Poisson Yang-Baxter maps with binomial Lax matrices

A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on $\mathbb{C}^4 \times \mathbb{C}^4$ with $3\times 3$ Lax matrices is also presented.Comment: 22 pages, 3 figure

3D compatible ternary systems and Yang-Baxter maps

According to Shibukawa, ternary systems defined on quasigroups and satisfying certain conditions provide a way of constructing dynamical Yang-Baxter maps. After noticing that these conditions can be interpreted as 3-dimensional compatibility of equations on quad-graphs, we investigate when the associated dynamical Yang-Baxter maps are in fact parametric Yang-Baxter maps. In some cases these maps can be obtained as reductions of higher dimensional maps through compatible constraints. Conversely, parametric YB maps on quasigroups with an invariance condition give rise to 3-dimensional compatible systems. The application of this method on spaces with certain quasigroup structures provides new examples of multi-parametric YB maps and 3-dimensional compatible systems.Comment: 14 page

Neural Network Methods for Boundary Value Problems Defined in Arbitrarily Shaped Domains

Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of complex boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function network. The later is used to account for the satisfaction of the boundary conditions. The method has been successfuly tested on two-dimensional and three-dimensional PDEs and has yielded accurate solutions

On Quadrirational Yang-Baxter Maps

We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads to some new families of Yang-Baxter maps corresponding to the geometric symmetries of pencils of quadrics.Comment: Proceedings of the workshop "Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems" (Glasgow, March-April 2009

Modelling mechanical percolation in graphene-reinforced elastomer nanocomposites

Graphene is considered an ideal filler for the production of multifunctional nanocomposites; as a result, considerable efforts have been focused on the evaluation and modeling of its reinforcement characteristics. In this work, we modelled successfully the mechanical percolation phenomenon, observed on a thermoplastic elastomer (TPE) reinforced by graphene nanoplatelets (GNPs), by designing a new set of equations for filler contents below and above the percolation threshold volume fraction (Vp). The proposed micromechanical model is based on a combination of the well-established shear-lag theory and the rule-of-mixtures and was introduced to analyse the different stages and mechanisms of mechanical reinforcement. It was found that when the GNPs content is below Vp, reinforcement originates from the inherent ability of individual GNPs flakes to transfer stress efficiently. Furthermore, at higher filler contents and above Vp, the nanocomposite materials displayed accelerated stiffening due to the reduction of the distance between adjacent flakes. The model derived herein, was consistent with the experimental data and the reasons why the superlative properties of graphene cannot be fully utilized in this type of composites, were discussed in depth.Comment: 29 pages, 12 figure

Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation

We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schroedinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its success probability and the size of the coarse grid.Comment: 4 page