dense set. We also discuss partial
results on the characterization of those finite point-sets with rational
coordinates that generate all of through iteration of
.Comment: 10 pages, 8 figures (low-res for the arXiv), Computational Geometry:
Theory and Application
Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates
We study the motion of a vortex dipole in a Bose-Einstein condensate confined
to an anisotropic trap. We focus on a system of ordinary differential equations
describing the vortices' motion, which is in turn a reduced model of the
Gross-Pitaevskii equation describing the condensate's motion. Using a sequence
of canonical changes of variables, we reduce the dimension and simplify the
equations of motion. We uncover two interesting regimes. Near a family of
periodic orbits known as guiding centers, we find that the dynamics is
essentially that of a pendulum coupled to a linear oscillator, leading to
stochastic reversals in the overall direction of rotation of the dipole. Near
the separatrix orbit in the isotropic system, we find other families of
periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the
guiding center orbits, we derive an explicit iterated map that simplifies the
problem further. Numerical calculations are used to illustrate the phenomena
discovered through the analysis. Using the results from the reduced system we
are able to construct complex periodic orbits in the original, partial
differential equation, mean-field model for Bose-Einstein condensates, which
corroborates the phenomenology observed in the reduced dynamical equations
Remarks on endomorphisms and rational points
Let X be a variety over a number field and let f: X --> X be an "interesting"
rational self-map with a fixed point q. We make some general remarks concerning
the possibility of using the behaviour of f near q to produce many rational
points on X. As an application, we give a simplified proof of the potential
density of rational points on the variety of lines of a cubic fourfold
(originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints
corrected, appendix modified
On the arithmetic sums of Cantor sets
Let C_\la and C_\ga be two affine Cantor sets in with
similarity dimensions d_\la and d_\ga, respectively. We define an analog of
the Bandt-Graf condition for self-similar systems and use it to give necessary
and sufficient conditions for having \Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0 where
C_\la + C_\ga denotes the arithmetic sum of the sets. We use this result to
analyze the orthogonal projection properties of sets of the form C_\la \times
C_\ga. We prove that for Lebesgue almost all directions for which the
projection is not one-to-one, the projection has zero (d_\la +
d_\ga)-dimensional Hausdorff measure. We demonstrate the results on the case
when C_\la and C_\ga are the middle-(1-2\la) and middle-(1-2\ga) sets
Backlund transformations and knots of constant torsion
The Backlund transformation for pseudospherical surfaces, which is equivalent
to that of the sine-Gordon equation, can be restricted to give a transformation
on space curves that preserves constant torsion. We study its effects on closed
curves (in particular, elastic rods) that generate multiphase solutions for the
vortex filament flow (also known as the Localized Induction Equation). In doing
so, we obtain analytic constant-torsion representatives for a large number of
knot types.Comment: AMSTeX, 29 pages, 5 Postscript figures, uses BoxedEPSF.tex (all
necessary files are included in backlund.tar.gz
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