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 $n$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 $n$th iterate of every Painlev\'e equation in sakai's list is at most $O(n^2)$ 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