42 research outputs found
Algebraic Entropy for lattice equations
We give the basic definition of algebraic entropy for lattice equations. The
entropy is a canonical measure of the complexity of the dynamics they define.
Its vanishing is a signal of integrability, and can be used as a powerful
integrability detector. It is also conjectured to take remarkable values
(algebraic integers)
Algebraic entropy for differential-delay equations
We extend the definition of algebraic entropy to a class of
differential-delay equations. The vanishing of the entropy, as a structural
property of an equation, signals its integrability. We suggest a simple way to
produce differential-delay equations with vanishing entropy from known
integrable differential-difference equations
On the algebraic structure of rational discrete dynamical systems
We show how singularities shape the evolution of rational discrete dynamical
systems. The stabilisation of the form of the iterates suggests a description
providing among other things generalised Hirota form, exact evaluation of the
algebraic entropy as well as remarkable polynomial factorisation properties. We
illustrate the phenomenon explicitly with examples covering a wide range of
models
Weak Lax pairs for lattice equations
We consider various 2D lattice equations and their integrability, from the
point of view of 3D consistency, Lax pairs and B\"acklund transformations. We
show that these concepts, which are associated with integrability, are not
strictly equivalent. In the course of our analysis, we introduce a number of
black and white lattice models, as well as variants of the functional
Yang-Baxter equation
Searching for integrable lattice maps using factorization
We analyze the factorization process for lattice maps, searching for
integrable cases. The maps were assumed to be at most quadratic in the
dependent variables, and we required minimal factorization (one linear factor)
after 2 steps of iteration. The results were then classified using algebraic
entropy. Some new models with polynomial growth (strongly associated with
integrability) were found. One of them is a nonsymmetric generalization of the
homogeneous quadratic maps associated with KdV (modified and Schwarzian), for
this new model we have also verified the "consistency around a cube".Comment: To appear in Journal of Physics A. Some changes in reference
Integrable Lattice Maps: , a Rational Version of
We give a rational form of a generic two-dimensional "quad" map, containing
the so-called case, but whose coefficients are free. Its integrability is
proved using the calculation of algebraic entropy
Industry influence on corporate working capital decisions
This paper provides evidence that corporate working capital decisions are affected by the industry/sector in which firms belon
Industry influence on corporate working capital decisions
This paper provides evidence that corporate working capital decisions are affected by the industry/sector in which firms belon
Scattering of cosmic strings by black holes: loop formation
We study the deformation of a long cosmic string by a nearby rotating black
hole. We examine whether the deformation of a cosmic string, induced by the
gravitational field of a Kerr black hole, may lead to the formation of a loop
of cosmic string. The segment of the string which enters the ergosphere of a
rotating black hole gets deformed and, if it is sufficiently twisted, it can
self-intersect chopping off a loop of cosmic string. We find that the formation
of a loop, via this mechanism, is a rare event. It will only arise in a small
region of the collision phase space, which depends on the string velocity, the
impact parameter and the black hole angular momentum. We conclude that
generically, the cosmic string is simply scattered or captured by the rotating
black hole.Comment: 11 pages, 2 figures, RevTe
Integrable lattice equations with vertex and bond variables
We present integrable lattice equations on a two dimensional square lattice
with coupled vertex and bond variables. In some of the models the vertex
dynamics is independent of the evolution of the bond variables, and one can
write the equations as non-autonomous "Yang-Baxter maps". We also present a
model in which the vertex and bond variables are fully coupled. Integrability
is tested with algebraic entropy as well as multidimensional consistencyComment: 15 pages, remarks added, other minor change