In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of Delauney type. In this note we conside...

Abstract We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system partial differential equations with constant coefficients. These rest on Fundamental Principle Ehrenpreis–Palamodov from 1960s. develop this further using recent advances in computational commutative algebra.

In this paper, we look at the notion of cohomological triviality fibrations homogeneous spaces affine algebraic groups defined over \({\mathbb {C}}\) and use topological methods, primarily theory covering spaces. This is made possible because structure groups. Further, generalize our results for arbitrary connected their As an application give a result quasi- reductive (i.e., whose unipotent ra...

The article surveys a number of potential theory results in the discrete setting of trees and in an application to complex analysis. On trees for which the associated random walk is recurrent, we discuss Riesz decomposition, flux, a type of potential called H-potential, and present a new result dealing with the boundary behaviour of H-potentials on a specific recurrent homogeneous tree. On gene...

A new type of optical fiber called heterogeneous multicore fiber (heterogeneous MCF) is proposed towards future large-capacity optical-transport networks and the design principle is described. In the heterogeneous MCF, not only identical but also non-identical cores, which are single-mode in isolation of each other, are arranged so that cross-talk between any pair of cores becomes sufficiently ...