10,199 research outputs found
Loop space homology associated with the mod 2 Dickson invariants
Peer reviewedPublisher PD
A new two-dimensional lattice model that is "consistent around a cube"
For two-dimensional lattice equations one definition of integrability is that
the model can be naturally and consistently extended to three dimensions, i.e.,
that it is "consistent around a cube" (CAC). As a consequence of CAC one can
construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted
a search based on this principle and certain additional assumptions. One of
those assumptions was the "tetrahedron property", which is satisfied by most
known equations. We present here one lattice equation that satisfies the
consistency condition but does not have the tetrahedron property. Its Lax pair
is also presented and some basic properties discussed.Comment: 8 pages in LaTe
The lattice Schwarzian KdV equation and its symmetries
In this paper we present a set of results on the symmetries of the lattice
Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point
symmetries and, using its associated spectral problem, an infinite sequence of
generalized symmetries and master symmetries. We finally show that we can use
master symmetries of the lSKdV equation to construct non-autonomous
non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE
VI
Radial Distribution Function for Semiflexible Polymers Confined in Microchannels
An analytic expression is derived for the distribution of the
end-to-end distance of semiflexible polymers in external potentials
to elucidate the effect of confinement on the mechanical and statistical
properties of biomolecules. For parabolic confinement the result is exact
whereas for realistic potentials a self-consistent ansatz is developed, so that
is given explicitly even for hard wall confinement. The
theoretical result is in excellent quantitative agreement with fluorescence
microscopy data for actin filaments confined in rectangularly shaped
microchannels. This allows an unambiguous determination of persistence length
and the dependence of statistical properties such as Odijk's deflection
length on the channel width . It is shown that neglecting the
effect of confinement leads to a significant overestimation of bending
rigidities for filaments
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
Difference schemes with point symmetries and their numerical tests
Symmetry preserving difference schemes approximating second and third order
ordinary differential equations are presented. They have the same three or
four-dimensional symmetry groups as the original differential equations. The
new difference schemes are tested as numerical methods. The obtained numerical
solutions are shown to be much more accurate than those obtained by standard
methods without an increase in cost. For an example involving a solution with a
singularity in the integration region the symmetry preserving scheme, contrary
to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure
Discrete derivatives and symmetries of difference equations
We show on the example of the discrete heat equation that for any given
discrete derivative we can construct a nontrivial Leibniz rule suitable to find
the symmetries of discrete equations. In this way we obtain a symmetry Lie
algebra, defined in terms of shift operators, isomorphic to that of the
continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page
Lie point symmetries of difference equations and lattices
A method is presented for finding the Lie point symmetry transformations
acting simultaneously on difference equations and lattices, while leaving the
solution set of the corresponding difference scheme invariant. The method is
applied to several examples. The found symmetry groups are used to obtain
particular solutions of differential-difference equations
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