We study the parameter space of a family planar maps, which are linear on each right and left half-planes. consider set parameters for every orbit recurs to boundary between These consist algebraic curves, determined by symbolic dynamics itinerary that connects points. geometrical properties these in relation such dynamics.

We construct a two-sided discontinuous piecewise linear minimal valid function for the 1-row Gomory–Johnson model which is not extreme, but which is not a convex combination of other piecewise linear minimal valid functions. This anomalous behavior results from combining features of Hildebrand’s two-sided discontinuous extreme functions and Basu–Hildebrand–Köppe’s piecewise linear extreme funct...

The Self-Organizing Map (SOM) is a vector quantization method which places prototype vectors on a regular low-dimensional grid in an ordered fashion. A new method to obtain piecewise linear models of dynamic processes is presented. The operating regimes of the local linear models are obtained by the Delaunay tessellation of the codebook of the SOM. The proposed technique is demonstrated by mean...

Abstract. We introduce a generalization of Liu-Yang’s weighted norm to linear and to nonlinear hyperbolic equations. Following an approach due to the second author for piecewise constant solutions, we establish sharp L continuous dependence estimates for general solutions of bounded variation. Two different strategies are successfully investigated. On one hand, we justify passing to the limit i...

A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal order a priori error estimates are obtai...