13,833 research outputs found
Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture
Geometry of the tracks left by a bicycle is closely related with the
so-called Prytz planimeter and with linear fractional transformations of the
complex plane. We describe these relations, along with the history of the
problem, and give a proof of a conjecture made by Menzin in 1906.Comment: 20 pages, 18 figure
Attractive forces in microporous carbon electrodes for capacitive deionization
The recently developed modified Donnan (mD) model provides a simple and
useful description of the electrical double layer in microporous carbon
electrodes, suitable for incorporation in porous electrode theory. By
postulating an attractive excess chemical potential for each ion in the
micropores that is inversely proportional to the total ion concentration, we
show that experimental data for capacitive deionization (CDI) can be accurately
predicted over a wide range of applied voltages and salt concentrations. Since
the ion spacing and Bjerrum length are each comparable to the micropore size
(few nm), we postulate that the attraction results from fluctuating bare
Coulomb interactions between individual ions and the metallic pore surfaces
(image forces) that are not captured by meanfield theories, such as the
Poisson-Boltzmann-Stern model or its mathematical limit for overlapping double
layers, the Donnan model. Using reasonable estimates of the micropore
permittivity and mean size (and no other fitting parameters), we propose a
simple theory that predicts the attractive chemical potential inferred from
experiments. As additional evidence for attractive forces, we present data for
salt adsorption in uncharged microporous carbons, also predicted by the theory.Comment: 19 page
Lie Symmetries and Exact Solutions of First Order Difference Schemes
We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.Comment: Minor changes and journal-re
Multiscale expansion and integrability properties of the lattice potential KdV equation
We apply the discrete multiscale expansion to the Lax pair and to the first
few symmetries of the lattice potential Korteweg-de Vries equation. From these
calculations we show that, like the lowest order secularity conditions give a
nonlinear Schroedinger equation, the Lax pair gives at the same order the
Zakharov and Shabat spectral problem and the symmetries the hierarchy of point
and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007
Conferenc
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl
Photoassociation dynamics in a Bose-Einstein condensate
A dynamical many body theory of single color photoassociation in a
Bose-Einstein condensate is presented. The theory describes the time evolution
of a condensed atomic ensemble under the influence of an arbitrarily varying
near resonant laser pulse, which strongly modifies the binary scattering
properties. In particular, when considering situations with rapid variations
and high light intensities the approach described in this article leads, in a
consistent way, beyond standard mean field techniques. This allows to address
the question of limits to the photoassociation rate due to many body effects
which has caused extensive discussions in the recent past. Both, the possible
loss rate of condensate atoms and the amount of stable ground state molecules
achievable within a certain time are found to be stronger limited than
according to mean field theory. By systematically treating the dynamics of the
connected Green's function for pair correlations the resonantly driven
population of the excited molecular state as well as scattering into the
continuum of non-condensed atomic states are taken into account. A detailed
analysis of the low energy stationary scattering properties of two atoms
modified by the near resonant photoassociation laser, in particular of the
dressed state spectrum of the relative motion prepares for the analysis of the
many body dynamics. The consequences of the finite lifetime of the resonantly
coupled bound state are discussed in the two body as well as in the many body
context. Extending the two body description to scattering in a tight trap
reveals the modifications to the near resonant adiabatic dressed levels caused
by the decay of the excited molecular state.Comment: 27 pages revtex, 16 figure
Integrability of Differential-Difference Equations with Discrete Kinks
In this article we discuss a series of models introduced by Barashenkov,
Oxtoby and Pelinovsky to describe some discrete approximations to the \phi^4
theory which preserve travelling kink solutions. We show, by applying the
multiple scale test that they have some integrability properties as they pass
the A_1 and A_2 conditions. However they are not integrable as they fail the
A_3 conditions.Comment: submitted to the Proceedings of the workshop "Nonlinear Physics:
Theory and Experiment.VI" in a special issue di Theoretical and Mathematical
Physic
Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
We consider multiple lattices and functions defined on them. We introduce
slow varying conditions for functions defined on the lattice and express the
variation of a function in terms of an asymptotic expansion with respect to the
slow varying lattices.
We use these results to perform the multiple--scale reduction of the lattice
potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur
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