We propose manifold piecewise constant systems (ab. MPC) and consider basic phenomena: the 2-D, 3-D and 4-D MPCs exhibit limit-cycle, line-expanding chaos and areaexpanding chaos, respectively. The righthand side of the state equation is piecewise-constant, hence the system dynamics can be simplified into a piecewise-linear return map which can be expressed explicitly. In order to analyze the p...

The application of the barycentric method for numerical solution Dirichlet and Neumann problems Helmholtz equation in bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. main assumption to set $\Omega$ boundary a piecewise linear representation. A distinctive feature order formation global system vector basis functions via coordinates. existence uniqueness by are establis...

Let P = fp1; : : : ; png and Q = fq1; : : : ; qng be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n2) between the rectangles which maps pi to qi for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear homeomorphism has size (n2). Introduction A homeomo...

The need to reduce the dimensionality of a dataset whilst retaining inherent manifold structure is key in many pattern recognition, machine learning and computer vision tasks. This process is often referred to as manifold learning since the structure is preserved during dimensionality reduction by learning the intrinsic low-dimensional manifold that the data lies on. Since the inception of mani...