320 research outputs found
Algebraic entropy and the space of initial values for discrete dynamical systems
A method to calculate the algebraic entropy of a mapping which can be lifted
to an isomorphism of a suitable rational surfaces (the space of initial values)
are presented. It is shown that the degree of the th iterate of such a
mapping is given by its action on the Picard group of the space of initial
values. It is also shown that the degree of the th iterate of every
Painlev\'e equation in sakai's list is at most and therefore its
algebraic entropy is zero.Comment: 10 pages, pLatex fil
Third-order integrable difference equations generated by a pair of second-order equations
We show that the third-order difference equations proposed by Hirota,
Kimura and Yahagi are generated by a pair of second-order difference
equations. In some cases, the pair of the second-order equations are equivalent
to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are
irrelevant to the QRT system. We also discuss an ultradiscretization of the
equations.Comment: 15 pages, 3 figures; Accepted for Publication in J. Phys.
Algebraic entropy for semi-discrete equations
We extend the definition of algebraic entropy to semi-discrete
(difference-differential) equations. Calculating the entropy for a number of
integrable and non integrable systems, we show that its vanishing is a
characteristic feature of integrability for this type of equations
On the complexity of some birational transformations
Using three different approaches, we analyze the complexity of various
birational maps constructed from simple operations (inversions) on square
matrices of arbitrary size. The first approach consists in the study of the
images of lines, and relies mainly on univariate polynomial algebra, the second
approach is a singularity analysis, and the third method is more numerical,
using integer arithmetics. Each method has its own domain of application, but
they give corroborating results, and lead us to a conjecture on the complexity
of a class of maps constructed from matrix inversions
Phosphoinositide-binding interface proteins involved in shaping cell membranes
The mechanism by which cell and cell membrane shapes are created has long been a subject of great interest. Among the phosphoinositide-binding proteins, a group of proteins that can change the shape of membranes, in addition to the phosphoinositide-binding ability, has been found. These proteins, which contain membrane-deforming domains such as the BAR, EFC/F-BAR, and the IMD/I-BAR domains, led to inward-invaginated tubes or outward protrusions of the membrane, resulting in a variety of membrane shapes. Furthermore, these proteins not only bind to phosphoinositide, but also to the N-WASP/WAVE complex and the actin polymerization machinery, which generates a driving force to shape the membranes
A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice
We introduce a tropical analogue of Fay's trisecant identity for a special
family of hyperelliptic tropical curves. We apply it to obtain the general
solution of the ultra-discrete Toda lattice with periodic boundary conditions
in terms of the tropical Riemann's theta function.Comment: 25 pages, 3 figure
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