We prove a Yau’s type gradient estimate for positive f-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces compact when infinite dimensional Bakry–Emery Ricci tensor and weighted mean curvature are bounded below. As an application, we give Liouville result condition. Our results do not depend any assumption potential function f.

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

Sensor-based discovery path planning is problematic because the path needs to be continually recomputed as new information is discovered. A process-based client-server approach is presented that permits concurrent sensor-based map updates, robot localization corrections , as well as concurrent path computation and execution. A harmonic function is constantly recomputed using an iteration kernel...

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in...

Sensor-based discovery path planning is problematic because the path needs to be continually recomputed as new information is discovered. A process-based clientserver approach is presented that permits concurrent sensor-based map and localization-correction updates as well as concurrent path computation and execution. Laplace’s equation is constantly solved (i.e., a harmonic function) by using ...

Abstract:Today, one of the important issues of power quality (PQ) in power systems is the current harmonics. Increasing expansion of nonlinear loads at different parts of the electric network makes harmonic distortion flow through the network. This causes the network to have background voltage and current harmonic distortion and even affect on the PQ of linear load performance. Therefore, it is...

The classic Poisson formula giving an integral representation of bounded harmonic functions in the unit disk in terms of its boundary values has a long history (as it follows from its very name). Given a Markov operator P on a state space X one can easily deene harmonic functions as invariant functions of the operator P, but in order to speak about their boundary values one needs a boundary, be...

Sensor-based discovery path planning is problematic because the path needs to be continually recomputed as new information is discovered. A process-based client-server approach is presented that permits concurrent sensor-based map and localization-correction updates as well as concurrent path computation and execution. Laplace's equation is constantly solved (i.e., a harmonic function) by using...